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Each oversize page Is also film ed as one exposure and Is available, for an add itio n al charge, as a standard 35m m slide or In black and w hite paper form at.* 4. Most photographs reproduce acceptably on positive m icrofilm or m icro fiche but lack clarity on xerographic copies made from the m icrofilm . Fbr an additional charge, all photographs are available in black and w hite standard 35mm slide form at.* ♦For more information about black and w hite slides or enlarged paper reproductions, please contact the Dissertations Customer Services Department Ujuiveraity Microfilms , International Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8609838 Gilmore, Rowan John NONLINEAR BEHAVIOR IN MICROWAVE GALLIUM-ARSENIDE MESFET AMPLIFIERS D.Sc. 1984 Washington University University Microfilms Internationa! 300 N. Zeeb Road, Ann Arbor, Mi 43106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PLEASE NOTE: In all cases this material has been filmed in the best possible way from the available copy. Problems encountered with this document have been identified herewith a check mark V . 1. Glossy photographs or pages_____ 2. Colored illustrations, paper or print______ 3. Photographs with dark background____ 4. Illustrations are poor copy______ 5. Pages with biack marks, not original copy_____ 6. Print shows through as there is text on both sides of page______ 7. Indistinct, broken or small print on several pages 8. Print exceeds margin requirements_____ 9. Tightly bound copy with print lost in spine______ >/ 10. Computer printout pages with indistinct print_____ 11. Page(s)__________ lacking when material received, and not available from school or author. 12. Page(s)__________ seem to be missing in numbering only as text follows. 13. Two pages numbered 14. Curling and wrinkled pages_____ 15. Dissertation contains pages with print at a slant, filmed as received________ 16. Other_______________________________________________________________ . Text follows. University Microfilms International Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. WASHINGTON UNIVERSITY SEVER INSTITUTE OF TECHNOLOGY NONLINEAR BEHAVIOR IN MICROWAVE GaAs MESFET AMPLIFIERS by ROWAN J. GILMORE Prepared under the direction of Professor F.J. Rosenbaum A dissertation presented to the Sever Institute of Washington University in partial fulfillment of the requirements for the degree of DOCTOR OF SCIENCE December, 1984 Saint Louis, Missouri [ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. WASHINGTON UNIVERSITY SEVER INSTITUTE OF TECHNOLOGY ABSTRACT NONLINEAR BEHAVIOR IN MICROWAVE GaAs MESFET AMPLIFIERS by Rowan J. Gilmore ADVISOR: Professor F.J. Rosenbaum December, 1984 Saint Louis, Missouri GaAs MESFETs are increasingly finding application in power amplifiers, as solid-state replacements for TWT amp lifiers, the output component in many microwave transmit ters. Nonlinear behavior of the amplifier can result in intermodulation distortion, with distortion of existing frequency components within the transmission bandwidth, and the creation of spurious components outside it. Such behavior affects both the capacity and quality of communications links; its analysis is necessary to be able to control these parameters. A large-signal, numerically efficient MESFET model together with a harmonic balance technique is used to examine the interaction between the MESFET and its circuit at different power levels. Under single-frequency excitation, it is found that the device can be represented Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT Continued in a quasilinear fashion through its large-signal S-parameters, which are functions of the incident power at both ports. These are found to give good characterization at the fundamental frequency for incident power levels up to the 1-dB compressed output power of the MESFET. The bandpass sampling theorem is used to modify the harmonic balance approach to allow efficient analysis of the MESFET when driven by two closely spaced, nonharmonically related frequencies. Aliasing, introduced by sampling the signal waveform below the Nyquist rate, is accounted for and effectively removed by a repetitive frequencyshift technique. This enables the two-tone intermodulation response of the amplifier to be studied, as well as gain suppression effects in limitihg amplifiers. Device-circuit interactions and the effects of bias and harmonic termina tions are studied, and compared with experimental results. A novel scheme for intermodulation distortion reduc tion is proposed. It is expected that large-signal S-parameters will prove useful in the design of oscil lators and amplifiers, and that the modified harmonic balance technique will prove to be a necessary analy tical tool in the design of highly linear MESFET amplifiers. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. iv TABLE OF CONTENTS No. 1. Page Introduction.................................... 1.1 1.2 Description of Nonlinear Effects in a Microwave Circuit......................... Existing Analyses of Nonlinear and Intermodulation Distortion in a MESFET... 7 10 1.3 The Standard Harmonic Balance Method 29 1.4 Review of Fourier Theory................. 36 1.4.1 2. 1 Properties of the Discrete Fourier Series..................... 42 1.5 The Madjar-Rosenbaum FET M o d e l .......... 44 1.6 Output Parameters from Analysis......... 52 1.7 Introductory Review...................... 53 The Applicability of Large-Signal S-Parameters to GaAs MESFET Circuit Design................. 55 2.1 Development of Uniqueness Criteria...... 56 2.2 Determination of S-Parameters......... 63 2.3 Experimental S-Parameter Simulation..... 75 2.4 Design Examples........................... 90 2.4.1 FET Amplifier...................... 90 2.4.2 FET Oscillator..................... 95 Discussion................................. 98 3. The Modified Harmonic Balance Method........... 102 2.5 3.1 Bandpass Sampling......................... 102 3.2 Nonbandlimited Signals.................... 108 3.3 Controlled Aliasing....................... 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V TABLE OF CONTENTS (Continued) o. Page 3.4 Solution for the Unknown Amplitude Components................................. 3.4.1 Modified Harmonic Balance Procedure.......................... 126 Achieving Convergence............. 130 Limitations of the Technique............. 134 Testing the Modified Harmonic Balance Method.. 139 3.4.2 3.5 4. 4.1 Linear Test...... 4.2 Fifth-Order Nonlinearity Test............ 141 4.3 Application to a Simplified MESFET Model. 144 5. Modelling of Nonlinear Amplifier Behavior 6. 140 153 5.1 Preliminary Matching of the Model to the NEC72089............................... 153 5.2 Amplifier Simulations and Measurements... 157 5.2.1 Intermodulation Distortion........ 161 5.2.2 Gain Suppression in Limiting Amplifiers......................... 190 Conclusions and Recommendations............... 196 6.1 The Modified Harmonic Balance Method 6.2 Intermodulation Distortion in MESFET Amplifiers................................. 198 Improving Intermodulation Distortion: A New Approach............................ 203 Acknowledgements............................... 213 6.3 7. 118 196 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vi TABLE OF CONTENTS (Continued) No. 8. Page Appendices...................................... 214 Appendix 8.1 Relationship Between the 1-db Compressed Power and Third-Order Intercept Point for a Third-Order Nonlinearity........... 215 Appendix 8.2 Error Analysis of S-Parameter Test Setup...................................... 217 Appendix 8.3 The Bandpass Sampling Theorem... 220 Appendix 8.4 Establishment of the Band-Edge Criterion....................................... 222 Appendix 8.5 Software for Time-to FrequencyDomain Conversion............................... 226 Appendix 8.6 Calculation of Coefficients Used in the Frequency-to Time-Domain Conversion...................................... 227 Appendix 8.7 Software for Frequency-to TimeDomain Conversion............................... 229 Appendix 8.8 The Modified Harmonic Balance (MHB) Algorithm................................. 230 Appendix 8.9 Test of Software Using a Fifth-Order Nonlinearity....................... 235 9. Bibliography.................................... 238 10. V i t a ............................................. 243 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vii LIST OF TABLES No. 2.1 2.2 4.1 5.1 Page Large-Signal S-Parameters as a Function of Incident Power at the Respective P o r t ...... 66 Errors in Reflected Voltage, Power, and Phase Using Both Small-Signal and Large-Signal S-Parameters...................... 73 Comparison of Analytical and Computer Generated Terms for a Test Nonlinearity...... 143 Device Performance as a Function of Geometric and Material Parameters............. 189 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. viii LIST OF FIGURES No. 1.1 Page Intermodulation-Distortion Spectrum and Measurement Setup..................... 9 1.2 Heiter's Nonlinear Channel Model......... 12 1.3 Relationship Between Power Parameters for an Amplifier........................... 17 1.4 Tucker's MESFET Circuit M o del ............ 21 1.5 Minasian's MESFET Circuit M o del.......... 26 1.6 Linear and Nonlinear Partitions for a General Circuit..................... 30 1.7 Partitioning of a MESFET Circuit......... 33 1.8 Fourier Transform Obtained from a Sequence of Periodic Samples............. 39 1.9 Madjar-Rosenbaum Basic MESFET M o d e l ...... 46 1.10 Three Terminal Nonlinear Capacitance 47 1.11 MESFET Model with Gate Charging Resistor Replaced with a Time Delay Element....... 48 Drain Current and Output Conductance for Modified Model........................ 51 Smith Chart Plot of Large-Signal S-Parameters............................... 67 1.12 2.1 2 . 2 (a,b,c)Relative Errors in Reflected Voltage Waves, Assuming Linear Superposition of Large-Signal S-Parameter Components...... 2.3 Large-Signal Experimental Test S e tup 69,70, 71 77 2.4 FET Output Power Versus Input Power...... 81 2.5 Smith Chart Plot of Reflection Coefficient Loci at the Drain............ 82 2 . 6 (a,b,c)Scaled Reflection Coefficient Loci at the Drain............................... 84,85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES (Continued) o. Page 2.7(afb,c) Scaled Reflection Coefficient Loci at the Gate............................... 2.8 88,89 FET Amplifier Equivalent Circuit Schematic................................. 91 FET Amplifier Output Power Versus Delivered Power.......................... 92 2.10 FET Oscillator............................ 97 3.1 Bandpass-Sampled Spectrum of Simplified Distortion Spectrum.......... 104 Fourier Representation of a Slowly Sampled Bandlimited Signal.............. 106 3.3(a,b,c) Bandpass Filter Spectral Characteristics.......................... 112 2.9 3.2 3.4 Bandpass-Sampled Spectrum After Frequency Translation.................... 116 3.5(a,b,c) Aliased Spectrum and the Foldback Technique................................. 4.1 4.2(a,b) Replacement of a Series Resistor by a Delay Element....................... 121 146 Output Power Versus Input Power for Optimum Gain and Optimum Power Loads.... 149,150 5.1 Complete FET M o del........................ 156 5.2 Complete FET Amplifier Circuit and Parasitics................................ 158 5.3(a,b) Power Curves for V GS=-0.25 V o lts........ 162,164 5.4(a,b) Power Curves for V GS=-0.50 V o lts........ 167,168 5.5(a,b) Power Curves for V Dg=4.0 V o lts.......... 170,171 5.6(a,b) Power Curves for RpB=100ft............... 174,176 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. X LIST OF FIGURES (Continued) No. Page 178,179 5.7(a,b) Power Curves for RpB=1000Mft.............. 5.8 (a,b) Power Curves at Optimum Bias P o i n t 5.9 Unequal-Level Tone Power Curves, 3 dB Apart............................... 192 Unequal-Level Tone Power Curves, 5 dB Apart............................... 193 5.10 181,184 6.1 Linear Output Characteristics............ 200 6.2 Novel Intermodulation Distortion Improvement Scheme....................... 6.3(a,b,c) Power Curves Using IMD Improvement Scheme.................................... 8.2.1 Signal Flow Graph of Large-Signal Test Setup............................... 205 207,208 211 218 8.3.1 Minimum (bandpass) Sampling Frequency... 221 8.4.1 Illustration of the Bandedge Criterion.. 223 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. NONLINEAR BEHAVIOR IN MICROWAVE GaAs MESFET AMPLIFIERS 1. INTRODUCTION In the last decade, the GaAs MESFET has become firmly entrenched as the major microwave solid-state device for use in amplifiers and has demonstrated its significance in applications ranging from oscillators to mixers. Within the last year, MESFETs have become commercially available in K-band (30 GHz), while devices have been experimentally tested at frequencies as high as 60 GHz. Initially used solely as components in microwave receiving systems, their increasingly higher power-handling capability has now enabled them to be used as the power amplifier stages in microwave transmitters. Solid state powers of 20 W in C-band, and 5 W in Ku-band are achievable with existing commercial devices. Furthermore, the advent of monolithic microwave integrated circuits has ensured that the GaAs Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2MESFET will remain the cornerstone of microwave solid-state technology for the foreseeable future. During this time, considerable effort has been expended * on developing models to describe FET behavior [1-6] . The need for device models is readily apparent, particularly at microwave frequencies where physical measurements, typic ally power based, are not only more difficult to make but reveal less information than can be found from observation of voltage and current waveforms at different circuit points as can be done in lower frequency circuits. Because of its physical size, it is often difficult to make measurements at intermediate points in a microwave circuit, and if the circuit could be broken to do this, its behavior would be quite different because of its distributed nature, and the altered loading effects thereby introduced. The pre sence of parasitic elements is an additional burdensome factor in microwave measurements; these are the critical, and often dominant, passive elements in microwave and millimeter wave devices. The continuity of the ground plane must also be assured in microwave testing. Finally, the expense of the device itself, and the cost and diffi culty of fabrication of microwave circuitry (microstrip, fin-line, stripline, etc.) often prohibits extensive test and bench development time. * The numbers in parentheses in the text indicate references in the Bibliography. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -3Simple linear device models at first sufficed for analysis of microwave circuits containing MESFETs and gave a good understanding of circuit operation and design that could not be achieved through measurement alone. For example, modelling of the FET gate-to-source input as a simple series resistor-capacitor combination gives insight to the design needed to develop a broadband amplifier. Such modelling enables existing linear, two port techniques to be used to model transfer character istics, input impedances, and even noise figures. The linear device models could be represented by compact, two port matrix elements (small-signal S-parameters), as linear circuit models suitable for use with nodal analysis programs (such as SPICE [7] or COMPACT [8]), or represented as analytical equations for use with the circuit mesh equa tions. Linear models in these forms enabled the develop ment of many low-noise receiver amplifiers and helped establish the MESFET as an important microwave device. The shortcomings of the linear FET model are fairly apparent. Many components rely on nonlinear effects for their operation-mixers, oscillators, and frequency doublers being three. In devices such as amplifiers, non linear effects limit the usefulness of the device. In order to characterize these traits and design for them, a nonlinear model and suitable analytical methods are needed. It is the purpose of this work to provide such tools. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -4In an imperfect communications channel, two types of distortion are present. caused by dispersion. The first, linear distortion, is Parameters such as gain and group delay (phase slope) have a sloped frequency response. This causes pulse-shape distortion, as the different spectral components comprising the pulse undergo different gains and transit times through the channel. Typically, this can be equalized at the receiver by introducing an inverse frequency characteristic. The second type of distortion is nonlinear distortion, which is of a more serious nature as new spectral components can be produced. This is caused when the channel gain is dependent on the level of the RF signal through it. In a microwave trans mission system, nonlinear distortion is a limiting factor in the information handling capacity of a given channel. Reduction of such distortion allows, for example, the replacement of FM by SSB AM for analog transmission, or an increase in the number of signal states for digital transmission. In a microwave transmitter, the microwave power amplifier is usually the major contributor of non linear distortion. Although the level of nonlinear dis tortion can be reduced by techniques such as feedback, feedforward, and pre- and post-distortion 19], these are extremely complex -to design. In addition to requiring a nonlinear model for the amplifier to be predistorted, the predistorter must be useful over a suitable dynamic range, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -5and must not apply frequency shaping to the input of the power amplifier. Yet again, the need for adequate device description in its nonlinear regime becomes apparent. The purpose of this dissertation is to provide a means to model and characterize nonlinear behavior in a GaAs MESFET circuit. There are several ways to describe the effect of a nonlinearity. The next chapter will investigate a quasilinear approach to describing the cir cuit, using a two-port, large-signal S-parameter matrix capable of describing single-frequency behavior of the FET. Such a characterization has proven useful in micro wave oscillator design [10,11], The remaining chapters will consider an alternate description, giving the dis tortion caused by the MESFET when driven with more than one spectral component. This multi-frequency input problem, while using the same nonlinear model, is much more complex to analyze because of the presence of nonharmonically related frequency components. The FET model remains the same; only the tools of analysis differ. The remainder of this chapter will be devoted to a review of the FET model to be used in this work, together with the consideration of a computationally efficient algo rithm (the harmonic balance method) which enables the single-frequency response of a circuit to be determined from the model. Chapter 2 then presents some results of this Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -6- analysis and investigates the description of large-signal, single frequency phenomena by quasi-linear two-port tech niques. The validity of such description is both the oretically and experimentally determined. In Chapter 3, a new analytical method is described which allows additional nonlinear parameters to be derived from a device model. Known as the modified harmonic balance (MHB) method, the level of new frequency components generated by mixing of two applied tones can be determined and both circuit and device effects investigated. Chap ter 4 compares the results of the MHB method with previous Volterra series analyses, using a simplified FET model. A single stage FET amplifier was then built to experi mentally examine the circuit, bias, and RF-level effects on the interaction of two input signals. Comparisons with simulations using the FET model and the MHB technique showed good agreement as different circuit parameters were altered. Finally, in Chapter 6, some conclusions are drawn about the design of microwave circuits, the limita tions of the modelling approach are discussed, and dis tortion reduction methods are assessed. General comparisons with previous experimental observations are also made. The Appendices contain portions of software that are unique to the modified harmonic balance method, with analysis which is pertinent to conclusions made in the text. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -7In the next few sections we shall examine the means of characterizing nonlinear behavior through measurement and review some analytical tools which can be used with a nonlinear model to portray nonlinear characteristics. 1.1 DESCRIPTION OF NONLINEAR EFFECTS IN A MICROWAVE CIRCUIT Consider firstly the case of single tone excitation applied to a GaAs MESFET. At low signal levels for which the device is fairly linear the output signal will be a linear reproduction of the input. However, as the signal level is increased and the device enters saturation, the gain will drop and the phase of the output may change. These effects, known as AM/AM and AM/PM conversion, re spectively, are direct manifestations of the nonlinearity of the device. The former is of concern in amplitude modulated systems, the latter in phase or frequency modulated ones. When two tones, typically equi-level and closely spaced in frequency are applied to a MESFET, another phe nomenon known as intermodulation distortion results. If the two tones are at frequencies of f^ and f2 separated by A = f 2 -f^, new sidebands at nf 1 - m f 2 are generated, where n and m are integers. If |n-m| = 1, these products will appear in the frequency band around f^ and f2 and are spaced from the original signals by a small frequency m A. i.1. They are caused by the (n + m) order term in the Taylor series expansion of the nonlinear transfer characteristic Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -8of the device. In a MESFET power amplifier, the predomi nant intermodulation products for most input levels are the third-order intermodulation terms at 2f 2 ~ f]_ and 2f^ - f 2 (n,m=l,2), arising from the third-order non- linearity of the MESFET. The power level of the third-order intermodulation products in a power amplifier is a useful performance measure in describing the linearity of its behavior at a given operating condition. It is a useful parameter for system design as it is indicative of the degree of line arity of the amplifier, its power handling capability, and its ability to handle modulated information at desired power levels. Because new frequency components are genera ted due solely to the action of the amplifier, it is of considerable interest to the communications systems designer that the level of third-order intermodulation products, at rated output power, be minimized. The measurement of the intermodulation products, as illustrated in Figure 1.1, is relatively simple. are fed through isolators one source to another) Two tones (to prevent injection locking of to a 3-dB coupler where the signals are combined and input to the amplifier. The output of the amplifier is viewed on a spectrum analyzer. In addi tion to the third-order intermodulation products and the two fundamental output signals, a DC component, sum and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. AMPLITUDE UH Amplifier Spectrum Analyzer Applied Signals Third-Order Intermodulation Products 0 f2-f, 2frt I '2 jJ L l y/y/- x , f2 2f2-f, 2f| f,+f2 2f2 3f| 3fj FREQUENCY Figure 1.1 Spectrum produced by a nonlinearity excited with two signals a frequencies f. and fp. The inset shows a typical system for measuring intermodulation and harmonic distortion products. -10*difference frequency components, and higher harmonics can be visible if the bandwidth of the amplifier is sufficient ly wide. At power levels approaching those needed to saturate the amplifier's power response, fifth and higher order intermodulation products also occur. Obviously, the inter modulation products are all level dependent, decreasing rapidly as the output power is reduced and the amplifier made to operate in its linear regime. This in fact is one, very common, means of controlling nonlinear distortion. Amplifiers are specified with saturated output power levels far in excess of the intended range of operation. The drawbacks of operating in the "backed-off" mode, high cost and high power consumption, rapidly worsen as the linearity requirements are tightened. is the GaAs MESFET. The source of the distortion As a first step in attempting to control device and circuit design to minimize nonlinear effects, we turn briefly to examine existing means of analysis of nonlinear distortion before considering a newer method. 1.2 EXISTING ANALYSES OF NONLINEAR AND INTERMODULATION DISTORTION IN A MESFET Numerous authors have developed models for the GaAs MESFET and used them to seek solution of a variety of problems. Various methods have been applied to analyze nonlinear distortion, each of which has its own advantages Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -11- and limitations. areas: They may be broadly classified into four analytical methods, physically motivated analytical circuit models, integral methods (including time domain integration and the Volterra series), and harmonic balance methods. This dissertation presents a new approach in the last category. The first two methods have a common element: the nonlinear equation or model is derived from consideration of device behavior under single frequency excitation. The application to the intermodulation problem with two, nonharmonically related frequencies as the excitation, follows as an extension. Several authors have derived analytical expressions for nonlinear distortion products by expansion of a non linear function in terms of its Taylor series most straightforward of these is due to Heiter [12-15]. [12]. The The nonlinear channel is assumed to be of the form shown in Figure 1.2. The transfer function for this channel is represented by an expansion of the form eQ (t) = cQ + c ^ (t-tj^) + c 2ei2 (t-t2) + c 3e i3 (t-t3), (1.1) where eQ (t) and e^(t) are the instantaneous output and input voltages, respectively. The presence of order- dependent time delays tj enables both amplitude and phase nonlinearities to be accounted for i.e. both AM/AM and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -12- in p u t LINEAR WITH MEMORY NONLINEAR MEMORYLE8S 8 C A L IN G AND 8U M M A TIO N Figure 1.2 Heiter's nonlinear channel model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -13AM/PM effects. For the case when time delays of all orders are equal, the model reduces to a conventional amplitude model, as it is then essentially memoryless. A memoryless approach will be inadequate to represent distortion at higher frequencies when the distortion is noticeably fre quency dependent. As stated earlier, excitation with a single frequency provides a simple means of separating amplitude and phase nonlinearities. By substituting a voltage e^(t) = A cos a t into (1.1) and expanding trigonometric products, the fun damental output component becomes e oa 3 3 (t) = [Ac. + t - A c, cos a(t, - t,) ] cos a t 1 4 3 3 1 (1.2) + 3 3 [j A Cj sin a (t3 - t^) ] sin a t = Aa cos a t + B Qsin a t, where the subscript a indicates the relevant frequency component. By defining aj°^ A cj cos a(tj-t^) and where the subscripts j indicate here the b1j°^ £ cjsina(tj-t1), order of the co efficient and the superscripts the relevant frequency, we can write A a = a lA + T a 3A and B a = bjA + J b 3A 3 (bx= 0) • Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -14- Thus eo a (t) = >/A a 2 + B a 2 cos <a t + V because B << A OL = A a cos (at + ' for c-j small, and where — (X $ =arctan a B (^-) a B . a Single tone tests can be made to allow determination of the unknown coefficients. The gain coefficients determined by plotting the oiltput power 1 2 can be 2 = A a versus input level A, and the phase coefficients bj by noting the phase change versus A. The sign of a^ determines the na ture of the amplitude nonlinearity, being "expansive" for a^ > 0 and compressive for a ^ < 0. Most practical devices are compressive and frequently specified in terms of the 1 - dB gain compression point Pa which is the output power level where A / a ^ A = 0 . 8 9 i.e a reduction of 1 - dB from linear power. The AM - PM conversion coefficient k a can in degrees/watt of input power. For third-order be expressed devices operated in the linear region where A a = a^A, k <j> - b~ ~ — S- = x — a ,2 4 a. A 1 = constant is independent of power level and a convenient measure of the phase nonlinearity of the device. For the case when Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -15(memoryless model) , <|»a = 0 and AM/PM effects are absent from the modelling. Heiter [12] then examines two-tone, three-tone, and noise loading tests as a means of classifying the nonlinearity, the intent being to probe the signal spectrum as closely as possible to that of the actual system load. For the purpose of microwave amplifier design, the two-tone intermodulation test is the most widely accepted (if imperfect) way of indicating amplifier performance in a communications system. The device is excited by two equal-level tones at radian frequencies a and 8: e (t) = A (cos at + cos 6t) . o By substitution into the nonlinearity and expansion as be fore, the output components are of the form e na + m 3 (t) = A„„ , ~cos (na + mg) t + B_„ , _ Qsin (na + mg) t na + mg na + mg with amplitudes + 9/4 a(3a) A 3 + .. . A a= A 2a±B = — a (2a±g)^3 4 3 ** * (1.3) (1.4) u ' and corresponding expressions for B a and B 2a+g* These results are highly instructive in yielding several useful described by rules of thumb for devices which can be a third order nonlinearity. The output Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -16amplitude of the fundamental A a increases linearly with input power A until the second term in Equation (1.3) causes gain expansion or compression (depending on the size of aj) . The third order intermodulation distortion products (henceforth abbreviated as IMD) increase as the cube of input power. On a logarithmic (dB) scale, this is a rate increase of 30 dB/decade. As shown in Figure 1.3, the intercept point is defined as that power level Pj at which the IMD power ^ 201-8 wo u 13 intercept results were extrapolated (dotted lines) power region. if low-level into the high- All power levels are referred to on a per-carrier basis (and not total power). It can be shown (Appendix 8.1) from these results that for third-order devices, PI = P a , l d B + where 10 *6 [dBm], is the output power level at which 1-dB gain compression has occurred for a single input tone. Further more, because the second term in (1.3) is now propor tional to 9/4 ajA^ (and not to 3/4 a^A^ as in (1.2)), the new ldB gain compression point at the output, determined by measuring the total power in the two output signals, will be log^g(3/2) = 1 . 7 6 dB lower than that for a single carrier. This is intuitively understood by considering the additional power "lost" in the intermodulation side bands when two carriers are present. The total (wideband) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -17- Px Figure 1.3 Relationship between fundamental output Pew third order intermodu lation product T?2 a_g/ an^ input power P IN per carrier for a thirdorder nonlinearity, showing the third-order intercept point Pj and single-tone 1 dB compressed point Pa>ldB. