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Order N u m b er 9032037 M icrow ave crystal oscillators Avanic, Branko, Ph.D. University of Miami, 1990 300 N. Zeeb Rd. Ann Arbor, MI 48106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. U N IV ER SIT Y OF M IAM I MICROWAVE CRYSTAL OSCILLATORS By Branko Avanic A DISSERTATION Submitted to the Faculty of the University of Miami in partial fulfillment of the requirements for the degree of Doctor of Philosophy Coral Gables, Florida May, 1990 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. U N IV ER SITY OF M IA M I A Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy MICROWAVE CRYSTAL OSCILLATORS Branko Avanic Approved : -— — i Guillermo Gonzalez Professor of Electrical and Computer Engineering Chairperson of Thesis Committee Pamela A. Ferguson^ Dean of the Graduate School and AssoSSfee-Trovost Tzay Y. ^Young Professor of Electrical and Computer Engineering Agustin A. Recio Associate Professor of Electrical and Computer Engineering Claude S. Lindquist Professor of Electrical and Computer Engineering George C. Alexandrakis Professor of Physics Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. AVANIC, BRANKO (Ph.D., Electrical and Computer Engineering) MICROWAVE CRYSTAL OSCILLATORS. (May, 1990) Abstract of a doctoral dissertation at the University of Miami. Dissertation supervised by Professor Guillermo Gonzalez. Number of pages in text : 151 In this dissertation we present a closed-form design method for negative resistance crystal oscillators. A full characterization is presented on every block of a two port oscilla tor. The oscillator design method is presented in a general form such that it is applicable to any resonator used today. Also, the design procedure is based on readily available mi crowave param eters known as, small signal S-parameters. The design m ethod is verified experimentally with two crystals, one with a fundamental frequency in the lower microwave region (i.e., 843 MHz), and the other with a fundamental frequency in the UHF region (i.e., 383 MHz). The crystals were selected since they represent a new type of resonators used in fundamental mode at the referred frequency bands. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CO NTEN TS page A ck n o w led g em en ts..........................................................................................................vi C hapter I IN T R O D U C T IO N .................................................................................. 1 1.1 M otivation and Background ............................................................................. 1 1.2 Dissertation Outline ........................................................................................... 2 C hapter II R F A N D M IC RO W AVE R E SO N A T O R S ................................5 2.1 In tro d u c tio n .......................................................................................................... 5 2.2 Resonator Equivalent Circuit ..........................................................................6 2.3 Helical Resonators ............................................................................................... 9 2.4 Transmission Line Resonators ........................................................................13 2.4.1 M icrostrip Resonator Im plem entation............................................... 18 2.4.2 Stripline Resonator Implementation ..................................................23 2.5 Dielectric Resonators ........................................................................................27 2.6 Y ttrium Iron Garnet (YIG) Resonators .......................................................32 2.7 Varactor Tuned N etw orks................................................................................ 35 C hapter III Q U A R T Z C R Y STAL R E SO N A T O R S ...................................42 3.1 In tro d u c tio n ........................................................................................................42 3.2 The Piezoelectric Effect ...................................................................................43 3.3 Q uartz Resonator Manufacturing ..................................................................44 iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.4 AT-Cut Crystal Characteristics ................................................................. 47 3.5 Resonator Model ............................................................................... 3.6 Quartz Resonator MeasurementS y ste m ....................................................... 52 50 C h ap ter IV N O ISE IN O SCILLATO RS ...........................................................54 4.1 In tro d u c tio n .......................................................................................................54 4.2 Noise Characterization in O scillators...........................................................56 4.3 Oscillator Phase N o is e .....................................................................................63 4.4 Oscillator Phase Noise Measurement ........................................................... 67 4.4.1 Direct SSB Phase-Noise Measurement ..............................................67 4.4.2 Phase Detector Method ........................................................................68 4.4.3 Heterodyne Frequency Measurement ................................................. 71 C hapter V N E G A T IV E R E SIST A N C E D E S IG N FO R C R Y STA L O SCILLATO RS ............................................................ 73 5.1 In tro d u c tio n ....................................................................................................... 73 5.2 Design P ro c e d u re .............................................................................................. 75 5.2.1 Oscillation C o n d itio n s........................................................................... 75 5.2.2 Resonator C haracterization..................................................................77 5.2.3 Transistor Feedback Synthesis .............................................................79 5.2.3.1 Series Feedback ..........................................................................79 5.2.3.2 Parallel Feedback ...................................................................... 81 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.2.3.3 Maximum Reflection Coefficient .......................................... 83 5.2.4 Terminating Net work S y n th esis..........................................................86 5.3 Design Example: 842.911 MHz Crystal Oscillator ..................................... 90 5.3.1 Oscillator Performance ......................................................................... 99 5.4 Design Example: 383.868 MHz Crystal Oscillator ..................................100 5.4.1 Oscillator Performance ....................................................................... 106 C h ap ter V I C O N C L U S IO N .................................................................................. I l l C hapter V II B IB L IO G R A P H Y ...........................................................................113 A P P E N D IC E S 1. Varactor Equivalent Q .....................................................................................123 2. Oscillator Phase Noise M odel..........................................................................125 3. Stability C riteria ...............................................................................................129 4. Series/Parallel Feedback Networks ............................................................... 134 5. Feedback Mapping ........................................................................................... 140 6. Optimum F eed b ack ...........................................................................................143 7. Argument of T i n .............................................................................................. 146 8. Instability Circles ............................................................................................. 148 9. Phase Angle Mapping ...................................................................................... 150 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOW LEDGEM ENTS I take this opportunity to express my sincere thanks and appreciation to Dr. Guillermo Gonzalez. Throughout my studies, he has not only been my academic advisor, but also a teacher, a bastion, and most of all a friend. His patience, guidance, and support have all contributed to the successful completion of this work. Special thanks and appreciation also go to the other members of the dissertation committee, Dr. Tzay Y. Young, Dr. Agustin A. Recio, Dr. Claude S. Lindquist, and Dr. George C. Alexandrakis for their support and advice during the course of this work. To my wife, Monique, my daughter Vanessa, I dedicate my love and gratitude for their sacrifice and patience throughout these years. Thanks to their motivation and understanding, this work has been a fulfilling and memorable experience. To my m other Desanka, my father Branko, I convey my eternal gratitude, for much of my accomplishments have been due to their encouragement, and the example they have set. Special thanks go to Dr. Kamal Yacoub, for his help, advice and guidance throughout my educational life. Dr. Yacoub has not only been my undergraduate advisor, but also a sincere friend. Finally, I would also like to express appreciation to the many friends whose opinions and discussions have been invaluable, and, of course, the support given by all my family members. Thank you all. vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER I * IN T R O D U C T IO N 1.1 M O T IV A T IO N A N D B A C K G R O U N D The number of communication systems using electromagnetic waves as a means of propagation is constantly increasing. At the same time the usable frequency spectrum remains constant or it is increasing at a much lower pace. This forces communications systems to have channel spacings that are closer to each other, and thus places increasing spectral requirements on the reference oscillator. The introduction of crystal oscillators with fundamental frequencies up to the microwave region provides the opportunity for research and development of verystable frequency sources. The crystal, as a resonator in the oscillator, exhibits a significant increase in the quality factor as compared to other presently used resonators at a given frequency and size. This high-Q characteristics of the crystal, translates into improved spectral performance of the oscillator. Most commercially available crystals operate at fundamental frequencies below 50 MHz. The majority of crystal oscillators, operating at frequencies above 50 MHz, rely on the harmonic generation of the crystal; th at is, they enhance the harmonic components at the desired frequencies. 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 Design methodology incorporating crystal resonators in the microwave region is nonexistent due to the infancy stage of these resonators. In this thesis a negativeresistance design procedure is developed for oscillator design using these crystal resonators. The general theory of negative-resistance oscillator design has been addressed in literature, but in most cases it lacks a unified and modern treatm ent for all types of resonators. Also, the theories presented concentrate only on particular sub-sections of the oscillator, relying heavily on computer aided optimization techniques; thus, in general, failing to provide a complete oscillator design methodology. This provided the motivation for finding a unified, coherent, and closed-form design methodology for negative-resistance oscillators. The unified design method developed in this thesis is of such a nature th at it is not only applicable to the new crystal resonator structures, but also to other resonator structures encountered in today’s oscillators. 1.2 D ISSE R T A T IO N O U T L IN E The design method presented uses readily available microwave parameters, known as S-parameters, in the determ ination of the oscillator circuit and topol ogy. The two port negative-resistance oscillator is shown in Fig. 1.1. As can be seen from Fig. 1.1, the negative resistance oscillator is composed of three main sections: the load network (resonator), the transistor network, and the term inating network. The load network is composed of the resonator, which is the frequency deter mining component. The resonator is considered an essential part of an oscillator, and as such, oscillators are named after the resonator being used. Being of such importance, and since a fairly new resonator is used in this dissertation, a complete review of typical resonators encountered in today’s microwave oscillators is pre sented in Chapter 2. Basic design formulas and complete references are provided Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 LOAD TRANSISTOR TERMINATING NETWORK NETWORK NETWORK F ig u re 1.1 Two port negative-resistance oscillator. for all the resonators so as to generalize the scope of the dissertation. Furthermore, in Chapter 2 typical Q ’s are presented for the different structures in order to have a comparison basis for the crystal resonators. Chapter 3 presents a short review of crystal resonators with emphasis on the new crystal resonators used in this dissertation. The nature of these crystals require new manufacturing techniques and testing methods. Both topics are discussed in Chapter 3. In Chapter 4 we introduce the terminology relevant to oscillator noise. We discuss a simple oscillator noise model in order to get a basic understanding of how some of the oscillator param eters affect the noise performance of the oscillator and how they are related to the resonator. Also, we present the different noise measurement systems used at microwave frequencies, one of which (i.e., the phase detector method) is used in Chapter 5 to evaluate the performance of the oscillators designed. In Chapter 5 we integrate the information of Chapter 2 and Chapter 3 in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 representation of the load network. We develop the closed-form design formulas for the transistor network which yield maximum negative resistance for all the feedback networks encountered, and we introduce new techniques in the design of the terminating network. As shown in Chapter 5, the design of the terminating network constitutes a key function in the negative-resistance th at the active circuit presents to the resonator, as well as being an interface to the external load. As a finale, the design procedure is verified with two oscillators, both of which use crystals having fundamental frequencies which are not commercially available. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H A P T E R II R F A N D M IC RO W AVE R E SO N A T O R S 2.1 IN T R O D U C T IO N In this chapter the various resonant structures th at are used in the design of negative-resistance oscillators are reviewed. The advantages and disadvantages of these structures are discussed and used as a comparison basis for the crystal resonators introduced in Chapter 3. Depending on the frequency of interest, a particular resonator structure might be more appropriate. In the lower MHz region, crystal resonators form the basis of highly-stable oscillators. As the wavelength is reduced, microstrip and stripline resonators become practical resonator structures. The standard sizes of these struc tures are A/4, more commonly known as quarter-wave resonators. These resonators form the main technology used in the upper megahertz and lower gigahertz regions. As the frequency is further increased the radiation losses in the microstrip lines be come a factor which in many cases limit its usability. For high-power applications, the resonators of choice are the cavity-type resonators. These types of resonators, which are bulky and expensive, will not be covered in this dissertation. 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A more recent advance in the m iniaturization of microwave circuits has been the appearance of the low-loss tem perature-stable dielectric resonators. At a particular frequency, the size of the dielectric resonator is considerably smaller than th at of the cavity resonator. This has been possible with the introduction of materials having relative dielectric constants greater than 30, and very-low dissipation factors (i.e., a high Q). In the area of tunable resonators we will discuss the varactor diode as a part of a resonant structure. Most tunable oscillators use the varactor diode due to its small size and easy integrability to MIC (Microwave Integrated Circuits). The main drawbacks of the varactor are its small tuning bandw idth and its low Q. At the higher frequencies, where larger bandwidths are required, the YIG (Y ttrium Iron G arnet) resonators is also used. 2.2 R E S O N A T O R E Q U IV A L E N T C IR C U IT Close to the resonator’s resonant frequency, a microwave resonator can be mod elled by an equivalent lumped series or parallel RLC circuit. The resistance R rep resents the losses of the resonant structure. This losses arise from metallization losses (i.e., a skin effect), dielectric losses, and in some resonator structures by radi ation losses. For most practical applications it is of interest to find the lowest-loss resonator. Figures 2.1a and 2.1b illustrate the resonator’s equivalent series and parallel lumped circuits, respectively. For the series configuration, the input impedance is given by Z i n ( iv ) = R + j wL — wC (2 .1) At resonance, the impedance is purely real and the resonant frequency is given by - = im ( 2 -2 ) The losses of the resonator are generally expressed in terms of the quality factor Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 ^IN ^IN /? (b) (a) F ig u re 2.1 (a) Series RLC resonator; (b) Parallel RLC resonator. Q which is defined by _ Energy stored in the resonator per cycle W° Energy lost per cycle When the circuit is not connected to other networks the Q is known asthe unloaded Q, and denoted by Qu\ while when the resonator is connected to other structures, the Q is known as the loaded Q, denoted by Q l . For the series RLC it is simple to show that the total energy stored is given by Energy stored per cycle = \-Li2 £ (2.4) and the energy dissipated per cycle is given by Energy dissipated per cycle = - i 2R (2.5) Substituting (2.4) and (2.5) into (2.3) gives Qu — W0 w0L R (2.6) Using (2.2) and (2.6) we can write (2.1) in terms of w0 and Qu, namely Z i n ( w ) = R + j Q uR w w0 w0 w Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.7) 8 For the parallel RLC in Fig. 2.1b the adm ittance seen at the input is YI N (w) = - + j wC wL Hence, the resonant frequency is given by (2.2). At resonance, Q u is given by Qu — w0R C Thus, }jjv(' J) can expressed as w w0 w — G + j Q uG w0 !'/« (« > )= w0 w wa w ( 2.8) It can be observed th at both (2.7) and (2.8) are identical in form. An alternate form, commonly used, for the Q is also presented. Using (2.1) — L - dw 1 ^ w2C (2.9) where X represents the reactance of Z j ^ . Letting w = w0 and substituting (2.2) into (2.9) results in the expression dX = 2L dw Using (2.6) we can write d X _ 2QUR dw wn or Qu = w0 d X 2R dw ( 2 .10 ) A dual form can be obtained for the parallel resonator, namely _ w 0 dB “ 2G dw where B represents the susceptance of Y i n . It can be seen from (2.10) th at tor a very-high Qu it is necessary to use resonators with the lowest losses possible and with a high reactive slope. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 In general these resonant structures are connected to other circuits, as shown in Fig.2.2. As can be observed from this figure the external circuits will load the resonant structure by presenting a resistance R l across it. W ith R> — R + R i it follows th at the loaded Q is given by w0 dX 2Rt dw Ql R, ( 2 .11) R, (b) (a) F ig u re 2.2 (a) Series RLC resonator coupled to a load; (b) Parallel RLC resonator coupled to a load. Similarly, for the parallel resonator shown in Fig 2.2b, it follows that 2 G t dw where Gt = G + G l . 2.3 H ELICAL R E SO N A T O R S Helical resonators are generally used in the VHF-UHF band. The helical res onator in its standard form is basically a quarter-wavelength coaxial resonator with the inner conductor coiled in the form of a helix. The helix is enclosed in a highly conductive shield of either circular or square cross-section. By coiling the center conductor a significant reduction in size is obtained with a relatively small reduction Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 'D/4 'D/4 F ig u re 2.3 Helical resonator. in performance. The application of this helical structure to oscillator design stems in the high-Q performance of this structure. Macalpine and Schildknecht (1959) presents closed form, approximate, formulas for the most im portant parameters. Figure 2.3 shows a sketch of a helical resonator. For the given notation, and for shields with circular cross-section, Macalpine and Schildknecht (1959) present the values of the equivalent parallel LC circuit as L = 0.025n 2d2 p H per axial inch and c = 0.75 p F per axial inch y/e^log(D/d) where n represents the number of turns, and the capacitance C is derived assuming th at a dielectric with dielectric constant er is filling the resonant cavity. The above equations yield accurate results when 1.0 < 3 < 4.0 a 0.45 < 0.6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 and for J 0.4 < — < 0 .6 r L for - = 1.5 a 0.5 < — < 0 .7 for = 4.0 r a d T<2 where r is the pitch (i.e., the separation between turns). For a given (d/D ) ratio, the total number of turns N , the characteristic impedance Z 0, and the pitch r are given by JV = nb = 1720 logiojD/d) f 0D ( d / D ) [ l - { d / D ) \ d 2 Z 0 = 183nd ( l - 5 ) r = fobd 1720 D logio(D/d) 1 - (d/D )* 2 As mentioned earlier, the above equations are valid for helical resonators that are enclosed in a circular cross section shield. Shields of square cross section are also commonly used. The above equations can also be used for the square cross section shield provided th at the length of one side of the square is set to 1.2 Losses in a helical resonator include the loss in the helix, a conductor loss influenced by: the skin and proximity effects, the loss owing to the currents in the shield, and a dissipation loss due to the dielectric. For a copper helix and a non-magnetic shield the resistance of the coil is Rn = 0.083 <j> -n 7r y / f 1000 nd0 ohms per axial inch and the resistance due to the shield R, = ' 9.37n2b2(d/2)4y f M W f >< 10-4 ohms per axial inch b[D2(b + d ) / 8]5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 where <j>is known as the proximity factor, which for the previous conditions is set to <b v = 3.7 nd,0 The unloaded Qu is given by — Qu - — Qm + 1 Qd where Q m represents the m etal losses, th at is _ 2 -irfo L ( R c + R 3) and Q d represents the dielectric losses, namely Qd — 1 tanO The quantity tanO is known as the loss tangent of the dielectric. For a resonator with copper coil and copper shield W ith air as the dielectric, an approximation for the unloaded Q is Q u « 5 0 D y / f o = 6 0 S \/7 o (2.12) which shows that a square shield yields a higher Q in the resonators. Another useful relationship is th at of Q u and the volume (v) of the shield. For 0.4 < d / D < 0.6 and 1 < b/d < 3, the relation is Qu « 5 0 ( v ) * y / f o Figure 2.4 shows typical values of unloaded Q (2.12) th at can be obtained from helical resonators with square shields. As can be seen from the figure, unloaded Q ’s in the order of 600 are attainable in the UHF band, the compromise stemming in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 Helical Resonator Shield Size -a-0.1'-Shield & 0.2'-Shield -o- 0.4'-Shield -* ■ 0.5'-Shield 0.3'-Shield 1000 900 800 U n 700 I o 600 a 500 d e 400 d 300 Q 200 100 300 400 500 600 700 800 900 1000 Frequency (MHz) F igu re 2.4 Unloaded Q of helical resonator. the size of the shield. Macalpine and Schildknecht (1959) found th at there exists an optimum value for the shield dimensions for maximum unloaded Q given by d/D ph 0.55. A deviation from this dimension results in a considerable degradation since the unloaded Q exhibits a fast decrease as a function of d/D. 2.4 T R A N S M IS S IO N LIN E R E SO N A T O R S Inductive and capacitive elements can be realized using transmission-line ele- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 x =0 *=i F ig u re 2.5 Transmission line term inated by Z[. ments. For a section of transmission line of characteristic impedance Z 0 terminated by an impedance Z{ (see Fig. 2.5), the input impedance at a distance x from the term ination is given by ^ Zi + j Z 0 tan fix ^ Zsn(x) = Z°z7+jZ,i^-/3x ( 2 -1 3 ) where (3 = 27r/A, or equivalently f> = wV l c = - vp = ^ c and Z0 = 'L Also, from the above expressions it is easy to show that the inductance and capac itance per unit length are given by: L = (H/rn) (2.14) C = f - (F / m ) £J o C Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 By properly selecting Zj and the length of the transmission line we can realize inductors, capacitors, and resonant structures. Setting Z/ = 0, (i.e., a shorted transmission line), (2.13) reduces to Z I N = j Z 0 tan fix (2.15) Equation (2.15) shows th at depending on fix, Z j n represents either an inductor (when tan (fix) > 0), or a capacitor (when tan fix < 0). W hen Z; = oo, (i.e., open circuit transmission line), the input impedance is Z i n = —j Z 0 cot fix (2.16) which can be used to represent either an inductor or a capacitor. For the realization of inductors and capacitors fix is set such th at small angle approximations can be used for the tangent and the cotangent. Using the small angle approximations, namely tand « 6, the angle 9 will have an upper bound for a given error. For an error of 10% or less, the angle 8 which in our case is fix will have to be bounded by fix < tt/6 or in terms of wavelength A x < — 12 Short circuited or open circuited quarter-wave or half-wave transm ission lines are equivalent to series resonant or parallel resonant structure under certain con straints. For the equivalence to be true , the behavior of the line and the corre sponding circuit in the neighborhood of resonance has to be the same. For the parallel resonant circuit shown in Fig. 2.1b it follows th at ^ = 2C aw Similarly, from (2.15) for the short-circuited quarter-wave line at an angular fre quency w0 (or an open-circuited half-wave line) the susceptance can be w ritten as B = Y0 cot(fix) = Ya cot Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Substituting x = A/4, A = v / f , and w = w 0 in the above equation, we obtain dB__Yo_ dw 4f 0 Therefore, for frequencies close to the resonant frequency, a short circuited quarter wave line has the same variation of susceptance with frequency as a parallel resonant circuit provided th at its characteristic impedance satisfies the relationship: Zo = 8cTo Plotting the susceptance variation versus frequency for the A/4 resonator and for the lumped-parallel resonator (see Fig. 2.6), we can see that the susceptance variation for the quarter-wave line follows a cotangent curve (solid line in Fig. 2.6), while the susceptance variation of a parallel resonant circuit follows the dotted line. Therefore, the representation of a parallel-resonant circuit by a transmission line is valid only in the region where the two curves have a common tangent (i.e., the straight line 2C, also shown in Fig. 2.6). In practice this equivalency is only valid over a band within a few per cent of the' resonant frequency. In the case of the series resonant circuit in Fig. 2.1, it can be shown th at a half wave transmission line term inated by a load resistance R behaves in an equivalent form as the series resonant circuit provided that the characteristic impedance of the line is set to Z0 = 4 L f 0 As in the case of parallel resonance, this equivalence is only valid in a frequency band within a few percent of the resonant frequency. In voltage-controlled oscillator applications, where tuning of the resonant fre quency over a large bandw idth is required it is common to combine lumped and distributed elements to form a resonator. Figure 2.7 illustrates a typical situation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 Transmission line B 2C (tangent) Resonant circuit Wo F ig u re 2.6 Transmission line and parallel resonant circuit susceptance variation vs. frequency. where the inductor of the parallel resonant structure is obtained using a transm is sion line. By proper selection of the capacitor and inductor, resonant structures from the VHF to the microwave region can be realized. ■o Zo F ig u re 2.7 Lum ped/D istributed resonant circuit. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 The previous configurations provide an improvement on the unloaded Q as compared to a fully lumped parallel resonant structure. This is possible since dis tributed inductors can have better Q than lumped inductors. The following two sections discuss the implementation of distributed inductors and resonators. 2.4.1 M IC R O S T R IP R E S O N A T O R IM P L E M E N T A T IO N In the present section the physical realization of transmission line components (i.e., inductors resonators) using microstrip technology is presented. Microstrip structures are the most common alternatives used to realize microwave integrated circuits. The photolithographic process is used do deposit a metallization pattern on one side of the substrate. Figure 2.8a illustrates the microstrip construction. At microwave frequencies all dimensions are critical and extreme care has to be taken to manufacture the structure to the desired specifications. In terms of the electromagnetic field distribution microstrips are very complex and approximate formulae are generally used. Strip conductor Th i F ig u re 2.8 (a) Microstrip construction; (b) Fields in a microstrip structure. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 The fields on printed microstrip structures extend over discontinuous regions, formed partly of dielectric and partly of air, (see Fig 2.8b). Electromagnetic waves propagating along the structure cannot he purely transverse and therefore exhibit dispersion. This dispersion effect is small for microstrip structures and can be neglected for practical applications. The dominant mode of the transmission line is called quasi-TEM. The problem would be much simpler if there was an unbounded homogeneous dielectric surrounding the strip. The velocity of propagation would then be defined unambiguously since the wave propagation would be strictly TEM; th at is, vp = c (2.17) Therefore, it is standard to analyze a microstrip structure by replacing the discontinuous microstrip structure line by a more tractable structure (see Fig. 2.9) with conductors having the exact geometry (w , h ,b ), surrounded by a single homo geneous dielectric of effective perm ittivity ee. For this geometry Hammerstad and Jensen (1980) derived the formulae er —1 er + 1 £e — — x-----1" , 1+ 10 h w —ab where a= 1+ (w / h ) A + [iy/(52/i)]5 749 a ln \ {wJh)2A + 0.432 ( and b= er — 0.9 er + 3 0 .0 5 3 The above equations are accurate within 0.2 % provided that w o.oi < - < 100 h and 1 < er < 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 The phase velocity vp and the wavelength A are then related to the effective perm ittivity by ( 2 ' 1 8 ) and A A0 y/^e where c„ is the velocity of light in free space and A0is the free-space wavelength (i.e., assuming air as the dielectric). lii F ig u re 2.9 Equivalent homogeneous microstrip structure. In order to realize inductors, capacitors, or resonators as determined by (2.15), and (2.16) it is necessary to find the relationship between the physical dimensions (w,h, and t) and the characteristic impedance (Z 0). An up to date review on the field of microstrips with extensive bibliography can be found in the paper by Gardiol (1988). Therefore, we only present the necessary synthesis formulae. Wheeler (1965) obtained the following synthesis formulae having a 1% accuracy, h " 1 TTCr ‘ - e - T 1. In ( B - l) + 0 .3 9 - M i] + % [ B - l - l n ( 2 6 -1 ) ] i f f 2 2; if f > 2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In many situations, a fast, and not so accurate result is sufficient; for this instances the width to height ratio required for a given characteristic impedance is read of graphs. A typical plot of Z 0 vs w /h , and er is shown in Fig. 2.10. 1000 100 G N o Oil 10 “ '1 2 ~ 10 1.0 W /h F ig u re 2.10 Characteristic impedance of microstrip lines, [Sobol (1971)]. As the frequency increases, the fields tend to concentrate within the dielectric substrate causing an increase in the effective perm ittivity ee. This frequency de pendence of the perm ittivity can be neglected if the frequency of operation is below th at of the dispersion frequency defined by Getsinger (1973) as f — ^° U ~ 2fi0h Losses in microstrip circuits arise from three different sources: conductor losses, dielectric losses, and radiation losses. Gonzalez (1984), presents a comprehensive Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 analysis for the losses of a microstrip structure. The overall Q of the microstrip can then be w ritten as J_ _1_ _1_ Qt ~ Qc + Qd + Qr A general form for the quality factor in terms of the loss factor a is given by 0 “ Ta Using <2' 19> the above formula we can the find the conductor Q (i.e.,Qc), dielectric Q (i.e., Qd) and radiation Q (i.e., Qr ) by replacing the associated loss factor. The conductor losses are due to ohmic losses resulting from the finite conduc tivity of the metal. An approximate value, sufficient in most situations is given by ac ~ 8.68 In f Ho \ Z 0w V where a is the conductivity cr ^ Qc = = * 17 T----- \a c of the metal and fi0= Anx 10-7 H /m . The dielectric losses are produced by energy dissipated within the substrate, an approximate relationship assuming cr = 0 is given by „ er ee —1 tan 6 & 2 7 .3 —f — y/ee 6r 1 Aq „ Qd = » = n AcVrf 7— where tan 8 is known as the loss tangent of the dielectric and is specified by the dielectric manufacturers. Finally the radiation losses are given by Qr Zo 4807t(/j/A o)F1 where TP = Ce( -f ) +1_ * '( f) (ee ( / ) - I ) 2 Jn 2 m / ) ] 2/ ’ / -\A e(/) + 1 \ \ - 1J It can be noted th at the radiation losses are strongly frequency dependent, and as the frequency increases so does the radiation. Bahl and B hartia (1980) find that for microstrip lines, radiation becomes significant at frequencies larger than / « 2 -1 4(f - )1/4 G H z Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 From the above presentation it is im portant to realize th at the unloaded Q of a microstrip component (i.e., inductor or resonator) is determined by several factors. A proper choice of low-loss dielectric m aterial is required for high-Q microstrip elements. Typical dielectric m aterial used due to its superior overall performance is Alumina. Alumina substrates exhibits a dielectric constant of er = 10 and a loss factor of tan 9 = 0.0001. From the above relationships, for a substrate thickness and frequency, a max imum unloaded Q can be obtained for a unique value of characteristic impedance. For example, using a 50 mils thick Alumina substrate at f 0 = 800 MHz a value of Qu ~ 350 is obtained for a characteristic impedance of Z 0 « 24 fh Using Fig. 2.10 this corresponds to a w / h ~ 2.5, thus the width of the microstrip line required is w = 125 mils. Similarly, for the same m aterial and at the same frequency, a line with Z Q = 80 Q, will only yield a value of Q u « 190, which is 54 % less than the optimum. It is also interesting to notice that if a thinner substrate i3 used (i.e, 25 mils) at f 0 = 800 MHz, the maximum attainable unloaded Q is approximately 220 which corresponds to a Z 0 = 15 fi line. Thus, for an oscillator, where the maximum unloaded Q is desirable, thicker substrates are desired. At higher frequencies (say f a > 5 GHz), where increased radiation is present, the fact th at higher Q are obtained with thicker substrate does not hold. At those frequencies the reverse is true; furthermore, lower Qs result due to the increased losses. 2.4.2 S T R IP L IN E R E S O N A T O R IM P L E M E N T A T IO N There exist many applications where the radiation losses encountered in mi crostrip lines are detrimental to the performance of the circuit (i.e. oscillators). For this situations it is standard to resort to stripline circuits. A stripline circuit is very similar to a coaxial line, since both use conducting ground plane separated by a di electric m aterial to confine the fields inside the dielectric thus no radiation is found Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 externally. The losses in striplines are determined by the conductor and dielectric losses. Thus, with the elimination of the radiation losses, stripline resonators have better unloaded Q values as compared to its equivalent resonator implemented in microstrip technology. The construction of a stripline is illustrated in Fig. 2.11. The wave propagates in a purely transverse electromagnetic (TEM) mode in a stripline structure. The phase velocity is given by (2.17). '///////////////////////////} . _w _ > //////////////////////;////; F ig u re 2.11 Stripline transmission line construction. The characteristic impedance Z 0 of a stripline is calculated by conformal m ap ping techniques. For the general case, where the thickness of the conductor is not neglected, the equations are very complex and it is common to use graphical plots to obtain starting points in a design. A good reference on stripline circuits can be found in the paper by Bahl and Garg (1978). Curves for stripline impedance for various values of t/b and w/b are shown in Fig. 2.12. In a synthesis problem, where Z 0, £ri and t/b, are given, the width of the strip can be found from the figure. As mentioned earlier, losses in a stripline are composed of conductor losses and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 220 ops 0.10 200 Us 180 Vi I o 160 Nf a 0 ^ 140 ‘0.05 ' 0 .1 0 120 0.15 100 0.1 0.2 0.8 0.3 1.0 2.0 3.0 4.0 w F ig u re 2.12 Characteristic impedance of striplines [Matthei, et.al (1980)]. dielectric losses. Thus, the Q of a stripline is given by J _ - _L _L Qt Qc Qd where Qc and Qa are found using (2.19) with the appropriate a (i.e., a c or aj). Bahl and Garg (1978) present the needed formulae for the computation of the conductor and dielectric losses of striplines, namely 0.0231er Zo irffi0 a ‘ = 30n ( b - t ) V — ( A + B) where . 2u; 16 + i (2 b — t A = 1+ : + - r-!— In b —t 7r b — t t Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 In the above equation B can be found using the following equations, namely if Z 0\/^r[l + 2.3(t/b)] > 120 then B is given by B=(b-wTut/by (l{17A5b+3 5 ro )-9w+585“nr) while if Z 0yf£r\)- + 2.3(t/b)] < 120 then 5 = 0. The dielectric loss a j is given by The limiting factor in the Q of a stripline resonator is generally the conductor loss, since the dielectric loss in a good m aterial such as Alumina is very small. At f 0 = 800 MHz, using a 50 mils thick Alumina substrate, a Z 0 = 50 line will yield a value of Q u ~ 350. This can be compared to a microstrip Z a — 50 ft, line which in the same substrate it will yield a value of Qu « 300. The difference is accounted by the radiation losses present in microstrip lines. The maximum frequency of operation for stripline transmission lines is limited by the excitation of the TE mode. For wide lines, the cutoff is given by _15_ /to U I ^ U 7 0 ,-1 J where to,b are in cm, and f c is in GHz. An interesting point isth at Qt and f c are strongly dependent on the ground plane spacing. A wider spacing willincrease Qt while reducing the cutoff frequency f c. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 2.5 D IE L E C T R IC R E S O N A T O R S W ith the development of stable, high dielectric constant materials with low loss and good tem perature characteristics, the applications of dielectric resonators in the microwave spectrum has increased considerably (Plourde, et.al. 1981). The high dielectric constant (er from 35 to 100) is crucial, since not only it represents a size reduction as compared to a metallic cavity by a factor of approximately l/y^e^, but also is needed to confine most of the electromagnetic fields within the resonator. Typical shapes used in dielectric resonators are circular and tubular (hollow center). The resonant mode used is denoted by T E qis. The T E qis mode is characterized by having no electric field component in the z-direction. The constant magnetic and electric lines distribution are shown in Fig. 2.13. Z _ X F ig u re 2.13 Field distribution in cylindrical dielectric resonators. The electric field lines are concentric circles around the z-axis. The magnetic field lines are contained in the meridian plane and are used to couple to microstrip circuits as illustrated in Fig. 2.14. For the configuration shown in Fig. 2.14 [Abe et.al (1978)] presents a simple relationship between the resonator shape and resonant frequency where the frequency is proportional to the dielectric constant er , diameter Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 D, and height L of the cylindrical resonator. T hat is, \ r f x( mn) \ n\ \ * D) where x ( m n ) is the n th zero of the Bessel function J m(x). For the T E 01s resonance mode, x(01)=2.404. The coupling between the line and the resonator is accomplished by having the magnetic lines of the resonator perpendicular to the microstrip plane such th at they interact with the magnetic lines of the microstrip line, as illustrated in Fig. 2.14. Metal Enclosure Substrate Microstrip F ig u re 2.14 Coupling of a dielectric resonator to a microstrip line. The equivalent circuit of a dielectric resonator coupled to a microstrip line Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 h -e rw IN r \j F ig u re 2.15 Dielectric resonator equivalent circuit. of characteristic impedance Z 0 and terminated in normalized input impedance at the reference plane Z 0 is shown in Fig. 2.15. The P P ' is given by * = 1+ T T 7 W ) ( 2 '2 0 ) where /? is known as the coupling coefficient between the resonator and the transm is sion line. It is a function of the distance between the resonator and the transmission line, and it is given by 3 - R R 2Z Rext 0 Sn S21 where S u and S 2 1 are the reflection and transmission param eters found experi mentally at the resonant frequency. Q u represents the unloaded Q factor of the dielectric resonator which is given by the manufacturer and 8= f-fo fo The value of 3 can also be found from the physical configuration. This approach is involved and requires finite-element techniques to find the electric and magnetic fields for a given configuration. A good reference source is found in the book “Di electric Resonators” by D. Kayfez (1986). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 The reflection coefficient at P P \ denoted by T r , is given by _ 2 —1 Tr = —— z+1 ( 2 .21) Substituting (2.20) into (2.21) gives T r P = 1 + The reflection coefficient ]?/# is P+ j ( 2 Q u <5) shifted by 6, namely T i n = T r e ^ 26 or T i n = —T ^ = e x p / —2j ( 6 + tan 1 ^ ( l + ^/?)2 1 + /? V ) 2 + (2 ( 2 Q u£)2 u5)2 (2.22) Equation (2.22) shows th at the dielectric resonator coupled to a microstrip line can be used to realize any impedance. If the dielectric resonator is used at its resonant frequency, namely f = f 0 then (2.22) simplifies to r ™ = y f i e_'i2# <2-23) we can see th at for constant coupling coefficients (i.e., /? = k), T i n will trace a family of concentric circles when the electrical length of the transmission line is varied from 0 to 2ir, thus enabling the synthesis of any passive impedance. It is from (2.23) th at one selects the desired coupling coefficient and length of transmission line required for a particular impedance. Once the coupling coefficient is known the physical dimensions are found using either of two methods which are quite involved and require numerical methods for the solution. We will only reference the methods and not attem pt to discuss them in full. The first of the two is known as the H / I method, where I is the current flowing in the microstrip line, and H is the magnetic field induced by the current I. A full development can be found in the paper by G arault and Guillon (1976). The second method is known as the Magnetic Flux method and is presented in full Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 by Guillon, et.al (1985). Both methods solve for the external quality factor of the coupled system, but the coupling coefficient is related to the external quality factor by Qu Qe A typical result presented by Guillon, et.al (1981) is shown in Fig. 2.16. It is interesting to observe that the closer the dielectric resonator to the microstrip line, the lower the Q of the equivalent circuit. 1500 1000 500 DR d|(mm) F ig u re 2.16 External Quality Factor variation as a function of distance, [Guillon, et.al (1981)]. From Fig. 2.16, it can be seen th at dielectric resonators can attain unloaded Qs in the order of Qu f« 1000. This type of resonators exhibit comparatively much higher Qs than the structures previously presented. The only lim itation of dielectric resonators is their size. Practical sizes can be found in frequencies above 3 GHz. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 2.6 Y T T R IU M IR O N G A R N E T (Y IG ) R E SO N A T O R S Y ttrium Iron Garnet has a crystal structure th at when magnetized with a direct magnetic field exhibits resonance at microwave frequencies. The most standard form of the resonator is the spherical YIG configuration, in this form the resonant frequency is controlled only by the magnetic field and not by its physical dimensions. Its resonant frequency is given by fo = l { H 0 + H a) (2.24) where H 0 and H a are the applied direct field and the internal field (known also as the anisotropic field), respectively. The param eter 7 is defined as the charge to mass ratio of an electron and is also known as the gyromagnetic ratio [ it is equal to 7 = 2.21 x 10-5 ]. The tem perature behaviour of the YIG resonator is determ ined by the anisotropic field (H a). By proper orientation of the crystallographic axes, the anisotropic field can be eliminated and thus have a tem perature independent res onator. Also for YIG structures the applied field H 0 is several orders of magnitude higher than the anisotropic field. The dielectric constant of a YIG crystal is er = 16 and the the dielectric loss tangent is tand < 0.0005. This low loss ferrite allows resonators with high unloaded Q-factors. The unloaded Q is determined by the ratio of the direct magnetic field H 0 to the uniform mode linewidth A H , namely Qu = (2-25) The linewidth isa frequency dependent constant. For example,for pure YIG crys tals in the X-band: A H = 19.9. Unloaded Q-factors of the order of 10,000 are realizable using highly-polished YIG spheres. When the YIG sphere is orthogonally coupled to rm external circuit Helszcilm (1985) derives the external Q-factor as Qext = % rjr2 fi0wmV f K 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.26) 33 where Ho = 4 k x 107 H / m w m = ■yAnMs Vf = sphere volume (m 3) K = Coupling coefficient The coupling coefficient is the ratio of the magnetic field H x at the position of the sphere without the YIG resonator to the current in the coupling circuit. 4?tM 3 is known as the saturation magnetization which for pure YIG crystals is 1750. C arter (1961) presents an extensive treatm ent of different coupling structures. One of the most popular structures in oscillator design is the wire loop shown in Fig. 2.17. The coupling coefficient for the wire loop is given by * = £ (2.27) substituring (2.27) into (2.26), the external Q for a wire loop coupled YIG is Qext — 4r*Zn 0 ° Hwmvf 0 Ferromagnetic sample F ig u re 2.17 YIG wire loop coupling. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 The YIG in a wire loop configuration can be modelled by a parallel RLC circuit in series with an inductor as shown in Fig. 2.18. The inductor Li is the inductance of the loop wire. The element values are given by £ = V o V ,K l R — w 0w mii0VfI{ Qu — w0L Q u C= (2.28) 1 w$L o and L t = 2 [In — ------- 1.76 w+1 where I is the length of the loop circumference, and w and t are the width and the thickness of the strip. Ll F ig u re 2.18 YIG resonator equivalent circuit. Using the element values given by (2.28), the input impedance of the resonator is Z l N (w) = 1 + Q I p 2 + 3 { u,£" “ l + Qlp* } and w ^ W 0 w0 W Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 2 '2 9 ) 35 The high unloaded Qs that a polished YIG sphere presents (i.e., 10,000), makes this type of resonators exhibit loaded Qs in the order of 2,000 at 6 GHz. These Q-values are considerably higher than those of dielectric resonators. However, the current required to generate the magnetic field H 0 is a big disadvantage in this type of resonators. Finally it is im portant to mention the YIG resonator tunability characteristics (variable resonant frequency) by the application of an external mag netic field. This property allows the YIG resonator to be used in voltage controlled oscillator applications. 2.7 V A R A C T O R T U N E D N E T W O R K S Up to now, with the exception of YIG resonators, all the resonators discussed have resonant frequencies which are not controllable; th at is, the resonant frequency can not be varied by an external stimulus. This section concentrates in tunable resonators, and more specifically in the means of tuning a resonator. In the area of voltage- controlled oscillators, the resonant frequency of the resonator is varied by an external dc-voltage, known as the tuning voltage. Present technology is inclined to use varactor diodes for this applications. The varactor is usually made part of the resonant network, as shown in Fig 2.19. The varactor acts as a variable capacitor Cv( V ), where the capacitance across its terminals depends on the applied voltage. The center frequency for the network shown in Fig. 2.19 is found using (2.2) with C replaced by Cv( V ), th at is f = -----2;Vy/LCv(V) (2.30) The equation relating the diode capacitance to the applied reverse voltage is C»(V ) = , C°...A (l + n Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2'31) 36 Z -rM Tv V TUNE Rp 5 in ( w ) l Cb, - h ^ V A Resonator F ig u re 2.19 Varactor tuned resonant circuit, where Co = diode capacitance at zero volts <f>= diode junction potential (0.6-0.8 for silicon) A = slope of the diode C vs V curve V = diode reverse bias voltage The slope of the diode C vs V curve is controlled by the doping profile of the P-N junction. For the linearly graded profile A = 1/3, forthe abrupt profile A = 1/ 2, and for the hyperabrupt profile 1/2 < A < 2. A typical C vs V curve isshown in Fig. 2.20. Using (2.31) and the different slopes for the various varactor types, it can be seen th at the hyperabrupt varactor will exhibit the largest capacitance swing for a given change in tuning voltage. Therefore, it is common to use hyperabrupt varactors in wideband applications. On the other hand, the abrupt varactor has a smaller capacitance swing for the same control voltage swing, but compensates w ith b etter Q-factors as compared to the hyperabrubt diodes. As in all physically realizable systems, the varactor exhibits losses. The losses are manifested as a series resistance, the equivalent circuit of the varactor including Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 60 40 30 20 10 8 6 5 4 3 2 1 1 2 3 4 5 6 8 10 20 30 40 VR- REVERSE VOLTAGE (Volts) 60 F ig u re 2.20 Typical Capacitance vs Tuning Voltage, [MSI Electronics Inc. Woodside, New York]. losses is shown in Fig. 2.21. The capacitance arises from the depletion region width, while the resistance is mainly associated with the undepleted region which exhibits a relatively low resistivity, some losses are also present in the contact resistance. Figure 2.21 also presents a series inductor in the model of the diode. This inductance which is mainly due to the lead inductance associated with the particular packages. This lead inductance, which is in the order of 0.5 n H to 3.0 n H , will provide a series resonance for the varactor. Operation at frequencies close to the series resonance of the device will further deteriorate the Q of the varactor. It is recommended to operate a varactor several orders of magnitude below its series resonance. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 Q-factor of the varactor is given by Qv(V) = (2.32) w R vC J V ) In general, the resistance R v is voltage dependent, decreasing with increasing reverse bias, but in practical applications it is customary to assume the resistance constant across the tuning voltage range. L A R /^C vlV ) F ig u re 2.21 Varactor equivalent circuit. The varactor Q v [see (2.32)] is inversely proportional to frequency, the higher the frequency the lower the Q-factor. This property severely limits the application of a particular varactor to any frequency range, and care must be taken to select the proper varactor for the desired frequency. Furthermore, the varactor has in general the lowest Q in a tuning network and will as a consequence limit the overall Q of the network. A common approach to circumvent this lim itation is to place a high-Q capacitor in series with the varactor as shown in Fig 2.22. Depending on the ratio of the series capacitor to the varactor capacitance at a particular bias, the Q of the combined network, (see Appendix 1), denoted by <3„(V) is given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 Cc C v(V ) c > 2S S C v(V ) F ig u re 2.22 Enhancing the varactor Q. 1+ o g a Qn{V) = Qc Q„ , C°(T\ , \ Vc where \ Cc (2.33) / is the varactor Q at bias V, Q c is the coupling capacitor Q, Cv( V ) is the varactor capacitance at bias V, and C c is the coupling capacitance. It can be seen th at if the coupling capacitor is much smaller than the varactor capacitance, the Q of the combined network will approach the Q of the capacitor, th at is Qn(V) Qc « while if the coupling is of the order of the varactor capacitance the Q of the combined network will approach the Q of the varactor. It is therefore necessary to select the coupling capacitor such that at all control voltages the capacitance of the varactor is much smaller than the capacitance of the coupling capacitor. For the circuit shown in Fig. 2.19 we can now determine the frequency range for a given control voltage range (i.e., Vi < V < V2 ). The varactor will exhibit a capacitance C\ at bias V\ C V(V 0 = C i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 and C’2 at bias V2 Cv(V2) c2 = Thu capacitance at low control-voltage will control the lowest frequency of the resonator, while the capacitance at high-control voltage will control the highest frequency of the :c. onator. The total capacitance of the tank at low control voltage can be found from Fig 2.19, namely Ct l = C p + { C c \ \ C x) (2.34) where Cp represents the parasitic capacitance associated with the resonator and can not be neglected. Substituting (2.34) into (2.30) we obtain the lower resonant frequency <2-35> Similarly, the total tank capacitance at high control voltage is given by Ct h = Cp + (C c || C 2 ) (2.36) and the resonant frequency at high control voltage is (237) In a design, (2.35) and (2.37) can be reversed to find the necessary capacitance swing for the required frequency tuning range. The Q of the tuned resonant circuit (see Fig. 2.19), denoted by Q t is given by Qr(vj = Q^ + QJY) ( 2 '3 8 ) It is also im portant to mention that excessive RF swing on the varactors, by poor selection of the coupling capacitor, can severly affect the performance of the tuned circuit. Large R F swings at low control voltage can forward bias the varactor. Also the presence of a large RF1swing is conducive to harmonic generation due to the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 non-linearities of the varactor. Recall that the Q of a resonant circuit is proportional to the energy stored at the fundamental frequency, and if some of the energy is used in the harmonic generation, a Q degradation at the fundamental will be exhibited. Finally, as can be seen from (2.33) and (2.38), the Q of the resonant structure is determined by the Qs of the varactor, coupling capacitor and the Q of the inductor. At f 0 = 800 MHz, typical value for the Q of the coupling capacitor (i.e., assuming high-Q components and Cc < 5 pF) is Qc ~ 500. Similarly, for a silicon varactor Q v « 100, and for a stripline inductor Qu « 270. These values yield an unloaded Q for the tunable resonant structure of Q t « 120, showing th at the overall Q is being limited by the Q of the varactor. Thus, if a GaAs varactor was used, where the Qv m 250 (i.e., higher th an silicon), the overall Q would be Q t « 160. In this case, the inductor Q is the limiting factor. From the above discussion it can be seen th at a compromise is necessary. The compromise stemming in the cost, since GaAs are ten times more expensive than silicon devices. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H A P T E R III Q U A R T Z CRYSTAL R E SO N A T O R 3.1 IN T R O D U C T IO N In this chapter a short review of crystal resonators is given; including a dis cussion of the new crystals th a t operate in the fundamental mode at microwave frequencies. A complete study of quartz crystal resonators is a very complex subject which has attracted the interest of many people over the past 100 years and is still being pursued. Crystals are divided into 7 crystal systems and 32 crystal classes. Only 20 of the 32 crystal classes exhibit piezoelectric properties. Quartz belongs to the crystallographic class 32. As mentioned earlier a variety of crystalline structures exhibit piezoelectric properties. However, quartz (S i0 2 ) has become the industry standard because of its superior stability with time and tem perature, and its high mechanical Q. Furthermore, quartz is physically strong, can w ithstand tem peratures in excess of 500°C, is insoluble in ordinary acids, and is relatively inexpensive. Quartz is found in natural and cultured form. Natural quartz is mined mainly in Brazil and its purity is its m ain drawback. Cultured (or synthetic) quartz is grown by a hydrotherm al process resulting in higher yields and quality crystal blanks. Todays quartz crystals are in general of the cultured form. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 The development of quartz crystal in resonator applications was first studied by Cady (1946). Shortly after World War I (1922), he introduced the first quartz crystal oscillator. W ith the proliferation of commercial broadcast stations, the expanding use of the frequency spectrum, and the extensive use of communications and radar technology in World War II, the need for stable frequency sources was a necessity. Quartz crystal resonators, which exhibit excellent Q-characteristics, found extensive uses in frequency generation and filtering. Finally, in this chapter we also review some of the sophisticated methods that are used to measure the crystal’s param eters and electrical representation. 