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -18power will be the same in both cases. Perlow [13] has also performed some interesting analytical third-order distortion analysis. He notes that the outputs at 2ot±£ appear to be caused by the mixing of the second harmonic of one signal and the fundamental of the other. This might result in the erroneous conclusion that if the second harmonic is eliminated, intermodulation distortion will be eliminated. Second harmonic reduction may be accomplished by decreasing the magnitude of C 2 , the second order term in (1 .1), as for example, by using a balanced configuration. However, if this decrease in the magnitude of C 2 does not affect the magnitude of 03 (hence a^), the outputs at the intermodulation distortion frequencies are not affected. A major strength of the modified harmonic balance technique presented later is that higher order effects, such as arising from impressing a second harmonic signal onto the PET with a fundamental, can be accounted for relatively easily. Perlow [13] also derives a very simple expression relating the intermodulation distortion ratio for a third order device to the AM/AM and AM/PM conversion characteristics measured for a single signal. The gain saturation, g c ' , is defined as the ratio of the fundamental output amplitude to the small-signal fundamen tal output amplitude, for a single signal of input level A^'. gc' is a complex quantity accounting for both Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. -19- amplitude and phase distortion. The intermodulation distortion ratio, imr, is defined as the ratio of the amplitude of the intermodulation distortion to the ampli tude of the fundamental output. If the total power of two signals of level A 1 each is equal to the power of the single signal i.e. 1 2 2 (Ax )2 = 2A1^ / the intermodulation distortion ratio is given by imr = gc I - 1. This is most easily determined graphically. Recalling Equation (1.4), the phase angle of imr must remain cont stant. The locus of points corresponding to gc - 1 is therefore a straight line extending radially from the origin at an angle |gc 'I sin A 0 = arctan (--- ;------ a--- ) , |gc |cos <f>g-l where |gc | and <}>g are the magnitude and phase of the gain saturation at any point. Takayama et al [14] have derived a fifth-order analytical expression relating input and output voltage which is curve fitted from single tone measurements of AM/AM conversion and AM/PM conversion. By using the same expression for two-frequency inputs, good agreement was Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -20achieved between the calculated and measured third-order intermodulation distortion spectra. Analytical approaches are useful in predicting gen eral trends of most of the nonlinear performance indicators for the GaAs MESFET. These approaches work particularly well for the Si bipolar transistor, which is accurately described by a third-order power series. such approaches are poorly motivated. Unfortunately, They provide no cause-effect relationship, and give no insight to either the semiconductor or circuit designer. They rely on empir ical measurements and curve fitting to quantify the nonlinearity chosen. Finally, higher order effects due to the presence of higher harmonics as voltage inputs, and self biasing, are not accounted for at all. By extending small-signal, linear models into the nonlinear realm it is possible to derive analytical ex pressions for intermodulation distortion products that are physically motivated [16,17]. The major problem in "nonlinearizing" a small-signal model is in determining which elements of the model should be nonlinear, and to what degree. The small-signal model used by Tucker [16] is given in Figure 1.4. of Minasian This is basically similar to that [18], which will be considered in more detail in the next section and in Chapter 4, when used to test the modified harmonic balance method. Tucker's model is a simplified unilateral model which incorporates Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -21- GATE Figure 1.4 DRAIN Tucker's extended small-signal model for the GaAs MESFET and embedding circuit. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -22nonlinearities in the gate (C^), the transconductance (gm ) , and the drain of the device (gD and CD ) . consists of two components, The drain current dependent on the input circuit voltage v^(t), and iy(t), dependent on the output circuit voltage. Third-order analytical expressions are used to represent these currents as functions of the relevant voltage. The coefficients of these expressions must be obtained either from low-frequency measurements of each of the nonlinearities, or from measured distortion data at microwave frequencies. Tucker obtained the linear terms from small-signal scattering parameter measurements; higher order terms were derived from a series of smallsignal intermodulation measurements as a function of load, and from measured gain compression data. Having found the coefficients empirically, analytical expressions for AM/AM, AM/PM, and third-order intermodulation distor tion products could be calculated at a number of different power levels, in a manner very similar to that used by Heiter [12]. The advantage of this approach, i.e. of incorporating analytical expressions into a conceivable device model, is that the mechanisms causing distortion are more readily apparent, and circuit effects can be accounted for. As before, the same parameters that affected the values of AM/AM and AM/PM also determine the intermodulation distortion, but no simple relation exists between them. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -23The work of Higgins and Kuvas [17] is also noteworthy, as the nonlinearities in gD , gM , and CQ (using the same model) were taken to the eighth power, and CQ was assumed linear. In this case, the coefficients were calculated for several different one-dimensional doping profiles. A modified Shockley model for the FET was used to establish the current, gate capacitance, transconductance, and output conductance over a range of bias conditions; from these, the polynomial expansion coefficients could be obtained directly. Tucker By using the analytical expressions derived by [16], predictions of IMD level for different doping profiles were obtained. The limitations of approaches such as these are clear. By attempting to extend a small-signal circuit model to the nonlinear regime, arbitrary assumptions must be made in choosing which elements are to be nonlinear. Minasian [18], for example, has derived a nonunilateral model, in which the gate-drain feedback capacitance is nonlinear, but the output drain-source capacitance is not. Gupta [20] assumes the contribution of the output conduct ance G q to the intermodulation distortion is small and neglects its effect, while Higgins et al [17] believe that this in fact provides the dominant contribution at low power levels. The second area of difficulty with such an approach lies in the extensive characterization needed to determine the coefficients in the Taylor series expansion. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -24While a useful technique for pinpointing the mechanisms causing intermodulation distortion and for providing broad circuit related guidelines, it is of limited use in circuit design due to the extensive device characterization needed to describe different devices at more than one bias level. The third category of nonlinear analysis methods is those that can be treated using integral techniques. Perhaps the most widely used steady-state dynamic nonlinear circuit analysis method is numerical time-domain integration [21 ,22 ]. State-variable differential equations are derived for the circuit, and initial guesses are assigned to the state variables. This results in an initial value problem. Numerical integration then proceeds until steady-state is achieved; that is, all of the state variables assume the same values at the beginning and end of a period. This method can be used for autonomous systems such as the oscillator, or for driven systems such as the frequency doubler [22]. It has not been used for intermodulation performance prediction because the modulation and beat frequencies are typically orders of magnitude smaller than the microwave carrier frequency. This would necessi tate integration over hundreds or thousands of microwave periods after steady-state has been achieved in order to extract the intermodulation response. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -25Another approach involving integration is the use of a Volterra series expansion [18]. The Volterra series can be regarded as a nonlinear generalization of the familiar linear convolution integral. The time domain output v(t) of a nonlinear circuit can be found from the nonlinear transfer functions hn (T]/T 2 '* *Tn^ usin9 *<«where v^t) = n l l vn (t)' | — 00 oo v 2 (t ) = | { h 2 (T1 ,T2)Vi n (t-T1)V.n (t-T2) dTLdT 2 — 00 OO vn tt)= | |... | V Tl'T 2' " Tn> ” — 00 A V i n (t' Tn,aTnJ. An equivalent frequency domain representation is achieved by using the Fourier transform to yield terms of the form 00 vn (t) = {}•*•} Hn (u)l,(1)2 /’ * — oo ^ ,(V i=1 v in (fi) exP .n (j2nfit)dfi* In order to obtain the higher order Volterra kernels H n (^i,... d)n ) , a large-signal circuit model is also re quired. The model used by Minasian is shown in Figure 1.5 and incorporates a feedback capacitance CD - The higher order transfer functions can be found recursively from Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -26- II bRflIN Co • t v LA5 1 T - Sate * i C f .a a a pF CbS»-067 pF Ri-- 4 f a Figure 1.5 Minasian's extended small-signal model for the GaAs MESFET. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -27the lower order transfer functions, using Kirchoff's law and writing nodal equations for the model. The expressions for the fundamental and third order intermodulation output voltage respectively are V0 l " Vi J Hl (“l) I cos ( w ^ 3 Vg3 _ j 3 +/&,(m,) ] (j^) I cos [(210^-0)2) / , enabling the intermodulation distortion to be determined. The limitations on the use of a Volterra series approach are that the Volterra kernels are extremely awkward to measure and nearly as difficult to calculate if a model is available. Furthermore, the degree of nonlinearity must be mild, as the representation otherwise requires an intract ably large number of terms for adequate modelling precision. In this instance the use of a Volterra series to analytical ly extract the intermodulation distortion from an analyti cal model (rather than with measured Volterra kernels) is of limited interest. The final category of analytical tools is the harmonic balance approach. This is a hybrid time-and frequency- domain approach, which allows all the advantages of a time domain device model to be combined with the strengths of steady-state frequency-domain techniques to represent the lumped and distributed circuit elements in which the device is embedded. The time-domain model can be completely Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -28general; laborious determination of coefficients by curvefitting over different bias levels is bypassed. The approach requires a large amount of software to be set up initially. However, like SPICE, the routines need not be devised around one particular model, and permit great flexibility in the choice of both embedding circuit and device model. The drawback of the harmonic balance approach is that it has been used principally for harmonically related frequency inputs i.e "single-tone" excitation. It has been useful in the design of amplifiers and oscillators [23,24], and in the modelling of harmonic distortion effects, such as gain compression and AM/PM conversion. Kawano In 1982, [25] devised a method of bandpass-sampling that allowed, for the first time, the efficient analysis of two nonharmonically related inputs to the MESFET model. An effective downconversion of the two signals was achieved by sampling the time domain output waveform at a slow rate, and, in the Fourier transform, translating the new frequency origin to a point exactly between the two driving tones. This enabled the third order intermodulation pro ducts to be effectively treated as a third-harmonic signal. However, the analysis was strictly small-signal, as the aliasing effects of higher harmonic signals were not considered; additionally, only equal-level input tones were possible. A contribution of the present work is to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -29- remove these restrictions and to allow the power of the harmonic balance technique to be applied to the general case of nonharmonically related inputs of any amplitude 1.3 [26]. THE STANDARD HARMONIC BALANCE METHOD The harmonic balance method [27,28] is an important means of analysis for nonlinear systems. Consider the system shown in Figure 1.6, in which a device model, including its parasitic circuit elements, is embedded in some linear circuitry, so that the total circuit performs some desired function. When partitioned as shown in the figure, all of the nonlinearities associated with the cir cuit are lumped into the "nonlinear" side of the circuit; the device parasitics which are constant, linear elements, and the embedding circuitry, are partitioned into the "linear" portion. In circuit analysis, nonlinearities are most usually represented by relating terminal quantities, such as vol tage or current, to each other by some nonlinear function g (•). The parameters of g (•) are typically functions of time. For example, for a simple Schottky diode, the diode current i(t) is related to the terminal voltage v(t) at the same instant of time by a relation such as i(t) = Ig exp [ ^ v(t)]. Similarly, for the MESFET, the gate current i^(t) and drain current i 2 (t) can be given (in matrix notation) by an explicit nonlinear function g(*), the parameters of which are the gate and drain voltages Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -30- NONLINEAR LINEAR Embedding Circuitry Device Porosities FREQUENCY DOMAIN Figure 1.6 Device Model TIME DOMAIN Division of a general circuit into nonlinear and linear partitions (frequency and time domains). Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. -31and their first order time derivatives at the same instant of time [i(t) ] = [g( [v(t) ], [ * ^ 1 ] ) ] . A similar time-domain representation could be found to represent the linear partition. If each branch of the linear circuit contained both inductors and capacitors, then the current in that branch could be related to some control voltage by a second-order, linear differential equation. The set of such equations representing the linear partition could be solved simultaneously, in terms of the variables at the partition interface, and time-domain integration then used to achieve a solution. The disadvan tage of this approach for steady-state, nonlinear analysis is that if the time constants associated with the linear partition are large (as with bias and blocking capacitors and RF chokes), many cycles of integration may be required until a steady-state solution is achieved. The harmonic balance method presents a much more efficient approach for steady-state nonlinear analysis whenever the nonlinear function g(.) relates several parameters at the same instant in time. The efficiency is achieved by treating the linear partition in the fre quency domain i.e. by using phasor representation. In this way, elements of differing time constant correspond simply to elements of differing amplitude. Furthermore, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -32a compact two-port matrix representation H[f] can be used to represent a linear network of any size by relating only the desired interface terminal quantities to each other. The partition between the time and frequency domains (corresponding to the nonlinear-linear interface) naturally suggests the use of Fourier theory to achieve a steady-state solution. A review of the discrete Fourier transform will be given in the next section; however, in order to motivate that section, the standard harmonic balance technique will be outlined here. The linear-nonlinear partitioning has been applied to the circuit in Figure 1.7. The linear circuit is de scribed, at some frequency u>, through a hybrid matrix H[<o] and its phasor input quantities (applied voltages) and currents i^/i 2 (assumed known): r “i V1 V2 T CM |-H 1 J .1 C “ H to V2 --- 1 L Z1 H(w) (All quantities are assumed to be phasor quantities unless specifically represented as functions of time). The left hand side of the equation represents the output variables. Similarly, the nonlinear MESFET model relates the terminal Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. STANDARD HARMONIC BALANCE -T ' *2^ Drain i 2 Linear Circuit And Parasitics Nonlinear • • Ji Gate MESFET Model l « ■ « (*!),“ ? ) wi Source I FREQUENCY DOMAIN I TIME , DOMAIN I I Figure 1.7A partitioned MESFET circuit. Applied gate and drain voltages and relevant terminal voltages and currents are indicated. I u> (.0 I -34- voltages and their derivatives to the terminal current, at each instant in time dv, (t) 2^ ' dt dv-(t) ' cFE (1 .6) ^' The problem is then one of finding a self-consistent set of parameters at the interface, I^,I 2 ,i^,i 2 ,v^,v2 , so that both the linear and nonlinear constraints are simul taneously satisfied. Once this has been achieved, the total currents 1 ^ and I 2 delivered from external voltage sources and V 2 are then easily found. The constraint may be expressed mathematically by [iv (t) ] = Re E [I. (nw) exp (jnoit) ] , n=0 K (1.7) which is just a statement of Kirchoff's current law at the interface nodes. A similar expression relates the phasor components v^(nui) and v 2 (nw) at the interface to v^(t) and v 2 (t). For simplicity, only the first N harmonic compo nents are considered (N=7 here). The harmonic balance algorithm then proceeds as follows: 1. Initial guesses are established for the terminal interface current phasors I(nw) at the DC, fundamental, and harmonic frequencies (n=0 ,l,...7). 2. The hybrid matrix H(nu)) at each harmonic, is calculated at DC and for the. linear circuit. This is used in Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. -35- (1.5), with I(noj) and the known applied voltages V.^ and V 2 , to calculate the phasor components v^(nu>) and V 2 (nw). 3. Using an expression similar to (1.7), and its derivative, time values for v(t) and ted from the phasor components. may be calcula Using (1.6), the corres ponding interface currents i(t) may be found. 4. Using the discrete Fourier transform the harmonic phasor components i(nw) may be extracted from i(t), if time samples of i(t) are available at the Nyquist rate. 5. An error function is formed to compare the pre vious current estimates with those just calculated N E ^i l ,i2 ,^l'^2^= n- 0 ^ i l^na)^ 6. 2 + !i 2 (na)) “ ^2 2 I ^ The error function is minimized by forming new initial guesses for the current phasors I(no) from the old estimates, and repeating steps (2 )-( 6) until the error function lies below some threshold. At this stage, the linear and nonlinear partitions give self-consistent results, since the currents in the interface branches are self-consistent. Transformation from the frequency domain to the time domain is easily achieved, since the phasor quantities are simply coefficients of a corresponding sinusoid, as given in Equation (1.7). The inverse transform, from the time to the frequency domain, is also needed to recover Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -36- the phasors from the waveform samples given by the non linear model. This necessitates calculating the value of the waveform at several instants of time. If the instants in time are periodically spaced, the discrete Fourier transform may be used to achieve the desired conversion. 1.4 REVIEW OF FOURIER THEORY Since extensive use of the discrete Fourier transform is made in the remainder of this work, a brief review of its major elements is in order. Consider an analog signal *a (t) with Fourier trans form X cl (jft). The signals are related [29] according to OO *a(t) -Er ) Vi«>e3ataB or, alternately OO The proof is simply by backsubstitution, interchanging the order of integration, and using orthogonality. We may then derive a sequence of points x(n) from x a (t) by periodic sampling every T seconds, where OO x(n) = x & (nT) = X a (jfi) e ^ nTdSi. (1 .8) An alternative approach to consideration of discrete time signals is to consider a linear system with transfer Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -37- response h(t). If a sequence x(n) = e^un for -<»<n«® is input to the system, then the output y(n) is given by convolution as y (n) = E* h(k)ejw(n~k) k=-~ = e^“n E°° h(k)e-^ ^ k=-» By defining Hgfe-^) = ^k=-«» k(k)e ^u k r (1.9) (sampled Fourier transform), one can write y (n)=Hg (e^a))e -1u n , where H ( e^ ^ b describes the change in amplitude and phase of a complex exponential as a function of frequency a). Equation (1.9) is simply a Fourier series, where the Fourier coefficients are h(k). Multiplying e -5**311 and integrating over w, and using (1.9) by orthogonality one obtains i; (1.10 ) h(n)=^ r j H s (eja))eja)ndu). — 7T Comparing Equations (1.8) and (1.10), which are two representations for the discrete time sequence, changing variables in (1.10), and letting ui=QT, yields „ , ju>. 1 v ,jw , .2irr. X s (eJ ) = = I X ( ^ - + D - f “ ). r=-<» ... (1.11) The dimensions of u are radians, so that ft corresponds to an analog frequency. Recall that T is the sampling interval. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -38- The importance of the result (1.11) is shown in Figure 1.8. An analog waveform x, (t) has the spectrum ci shown at the top of the figure, which is just X_(jft). a For the case shown, the signal is bandlimited to If x (t) is not known at all instants of time, but known a only at x(n), using (n=-«to n=+~) , then Xg(e^a>) may be calculated (1.9), and is shown at the bottom of Figure 1.8. Because of the periodicity of the phase angle u», the re sultant waveform obtained is a scaled, periodically repeated version of the original frequency spectrum X (jft), (2q as given by Equation (1.11). For -j-< ir/T i.e. fsampie= 1/T > , then no aliasing, or overlap of the periodically repeated waveforms, occurs. Nyquist sampling: This is just a statement of if time domain samples of an analog waveform are taken at a rate at least twice the highest frequency component of X a (jfl) , then Xg(e3ai) is identical to X (3“/T) easily shown in the interval -ir<a)<ir. Furthermore, it can be [29] that the original analog time waveform may be reconstructed from the samples using x (t) = a Z* x (kT) si.n[(VT) (t-kT) ] - oo a (tt/t ) (t-kT) * jU The problem with finding the Fourier transform of a lowpass signal x_(t) from its samples using Equation (1.9) a is that an infinite number of samples are required in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -39- SL -27V Figure 1.8 The bottom curve shows the Fourier transform of the sequence obtained by periodically sampling the waveform whose spectrum is shown at the top. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -40summation (but the frequency spectrum is then continuous and may be found at all values of to) . In the special case of a periodically repetitive waveform, which has only a finite number of spectral components, a discrete Fourier series may be used. The discrete Fourier transform (DFT) is similar to this, and is a Fourier representation of a finite length sequence which is assumed periodically repeated. The discrete Fourier transform corresponds to samples, equally spaced in frequency, of the Fourier transform of the signal. Because we are restricting our attention only to certain values of frequency, the summa tion need no longer be performed over an infinite number of time-samples. The construction of the discrete Fourier transform is most easily achieved by applying the same approach as above to an infinite length sequence. sequence Now, however, the is periodic, with period equal to the finite length of the desired waveform interval. Consider an N-point sequence periodically repeated: x(n) x(n) = x ( n + kN) If x(n) n = 0 , 1 , 2 . . .N-l, with period N k an integer. is represented as a Fourier series, with radial terms that are integer multiples of the fundamental angle 2tt/N associated with the periodic sequence, then the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -41- highest harmonic angle that need be considered is the (N-l)st, due to the periodicity of angles greater than 2tt . Thus let i N—1 x(n)=±-Z N k=0 • / 2 tt« . X (k) e^ N , (1. where the desired Fourier coefficients are X ( k ) . multiply N-l Z (1.12) by e“ ^ ( ^ ) nr and sum from n = 0 .,2tr. . N-l x(n)e'](T )nr = h n=0 N-l . Z n=0 N-l = Z X(k) k=0 ,N-l Z Nn=0 12) Now, to N - l , ,2tk Xfkje3 ( N )n(k”r) k=0 . ,2ir. , , eD ( N )(k-r)n] N-l = Z X (k) 6. k=0 Kr N-l ..2irv X (r) = Z x(n)e-:j N n=0 or N-l .,2ik X(k) = Z x(n)e-^ N n=0 . (1 .13) Comparison with (1.9) reveals that this is equivalent to the sampled Fourier transform given there; now, however, the summation is finite, and only discrete angular fre quencies to are considered. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. -42- Note that the Fourier coefficients X(k) given by (1.13) are periodic in N, X(k) = X ( k + i n t e g e r x H) / so that there are only N distinct coefficients in the Fourier series representation of a periodic sequence. _ -i ( I Z . ) By letting WN = e J N , the discrete Fourier series pair may be written from (1.12) and X(k) (1.13) as N-l . = I x(n)W„ n=0 (1.14) x(n) 1.4.1 1 N_1 -kn X(k)W„ Kn N k=0 N Properties of the Discrete Fourier Series In practice, equation (1.13) is not directly used to extract the desired fundamental and harmonic components of frequency from the time samples x(n)(n=0,1,...N-l), 'obtained from the nonlinear model. algorithm Rather, a Goertzel [29] is used, which exploits some of the symmetry properties of the transform, thereby enabling considerable improvement in computational efficiency. However, the basic properties of the discrete Fourier series may be related directly to Equations (1.13) and (1.14). Although elementary in nature, their understanding is essential for later development when aliasing will be considered in the case of band-pass sampling. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -43- (1) Shift of a sequence: Because the Fourier coefficients are a periodic sequence, the values of the periodic sequence X(k+1) with 1 an integer, are just the Fourier coefficients of the sequence W Nn ^x(n) (using Equation 1>N (1.14)). Any shift 1 greater than the period cannot be distinguished in frequency from a shorter shift 1- = lmodulo (2) Symmetry properties: Re[X(k) ] = Re[X(-k) ] Im[X(k) ] = - Im[X(-k) ] (from (1.14), using real and imaginary parts of WN ) . Thus, once N/2 Fourier coefficients of the N-point sequence x(n) have been obtained, the remaining N/2 coefficients may be obtained using the symmetry property. This is simply a statement that the frequency spectrum of a waveform is even in real part and odd in imaginary part about the origin. Due to the periodicity introduced by the sampling process, aliasing results in exactly the same way as before, when an infinite-sequence length was used and the sampling interval chosen was too long. By using a discrete Fourier transform, the frequency spectrum can be found at discrete frequency points, with fundamental radian angle w= 2ir /N, corresponding to fundamental radian frequency 2 tt /NT, with T = sampling interval. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -44- In the case of the standard harmonic balance method, where a single frequency excitation fo is used [27], the resulting time waveform at the linear-nonlinear interface will also have fundamental frequency fo. However, due to the nonlinearity, higher harmonics may be generated. Since the resulting frequency spectrum of the interface waveform will be discrete, with frequency components at DC, fo, 2fo, 3fo, ..., choosing T = sampling interval= 1/15fo and N = 15 will be satisfactory for DFT representation if harmonics greater than the seventh are negligible. This is because the sampling rate is greater than the Nyquist rate (to pre vent aliasing), and the discrete Fourier transform will yield coefficients with fundamental frequency its first seven harmonics. an^ Note also that the applied excitation could have harmonic content as well, with no change needed in the time to frequency algorithm. In this case, the discrete Fourier transform gives the total frequency spectrum (a line spectrum) as only the funda mental and its discrete harmonic frequencies are present. 1.5 THE MADJAR-ROSENBAUM FET MODEL A large-signal, physically based, time domain model for the FET has been extensively studied by both Madjar [1,2] and Green [22]. The input data required by this model includes gate length and width, active layer height, built-in potential of the Schottky barrier, and physical Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -45- constants associated with GaAs. In addition to the drain conduction current, the Madjar-Rosenbaum model calculates transconductance gm, GD0M= d /^ v d s ^V D S = e lectron transit time t , and R , an effective series gate-charging resistance. The equivalent circuit is illustrated in Figure 1.9. An incremental capacitance matrix is also determined, to allow calculation of the gate and drain displacement cur rents from the first order time derivatives of v^ and v 2 . The current-voltage relationships for this three terminal, nonlinear, nonreciprocal capacitance are shown in Figure 1.10. A more complete model, which includes gate conduction current diodes to account for forward and reverse breakdown effects, is shown on the top of Figure 1.11. By modelling the effect of the series gate-charging resistor R as a time delay element in the gate voltage applied to the three terminal capacitance, as in Green [22], explicit calculation of total gate and drain currents i^ and i2 is now directly possible from the Fourier components of the time delayed voltage v^, and the drain voltage v 2 . modelling change is shown in Figure 1.11. This Such a step is essential in efficient calculation of network response, because it enables the interface current arising from the nonlinear MESFET model to be calculated explicitly as functions of the applied voltages at the interface Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. -46- MESFET CROSS SECTION "ACT IVE* REGION SOURCE GATE Jjn^n£ DRAIN J EPI SUBSTRATE EQUIVALENT CIRCUIT DRAIN ;dk GATE # SOURCE Figure 1.9 Equivalent circuit for the MadjarRosenbaum basic MESFET model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -47- te l — [V I — Pcn Cizl A. f"vi UoJ “ LfeJ ~ L?zi CzzJ" lVs. WHERE: C«=C^(v,,Vj) /^--V2 Figure 1.10 Terminal currents and voltages for the non linear, nonreciprocal three terminal capaci tance of Figure 1.9. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -48- iw r tv? W W BMW* ISPLACEMENT "H lu CO*V GO X3 fv? y n ( M > GATE CONDUCTION CURRENT DIODES Figure 1.11 GATE VOLTAGE DELAY <r seconds) THREE TERMINAL NONLINEAR,NONRECIPROCAL CAPACITANCE MODELLING DISPLACEMENT DRAIN CONDUCTION CURRENT The nonlinear MESFET model shown with a gate charging resistor R , and an equiva lent representation in wnich the effect of R c has been accounted for by a time delay of t seconds in the gate voltage V^. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -49- terminals. Previous authors [28] had to use numerical integration to calculate the currents arising from the terminal voltages due to the use of more complex model topologies. This represented a considerable burden since integration was required every iteration of the harmonic balance loop. The explicit representation of the currents i in terms of the applied voltages v and their time de rivatives enables this model to be readily adapted to frequency-domain simulation techniques such as harmonic balance. As shown on the figure, the drain current consists of three components: the displacement current through the nonlinear capacitances, which are calculated by the Madjar model, the conduction current, and the current due to re verse breakdown of the gate-drain diode. In this work, an alternate expression for the conduction current component was used, similar to that developed by Curtice [4], It consists of two t e rms: I ('V c ^ - V J * [8tanh (nV_Q )+ (1.15) 0 The first term of the sum accounts for behavior at low V D S , and the saturation effect of the current as the drain voltage is increased. The second term is similar Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -50- to a leakage current term, and accounts for the finite incremental output conductance GD of the FET (i.e. it measures the slope of the current characteristic). The square-law relationship of current to gate voltage is typical of FETs in general. These terms are illustrated in Figure 1.12. The coefficients B and o are obtained indirectly from the Madjar model, which yields values of saturated drain current ID S S » pinchoff voltage Vp , and GD Q M . Matching at the condition that v s G = ®' p „ IDSS P 2 vp and „ g dom I DSS The term G SUBST be .045 xG„_„, DOM in (1-15) is taken in this work to from measurements made of the small-signal output impedance of the FET. This factor also accounts for leakage current through the bulk semiconductor. Agreement with measured DC drain current was found to be good over a range of biases. The Madjar model could not be called directly from the harmonic balance software because of its large memory requirements. Rather, it is called once first to enable Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. -51- Figure 1.12 Drain current predicted by the Curtice expression (1.15) as a function of V Dc showing both components, ana output con ductance Gn as a function of V SG* Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -52- the creation of a data look-up table, which contains the basic model parameters. With this modification, quasi-static large signal analysis could be efficiently achieved by means of the harmonic balance loop, which accesses the data look-up table. 1.6 OUTPUT PARAMETERS FROM ANALYSIS The purpose of the harmonic balance loop is to equalize the nonlinear and linear current flowing in the (same) branches at the partition interface. Once these are known, the current in any branch of the circuit may be found from the linear circuit matrix. Usually, the current sourced by the generator, and the current flowing in the load are required. These are easily found by using the hybrid matrix H(w) which is used to model the linear circuit in the harmonic balance algorithm. If V GS is the applied external voltage at the gate and IG the current sourced through source characteristic impedance ZQ , then V„ = V_„ - I„Z^ is the voltage at the terminals of the b bb b O physical FET circuit (including matching). The impedance presented by the FET and its circuit at this point is Z« = VG/I-, so the reflection coefficient at the gate matching network is p = known quantities. ZG-Zo + z , and may be calculated from G o The desired, observable parameters of interest at the gate are then Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -53- 1 * Pdel = delivered power = ^ Re (VQ IG ) 2 reflected power = incident power = ■*p 1- p 2 del (VG S )2 . o Similarly, the current ID in the drain load Z i s easily calculable from the hybrid matrix, so that the output power is simply pout = Re I ^ d ^ d I21Although the currents and voltages are of interest in determining the operating regime of the FET, the parameters of interest to compare with measurement are the powers calculated above. These will be of primary concern in later analysis. 