3.2 T H E PIE Z O E L E C T R IC E F F E C T Piezoelectricity is derived from Greek where piezo means “to press” . In elec trically uncharged crystals, the positive and negative charges are balanced and no piezoelectic effect can be observed. To observe the piezoelectric phenomena it is necessary to unbalance these charges by the application of an external force. The piezoelectric effect was first observed by the Curie brothers in 1880. They found that by applying mechanical force to certain crystalline materials an electrical charge was detected on the surface of the material. A year later Lipman showed the reverse effect, namely, by applying a voltage to the crystal a deformation proportional to the voltage was observed. Details of these early efforts in the development of the quartz crystal technology are given by Cady (1946). A good understanding of the piezoelectric effect is obtained by looking at the distribution of charges at the molecular level. Figure 3.1 illustrates unit cells which are considered as the building block of a crystal. Figure 3.1a shows a cell where the distribution of charges have a center of symmetry, while Fig. 3.1b is one without a center of symmetry. The edges of the unit cells are parallel to a set of lines called the crystallographic axes, identified as X, Y, and Z (Fig. 3.3). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 F ig u re 3.1 (a) Molecular structure with a symmetric charge distribution, (b) Quartz crystal molecule, (no center of symmetry) Observe th at crystals possessing a center of symmetry can not be electrically unbalanced by the application of a force, since the same displacement of positive and negative charges occurs. On the other hand, for a quartz crystal molecule (asymmetric structure), depending on the direction of an applied external force a displacement of the charges occurs and the center of each set of like charges shifts in opposite directions. This results in the separation of the centers of gravity of the positive and negative charges and therefore the production of an electric dipole moment or electrical polarization. The piezoelectric effect is illustrated in Fig. 3.2. 3.3 Q U A R T Z R E S O N A T O R M A N U F A C T U R IN G For application in electronic circuits, the raw quartz crystal bar must be first cut into different shapes and sizes. By wafering (slicing) the quartz bar at different angles with respect to the crystallographic axis, resonators with different charac teristics can be obtained. This quartz wafers are also known as blanks. Figure 3.3 shows the crystallographic axes in a quartz crystal. Wafering is generally done on circular or slurry saws. After sawing, the blanks Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 0-.( TnL . Y (b) (a) F igu re 3.2 (a) Longitudinal force; (b) Shearing force. Z axis X F igu re 3.3 Crystalline quartz axis orientation are lapped in double-sided planetary lapping machine to a desired thickness. The blanks are classified into two groups; The X-Group, where the thickness dimension of the blank is parallel to the X axis, and the Y-Group, where the the thickness Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 dimension of the blank is parallel to the Y axis. The resonant frequency of quartz is determined by the size of the plate com bined with the mode in which it vibrates. The three basic modes of vibration are the flexture mode, the extensional (or longitudinal) mode, and the shear mode. Each type of vibration can occur in a fundamental mode or in an overtone (harmonic) mode. The most common cuts in the X-Group are the: X-cut, 5° X, —18° X, MT, NT and V-cuts. Similarly in the Y-Group the most common are: Y-cut, AT, BT, CT, DT, ET, FT, GT, and SC-cuts. Figure 3.4 illustrates a Y-cut blank and an AT-cut blank. Notice the AT cut is a Y-cut rotated by 35.25°. The AT-cut is one of the more popular crystal cuts used. A selection of the appropriate cut will depend on the frequency of operation, tem perature characteristics, aging characteristics, and size. An excellent summary of the characteristics of each cut is listed in “Precision Frequency Control” by Gerber and Ballato (1985). ' Crystal Blank Crystal Bar (a) (b) F ig u re 3.4 (a) Y-cut blank; (b) AT-cut blank 6 = 35.25°. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 For high frequency applications the AT and SC-cuts are generally used. Using the standard lapping and polishing methods, an upper frequency limit for the fun damental mode of an AT-cut resonator is approximately 50M H z (Gerber, et. al 1986). This corresponds to a plate thickness of about 30 \xm. At this thickness the plates are to fragile to handle. Crystals operating beyond this frequency require a new plate configuration introduced by Guttwein, et. al (1972) as illustrated in Fig. 3.5. In brief, a thin membrane (1.6 //m for a 1 G H z resonator) is produced in the center of a wafer with an outer ring thickness of > 50 fim which provides the mechanical strength. Two techniques are extensively being investigated th at produce this new configuration. The first one is a chemical etching and polishing method which is outlined in a paper by Hunt and Smythe (1985). A second method uses ion beam milling and (reactive) ion etching and is outlined in Berte and Harteman (1978), and Wang, et. al (1984). After the final frequency adjustment the crystal is sealed in a hermetic can, see Fig. 3.6. Sealing is done in a controlled atmosphere to eliminate oxygen and moisture which are the main culprits of aging of the crystal. 3.4 A T -C U T C R Y STAL C H A R A C T E R IS T IC S The AT-cut is generally used for applications where higher frequencies are required. As mentioned earlier the AT-cut is a Y-cut rotated by 35.25°. Small variations from this angle are still considered and analyzed as AT crystals. The AT crystal operates in a thickness shear mode. The thickness shear mode can also be exited at the odd multiples of the fundamental frequency. The resonant frequency is given by 1 h i ~ 2v 'p V [cM2 t2 where p, c, M, and t represent the density, effective elastic constant, resonance order and the blank thickness, respectively. Using the values given in Table 3.1, we arrive Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 50 mils 400 mils 80 mils F ig u re 3.5 U ltra-thin UHF resonator, [Gutwwein, et. al. (1972)]. F ig u re 3.6 High frequency crystal mount, [Piezo Technology Inc. catalog]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 to the familiar expression for AT-cut quartz, namely; . (1.65465)M /« = — where /m is in MHz and t in mm. j— Present crystal oscillators operating above 50 M H z use the odd overtones of the crystal, in general the highest used is the 1th overtone. The acoustically active region of the AT resonator is defined by the electrode pattern. The acoustic energy level falls of exponentially with increasing distance from the electrode area. If the blank diameter-to-thickness ratio is large (i.e. greater than 50) , the resonator can be edge mounted with negligible damping effects from the mounting system. As the diameter-to-thickness ratio is reduced, the crystal Q is also reduced. The mounting of an AT crystal is shown in Fig. 3.6. Table 3.1 lists the values of the key physical parameters for the AT-cut res onator. The values listed are for the nominal AT angle (i.e. 35.25°), for cuts th at deviate from the nominal angle, small variations in the parameters are also ob served. However, the errors resulting from the thickness mode approximation are generally of greater importance. Density Effective Elastic Effective Piezoelectric Effective Dielectric Frequency constant p = 2,649 c = 29 x 109 e = 0.095 e = 40.3 x 10"12 1,660 kg/m 3 N /m 2 C/m 2 F/m kH z/m m T a b le 3.1 Physical constants for AT-cut Quartz resonators [Gerber and Ballato (1985)]. The best known property of the AT crystal is it’s frequency-temperature char acteristics. The tem perature response of the resonant frequency can be modelled by a third-order polynomial known as Bechman’s equation, that is = a(T - T0) + b(T - t 0)2 + c(T - T0)3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 where / is the resonant frequency at tem perature T, and f 0 is the resonant fre quency at the reference tem perature T0 which for the numbers below is 20° C. The coefficients a, b, and c are constant, and exhibit a linear dependance with the angle increment from the nominal, namely a = (-5 .1 5 x 1O_6)(A0) b = (0.39 x 10“ 9) - (4.7 x 1O“ 9)(A0) c = (109.5 x 10“ 12) - (2 x 1O_12)(A0) For the AT crystal the coefficient of the cubic term in the polynomial is the dominant one, thus giving rise to the typical S shaped characteristics. These curves are also known as Bechman’s curves. Several examples are shown in Fig. 3.7. As the angle of cut of the blank is shifted, the frequency-temperature characteristic is rotated about the inflection tem perature. By choosing the proper cut, the frequency drift can be minimized over a specific tem perature range. 3.5 R E S O N A T O R M O D E L An exact lumped param eter equivalent circuit for a quartz resonator can not be obtained. However, in the neighborhood of a particular resonance (i.e. fundamental or harmonics) a simple series RLC lumped resonant structure accurately models the crystal behavior. Figure 3.8 illustrates a crystal resonator symbol and equivalent circuit model. In this model a shunt capacitance C0 is included which accounts for the capacitance formed by the electrodes of the crystal and the crystal dielectric. The circuit elements Ci, L x, R x are the electrical equivalents of the inertia, stiffness, and internal losses of the mechanical vibrating system. C\ , L x, and R x are known as the motional parameters of the crystal. Cady (1946) presents the values of the equivalent lumped elements as a function Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 +20 _i +10 a. 10 a. u. co > -10 -20 u. -34 -14 +6 +26 +66 +46 +86 TEMPERATURE <°C) F ig u re 3.7 Frequency-temperature characteristics of AT-cut crystals [EG&G CINOX catalog, Cincinnati, Ohio]. Ll — — 1 Cl Rl -----1(------ V\A— Co If- F ig u re 3.8 Crystal symbol and model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 of the quartz param eters given in Table 3.1, namely: r - eA Co~ 2 T C l= U = c t M 21T2 ft e2A pi3cum Ri = 4e2A where A, t, M, and a represent the electrode area, quartz thickness, resonance order, and damping constant (generally equal to 1), respectively. The above values give a good approximation to the crystal model. In practice the crystal param eter model is extracted from experimental measurements. 3.6 Q U A R T Z R E S O N A T O R M E A S U R E M E N T SY S T E M In the UHF and microwave range of frequencies, new techniques are neces sary for the measurement of the crystal equivalent circuit. A system which uses a reflection coefficient bridge introduced by Smithe (1981) is presented. The system is based on the HP 4191A RF Impedance Analyzer that mea sures the reflection coefficient, with respect to 50 Q, of an unknown impedance in the range from 1 MHz to 1 GHz. The instrument is calibrated against reference impedances, open circuit, short circuit, and a 50 Q termination. These impedances and an appropriate test fixture are needed for the resonator measurement. An internal microprocessor converts the reflection coefficient into a series or parallel impedance (i.e., R + jX or G+jS). To obtain the equivalent circuit at the fundamental resonance it is necessary to find the series resonance of the unknown, this can be done by simply performing a frequency scan and monitoring the reactive part of the impedance; At the series resonance the HP 4191 reading in the impedance mode will yield (wo) — R-i + j(0 ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 From this point we can extract the information for the resonant frequency f 0 and the value of R i The reactance of the motional branch of the resonator equivalent circuit given in Fig. 3.8 can be w ritten as X,(w) = v,Ll - - ^ r = L1 ( t°2 wC\ \ w J where "•= 7 m For the narrowband approximation w ( 3 -1 ) XV q X i(ty) « 2Li(w —w0) (3.2) Evaluating (3.2) at w = w0 + A w and at w = w0 —A w , and then adding the results gives the value of L\ in terms of the reactance of the motional arm, namely r X 1( f 0 + A f ) - X 1( f 0 - A f ) u = --------------- S a T --------------- (3-3) The value of L\ is independent of A f within the accuracy of the narrowband ap proximation. The capacitance C\ is found from c a= 1 w2 0L x Smithe (1981) shows th at C 0 is determined as the average of shunt capacitance measurements at / = l .l / „ and / = 0.8888/o. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H A PTER IV N O ISE IN O SCILLATO RS 4.1 IN T R O D U C T IO N The number of communication systems using electromagnetic waves as a means of propagation is constantly increasing. At the same time the usable frequency spectrum remains constant or it is increasing at a much lower pace. This forces the communications systems to have channel spacings th at are closer to each other and thus places increasing spectral requirements on the reference oscillator in both the transm itter and in the receiver. When talking about noise in an oscillator it is standard to refer to it as the frequency stability of the oscillator. In this chapter we introduce the terminology relevant to oscillator noise. We develop a simple oscillator noise model to get a basic understanding of how some of the oscillator param eters affect the noise performance of the oscillator. Also, we present the different noise measurement systems, one of which (i.e., phase detector m ethod) is used in Chapter 5 to evaluate the performance of the oscillators designed. 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 The spectral characteristics of the oscillator are determined by the noise pro cesses present within it. The sources of noise in an oscillator include therm al noise, shot noise, and flicker noise. Depending on the offset frequency from the funda mental, the oscillator will exhibit a power spectral density which is proportional to / “ “ , a = 0 represents white noise, a = 1 represents flicker noise, and a = 2 represents random walk noise. Flicker noise, also commonly referred to as 1/ f noise, is always present in oscillators. It is standard to neglect its effect only when the offset frequency is larger than the 1/f cutoff frequency, which can be approximated from where Q i represents the loaded Q of the oscillator. The frequency f c is also known as the noise floor corner frequency. In a low frequency crystal oscillator (e.g., f Q = 10 M H z ) the noise floor corner frequency is around 500 H z or less, while for a high frequency oscillator (e.g., f 0 = 800 M H z ) the corner frequency can be greater than 5 M H z . A typical single sideband noise (SSB) curve of a microwave oscillator is shown in Fig. 4.1. The phase noise of signal sources is a concern in frequency conversion applica tions where signal levels span a wide dynamic range. The sideband phase noise can convert into information in the passband and limit the overall system sensitivity. Tolerable levels of phase noise vary greatly in different microwave systems. Figure 4.2 illustrates a typical situation. Suppose two desired signals f i and /2 are applied to the input to of a mixer, where they are to be mixed with a local oscillator signal f i o down to an intermediate frequency IF. The phase noise of the local oscillator will be directly translated onto the mixer output products. Note that although the system ’s I F filtering may be sufficient to resolve the larger signal’s mixing product ( /i — f l o ), the smaller signal’s mixing product (/2 —/ l o ) is no longer recoverable due to the translated local oscillator noise. The noise on the local oscillator thus degrades the system’s sensitivity as well as its selectivity. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 Measured SSI Phase Noise at 1 GHz -2 0 -4 0 8-100 a .-120 -1 4 0 -160 10k 100k Offset Frequency, Hz 1k 1M 10M F ig u re 4.1 Typical noise sideband curve of a microwave oscillator [Hewlett Packard Instrument Catalog]. 4.2 N O IS E C H A R A C T E R IZ A T IO N IN O S C IL L A T O R S Frequency stability can be defined as the degree to which an oscillating source produces the same frequency throughout a specified period of time. This stability can be broken down into two components: long-term and short-term frequency stability. Long-term frequency stability describes the frequency variations th at occur over long time periods, from several seconds and up to several days or even months. It is mainly associated with the aging processes of the materials used in the resonant structure. Long-term stability is predictable and well defined, it is expressed as a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 A f 'lo F ig u re 4.2 Effect of local oscillator noise in a frequency conversion system. linear frequency drift (i.e., change of frequency per unit time) in typical units of parts per million per hour, day, month, or year. On the other hand, short-term frequency stability represents frequency changes about the nominal frequency in periods less than a few seconds. Short-term stabil ity is further divided into two categories: The first kind encompass deterministic variations; th at is, changes th at are caused by known phenomena such as power line modulation and microphonic vibrations (mechanical vibrations inducing frequency changes). The second type of short term stability is associated with random vari ations (e.g., internal noise) in the oscillator which modulates the carrier. We are primarily interested in characterizing this type of short term frequency stability. To better understand the concept of noise in oscillators consider an ideal oscil lator which will exhibit the following output: Vo(f) = Acos(w0t ) (4.1) where w0 = 2irf0 is the frequency of oscillation. On the other hand the output of an actual oscillator is of the form V0(t) = A(t )cos[w0t + 0(f)] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.2) 58 where A(t) represent the am plitude variations as a function of time (i.e., AM noise), and 6{t) represent the random phase variations. 6{t) is referred to as the PHASENOISE of the oscillator. Figure 4.3 illustrates the spectral distribution of an ideal and actual oscillator. In well designed oscillators the amplitude variations are neg ligible and the noise sidebands arise from the random-phase noise contributions only. SSB I*—fm— * (a) (b) F ig u re 4.3 (a) Spectral density of an ideal oscillator; (b) Spectral density of an actual oscillator. In Fig 4.3, for the actual oscillator we see th at in addition to the desired component termed the carrier, there is a concentration of power surrounding the carrier. These spectral distributions on opposite sides of the carrier are known as the noise sidebands. It is useful to separate the noise sidebands into AM and FM components, and to discuss noise as a modulation phenomenon, where the noise sidebands can be thought of as arising from low-frequency signals m odulating the carrier. The AM components are generally neglected since they are very small in a well designed oscillator. From communication theory a carrier signal frequency m odulated by a sine Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 wave of frequency f m is represented by v(t) = Re \AC eKw0t+epsinwmt) (4.3) = A c cos [wat + dpsin wmt] where er = P = ¥ Jm is known as the modulation index and represents the peak phase deviation. For dp < 1, using Taylors expansion (4.3) can be w ritten as v(t) = Re \ A ceiWot 1 + £ 1 n = l n- since 6p < 1 the series can be truncated after the second term and v(t) = Re [Ace*Wot (1 + jd psin w mt)] or v( t) = A c [cos w Qt —dpsin w0t sinwmt ] Finally, after expanding the product of two sines we obtain v(t) = A c jco su ;0i - ^ [<cos(w0 + w m)t - cos(w0 - wm) i ] | (4.4) Equation (4.4) indicates th at for small dp, the phase deviation result in frequency components having an offset of f m Hertz from the carrier on each side of the carrier. The complete oscillator spectrum is then represented by the summation of a large number of strips of w idth B = 1 H z (i.e., components) f m Hertz away from the carrier. The energy in each strip B is modelled by a sinusoidal frequency modulating signal as determined by (4.4) and shown in Fig. 4.4. The most common characterization of the phase-noise of a source is the fre quency power density, denoted by £ ( / m). The National Bureau of Standards (NBS) defines £ ( f m) (also known as the sideband noise) as the ratio of the signal sideband Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 power of phase noise in a 1.0 H z bandwidth, f m Hertz away from the carrier fre quency to the total signal power; th at is, . Power density in one phase modulation sideband £ { fm ) = --------------------- , , . .