1.7 INTRODUCTORY REVIEW This chapter has laid the foundations for solving sys tems with excitations which are harmonically related. We have reviewed all the necessary tools needed to proceed: measurement descriptions, existing analyses, the harmonic balance method, a complete FET model, and simple Fourier theory. We are in a position to move on to consider the more difficult case, but one of great interest, when two signals, closely spaced in frequency, interact with a system nonlinearity to produce new, nonharmonically related Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -54signals which fall inband. This "crosstalk in the fre quency domain" is potentially troublesome, and is not readily handled by the standard harmonic balance technique. This is because the discrete Fourier transform yields a point spectrum best suited to handling harmonically re lated spectral lines. However, a modification to allow analysis of this more complex case is possible through the use of controlled aliasing, which will be discussed in Chapter 3. Before this complete, nonlinear method is tackled, however, we digress to consider an alternative representa tion for describing the nonlinear behavior of the MESFET under single signal excitation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -552. THE APPLICABILITY OF LARGE-SIGNAL S-PARAMETERS TO GaAs MESFET CIRCUIT DESIGN Scattering parameters are a matrix representation of a linear circuit, relating transmitted and reflected voltage wave information at the have found wide afforded passive and active As a by this desire to use result of the ease formalism, there and is an them for the large-signal The purpose convenience understandable single frequency are a linear system validity for describing must be established. design, and such as power FETs However, since S-parameters concept, their They microwave small-signal characterization of nonlinear devices, [30,31], ports. application in the analysis, measurement of components. circuit's external nonlinear systems of Sections 2.1 and 2.2 is to examine the assumptions inherent in the extension of the linear circuit S-parameter concept to levels, to consider the circumstances extension is meaningful present a rigorous obvious modifications, applies to under which such an in the case of GaAs basis the large-signal for following impedance of Sections 2.3 experimentally the FETs, and to their application. representations for nonlinear two-ports The purpose large-signal discussion or also admittance [32]. and 2.4 is applicability of With to establish the characterization Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -56- to microwave MESFETs and to present experimental data used in design examples. 2.1 DEVELOPMENT OF UNIQUENESS CRITERIA For a linear two-port, reflected voltage waves V ^ and V 2 “ can be related V^+ and V 2+ , 311 to incident voltage S-parameter matrix at waves, each < m + N ' S2 2 (u) m (2 .1 ) ---- 1 S2 1 (u) S12 (u) sl2(u) — Sn (w)• * 1 ... " ■1 1 < fo< 1 H* 1 1 1 1 1 frequency of interest: This formalism implies the linear superposition of V 1+ and V,+ terms. Z Moreover, the S • .(u) *s lj are not, in general, uniquely defined by arbitrary values of V 1+ , v2+ , v,-, because Equation and V 2_ . (2.1) represents unknown This is only two equations matrix elements small-signal case, the matrix additional constraints setting must be from which determined. elements are which are obtained the four In the defined from by alternately and V2+ to zero: v‘ i« fa) ■ ^ (2 .2 ) o, k i j j This procedure two-ports V because presents the no difficulty j ( w ) 1s are with linear independent of or V 2+ * Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -57For nonlinear two-ports under large-signal conditions, incident and reflected voltage waves simple sinusoidal time dependence, by sums of harmonically single-frequency input). Fourier coefficients of for each harmonic describing of (for the these voltage waves can frequency component functions [33]. discussion, only the fundamental a These For the a complex be taken to derive are also known sake of this frequency component will be considered, as is customary. applied to sinusoids ratios large-signal scattering parameters. as have but may be represented related The need not Similar analysis could be the harmonic components provided the harmonic terminating impedances are known. For nonlinear two-ports in general, the relationships among reflected and incident voltage waves are expressed: 1 1 *H > * " F1 (2.3) . v Furthermore, V VP for representation, V2>_ a large-signal it is necessary to S-parameter decompose V^- and V 2” into unique components according to v vl- vP VP sl2<vp VP 21^1* vp S22(Vl ' 1 x-s V* V2(V1’ V2> 11^’ (2.4) 2 J In general such a decomposition m a y not be possible. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -58- Consider the first row of Equation (2.4): V V1’ V? " S11(V1* V2)V1 + si2(Vr v2)v2 To define equation. S1 1 , set V 2+ to zero in (2*5) the above The S12 term is eliminated and (2.5) yields: . V"(V+ 0) su tvi> 0 > " ' ^ — 1 This determines respect l2-6) the functional dependence of with to V^+ , but gives no information regarding its dependence on V 2+ . Similarly, one cannot directly determine the functional dependence of S 12 on V 1+ . Now consider the matrix special case elements in incident voltage wave; the row Assuming that the one of the a function V 1+ if this element first column, or V 2+ if this column. is in which of only one is in the element is in the second first column element satisfies this requirement we have: v - ( v j, v+> - sl l ( v X Now setting V 2 + s12(v+ v X (2-7) to zero yields: , . Vi(TI l ^ SlltVl, V+ V1 (2.8, which is independent of V 2+ . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -59- Knowing sn ( v i+ ) one can indirectly determine S1 2 : + *!»< + v 2> - su (vi ,v i V vr V2> - V ■ -1 (2.9) 2 The problem with is now no such a two-step procedure way of determining whether assumption of the independence of valid solely on the basis as the measurement example presented V 2+ would be S11 not the initial on V 2+ is of the voltage wave information of S^1 assumed V 2+ =0. For the here, whatever dependence attributed to S12 as using Equation (2.9). this case or is that there has on it is determined Thus the matrix that is obtained in depends upon the assumptions made in deriving it; consequently it is a non-unique representation. To be S-parameter able to construct representation, we a unique must large-signal first consider the restrictive case in which the matrix elements in the first column are not the functions of V 2+ , and second column are not dependence is sufficient large-signal S-parameter the elements in functions to be able to matrix. We of V^+ . Such uniquely define a will call this condition the uniqueness criterion. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -60- If this condition applies, then setting V 1+ to zero in Equation (2.9) yields: c fv+. _ S.« (V~) — 12 2 Now, V^+ is no longer validity of the nature of V I (V2> v+ 2 used to determine assumption the output of V 2+ ) should unchanged concerning the the to the input and the device, and our , at s^2* and S12 may now be checked. simultaneously apply V^+ + (2.1C) by the and For if we V 2+ to application of s i2^V 2+ ^ application of functional assumption is correct, be unchanged by the output, T^e should V 1+ be at the input; v iz .: V- - su ( v X « i 2( v (2al1 X for all V 1+ and V 2+ . If, in fact, the superposition V^~ is not the same technique of as predicted by equaton (2.11) when both V 1+ and V 2+ are non-zero, then S11 or S12 must differ from the values postulated by the assumption of functional dependence be incorrect. measurement, i.e., has been shown to Similar arguments apply to the output, with V 2” predicted by V2 ' S21<V1)V1 + S22(V2)V2 (2-12) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -61- An exception is in the special case of device when s^2=0, or w ^en in a row is as zero. In that sn ^ v i+ 'v 2+ ^ may be directly one a unilateral the S-parameter terms case, the representation of alwaY s uni<jue, as it determined from Equation (2.5), with the last term identically zero. Another exception is the special case where the S-matrix elements are not functions of the relative phases of the incident voltage waves, i.e. only to the incident powers. and van der Puije [341, the uniqueness element is and, for the )V^+ |, and only of uniquely the first both |V1+ 1 defined however, the determine degenerate are every these case in functions only of column elements is assumed [30,31]. in which and |V2+ \> case, column elements the second |V2*|/ Using the method of Mazumder S-parameter matrix Generally, sensitive one could conceivably demonstrate a function of elements. which of an they are are functions Such a case does satsify the uniqueness criterion; however, the validity of such a representation to a particular device must still be established. In the this representation application to a GaAs which will 2.3, since it is from this MESFET, it is be considered in Section special case that the historic usage of the term "large-signal" S-parameters prevalent in the literature [30,31] is derived. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -62- As we have just noted, large-signal determined from incident and reflected information by alternately setting zero, as is done in S-parameters. the S-parameters are voltage wave v ^ + and V 2+ to measurement of However, additionally, the small-signal validity of the representation in the large-signal case must be checked by comparison with results when both simultaneously locus method used. A non-zero. Alternatively, of Mazumder measurement large-signal and Van der technique S-parameters was applied to Si will b e used for GaAs MESFETs applicability v ^ + and V 2+ are for MESFET will be the circular Puije [34] may be determination presented bipolar transistors. of the by them, of and The same technique in Section 2.3, where the quasilinear representation investigated for different levels to a of bias and drive. It is worthwhile assumptions made. frequency domain conditions, at this point The device has to the been represented in the in a quasilinear manner. its response to state Under certain a single frequency excitation can be completely described by the S-parameter matrix, at the same frequency. intermodulation, and other nonlinear effects are the have been advantages representation circuit Harmonic, design for of lost. a Offsetting this loss frequency-domain two-port the device: techniques (but availability at of applicable linear power Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -63- levels), ease of use, and a compact representation. Whilst an alternative characterization would be to nonlinear time-domain model to characterize the thereby allowing complete description at multiple frequencies, device, of device behaviour the assumption considering only fundamental use a made here in response allows considerable simplification. Instead, the circuit order frequency terms have been effects of included in higher the overall single-frequency response to the same order as they appear in the measurement circuit. Furthermore, device impedance changes in in the of level changes gain compression manifest S-parameters, However, S-parameters can unique not be found uniqueness described purpose of the next simulations) how a above which set of level large-signal conditions for are satisfied. closely these as and hence are now unless the section to and themselves the fundamental frequency response, the large-signal dependent. effects It is the determine (by numerical conditions are satisfied for the GaAs MESFET. 2.2 DETERMINATION OF S-PARAMETERS When the large-signal model for the MESFET [11 is used with the standard functional dependencies Using appropriate harmonic-balance technique, discussed above can the be examined. external circuits, S-parameters can be determined from the model under varying drive voltages, as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -64- given in Equation (2.2). An NEC type 869177 Ku-band modelled at 10 GHz. and the common-source GaAs FET was I(jss for this transistor is 230 mA, output power specification compression point is 22 dBm at 11 GHz. internal impedance internal matching matching stubs the 1 the FET chip could be directly gain However, removed modelling and the measurement, so dB The transistor has elements. were stubs reduced the measured 1 at for the both the that incident powers on noted. The removal of the dB compression point to 20.8 dBm. In order large-signal to investigate S-parameters, S-parameters satsify the the applicability initially assume of that the required relationships for their unique determination by measurement: S12 = S12 *V2+) (2.13) S22 = S22 *V2+) Furthermore, and V 2+ assume do not S-parameters i.e. the incident assumption is that the determine is the values of V ^ + of the the S-parameters are functions power on certainly uniqueness criterion. case relative phases interesting the respective not necessary However, such because, experimental determination if to only of ports. This satisfy the a highly restrictive valid, of the S-parameters it allows by direct Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -65- measurement. Under these conditions, voltage wave need be present at relative phase is now arbitrary. such a representation also accepted (but by only one any instant, incident since its It should be noted that corresponds to the generally no means established) usage of the term 'large-signal S-parameters1 prevalent in the literature. With such a representation, S-parameters S ^ and simulated using the large-signal S12 and S21 can be uniquely S22 are unique and incident the power levels defined and model [11. can be found on the large-signal output Similarly, for various port. The values obtained from the simulation at power levels between 0 dBm and 24 dBm incident on the respective ports Table 2.1, and are plotted on 2.1. Their power the Smith Chart dependence large-signal S-parameter are shown in is in Figure similar results reported by to the Tucker [30] and Johnson [31]. The most evident variation in the S-parameters is the gradual reduction in S21 along a line of constant phase, with the powers change becoming in excess of quite rapid 16 dBm. practically independent of drive, movement results, towards a match only indicating small large-signal S-parameters, recessed gate structures at incident S^ and S12 input are and S22 exhibits some at high drive. power-variation appear typical These of the for FETs with [35]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -66- Table 2.1 Simulated values of FET s-parameters as functions of incident power. Pin dBm S11 0 4 8 12 16 20 24 .9106 A 162 .9106/-162 .9107 /-162 .9110/-162 .9117/-162 .9134/-162 .9171/-162 S12. •1032/-21.4 .1032/-21.4 .1032 /-21.4 .1032/-21.4 .1032/-21.4 .1034/-21.4 .1020/-21.Q S21 S22 1.089 /48.5 1.088/48.5 1.085/48.5 1.078/48.5 1.060/48.5 1.014/48.4 .9063/48.3 .7284 /-76.3 .7284/-76.3 .7284/-76.3 .7285 Z-76.3 .7286 /-76.3 .7320/-76.2 .7050/-76.3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -67- Figure 2.1 Smith chart plot of the FET S-parameters for incident power between 0 and 24 dBm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -68- The assumption stated in necessary condition in order definition, but is It can for a predicted valid, particular MESFET S-parameters previously the V^“ (2.11) level and the deviations from at V 2“ the appropriate using the FET model incident power at both a comparison is superposition at a large-signal nonlinear assumed simulation. functional (2.13) will then cause deviations Any dependence in in the calculated value and V 2~SP comPare(* to t*ie true values of V 1~SIM to the Effectively, "correct" the If the identical to those given application of being made between linear by applying (2.12), using b y the complete nonlinear simulations the input and output ports. and and calculated incident power levels, should be with simultaneous a the MESFET. at the two ports. then from Equations is to permit unique S-parameter simultaneous incident voltages is (2.13) not necessarily valid for be verified assumption Equation V 2~SIM* the calculations Equations (2.11) (T^ e subscript performed or (2.12); by into a be expressed in matched load; difference. power, and phase angle, to the The relative error in also be presents the in both various drive level combinations. in complete terms of voltage there will Figure 2.2 superposition, as SIM refers nonlinear simultaneous simulation). magnitude can SP refers or power a phase angle relative error in and V 2 “ , for The relative powers are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -69- INPUT eDBM} <?° Uji Figure 2.2 Relative errors in V^~ and T f ~ when obtained by assuming linear superposition of largesignal S-parameter components as functions of incident power,(a) relative error in the magnitude of |V^~|2 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -70- PINPUT (DBM) 0 4 8 12 16 2 0 24 i I I Figure 2.2 (b) relative error in magnitude of |v2” |2 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. P INPUT (D B M ) Figure 2.2 (c) phase angle error in V 2 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -72- plotted as functions of input- (Figure input and output power 2.2a) and output-ports (Figure for the 2.2b); the phase angle error at the output is plotted as a surface in Figure 2.2c. plotted, The phase-angle error at the input is not as the error is within one-half a degree over all combinations of incident power. The relative error in power _ 2 |V^ | , is less than 2 percent. since S11 and S12 reflected from the input, This is to be expected are practically constant. However, 2 the relative error in |V2 I exceeds 40 percent when both input and output are simultaneously incident powers of 24 dBm. Table relative errors in V^” and V 2 “ at condition, when V ^ g p using incident V 1~SIM S-parameters, large-signal powers, the V 2~SIM are when small-signal the this drive rather S-parameters. errors aP P r°ximately At While the S-parameters some the these with double used rather large-signal S-parameters. in than on comparison S-parameters are results 2.2 compares with and V 2”gp are calculated small-signal corresponding driven than the use of large-signal improvement over the small-signal case, it has been demonstrated that the often tacitly made assumption of Equation valid beyond a certain range (2.13) ceases of drive levels, to be for which the errors increase significantly. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -73- Table 2.2 Comparison of relative errors in V “ and V ” expressed as voltage, power, and pnase-angle, when calculated (i) assuming linear superposition of smallsignal S-parameter components, and (ii) assuming linear superposition of largesignal S-parameter components. Reference values are calculated from nonlinear simulations at the + 2 4 dBm drive level. 24 dBm IN 24 dBm OUT Simultaneous drive TERMINAL ANtt AP% A e Small signal S-parameters GATE DRAIN 1.71 -39.83 3.387 -95.51 .237° _ -12.87 Large signal S-parameters GATE DRAIN .899 . -19.54 1.792 -42.89 .364° -5.38° Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. -74- Were (2.13) the necessary exact, the defined would condition use of For dBm. Since compression point of this is either less than two port does close to not the quoted the FET, this gives level, below which the we two and a half degrees in if the incident power at exceed 20 so the transistor relative errors will be percent in power, and less than angle, Equation large-signal S-parameters give zero errors. have modelled, the expressed in ldB a maximum power (unique) large-signal S-parameters so defined can be successfully used. The importance of representing the device as a function of its incident power levels is essential in this characterization. near-conjugate In the matching typically never zero of the at in a the direct wideband undesirable at power in an S-parameters that Takayama [36], these should due to reflection matching circuit. where or in the case amplifier, at the FET The levels of both are thus important in the is some for gain flatness (although an oscillator, V 2+ will be nonzero. requiring V 2+ amplifier the output), interstage matching design of output, the reactive reactive mismatch is required this is of amplifiers at the FET drain, output power from Similarly, case or in of the output port V 1+ and V 2+ determination of the applicable be used. As voltage waves and the described by phase between them effectively set the impedance of the circuit in which Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -75- the device is embedded. large-signal The representation of the FET by S-parameters, as possible means presented by which device herein, is effects may one be compactly represented in the frequency domain. 2.3 EXPERIMENTAL S-PARAMETER SIMULATION In order to experimentally verify the model and demonstrate the quasilinearization techniques discussed, a set of independent experiments intended to prove (2.12) are valid. Equation directly that Equations They are (2.11) and Dividing Equation (2.11) by V^+ and (2.12) assumption that was performed. by V 2+ , and using the S-parameters the additional are phase independent, gives (2.14) 12 = S22(|V2+ |) + S21(|V1+ |) v + where is the input e -j<}> (2.15) reflection coefficient the output reflection coefficient, and and the angle between the incident waves V^+ and V2+ . Equation (2.14) is the locus coefficient when the gate and drain. circle with S1 2 ^ V 2+ ^ times gate reflection FET is simultaneously driven Provided Equation + for constant |V1 [ locus of of the center and S1 1 ( |V1+ 1) |V2+/ V 1+ |, as the (2.11) is at the valid, the + lv2 I is a and radius phase Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -76- between locus V^+ and of the |V1+ I and V 2+ is drain varied. reflection IV2+ I are held between them s22(lv2+ l) is radius a the coefficient constant and varied) and Similarly, (as the phase circle, S2 1 (|V1+ |) with center times |vi+/v2+ 1* A measurement technique to obtain the loci of r2 has been previously reported for characterization [ 34 1 and for of power FETs in our [36]. the phase A load-pull characterization Rectangular shifter and available. bipolar transistor Figure 2.3 shows experiments. the apparatus used waveguide was variable attenuators waveguide and bandpass filter used, as were readily was used to prevent propagation of harmonically-generated higher order modes; the loci were then obtained directly from a standard harmonic convertor and network analyzer test set, using suitable attenuation in the reference and test channel arms. Although the determine all V^+ , the phase sources calibration usefulness could four S-parameters v 2+ / and several system of of the in a arise and method applied somewhat dependent to directly linear system make would in (if known), which which Principally, the harmonic convertor analyzer is used are accurately errors difficult, be [34] accurate limit and the [ 36] . output to the network on its reference-channel Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -77- HP-84J0B Network Analyzer Test Gunn Diode Channel V L 0-20dB Atten TWT Amp Plotter Device Under Test Isolator Dir Coupler BPF 8 -12GHz Figure 2.3 Experimental test setup used to obtain large-signal loci at independent levels of incident drive V^+ and V 2+ e ^ . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -78- input level. As the incident power on the device increased, the reference-channel power level and a slightly result in coefficient, even linear (i.e. constant P w i t h individual locus of will change different measured if the device However, since each was made at a constant j. reference-channel I 1 level of either (according to whether or measured), this is not a problem. between two loci at different be observed due to the reflection under test is perfectly drive). or is ^ 2 |V2 | or was being However, a slight shift reference power levels will action of the harmonic convertor under different input levels. A second source directivity of through of error the directional couplers. piece of waveguide replacing test, perfect circles with the obtained, since the linear and arises due to the For the a straight device under origin as center should be "device under test" is matched. finite Calibration now perfectly curves under this condition yield circles which are slightly offset from the origin. The radius of each circle was constant to within a maximum deviation of 3% (possibly also due to the effect of nonlinearities within the network analyzer itself). The radius of each circle should also be proportional to a factor depending of 1V1+/V2+ | upon which When the radii of the (or channel |V2+/ V 1+ 1, is chosen as reference). calibration curves were compared to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -79- the respective incident voltage ratio, the was less than 1% over all a lower power levels. bound on the accuracy maximum error These errors set of results to be expected from the measurement system. As a consequence circles, accurate ANAs) were to of algorithm (as be used method. shifting center S 22 and using this system. to large-signal S-parameters by this the values presently be obtained error correction of Since account for the cannot However, employed in could be of if an small-signal the shifts, all determined accurately the purpose of the measurements here is only to observe the circularity of the loci as the reference power the error is changed, and sources noted about S12 this is (see Appendix invariant under 8.2), information S21 can still be easily extracted, as an indication of any device departure from linearity. Common-source measurements were made at 10 GHz at three different gate bias levels, corresponding to Class A and Class C operation. were used at matching stubs limited by the both Incident power levels up to 24 dBm the gate and were removed from the drain (internal the dev i c e ). This was maximum safe FET DC gate current (set by either the reverse breakdown of the drain-gate junction or forward conduction of the gate-source junction). At all bias levels measured, the maximum safe gate current rating of 1.3mA DC was attained for some combination of phase and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -80- incident power levels less than the 24 dBm available from the complete setup. Consequently, the useful dynamic range of the FET could be tested. Scalar power later measurements of comparison with The experimental the FET the large-signal gain saturation were made value of S2 1 . characteristics of unmatched FET are shown in Figure for the 2.4, for the three bias levels investigated. As can be seen, at a gate -3V, the small-signal value of S21 calculated bias of from the power gain is approximately unity (in close agreement with that measured on an ANA). The power out at the 1 dB gain compression point is 20.8 dBm corresponding to an incident input power of 22.2 dBm. compression point (with bias of -IV, the 1 dB gain occurs at an input power considerable At a gain of 19.4 output power of dBm). expansion At a occurs estimated 1 dB compression point. 18.6 dBm bias of right up -5.5V, to the However, this point is an estimate only, because the DC gate current was limited to its maximum safe rating before the incident power could be raised to a suitably high value. The locus of f 2 as the shown on the Smith Chart of . at the drain 12 |V2 dBm, while levels. + Smith Chart Figure 2.5. is driven extend beyond because under varied is The signal input 2 • | is held constant at the gate The loci phase shift ^ is a level of at several the different boundaries of certain phase the conditions more Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. / / d 20 J i- 'b Volts -I V o lts // •o z N cc tii £ a. o K- 3 Q. h 3 O " 5.5 20 INCIDENT POWER lv,+l2 IN dBm Figure 2.4 Plot of FET output power in dBm vs. incident input power for DC gate bias voltages of -1 V, -3 V , and -5.5 V. The 1-dB compression points are indicated for the -IV and -3V bias levels. 25 -82- 22 v. Figure 2.5 Smith chart showing reflection co efficient loci at the drain {T under simultaneous drive conditions, as the relative phase between drive signals is varied. |V2+ |2 is 12 dBm; |v^+ |2 is the parameter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -83- power is returned from the drain than was incident, due to drive at the gate. These loci loci, which The are generally loci in made of a for +2 and |V2 I the two distinguished Figures 2.5-2.8 measurements + 2 |V1 | should be from far more complex shape. are reflection constant pull measurements, loci of powers to as the phase is however, are coefficient incident simultaneously applied ports of the FET, load-pull varied. Load constant output power as the terminating impedance of the FET is changed. Complete 2.6a-c for scaled loci are shown in three different drain drive first Figure 2.6a, where small incident power on seen, a this is circles. of f2 levels. at a relatively the drain (12 dBm). case, as As can be the loci with zero incident power (corresponding to a Consider [*£ is plotted nearly linear Note that Figures are at the gate 50 ohm termination t h e r e ), the locus collapses to a single point, which is just S22< Even at a gate loci from drive of +24 dBm, the deviation of the the average circle it represents is within the of S 21 times measurement system error bound. is then given IV2+/Vi+ I. magnitude of dBm; b y the With 1.0 at gives s ^ d V ^ I ) atlv-j4]2 =20 is less than 2.5%, which S21 12 dBm radius of the normalized incident on as .991 at I v ^ 2 = 16 dBm; .856 at J v ^ 2 = 22 The value circle, R, to a the gate, this .931 dBm; and .755 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -84- |VJ|2 = Z4dBm 22 20 22 Figure 2.6 Scaled reflection coefficient loci at the drain (r2 ) under simultaneous drive con ditions as the relative phase between drive signals is varied. |'Vi+ |2 is the parameter, and increases from 12 dBm to 24 dBm. (a) |V2+ |2 = 12 dBm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2.6 (b) |Vj"*”|^ = 20 dBm. 22 K l 20 22 24 Figure 2 + 12 = 24 dBm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -86- at !v ll2 = 24 dB®* Since the 1 dB gain compression point using this method is then just under 22 dBm incident gate power, good correspondence 2.4, which was measured by an is obtained with Figure independent, and simple, power measurement. As the reference-channel (i.e.drain) increased to 20 dBm and respectively), the becomes more dBm (Figure degree of pronounced. reasonably circular (to up to 22 24 power level 2.6b and departure from The locus dBm incident power on remains average circle) the drain, but at incident power, there is marked deviation, the use of the S-parameters S21 2.6c circularity of within 6% of the is 24 dBm indicating that and S22 is invalid at this drive level. The loci information of (Figures concerning the 2.7a,2.7b, and values of 2.7c) give and S1 2 . Figure 2.7a shows the locus of T^, recorded on the Smith chart. Figures 2.7b and 2.7c, drive, are scaled at higher levels accordingly, examination of their shape. to allow more of gate detailed As might be expected, the FET gate exhibits substantially greater nonlinearity than does the drain. At 12 dBm incident input power (Figure 2.7a), the loci are circular, although drain drive the circularity error of S1 2 ( \^ 2 when 1) *s 9iven bY R times normalized onto the gate to 1.0 at 12 is 7.5%. at 24 dBm The magnitude lv 1+/v 2+ dBm, the values are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -87- 1.005,1.016,1.040, and 1.178 respectively, at powers on the drain of 16, 20, 22, and 24 dBm. of gate nonlinearity is seen b y the incident little effect on the locus shape. define, but is of input of 24 drain power has At an input power of 20 average deviation from circularity is harder to drive The degree to be prinicpally controlled input power on the gate; dBm (Figure 2.7b), the incident dBm obviously so noncircular the order of 13%. (Figure 2.7c), the At an locus that S-parameters are of is no use in this case at all. Finally, the loci were observed -IV and -5.5V, beyond the at bias pinchoff voltage. voltages of The limiting factor in this case was the gate current when large drives were applied. terms of However, arcs At both voltages, nonlinearity in Class of circles observed (i.e. even up At -1 essentially C operation, the loci in the areas with DC gate in V gate biases. remained they could of 22 dBm on the loci corresponding loci of at -3V be 1.3 mA), the gate. were less and -5.5V gate The error at a relatively low incident gate power 16 dBm expected bias however, of which in unaffected. currents less than to incident power levels circular than of was the behaviour of was 18% since the at this onset of bias level. the 1 This might dB gain be compression point occurs earlier at this bias level (Figure 2.4). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -88- 0.1 10.2 - 0.5 20 0.2 22 *24 0.5 Figure 2.7 z Scaled reflection coefficient loci at the gate (r^) under simultaneous drive condi tions, as the relative phase between drive signals is varied. |V2 I is the parameter, and increases from 12 dBm to 24 dBm. 24dBm 22-20-- Figure 2.7 (b) |vn+ |2 = 20 dBm. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. Figure 2.7 (c) |V1+ |2 = 24 dBm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -90- 2.4 DESIGN EXAMPLES Having justified the quasilinearization through the use worthwhile to of large-signal now illustrate of the MESFET S-parameters, ways in it which they is may be used in the modelling and design of microwave circuits. Two examples serve first example shows the fundamental output may to b e derived demonstrate their use. analytical ease power response using simple of a with which the MESFET amplifier linear circuit while the second example notes their The techniques, use in the design of an FET oscillator. 2.4.1 FET Amplifier Using the FET described above, constructed with 2.8. the equivalent The measured a 10 GHz amplifier was circuit shown input-output in Figure characteristics of amplifier are plotted on Figure 2.9. the Shown for comparison is a simulation of the amplifier using the time-domain FET model and the standard harmonic-balance routine previously used in the theoretical that was analysis to derive the large-signal S-parameters. In order S-parameter to analyze the linearization, partition the device from system using it from this at the FET first necessary to the surrounding linear network. Although the incident power from incident power is the simplified the source is known, the terminals must first to determine the applicable be found s-parameters. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -91- Zl - •24 pF Figure 2.8 s o n Lumped-element equivalent circuit schematic of FET amplifier. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -92- MEASURED S PARAMETER SIMULATION CmW) 700 80 - GO OUTPUT POWER .7 — FULLY / / SIMULATED VGS - -3.0 V 20 YDS - 7.5 V 20 DELIV ERED Figure 2.9 40 POWER 50 80 (m W) Plot of FET amplifier output power as a function of delivered power, showing measured and simulated results. Simulated results were obtained using a complete nonlinear analysis, and also by using large-signal S-parameters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -93- following procedure outlines the modelling process. For a known incident circuit power: 1. Determine the incident power not the incident circuit at the gate. power if (This is matching elements are used). 2. From an estimate of output the system gain, determine the power reflected back onto the drain by the output matching elements and load. 3. Determine the applicable S-parameters to use to describe the FET. 4. Use linear elements with analysis the to cascade FET, using determine a system S-matrix. reflection coefficient, its the matching S-parameters to From this, determine the p, and power gain, Ap for the entire system. 5. Check and drain the calculated incident using the values of necessary, revise the values of powers at £ and A^ the S-parameters used incident power the gate found. If according to just calculated, and repeat step (4). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -94- 6. Determine the output power from Ap and the known delivered input power. Linearization by step (4), as S-parameters is extremely linear system concepts can for optimization and analysis. noted that if elements, it the device However, is is necessary to useful in then be employed it must still be surrounded de-embed these by matching elements in order to determine the incident power on the FET itself. This procedure was carried out experimental single-stage FET amplifier. relative insensitivity of and a good estimate of system made. device itself. S-parameters Because of the S12 to drive level, reflection coefficient could be This enabled relatively correct values of in the analysis of the |V1+ | and to the JV2+ I incident on the Consequently, are readily quick convergence the applicable large-signal selected, allowing a complete analysis of the system to be made. The results of this analysis are shown as labelled "S-parameter simulation" on shows the results results of the values of S-parameters. S-parameters of the second As iteration, using can be does not Figure 2.9. first iteration; )V1+ 1,|V2+ 1, and seen, the give strictly higher power levels; this was the curve Curve 1 curve 2 the self-consistent large-signal use of large-signal correct results at foreshadowed in the earlier Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -95- modelling analysis. error will be acceptable approach, and analysis. As can the becomes great at 1-dB considering accuracy be compared seen from the to computer-aided point. design of small error power, or is well beyond Applications schemes the signal figure, the power on the FET, which compression such an the ease approximately 28 mW delivered 22.7 dBm incident its However, in many instances, relying on include matrix manipulation techniques (such as COMPACT [8 ]). 2.4.2 FET Oscillator A second example of the the application S-parameters is design Intuitively, the oscillator of an works by of large-signal FET oscillator. feeding back some output power to the input, to act as drive for the device. If not enough power is fed back, the device is not driven hard enough, power is output power results. fed back, the device power feedback. 1V^+ 1 and low output and again Because 1V 2+ | is in saturation, suffers due of is the If to feedback critical that lost loop, to obtain too much but the through control of maximum power and to sustain oscillation. In the example |V1+ J and |\?2+ 1 given previously [10] , an is used S-parameters. The actual chosen can then |V1+ | S-parameters) from to set value of be the the oscillator device |V2+ I obtained estimate of for the (given the design equations. If Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -96- the value estimate, of IV2+ | does not |V2+ | and the agree with the initial selected S-parameters must be adjusted until a consistent result is achieved. of output power self-consistent varying the the power at for |V2+ 1 the may then IV^*! chosen given be The value |V1+ | and calculated. and repeating each operating point may By the process, be determined, and an optimum found. The predicted obtained by power + 2 |V1 | curve versus repeating the process obtained experimentally using so is very close to that power-added considerations, indicating the usefulness of the large-signal S-parameters in predicting output p o w e r . By oscillator at the optimum power 23.9 dBm was of 24.8 dBm. the level, an output power of obtained, very close to The oscillator constructing the predicted level circuit design is shown in Figure 2.10. In this instance, used to set below the compression effect of the operating saturation but point to above the S21 is the optimum point, small-signal area where output power is low (i.e. close to the maximum power-added efficiency p o i n t ) . + 2 |V2 | was 19.3 The optimum value of + 2 |V1 | and and 21.1 dBm respectively, for the same FET discussed earlier. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Zo=75a Hh L = .073x0 II - Z0= 50a L «.076Xg Zquj = 50a =t=z0 =50a l = .052x9 RFC .In H '6S Figure 2.10 FET oscillator designed using largesignal S-parameters. R eproduced with permission o f the copyright owner. Further reproduction prohibited without permission. -98- 2.5 DISCUSSION In general, a unique, large-signal S-parameter matrix for a nonlinear two-port can reflected voltage wave column elements are second column Using this be derived from incident and information only not functions elements are not of setting Alternatively, V 1+ functions and V 2+ only incident voltage wave the circular locus method of be verified by are zero. magnitudes, then Mazumder and van comparison with results where must S-parameters determined by be are to der Puije either case, these assumptions must simultaneously criterion V 1+ . are assumed to be functions of In first the of to if the S-matrix elements (34) may be used. the V 2+ , and assumption, matrix elements are alternately V 2+ if non-zero. satisfied be used The if V 1+ and uniqueness the large-signal with circuit-analysis and design techniques developed for linear circuits. It should be also be noted that the uniqueness applied determined terminals from voltage of a large-signal S-, be uniquely to large-signal Y- and current nonlinear If of the the representations can a nonlinear be converted to the other using standard linear techniques, even though the other parameters one measurements of two-port, then it can generally two representations Z- information at two-port. Y-, or Z- parameter determined from or criterion can circuit theory two m a y not be uniquely Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -99- determinable directly from measurements. However, for nonlinear circuits characterized by S-parameters which are obtained independently of phase (i.e. as functions only of incident p o w e r s ), the Y- and Z-parameters obtained through conversion will be meaningless, voltages and currents can no solely by the argument that only of the while definable, must be functions their voltages. magnitudes magnitude and phases of the nonlinear device can be varied incident power. Y-parameters large-signal However, of the sources supplying keeping This that terminal by a similar earlier, respective terminal varying the phases power, For example, presented Y-parameters, to be uniquely the longer be uniquely specified incident power. to because should the incident constant, terminal voltages will at will, affect be used; by for the at the the constant applicable however, the S-parameters, which are postulated to be dependent only on the magnitude of the incident In this case it is powers, will be unaffected. meaningless to convert between large-signal parameter matrices. When using Y-parameters the in terms of terminal voltages power flow. A high incident and currents, waves. To or a low terminal voltage, relative phases of the define a rather than power flow can correspond to either a high terminal voltage depending on the designer must work solely linear regime incident voltage for large-signal Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -100- Y-parameters in terms o£ incident power levels is both demonstrated that inappropriate and incorrect. In this chapter, computer simulation GaAs MESFET has been and experimental verify S-parameters (when it the applicability large-signal modelled as functions of the incident up to 20 dBm for this accordance with theoretical that level, a of power at each port) at drive levels FET, in measurements for predictions. higher order nonlinear effects Beyond will manifest themselves as large deviations from the circular impedance loci presented of in Section 2.3. greatest loci peak DC sharp noncircularity correspond gate current. peaks correspond at to the For left -1mA nonlinearity and and conduction of it is high DC is to areas of of 2.7c, the the figure respectively. This mechanism controlling severe of the the gate-source Schottky gate current, loci, which bottom +1. 3mA breakdown highly undesirable the regions example, in Figure indicates that the principle FET Furthermore, to operate in operation at correspond to lower gate-drain junctions. an area Since of such safer points DC gate current and on the and are also more circular, will result in the device behaving in a At these will be more linear S-parameter fashion. description Generally, however, incident powers the representation operating more applicable. is not (after removing the effects points, valid for of matching) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -101at either port greater than the 1-dB compressed output power of the device. As seen in the design examples, the approximation made in linearizing the MESFET entirely satisfactory in response, if the about its operating point is predicting fundamental frequency uniqueness conditions remain valid at that operating point. This chapter simulating) a and output has shown that by particular MESFET under drive conditions, determined and a set a measuring (or simultaneous input power range can be of large-signal S-parameters defined that are valid for use at that power level. We have examined one way in which the device can be represented in the nonlinear realm when driven b y a single tone. but The extension to considerably more harmonic excitations is possible, complex. complete nonlinear analysis is accounts for In the next chapter, introduced. nonharmonically related This analysis excitation and additional frequency components generated, a the and enables the device to be studied under multiple-signal excitation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -102- 3. 3.1 THE MODIFIED HAEMONIC BALANCE METHOD BANDPASS SAMPLING In the analysis of intermodulation distortion using the standard harmonic balance method, excitation is applied at frequencies f^ and f 2 . Since the output intermodulation components at (2f1~f2) and (2f2~f1) are also desired, the Fourier components of the interface current at these fre quencies must be found. For the present, we will neglect the components at (3f^-2f2) and (3f2-2f^) (fifth-order intermodulation products), as well as those at 2f^, 2f2 , 3f1 , 3f2 , etc. (higher harmonics), although they will be present in the time-domain waveform generated by the nonlinearity g(v(t), Let a = ^2” ^1 k e the frec3uencY spacing of the two applied signals. In order to include the components at f^-A, f^, f 2 , and f2 + A as consecutive coefficients in the discrete Fourier transform, A must be the "fundamental", frequency for the Fourier series. Additionally, if (2f2-f1)= f 2+ A is the uppermost fre quency component to be considered, then to prevent alias ing in the transform, sampling at the Nyquist rate would 2f2~f require an N-point sequence, where N = 2 ( — ~ — — ) . For a typical case of f 2 = 2 GHZ and A = 2 M H Z , approximately 2000 time-domain samples would be needed in the sequence to enable all the frequency components up to (f2 + A) R eproduced with permission o f the copyright owner. Further reproduction prohibited without permission. -103- to be found. This is obviously an enormous number of samples, particularly since the number of operations 2 needed in the DFT increases as N . The numerical reso lution of the computer would also rapidly be approached if such a high number of samples were used. If one were able to obtain the point spectra from such a large number of time samples, most of the consecutive fre quency "slots" spaced by A would be zero. For the simpli fied intermodulation case just considered, the only frequency components present are at DC, f2-f^(the beat frequency), f1~A, f^, f 2 , and f2 + A, as shown at the top of Figure 3.1. at With the exception of the beat frequency s^9na^- bandlimited i.e. all frequency components lie between f^-A and f2 + A . This observation suggests the use of bandpass sampling. The bandpass sampling theorem states that if a bandlimB B ited signal lying between f - -j- and fQ + - y is suitably centered above the origin at fQ , then the waveform may be completely reconstructed from time-domain samples of the signal taken at a rate 2B. The advantage of using this property is that the sampling rate 2B is now very much less than the Nyquist rate 2(fQ + /2). The proof follows easily from Equation (1.11) and the properties of the sampled Fourier transform. A complete statement of the theorem is given in Appendix 8.3. Qualitatively, if T = ^ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. AMPLITUDE -104- L I fi-A Periodic Sampling f, f2 f2+A FREQ. AMPLITUDE O ET b+c £ 1 A Figure 3.1 2A 3A 4A B FREQ. Simplified intermodulation distortion spectrum, with higher frequency terms omitted for clari ty. The spectrum at the bottom of the figure results from bandpass sampling of the waveform at the top. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -105is the sampling interval, the effect on the Fourier trans form of sampling the analog waveform xa (t) is to frequency translate the analog frequency spectrum X (jfi) along the CL u axis in multiples of 2tt, and to alter the scales accord ing to n = w/T. From Figure 3.2, no aliasing occurs if (2irB)T = ir, so the sampling frequency V t must equal 2B. The sampled Fourier transform X g (e^w) (Equation 1.9) thus yields the Fourier coefficients of the spectra in Figure 3.2. Since the spectra is periodic, we need only consider the baseband interval - ir<o)<u. Because the baseband spectra of X g (e^a>) is simply a frequency translated version of Xa (jft), the Fourier coefficients are identical to those of the original waveform. The above representation applies to a continuous line spectra for which an infinite number of samples can be taken to determine the function at any frequency within the band. However, in exactly the same way as before, a discrete point frequency spectrum can be obtained for a periodic band-limited waveform by considering a finite number of samples equally spaced within the period. In our case, a discrete Fourier transform ma y be used in the same way to extract the frequency components at f ^ - A , f2 , and f 2 + A. at a rate ^ f^, By the bandpass sampling theorem, sampling = V 9 A will ensure no overlap between periodically repeated spectra resulting from the Fourier Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ANALOG FOURIER TRANSFO RM Figure 3.2 SAMPLED FOURIER R EPRESENTATION A bandlimited spectrum is shown on the left, with bandwidth B. VJhen this waveform is sampled at a rate 2B and represented as a Fourier series in (e3“>) t the periodic spectrum on the right is obtained, fl is a radian frequency; w a radian angle. -107- transform. The discrete Fourier transform then allows us to use only a finite number of samples. of 9 samples cj= (N=9) , By taking a total the fundamental radian angle will be 2 tt /N, corresponding to a fundamental radian frequency of n = the point frequency spacing required. In this way, the sampling and DFT process will yield the frequency coefficients of the terms at A, 2A, 3A, 4A, ...9A, which are just the downconverted terms due to f-^-A, f^, f2 r and f 2 + A respectively, with the coefficients at 5A to 9A being just the complex conjugate of the terms 4A to A (symmetry property) . In this way, the number of samples needed to find the intermodulation products has been reduced from 2000 to 9, through use of the bandpass sampling theorem. One criterion of the bandpass sampling theorem thus far glossed over is the need for suitable centering of the waveform above the origin. The bandlimited signal shown in Figure 3.1 was assumed to be an exact integral multiple of 2B above the origin. The dot in the figure at f ^ - 2 A , corresponds to a vacant frequency "slot" so that, upon shifting of the waveform along the frequency axis in integral multiples of 2B, the component at DC aligns exactly with this slot. This is known here as the band- edge criterion, and enables the DC to be reconstructed from the signal, as no overlap occurs at this frequency. The band-edge criterion is achieved in the software by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -108- slight ly adjusting A so that the bandedge at f 1 - 2A is an exact integral multiple of the sampling rate 2B = 9A (Appendix 8.4). In this way, the lowest intermodulation product at f^ - A is downshifted to fall at the fundamental frequency A. Effectively, nine time samples are taken within one cycle of oscillation of frequency A, to obtain the Fourier coefficients of the first four (downshifted) harmonics and DC. 3.2 NON BANDLIMITED SIGNALS It should be noted that the beat frequency at f2 ” ^1 will also be translated by integral multiples of 2B upon sampling. For the case of zero translation, the beat component will occupy the frequency slot at A. Similarly, f 1_2A when downshifted — — times, the component at f^ - A will also occupy the frequency slot at A. Since, by Equation (1.11), the sampled frequency spectrum is given by the sum of the analog spectra over all possible periodic translations, the component given by the DFT as the funda mental frequency at A will not be the lowest order inter modulation product (that at (f^-A) downshifted) alone, but will be this component summed with the beat frequency component. The bandpass sampling theorem is seen not to be truly applicable in this instance because the waveform being sampled is not truly bandlimited i.e. it has a component at fj - f^/ which lies outside our assumed Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -109- bandlimited region extending from f ^ - 2A to f2 + A. Although we effectively overcame this problem for the DC component by leaving a "vacant" slot in the band window (at f^-2A) into which the out-of-band component fell after periodic translation, this way of avoiding the problem becomes increasingly complex as soon as the other out-of-band components at f^ + f2 , 2^2' e t c * are included in the analysis. It is necessary to note here that although the higher harmonics and beat frequencies may be negligible in the final physical solution (e.g. such components may be shorted out by the linear embedding network), they are of appreciable magnitude within the harmonic balance loop at the stage at which they are considered. As an example, a single sinusoid V cos oi^t applied to a square law device 2 with a nonlinear functional relationship i = g(v) = v will 2 give phasor currents of value V / 2 2 at DC and V / 2 at 2oi^. The next step in the harmonic-balance algorithm is to apply these phasor quantities to the linear network, where there they may be effectively blocked. However, the DFT must first find the phasor currents developed by i = g(v), with v known, before any bandlimiting effects are introduced by the external, linear circuit. For this reason, nonlineari ties of high order become increasingly difficult to handle Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -110- analytically, due to the large number of new frequency com ponents generated by the- nonlinearity, even though very few new frequency components may be observed externally. The effect, then, of the new frequency components generated by any nonlinearity is to extend the bandwidth of an applied signal, even if the desired output components and input signals are eventually bandlimited by a linear embedding network. In this work, a nonlinearity of up to fifth-order (when expanded in a Taylor series) was assumed. This was felt to be of high enough order to examine the third-order intermodulation properties of any general device (up to a certain operating level) without introducing ex cessive complexity. Previous analytical attempts at describ ing the microwave MESFET have principally used a third-order model [16,18], Given that higher-order components are present in the waveform generated by the raw nonlinearity, some means must be found to enable the desired components to be correctly found. There is no problem with a single frequency excitation, as in that case the only new frequency components generated are those at integrally related higher harmonics. The standard harmonic balance technique may be used in this case. The more difficult problem is for two applied signals at frequencies f^ and f2 which differ by A i.e. the inter modulation case. In this case, for a fifth-order Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. -111- nonlinearity, there are 31 distinct frequency components present in the waveform generated by the nonlinearity. highest of these components is at 5f2* The In the prior example, where f ^ = 2 GHz and A = 2MH2., the standard harmonic balance technique could be used bu'c with a fundamental frequency of 2 MHz. However, 10,000 time samples would be needed to include all the new frequency components up to 5f 2 . To perform a DFT on a sequence of this length would be staggering. As discussed previously, the use of bandpass sampling greatly reduces the number of time-samples required, but aliasing will result because the true bandwidth of the nonlinear partition signal is not confined solely to the eventual band of interest. The out-of-band components of 1(f) by convolving i(t) could be eliminated in the time-domain with a bandpass filter characteristic prior to sampling: 00 iBANDLIM (t) = J i (t ) h ( t - x ) d x — 00 00 i.e. i(k) = I (3.1) i(n)h(k-n). For a bandpass characteristic as shown in Figure 3.3 h Dr (t) = 2H_B sine B(t-t_) cos oj (t-t_) . O O O O Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -112- t*— -fc fi- (a) (b) (c) Figure 3.3 (a) The bandpass filter spectral characteristic. (b) The impulse response of the bandpass filter shown in (a). (c) The impulse response of a bandpass filter with bandwidth B = fQ . Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. -113- By choosing B = fQ , all out-of-band components are elimina ted, and the sine function in hgpCt) oscillates only several times before dying off. There are several drawbacks to performing such a convolution. One is the additional integration step re quired to find each new time sample. integration in Equation (3.1) To perform the (particularly with sufficient resolution to extract a weak intermodulation response from a strong fundamental signal), many additional time samples i(n) are needed to calculate each i(k). A second, and perhaps more severe restriction, is that the effect of any harmonics or difference frequencies outside the band of interest is lost. By including the numerical bandpass filter, a physically unrealistic step has been introduced into the model. Some studies have shown [18,19] that the effect of second harmonic source termination has signifi cant effect on the level of intermodulation product. Such an effect would be completely lost if the time-domain con volution were used to bandlimit the signal of interest. 3.3 CONTROLLED ALIASING It was noted in the previous section that if the standard harmonic balance technique were to be used, a 10,000 point sequence would be required to include all 31 frequency components generated by the fifth-order linearity. Since 5000 independent frequency coefficients Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -114- can be obtained from this sequence, each spaced by A, signals in most of the possible frequency "slots" will be zero. Although considerable aliasing would occur by selecting a band window B = 4 . 5 A followed by bandpass sampling at the rate of 1/2B, most of the aliased com ponents would be zero and thus not contribute to the sampled spectrum. Furthermore, the exact frequency of the non-zero components causing aliasing can be calculated, and their location in the aliased spectrum easily deter mined. This suggests the use of the controlled aliasing technique to determine the magnitude of these components. The waveform in Figure 3.1 has previously been con sidered in the application of the bandpass sampling theorem. Periodic sampling results in the aliasing shown in the bottom of this figure, where the components labelled b and c overlap in the spectrum obtained after sampling the time-domain waveform. x s (eja)) E°° xa ( ^ r=-~ Using + j^r>, (3-2) setting r = 0 yields the components labelled a and b at the angle corresponding to frequency 0 and A. Setting (f1-2A) , r = + -- — --- (which is an integer because we have ensured that the bandedge criterion is satisfied) translates the components c, d, e, and f into the frequency slots Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -115- corresponding to A, 2A, 3A, and 4A, respectively. sampled spectrum X The at baseband is the sum of the spectra corresponding to these two values of r (all other values of r translate X_(jfl) outside the baseband interval under cl consideration: the periodic repetition in X s results from this), as shown at the bottom of the figure. The magnitudes a, d, e, and f can be directly determined from the Fourier coefficients, but only the total magnitude of (b + c) is known. The heart of the controlled aliasing technique is to solve for the remaining unknown magnitudes by simple, linear algebra. Suppose the same two input signals of relative magnitude d and e are applied, as before, to the system, but at frequencies shifted down by A to f as shown in Figure 3.4. and For a low-Q device model (i.e. one in which the output signals are not greatly frequency dependent, such as for a resistive nonlinearity), the magnitudes a through f generated by the nonlinearity will be essentially unchanged; however, their frequencies will be altered according to the order of their product. The beat frequency at A will be unchanged; however, the intermodulation products will lie an amount A lower than in the first case. Furthermore, because the position of the applied signals is different relative to the bandedge, (fl-2A) different aliasing will occur. Now , when r = + --- — -— Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. AMPLITUDE -116- f. Periodic Sampling FREQ. AMPLITUDE O b*d ,0+C I I I I' Figure 3.4 J L FREQ. Simplified intermodulation distortion spectrum when the two applied fundamental tones are shifted down by A to f-.-A and f2"A = f1 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -117- in Equation (3.2), the band edge, occupied by the component c, will overlap with the DC component, to give a total component (a + c) at zero frequency (DC) , and the component at f^ - A, now of value d, will overlap with the beat component at A, b, to give a total Fourier coefficient at A of amplitude (b + d) . Since a and d were found from the first application of the DFT, the value of the components b and c can be directly obtained by simple subtraction. What has effectively been achieved here is the follow ing: by sampling the time-domain nonlinear current wave form at a very slow, bandpass rate, a total of N Fourier coefficients have been obtained from the DFT. By frequency shifting the two input signals a total of A, recalculating the time-domain waveform, and resampling, N additional Fourier coefficients can be obtained from the DFT. If the frequency translation, sampling, and DFT are repeated a total of m times, a total of mN Fourier coefficients are obtained. Because the way in which the new frequency com ponents will alias is known beforehand, a system of mN linear algebraic equations can be obtained, with right hand side equal to the DFT coefficients. By inverting the system of equations just once, the unknown amplitudes of any frequency component can be obtained from the (universal) inverse matrix and the DFT coefficient column matrix. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1183.4 SOLUTION FOR THE UNKNOWN AMPLITUDE COMPONENTS In the example just given, only two frequency shifts were required, since only 6 frequency components were present. The formulation and selection of independent equations and their solution for the unknown amplitudes was trivial in this case because all higher frequency components were ignored. In the case of two input signals to a fifth order nonlinearity, in which 31 frequency com ponents are present (including D C ) , the bandwindow of interest will consist of 6 signals each spaced by A (two fundamental and two each of third and fifth order inter modulation products) so that for bandpass sampling, a 13-point sequence of time samples spaced V l S A in time may be used. Since 7 real and 6 imaginary Fourier coefficients will be obtained from each 13-point sequence (DC, A, 2A,... 6A), a minimum of 5 frequency shifts of the input signals will be needed to ensure a sufficient number of equations is obtained to enable solution of all 31 unknowns. Two sets of equations are thereby obtained; a set of 35 equations relating their real parts and DC; and a set of 30 equations relating their imaginary parts. Selection of a 3 1 x 31 matrix set (or 30 x 30 matrix set to define the imaginary parts of the coefficients) from the equations to be inverted is not as trivial as in the example given, and constituted much of the work of the present dissertation. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. -119- We have elected here to choose a bandwindow of 6.5A and calculate the Fourier coefficients from a sequence of 13 time samples sampled at a bandpass rate. components are aliased. The resulting However, by shifting the input frequencies slightly, different aliasing products will result. By repeating the shifting 5 times, it is possible to deembed the desired components from the aliased products. However, it is worthwhile to note at this stage that several alternative schemes are possible to deembed the aliased spectra. The bandwindow of interest could be made wider than the 6.5A chosen. This would open up more vacant slots and simplify the aliasing, but require more time samples and be less computationally efficient. Alterna tively, certain symmetry properties might be used - if the two input signals were to be always of equal level, the intermodulation products and harmonics would also be the same amplitude, and the number of unknowns nearly reduced by half. This was not done so that the gain of a small signal in the presence of a closely spaced, but larger, saturating signal could be studied. Finally, for certain nonlinearities, the sum and difference frequencies generated will be equal in magnitude. This fact could again be used to reduce the number of unknowns,- and simplify the matrix set. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -120- At first glance it might appear difficult to determine where each frequency component will fall after aliasing i.e. after periodic translation to baseband. However, by recalling the symmetry properties of the DFT covered in Chapter 1, and noting that the original band-edge frequency satisfies the band-edge criterion i.e. always translates to DC, a simple "foldback" technique can be used to deter mine the baseband position of each component after aliasing has occurred. Consider the spectrum shown in Figure 3.5. The com ponents at d and e are the two tones at f1 arid f2 = f1 + A. a is the DC component, b and g the fifth-order intermodula tion products, and c and f the third-order products. When (f1-3A) r = + -------- in Equation (3.2), and is an integer because 2B of the band-edge criterion, the components a-g will occupy the baseband frequency slots OA, A, 2A,...6A after sampling. Consider now the second harmonic of f^, at a 2 (fi~3A) 2 ( f ^ - 3 A ) + 6A. When r = r^ = + — — ----, which is again an integer, the component labelled h will fall into the base band slot at 6A. The second harmonic of f2 , labelled j, is at 2(f1-3A) + 8A. When r = r^ this component does not fall into the baseband window, since 8A is outside this window. When r = r 1 ~l, the component j falls into the slot at (8A-2B), or at -5A, and again is not in the baseband window for the DFT. However, when r = - r ^ + l , Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -121- a) k Xa (j’A) d*e. AflNOCtfF AT 2 f £ - ^ c .. . V & , ■If Kfi-65^ Xs(e^) fr a ia tu c & b) S h - ^ i J I aTSfe' a m p l in g d c |f . b A jl - _ O L -/ 2A 3A 4& f A £A 4 . fi c) A» 4. e | Reflection i. 0 t I. L 3 + S 6176 K .M Figure 3.5 3 10 11 12 13 ft fS * + + * + * (a) A more complex distortion spectrum. (b) Spectrum showing aliasing that has occurred due to bandpass sampling. (c) The simple foldback technique that assists in the determination of this aliasing. The frequency scale in (b) and (c) has been increased to allow greater resolution. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -122- the negative frequency component of the second harmonic component of f2 (i.e. at -2f2) is translated in the positive frequency direction and falls into the slot at + 5 A, which is now in the DFT baseband window. Because of the symmetry properties of Xa (jft), the component at -2f2 has the same real but inverted imaginary part to that at * 2f2 . Thus the aliased component is labelled j because of the inversion of its imaginary part (see Figure 3.5(b)). A simple foldback technique allows quick recon struction of the aliased spectrum. With the original bandedge labelled slot 0 (since it translates to 0 fre quency) , consecutive slots are labelled 1-13 within the total bandwindow (see Figure 3.5(c)). The slot 13 will also translate to zero frequency upon aliasing because it is a unit multiple of 2B in frequency above the bandedge (the DFT is periodic with frequency period 2B ) . Slots greater than 13 are again labelled 1, 2,...13. Within the total bandwindow, slots labelled 7 through 13 can be treated as if they reflect in an imaginary mirror located at 6.5A, with inversion of the imaginary quanti ties occurring upon reflection in the mirror. Physically, these slots correspond to negative values of r, in which the negative frequency components of the signal will alias into the baseband window 0 to 6.5A. Alternatively, they can be thought of as periodic with slots -6 to -1, located 2B = 13A further down the frequency axis. Because they Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -123- then represent components of frequency on the negative axis, the complex conjugate of their component amplitudes is taken when they appear in the baseband window 0 to 6.5A. Now, because the periodicity of the band has been accounted for by resetting the bandedges to an origin of 0, frequencies such as 2 f 2 , 3f2, 2f 2~f^, etc. are found simply by using the new coordinate system. Thus in Figure 3.5(c), with f 2 in slot 4, 2f2 falls into slot 2 x 4 = 8 which reflects into slot 5, taking the conjugate; this is the same as derived from the first principles Equation (3.2). • The component 2f^, on the other hand, falls directly into the slot at 2 x 3 = 6, and is already inband. This enables the total positive and negative frequency axes to be wrapped around onto the baseband window of interest. Using this technique, five sets of translations of the two input signals were made with the lower input signal lying in turn at f^-2A, f^-A, f^, f^+A, and f^+2A. The resulting aliased spectra were found by the foldback tech nique in each case, as a function of the unknown amplitudes of the 31 frequency components (31 real and 30 imaginary unknown amplitudes, as the DC is real only) . On applica tion of the discrete Fourier transform to the output waveform, each translation set yielded 7 real Fourier coefficients and 6 imaginary Fourier coefficients Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -124- (the DC slot yielding only a real number). As a result of the 5 translations, 35 real equations and 30 imaginary equations were obtained. The resulting equations were sparse with coefficients of either + 1 or - 1, as each amplitude appears only once for each set of translation. The 30 x 30 matrix relating imaginary amplitudes to the imaginary parts of the Fourier coefficients is uniquely invertible. Selecting 31 independent, real equations from the 35 options and trying to solve for the 31 real ampli tudes proved to be far more difficult. The inversion procedure was first tried for a thirdorder nonlinearity. Three translations were required to solve for 13 real unknowns, as each translation set yielded five equations. A linearly independent set of equations could be obtained by deleting one equation from two of the translation sets, and using the remaining thirteen equations as a basis. For the fifth-order problem each translation set yielded seven equations in the real amplitudes. One equa tion in turn of the total of seven was dropped from four of the translation sets to reduce the total set to 31 equations in all. Numerous equations were deleted in turn, but no matrix of sufficiently small condition number was ever found to give a reasonable inverse matrix. Using the APL compiler running on an I B M - P C , determinants Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -125of the 3 1 x 31 matrices tried ranged between 10- ^® and 3 x i o ® , but in all cases the conditioning was so poor that the inverse was useless. The inversion process was also attempted on an IBM - 370 mainframe computer using a sophisticated IMSL matrix routine [37]; error messages always showed an "algorithmically" singular condition. The problem of sparse matrices and the selection of an independent basis is an interesting mathematical puzzle that arises naturally here. In system theory, the problem can be solved by a least squares approach [38]. For a linearly dependent system of equations, [A] [x] = [b], where A has more rows than columns, a least squares fit can be obtained by premultiplying both sides of the equation by A T and solving the resulting equation [AT ] [A] [x] = [AT ] [b]. Here, m [A ] is the transpose matrix. T The new matrix [A A] is now square and under certain conditions will be inverti ble. The additional cost is that the known column vector T [b] must now first be premultiplied by A . In our case, because of the very poor condition of any equation set, an additional orthogonal equation was obtained from a sixth translation, which positioned the lowest input tone right on the bandedge (f^-3A) . In this translation set, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -126- the term at 5f2 fell into a slot alone. This equation was obviously orthogonal to any of the other 35 equations. By then removing 1 equation from each of the other trans lation sets, an invertible, well conditioned, 31x31 matrix set was obtained. The inverse matrices could then be used to operate on the column vectors of Fourier coefficients obtained from the six translations, to yield the desired amplitudes of all 31 unknown frequency components. 3.4.1 Modified Harmonic Balance Procedure Because of the additional frequency shifts required to deembed the aliased amplitudes, several modifications were made to the standard harmonic balance software. Surprisingly, no changes were required in the DFT Goertzel algorithm that performed the time-sample to frequency conversion. Given the fundamental angle, 2ir /N, and the time samples, the routine simply performs the summation given by Equation (1.13), (but in a more efficient manner), in order to calculate the Fourier coe fficients. This routine is effectively blind to the aliasing and frequency shift ing of the time-domain waveform (Appendix 8.5). On the other hand, the routine performing the phasor (frequency) to time transformation was considerably more complex. In order to obtain the nonlinear current Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -127time-samples from the model, the inputs of gate and drain voltage and their derivatives, at the same periodic time instants, had to be calculated from their phasor values. At this point, a total of eleven phasor voltages were impressed upon the model, to allow for higher order effects. These were the basic input voltages at DC, f^, and ± 2 > addition to the self-generated voltages at f^-2A, f^-A, ^2+ 2f^, 3f^, 2f2, and 3f2 (caused by the nonlinear .currents in the linear circuit and the developed voltages across the model terminals). The frequency-to-time-domain conversion necessitated the calculation of terms like cos 2ir(fn-2A)t , l s sin 2ir (f^-2A) t g , cos 2tt (f^-2A) (tg -T) , and sin 2tt (f1-2A) (tg -T) etc. to transfer the phasors to the time domain. These sinusoidal and delayed sinusoidal terms were stored in 6x11x13 arrays, calculated once at the beginning of the program: 6 frequency shifts, 11 phasor components, and 13 sampling time-instants. Because the frequency shifts were in multiples of A, many of the frequency components had common values. This overlap was used to improve the efficiency of setting the arrays (Appendix 8.6). The vol tage derivatives were obtained by simple analytic differentation (multiplying by jw). The total time sample was then obtained by summing the contribution from each of the phasor components (Appendix 8.7). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -128- The process of deembedding the aliased components also necessitated software modification to the standard harmonic balance method. The inverse matrices were read from a data file and stored in 3 1 x 31 and 30 x 30 arrays. The 7 Fourier coefficients from each of the 6 frequency translation sets were held in 6 * 7 arrays. The deembedding process then simply consisted of matrix multiplication, taking care to eliminate, in the multiplication routine, those elements of the Fourier array corresponding to equa tions not used in calculating the inverse matrix. However, all 30 equations due to the first five translation sets were used in finding the inverse matrix for the imaginary components of the phasor amplitudes, so all imaginary Fourier components were used. Additional changes were also needed in the harmonic balance loop. Although a total of 31 frequency components are generated in the current waveform by the nonlinearity in response to an excitation at f^ and ± 2 , only 11 of these (stated earlier) were of sufficient magnitude to generate a voltage worthy to be reimpressed as an input to the model. This enables the effect of DC, fundamental, second, and third harmonic terminations to be examined (while any fourth and fifth harmonic voltages would effectively be shorted o u t ) . For a solution to be achieved, consistency had to be obtained in both the real and imaginary parts of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -129- all eleven phasor currents, at both the gate and the drain on both sides of the linear-nonlinear interface . This solution is obtained by the iterative nature of the harmonic balance method. During each iteration of the loop (i.e. steps 2-6 in Section 1.3), step 3 must now incorporate the six frequency translations to calculate the six sets of time samples sampled at the bandpass rate. Step 4 must perform a DFT on each set to calculate the Fourier coeffi cients, and finally perform the inverse matrix multiplica tion to deembed the desired phasor current components. The error function in step 5 is calculated as a squared ampli tude error sum based on the eleven predominant phasor components, and the minimization is attempted by adjusting these values accordingly (see Section 3.4.2) . Looping to step 2, these new currents are then fed back to the linear circuit to calculate new phasor voltages predominant frequencies) (at the eleven to reimpress upon the model (Appendix 8.8). The savings are quite substantial using the MHB tech nique. Six sets of frequency translation, each requiring 13 time samples are needed i.e. 78 time samples are re quired to deembed all 31 frequency components. This is several orders of magnitude improvement over the crude Nyquist method, which would have required 10,000 time samples to achieve the same result. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1303.4.2 Achieving Convergence The aim of the harmonic balance loop is to minimize the error function. For zero error function, the currents sunk by the nonlinear model are exactly equal to those sourced by the linear partition at every point in time (or frequency), given equal node voltages at the interface. Usually, convergence is deemed to occur whenever the error function falls below a certain threshold. At this point, the gate and drain current phasors (at the eleven predomi nant frequencies) on the two sides of the partition are deemed to be equal for all practical purposes. What forces the error function to zero on successive iterations of the harmonic balance loop? Basically, the initial guesses made for the terminal interface current phasors at the beginning of each iteration are made closer to their true value each time. As the current phasors are complex quantities, there are a total of 44 real variables that must be adjusted (accounting for both gate and drain branches) on every iteration of the loop. Initially, an IMSL routine the error function. [39] was used to minimize This routine uses a quasi-Newton optimization algorithm to alter the input variables current phasors) to minimize the error function. (the However, it was found that this gradient method would not converge for more than 7 or 8 variables and that the number of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -131- iterations through the loop exceeded 1000 for this many variables. Convergence 10 variables, was never achieved for more while the number than of loop iterations increased dramatically as extra variables were added [pg. 121, r e f . 22]. The fixed-point iterative method of Hicks and Khan [40] was found better to adjust the current phasors to ensure convergence. Because the FET model solves for gate and drain current as implicit functions of the terminal vol tages, the current was always the dependent variable; however, the current was also used as the variable updated each iteration. After the k ^ 1 iteration of the loop, consider a general current i _I v a x^lt) = x^ e J _j1{noj,+ma)9) 2 t , n, m nro so that i. is one of the eleven phasor currents n,m arising from the nonlinear partition, with corresponding Ik (t)=Z I ej(na)i+ma)2}t n,m Urn arising from the linear partition. then carried out with The next iteration is (t) having components I <k+1>™n m nm = P i *„,n nm + a ~ P ) l * rnm where p is determined by convergence considerations and 0<p^l. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -132Hicks and Khan [40] show that the rate of convergence is dependent on the ratio of the impedance of the nonlinear partition to that of the linear partition at each harmonic. This may be altered by inserting an "identity" network at the interface. Because of the implicit form of the non- linearity, adding a shunt impedance element to the non linear partition, and a corresponding negative shunt element to the linear partition, will alter this ratio. This alters the calculation of the hybrid matrix H(nu), and consequently the magnitudes of the phasors being updated, without altering the overall observed circuit performance. Furthermore, the loop was modified slightly so that initial ly the terminal voltage phasors were estimated and consequently updated rather than the current phasors. This would enable the error magnification factor defined in [40] to be controlled by the choice of shunt element used in the identity network, and the loop would then correspond directly to the voltage update method described therein. Both the current update (guess the initial current) and voltage update (guess the initial voltage) methods were applied to the same problem, the latter incorporating a variable identity network for convergence acceleration. Both methods were found to give the same, externally observable results whenever convergence was achieved. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -133The shunt identity element was found to be of no assistance in accelerating convergence for this problem. The shunt resistance in the identity network was varied from -100 ohms up to +50 ohms before convergence was finally achieved. Even with a shunt impedance of 1000ft, conver gence was much slower than for the current update method for which no identity network was present. The value of the first guess for the current or voltage phasors was invariably zero. If some other value was used as a uniform guess for the 44 unknown phasor quantities, the error function was generally too large for convergence to ever be achieved. In an attempt to equalize the rela tive contribution of the phasors to the error function, the DC and two fundamental terms were weighted more lightly than the other contributions because of their larger magnitudes. This, of course, did not affect the rate of convergence as the error function is used only as a means of establishing a convergence criterion. However, by slightly adjusting the point at which convergence was deemed to have occurred, the relative accuracy of each of the phasor terms can be altered. For typical intermodulation simulations, a value of the iteration constant p between .2 and .3 proved to have the quickest convergence. was set at 1 0 ~ ^ . The error function threshold At low signal levels, approximately Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -13430 iterations of the harmonic balance loop were required per simulation. As the signal level approached saturation and the model became more highly nonlinear, up to 120 itera tions were required. If greater than 150 iterations were needed, or the error function grew to exceed 10^°, cutoff occurred and a new value of the iteration constant p was requested. In all the similations presented in the follow ing chapters, convergence was achieved within 150 iterations. With 44 variables to equalize, this system is one of the largest applications of the harmonic balance technique yet reported. 3.5 LIMITATIONS OF THE TECHNIQUE The modified harmonic balance technique is the first reported application of the harmonic balance method which is able to handle nonharmonically related signal excita tions in a nonlinear circuit of potentially any complex ity [26]. As such, it is a tool capable of handling, in considerable detail, the analysis of nonlinear and intermodulation distortion, crosstalk on communication channels, the suppression of small signals in the presence of larger ones in a nonlinear system, and the analysis of other parametric devices such as mixers. One assumption and one limitation have so far been made. Both of these are critical only because of the dynamic range we are demanding of the system. In the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -135- case of intermodulation distortion in amplifiers, we need to be able to simultaneously observe both the funda mental output signals, and the intermodulation products, which may be 60 dB below the fundamental. The dynamic range of the system must thus be at least 60 dB, or the accuracy with which all computations must be performed is one part per million. It is this extreme resolution that ultimately limits the performance of this analytical technique. The assumption we have made, and carried through the work, is that the nonlinearity is of no more than fifth - order. Any higher order frequency products generated by a nonlinearity of order higher than the fifth will cause aliasing that has not been accounted for and will not be deembedded. In the FET model, for instance, the drain current is modelled as a hyperbolic tangent function of the drain voltage, and as square law in the gate voltage. Expanding ID as a power series, ID^VDS^ ~ tank(nVDg) - I ^ d s ' - 1 ^ D S 1 3 + T S ( ' " W 5 * ^ D S 1 7 + • • • At small values of V D g , the seventh order term makes negligible contribution to the current ID , so that fifth order behavior is a fair assumption. However, as the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -136- signal swing V Dg is increased, the seventh order term contributes to the total drain current; this term generates additional frequency components (such as the'seventh har monic and seventh-order intermodulation terms like 4f^- 3f 2 ) , which will cause additional aliasing. Fortunately, when the aliasing becomes more complex than that predicted by the fifth-order behavior, the failure of the deembedding is readily detected. Two equal signal levels as input will no longer produce symmetrical intermodulation sidebands (as they should) if extra aliasing terms are present. This phenomenon is observed in the FET simulations for two sig nals as saturation is approached, and the model behavior extends past the "well-behaved" fifth-order regime. This behavior however, is not a fundamental limitation on the method. The deembedding procedure could be extended, at the cost of some complexity, to account for sixth and seventh order products, and so on; alternatively, as done in the next chapter, a FET model could be fit to a fifthorder nonlinearity, or less, so the method could be used as it presently exists. For the complete FET model used in Chapter 5, however, our simulations will be limited to that regime below the point at which assymmetry effects due to aliasing are seen to occur. The major limitation of the method is that the device model be low Q, so that the frequency translation (which Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -137is applied only to the nonlinear partition of the circuit and not to the external linear circuit) does not affect the magnitude of the frequency components that we seek. This presents no problem at all if the nonlinearity is purely resistive, or if any frequency dependent elements in the model can be linearized and partitioned into the linear circuit. However, for reactive nonlinearities, the frequency translation can produce minute changes in ampli tude of the signal, with devastating results. the simplest reactive nonlinearity of all: capacitor. Consider a shunt, linear For this element, with a sinusoidally varying terminal voltage of amplitude V, the resultant current is i = jwCV, so that Ai = A uj. A frequency shift of 2 MHz on a 2 GHz signal will produce a relative amplitude change in the fundamental current phasor of 10- ^. Unfortunately, this is comparable to the signal levels of the intermodula tion terms we are seeking to deembed, and the controlled aliasing procedure is rendered worthless. This phenomenon predominates at extremely low level signals when the need for resolution is greatest, and can again be detected by asymmetry in the spectrum produced by two equal-level signal inputs. Fortunately, it may be eliminated by choosing an extremely narrow signal spacing relative to carrier frequency. ratio of 10 —9 A signal spacing to carrier frequency was used in testing with no adverse effects Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -138on accuracy; for the ensuing simulations, a frequency spacing of 1 kHZ was used. This entirely eliminated any effects due to varying amplitude level during frequency translation of the carriers. Several comments about the narrow spacing should be made here. Firstly, such a narrow spacing is never physically used during testing only because of the difficulties associated with main taining two sources frequency-locked at this spacing, and because of phase noise problems that arise so close to the carrier. If, in fact, it could be used, it would permit much more accurate use of the spectrum analyzer to measure intermodulation products as smaller resolution-bandwidths and slower sweep speeds could be used. Secondly, the intermodulation and other distortion products have been found experimentally to be independent of frequency separa tion of the two carriers Volterra series [14]. Finally, neither the [18] nor analytical approaches [12-17] can account for frequency separation in their analyses. Having stated the assumption and major limitation of the modified harmonic balance technique, we turn our attention to testing the performance of the technique and verifying its analysis by comparison with the Volterra series approach, before undertaking complete experimental comparisons with the Madjar-Rosenbaum FET model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -139- 4. TESTING THE MODIFIED HARMONIC BALANCE METHOD In the development of much of the software for the modified harmonic balance method, many observations p r e ceded the unearthing of pre-existing theory* For instance, both the band edge criterion and the foldback technique came to light only through persistence in debugging fre quency transformation software that wouldn't work! As a result, the fait accompli presented here has in fact been pieced together from subassemblies that have been extensively tested, and in fact, individually serve as illustrations of much of theory already presented. The final testing of the program consists of three phases: (i) test of a linear model, to check deembedding, derivatives, time delay, and time-to frequencyto time-conversions (ii) test of a fifth order nonlinearity, with un equal, (iii) real, and imaginary inputs comparison with previous analytical techniques. The only facet of the software not tested here was the performance of the linear circuit partition (calculation of the hybrid matrix relating voltage to current for the linear circuit), which was extensively tested in previous work as part of the standard harmonic balance technique [41] . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1404.1 LINEAR TEST The first test is relatively straightforward and simply checks the performance of the DFT and ensures that the band edge criterion is successfully applied. For performance checks such as these, the parasitic and linear circuits were effectively removed by judicious choice of element values, in order that the applied voltages were impressed directly upon the nonlinear partition. nonlinear subroutine, with and In the the current and vol tage at the partition gate node and C 2 and V 2 those at the drain node, by setting C1 = V1 and C2 = V2* a one ohm resistor effectively serves as the nonlinear element between the gate and source and the drain and source. The first simple check is to verify that the real and imaginary parts of the applied voltages are reproduced as the relevant fundamental output currents, and that all other intermodulation and harmonic terms are zero. A second check is to verify that the delayed sine and cosine terms preserve the delay. A time delay of t = .02 n sec was used, which corresponds to I/5 of a cycle delay at 10 GHz, O or 72 in phasor terms. The current phasor was then found to lag the applied voltage phasor by this amount, as required. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1414.2 FIFTH-ORDER NONLINEARITY TEST The most useful check of nonlinear performance is to apply an input signal of the form V = a cos w^t + b sin u^t to a fifth-order nonlinearity: I = V + V2 + V3 + + V3. Such a nonlinearity generates all possible 31 frequency components. The use of two unequal signal levels a and b forces asymmetry in the spectrum, and the use of cosine and sine terms checks both the real and imaginary deem bedding. Analytical substitution of V into the expression for I is surprisingly complex. The following are the DC and three of the terms of interest generated: DC: cos (301^- 20)2)t: “ ^8 sinUo^-oij) t: - f [f a2b (a2 + b 2) + ab (a + f a 3 + f a b 2 ) + f 2 (b + f a 2b + f b 3) C O S {Uj) t Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -142A computer listing of the results for the case where a = l and b = 1 is given in Appendix 8.9. Table 4.1 gives a comparison of the computer-generated terms and those generated using the analytical expressions above. Note that the computer calculates residual complex components which are at least three orders of magnitude smaller ( - 60 dB in power terms) than the principal value, due to the accumulation of numerical errors. Such residual values are negligible compared to the total component. Agreement is seen to be good. It is interesting to note that in examples such as these, where there is no linear circuit network to alter applied voltages at the interface on successive iterations, convergence was achieved with any value of the iteration constant p. by 10 —2 With p = 0 . 9 , the error function is reduced each iteration (because the linear circuit contri bution is effectively absent and the interface voltage remains constant) so that just 9 iterations were needed for convergence to be achieved. The presence of parasitic elements, however, often gave an unstable solution, which diverged for any value of iteration constant. As a final test, a reactive nonlinearity given by I=x+ x 2 +x 3 +x 4 + x 5 was used, with x = k ^ V ( t - x ) , and k a constant scaling Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -143- Table 4.1 Comparison of analytical and computer generated components for a fifth-order nonlinearity COMPONENT COMPUTER ANALYTICAL DC 3.25 3.25 3RD ORDER -3.875 + j 3 .5 x 1 0 ” 4 -3.875 5TH ORDER -5 x 1 0 -4- j .625 FUNDAMENTAL 1.67 x 10~3+ j 9.50 2ND HARMONIC -2.5+j7.7 x 10 4 3RD HARMONIC 1.813-j1.23 x IQ” 3 -0.625j 9.500j -2.500 1.8125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -144factor. The presence of the derivative adds frequency dependence to the nonlinearity (up to fifth o r d e r ) , and the presence of a time delay ensures mixing of the real and imaginary parts of the phasors. Using V of the form V = cos w1t + cos u)2t , the output current components at the fundamental as well as the third and fifth order intermodulation products, were found to be both symmetric, and lagging the voltage by a constant phase angle. The reactive dependence of the nonlinearity was rendered invisible to the frequency translation sets by choosing a frequency spacing of 1 kHz (with 10 GHz carrier). 4.3 APPLICATION TO A SIMPLIFIED MESFET MODEL Numerous authors have approached the problem of intermodulation distortion in a MESFET amplifier in a variety of ways. While analytical approaches [14,16,17,18] have been used, they involve considerable manipulation to extract the intermodulation response. Minasian [18] has applied a Volterra series approach to a weakly nonlinear model for the FET, and because his model is very nearly a third order one, represents a prime candidate for appli cation to the modified harmonic balance technique. Because the modified harmonic balance technique is capable of describing grosser nonlinear behavior than Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the Volterra series, it is also interesting, by way of comparison, to carry the simulations to a higher degree of saturation (and also, it should be noted, into a range exceeding the validity of the model) than possible with the Volterra series. The large-signal circuit model used in [18] for the 2ym Plessey 360ym-FET is given in Figure 1.5. The three nonlinear elements used in the model are the transconduct ance term, gm , the gate-source capacitance CG g , and the output conductance gQ . The voltage dependence of these terms is simple, and described by the closed-form ex pressions below: gm(v i> = °-12 + (.0°24)Vi- (.°016)vi2 cG (Vi) = .364 + ( . 0 8 2 ^ * o (V2> = , 'Til' (v2) (8) (4.1) (pF) (4.2) <ts) (4-3) where V^ represents gate-source voltage and V 2 the drain-source voltage. The problem with the model presented in Figure 1.5 is that the control voltage v across the gate capacitor is implicitly contained within a branch of the model. As the harmonic balance software requires terminal voltage as an input, and calculates current into the gate and drain nodes as an output, some simplification is necessary Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -146- to reduce the model to one consisting of just two input nodes. The approach is identical to that used to eliminate the gate charging resistor in the Madjar-Rosenbaum FET model [22]. Refer to Figure 4.1 below. Figure 4.1 Simple series resistorcapacitor model for the FET input. If v x = R e t V ^ e ^ * ) , then v g = v.. 1 . . ■■ -p1 + 3 “c g r i Using the average value of in Equation 4.2 as .36 pf, R as 45 ohms, and w as 1.5 * 10^°, v g = v l ,971 / " 13.7° k v lD (vx delayed) Thus the control voltage is simply the input voltage o . delayed by 13.7 , and scaled in magnitude. In this way, Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. -147- the current flowing in the R - C branch is given by , XG jwCG v l“ 1 + DwCqRj = (jaCG ) v lD . As Minasian [18] does not give any value of transit time delay t to be used with the transconductance, no delay was assumed other than that arising from the charging resistor and capacitor combination just considered. The nonlinear nodal equations may thus be expressed analytically, from Figure 1.5, as These are easily programmed into the harmonic balance software, as the nonlinearity is now of the desired form i = g [ v , ^ ] . The principal nonlinearity of the model is that due to the transconductance current term gm v^D . The current generated by this term is to third-order power in the input voltage v^. As a result, third-order intermodula tion products will be generated in the output waveform. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -148The external linear circuit was set up as described in [18]. A 50ft source impedance was used (no matching). a) or The output load was chosen input to be either 120 + j80ft for optimum gain b) 110 + j40ft for optimum power. The load impedances were estimated from Smith Chart plots in [18]. There, the optimum load impedance for maximum power was deterimined empirically, as the range of validity of the model (and the Volterra series analysis) extend into the compression region of the FET. did not As a con sequence, only the reduced gain due to using an optimum power load is observable in the simulations. Graphs showing output power versus input power are shown for the two different load terminations in Figures 4.2a and 4.2b. Three sets of data points are shown for each of the two curves, representing measurement, simula tion by Volterra series, and simulation by the MHB technique. The upper set of curves in each graph shows the fundamental output power (per carrier); the lower set shows the third order intermodulation products. In view of uncertainties about the exact values of time delay t , the true load impedances, and higher harmonic terminations, agreement is surprisingly good. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -149- HHB OUTPUT POWER (dBm) X VOLTERRA SERIES -20 -50 -60 INCIDENT POWER (dBm) Figure 4.2 (a) Measured and simulated output power versus input power for optimum gain output load. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -150- 10 - MHB VOLTERRA SERIES - -10 - -20 - OUTPUT POWER (dBm) 0 -3 0 - -4 0 - -5 0 - - 60 - INCIDENT POWER (dBm) Figure .2 (b) Measured and simulated output power versus input power for optimum power output load. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -151It is interesting that saturation effects are illus trated remarkably well by the MHB method in Figure 4.2b. In fact, as the input power continues to increase into saturation, the cubic term (which is negative) in the expression gm Vj_ dominates and the output power decreases. However, over most of the useful dynamic range of the FET, a third-order nonlinear model was sufficient to predict the principle nonlinear effects. As the input driving voltage of the two signals was increased, the presence of fifth-order intermodulation products was noted. At first, these products were thought to arise from the higher order nonlinearity present in the expression for gQ , Equation (4.3). However, principally because the exponent of v ^ in the denominator is so close to 1, and the term is then multiplied by V 2 to obtain the conductance contribution to current, elimination of this component did not eliminate the observed fifth-order sideband. Next, by changing the cubic term in gmv lD ( - . 0 0 1 6 in Equation (4.1)) to zero, all third and fifth order products were lost remaining. (i) with only the fundamental term This illustrates two important points: in the Minasian model, the cubic term in the transconductance current g v. is extremely m x sensitive in controlling the level of the third order intermodulation distortion, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -152and (ii) the fifth-order intermodulation terms arise due to reinteraction of the third-order intermodula tion products with the nonlinearity. This is a higher order effect, and accounts in part for the effect of the external circuit on the level of intermodulation products. The currents at 2f2~f^ and 2f^-f2 cause voltages in the linear partition, which, when reimpressed upon the non linear partition together with the applied external voltages at f-^ and f 2 * create new, third-order sidebands at 3f^-2f2 and 3f2~2f^. In summary, the MHB technique has been shown to be compatible with existing models and equal in performance to previous analytical tools capable of handling multiple frequency excitation. However, the advantage of the MHB method over the Volterra series approach is that any model, expressible in the form i = g(v),may be used in the loop. The complicated task of model analysis to determine the Volterra kernels or nonlinear coefficients can be bypassed. Additionally, a greater number of interaction terms may be considered simultaneously (eleven in our case), and finally, a higher order nonlinearity may be used once- the deembedding software is in place. All of these advantages will be seen in the next chapter where an analytically intractable model is used with a complex parasitic and embedding network, to model intermodulation distortion in an FET amplifier. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1535. MODELLING OF NONLINEAR AMPLIFIER BEHAVIOR 5.1 PRELIMINARY MATCHING OF THE MODEL TO THE NEC72089 As described in Chapter model was modified to give the FET characteristics 1, the an efficient representation of capable computer memory. The modified table, a short which is the storage within data table pinchoff transconductance gm , of the approach creates a look-up electrical properties such as ID S g , Mad j ar-Rosenbaum FET containing the FET the saturated drain current voltage as well Vp/ as the and the endpoints of the four nonlinear capacitances which are interpolated for any given (VG S ,VD g ) combination to matrix valid at that operating yield point. harmonic balance program, the look-up created. which capacitance Before running the table must first be This is done by calling the Madjar routines require, semiconductor dissipation. the a as inputs, properties The of FET geometrical the FET, interpolation routines look-up table are then and [1], factors, the power and reading performed from within of the harmonic balance program. Additional modifications [22] are also used. In made to particular, the model the by Green addition of Schottky diodes between the gate and source, and drain and source, illustrated in Figure 1.11, permits the modelling Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -154- of gate breakdown current in effects. this way, By allowing forward clamping for DC and self gate biasing through a series gate resistor can be fully modelled. diode characteristics are represented segments, with voltages. The adjustable forward disadvantage of however, is that at large of the piecewise The b y piecewise linear and such reverse a turn-on representation, signal levels, the nonlinearity breakpoints can be represented as a Taylor series to very high order powers. The transistor chosen NEC72089, principally lower drive-power usage in industry. working at a low for modelling because of its frequency the measurement parasitics could harmonic measurements can was the ready availability, requirements, high Although it is a here gain, and common packaged device, by effect of package be minimized. be carried and Additionally, out with existing equipment whose bandwidth can accomodate several harmonics of a low frequency. In creating this device, the FET geometry length is pm. 1.0 pm (usable Because of the very no data was layer height. 1 7 - 3 cm table for was fairly well known: gate to X-band), and gate-width 400 large variation between devices, available on doping density However, in order to IDSS and Vp from the Madjar model, 1 x 10 the look-up or the epitaxial obtain the correct a doping density of and epitaxial layer height of .16 pm was used, with satisfactory results. These values lie in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -155- range used for such a transistor. Because of the difficulty in obtaining consistent devices, considerable time was taken in selection parasitic order elements representation of was expended in in this transistor, not blowing during experiment! simple and even a valid more effort device in question shows the linear-nonlinear parasitics, and the allowable circuit topology linear program obtain up the Figure 5.1 partition of the model, the external to of the in was the linear partition. first used characteristics (through Rg and to match the A DC in particular); the small-signal AC characteristics were then matched with the measured S-parameters by adjusting CDSC' L S' LG' L D' and RG* Physical characteristics assistance in For CG S , Cpg , are of estimating many of the instance, the 0.8mm bonding 2 GHz. leads inside Similarly, the reflection coefficient S 22 is considerable parasitic elements. package are approximately equivalent to inductance at C GSC, ID -VDS required Since this term characteristic, for also its accurate small-signal output predominantly controlled sets the judicious modelling. optimization programs such as COMPACT to alter the parasitic elements FET 0.35 nH of series Secti°n by the choice of output conductance ^sUBSTo 1.5). the slope choice of the is Small-signal [8] can also be used until a match to measured Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -156- EXTERNAL CIRCUIT NONLINEAR MODEL PARASITICS DRAIN OS DG DSC GS R e = 6R L s = .1 nH = 20 L q = .35nH c; VO II LD = .5 nH CGS = *3pF CGSC = 'lpF CDS = *3P F CD S C = *lpF Figure 5.1 The complete FET model, showing the nonlinear/ linear partition, the external circuit, and the parasitics used to model the NEC 72089. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -157- S-parameters has been achieved over a broad band of frequencies. 5.2 AMPLIFIER SIMULATIONS AND MEASUREMENTS Part of the linear partition network contains series and in the these elements balance shunt elements that represent the effect of impedance matching Although harmonic at the device terminals. are all lumped (for ease of analysis), any distributed network ma y also be referred to the device terminals its effect there. at the frequencies of represented by the equivalent This enables the simulation interest, and lumped elements of generator and load impedance, and enables any desired their effect on circuit performance to be investigated. In the simulations to follow, designed, simulated, operating points. which the This is nonlinear amplifier has and a feedback amplifier was measured at the first behavior been simulated. of a a variety of known instance in MESFET Several feedback different circuit configurations were investigated with the aim of comparing the MHB simulations with measurement, and of consequently using the method in the design of high performance, low distortion amplifiers. The amplifier built corresponds equivalent circuit shown actual generator and load in Figure to the lumped element 5.2. Although the characteristic impedances are both 50 ohms, Figure 5.2 shows a generator impedance of 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -158- 2 d& - 5o o ^ Figure 5.2 * Rpa Lumped element AC equivalent circuit, showing the circuit external to the FET that was used to model the amplifier. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -159ohms; this was achieved in the circuit distributed single stub tuner. Analysis by using a of the input match of Figure 5.2 on the Smith Chart reveals the effect of the shunt 10 nH inductance and impedance is to Similarly, on the effective 75 ohm generator achieve a the good output, input match additional at 2 GHz. capacitance was necessary in order to obtain the desired, measured gain. Adding the 500 ohm feedback resistance lowers the gain from the maximum available value. However, considerably simplifies the task particularly frequencies where at lower it of impedance matching, the FET gate appears highly reactive. Bias was brought in to the amplifier through RF chokes consisting lines. The of ac-grounded, gate voltage chip resistor, just this is to quarter-wave, high was applied through a prior to the RF choke; quell any tendency for impedance 100 ohm the effect of oscillation, and to limit the DC gate current under conditions of heavy drive, by self-biasing the FET towards pinchoff. Several frequency measurements were tone was applied power-transfer characteristic. power vs. output capability NEC72089, maximum between 14 dBm at 2 Firstly, a GHz and 17 of to the saturated a of input and reveals the device. output dBm, depending single obtain This is a curve power up to saturation, power-handling the made. For power on the the varies operating Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -160- characteristics of the particular device. The modified harmonic balance technique has no limitations when used to simulate single-frequency of the tests, regardless of nonlinearity, because components unaccounted any higher-order for in the smaller than the fundamental be neglected. the output interest intermodulation products give cause for made under be much and consequently may two tones are applied, is sidebands, frequency aliasing will output, It is only when of the order the that concern. and relatively low-level additional aliasing Additional measurements single-tone excitation were reflected power, drain current, and second-harmonic power. The second type application of two intermodulation of measurement involved closely spaced tones at 2 performance fundamental output could be power per carrier was the GHz so that found. The recorded, along with the level of the third-order intermodulation product. Because of the presence input power 4.77 dB needed in either of less than the power in needed to reach the same is a factor all, the two tones a single applied signal measured of measurments and This can be analytically with third order behavior, Eight series to compare was about degree of gain saturation. of three times less, and established for devices Heiter [12]. of intermodulation sidebands, the as in were made simulated behavior. in The results will be presented in Figures 5.3 to 5.10. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -161- 5.2.1 Intermodulation Distortion The first series of measurements 5.3(a) and 5.3(b). drain-source voltage voltage The amplifier VD S = +3 V Gg= -0.25volts. both measured excitation. biased and with gate-source Figure 5.3(a) points for are single-frequency The uppermost curves (i) show the fundamental for incident powers of sufficient level Agreement was volts, Shown in and simulated output power +10 dBm, are shown in Figures is excellent. (ii), with ranging from -7 to saturate The reflected the amplifier. power is the error between measurement being 2 dB. dBm to curve and simulations Below this, in curve (iii), the simulated and measured values of second harmonic output power are shown. Small-signal agreement predict levels large 5 dB too input power approached, the itself (across pinchoff. is good, but high for the levels. FET began to As simulations second-harmonic at these levels draw gate current the series DC-blocking The effect the of this was to were and bias capacitor) towards symmetrize the bias point between saturation and cutoff, reducing the level of the second-harmonic component. piecewise diode characteristics would affect consequently the the Errors of the DC gate in the modelled gate-source Schottky current degree of self bias drawn and and second-harmonic generation. Furthermore, the behavior of the distributed circuit at the second-harmonic was not known accurately. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -162- 3IMULATED OUTPUT POWER (dBm) MEASURED -5 0 \ h ~ -T~ -1 0 -6 -2 2 6 10 INCIDENT POWER (dBm) Figure 5.3(a) Single tone measurements on an FET feedback amplifier. v g s =-°-25v' V _ C=3V, ,=500J). From the top, the m e S s u r e d ^ n d simulated groups of curves show (i) fundamental, (ii).re flected and (iii)second harmonic out put powers, and (iv)drain current, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -163- As will be presented to seen shortly, the harmonics the terminating substantially impedance affects the degree of observed nonlinear behavior. The bottom-most curves (iv) drain current of the these simulations the model because amplifier. was that the was too high a limitation adjustment of of Figure 5.3(a) show the (by of A consistent trend DC gate current several m A ) . the Madjar IDSg through FET given by This arose model is doping-density of that level and epitaxial-layer height estimates also affects the pinchoff voltage Vp in one-volt steps (with values of V p ), restricting somewhat obtainable. However, fundamental output accurate the DC give confidence intermediate the range of values accuracy powers were variation of power level, the no with which always modelled, drain the and the current with to the method, and input to the modelling of this FET. Two-tone measurements shown in Figure between 5.3(b). the simulated saturation. The under the Perfect and measured bottom set same conditions tracking is observed output powers, of curves third-order intermodulation distortion. are into shows the The measurements, made o n a spectrum analyzer, indicate the power per signal (and not the power per observed total power). signal is 11 dBm, in the single-tone Thus the saturated roughly 5 dB less measurements, output than that as would be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -164- SIMULATED OUTPUT POWER (dBm) MEASURED -1 0 -6 -2 2 6 10 INCIDENT POWER (dBm) Figure 5.3(b) Two tone measurements on the same FET amplifier, showing measured and simulated fundamental output (top) and third order intermodu lation distortion (bottom) powers. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -165- expected from third-order analysis. The slope of the intermodulation characteristic is approximately 3:1 at the small-signal level. certainly However, this third-order behavior is not maintained range of the device. throughout the useful A t an input power level of 1 dBm for instance, the slope is about 5.7:1. observed this phenomenon in FETs Numerous authors have [14,17,42]. The modelling also incorporates At an incident power level of 6 FET was measured power level in dynamic self biasing effects. dBm, the gate bias on the to have fallen to -0.32 the simulations, the DC volts. At this gate current into the gate through the 100 ohm bias resistor was modelled to b e 1170 uA. The net effect of this current is to increase the gate bias from -0.25 volts effects are determining to -0.36 volts. extremely important at correct operating high power levels Such bias levels in and circuit interaction. Finally, third-order it can be seen intercept point Pj intermodulation behavior shown is that for the concept devices with not reasonable. of a the A Pj level around 26 dBm might be estimated from generous curve fitting, which gives, coincidentally, to 10.6 dB above the predicted by Heiter Pj which is close single-tone ldB compressed point, as [12]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -166- The second series of measurements, 5.4(a) and 5.4(b), are the shown in Figures same as those just considered, but at a bias level of VGg= -0.5 volts. The fundamental output, reflected, powers shown and second-harmonic for single-tone excitation in Figure 5.4(a) agree closely with measurement, The as two-tone does the form of results, in Figure modelling of the fundamental comparison with the In this occurs due 5.4(b), between show good output component, but poorer in the level both bias and tuning by the the of third-order The occurence of such to partial correlation in components produced coherence dip observed. highly dependent on drain current. third-order intermodulation products. instance, a product can be the DC and conditions, and the intermodulation nonlinearity, gate a dip is drain because of the voltages. The correlation can result in a cancellation effect observable as dips (17 ]. Such dips conversion curve masked in the [14]. may also be seen in In our case, this simulation by the AM/PM correlation is higher-order terms causing aliasing at a level comparable with the desired component. Several useful facts emerge from the analysis however, with regard to increasing of the gate bias. For the same output power level, drain current is reduced, the gain is reduced, the reflected power second-harmonic and are increased, the saturated output the gain is more linear. levels power is similar, and At small-signal power levels the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -167- SIMULATEO OUTPUT POWER (dBm) MEASURED 29099994 -5 0 anm -1 0 -6 -2 2 6 10 INCIDENT POWER (dBm) Figure 5.4(a) Single tone measurements on the FET feedback amplifier. V_S=-0.5V, V ps=3V, RpB=500S2. Curves are as for Figure 5.3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -168- SIMULATED OUTPUT POWER (dBm) MEASURED 62999999999^ 550 D9A -10 6 - 2 2 6 INCIDENT POWER (dBm) - Figure 5.4(b) Two tone measurements on the same FET amplifier, showing measured and simu lated (i)fundamental output and (ii) third-order intermodulation distortion powers. The bottom curve (iii)shows the measured fifth-order intermodula tion distortion product. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -169- third-order imr is worsened; however, because of the dip, the third-order imr is improved near compression and up to the onset of saturation. There is AM/AM with considerably less curvature in (gain compression) that in lower at higher this curve of Figure 5.4 (b) compared Figure 5.3(b) ; the third-order intermediate power levels. order intermodulation case, and "best 11 the two-tone linear it However, products occur is therefore amplifier. imr is The difficult also several as well in to define optimization a criterion clearly depends on the systems application e.g. modulation format for a communications system. The curves in Figures comparison of measured 5.5(a) 5.5(b) and simulated results for feedback amplifier operating with a bias of volts, and an increased drain-source volts. and show the FET V QS= -0.25 voltage of VQS= +4 Figure 5.5(a) shows, from the top, the fundamental output and reflected powers for a single-tone measurement, and the DC measurements drain current. and simulations Figure of 5.5(b) shows the fundamental output power and third-order intermodulation products. Agreement between all the measurements and simulations is very good, with the exception of the intermodulation product. Several factors affect the modelling accuracy for the intermodulation product. extremely sensitive to Because of its model low level, it is parameters. As noted in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -170- SIMULATED OUTPUT POWER (dBm) MEASURED hi i ........... i ii 111 iiiiiip i iiiiiiim 66 C./A -1 0 -6 -2 2 6 10 INCIDENT POWER (dBm) Figure 5.5(a) Single tone measurements on the FET feedback amplifier V Gg=-.25V, v Dg= 4V, RpB=500n. From the top, the measured ana simulated groups of curves show fundamental output power, reflected power, and drain current, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -171- SIMULATEO OUTPUT POWER (dBm) MEASUREO 5141 -1 0 -6 -2 2 6 10 INCIDENT POWER (dBm) Figure 5.5(b) Two tone measurements on the same FET amplifier, showing measured and simulated (top) fundamental output and (bottom) third-order intermodulation distortion power. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -172- Chapter 4 for the Minasian model, slight variation in the third-order transconductance term affects the magnitude of the intermodulation more difficult be effects Finally, of the nonlinearities as as as infinite and dips in will terminations problem is Additionally, most aliasing such The numerous contributing expanded "residual 11 dB. here because of the nonlinearities. can product by many be (which masking the do thereby extremely shortly, low-level not product. the harmonic necessarily fit circuit assumed for fundamental) can alter the intermodulation product causing intermodulation seen lumped-element equivalent partition at the series, by over 5 dB. the the linear level of the Nonetheless, the trends predicted by the simulation with respect to circuit and power variation agree well with those measured. The effect of observed The by comparison gain and slightly, by when the voltage has match of the FET. few we as output current-voltage power beyond are little effect the characteristic. 5.3(b). increased is increased. unchanged, indicating The DC drain current are FET can be 5.3(a) and drain bias power is essentially the drain mA with Figures saturated 0.5 dB, The reflected that increased drain bias on the on the input increases only a "knee" of Importantly, the at FET a constant output power, the level of the third-order IMD is reduced by up to 5 dB. The reduction is particularly Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -173- noticeable for medium signal levels, with outputs range of +5 to +10 dBm. compression curve this phenomenon The curvature noticed in the gain of Figure 5.3b thus appears gate bias voltage. in the is also to be apparent here; associated with the Finally, the simulated DC gate current was less, indicating the onset of voltage clipping during the negative half-cycle of the input waveform, allowing DC current to flow out of the gate when the gate-drain diode enters reverse breakdown. In order variation reduced to on the to 100 VQS= -0.25 investigate amplifier, ohms. The the the effect circuit feedback resistor original volts and VDg= of was bias conditions +3 volts were of used again. The effect of reducing the feedback resistor was to reduce the power reflected b y the amplifier below the incident power. effect of the 100 to more than 20 dB This arises due to the shunting ohm resistor at the input; a natural consequence of this is the considerable reduction in gain. This technique is often used in amplifier front-ends where gain is cheap and input match is important. The results can Figures excellent 5.6(a) and be seen in the 5.6(b). agreement, the Figure measurements shown in 5.6(a) shows, measured and with simulated fundamental output power and DC drain current for a single frequency tone. are presented Intermodulation distortion in Figure 5.6(b), which measurements shows the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -174- SIMULATED MEASURED OUTPUT POWER (dBm) 86 ..I'wugga - r l' P 5851 -5 0 -1 0 -6 -2 2 6 10 INCIDENT POWER (dBm) Figure 5.6(a) Single tone measurements on the FET feedback amplifier, V Gg= -.25V, V d s = 3V, RpB = 1000. The top curves show measured and simulated fundamental output power; the bottom curves show DC drain current. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -175- fundamental and third-order output powers per carrier for two-tone excitation. The reduction in power-handling seen in Figure 5.6(b) where The 1-dB gain point at +6 dBm output to a (4.77dB higher) one-tone 10.5 dBm, as would be represents much reduced obtained power dBm from power, corresponding 5.6(a). This performance (14 to than can 17 be The product has very close to the levels, enabling a good Pj the measured approximately 10.5 two-tone dBm). ideal 3:1 slope at low power falls the indicated on Figure intermodulation distortion of 21 for compression power of around without feedback estimate best the input loading is greater. compression application occurs capability is dB above data. This then the single-tone 1-dB compressed point. It can be seen that at lower power amplifier has very good third-order single-tone compression third-order can be observed to 4:1. reflected in to third-order at the lower power with Figure slope rises very levels. levels appear to be 5.3(b) reveals its Above behavior is the simulations, which show the third-order distortion in dBm, a breakpoint at which the intermodulation additional aliasing comparison and characteristic. per carrier of 2 The closeness this behavior, both in its characteristics intermodulation incident power levels levels, that also little Although very low, at a given Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -176- SIMULATEO MEASURED OUTPUT POWER (dBm) 1299031 2999 57 -10 - 6 - 2 2 6 INCIDENT POWER (dBm) Figure 5.6(b) Two tone measurements on the same FET amplifier. Curves are as for Figure 5 . 5 (b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -177- output power, an increase of between 10 and 15 level of the third-order product reduction of the dB in the has actually resulted by feedback resistor to 100 increase could be anticipated b y ohms. Such an the lowering of the 1-dB compression power level from 15 dBm to around 10 dBm. fact that the simulations can The predict a similar reduction in the third-order intermodulation performance augurs well for their use as a circuit design aid, and in predicting circuit trends. The curves performance for VD S = +3 Comparison in Figures the same volts) with the with Figures 5.7(a) amplifier (VGg= feedback 5.3(a) improved power-handling capability. compressed output the power in Figure two-tone compressed removal of the match. power feedback has reflected power, and making the is 5.7(b) -0.25 volts, resistor and removed. 5.3(b) indicate The single-tone 1-dB 5.7(a) is 15 11 dBm). dBm (and Note considerably increased amplifier more However, the direct tradeoff show that the difficult to is that the gain has increased by 3 dB. With such a high gain, the compression into saturation is particularly gradual. Similarly, the slope of the small-signal third-order intermodulation characteristic is closer to 3.5:1 in spite of than the commonly assumed these "nonidealities", handling capability 3:1. However, the increased still translates directly power to reduced Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -178- SIMULATED MEASURED 8 9 OUTPUT POWER (dBm) 0 ^9414752 9 -5 0 -1 0 -6 -2 2 6 10 INCIDENT POWER (dBm) Figure 5.7(a) Single tone measurements on the FET amplifier, V =-.25V, V__=3V, R-^lOOOMfi. Curves are as for Figure 5.5(a) . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -179- SIMULATED OUTPUT POWER (dBm) MEASURED 20 -5 0 D9+:/43/16D 10 -6 -2 2 6 10 INCIDENT POWER (dBm) Figure 5.7(b) Two tone measurements on the same FET amplifier. Curves are as for Figure 5.5(b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -180- third-order intermodulation product. Comparison Figures 5.3(b) and 5.7(b) reveal that, powers, the intermodulation product for the case of no between for the same output is slightly feedback resistor. smaller Unfortunately, the FET becomes much harder to match and more unstable (due to increased gain) under these conditions. Part of the reason for the gradual noticed above is related to the rectification of the signal. DC gate current caused by In both measurements simulations at this bias resistance was observed to increase power, with the slowly increase level, increasing consequence of current into the FET. gain compression the feedback the gain increasing The observed and output the DC the gate-bias voltage, which also slowly A slight error in characteristics of the gate-source diode resulted simulated value account in part of DC the forward turn-on gate current. for the pessimistic value output power at the higher gate effect of this was to reduces the gain. high a and in too This would of fundamental incident power levels shown in Figures 5.7. The final involved biasing series of intermodulation the amplifier above (Rpg experiments removed) for minimum distortion product at some operating output power. This was found to be at a (VDS= +3 volts). gate bias of V QS= -0.57 volts The single-tone output and reflected power curves are given in Figure 5.8(a), with the DC drain Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -181- SIMULATEO OUTPUT POWER (dBm) MEASURED -1 0 -6 -2 2 6 10 INCIDENT POWER (dBm) Figure 5.8(a) Single tone measurements on the FET amplifier, V g=-.57V, V_g=3V, RpoSlOOOMn. Curves are as ror Figure 5.5(a). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -182- current:. The power Note superb the indicating low linearity output is simulations in in excellent the at small Figure 5.7 Compared where the curve, signal levels, performance. and the measured 15.8 dBm. agreement. compression good intermodulation is 8.9 dB, power of AN/AM conversion and foreshadowing linear gain curves are The 1-dB compressed to the previous gain was 13 dB, a tradeoff has been made in the operating point, sacrificing gain for linearity. achieved partly The improved linearity has through symmetrizing the gate been bias point between pinchoff (-1 volt) and forward saturation (above 0 volts). A manifestation of this is the low amount of self bias observed as saturation is approached - raising the incident power from +5 dBm to +10 dBm caused the gate bias to increase from -0.57 volts to -0.62 volts; a similar action when biased at -0.25 volts caused it to change from -0.25 volts to -0.36 volts. gate bias total change was DC gate positive current coupled with current in as the In the simulations, the small a direction incident power components due a fluctuation to was reverse of the from negative increased. breakdown to The DC of the gate-drain diode and forward conduction of the gate-source diode (on negative voltage and positive half-cycles of respectively) balanced forward conduction out the input initially, with process eventually dominating the at high input powers. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -183- Intermodulation measurements 5.8(b), showing measured third-order products, output power. dB. and with the Figure -0.25 volts, the to output power is 12 dBm. substantially Whereas at a gate bias level of small-signal gate bias remains 8.9 reveals third-order (as indicated behavior was b y a 3:1 IMD -0.57 small-signal levels. reasonably slope), reduction volts (Figure actually flattened the slope at linear gain 5.7(b) different IMD behavior. and simulated second-harmonic The small-signal with in Figure simulated fundamental The two-tone 1-dB compressed Comparison of are presented 5.8(b)) has of the intermodulation curve The behavior at higher power levels is also "nonideal", rising from a dip of -28 dBm at 0 dBm incident to a level of -3.5 dBm at 7 dBm incident, an average slope of 3.5:1. Comparisons 5.8(b) with of the those in third-order Figure 5.7(b), power levels, reveal that product level curve shows from a dip at at this Figure at constant output lower here for output powers The shape of the intermodulation around this power improved cancellation nonlinearities in the third-order intermodulation is somewhat above 8 dBm per carrier. product between power level. level, resulting the gate and At lower drain incident power levels, however, the IMD is made worse by increasing the gate bias. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -184- SIMULATEO OUTPUT POWER (dBm) MEASURED -1 0 -6 -2 2 6 10 INCIDENT POWER (dBm) Figure 5.8(b) Two tone measurements on the same FET amplifier. The top curve shows measured and simulated fundamental output power; the bottom curve shows third order inter modulation distortion. The middle curve shows simulated second harmonic output power (per carrier). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -185- The behavior power of the simulated is shown for comparison behavior of this component with much more from slowly than second order expected slope, no second-harmonic output in Figure The incident power also rises 20 dB/decade the simple terms. 5.8(b). Apart from the expected lower sudden variations are observed than in its character. In order to investigate the contribution of the higher order terms to third-harmonic shorted the currents were out by the linear This was achieved higher observed IMD, all allowed circuit to be generated in and effectively in the by not reimpressing upon order voltages second simulations. the model any the linear circuit, caused by the second or third harmonic currents generated by model. the nonlinear drastically presented FET reducing to the the This linear model at is equivalent circuit frequencies to impedance away from the fundamental. Very little product was alone. observed by However, second-harmonic model. change in several voltage the level of the shorting out effects were was As might be expected, not I third-order the third-harmonic noted when the reimpressed upon the the effect of removing this additional source of nonlinearity was to reduce the of third-order distortion products. level Just as the effect of reimpressing the third-order distortion products upon the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -186- third-order model fifth-order in Chapter 4 to introduce terms, the higher order interaction of second-harmonic signal with the additional third-order distortion. reduced was by approximately configurations ( i.e. with the fundamental here The level of 3dB for both Rpg 3 500 ohms, causes IMD was amplifier and with feedback removed) over most of the input power range. On the other hand, the level of the fundamental signal was affected very little by the termination. At -7 dBm output power increased from 1.59 dBm the shorted incident power, harmonic termination, second-harmonic the fundamental to 1.66 dBm and in due to saturation, no meaningful difference was found in the output power. is of considerable neglect of fundamental consequence, harmonic frequency as terminations response it when using This justifies our considering large-signal S-parameters (Chapter 2). The use of open-circuit terminations at the gate and second- and drain was also investigated, b y directly seting the external ports to a large value. the parasitic is third-order circuit elements at those Some harmonic currents elements, however, such terminations is reduced. power basically so that flow in the effect of As before, the fundamental unchanged, distortion third-harmonic products bu t is the now level of increased the by several additional dB compared to the case of typical load Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -187- terminations at both ports. Knowledge of the thus important harmonic behavior of the in linear third-order product amplifier may be obtained circuit is design. A by presenting low a low impedance shunt termination to the second-harmonic at the gate and that higher order effects voltages are minimized, by keeping their drain. due to harmonic This ensures level as low as possible at same token, however, the complicated, as impedance of the task of analysis is an incomplete knowledge of the linear circuit several dB in modelling though the device terminals. By the made more the harmonic can result in errors of intermodulation performance, even fundamental frequency response will be relatively accurate. Fundamental load impedance critical parameter high power in amplifier levels. load-pull for (without termination changes relatively changes determine and such easily using the as the output ohms, or increasing it to 120 MHB can higher be harmonic to the Load investigated For example, drain to ohms, reduced the reduced the imr at load varied). method. load presented at intermodulation was these reducing the device, and for a laborious optimum minimum load the is also performed accounting as drain design, particularly [42] to power distortion impedance Sechi experiments terminations at the 20 gain of constant output power Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -188- by approximately 2 dB. As a final example of the applications described here, Table 5.1 consider now the shows various of the method case of device design. device parameters (ND , doping density; a, epitaxial layer height; LG , gate length; and W, gate about width). their created. t, and As these parameters nominal incremental look-up capacitances, for Simulations feedback amplifier table shows power, P0 U T ; the new 25 % table was the MHB This yielded new values of ID S S # V p , algorithm. The values, a were varied ± were then at a fixed the new use run equivalent IM3 for levels of the same basic -4 dBm. fundamental output power, P 2n d ? intermodulation product, ( at the incident power of the second-harmonic output third-order in IM3, P0 U T ) for and the t*ie unchanged feedback amplifier with the original device. Such a design doping table is relatively (e.g. manufacturers profiles to crude in terms such as improve of device Fujitsu use linearity, graded which are unaccounted for in this model), but, by indicating trends, the amplifier designer is assisted in the task of device selection. For example, a IDSS and increased capability) has wider gate-width improved improved output device (with its power-handling intermodulation performance. Gate length basically controls device transit time and has Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. transistor of Effects co ro 00 i—i . 00 . in ov CO m -3 8 .5 -3 5 .2 -2 3 .8 -3 2 .5 -3 3 .7 -2 4 .8 -3 3 .0 CM . in • ro .8 7 CM • 00 ro -3 3 .3 -3 5 .5 -3 4 .9 o\ o • r- r- CM O rH . in rH . CO CM CM • o • M" CM • CM • VO CM • (A ) 00 VO H • CTV *s> • rH • CM 1 1—1 o 1—1 . rH CM rH CM rH rH •H rH rH C/1 ~ to Q E H — rH • co ID • 1—i • <T> 00 . O CM *!• . © '3' rH • 00 *3* 00 • ro ro 3. in CM • rH 3. m r* * o II O o s 2 Q ^K) — dA 5.1 r- GO 00 . .8 7 H in o • CM .7 5 a r00 • rH •M* CM 1 .7 3 im ro • co .7 6 changes design J >^ 8 a •H VO • *a* S 'S w Table 1 -4 2 *a § <N 1 VO • CO -2 4 -3 9 .0 -4 3 .9 iH • 1 -5 0 -4 2 .7 Q in • -2 4 -3 9 .6 GO CO 1 3n e w -4 3 .9 i -3 4 .2 _ 'p | *2 2 (d B m ) Q i-* o PO s M on amplifier performance -189- aj >o 4J Q) (I) Di E a IQ IQ m .c <Q o di ov cm ro VO i o •H 0] (0 m CM ro CO CM CM © o 1—1 1—1 X X in in CM r-' • • rH o ii II Q Q Z Z m in VO 3. i—1 • II 3. CO i—1 . It < < o\ co ii u in o o in II 5 CM ro 2o o ro II 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -190- little effect frequency). doped on intermodulation performance (at Similarly, the use of epitaxial layers is able a fixed higher or more heavily to improve amplifier harmonic balance method is linearity. In summary, able to the modified assist both evaluating amplifier trends the circuit which linearity and device enable and the the designer in improvement reduction of of third-order intermodulation produ c t s . 5.2.2 Gain Suppression in Limiting Amplfiers Having demonstrated the harmonic balance technique, useful in distortion ability the modified with a suitable model, predicting the fundamental performance of of the to be and intermodulation MESFET amplifier, we consider now two examples of gain suppression. A phenomenon time in amplifiers that has been limiting amplifiers power [43,44] ratio of stronger than the noise. that, a considerable and nonlinear [45] is the possible improvement of 3 dB in the output signal-to-noise is known for for large ratio of a A a signal consequence signal-to-noise weak signal to which is much of these results ratios, the a much output stronger (simultaneous) signal can be as much as 6 dB less than the same ratio at amplifier. the This is weaker signal. input of known the as limiter or saturating gain suppression Such results can be of the analytically derived Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -191- using various nonlinearities probability theory g (*), to represent the noise calculations are extremely tedious selection of terms signal, to using The and ultimately involve the inband output itself is in the by process. from a series expansion represent of the output products. Since the form of hypergeometric functions, the derivation is by no means trivial. of nonlinearity and g(«) can be handled, However, any order either analytically or numerically. The modified harmonic balance method can also be used to demonstrate this phenomenon for nonlinearities of up to fifth-order. The advantages nonlinearity is together dynamic with the of this method are in the that instantaneous operating device (enabling the use of a frequency sense components are that the it varies point of the device model), and that the obtained numerically. The limitations compared with the analytical treatment are the restrictions on the order of the nonlinearity and the absence of signal noise. Figures 5.9 and 5.10 show the fundamental output power and third-order product in each of the two carriers and their sidebands. The horizontal axis represents the power incident strongest signal. in the smaller signal is it is 5 dB weaker. In 3 dB weaker than this; Figure 5.9, the in Figure 5.10, In both cases the small-signal gain is 9 dB (for both signals). At an incident power of +10 dBm Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -192 SIMULATED OUTPUT POWER (dBm) MEASURED 4 3 -1 0 -6 -2 2 6 10 INCIDENT POWER (dBm) Figure 5.9 Two-tone tests of the FET feedback amplifier of Figure 5.3(a). Two unequal tones of level P IN and Pjn -3 were applied to the ampli fier. The top set of measured and simulated curves show the output power in each funda mental carrier; the bottom set show the level of each third order intermodulation sideband. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -193- SIMULATED OUTPUT POWER (dBm) MEASURED 66 -1 0 -6 -2 2 6 10 INCIDENT POWER (dBm) Figure 5.10 Two-tone tests of the FET feedback ampli fier of Figure 5.3(a). Two unequal tones of level P-j. and P IN-5 dB were applied to the amplifier. Output curves are as for Figure 5.9. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -194- £or the strongest power is 5.10. signal, the 14.3 dBm in Figure At this 5.9, and 14.5 dBm point, the amplifier is dB into compression and almost the output power power) strongest signal is just in Figure approximately 4.5 saturated. In Figure 5.9, of the weaker signal (at 7 dBm incident 9 dBm. weaker signal of output Similarly, in Figure 5.10, 5 dBm incident power power of just 7 dBm. the produces an output In both cases, the gain of the small signal has been suppressed by 2.5 dB. The behavior of the third-order sidebands is also interesting. between two the intermodulation The separation intermodulation carriers in levels is directly proportional to the level separation between the two input signals (3 also dB and show that 5 dB the separation harmonic signals is separation, The between twice (in dB) the simulations the two second- input signal level and three times for the third-harmonic [13]. Agreement between is good. respectively). the measured and The amplifier used in simulated results the simulations was not a true limiting amplifier, but the onset of gain compression and saturation Unfortunately, gave a true it limiting terms of infinite order, and method would be By technique "soft limiter", could be characteristics. amplifier has nonlinear the results obtained by this in error due to considering a balance similar the additional aliasing. the modified useful in the harmonic design of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -195- nonline ar amplifiers characteristics. Furthermore, analysis the and straightforward, A tailored for predistorter is application design and is one such of the of similar suitable to component. technique resistive the gain to mixers the is unequal-level signal cases just considered. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -196- 6. CONCLUSIONS AND RECOMMENDATIONS The purpose of this work was to improve nonlinear circuit simulation methods in order to facilitate the design of circuits involving the large-signal operation of microwave MESFETs. 6.1 THE MODIFIED HARMONIC BALANCE METHOD In particular, the goal was to implement a technique that would enable two-frequency analysis of any circuit, given a nonlinear time-domain model for the device within the circuit. model Using modifications to an existing FET [1 ,22 ], a harmonic balance method was devised which allows efficient implementation, and which overcomes the problems associated with a small difference frequency rela tive to the carrier frequency. By bandpass sampling of the time domain waveform generated by the FET model, and by small input frequency changes, the desired fundamental and intermodulation components can be reconstructed from the aliased spectrum. By achieving a harmonic balance between the frequency components present in the nonlinear model and the linear circuit, Kirchoff's laws are satisfied. This enables the effect of circuit changes and device parameters to be determined for both singleexcitation. (i) and two-tone The method is numerically efficient because the linear circuit is analyzed by a two-port matrix in the frequency domain, where differing time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -197- constants correspond only to different amplitude phasor components, and additional branches do not increase the size of the matrix. (ii) the time-domain waveform from the model is sampled not at the Nyquist rate for two closely-spaced signals, but at the bandpass rate. (iii) the time-domain waveform is obtained as an explicit function of the input nodal voltages and their time derivatives. The method is a complete analytical method because (i) any nonlinear time-domain model can be used. (ii) the nonlinearity can be a function of the instantaneous (unknown) operating poing i.e. it can account for changing bias conditions. (iii) harmonics and higher-order frequencies gener ated are reimposed as additional inputs to the model. (iv) a solution is obtained for the phasor value of every frequency component present, including DC components. (v) operating conditions are completely known (bias level, DC efficiency, etc.). The restrictions on the method as implemented are that the nonlinearity be no greater than fifth-order, and that if the nonlinearity is frequency dependent Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -198- (i.e. nonresistive), the frequency spacing of the two input signals be much less than the carrier frequency. 6.2 INTERMODULATION DISTORTION IN MESPET AMPLIFIERS A survey of the results of Chapter 5 indicates that we have characterized and modelled FET behavior for both single and two-tone excitations reasonably accurately. While all other approaches [14,16,17,18] have initially characterized the FET on the basis of single-tone largesignal measurements in order to obtain insight into intermodulation distortion properties, only DC and smallsignal (linear) measurements were performed here before large-signal single-and two-tone comparisons were made. Some improvement could undoubtedly be made to the model ling accuracy of the two-tone measurements here if fine model adjustments had been made on the basis of singletone large-signal measurements, and second-harmonic terminations determined more accurately. The improvement of model accuracy represents a major area for future work, to allow the power of the analytical tools that have been developed here to be more fully exploited. We have investigated the effects of input power level, bias variation, circuit variation, and device parameter changes on the level of the third-order intermodulation product, and are now in a position to draw some conclu sions, and compare them with previous work. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -199Cons ider a perfectly linear device whose output characteristics are given in Figure 6.1. for class-A operation at point A When biased and operated into a resistive load R^ , no distortion products occur until saturation. At this point, the peak voltage and current swings are Vq and *1)35/ 2 ' respectively. Further increase in drive causes clipping of the waveform with the result ant onset of distortion products. In a real FET, the drain-current lines are bunched together at larger gate bias, and spread apart at smaller gate biases. This is the result of the nonlinear trans conductance gm (and is a "gate-side" effect). Additionally, the slope of the drain-current lines is not constant, but has a small value gD , which is the output conductance of the FET (a drain-side effect). Both of these nonlineari ties cause the input sinusoid to be transformed even at small-signal levels, with consequent distortion of the output signal. Additionally, the load line need not be straight, but may be elliptical (or even more highly distorted) due to the presence of load reactance and harmonic signals. The distortion level will obviously be sensitive now to both tuning and loading, as these will affect the peak RF swing and contour. The extended small-signal model of Higgins, et.al. [17] indicated that at lower power levels, the MESFET is drain-side dominated i.e. gD is dominant, and that Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -200- &LOPE V d s Figure 6.1 The output characteristics of a perfectly linear device. The horizontal axis is drainsource voltage V Dg; the vertical axis is arain current I_. Optimum bias point A is shown, with the operating load line. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -201- this causes gain expansion. At medium power levels, the device is gate-side dominated i.e. gm is dominant, and this causes gain compression. The authors also found by curve fitting that the fifth-order term in the non linear expansion for gm gave a greater contribution to third-order intermodulation distortion than did the third-order term. The nonlinear gate input capacitance was found to have little effect on the intermodulation products. Many other authors have also examined, through extended small signal models or measurement, the interre lation between intermodulation ratio, input power, circuit impedance, and device operating point [42,46-48], Con clusions drawn by them are that the second harmonic impe dance at the drain has little effect on the level of intermodulation product [46,47], that the higher order effects of harmonics and third-order intermodulation products can be neglected in calculating the imr (for ratios less than 20 dB) [46], and that the least inter modulation distortion occurs when the AM/AM conversion is minimized [47]. Other authors [48] have found that the second harmonic impedance at the gate has significant effect. Some authors have noted that as the output load * is detuned from optimum small-signal match at S 22 towards the optimum power load, the IMD is reduced [42,46] due to the increased shunting effect of the load across the nonlinearity of the output conductance [48]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -202- It is the view of this author that such conclusions can be rash due to the complex interaction of circuit and device effects. The above statements are made without reference to either input power level or to bias point. However, for the resistive feedback amplifier studied, some general conclusions can be shown. It was found that (a) an optimum bias point could be found. Increasing V DS improved the 1-dB compressed power and imr (inter modulation distortion ratio). Symmetrical placement of V QS between V p and forward turn on (as evidenced by reduced DC gate current) improved the distortion perform ance of the amplifier. (b) for a given AM/PM characteristic and 1-dB com pressed power# a linear AM/AM conversion curve into com pression corresponded to improved imr. (c) decreasing the feedback resistance lowered the gain, the 1 dB compressed power, and the imr, at all power levels. (d) the effect of second harmonic terminations was appreciable at all power levels. The third harmonic termination had little effect on the level of the dis tortion product. (e) the imr was improved by changing the following device characteristics: increasing doping density, epitaxial layer height, and gate width. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -203(£) the intermodulation distortion is a complex function of operating point, circuit, and device para meters that does not obey simple third-order behavior. In view of the ease with which the operating point, embedding circuit, and device parameters may be inter changed, the modified harmonic balance approach to modelling intermodulation distortion presented here rep resents a new, complete, and integrated approach to the circuit simulation problem. 6.3 IMPROVING INTERMODULATION DISTORTION: A NEW APPROACH There are several obvious means by which one may attempt to reduce the level of intermodulation distortion in a feedback amplifier. Many of these are suggested by the conclusions above, and most involve tradeoffs. For instance, the use of higher drain-source voltage or a larger feedback resistor may involve allowable compromises, and be suitable solutions in some applications. The use of DC gate current as an indicator of gross FET breakdown is one way in which low intermodulation product amplifiers could be designed. High DC gate current, indicative of rectification of the signal waveform, commences at the onset of saturation and clipping, and is always accompanied by distortion products. Minimization of the DC gate current by symmetrizing the bias point produces odd order harmonics in the output waveform, rather than even order harmonics produced by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -204asyinmetrical clipping. At certain power levels, these counteracting effects can cause the often observed dip in the third-order intermodulation power Figure 5.4b). Gupta et.al. (see, for example, [49], noting a minimum in the IMD at one gate bias level, used a scheme which coupled some of the input power to a diode detector. The detected output was amplified and subtracted from the gate bias voltage, so that as the input power was increased, the gate-bias became more negative. This resulted in a decrease in amplifier gain as well as of the IMD. Because the IMD reduces more rapidly than the gain, a net improve ment in the distortion product was obtained. A 10-dB increase in imr over a very narrow range of input powers was achieved. However, the output power was maintained constant due to the reduction in gain, and at large signal levels, the FET was driven close to pinchoff, thereby increasing the IMD at these power levels. Such a scheme, utilizing gate-bias control, allows some control of the intermodulation distortion. A variation of such a scheme would be to control the gate-bias point in such a way that the DC gate current was minimized. A completely novel scheme is presented in Figure 6.2. The top part of the figure shows a basic matched FET amplifier A^ driven simultaneously by both and V 2 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -205- 2 l Figure 6.2 Novel intermodulation disortion im provement scheme. The top figure shows the configuration used in sim ulations; the FET and its matching elements make up amplifer A^. The bottom figure gives a practical im plementation: V 2 is added in series by the action or a second amplifier Aj and the circulator. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -206- Simulations of this configuration were run using the MHB approach (note that both the signals V 1 and V 2 actually consist of two tones e a ch). A practical implementation using a single driver is shown in the bottom of the figure. There, the signal V 2 is generated by coupling some of the input signal into a secondary amplifier A 2, and injecting the new signal into the output of the original amplifier A^. In many ways, this circuit appears to be a form of feed-forward distortion reduction [9], in which a sample of the intermodulation distortion is fed, out-of-phase, into the output to subtract out the distor tion component. However, in this case, a linear in-phase reproduction of the input signal (assumed pure) is instead added to the output voltage produced by the FET. two effects. This has The first is to increase the output power, because the two voltages are added in-phase. The second is to force amplifier 1 to operate in a more linear mode, thereby reducing the level of third-order intermodulation distortion. Comparisons with circuits used to achieve injection locking of oscillators could also be made. Simulations were performed using the modelled system of Figure 6.2, and are shown in Figure 6.3. Both V.^ and V 2 were set equal in amplitude and phase in Figures 6.3a and 6.3b, which are one-tone and two-tone power curves, respectively. The uppermost curve (dashed) in each graph Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -207- OUTPUT POWER (dBm) SIMULATED w m 3,,, lUliiriiiOB H -1 0 -6 -2 2 6 10 INCIDENT POWER (dBm) Figure 6.3(a) Single tone simulations of the amplifier system of Figure 6.2, with v2 = v i* T^e solid lines show fundamental output and drain current data with amplifier 2 dis connected; the dashed lines show the same data with amplifier 2 connected. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -208- SIMULATEO OUTPUT POWER (dBm) 50 -1 0 - 6 -2 2 6 10 INCIDENT POWER (dBm) Figure 6.3(b) Two tone simulations of the amplifier system of Figure 6.2 (V2 =V.). The solid lines show fundamental output and third-order intermodulation distortion data with amplifier 2 disconnected; the dashed lines show the same data with amplifier 2 connected. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -209- is the fundeunental output power in the new system. The solid curve indicates, for comparison, the response of the original system studied in the previous chapter (i.e. V 2 = 0). As can be seen, the distortion reduction effect occurs at high power levels, where the distortion products are reduced by up to 10 dB. Furthermore, because of the increased output power, the imr at constant power is considerably increased. At an output power of 5 dBm for example, the imr has im proved from 39 dB to 45 dB. At higher power levels, where the third-order product dips, the improvement is substan tially greater. The implementation of this scheme, as shown in the bottom of Figure 6.2, uses a circulator to add the voltage produced by A 2 into the output of A^, after its matching. The resultant combined voltage is circulated to the output load. This would require that A 2 have zero source impedance and produce an internal voltage swing of V 2 equal to V^. However, the output impedance of amplifier 2 is matched to 50ft, and the input impedance at the output of amplifier 1 is also matched to 50ft. This effectively splits the generator voltage from A 2 added into A^ in half, when representing V 2 in the zero-source impedance scheme of the model. Thus, in order that V 2 = V^, the power handling capability of the FET in amplifier 2 must be 6 dB greater them the driver for amplifier 1 i.e. of comparable drive capability to amplifier 1 itself. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -210A cheaper scheme would be to choose V 2 = JjV^, for then the FET in amplifier 2 need produce a total (internal) swing of magnitude V^, the same as the driver for the system. This also enables V 2 to be relatively distortion- free compared to the voltage produced by amplifier 1 , as A 2 is then run in a backed-off mode. The physical require ments could also be met in a variety of other ways; for instance, amplifier 2 could be omitted and a simple 3-dB power splitter used to derive V 2 . gain then suffers, However, the system loading problems arise, and a different set of simulations would be necessary. The results of the simulations for V 2 = ** V1 are Pre“ sented in Figure 6.3c. The solid lines indicate the simulated fundamental output and intermodulation distortion data previously presented in Figure 5.3 for the resistive feedback amplifer and V DS = + 3volts. (RpB = 500(2) biased at V G g - 0 . 2 5 volts The dashed lines indicate the same data with the intermodulation distortion improvement scheme in place. It can be seen that the output power is increased, the intermodulation distortion reduced, and the imr at large output power levels is improved by at least 6 dB. power-combining two equal FETs with hybrid By couplers, an imr improvement of 6 dB is also obtained, but at double the cost. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -211 SIMULATED oc -10 o -20 b -30 0 -1 0 -6 -2 2 6 10 INCIDENT POWER (dBm) Figure 6.3(c) Two tone simulations as in Figure 6.3(b), but with V2 . % v 1 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -212- As the phase of V 2 is varied through 90° to 180°, the opposite effects occur: lower output power, higher dis tortion products, and severely deteriorated imr result. It is anticipated that a simple variable line length, in the form of a microstrip trombone at the input of amplifier 2 could be used to adjust for zero overall phase length and account for phase lengths in the amplifier, coupler, matching networks, and circulator. In one sense, this scheme achieves its increased power handling capabilities through power combining. A typical power-combined amplifier, employing a 3-dB hybird at its output, combines the output powers of two equal FETs. Consequently, a 3 dB improvement in intercept point is achieved, with an improvement in imr of 6 dB due solely to the back-off possible in operating with lower output power. The scheme presented here also employs a type of power combining, but the FETs need no longer be of equal power handling capability. For the additional cost of a coupler and circulator, considerably greater improvement in imr can be achieved due to the increased linearity obtained from the FET. Such a scheme could be more efficient and more generally applicable than either current predistorter or "backoff" approaches. Through further use of the modified harmonic balance method, additional improve ments in the intermodulation distortion performance of amplifiers should be possible. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2137. ACKNOWLEDGEMENTS Numerous people were of considerable assistance in the development and production of this dissertation. Stacie Kawano was particularly generous with her pre vious work on the subject and her collection of papers. My colleagues in the Microwave Laboratory, Rimmon Sachs and Patrick Roblin provided useful assistance with numer ical analyses involving the IBM personal computer, as well as encouragement and patience in bearing with the constant clatter of my terminal! Janet Gittelman was also inval uable for her typing and attention to detail during the compilation process. At Central Microwave Company, the author is par ticularly indebted to Dr. Robert Goldwasser whose prac tical and theoretical insight never ceases to amaze, and who made possible much of the work herein through the support of his company. Rick Kiehne and Bill Lazechko are two highly capable individuals who also deserve praise for their cooperation and efforts on my behalf while I was employed at CMC. To my friends in St. Louis who contributed to the enjoyment of my stay here, I owe a very big note of thanks. Finally, to my parents, who never cease to encourage and provide unending support, my gratitude is immense. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -214- 8. APPENDICES Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -215- APPENDIX 8.1 Relationship between the 1-dB compressed power and thirdorder intercept point for a third-order nonlinearity With the notation of Chapter 1, the fundamental output ,tud€ A a for a single input signal of A cos a t amplitude is given by A a = a lA + T a 3A3 At the 1-dB compressed point,. ^ = . 89 = 1 + | ^ A 2 so that A 2 4 al = (-.11) T — 3 3 a 3 and A 2 = (. 89) 2 x i x (-. H ) a j -± - a3 The output power at the 1-dB compressed point is thus (in dBm in a 50 ft system) 3 'a, l d B = 1 0 l o g ( | x ( . 8 9 ) 2 = < | x . n x i 0 » 2 . ) + 10 1 o g ^ j = .65 + G where G Q o Ia3 I - 10 log ■!