-----:-----------------------------total signal power (4.5) The Fourier Transform of (4.4) results in the spectrum of v(t), namely v(f) = + 4sW 6 A f - f W - fo) ± + Kf (fo + f o) } + f m)]) ~ 6\f ± (4-6) (fo ~ fm)]} The magnitude spectrum |F ( / ) | is plotted in Fig. 4.4 from which the single sideband is found using (4.5) and (4.6) with the assumption that the total signal power is the carrier power. This assumption is valid, since the sidebands are very small as compared to the carrier, thus f0 - fm fo ^ fm F ig u re 4.4 M agnitude spectrum. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 A A '2 oi = e2 (4.7) The spectral density of phase fluctuations is defined as the total noise power in both sidebands to carrier power and denoted by Sg. From (4.6) we obtain 2 2 { 6f L) Sg(wm) = —^ = 2£ ( w m) (4.8) For reasons of convenience sideband noise is presented logarithmically; (i.e., in dBs relative to the carrier per H z (d B c / H z )). Another common way of specifying frequency stability is by residual FM, com monly known as hum and noise in the communication industry. Residual FM is defined as the total rms frequency deviation within a specified bandwidth. Regu larly used bandwidths are: 50 Hz to 3 KHz, 300 H z to 3 k H z and 20 H z to 15 kHz. It is related to single sideband noise by the following relation A /res = V 2 . J J f h Many other definitions exist in the specification of frequency stability. Barnes, et.al. (1971) presents an extensive an complete synopsis of the different m eth ods used and approved by the National Bureau of Standards. The two methods presented above are the most universally used, and represent frequency domain methods only. Frequency stability in the time domain consists of characterizing the frequency changes as obtained from a frequency counter. The instantaneous (angular) fre quency is defined as where is the total phase of the signal. From (4.2) we obtain <f>(t) = w0t + 6(t) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 therefore w (t) = ^ [w»t + #(<)] = w 0 + or + <4-9> In equation (4.9) we axe interested in characterizing the tim e varying component. Howe (1976) defines the instantaneous fractional frequency deviation from the nom inal frequency f 0 as (-0 ) Characterizing fractional frequency fluctuations allows better comparisons between sources at different carrier frequencies. The characterization is based on the sample variance of fractional frequency fluctuations. The National Bureau of Standards defines the Allan Variance as The brackets < > indicate an average over an infinite time, r represents the fre quency counter measurement duration, and y k is the average fractional frequency difference of the kth sample. Recall, conventional frequency counters measure the number of cycles in a period r , th at is, they measure / Gr ( 1 + y k). In the above def inition the sample period is assumed to be r , no dead time between measurements. From the above definition it can be seen that the Allan Variance can not be precisely determined since an average over an infinite time period is not realizable. Howe (1976) presents that the confidence of the estim ate of the Allan variance improves nominally as the square root of the number of data values used. For example, if 8 samples are used, the confidence can be expressed as being no better than l / \ / 8 x 100% = 35 %, this represents the allowable error in the estimate. An estim ate is then obtained from 1 M—1 ^ (r) * 2 ( M - 1 ) S ^ ' +1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4‘n ) 63 where M represents the number of data values. 4 .3 O S C IL L A T O R P H A S E N O IS E An exact relationship of the oscillator parameters and its effects within the noise processes of the oscillator is still under investigation. Several authors have presented many theories, and all basically converge to the simple relationships found by Lesson (1966). Lesson in his work, presented the theory of oscillator noise working with a feedback model for the oscillator. Kurokawa (1968) arrives to very similar results considering the oscillator as a negative resistance device. A v = 1.0 H r (Jw ) F ig u re 4.5 Feedback model of an oscillator. Consider the feedback model shown in Fig. 4.5. The input signal n* will represent noise which is internally generated in the oscillator. The transfer function H n ( j w ) will represent the transfer characteristics of the resonator (series RLC), see Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 Appendix 2 for the derivation. The overall transfer function H r { j w ) is then HMw) “ l-H ndv,) <4'12) where 1 H r U io) W and wn W L : 2Q l The feedback system shown in Fig. 4.5 can then be modelled by a linear system with transfer characteristics given by H t {j w ) as shown in Fig. 4.6. The transmission of white noise through a linear time-invariant system can be found from Sn Sn„ ( H t (j w ) F ig u re 4.6 Linear model of oscillator. Sn0M = Sni(™ )|iM u>)|2 (4.13) where S ni represents the spectral density of phase fluctuations at the input and S no will represent the spectral density of phase fluctuations at the output, th at is Se,0(w ) = S n0(w ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 The expression for S ni is derived in Appendix 2 (see A2.9),where the noise is considered to be composed of thermal noise and flicker noise,namely _ . . FKT / w c\ ^ ,- ( ^ = - ^ ( 1 + - ) . (4.14) Therefore, the spectral density of phase fluctuations of the oscillator is obtained from (4.12), (4.13), and (4.14); th at is, S»(wm) = ^ (l + ^ ) (l + ^ ) (4.15) Using (4.7), the single sideband phase noise of an oscillator is given by { ^ t ) { +m h ) = 1 6 dB/oct (416) 0 dB/oct F ig u re 4.7 Phase noise model. A plot of (4.16) is shown in Fig 4.7. For an offset frequency larger than the half bandw idth of the resonator (i.e. f i ) the oscillator noise has a white spectrum and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 Ql 1 10 20 40 80 100 200 500 1,000 10,000 £(25 k H z ) [dBc/Hz] -8 5 -9 5 -9 8 -1 0 1 -1 0 4 -1 0 5 -1 0 8 -1 1 2 -1 1 5 -1 2 5 T a b le 1. SSB-phase noise is limited by the noise figure of the active device and the power of the oscillator. These two param eters basically set the noise floor of the oscillator. For example, for an oscillator whose power is -10 dBm, and with an active device that at the operating point has a noise figure of 4 dB, the noise floor of the oscillator (NF) will be N F = lOlog(fcT) + 10 log(.F) - 101og(2) - 101og(Pa) = -1 7 4 + 4 - 3 + 10 = -1 6 3 dB In the communication industry the channel spacing is generally 25 k H z which cor responds to an offset frequency below the half-bandwidth frequency, then the SSB phase noise can be approximated from C(fm) = -1 7 4 + 10 lo g P + 20 log r j - - 10 logP s - 10 logQL £jm From the above expression it is evident th at better noise performance will be achieved with high-Q circuits, but it is im portant to notice that use of high noise figure devices can offset gains obtained by improved Q resonators. Table 4.1 illustrates the dependance of £ ( / m) on the loaded Q for a typical oscillator at f 0 = 800 M H z , F = 5 dB, Pa = 0 dB, and f m = 25 k H z Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 4.4 O S C IL L A T O R P H A S E N O IS E M E A S U R E M E N T As mentioned earlier, when referring to the phase noise of the oscillator it is generally the single-sideband phase noise th at one addresses, thus the techniques used to characterize the oscillators noise are in general geared towards the mea surement of £ ( / m). Several methods for the measurement of SSB-phase noise can be used. Depending on the noise characteristics of the oscillators to be tested a particular method can be selected. In the following discussion we will address the direct measurement method, the phase detector method, and for the time domain characterization of cry(r) the heterodyne frequency measurement. 4 .4 .1 D ire c t S S B -P h a se -N o ise m e a s u re m e n t The direct measurement m ethod is the simplest technique. Figure 4.8 illus trates the setup used for this method. The oscillator under test is fed directly into a spectrum analyzer which measures the spectral density of the oscillator. DUT Spectrum Analyzer F ig u re 4.8 Direct SSB phase-noise measurement set-up. This m ethod is the most straight forward and logical, yet due to the spectrum analyzer’s dynamic range, resolution, and LO phase noise it is limited to oscillators with relatively high noise sidebands. T hat is, oscillators th at have higher noise sidebands as compared to the spectrum analyzer SSB-phase noise at the particular offset frequency. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 4 .4 .2 P h a s e D e te c to r M e th o d The most universally used technique in the measurement of SSB phase noise is the phase detector-method. This particular method eliminated the problems associated with the direct measurement technique. Figure 4.9 shows the basic block diagram of the phase detector measurement set up. In this technique, a reference source with b etter noise characteristics than the oscillator under test is needed. Source Under Test Phase Detector I(Mixer) Low Pass Filter Low Noise Amplifier 90° Reference Source Phase Lock Loop Baseband Analyzer F ig u re 4.9 Phase Detector SSB phase-noise measurement set up. The basis of this method is the double-balanced mixer which is used as a phase detector. In this mode of operation, and when the input signals are set at the same frequency and in phase quadrature, the mixer output voltage is proportional to the fluctuating phase difference between the two input signals \(f>Lo(t) ~ <l>d e v ( t )]• I t is therefore evident th at the reference signal SSB-phase noise must be better than the device under measurement. In general as a rule of thumb the phase noise of the reference source should be at least 10 d B c / H z better than the device under test. The phase quadrature is im portant since it places the operation of the mixer in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 ov 20- 40* 00* B o y 100* 120* 140*- 100" ISO* F ig u re 4.10 Phase characteristics of a double-balanced mixer, linear region as can be seen in Fig. 4.10. To better understand this method let the inputs into the mixer be Ure /O 0 = Vdev{t) Vr C O S ( w rt) = Vo cos [ w 0t + <j)(t)] when the two signals are set at the same frequency and in phase quadrature, the output of the mixer will have a signal at baseband and a signal at twice the fre quency which after the low pass filter will be filtered leaving only the component at baseband. T hat is, the output after the low pass filter is v(t) = K o sin[</>(t)] where K q is defined as the phase-detector constant and represents the (4.17) peak of voltage of a beat signal (i.e. the to signals not in quadrature) v(t) - Vb-peak C O S [(ltJr - Wa)t + This beat signal is used to set up a reference level at the desired offset frequency (such as: 12.5 k H Z , 25 k H z , 100 kH z , etc) for the measurement of SSB phasenoise. In (4.17) the phase fluctuations <f>{t) as discussed earlier are less than one Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 A < < 1 ; hence, under such circumstances the small angle approximation can be used and (4.17) can be re-written as A V(t) = K DAcj>{t) = ± V b- p a k & m (4.18) Transforming (4.18) into the frequency domain gives A V (/) = K D The spectrum analyzer provides the rms value of the signal, thus A <j>(f) as measured by a spectrum analyzer is Kd v2V j_rma Then, the spectral density of phase fluctuations is SA*(/) = A 6—rms and from (4.7) for A <f>« 1 the SSB-phase noise can be found, and £ (/m ) = Z ^ (4.19) b—rms Equation (4.19) can be expressed in log form as £ (/m ) = 20 log AVrma - 20 log Vj-rms - 6 dBc in (4.20) Equation (4.20) shows th a t in order to obtain the SSB phase noise of a signal using the phase-detector method it is necessary to obtain a reference level from a beat note at the desired offset frequency and subtract it from the level obtained when the two signals are in quadrature. Figure 4.11 illustrates a typical spectrum analyzer output for the phase-detector method. In the figure, the beat note level has included already the 6 dB conversion obtained in (4.20). For an explanation of the reading refer to Chapter 5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 - 69 dB /H z -133 dB/lIz CENTER 100.0 kHz RES BW 300 Hz VBW 10 Hz SPAN 200.0 kHz SWP 30 sec F ig u re 4.11 SSB-phase noise measurement using phase detector method 4 .5 .3 H e te r o d y n e F re q u e n c y M e a su re m e n t The heterodyne frequency measurement is a m ethod for characterizing the frequency stability of an oscillator in the time domain, namely obtaining an estimate for the Allan Variance (i.e. cry( r )) for the oscillator under test. Figure 4.12 shows the heterodyne frequency measurement set up. Basically the signal under test is downconverted to an interm ediate frequency which is fed into a frequency counter. The time period between each measurement Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 DUT Period Counter Reference F ig u re 4.12 Heterodyne Frequency measurement set up. is kept constant and a large sample of frequency measurements is obtained from which using (4.11) an estim ate for the Allan Variance is obtained. This method is used mainly for close-in measurements of frequency standards. Readings for offsets greater than 10 k H z are not reliable since the measurement period required in the frequency counter would be less than r < 100 fxs. Formulas to convert from the Allan Variance to SSB-phase noise are presented by Barnes, et.al (1971). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER V N E G A T IV E R E SIS T A N C E D E S IG N FO R CRYSTAL O SCILLATO RS 5.1 IN T R O D U C T IO N The concept of negative resistance in the design of oscillators has been ex tensively investigated. Kurokawa (1960) has presented an extensive development of negative resistance oscillators which includes stability and noise considerations. Re cent work in this field (Basawapatna and Stancliff 1979, Besser 1973, El-Kamali,et. al. 1986, Esdale and Howes 1981, Gilmore 1983, Johnson 1979, Kotzebue and Parrish 1978, Mitsui, et. al. 1977, Niehenke 1979, Pucel, et. al. 1975, Ollivier 1972, Scherer 1981, Trew 1979, and Wagner 1979) present different approaches to the design of microwave oscillators with emphasis on a variety of param eters, such as, power, noise, tuning bandwidth, etc. Generally, these approaches rely on mea surements of device or oscillator param eters which are very limited to the specific oscillator, and in general, are not easily obtained. 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 In this chapter we present a closed-form design method for negative resistance crystal oscillators. A full characterization is presented on every block of a two port oscillator. The oscillator design method is presented in a general form such that it is applicable to any of the resonators introduced in Chapter 2. The design method is verified experimentally with two crystals, one with a fundamental frequency in the lower microwave region (i.e., 843 M H z ) , and the other with a fundamental frequency in the UHF region (i.e., 383 MHz). During the design process, the results are compared to results obtained by optimization techniques, (e.g., CAD method utilized in oscillator design). The crystals were selected since they represent a new type of resonators used in fundamental mode at the referred frequency bands. Operation of commercially available crystal oscillators are limited to funda mental frequencies up to 50.0 M H z (Gerber, et. al. 1986). This is basically due to standard lapping and polishing methods used in the manufacturing of BAW (Bulk Acoustic Wave) crystals, which limit the upper frequency of the crystal. For exam ple, for an AT cut quartz crystal the approximate fundamental resonance is given by fo = 1.661 /t , where t is the plate thickness in millimeters, and fo is the funda mental resonance frequency in M H z . For a 10.0 M H z crystal, a plate thickness of t = 166.1 iim is required, while for an 800.0 M H z crystal, a plate thickness of t = 2.076 fim is needed. W ith standard processing techniques, the latter thickness is practically impossible to obtain. Recent years have seen new technologies being applied in the manufacturing of BAW crystals. Using one of these new techniques (reactive-ion beam milling), BAW resonators operating at fundamental frequencies as high as 4 G H z have been reported (Bidart 1982). Chemical etching is another technique presently being in vestigated. Hunt and Smythe (1985) report resonator plate thicknesses of just under 1.0 fim using this technique. For this thickness, the fundamental resonance is ap proximately at 1.6 GH z. W ith these new developments, the crystal can become one of the leading devices used in the design of highly stable microwave oscillators. This is apparent due to the many advantages a crystal offers over presently used technolo Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 gies such as, m icrostrip/stripline resonators, YIG resonators, dielectric resonators, and cavity-tuned resonators, etc. Some of these advantages are: - a crystal resonator is very small, thus reduction in size and weight is possible, - a crystal resonator is less vibration sensitive, - excellent frequency vs tem perature characteristics can be obtained, - excellent aging characteristics due to environmental changes, - high Q of a crystal resonator. Aubry (1983) reports Q’s of 30,000 for a BT cut quartz crystal at 1.0 GH z, also Hunt and Smythe (1985) report loaded Q’s of 1,600 at 1.4 GHz. The design presented in this paper is based on the small signal S-parameters, and therefore the prediction of the oscillation frequency is our main concern. When small signal S-parameters are used it is not possible to properly predict output power or noise performance without some further oscillator measurements. A systematic negative-resistance method, based on the small signal S-parameters, for the design of microwave oscillators is now developed and used to implement the design of a crystal oscillator whose fundamental frequency is at 842.911 M H z . 