— = -!a^ (i) ^ 2 0 log a^ , and is the small signal gain in dB. The intercept point is that point at which the (extra polated) small-signal fundamental power, p a would equal that Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -216- of the third-order intermodulation product, P 2 a _g* Equation (1.3) P a = ^ (a-jA)2 and Equation (1.4) yields P 2 a_g = 7 Using (mW) , (f-a 3A 3 ) ^-gy0, (mW) , and at Pj = P * = P 2a-e, „ 2,2 _ 9 a 26 al A ~ TS" 3 a2 - 4 al _ 3 ~T^~ a 3 a l x 1000 50 * and _ 1 ,4 Pi_ 2 Thus al P T = 10 log p T = G + 10 log -r~-— T + H * 25 i x o Ia 3 j Comparing * (dBm) * (ii) (i) and (ii) , p i = po,iaB + 10-6 . (dBm) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -217- APPENDIX 8.2 Error Analysis of S-Parameter Test Setup The finite directivities of the directional couplers used to sample the reference and offset that must be accounted required. This was observed channels leads toan for if extreme precision is experimentally with a linear system (consisting of a waveguide FET amplifier). test section in place of the The loci were indeed circles with centers offset from the origin, dependent on the reference channel power level. The system shown in in Figure 2.3 may the signal flow graph measurement signal, shown of f^, where be modelled of Figure 8.2.1, V 2+ is the and V 2“ the test-channel signal. as for the reference Let C be the coupling coefficient (-30 dB for the HPX750E couplers used here), and L the leakage between oppositely travelling the detector arm and the wave, with S the reflection coefficient (assumed equal in the phase shift of the transmitted detector assembly both arms). 9 is wave through the coupler. By signal flow graph theory, REF = V 2+ej<hc+LS(l-C)eje] + V 2"[LtCS(l-C)eje] ------------------------------ (i) 1 - S2 (l-C)2e2^e Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -218- s(l-c)e Ref Test Figure 8.2.1 s(l-c)e i« Signal flow graph for the system of Figure 2.3 showing errors arising due to finite coupler directivities. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -219- and TEST = V 2+e ^ [I/+CS(1—C) e^e] + V 2 IC+LS (1-C) e^] (ii) 1 - S2 (l-C)2e2je Thus the measured reflection coefficient, TEST/REF, is given by = V 2+k l e ^ + V2“k2 f MEAS (iii) V2"kl with kl= I> + + V 2+k 2 e ^ CS(l-C) exp(j 6 ), and k2=C+ LS(l-C) exp(jQ). Now, the desired reflection coefficient is given by: Civ) Using this in (iii) with z a complex variable and K=kl/k2 a complex constant gives: Sn (Z) + K f (Z) Cvl KSU (Z) + 1 This is Sii<z) is a of circle (as for a linear the measured circle. this a bilinear transformation. £(z) due to the cross-ratio transformation, the relative locus of device), the locus reflection coefficient Furthermore, If the size is also a property of (radii) of the transformed circles to each other is unchanged. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -220- APPENDIX 8.3 The Bandpass Sampling Theorem [50] A signal which lies within a band extending from f 1 to f ^ + W could be translated to the range 0 to W by stan dard modulation techniques, sampled at a rate 2W, and restored to its original range by an inverse translation. In order to avoid shifting the band, the bandpass sampling theorem may be invoked: For uniformly spaced samples, the minimum sampling frequency is given by £s =2M(1 + S> (i) where f 2 = highest frequency in band f W s = minimum sampling frequency = width of band m = largest integer not exceeding fj/W and k = ( f 2/ w ) - m . The value of k in (i) varies between zero and unity. When the band is located between adjacent multiples of W, we have k = 0 and f range may be. S = 2W, no matter how high the frequency As k increases from zero to unity the sampling rate increases from 2W to 2W (! + =•). The curve Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -221- of minimum sampling rate versus the highest frequency in a band of constant width thus becomes a series of sawteeth of successively decreasing height as shown in Figure 8.3.1. > vl 1 a# s 2 2 w aw Hig h e s t Figure 8.3.1 7W aw aw FK eausN C r -f* Minimum sampling frequency for band of width W. The highest sampling rate is required when m = 1 and k approaches unity. This is the case of a signal band lying between W - Af and 2 W - A f , with Af small. The sampling rate needed is 2 ( 2 W - A f ) which approaches the value 4W as Af ap proaches zero. case of m = 2 , When Af actually equals zero, we change to the k=0, and f = 2W. By adjusting W so that the band edge is an integral multiple of W above the origin, the minimum sampling frequency of 2W may be used. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -222- APPENDIX 8.4 Establishment of the Band-Edge Criterion Let the sampling frequency of the time-domain waveform be f , and N be the number of spectral lines desired within s the bandlimited region (6 in our case, to include both fifth-order components). Refer to Figure 8.4.1. Let NORD be an index parameter equal to N/2. Suppose the bandedge, which will align with the origin, is at fB.E. = fl " <N0RD> A • Then the bandedge criterion requires that fB.E. = integer x £s (m - NORD) A if f^ I mA . Thus set (m - NORD) A = integer. f s Furthermore f = s (i) (ii) SAMPLING FREQUENCY = (2N+ 1) A for bandpass sampling A NS x A . i.e. set (iii) m - NORD = integer, NS from (ii) and (iii). i.e. set m = N S x integer + N O R D . (iv) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -223- BANDEDGE Figure 8.4.1 The desired bandpass window about and showing the relative positioning of the bandedge needed to achieve the desired downconversion. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -224A is adjusted to ensure that (i) and (iv) are simultaneously satisfied. For the minimum change to A, use (ii) to set integer = closest integer to Then, fJS E| —— — - . s find m from (iv) , and use (i) to adjust A = f^ym . An alternative approach to understanding the band-edge criterion is to consider the sampling instants of the band-edge frequency. fB £ Its carrier frequency is = fi - (NORD) A = (m - NORD) A . At the sampling instants, t s =l/f x k rs k = 0, 1, . . .NS - 1 and the carrier phase is related to exp j2lr s ,(m - NORD) A • k, ns~A ] exp d 2tt [ = exp j 2 tt [integer x k ] , if the band-edge criterion is satisfied, = 1 . Consequently, the sampling instants always occur at the peaks of the carrier at this frequency; as a result a wave form of this frequency is interpreted as a constant (DC) value. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -225- "0'22 0022 0023 0024 0025 0026 0027 r-t 50 WRITER *100) 50 WRITE(6.100) 100 FORMAT(' ENTER THE LOWER OFTHE TWOFREQUENCIES IH MHz;-) READ<9»*) FREQ1 WRITE(c*101) 101 FORMAT(' ENTER THE DIFFERENCEFREQUENCY IN MHz 5') READ<9**> DFREQ 0029 102 0030 103 0031 0032 0033 0034 0035 003c C* 0037 C* 0038 C* 0039 C# 0040 Ct 0041 0042 0043 URITE(6?103) FORMAT<' ENTER THE TOTAL NUMBER OF SIGNAL COMPONENTS TO'* #' CONSIDER (MUST BE EVEN).*') READ(9** > N N0RB=N/2 NN=N+S NS=2*NH Calculate the radial freau'encies* butfirst 3djust the difference freauencs to be an integral submultiple of the lower freuerics. Their ratio* M* Bust also satisfy the band-edgecriterion. ADJN=FREQ1/(DFRECI*NS) M=NS*IDINT(ADJN) + NORD DFREQ=FREG1/M The software above is a portion of the main program and is used to achieve the band-edge criterion. The integral value of the parameter ADJN (corresponding to "integer" in the previous equations) 41. is obtained in line With the same notation as before, M is set in line 42 according to Equation (iv), and in line 43, the difference frequency DFREQ is adjusted according to Equation (i). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -226- APPENDIX 8.5 Software for Time - to Frequency - Domain Conversion SUBROUTINE FQRTX <FNT»A>B>IEK; 0092 .'1A07 0094 0093 0096 0097 0098 0099 0100 0101 d ts p* C* Of C* C* C* C# C* C* r»o» PURPOSE! This routine converts from the time domain to the freauencs domain usms a standard Goertzel algorithm DFT. Subroutine SETFQN must be called once> to load the COMMON block FQNSET This routine is derived from subroutine F0RIT. BATE! 20 October.1983 AUTHOR! Rowan Gilmore IMPLICIT REALX8 (A-Hf0-Z) 0103 C0MM0N/F0NSET/ C0EF>81> Cl>FC0S<6>11>13)> FSIN <6>11>13)t 0104 *FC0SD(6*ll»13)»FSIND<6fll»13)»RNW(12)»N»NSAMP»NPl 0105 DIMENSION ACS)»B(3)?FNT(13) 0106 0107 C* 60 C=1.0 0103 010? 0110 0111 0112 8=0,0 J=1 FMTZ=FNT(1) ; p u t contains the input time-samples of 70 U2=0.0 current and is a vector of length 13. U1=0.Q I=NSAMP 0113 0114 0115 C3 FORM FOURIER COEFFICIENTS RECURSIVELY 0116 C* 0117 C» 73 U0=FNT<m2.0*C*Ul-U2 0118 U2=U1 0119 U1=U0 0120 0121 0122 1=1-1 IF<I—1) 80780>75 30 A(J;=COEF4t(FNTZ+CiUl-U2) ; A (J) and B(J), J=l,...7 contain £(3)= -CQEF#S#U1 the real and imaginary Fourier IF(J-NPl) 90>100>100 coefficients as output. 0123 0124 0125 0126 C* J=1 corresponds to the DC case. J=2 is the lowest freauency 0127 C* bandpass component. 0128 C* 0129 C* 0130 90 a=Cl*C-31#S ? COEF,Sl,Cl are trigonometric constants 0131 S=C1*S+S1#C related to the fundamental angle ^^/N. 0132 (See Appendix 8.6) .• J=J+1 C133 GO TO 70 0134 m ir 100 A(1>=A(1>*0.5 RETURN 0136 END 0137 V J.WW Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 227- APPEKDIX 8.6 Calculation of Coefficients Used in the Frequency - to Time - Domain Conversion IMPLICIT REALX8 (A-HrO-Z) V1S3 0154 COMMON/FQNSET/ COEF>SI>C1»FC3S<6»11>13 >»FSIN <6»11»lo >» XFC0SD<6»ll»13)fF3IND(6»ll»13)»RNM(12)»NMAXfNSAMP»NPl 0155 COMMON/PHSVLT/ OMEGAiV1REAL(11> >VIIMAG <11)>V2REAL(11)* 015a XV2IMAG(11)»FREQ1 0157 C0MM0N/MYN2IN/ RJUNK1 (4) »TAU?RJUNK2(10) 0158 DIMENSION RNUTAU(12)»RN(12> 015? 0160 c* Set maximum number of harmonics* 0161 C* 0162 C* NMAX=N N=6 0163 0164 NF’1=N+1 N0RD=N/2 0165 0166 C* Calculate constants for sampled current calculations. 0167 C# 0168 C* NSAMF=2*NMAX + 1 0169 RNSAMP=DFL0h T(NSAMP) 0170 TSAMP-2.D0 * DARCOS<-1.DO)/(RNSAMPXOMEGA) (OMEGA=A, the difference 0171 frequency). 0172 CX T3AMP is the sampling time interval 0173 c* RHU will give the different RF freauenca components 0174 c* RNWTAU gives the gate delaa time at each component 0175 c* 0176 c* 0177 c* M=l-N 0173 017? c* N2=2*N 0180 DO 50 L=2»N2 0181 RNU(L)=FREQ1 + MXOMEGA 0182 M=M+1 0183 50 RNUTAU(L>=TAU*RNU(L) 0184 0135 cx DO loop for each time sample. I =time interval considered 0136 c* T is the current sampling instant 01S7 c* 0138 c* 018? DO 100 I-lrNSAKP (Set up loops to calculate the sine andcosine) T=TSAMP#DFL0AT(I-1)(values at the sampling instants foreach ) 0190 CTEMP=DC0S(FREQ1XT)(phasor frequency and time, and each ) 0191 STEMP=DSIN(FREQ1XT)(frequency-shift set. ) 0192 M=l-N 0193 0194 INBEX=1 0195 ex DO loop for each freauenca component that is needed, 0196 c* 0197 c* fi- the freauenca component considered. ■no Use lire c o s in e angle form ula to improve a ccu ra cy. •S.U Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -228019? C:t' 0200 0201 0202 0203 0204 0205 0206 020? 0209 020? DO 90 L=2»N2 TOELT= M*OMEGA *T CD£L=DCOS(TDELT) SDEL=DSIN<TDELT) FCS=CTEMP*CDEL - STEMP*SDEL FSN=STEMP*CDEL + CTEMP*SDEL RNWT=RNW(L)*T RNWTMT=RNWT - RNUTAU(L) FCSB=DCOS(RNWTMT) FSND=DSIN(RNWTMT) 0210 C* 0211 C* Ne::t assign FC0S(K»J»I) so that K=stuft no» J=RF component no.? 0212 C* and I=time instant. J=1 is the DC case» so is not included here. 0213 C* 55 K=INDEX 0214 60 J=L-K+1 0215 These lines improve efficiency due to 0216 IF(J.LE.l) GO TO 90 overlap of adjacent frequencies when 021? IF(J-NPl)70»70f65 components are shifted by A in different 021S 65 INDEX=INDEX+1 translation sets. The desired outputs 021? GO TO 55 sure aligned correctly into 6x11x13 arrays 0220 C* FCOS and FSIN. FCOSD and FSIND are time0221 70 FC0S(K»JiI)=FCS delayed coefficients. 0222 FSIN(KfJ?I)=FSN (See Appendix 8.7). 0223 FCOSD(K rJ >I)=FCSD 0224 FSIND(K»Jfl)=FSND K=K+1 0225 0226 IF(K-N)60f60»90 90 M=M+1 0227 0228 100 CONTINUE 0229 C* 0230 C* Calculate the fundamental fpeouencies to pass across in COMMON 0231 c? 0232 DO 200 J=2»NP1 0233 M=J-N+2 0234 RNW(J)= FREQ1 + M*0MEGA 0235 CUc 0236 C* Calculate the remaining fpeouencies of the harmonics 0237 C* 0238 RNU(8)=2.*RNW(4) 0239 RNW(9) = 2.*RNU<5) 0240 RNW(10)= 3.*RNW<4) _0241 200 RNW(11)= 3.*RNW<5> 0265 C* Calculate constants for DFT's. J266 C* 0267 C* (Needed for DFT;) CCEF=2»/RNSAMP 0268 (See Appendix 8.5) CGNST=DARC0S(-1.DO)*C0EF 026? S1=DSIN(CONST) 0270 C1=DC0S(CONST) 0271 0272 C* WRITE(6»1000) V/4. w 0274 1000 FORMAT'.' *** /FQNSET/ LOADED***') 0275 RETURN t 02?6 END Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -229- APPENDIX 8.7 Software for Frequency - to Time - Domain Conversion In the software below, time samples of the interface v o l tages are calculated using the phasor frequency components, and the values of the sine and cosine terms at each sampling instant At (stored in FCOS, FSIN, etc., see Appendix 8 .6). any one time instant, V1D is the time-delayed gate voltage, and DV1D its derivative. DV2 is the derivative of the drain voltage. These time-sample voltages are calculated from the 11 phasor frequency components contained (according to their real and imaginary parts at the gate and drain, respect ively) VTREAL, V1IMAG, V2REAL, and V2IMAG. : )0'-8 C* ■'JO 4? C* 0050 C* 0051 C* 0032 0053 Calculate voltade* delayed voltade* 3nd time deriv. samples, The first DO loop (150) is at each time sample. DO 150 1=1»NSAMP 01=01REAL(1> 0054 010=01 0055 0056 02=02REAL(1> 0010=0, 0057 0058 005? 0060 0061 0062 D02=0, C* C$ C* C* Calculate the conipcnents to this time sample frum every freauency component. DO 100 J=2>NN 0063 01=01 + 01REAL(J)*FC03(KfJ»I> - 01IMAG(J5*FSIN(K>J»I) 0064 0065 01D=01D + 01REAL(J)*FC0SD( J>I) - OlIMAGtJ)*FSIND(KrJ»I> 02=02 + 02REAL(J)JtFCOS(K»J»I) - 02IMAG(J)*FSIN(K»J>IJ 0066 0067 0068 D01D=D01D + RNW(J)*< -01REAL<J)#FSIND(KfJ»I) * - 01IMAG(J>*FCOSD(K»J>I)) 100 D02=DV2 + RNW<J>*< -V2REAU J)*FSIN(KpJ»I> - V2I»AG( J)*FC0S(K» J* i>J 006? C* Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -230APPENDIX 8.8 The Modified Harmonic Balance (MHB) Algorithm The following three pages show the software that is the heart of the harmonic balance program, i.e. the loop in which the balance between the linear and nonlinear par titions is achieved. The input vector C contains the initial phasor current estimates at the gate and dra,in. The output vector WORK contains final error and convergence criteria. Upon con vergence, the routine is exited, and the desired phasor voltages and currents at the interface are contained in the common blocks /PHSVLT/ and /PHSCUR/, respectively. Initialization occurs in lines 1-50; the main itera tion loop is in lines 51-116; and the error messages for failed convergence are in lines 116-140. For each iteration of the main loop, the phasor inter face currents are loaded into CUR from the estimates in C (lines 54-60). and outputs V(K,1 or 2). Subroutine VNODE calls the linear circuit, (in COMMON) the phasor interface voltages These are loaded into the nonlinear phasor voltage vectors VlEEAL etc. (lines 68-74), and the nonlinear model uses these upon calling subroutine FRQNON. That rou tine also performs domain conversions, the frequency transla tions, and aliasing deembedding. Current phasors ClREAL etc. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -231are generated as output (via COMMON), enabling the error function to be formed (lines 88- 100) and used in con vergence criteria (101-106). Finally, new current esti mates are formed as a weighted average of the old estimates and loaded into the vector C (lines 107-112), and the process repeated. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -232- |'-'Uh ; l .L 0001 cx 0002 SUBROUTINE Bm LN(C?WORK»J) 0003 CX., .............. .......... *...... ...»................ 0004 CX 0005 CX PUSPOSEJThis routine calculates the linear voltages from 0006 CX current estimates (in COMMON LINET)* finds the resulting 0007 CX non linear currents* and finds the error function and then iterates 0008 CX usins a fixed point iteration method. 000? C# This file contains all harm, balance procedures that remain 0010 CX constant throughout all such procedures. 0011 CX 0012 C* FILENAME: RJG.NBALN.TEST 0013 CX 0014 CX BATE:13 Octoberrl?33 • 0015 CX MODIFIED TO INCLUDE HIGHER HARM0NICS( 2ND*3RD) 9JULY 1984 0016 CX AUTHOR:Rowan J. Gilmore 0017 CX 0013 CX SUBROUTINES CALLED!VNODE* FRQNON C-019 CX 0020 CX IMPORTANT: This routine is called from the main program and 0021 CX reauires current estimates for the first NC harmonicsrand 0022 CX voltages calculated from the linear analysis subroutine. 0023 CX 0024 CX.,,........... .................... .................. 0025 COMMON /LINET/V*CUR*N*NC /0MEGA/SC*WK 0026 COMMOH/PHSVLT/DFREQ»V1REAL*VIIMAG *V2REAL*V2IMAG *W 0027 C0MM0N/PHSCUR/C1REAL *ClIMAG *C2REAL tC2IMAG 0023 COMPLEXX16 V<11,4),CUR<11»4)*DCMPLX 002? REALX8 C(44),F<44>*W*MFREQ*SC(11)iP*WK 0030 REALX8 C1REALU1 >*C1IMAG( 11) *C2REAL< 11) »C2IMAG<11)* 0031 * VlREAL(ll)*VlIMAG(ll)*V2REAL(11> >V2IMAG(11)»DREm L»DIMAG*wORK(3) 0032 CX 0033 CX The primary variables to alter to achieveoptimization at the 0034 CX interface are CUR(K»1) and CUR(K»2). These must be set from the 0035 CX single input vector C to allow calling further subroutines* 0036 WRITE(6*50) 0037 50 FORMAT?///' Enter the iteration constant p ? and relative weight;') 0038 READ(9*X)?»HEIGHT 003? IF(P.LT.O.DO) GO TO 490 0040 CX 0041 CX Set the weighting coefficients for each harmonic component in the 0042 CX total error sum. 0043 MN=N+5 0044 DO 60 I=1*NN 0045 60 SC(I)=1.0 0046 SC'1)=UEIGHT 0047 SC(4)=10*UEIGHT 004S SC',5)=SC(4) 0049 CX 0050 RA0FRQ=« Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -233C'05: 0052 0053 0054 -.'035 0056 0057 0058 0059 0060 DO 400 KGUNT=1»150 75 DO 100 1=1>NN K=4*I C* CX CX Rescale variables back to true currents CUR(I»1)=BCMPLX(C(K-3)»C(K-2)) CUR <1>2>=DCMPLX(C(K-l).C(K)) C ,100 CONTINUE 0061 CX 0062 CX Call the linear network with currentestimates in CUR(I..)j 0063 CX which are obtained from thelatestiterates. C(K). 0064 CX Return with voltages at linearinterfacein V(I».) 0065 CALL VNODE 0066 CX C067 CX 0068 DO 200 K=1»NN 0069 V1REAL(K)=DREAL(V(K»1)) 0070 V1IMA6(K)=DIMAG(V(K»1)) 0071 V2REAL(K)=BREAL(V(K»2)) 0072 V2IMAG(K)=DI(1AG(V(K»2>) 0073 C 0074 200 CONTINUE 0075 CX 0076 CX Call the nonlinear network with the new voltage estimates at interfac L* 0077 CALL FRQNON 0078 CX 0079 CX New nonlinear currents C1REAL etc. are calculated in FRQNON. 0080 CX The old iterates for current are kept in the vector C(J).used above. 0081 CX Higher harmonic currents are also generated. 0082 CX Bandpass sampling and aliasing deembeddirig are performed in 0083 CX FRQNON by freouency shifting. 0084 CX In V1REAL etc. K=1 is DC 0085 CX In V1REAL etc.. K=1 is DC? K=2>7 are about the fundamental; 0036 CX K=S»9 are the second harmonics; K=10.11 are third. 0037 CX 0088 CX Calculate the error function from the last current values. COS? ERR0RG=0.0 0090 £RR0RB=0,0 0091 290 DO 300 1=1.NN 0092 K=4XI 0093 F(K-3)=SC(I)X(C1REAL(I)-C(K-3 >) 0094 F(K-2>=SC(I)X(ClIMAG(I)-C(K-2)) 0095 F(K-1?=SC<DX< C2REAL <I>-C(K-l>) 0096 F(K) =SC<I)X(C2IHAG(I)-C(K)) 0097 ERRCRG=ERRORG+F(K-3)XX2+F(K-2)XX2 0098 ERR0RB=ERR0RB + F(K-1)XX2 + F(K)XX2 0099 ERROR=ERRGRG+ERRQRD C-100 300 CONTINUE 0101 IF(KOUNT.LE.IO) GO TO 460 0102 295 IF(ERROR.GT.IEIO) GO TO 475 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -234- d04 Ct For® the new updates of the current for the next iteration* C-105 C* if the error is still too larde. 010a C* 010? DO 380 1=1*NN 010S X=4*I 010? C(K-3)=P*C1REAL(I) +<l-P)*C(K-3) 0110 C(K-2)=P*C1IMAG(I) + (l-P)*C(K-2) 0111 C(K-1)=P*C2REAL(I) + (1-P)*CCK-1> 0112 C(K)=P*C2IMAG(I> + (1-P)«C(K) 0113 330 CONTINUE 0114 TOL=(1.E-10)*SC(1)*SC<1> 0115 IF(ERROR.LT.TQL) GO TO 500 0116 400 CONTINUE 0117 WRITE<6*450) 0113 450 FORMATdX*'ITERATION LIMIT EXCEEDED') 0119 GO TO 500 0120 460 WRITE <6*461)ERRORG *ERRORD 0121 461 F0RMAT(1PE?.2»5X*1PE?.2) 0122 IF(KO'JNT.EQ.IO) GO TO 470 0123 GO TO 295 0124 470 WRITE(6*472) 0125 472 FORMAT(IX*' Enter 1 to CONTINUE:') 0126 READ (9.4!)IGOON 0127 IF(IGOON.EQ.l) GOTO 295 012S 475 WORK(3)=1.0 012? WRITE(6*480)KGUNT 0130 480 FORMAT(' K0UNT='*I5) 0131 RETURN 0132 490 UGRK(3)=-1.0 0133 RETURN 0134 C* 0135 C* The error function is snail enouah now. The currents adree. 0136 C* 0137 0138 0139 0140 500 W0RK(2)=K0UNT W0RK(1)=ERR0R RETURN END Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -235- APPENDIX 8.9 Test of Software Using a Fifth - Order Nonlinearity Section 4.2 evaluates a test drain current C 2 as a nonlinear function of the applied drain voltage V 2 . The following page shows the computer output of the results of the MHB technique applied to this nonlinearity, in which the drain current at each of the desired frequency components is the output of interest. Harmonics labelled 3 and 4 give the desired funda mental outputs; those labelled 2 and 5 give the t h irdorder intermodulation components. Similarly, labels 1 and 6 are the fifth-order terms; 7 and 8 the second harmonic terms; and 9 and 10 the third harmonic terms. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -236- xxxxxxxxxxxxxxxxxxxxxxxxsaxxxxxxxxxxxsssx*********?^******#.****** Total power dissipation- 1.070+04 mW, DC gate current *-4.430-04 mA and DC drain current- 3.25D+05 mA ♦ 1 HARMONIC: » Pincident*-9.99D+01 dBm. Pref--8.91D+01dBm» Pdelivered*-1.00D+02dBmr and Pout--6.710+01 dBm* RH0in*222222222 at 222222 decrees GATE CURRENT* 1.560+02 + J-l.900-02 mA PRAj W CU.RWEWT"-A; 2fflt02 + J 3.110-01 mA V G S i n t e r n a l = - 4 . 8 8 D - 0 8 + J-8.89D-07 Volts VOSinternal* 1.070-06 + J 3.26D-06 Volts 2 HARMONIC: Pincident*-9.99D+01 dBm. Pref--7.43D+01dBm» Pdelivered=-1.00D+02dBmr and Pout*-3.12D+01 dBm. RH0in=X22X22X*X at 22X822 degrees GATE CURRENT* 8.590+02 + J-5.51D-02 mA DRAIN CURRENT=-6♦26D-01 + J-3.870+03 >> VGSinternal=-4.700-08 + J-5.76D-06 Volts » V0Sinternal=-2.290-03 + J 5.68D-06 Volts 3 HARMONIC: Pincident* 1.21D+02 dBm? Pref* 1.21D+02dBmr Pdelivered* 2.920+OldBmr and P out*-4.350+01 dBm. RHOin* ' 1.000 at' 0.0 degrees GATE CURRENT* 1.680+03 + J-4.96D-C2 mA DRAIN.CURRENT* 9.30D+03 + J-6.38D-01 mA A O O l ifrCK VGSinternal* 1*000+00 + J-l.090-03 Volts ArrLlfeU ttDSinternal* 1.000+00 + J-5.77D-05 Volts 4 HARMONIC: Pincident* 1.150+02 dBm* Pref* 1.15D+02dBm t Pdelivered* 2.:55D+01dBm» and Pout*-4.35D+01 dBm. RHQin* 1.000 at 0.0 degrees GATE CURRENT* 1.43D+03 + J-1.33D-02 mA DRAIN CURRENT* 1.670+00 + J 9.50D+05 n.A n n , >2.” VGSinternal* 5.000-01 + j - 7713D^06 Volts VDSinternal* 5.320-03 + J 1 .000+00 Units A P rU £ D 5 HARMONIC: » » Pincident=-9.y90+01 dBm. Pref*— 7.83D+01dBm» Pdelivered=-1.00D+02dBm» and Pout=-5*120+01 dBm. RH0in*888282288 at 222222 degrees GATE CURRENT* 5.47D+02 + J-1.23D-02 mA DRAIN CURRENT*-3.87D+03 + J 3.49D-01 mA VGSinternal=-4.28D-07 + J-2.97D-06 Volts VDSinternal* 6.960-06 + J 2.11D-05 Volts 6 HARMONIC: P i n cident=-9.990+01 dBm* Pref*-9.52D+01dBm» Pdelivered=-1.00D+02dB*» and Pout*-6.710+01 dBm. RH0in=828888888 at 222222 degrees GATE CURRENT* 7.81D+01 + J-l.790-03 mA DRAIN C U R R E N T S .970-01 + J-6.25D+02 mA » VGSinternal*-2.880-08 + J-5.66D-07 Volts » VDSinternal=-3«640-06 + J 1.040-06 Volts Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -237- 7 HARMONICt » » Pincident*-?«99D+01 dBa» Prer*-1.23D+02dBa» Pdelivered=-1.00D+02dBm> and Pout=-5.51D+01 dBm. RHOin********** at ****** degrees GATE CURRENT* 3.29D+00 + J-1.72D-03 aA J1RAIN CURRENT* 2f49Df03 + J-9.99D-01 «A voSinternal* 1.73D-$7 + J-2715D-07 Volts VDSinternal*-l .45D-06 + J-3.09D-05 Volts 8 HARMONIC? Pincident*-?.9?D+01 dBm? » » 9 HARMONIC: » » 10 HARMONIC: » » Pref*-1.74D+02dKm, Pdelivered=-1»OOD+V2dB*r and Kout--5.51B+01 dBm. RHOin********** at ****** degrees GATE CURRENT* 1.55D-03 + J-8.37D-03 aA DRAIN C U R R E N T = - 2 ♦50D+03 + J 7.69D-01 aA VGSinternal*-1.73D-07 + J 1.74D-07 Volts VDSinternal* 1.65D-06 + J 3.10D-05 Volts Pincident*-?.99D+01 dBa» Prer*-7.71D+01dBa» Pdelivered=-1.00D+02dBa* and Pout*-5.78D+01 UB». RHOin********** at ****** degrees GATE CURRENT* A.25D+02 + J-7.A1D-02 aA DRAIN CURRENT* 1.81D+03 + J-1.23D+00 aA ^ + J-l.18D-05 Volts VDSinternal* 8.&3D-08 + J-3.51D-05 Volts Pincident*-9.99D+01 dBa> Pref*-8.86D+01dBa> Pdelivered=-1»0OD+02dBm? and Pout*-5.73D+01 dBm. RHOin********** at ****** degrees GATE CURRENT* 1.66D+02 + J-2.29D-02 aA DRAIN C URR E N T = - 5 .9GD-01 + J-1.81D+03 aA VGSinternal=-2.01D-08 + J-3.12D-06 Volts VDSinternal*-3.41D-05 + J-2.45D-07 Volts x************************************************************ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -238- 9. BIBLIOGRAPHY 1. A. Mad jar and F.J. Rosenbaum, "An AC Large Signal Model for the GaAs MESFET", Report N00014-78-C-0256, Washing ton University, St. Louis, Missouri, August 1979. 2. A. Madjar and F.J. Rosenbaum, "A Large-Signal Model for the GaAs MESFET", IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-29, No. 8 , pp 781-788, August 1981. ------------- 3. Y. Tajiina, B. Wrona, and K. Mishima, "GaAs FET LargeSignal Model and its Application to Circuit Designs", IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-^ 8 , P P 171-175, February 1981. 4. W.R. Curtice, "A MESFET Model for Use in the Design of GaAs Integrated Circuits", IEEE Transactions on Micro wave Theory and Techniques, ~Vol. MTT-28, pp 448-456, May 1980. 5. C. Rauscher and H. Willing, "Simulation of Nonlinear Microwave FET Performance Using a Quasi-Static Model", IEEE Transactions on Microwave Theory and Techniques, Vol. MTT - 2 7 , No. 10, pp 834-840, October 1979. 6. C.L. Chen and K.D. Wise, "Transconductance Compression in Submicrometer GaAs MESFETS", IEEE Electron Device Letters, Vol. E D L - 4 , No. 10, pp 341-343, October 1983. 7. L.W. Nagel, "SPICE 2: A Computer Program to Simulate Semiconductor Circuits", Memo ERL-M520 1975, Electron ics Research Lab, University of California, Berkeley. 8. COMSAT General Integrated Systems, Inc., SUPER-COMPACT User Manual, 1982. 9. D.R. Green Jr., "Characterization and Compensation of Nonlinearities in Microwave Transmitters", 1982 P r o ceedings of IEEE Global Telecommunications Conference, (IEEE Catalog No. 82CH1819-2), Miami, Fla., Section A7.5, pp 213-217, 1982. 10. R.J. Gilmore and F.J. Rosenbaum, "An Analytic Approach to Optimum Oscillator Design Using S-Parameters", IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-31, No. 8, pp 633-639, August 1983. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -239- 11. R.J. Gilmore and F.J. Rosenbaum, "GaAs MESFET Oscil lator Design Dsing Large-Signal S-Parameters", 1983 IEEE MTT S-International Microwave Symposium Digest, Boston, Mass., pp 279-281, June 1983. 12. G.L. Heiter, "Characterization of Nonlinearities in Microwave Devices and Systems", IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-21, No. 12, pp 797-805, December 1973. 13. S.M. Perlow, "Third-Order Distortion Products in Amplifiers and Mixers", RCA Review, Vol. 37, pp 234-266, June 1976. 14. Y. Takayama and K. Honjo, "Nonlinearity and Inter modulation Distortion in Microwave Power GaAs FET Amplifiers", NEC Research and Development, No. 55, pp 29-36, October 1979. 15. O. shimbo, "Effects of Intermodulation Distortion, AM-PM Conversion and Additive Noise in Multicarrier TWT Systems", Proceedings of the IEEE, Vol. 59, pp 230-238, 1 9 T T 16. R.S. Tucker, "Third Order Intermodulation Distortion and Gain Compression in GaAs FETs", IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-27, No. 5, pp 400-408, May 1979. 17. J.A. Higgins and R.L. Kuvas, "Analysis and Improvement of Intermodulation Distortion in GaAs Power FETs", IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-2 8 , No. 1, PP 9-17, January 1980. 18. R.A. Minasian, "Intermodulation Distortion Analysis of MESFET Amplifiers Using the Volterra Series Representation", IEEE Transactions on Microwave Theory and Techniques, Vol. M T T - 2 8 ,' No. 1, pp 1-8, January 1980. 19. R.A. Minasian, "Analysis of Intermodulation Distortion in GaAs MESFET Amplifiers", Electron.Letters, Vol. 1 4 , 31 August, 1978. 20. R. Gupta, C. Englefield, and P. Goud, "Intermodulation Distortion in Microwave MESFET Amplifiers", 1979 IEEE MTT-S International Microwave Symposium Digest, pp 405-407, June 1979. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -24021. C.W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations , Prentice-Hall, Inc., Englewood Cliffs, N.J. 1971. 22. D.R. Green, Jr. and F.J. Rosenbaum, "Performance Limits on GaAs FET Large-and-Small-Signal Circuits", Report N00014-80-0318, Washington University, St. Louis, Missouri, October 1981. 23. D.L. Peterson, A.M. Pavio, Jr., and B. Kim, "A GaAs FET Model for Large-Signal Applications", IEEE Trans actions on Microwave Theory and Techniques, Vol. MTT-32, No. 3, pp 276-281, March 1964. 24. V. Rizzoli and A. Lipparini, "A Computer-Aided Approach to the Nonlinear Design of Microwave Transistor Oscilla tors", 1982 IEEE MTT-S International Microwave Sympo sium Digest, Dallas, Texas, pp 453-455, June 1982. 25. S. Kawano and F.J. Rosenbaum, "Modelling of Third-Order Intermodulation Distortion in GaAs MESFETS", MSEE Thesis, Washington University, St. Louis, Missouri, August 1983. 26. R.J. Gilmore and F.J. Rosenbaum, "Modelling of Non linear Distortion in GaAs MESFETs", 1984 IEEE MTT-S tsium Digest, San Fran- 27. V. Rizzoli, A. Lipparini, and E. Marazzi, "A GeneralPurpose Program for Nonlinear Microwave Circuit Design", IEEE Transactions on Microwave Theory and Techniques, Vol. M T T - 31, No. 9, pp h 6 % - 7 i 0 , September 1983. 28. M.S. Nakhla and J. Vlach, "A Piecewise Harmonic Balance Technique for Determination of Periodic Response of Nonlinear Systems", IEEE Transactions on Circuits and Systems, Vol. CAS-23, No. 2, February 19~7£. 29. A.V. Oppenheim and R.W. Schafer, Digital Signal Pro cessing, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1975. 30. R.S. Tucker, "RF Characterization of Microwave Power FETs", IEEE Transactions on Microwave Theory and Tech niques, Vol. MTT-29, N o 8 , pp 776-781, August Id&l. 31. K.M. Johnson, "Large-Signal GaAs MESFET Oscillator Design", IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-27, No~. 3, pp 21^-237, March 1979. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -24132. L.S. Houselander, H.Y. Chow, and R. Spence, "Transistor Characterization by Effective Large-Signal Two-Port Parameters", IEEE Journal of Solid-State Circuits, Vol. SC-5. pp 77-79, April 1970. 33. L. Gustaffson, G.H. Bertil Hansson, K.I. Lundstrom, "On the Use of Describing Functions in the Study of Nonlinear Active Microwave Circuits", IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-20, No. 6 , pp 402-409, June 1972. 34. S. Mazumder and P.D. van der Puije, "Two-Signal Method of Measuring the Large-Signal S-Parameters of Transist ors", IEEE Transactions on Microwave Theory and Tech niques, Vol. M T T - 2 6 , No. 6 , pp 41^-420, June 1978. 35. T. Sakane, Y. Arai, and H. Komizo, "Large-Signal Dynamic Behavior of X-Band GaAs FETs", IEEE 1980 International Solid-State Circuits Conference D i g e s t , pp 160-161. 36. Y. Takayama, "A New Load-Pull Characterization Method for Microwave Power Transistors", N E C Research and Development, N o . 5 0 , pp 23-29, April 1978. 37. International Mathematical Subroutine Library, LEQT1F, IMSL Library Users' Manu a l -, Houston, Texas, June 1952. 38. C. Lawson and R. Hanson, Solving Least Squares Problems , Prentice-Hall Inc., Englewood Cliffs, tf.J.',' 15P74. 39. International Mathematical Subroutine Library, ZXSSQ, IMSL Library Users' Manual , Houston, Texas, June i r a n ------------------------- 40. R.G. Hicks and P.J. Khan, "Numerical Analysis of Nonlinear Solid-State Device Excitation in Microwave Circuits", IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-30, Nol 3, pp 251-259, March 1982. 41. R.J. Gilmore and F.J. Rosenbaum, "Large-Signal Circuit Design Using GaAs MESFETs", Report N0014-81-C-2343 Washington University, St. Louis, Missouri, September 1982. 42. F. Sechi, "Design Procedure for High-Efficiency Linear Microwave Power Amplifiers", IEEE Transactions on Micro wave Theory and Techniques, Vol. M T T - 2 8 , No. 11, pp 1157-1163, November 1980. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -24243. F.E. Emery, "Solid State Limiting Amplifiers", Watkins-Johnson Co. Tech-Notes, Vol. 5 , N o . 5, September/October 1978. 44. J.J. Jones, "Hard Limiting of Two Signals in Random N o i s e " , IEEE Transactions on Information Theory, Vol. IT-9, pp 332-340, January 1963. 45. J.A. Roberts, E.T. Tsui, D.C. Watson, "Signal-to-Noise Ratio Evaluations for Nonlinear Amplifiers", IEEE Transactions on Communications, Vol. COM-27, N o . 1, pp 197-201, January 1979. 46. C. Rauscher and R. Tucker, "Modelling the Third-Order Intermodulation Distortion Properties of a GaAs FET", Electronics Letters, Vol. 13, No. 17, 18 August 1977. 47. E.W. Strid and T. Duder,> "Intermodulation Distortion Behavior of GaAs Power FETs", 1978 IEEE International Microwave Symposium Digest, pp 135-137. 48. J.A. Higgins, "Intermodulation Distortion in GaAs FETs", 1978 IEEE International Microwave Symposium Digest, pp 138-141. 49. R.K. Gupta, P.A. Goud, and C.G. Englefield, "Improve ment of Intermodulation Distortion in Microwave MESFET Amplifiers Using Gate-Bias Compensation", Electronics Letters, Vol. 1 5 , No. 23, 8 November 1979. 50. C.B. Feldman and W.R. Bennett, "Band Width and Transmission Performance", The Bell System Technical Journal, Vol. 2 8 , No. 3, pp 490-595, New York, July 1949. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -24310. VITA Biographical items on the author of the dissertation, Rowan J. Gilmore 1) Born October 26, 1955 2) Attended the University of Queensland from February, 1973 to December, 1976. Received the degree of Bachelor of Engineering with First Class Honors in Electrical Engineering. Awarded the University of Queensland Medal, 1977. 3) Electrical Engineer, The Overseas Telecommunications Commission, Sydney, Australia, 1977. Field Engineer, Schlumberger Technical Services, Inc., 1978-1980. Senior Engineer, Central Microwave Company, August, 1982 to present. 4) Attended Washington University from January, 1981 to the present date. 5) Membership in Professional Societies: I.E.E.E., I.E.E. December, 1984 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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