5.2 D E S IG N P R O C E D U R E 5.2.1 O SC ILLA TIO N C O N D IT IO N S Figure 5.1 illustrates the standard configuration of a two-port negative resis tance oscillator (Gonzalez 1984). This network can be reduced to a one-port nega tive resistance network as shown in Fig. 5.2. The impedance Z j n is the impedance presented by the active device, it exhibits a resistive part R j n , which for oscillations to occur m ust be negative. Furthermore, for proper start-up of the oscillations, it Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 is required that \R i n \ > R l rT TERMINATING TRANSISTOR LOAD Rl NETWORK NETWORK NETWORK Z IN = R i n + j Zl X lN = R l + j X l F ig u re 5.1 Two port negative resistance oscillator. The oscillation frequency is determined by the reactive p art of Z j n , denoted by X l N , and the reactive part of Z i (i.e. of the resonator), denoted by X l - Kurokawa (1960) also shows th at stable oscillations will occur at a frequency wo, where X L(w0) = X I N (w0) and d R i N (V,w) dV v= va d X L(w ) dw 9X w=wa in (V,w ) dV 0 R l (w ) v= v0 dw are satisfied. The above expressions are derived in Appendix 3. In what follows a design procedure for Z j n and Z l >based on the small signal S-parameters, is developed. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 F ig u re 5.2 One-port negative resistance oscillator. 5.2.2 R E S O N A T O R C H A R A C T E R IZ A T IO N The load network, also known as the resonator, is represented by Zi,. The resonator constitutes the essential component of a microwave oscillator. In fact, oscillators are usually named after the type of resonator being utilized, (e.g., crystal resonator, YIG resonator, cavity-tuned resonator, etc.) Depending on the resonator type being used, it is common to define an equiv alent series or parallel RLC circuit, as shown in Fig. 5.3. The noise and frequency stability of the oscillator can be related to the loaded quality factor of the resonator. Montgomery, et. al. (1965) shows that for a series RLC network with impedance R + j X , the unloaded Q is defined by w 0 dX Q ' = 2R /r (51) where R represents the losses of the series network, and w a is the resonant angular frequency. From (5.1), we can see that the frequency change around resonance is Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission. 78 R x (a) (b) F ig u re 5.3 (a) Equivalent series resonant circuit; (b) Equivalent parallel resonant circuit. directly related to the reactance change, namely df=A / x from which we can observe th at a high-Q circuit will be less susceptible to frequency changes as compared to a low-Q circuit. This constitutes one of the basic rules in the resonator design, namely “maximization of the unloaded Q-factor” . Similar considerations apply to the parallel equivalent circuit in Fig. 5.3. In Figure 5.3, the resistor R x for a negative resistance device represents the input resistance of the active circuit (i.e., R i x ) and shown in (A3.13) for oscillations to occur it is negative. The loaded Q factor using (5.1) is then defined as Wo_dX 2R td w where R t=R+ R x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 it is the loaded Q, that is used in (4.16) for the prediction of SSB phase-noise. 5.2.3 T R A N S IS T O R F E E D B A C K SY N T H E S IS In this section we develop the design procedure for the feedback network which forms part of the synthesis for Z j n . At microwave frequencies, the parasitic ca pacitances of the device provide some or all of the feedback needed for oscillation. However, a properly designed series or parallel feedback network can substantially increase the negative resistance associated with the input or output port. Two typical feedback networks are shown in Fig. 5.4, and Fig. 5.5. The design of a series or parallel feedback network in the configuration shown in Fig. 5.4 or Fig. 5.5 is usually done using CAD methods. In this paper, an analytic procedure is presented that provides the value of the series or parallel feedback element th at will produce the largest negative resistance at the input or output port. 5.2.3.1 SE R IE S F E E D B A C K Consider the network shown in Fig. 5.4 where the impedance matrices of the transistor and series feedback network are [Z a] and [Z&], respectively. We can express [Za] and [Zb] in the form [Za] = Z11,a _z21,a z 12,a z22,a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 S 11, c S 2 2 ,c TRANSISTOR FEEDBACK NETWORK [Z b] F ig u re 5.4 Series feedback network. where Zll,a — "T— [ ( 1 + 5 ll,a )(l ~ S22,a) + 5 l2 ,a 5 2 1 ,a] 212,a = ~7~~ [2 5 i2 ,a ] z21,a = [ 2 5 2 1 , a] 2 2 2 ,a — 2 ^ - [ ( 1 ~ ‘S ' l l , a ) ( l + 5 2 2 , a ) + 5 i 2 , a 5 2 i , a ] and z ll,b [Zb] = .221,6 2 i2 ,6 222,6. where 211,6 = 212,6 = 221,6 = 222,6 = Z 0 ( r + j x ) ■S'li.aj <S'i2,a) 521,0> and S22ia are the small signal S-parameters of the transistor measured at / 0, Z 0 is the normalizing impedance (usually 50 Cl ) and A a = (1 — 5 n , a ) (1 — 5 2 2 ,0 ) ~ 5 l2 ,a 5 2 1 ,a The overall impedance m atrix of the composite network in Fig. 5.4 denoted by [Zc], Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 is given by r + j x + z'Ua r + j x + z'12ja [Zc] = z 0 r + j x + z'2ha r + j x + z'22ta where , _ ^ij,a z ij,a i , j = 1,2 Z0 ’ Using the conversion relations from z-param eter to S-parameter, it can be shown th at [see Appendix 4, equations (A4.1) to (A4.7)] (r + j x ) A s + B s (r + j x ) E s + D, (r + j x ) A s + F. 5 ,2 2 , c = (r + j x ) E s + D 3 2 S l2 ,c= [ r + j x + z[2,a] (5.2) 'S 'll.c — (5.3) (5.4) 2 S 2 i,c [r+ jx + z = (5.5) '2 l j a ] where A3 = T> _ z\ l <a J + J z22a —z[2a —z21>a J J I J _/ n 3 — z n,az22,a ~ z 12,az21,a T zll,a ~ z22,a ~ D a = Zn,az22,a ~ z 12,az21,a ~ Z\1,a + 1 1 ^ 22 ,a + 1 (5.6) Es ~ ZU,a ■^3 = + z22,a ~ z 12,a ~ z21,a + zn,az22,a ~ z 12,aZ21,a ~ z ll,a ^ 1 = ( 2 ll,a + l)(2 2 2 ,a + 1) ~ 2 + ^2 2 , 0 ~ 1 z 12,az21,a 5.2.3.2 PARALLEL F E E D B A C K Consider the network shown in Fig. 5.5 where the adm ittance matrices of the transistor and parallel feedback network are [Fa] and [Yj], respectively. The adm ittance of the combined network is [Fc] = [Fa] + [Y&]. Following a similar procedure as for the series feedback case [see Appendix 4, equations (A4.8) to (A4.15)], we arrive to the S-parameters of the composite network given below: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 FEEDBACK NETWORK [Yb] S 22, S 11 TRANSISTOR [Ya] F ig u re 5.5 Parallel feedback network. 5 u )C (r + j x ) A p fi + B p (r + j x ) E p + D p (5.7) £>22,c ( r + j x ) A p , 2 + Fp (r + j x ) E p + Dp (5.8) 2 l_ \ / (5.9) S l2,c S 2l,c L _ A 2 lJl2’a r + jx _2_ , A 2 V n 'a 1_ r+ jx (5.10) where -dp, 1 — -d p ,2 = 1~ Vu,a + 2 /2 2 ,0 — 2 /ll,a 2 /2 2 ,a + 2 /l2 ,a2 /2 1 ,a 1+ 2 / l l , a — 2/22,a — 2 /ll,a 2 /2 2 ,a + 2 /l2 ,a2 /2 1 ,a Bp = Fp = —y n ta ~ 2 /2 2 ,0 ~ 2 / l 2 , o — 2/21,0 Dp - 2+ y'l l a + 2/22,0 + 2/12,0 + 2/21,0 Ep = 1+ y'iita+ 2/22,0 + 2 /n ,a2 /2 2 ,a ~ 2/l2,a2/21,a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.11) S3 and ( 1 V n ,« — 5 n , g ) ( l + = ^ 22,0 ) + Si2,gS21,g A ft / 25i2 h 2 '* “ a T" / 2S21 a ^ = " “ aT ,a ( 1 + ‘S ' l l , o ) ( l — S 2 2 , a ) + ..1 - •5'l21a ‘S,2 i Ia ^ A ft = (1 + S i i )a) (1 + A2 = (y'n ,a + 5 2 2 , a ) — 5 i 2 , a 5 2 i ia l) ( y 2 2 ,a + 1) ~ V li.aV il.a 5 .2.S .3 F E E D B A C K F O R M A X IM U M R E F L E C T IO N C O E F F IC IE N T Equations (5.2) to (5.6) express the relationship between the series feedback network and the overall S-parameters of the composite network. Similarly (5.7) to (5.11) express the relationship between the parallel feedback network and the overall S-parameters of the composite network. It is observed th at (5.2) and (5.7),and (5.3) and (5.8), have identical form, differing only by the value of the constants. Therefore the analysis th a t follows is applicable to both series and parallel feedback networks. It should be noted th at the subscripts of the constants, which indicate series or parallel, have been om itted from the following analysis. In order to find the value of r + j x that maximizes |S n lC|, we analyze how constant values of r and x m ap in the 5 n iC plane. The analysis with ^22,0 is similar. From (5.2) and (5.7), (see Appendix 5), it is shown th at constant values of r, m ap onto circles in the S n tC plane. The center of the circles (Cr ) is given by (A5.6), namely ^ r 2rAE* + AD* + BE* 2 [r|E |2 + Re(E*D)\ 1 j and the radii of the circles (R r) are given by (A5.7), th at is R = ~ r 2[r\E \2 + Re(E*D)\ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5 13') 1 j 84 Similarly, constant values of x map onto circles in the Sn,c plane. The centers and radii of these circles, [see (A5.8) to (A 5.ll)], are given by Cx = 2xA E* + A*D — B * E 2 [ x \E \2 + Im(E*D)\ Rx = |AD - BE \ 2 [ x \E \2 + Im(E*D)\ and The r = 0 circle forms a natural boundary to the constant .T-circlcs. The center and radius of the r = 0 circle, called C r=o and R r=o 5 are obtained from (5.12) and (5.13) when r = 0. It also follows from (5.13) th at the circle with the largest radii occurs when r = 0. A typical r = 0 circle is shown in Fig. 5.8. r=0 C ircle F ig u re 5.6 r= 0 circle in S n plane. The maximum value of |5 n )C| occurs on the 7’ = 0 circle at the point that is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 farthest from the origin, namely = R r= 0 + \ C r —Q| \AD - B E \ + IAD* + BE * | 2Re{E*D) (5.14) We can now substitute (5.14) into (5.2) and (5.7) (with r = 0) to obtain the value of x th at produces |5 'n iC|mai. The substitution results in a quadratic equation in x, namely \Sn,c \ m a x \E \2 ~ W 2J *2 + [2Im (A B * ) - 2 I m (E D * )\S u ,c (5.15) + \Sn,c\'max\D\> - \Bf = 0 From (5.15), it appears that there are two solutions of x th at maximizes |5 n )C|. However, only one value from a constant x-circle can touch the maximum point in Fig. 5.6. In fact, after some manipulations [see Appendix 6, equations (A6.1) to (A6.8)], it follows th at this solution, called x optt\, is given by \ S } l , c \ m a z I m ( E D *) Similarly, \ S 22,c\max is - I™(AB*) Siven I 2 2 ,c |m a i 2R e(E*D) ' ' ' and the reactance that produces |*S22,c|m ai) called x opt , 2 is given by (5 .18) ■ f e L i s p - w The design procedure for the feedback network can be summarized as follows: 1 - From the S-parameters of the transistor, calculate the constants A 3, B s, D 3, E 3, Fs using (5.6) for series feedback, and calculate A p, B p, Dp, E p, Fp using (5.11) for parallel feedback. 2 - Using (5.14), (5.16), (5.17), and (5.18) obtain |S ii,c|mBX» x opt,l, | 5 ,22 )c | m a i , and x optt2 for both series and parallel case. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 3 - Depending on the configuration desired, select the value of x opt, i or x opt ,2 th at provides the largest negative resistance. 4 - Using the selected x optji i — 1,2 and setting r = 0 use (5.2) to (5.5) to find the series feedback S-parameters of the composite network, or (5.7) to (5.10) to find the parallel feedback S-parameters of the composite network. 5.2.4. TER M IN A TIN G N ETW O RK SYNTH ESIS Once the feedback is designed, the value of Z j n requires the selection of a term inating network. We now proceed to develop an analytic approach for the design of the term inating network. For the network shown in Fig. 5.1, T i n is related to T t by t-, c, , I I N = * 1 1 ,c + S i 2 ,cS2 1 ,cTt ------5 ----- 7T— ‘-’22,<4 T also ^ ^IN — ^ 0 1 + Tin -rn 1 - 1 jjv. - It is of interest to plot in the V j n plane the circle |F r | = 1.0. This gives information of all possible passive impedances (i.e. |T t| < 1.0) that will make |Fjjv| > 1.0 (i.e., Re[ZjN] < 0). As shown in Fig. 5.7, the ir^ l = 1.0 circle maps onto a circle in the T/iv plane (Gonzalez 1984) centered at Ci = ( 5 „ , c - ASJ2,c)‘ |Sll,c|2 - |A |2 (5.19) and having a radius of Ri = ‘S'l2Ic‘S21,c |Sn,cl2 - |A|= (5.20) where A = S n , c S 2 2 , c — S i 2 , c S2 1, A careful examination of Fig. 5.7 yields the following key-points on the term i nating network design. The maximum |F jjv| is obtained if we select the point on Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 0 LO F ig u re 5.7 |IY | = 1 in the |Fj-jv| plane. the |r T| = 1 circle (point A in Fig. 5.7) th at has the same phase angle as C{. This point gives the largest reflection coefficient for T//v realizable with a passive load. The value of T j n at this point is given by \T iN \ = \Ci\ + Ri ^T/iv = £Ci It is interesting to note th at this procedure yields identical results to those obtained by Wagner (1979). However experimental results show th at it might not be the optimum approach for the design of stable oscillators. By selecting T i n at point A, T l (i.e. the resonator impedance) will have to be matched to it. T hat is select R l ~ ^ R i n and X l = X j n (Gonzalez 1984). In this paper a design procedure is proposed where the design for the active network (i.e. of Tj/v) is made to match Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 the resonator and not vice-versa. The reasoning behind the proposed method is to m aintain the resonator intact in order not to degrade its Q. We then select T i n as follows, \Ti n \ > , rL R l < 50 ft for or , Tl is selected such that i.o < \t i n \ < and the angle of T i n R l > 50 ft for (5.21) I m [Z IN] = - I m \ Z L] is satisfied. The procedure for obtaining IV , once T i n is given by (5.21), is now developed. A constant value of |TjjvU |T i n \ = k where k > 1 represents a circle concentric to the |T/iv| = 1 circle, (see Fig. 5.8). Using (5.19) and (5.20) we also plot the |r T| = 1 circle in the T in plane (see Fig. 5.8). The \Ti n \ = k (k > 1) circle and the IT^I = 1.0 circle intersect at points (® i,yi) and (22,2/2)- This set of points will define the region where T i n can be selected; th at is, the range of phase angles that T i n can have for the particular magnitude. Since we know the center and radii for both circles, the location of these points satisfy the relation (see Appendix 7), 1+ ta n 2 ( L C i ). x — k 2 + |C,-|2 - R \ |C,-| tan (L C i)sin (ZC,-) x+ 2\2 {k2 + |C ,f - fl?) 4 |C , |2 s i n 2 ( Z C .) = 0 (5.22) Equation (5.22) will yield the values of x\ and x2 which then can be used to solve for yi and y2- The associated values of y are obtained from Vj ~ \ J k 2 — x j, j = 1,2 The values(® i,yi) and (5.23) (a;2,y 2) can now be used to find the range of phase angles th at T i n can have forthe given magnitude, namely &i = ta n - 1 — ; * = 1,2 X{ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.24) 89 I R i r Tl = i.o F ig u re 5.8 \Ti n \ = k, |r y | = 1 in the \Ti n \ plane. Therefore, if the phase angle of the resonator (Z T l) is not between 6 \ and #2, a new m agnitude for T i n (he. another k ) will have to be selected. It should be noted th at there might not be a value for the magnitude of T i n th at satisfies the previous requirement. In this case a change in the feedback network criteria or even a change of transistor, might be required. Another alternative is to add matching elements to the resonator, but this goes against the proposed design principles since the Q of the resonator will, in general, be degraded, and can affect adversely the oscillator performance. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 Once the design of T j n is done, we proceed to develop a graphical procedure to find Ty. We are interested in mapping onto the Ty plane, the semi-circle |T /^ | = k (k > 1) in the T/at plane, bounded by the angles 9\ and 02. This is easily accomplished by mapping the [T/jv | = k circle onto the ITy plane and selecting the values for I V such th at | I V | < 1.0. The ITj^v| = k maps onto a circle in the I V plane with center C0 = (k*S22,c F | 5 22jCp - |A |2 (5.25) and radius R0 = k S \2 .c ^ 2 \,c (5.26) ^2|*S'22>c|2 — |A |2 A typical example is shown in Fig. 5.9. Notice th at this mapping only considered the magnitude of T i n , thus we also need to address the phase angle of T i n - T hat is for T j n = k La where 9\ < a > #2, it can be shown that a maps onto an angle (f>in the IV plane given by <f>= ta n -1 G -H l + GH. G _ I m [ S n , e] - k sin (a) -Re[SntC] —k cos (a) H _ I m [ A] - |522|C|fc sin (ct + Z522|C) jRe[A] - |5 22|C|fc cos (a + Z522)C) (5.27) This corresponds to a line from the origin at an angle <f>which intersects the |T / i v | = k (k > 1) circle at two points, (see Fig. 5.9). Only the point th at satisfies the T t < 1.0 is a solution, (i.e., point ”A” in Fig. 5.9). At this point one can read the magnitude of Ty. Now th at is known, namely Ty = \T t\L (}> , one can use standard matching techniques to synthesize Ty- 5.3 D ESIG N EXAM PLE: 842.911 MHz CRYSTAL OSCILLATOR In this section we will use the proposed method to design a crystal oscillator having a fundamental resonant frequency in the microwave region. We selected such Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 Ro 0.1— 0. •3 .0 - F ig u re 5.9 |P/;v| = k in the |IY| plane. a crystal because it allows us to test the proposed method with a new device, where development work is in its infancy stage. A crystal developed by Piezo Technology Inc.f having a fundamental frequency at 842.911 M H z will be used in the design. Due to the high Q-characteristic of the crystal, special measuring equipment was used by Piezo Technology Inc. to measure the equivalent circuit param eters of the crystal. For this crystal, the equivalent circuit param eters, (see Fig. 5.10), are: R = 34.12 Q,, L = 40.0 pH , C = 0.895 f F , and Cp = 1.14 pF. The Q is 6,200 at f 0 = 842.911 M H z . f Piezo Technology Inc., Orlando, Florida Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. At the fundamental series resonance ( / 0 = 842.911 M H z ) the crystal impedance is Z i = 34.12 + j 0.0, which corresponds to T/, = 0.6524Z1800. Us ing (5.21) we need to select 1 .0 < \Ti n \ < 1 and I m [ Z iN ] = 0.0 0.6524 1.0 < \Ti n \ < 1.54 and /.VIN = Z0° from which we select T j n = 1.5/0°. L=40 |XH R= 34.12£2 — TOP 1( V\A— C = 0.895 fF — 1(-------C = 1.14 pF F ig u re 5.10 842.911MHz crystal resonator equivalent circuit. The Motorola M R F 901 bipolar transistor was selected for this design. The common-emiter S-parameters at / = 843 M H z , V c e = 2.6 V, and I c = 1.7 m A are: Sn,a = S 2i , a = 0.473Z - 106° 2.07Z - 81° S i2,a = 0.136Z41 O o S 22 , a = 0.66Z - 45 Using standard common-emiter to common-base transformations, the commonbase S-parameters at / = 843 M H z, Vc e = 2.6 V, and I c = 1.7 m A are found to be: S n ,a = S 2i , a = 0.386Z1420 1.380Z —45.4° 5 i 2,« = S 22iCl= 0.147Z81.3 O 1.1 1 0 Z -2 8 .5 o Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 We now follow the procedure outlined in Section 5.2, using the S-parameters to z-parameters conversion relation, we obtain: * ii,« = *2i[a = 0.875Z —7.19° 4.870Z - 136.42° z[2ta= z'22>a = 0.517Z - 9.72° 4.959Z - 112.74° the constants ^4a, B a, D a, E a, and Fa are found using (5.6), th at is A a = z'll a + z22a —z \ 2 a —z2l a = 2.326 Z —32.15° Es = *11,0*22,a “ *12,o*21,a + *11,a~ *22,o — 1 = 2.715Z50.91° 7^3 = * ll,a*22,a ~ *12,0*21,a —*11,o4" *22,0 + 1 = 7.041Z —90.99 Es = * ii,a + *22,0 - *12,0 - *21,o + 2 = 4.158Z - 17.32° E» — —* 1 2 , 0 * 2 1 , 0 —* n , a + *11 ,a * 2 2 ,a *2 2 ,0 —1 = 7.837Z —119.5° Using the above constants and (5.16), (5.14), (5.18), and (5.17) we can proceed to find x 0pt,i) l^n^lm oi) x opt, 2 und |‘S,22,c|mai5 th at is Ic I I HiclTOax ,x _ \ A D - B E \ + \AD' + BE* 2Re(E*D) |(2.3Z - 32)(7.0Z - 91) - (2.7Z 51)(4.2Z - 17)| + |(2.3Z - 32)(7.0Z 91) + (2.7 Z 51)(4.2Z 1 7)| 2jRe[(4.16Zl7)(7.04Z - 91)] = 3.32 similarly . . I 22,c|m a x \ A D - F E \ + \AD* + FE*\ 2 R e(E*D) |(2.3Z - 3 2 ) ( 7 L - 91) - ( 7 .8 L - 120)(4.2Z - 17)| + |(2.3Z - 3 2)(7Z 91) + (7.8Z - 1 2 0 ) (4 .2 Z l7 ) | 2/?e[(4.16Z 17)(7.04Z - 91)] = 2.128 From the above results, since |S,22,c|maI is smaller than |5'ii,c|max we select the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 feedback th at optimizes the input port, that is \S u A L j™ (E D ')-Im (A B ') I 'V e ! 2™J E I 2 - W (3.32)2Im[(4.16Z - 17)(7.04Z91)] - Jm[(2.33 Z - 32)(7.84Z120)] (3.32)2(4.16)2 - (2.33)2 = 1.71 which corresponds to an inductor of value (1.71X50) 2 jt(843 (10s )) the feedback network is shown in Fig. 5.11. M ~ R F 9 0 1 \ y L = 1 8 . 8 n H F ig u re 5.11 Series feedback network. The realization of the inductance will be attained using a m icrostrip short circuited stub. The m aterial selected for the assembly is G-10FR board with relative dielectric constant er = 4.8 and dielectric thickness of h = 0.150 cm. From Chapter 2, we find th at the effective dielectric constant for a Z 0 = 50 Cl is ee = 3.64, and the width to height ratio is w / h — 1.775. The wavelength at f 0 = 843 M H z in this m aterial is c 3(i08) , oz>e A = —— — = ------------- t = = 18.65 cm foy/Z 843(106)\/^ 6 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 To realize a inductive impedance of x optt\ = 1.71 the length of the short circuited stub required is I = 0.165A = (0.165)(18.65) = 3.07cm W ith r = 0 and x opt = 1.71, using (5.2) to (5.5), we find the S-parameters of the transistor with series feedback to be Su,c = S'2 i , c= 3.32/63° 3 .8 8 / - 147° Si2,c = S 22, c = 1.69/80° 1 .9 3 / - 109° It is interesting to note that these values where also obtained independently using the CAD program, COMPACTf. We next proceed to find out if there is a T t th at can give T j n = 1.5/0° which is realizable. The |!TV| = 1.0 m aps onto a circle in the T/jv plane with center given by (5.19), th at is (Sn>e - A S 2*2iC)* [(3.32/63) - (2.33Z37)(1.93Z109)]* i _ |S11)C|2 - |A|2 “ (3.32)2 _ (2 .3 3 )2 = 0.937/ - 4.66° and having a radius given by (5.20) of Sl 2 ,cS2 l,c l^ll.cl2 - |A|2 (1.69Z80)(3.88Z - 147) (3.32)2 - (2.33)2 = 1.19 the mapping is shown in Fig. 5.12. Using (5.22), (5.23), and (5.24) we find th at the angles th at bound the phase of T t for [Tjiv | = 1-5 are 9i = 60° and 92 = —47° which can also be seen in Fig. 5.12. Since the desired phase of T/w, / r , n = 0° is encompassed by 9\ and 92, we can now find the value of T t f Communications Consulting Corp./Com pact Software Paterson, New Jersey Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 | Tin | = 1.5 | T t | = 1.0 F ig u re 5.12 lIVl = 1.0, |r//v | = 1-5 in the T/yv plane. Using (5.25) and (5.26) we can now map the circle |T/Af| = 1*5 onto a circle centered at {k 2 S 22>c - AS*i,c)* C° = k 2 \S2 2 ,c\ 2 ~ |A |2 [(1.5)2(1.93Z - 109) - (2.33Z37)(3.32Z - 63)]* “ (1.5)2(1.93)2 - (2.33)2 = 2.844Z175.60 and radius k S i 2 lCS 2 i tC k 2 \S2 2 , c \2 - |A |2 (1.5)(1.69Z80)(3.88Z - 147) (1.5)2(1.93)2 - (2.33)2 = 3.33 in the T r plane, where k=1.5. This mapping is shown in Fig. 5.13. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 R 0=3.33 r i N| =1 . 5 I Co 1=2.84 | C o= 176°^ P f F ig u re 5.13 [Tj^vI = 1-5, £T i n = 0 in the Ty plane. Notice the arc IT/jv | = 1-5 bounded by 9\ = 60° and 62 = —47° in the T i n plane, maps onto the arc \Tjn\ = 1-5 that is encompassed by the ITx1| = 1-0 circle. Also using (5.27) an angle of 0° in the T i n plane maps onto an angle of 34° in the T r plane. Both mappings are shown in Fig. 5.13. The intersection of the line with angle 34° with the circle |r//y | = 1.5 gives the value for T t th at produces T i n = 1.5Z00, namely T t = 0.59Z350. As a final step in the design, we have to implement Ty and a dc biasing scheme. Utilizing standard matching techniques, the 50 load must be transformed to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 T t = 0.59L35°, this is achieved with a series capacitor and a short circuited stub. The ac schematic is shown in Fig. 5.14. In this design we opted for a passive biasing scheme. The complete crystal oscillator schematic is shown in Fig. 5.15. 3.5 pF MRF 901 ~© © - H F 100 pF 34.12 £2 4 = 100 pF Lh O : .895 fF : T o 0.165X ID 40 pH ~ r 0.1191 50 Q. J i F ig u re 5.14 843 MHz-oscillator ac-model. MRF901 430 nH 100 pF so a i---- 1 T 4.35 k Q o 0.1651 445 k Q £> F ig u re 5.15 843 MHz crystal oscillator schematic. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 5.3.1 O S C IL L A T O R P E R F O R M A N C E Using the HP8568B Spectrum Analyzer the oscillator frequency and power were measured and found to be: f 0 = 842.669 MHz and P 0 = -5.38 dBm. The SSB phase noise of the oscillator was measured using the “phase detector m ethod” w ith the HP11729C/8642A . The phase noise measurements are shown in Table 5.1. Measurements were taken at 12.5 kHz, 25.0 kHz, 100.0 kHz, and 500.0 kHz offsets from the carrier. - 69 dB/Hz CENTER 1C0.0 kHz RES BW 300 Hz VBW 10 Hz SPAN 200.0 kHz SWP 30 sec F ig u re 5.16 Oscillator phase noise. Figure 5.16 illustrates the measurement at an offset of 100.0 kHz. The beat note at the center (100.0 kHz from the carrier) sets the reference level at —69 dBc/Hz. The crystal oscillator sideband crosses the axis at —133 dBc/Hz which is the third entry in Table 5.1. It should be noted th at the noise floor of the measuring set-up is at —150 dBc/Hz. The crystal oscillator draws I = 0.910 mA from a 7.82 V Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 regulated battery source. fm [kHz] 12.5 25.0 100.0 500.0 C {fm) [dBc/Hz] -111 -119 -133 -146 T a b le 5.1 SSB phase noise 5.4 D E S IG N E X A M P L E : 383.868 M H z C R Y S T A L O S C IL L A T O R In this section we will use the proposed method to design a crystal oscillator having a fundamental resonant frequency in the UHF region. A crystal developed by Piezo Technology Inc. having a fundamental frequency at 383.868 M H z will be used in the design. For this crystal, the equivalent circuit parameters are, see Fig. 5.17, R = 67.92 Q, L = 220.0 pH , C = 0.7703 f F , and C = 0.79 pF. The Q is 8,000 at Jo = 383.868 M H z . At the fundamental series resonance (f 0 = 383.868 M H z ) the crystal impedance is Z i = 67.92 + J0.0, which corresponds to T l = 0.15197/0°. The m ajor difference in this design is the level of negative resistance th at is required. For this particular crystal the active network must be capable of providing a nega tive resistance of at least —67.92 Ft. From this information the required reflection coefficient required will be: i.o < |r / N | < 0.15197 and Im [Z iN ] = 0.0 1.0 < |r / N | < 6.58 and I F IN = Z0° Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 L= 220 flH R= 67.9212 — 10000'----- 1(— w — C = 0.770 fF ---- — If C = 0.79 pF Figure 5.17 Crystal equivalent circuit. from which we select T i n = 2.5/0° which corresponds to an impedance Z j n = -1 1 7 + j0 . The Motorola M RF 901 bipolar transistor was selected for this design. The common-base S-parameters at / = 380 M H z , Vc e — 3.0 V, and I c = 2.0 m A are: Sn,a = S2i,a = 0.579/169° 1.56/ - 19° S12,a = S22,a = 0.021/91° 1.02/ - 12.33° Using the procedure outlined in Section 5.2, for the series feedback case, we obtain x opt,l = 3.33 which maximizes |S n ,c| to a value of |S n iC|mai = 6.73, similarly we obtain x o p t t 2 = 3.21 which maximizes |5,22,c| to a value of \S2^tC\max — 3.44. For this design we also consider the possibility of using parallel feedback, thus following the procedure outlined in section 5.2. For the parallel feedback circuit, we obtain a;0pqi = 3.55 which maximizes |Sii,c| to a value of |5 n iC|mai = 0.584, similarly we obtain x opt i2 = —.863 which maximizes |S,22,c| to a value of \S22}C\max = 1.25. Analyzing the results, we can see th at in the parallel feedback case, with x opt,2 = —.863 the feedback network will only enhance the reflection coefficient Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 to a maximum of 1.25 for £ 22,0• This is compared to the series feedback case which for the same port, th at is 522,c a maximum of 3.44 is attainable. This information is not sufficient to make a decision, since the phase angle required is not considered. To have a full picture we m ust plot the \Tt\ = 1 circle in the Tout plane, as shown in Fig. 5.18. For the parallel feedback case, with r = 0 and x opt = —.863, using (5.19) and (5.20) where 5 n )C is replaced by 522,c since we are looking at the output port, the | r T| = 1 circle maps onto a circle in the Tout plane centered at C{ = 1.22/63° with a radius i2:- = 0.83, see Fig. 5.18. IRol = 0.83 C ol = 1.22: F ig u re 5.18 iFy] = 1.0 in the Tqut plane for the parallel feedback network. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 Careful exam ination of Fig. 5.18 shows th at a phase angle of 0 degrees, which was the requirement for T o u t (i-e>T ou t = 2.5/0°), is not possible for this circuit. Either the crystal must be combined with some external circuitry, which is not desirable, or a different feedback network must be considered. The later of the two is the preferred choice, since adding external circuitry to the tank circuit (i.e., crystal) will in general degrade it’s Q. It is therefore necessary to analyzed the series feedback network. For the series feedback case, with r = 0 and x opt = 3.21, using (5.19) and (5.20) where Sn.c is replaced by 522,c since we are looking at the output port, the | r r | = 1 circle maps onto a circle in the Ri T o u t plane centered at C; = 14.9/9° with a radius = 13.95 (see Fig. 5.19). Analyzing Fig. 5.19, it is observed th at T o u t — 2.5/0° is encompassed by the |IV | = 1 circle which was a necessary condition for T o ut- It is also interesting to comment, that in the series feedback case, the crystal can also be placed in the input port. W ith r = 0 and x opt = 3.21, using (5.2) to (5.5), we find the S-parameters of the transistor with series feedback, namely 5 ii,c = 5 21’c = S i 2 ,c = 522,c = 6.58/95° 6 .8 3 /- 9 9 ° 3.14/101° 3 . 4 4 / - 83° As was the case in the 843 MHz oscillator, these values where also obtained inde pendently using COMPACT a microwave analysis/optimization CAD package. We next proceed to find out the T out phase of T T t that is necessary to yield T o u t — 2.5/0° in plane. Using (5.22), (5.23), and (5.24) we find that for |r o c /r| = 2.5, the o u t is bounded by 9\ = 90° and 92 = —90°, see Fig. 5.19. Since the desired phase of T o u t is limited by 91 and 9 i, we can now find the value of rV . Using (5.25) and (5.26) we map the circle IToi/tI = 2.5 onto a circle centered at CQ = 0.15/ —77° with radius R a — 0.21 in the T t- Also an angle of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 Cil =14.9' F ig u re 5.19 | T = 1.0, |T ot/rI = 2.5 in the T out plane for the series feedback network. 0° in the T i n plane maps onto an angle of —41° in the IV plane. Both mappings are shown in Fig. 5.20. The intersection of the line with angle —41° with the circle |r o t / r | = 2.5 gives the value for T t that produces T o u t = 2.5Z00, namely Tt = 0.31Z - 4 1 ° . As a final step in the design, we have to implement x o p t i 2 and T t- We opted for a microstrip design using G-10FR board with relative dielectric constant er = 4.8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 \ \ >- | T o u t F ig u re 5.20 \T o u t\ = 2.5, 1= 2.5 IT o u t — 0 in the T t plane. and dielectric thickness of h = 0.150 cm. For this frequency, the wavelength is A= c fo ^ e 3 (1 * ) = 41.0 7 c m 383(106) v /O 4 to realize a inductive reactance of £ 0pt,i = 3.21, the length of the short circuited stub required is I = 0.202A = (0.202)(41.07) = 8.29 cm this length is prohibitive, thus it will be necessary to obtain the inductive reactance °f x opt,l = 3.21, which corresponds to an inductance of _ (3.212)(50) = 67.28 n H 2tt(383 106) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 This inductance L f = 67.28 n H is realized with a combination of both lumped and distributed elements. T t = 0.31Z —41° is obtained with a shunt inductance of 84 nH cascaded to a series capacitor of 10.5 pF. The complete crystal oscillator schematic is shown in Fig. 5.21. Notice also in this design, an active circuit biasing is used. This configuration enables a good tem perature compensated design for the bias circuit with no degradation of the oscillator ac-components. + Vce 10 Q MRF 901 A A / V - a / ----- 10.5 pF 50 Q + vcc 430 nH 430 nH s f; 330 pF 55 nH 413 Q 150 Q 67.3 n H > 150 kQ. to U ) 0.165 X _L. F ig u re 5.21 383 MHz Crystal oscillator schematic. 5.4.1 O S C IL L A T O R P E R F O R M A N C E Using the HP8568B Spectrum Analyzer the oscillator frequency and power were measured and found to be: f 0 = 383.8638 MHz and P 0 = —11.85 dBm. The SSB-phase noise of the oscillator was measured using the “phase detector m ethod” with the HP11729C/8642A . The phase noise measurements are shown in Table 5.2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 £ ( / m) [dBc/Hz] -1 3 2 -1 3 8 -1 4 4 -1 4 6 fm [kHz] 12.5 25.0 100.0 500.0 T a b le 5.2 SSB phase noise Measurements were taken at 12.5 kHz, 25.0 kHz, 100.0 kHz and 500.0 kHz offsets from the carrier. -1 0 0 N -105 SID E BRND N O I S E jP H flS ir N O ISE Q ~2S j K H z " i s" - T l S f d H c / H Z Fo r a C a r r i e r F r e q o f : 3 8 3 . 3 3 8 MHz : X \ o "D BQ m -120 o z Ld -125 -130 X -135 -140 - 150 . FREQU EN CY (K H z) F ig u re 5.22 Oscillator phase noise. Figure 5.22, illustrates the measurement at an offset of 25.0 kHz. The crystal oscillator sideband crosses the axis at —138 dBc/Hz which is the second entry in Table 5.2. It should be noted that the noise floor of the measuring set-up is at —146 dBc/Hz. The crystal oscillator draws I = 4.15 mA from a 7.82 V regulated battery source. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 Temperature data was also obtained for the oscillator. Figure 5.23 illustrates the tem perature behaviour of the center frequency cf the oscillator. It is interesting to observe th at it follows the typical Bechman shape (e.g., S-shape) of AT-cut crystals. 383.885 383.88 F r e 383.875 q 383.865 N 383.87 u e n c y \ 383.86 383.855 383.85 \ / V 383.845 383.84 -6 0 -4 0 -2 0 0 20 40 60 80 100 120 140 Temperature (Celcius) F ig u re 5.23 Oscillator center frequency tem perature behaviour. It is standard to show the tem perature variation of the center frequency in terms of parts per million (i.e., ppm), this is shown in Fig. 5.24. Figure 5.25 shows the output power behavior. It can be seen th at the power does not change considerably, which indicates th at the current is fairly constant over tem perature. Table 5.3 illustrates additional data th at was taken over tem perature. The noise performance of the oscillator (i.e., Hum and Noise) over tem perature is illustrated in Table 5.3. It is the belief of the author th at this oscillator has a noise performance th at is not matched by any device in todays market. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 60 40 20 P P U -4 0 -6 0 20 40 60 80 100 120 140 Temperature (Celcius) F ig u re 5.24 Oscillator center frequency in ppm vs tem perature. -11 •1 1.2 •11.4 ■1 1.6 11. 8 ■ Output Power (dBm) ------- -E !----- -1 2 ■ 12.2 -12.4 - 12 .6 - 1 2 .8 -1 3 -6 0 - 4 0 -2 0 0 20 40 60 80 100 120 140 Temperature (Celcius) F ig u re 5.25 Oscillator output power tem perature behaviour. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 Temperature [°C] -47 -20 -14 25 50 80 110 130 140 I c [mA] 3.21 3.53 3.68 4.15 4.21 4.53 4.77 4.96 5.08 Hum & Noise [dB] -60 -64 -69 -75 -75 -75 -75 -75 -75 T a b le 5.3 Oscillator tem perature behavior. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H A P T E R VI C O N C L U SIO N A unified method for the design of a negative-resistance oscillator has been de veloped. The method was successfully applied to the design of two crystal oscillators having fundamental resonances at f 0 = 842.911 M H z , and at f 0 = 383.86 M H z , respectively. As the experimental results indicate, the negative-resistance design procedure developed is well suited for this type of oscillators. In the crystal oscillators presented, the SSB-phase noise performance can be improved further. However, it should be noted th at in the negative resistance ap proach, phase noise is not a param eter in the design. The potential of crystal oscillators with fundamental frequencies at RF and microwave frequencies is im mense. For example, these crystal oscillators can be used as reference oscillators in the RF and microwave region, and as volt age-controlled oscillators with excellent spectral characteristics. In the 843 MHz design, the SSB phase-noise performance has an average performance as compared to presently available oscillators, while in the 383 MHz oscillator, the noise performance is superior to presently available oscillators. The 843 MHz crystal oscillator discussed in this thesis is believed to be be the first of its kind. Ill Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 The design presented in this thesis is based on the small signal S-parameters, and therefore the prediction of the oscillation frequency is the main goal. It is im portant to mention th at the oscillator under normal operation is a non-linear device; thus, prediction of oscillator power or noise performance based on the small signal S-parameters is not possible. This is not a problem, since noise and power adjust ments can be accomplished after a prototype is built using pieces of information such as device line and oscillator power contours. The design methodology is well suited for this applications since it provides the oscillator at the desired frequency for a subsequent optimization. F U T U R E R E SE A R C H The area of oscillator design at microwave frequencies can be considered as an eternal fountain of research. As mentioned earlier, this thesis focused on a closedform design procedure for negative-resistance oscillators. The closed-form method developed in the thesis does indeed yield working oscillator modules at the desired frequencies. Future research should focus on closed-form design techniques for: - Oscillators with prescribed noise performance, - Oscillators with prescribed power performance, and - Voltage controlled oscillators with prescribed tuning range and noise perfor mance. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H A PT E R VII B IB L IO G R A P H Y Abe, H., Y. Takayama (1977), “A High-Power Microwave GaAs F E T Oscillator,” NEC Reasearch and Development, Tech. Note 45, pp. 58-65. Abe, H. (1986), “A GaAs M ESFET Oscillator Quasi-Linear Design M ethod,” IEEE Trans. Microwave Theory Tech., Vol. MTT-34, pp. 19-25. Abe, H., Y. Takayama, A. Higashisaka, H. Takamizawa (1978), “A Highly Stabilized Low-Noise GaAs FET Integrated Oscillator with a Dielectric Resonator in the C Band,” IEE E Trans. Microwave Theory Tech., Vol. MTT-26, pp. 156-162. Alley, G., H. Wang (1979), “An Ultra-Low Noise Microwave Synthesizer,” IEEE Trans. Microwave Theory Tech., Vol. MTT-27, pp. 969-974. Arai, Y., H. Komizo (1976), “ GaAs FET applications Expand,” JEE, pp. 51-58. Aubry, J. (1983), “Q uartz and LiTaOs VHF Resonators for Direct Frequency Gen eration in the GHz Range,” IEE E Ultrasonic Symposium, pp. 487-490. Bahl, I.J., R. Garg (1978), “A Designers Guide to Stripline Circuits,” Microwaves, Vol. 17, pp. 90-96. Barnes, J., A. Chi, et al (1971), “Characterization of Frequency Stability,” IEEE Trans. Instrumentation and Measurement, pp. 105-120. 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 114 Barnes, J. (1976), “Models for the Interpretation of Frequency Stability Measure ments,” National Bureau of Standards, Tech. Note 683. Bajeu, G. (1976), “Design Transistor Oscillators,” Electronic Design, pp. 98-102. Basawapatna, G.,R. Stancliff (1979), “A Unified Approach to the Design of WideBand Microwave Solid-State Oscillators,” IEEE Trans. Microwave Theory Tech., Vol. MTT-27, pp. 379-385. Berte, M., P. Hartemann (1978), “Quartz resonators at fundamental frequencies greater than 100 M H z ,” IEE E Ultrasonics Symp. Proc., pp. 148-151. Besser, L. (1973), “Computerized Optimization of Microwave Oscillators,” Wescon Tech. Paper, Vol. 17, pp. 11-14. Bidart, L. (1982), “New Design Of Very High Frequency Sources,” Ferro electrics, Vol. 40, pp. 231-236. Bodway, G. (1968), “Circuit Design and Characterization of Transistors by Means of Three-Port Scattering Param eters,” Microwave Journal, pp. 55-63. Cady, W. G. (1922), “The Piezo-Electric Resonator,” Proc. IRE, 10, pp. 83-114. Cady, W.G. (1946), “Piezoelectricity. Dover, NY. Chan, Y., S. Kazeminejad (1988), “Voltage Controlled Oscillator Using Dielectric Resonator,” Electronics Letters, Vol. 24, pp. 776-777. Chen, S., L. Chang, Y.Chin (1986), “A Unified Design of Dielectric Resonator Oscillators for Telecomunication Systems,” IEEE M T T -S Digest, pp. 593-596. Clarke, K., D. Hess (1978), “ Communication Circuits: Analysis and Design.” Addison- Wesley , Reading, MA. Day, W. (1970), “Dielectric Resonators as Microstrip-Circuit Elements”, IEEE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 Trans. Microwave Theory Tech., Vol. MTT-18, pp. 1175-1176. Driscoll, M. (1973), “Two-Stage Self-Limiting Series Mode Type Quartz-Crystal Oscillator Exhibiting Improved Short-Term Frequency Stability,” IEEE Trans. In strumentation and Measurement, pp. 130-138. El-Kamali, W., J. Grimm, R. Meirer, C. Tsironis (1986), “New Design Approach for Wide-Band F E T Voltage-Controlled Oscillators,” IEEE Trans. Microwave Theory Tech., Vol. MTT-34, pp. 1059-1063. Esdale, D., M. Howes (1981), “A Reflection Coefficient Approach to the Design of One-Port Negative Impedance Oscillators,” IEEE Trans. Microwave Theory Tech., Vol. MTT-29, pp. 770-776. Galwas, B. (1983), “Scattering M atrix Description of Microwave Resonators,” IEEE Trans. Microwave Theory Tech., Vol. MTT-31, pp. 669-671. Gardiol, F. (19SS), “Design and Layout of Microstrip Structures,” IE E Proceedings, Vol. 135, P t. H, No. 3, pp. 145-157. Gerber, E. A., A. Ballato (1985), “Precision Frequency Control.” Academic Press, Orlando, FL, vols 1 and 2. Gerber, E., T. Lukaszek, A. Ballato (1986), “Advances in Microwave Acoustic Fre quency Sources,” IEEE Trans. Microwave Theory Tech., Vol. MTT-34, pp. 10021016. Getsinger W. H. (1973), “Microstrip dispersion model,” IEEE Trans., MTT-21, pp. 34-39. Gilmore, R. (1983), “An Analytic Approach to Optimum Oscillator Design using S-Param eters,” IEEE Trans. Microwave Theory Tech., Vol. MTT-31, pp. 633-639. Golio, J., C. Krowne (1979), “New Approach for FET Oscillator Design,” Microwave Journal, pp. 59-61. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 Gonda, J. (1972), “Large-Signal Transistor Oscillator Design,” IEEE M T T -S Di gest., pp. 110-112. Gonzalez, G. (1984), “Microwave Transistor Amplifiers Analysis and Design.” Prentice-Hall Inc, Englewood Cliffs, N.J. Gonzalez, G., B. Avanic (1987), “Design of Crystal Oscillators with Fundamental Frequencies in the UHF and Microwave Regions,” Electronic Letters, Vol. 23 , pp. 1379-1381. Guillon, P. Y. G arault (1976), “ Coupling between a microstrip transmission line and a dielectric resonator,” IEEE M T T -S Int. Microwave Symp. Dig., pp 200. Guillon, P., Y. G arault (1977), “Accurate Resonant Frequencies of Dielectric Res onators,” IEEE Trans. Microwave Theory Tech., pp. 916-922. Guillon, P. B. Byzery, and M. Chaubet (1985), “Coupling param eters between a dielectric resonator and a microstrip line,” IEEE Trans. Microwave Theory and Tech., vol. MTT-13, pp 222-226. Guttwein, G., A. Ballato, and T. Lukaszek (1972), “VHF-UHF piezoelectric res onators,” U.S. Patent 3 694 Hammerstad, E. O., Jensen, O. (1980) “Accurate models for microstrip computer aided design,” IEEE Mirowave Symp. Digest, MTT-5, pp 407-409. Hamilton, S. (1978), “FM and AM Noise in Microwave Oscillators,” Microwave Journal, pp. 105-109. Helszajn J. (1985), “YIG Resonators and Filters.” John Wiley, NJ. Hewlett Packard (1985), “Phase Noise,” R F and Microwave Phase Noise Measure ment Seminar. Hewlett Packard Application Note 975, “A 4.3 GHz Oscillator Using the HXTR- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. il7 4101 Bipolax Transistor” . Hewlett Packard Application Note 978, “The Design of GaAs FET Oscillators” . Hewlett Packard Product Note 11729B, “Phase Noise Characterization of Mi crowave Oscillators” . Howe, D., (1976), “Frequency Domain Stability Measurements: A Tutorial Intro duction,” National Bureau of Standards, Tech. Note 679. Howes, M., R. Pollard (1979), “Large Signal Characterization Plus Computer Graphics Aid Oscillator Design,” Microwave System News, pp. 42-57. Hunt, J., R. Smythe (1985), “Chemically Milled VHF and UHF AT-Cut Res onators,” Proc. 39th AFCS, pp. 292-300. Ishihara, O. (1980), “A Highly Stabilized GaAs FET Oscillato Using a Dieleectric Resonator Feedback Circuit in 9-14 GHz,” IEEE Trans. Microwave Theory Tech., Vol. M TT-28, pp. 817-824. Jaeger, R., A. Brodersen (1970), “Low Frequency Noise Sources in Bipolar Junction Transistors,” IEE E Trans. Electron Devices, Vol. ED-17, pp. 128-133. Johnson, K. (1979), “Large Signal GaAs M ESFET Oscillator Design,” IEE E Trans. Microwave Theory Tech., Vol. MTT-27, pp. 217-227. Kajfez, D., P. Guillon (1986), “Dielectric Resonators.” Artech House Inc., Dedham, MA. Khanna, A. (1984), “Network Analyzer Doubles as Oscillator Diagnostician,” Mi crowaves & RF, pp. 106-112. Khanna, A., J. Obregon (1981), “Microwave Oscillator Analysis,” IEE E Trans. Microwave Theory Tech., Vol. MTT-29, pp. 606-607. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 Khanna, A., Y. G arault (1983), “Determination of Loaded, Unloaded, and External Quality Factors of a Dielectric Resonator Couppled to a Microstrip Line,” IEEE Trans. Microwave Theory Tech., Vol. MTT-31, pp. 261-264. Khanna, A., J. Obregon, Y. Garault (1982), “Efficient Low-Noise Three Port XBand FE T Oscillator Using Two Dielectric Resonators,” IEE E M T T -S Digest, pp. 277-279. Kayfez, D. 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R am ratan (1980), “FE T vs. Bipolar: Which Oscillator is Quieter,” Microwaves, pp. 82-83. MacDonald, B. (1970), “Avalanche-Induced 1/f Noise in Bipolar Transistors,” IEEE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 Trans. Electron Devices, Vol. ED-17, pp. 134-136. Mw.-da, M., K. Kimura, H. Kodera (1975), “Design and Performance of X-Band Oscillators with GaAs Schottky-Gate F E T ,” IEE E Trans. Microwave Theory Tech., Vol. MTT-23, pp. 661-667. Malcolm, B. (1975), “Frequency Pulling: W hat Happens to Oscillator Stability with Load Variations?,” Microwave System News, pp. 75-78. Mansour, N., W. Tinga (1987), “Generalized Nonlinear Theory for One-Port Mi crowave Active Circuits,” IE E Proceedings, Vol. 134, P t. H, pp. 411-422. Mitsui, Y., M. Nakatani, S. Mitsui (1977), “Design of GaAs M ESFET Oscillator Using Large-Signal S-Param eters,” IEEE Trans. Microwave Theory Tech., Vol. MTT-25, pp. 981-984. Montgomery, C., R. 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Microwave Theory Tech., Vol. MTT-35, pp. 657-662. Ohtomo, M. (1972), “Experimental Evaluation of Noise Param eters in Gunn and Avalanche Oscillators,” IEEE Trans. Microwave Theory Tech., Vol. MTT-20, pp. 425-437. Ollivier, P. (1972), “Microwave YIG-Tuned Transistor Oscillator Amplifier Design: Aplication to C Band,” IEEE Journal Solid State Circuits, Vol. sc-7, pp. 54-60. O ndria, J. (1968), “A Microwave System for Measurements of AM and FM Noise Spectra,” IEE E Trans. Microwave Theory Tech., Vol. MTT-16, pp. 767-781. Osborne, M., G. Morgan (1985), “A Load-Pull Technique for the Determination of Transistor S-Parameters Under Large Signal Conditions,” IE E Proc., Vol 132, pp. 419-423. Parker, T. (1978), “SAW Controlled Oscillators,” Microwave Journal, pp. 66-67. Pengelly, R.(1986), “Microwave Field Effect Transistors Theory. Design and Appli cations.” Research Studies Press, England. Pergal, F. (1979), “Detail A Colpitts VCO As a Tuned O ne-Port,” Microwaves, pp. 110-115. Plourde, J. 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A P P E N D IX 1 V A R A C T O R E Q U IV A L E N T Q To overcome the low-Q characteristics of a varactor it is standard to couple the varactor to the inductor through a series capacitor. As shown in Fig A l.l, the varactor will be represented by resistance R v in series with a capacitance Cv, and the coupling capacitor is modelled with a resistance R c in series with a capacitance Cc. The varactor Q, denoted by Qv is given by Qv = w R vCv Rc Cc Rt Cc Rv i Cv = 4 = : CT Cv F ig u re A l . l Varactor Equivalent circuit representation. 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 124 The coupling capacitor Q, denoted by Qc is given by 1 Qc - (A1.2) W cC c The varactor-capacitor network can then be reduced to an equivalent capacitor C t in series with an equivalent resistance R T , as shown in Fig. Al.2. The element values are given by CT = Cv II Cc = CvCe C v + Cc (A1.3) Rt = Rv+ Rc The equivalent Q of the combined network, denoted by Qn, can then be found from 1 Q n = (Al.4) R t Ct Substituting (A l.3) into (A l.4) we get Qn = w (R v + R c) (A1.5) c vc e Cv + Cc It is of interest to express (A l.5) in terms of the varactor and capacitor Q. Thus, (Al.5) can be w ritten in the form Q n = Cc + C v w R vCvCc T w R cCcCv (.41.6) Then, substituting (A l.l) and (A l.2) into (A1.6) gives = Cc + C . = ( ^ ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (.41.7) A P P E N D IX 2 O SC ILLA TO R P H A S E N O ISE M O D EL The transfer function of a series RLC circuit is given by HrUw) = R + (wL-l / wC) (A2A) Since we are interested in noise around the center frequency, it is common to rep resent (A2.1) in terms of w 0 and Q. For the feedback model presented in Chap. 4, the resistor R in the resonator will represent the combined losses of the resonator and the amplifier, thus the Q represents the loaded Q. Using (2.2) and (2.6) as definitions for w 0 and Q, (A2.1) can be simplified to H r {Jw ) = --------------------------- ST .w0L f w w0 \ ------------K \w 0 w J For w = w 0 + w m, where w m represents the offset frequency (such th at w 0 » we can make the approximation w w0 w 2 —w 02 W0 IV w 0w 2 wrn — —— w0 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A 2 .2 ) w m), 126 since 2w 0 w m » wT and Wg » w Qw Finally, using the above approximation, (A2.1) can be w ritten as 1 ■„ 2 w tn 1 + jQ H R(jw) (A2.3) W n Defining wn 2Q = wl (A2.3) simplifies to H R{jw ) • (A2.4) w l + J — WL U I= U )m Substituting (A2.4) into (4.12) yields the following expression \H r ( jw ) \ 2 = 1 1 - H R(jw) 1 1 +Jw/WL 1+j w 2 W L (-42.5) .w ?— wL 1+ w W L (*)■ = 1+ w‘ = 1+ 4 Q 2 w '1 To obtain the output spectral density of phase fluctuations as defined by (4.13) it is necessary to derive S n; which defines the spectral density of phase fluctuations of the active device translated to the input. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 127 V n,rms V s,rms F ig u re A 2.1 Amplifier noise voltage relationship. Figure A2.1 illustrates the amplifier additive noise voltage relationship where v n,rms represents the noise voltage originating from thermal noise. It is given by V n ,r ms = '/F kT R (A2.6) The signal in terms of power and resistance can be expressed as v3,rms = y/p sR (A2.7) From Fig. A2.1 the peak phase deviation due to the additive noise can be derived from , rms tan A 6 „ = P but in oscillators vn rms « V. y a , rms v3 rm3, therefore vn , r m3 « 1 Using small angle approximations, namely tan A 6 m A 9 we then obtain A /i L^Up ^ . v n,rms VFkTR \/Fs,rmsR / FkT V Ps,r Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 128 Then, the power spectral density of phase fluctuations at the input originated from therm al noise is given by F'h'T Se,i = (A 6 pf = (.42.8) * 3 ,rm a As expected the noise spectrum is independent of frequency, presenting a flat spec trum . To incorporate the effects of flicker noise in (A2.8), the noise figure F is modified to account for the 1 /f corner defined by / c, namely F = F0 ( 1 + — V wm Then, (A2.8) becomes 2 F 0k T ( wc So,i = (A 9PY = - 2 — 1+ - ^ ) a.rm a \ tU r Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A2.9) A P P E N D IX 3 ST A B IL IT Y C R IT E R IA For the derivation of the equations presented in section 5.2.1, consider the oneport network shown in Figure A3.1. The method discussed was first presented by Kurokawa (1969). The following analysis requires that the current in the oscillator be sinusoidal, and th at the effects of harmonic terms can be neglected. i (t) RL -R d XL Xd © e (t) F ig u re A 3.1 Negative resistance model. For the circuit in Fig. A3.1, the current is assumed to be, i(t) = A(t) cos [wt + 0(f)] (A3.1) where the am plitude A(t) and the phase 6 (t) are slowly varying functions of time. 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 130 The impedance Z d = R d + j X o is the impedance presented by the active de vice which is dependent on the amplitude and frequency, namely Z d ( A , w ). The ini] >'-dance Z l represents the resonator and is a function of frequency only. In the case of free running oscillators e(t ) is zero; e(t ) is mainly used in the analysis of noise and injection locking phenomena in oscillators. From network theory, it follows that VD(t) + = e(i) (.A3.2) where vo(t) = RD-A(t )cos [wt 0(f)] + — X D A ( t ) s i n [wt + 6 { t )] (A3.3) and v L(t) = R e [ZLI] (A3.4) The frequency w attains a different form due to the nature of i(t). To find w we need to find d i ( t ) / d t , namely = Jtm c“ [“ " +e(i)1 = j R e [yl(()e(i" ‘+*<‘» dt J A(t) dt A(t)e From the above we can conclude th at (A3.4) represents the voltage vi{t) provided th at w is defined by [u> + dd(t)/dt —j[l/A(t)\dA(t)/dt\. Furthermore for a stable oscillator it is required that d0(t) w » ~ ir 1 dA(t) W >> A(t) ^ dt Then, we can approximate Z l using the first two terms of Taylor’s series expansion, namely Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 131 Substituting (A3.6) into (A3.4) and recalling that Z l (w ) = R l (w ) + j X ^ w ) we obtain VL(t) = Re [ZlI] B ( 71A I d R L{w ) de{t) | d X L(w ) 1 dA( t y A(t)cos [wt + 0(f)] M dw dt ^ dw A(t) dt ( d X L(w) dOjt) _ dRijio) 1 d A jt) ' A{t)sin [wt + 0(f)] L dw dt dw A(t) dt (A3.7) Substituting (A3.3) and (A3.7) into (A3.2) we get dR L(w)d9(t) J------------dw at d X i ( w ) dd(t) X j) + X i ( w ) + dw dt e(f) = RJ?d i i? (i,A i + R L\W) H d X L(w) 1 cL4(f) A(i)cos [wt + 0(f)] dw A(f) dt d R i { w ) 1 dA{t) A(t)sin [wt + 0(f)] dw A(t) dt (A3.8) 1-------- J---------- 7 7 7 7 — 71— Using the orthogonal properties of the sin and the cos, if we multiply (A3.8) by cos[wt + 0(f)] and integrate over one period to eliminate the second harmonic term we obtain (nA +i — d R L{w)d 6 {t) R7?D +I R7?L(w) ------— + d X L(w) 1 dA{t)' A(t) ec{t) (A3.9) Similarly, if we multiply (A3.8) by sin[wt + 0(f)] and integrate over one period we obtain: X p + X l (w ) + </Xl(w) dd(t) dw dt d R i ( w ) 1 dA(t) dw A(f) dt 1 A e3(t) (A3.10) ( f ) where :(f) = Tjr [ e(t)cos [ruf + 9(t)]dt o Jt-To 2 ,(t) = — I e(t)sin [tuf + 9(t)\dt To J t - T 0 Equations (A3.9) and (A3.10) have to be satisfied simultaneously. As can be seen, both equations contain d9(t)/dt and dA(t)/dt. It is of interest to the find two new equations in terms of the am plitude variation or the phase variation only. By Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 132 multiplying (A3.9) by X'L(w) and (A3.10) by —R'L(w) and adding we obtain [JJi + Rd ) ^ - IXL + X D] ^ + dw dZi(w ) dw A(t) 1 dA(t) A ( t ) dt (A3.11) dw Similarly, by multiplying (A3.9) by R'l (w ) and (A3.10) by —X'L{w) and adding an expression with the phase variation only is obtained: [Rl + R d ] ^ + [XL + X D\ ^ + dw dw dZi{w) dw A(t) 1 dA(t) A ( t ) dt <1Rl i (A3.12) dXL For a steady-state free-running oscillator: e(t) = 0 => ec(t) = es(t) = 0 , dA(t)/dt = 0 and d0{t)/dt = 0. Then (A3.11) and (A3.12) yield the well known conditions for oscillation, namely Rl + Rd = 0 (A3.13) XL+ X D = 0 Equation (A3.13) determines the frequency and the amplitude of oscillation, this equation constitutes the basis of negative resistance oscillator design. For stability it is of interest to find what happens if the amplitude deviates by a small amount A A. Equation (A3.13) then becomes: OR d dA (A3.14) ,dI X /\ D T~i X L + X D = A AdA Substituting (A3.14) into (A3.11) we obtain a differential equation in AA, namely: R l 4- R d = AA dw “ 1 d(AA) + A0 dt dXL 8 Rd dw dA A=A0 dRL d X o dw dA AA = 0 A=Aa For a stable oscillator the deviation A A is expected to decay so that the oscillator stabilizes at its original am plitude A0, from the above differential equation A A decays with time only if d X L dRo dw dA A=A0 dRL dX D dw dA > 0 (A3.15) A=Aa Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 In conclusion, the operating point of an oscillator is determined by equation (A3.13). Furthermore w 0 and A 0 will constitute a stable operating point if and only if equation (A3. IF) is satisfied. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A P P E N D IX 4 S E R IE S -F E E D B A C K A N D P A R A L L E L -F E E D B A C K N E T W O R K S A 4.1 S E R IE S -F E E D B A C K N E T W O R K Consider the network shown in Fig. A4.1 where the impedance matrices of the transistor and series-feedback network are [Za] and [Z&], respectively. We can express [Za] and [Zi] in the form S22.c S ll,c TRANSISTOR [Z a ] FEEDBACK NETWORK [Z b ] F ig u re A 4.1 Series-feedback network. 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 135 *11, a .*21,a [3 -] = z 1 2 ,a z 2 2 ,a . where z ll,a — Z_Aa *12,a = [(1 + S U>a)(l — 522,a) + 5i2,a52l,a] [25i2,a] z 2 1 ,a = -7~ [252l,a] z 2 2 ,a = [(1 —5 n )a)(l + 522,a) + 5i2,a52i,a] and z ll,b z 1 2 ,b mz 2 1 ,b z 2 2 ,b [Zb) = where z n,b = z i 2 ,b = z 2 i,b = z 22 ,b = Z 0 ( r + j x ) •S’li.aj 5i2,aj 52i,at and 5^2,a are the small signal S-parameters of the transistor measured at f a, Z Q is the normalizing impedance (usually 50 Q, ) and A a — (1 —S n >a) (1 —5 22,a) —5i2,a52i,a The overall impedance m atrix of the composite network in Fig. 4a denoted by [Zc], is given by r + j x + z'n>a r + j x + z[ 2>a [Zc] = z 0 r + j x + z 2' j,a r + j x + z22a (A4.1) where / _ Zjjtg z tj,a 7 1 ZJ q Using the conversion relations from z-parameter to S-parameter, it follows th at 5 n iC is given by ’11,c ~ 1)(*22 + !) ~ (* ii + l)(* 2 2 + 1) “ *12*21 (* n ^12^21 Substituting (A4.1) into (A4.2) we get Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A4.2) 136 _ [(r + i a!) + zn ,a - 1] [ir + 3X) + Z2 2 ,a + 1] - [(r + •7*) + 212,g] [(r + 3X) + 5 4 l,a ] [(»• + J * ) + *1 1,a + l ] [(»■ + i * ) + *22,a + * ] “ [ ( r + i x ) + * i 2,a ] [ ( r + J®) + 4 l , a ] or ■5ll,c — (r 3X) H i , a d~ *22,a ~ 212,a ~ 221,o] ~t~ [2n , a 222,a ~ 212,az21,a (r + j x ) [2 jl fl + 2 2 2 ,0 ~ 212,a ~ 221,a + 2ll , a ~ 222,a ~ *] + [ * n , a 222,a ~ Z 1 2 , a Z21, a ~ 2l l , a + Z22, a ~ 1 Following the same procedure we obtain the remaining S-parameters of the composite network, A convenient form of expressing the S-parameters of the seriesfeedback network is (r + j x ) A 3 + B 3 (r + j x ) E 3 + D s s ^ ,c = (r (r + j x ) E s + D s _2_ 5l2,c = XT A i [r + ^ + 212,a] _2_ S 21 ,C = -T- [r + j x + z'21>a] Ax (44.4) (A4.5) (A4.6) where As = 2 n ,a + z'22ta ~ B3 = ^ 1 1 ,0 ^ 2 2 , 0 — ^ 1 2 ,a — z 2 1 ,a z 12,az21,a + z l l a — z 22,a — 1 Ds = z ll,az22,a ~ z 12,az21,a ~ zll,a + ^ 2 2 ,a + 1 (A4.7) E s = 4 l,a + 222,a — z12,a ~ Z21,a + 2 Fs = z[Xaz22a —z[ 2 az21a — z'l l a + z22a — 1 A l = (z[ i , a + l ) ( z 2 2 ,a + 1 ) “ z 12,az21,a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 137 A 4.2 P A R A L L E L -F E E D B A C K N E T W O R K Consider the network shown in Fig. A4.2 where the adm ittance matrices of the transistor and parallel-feedback network are [FQ] and [Yj] FEEDBACK NETWORK [Yb] S22.c Sll.c TRANSISTOR [Ya] F ig u re A 4.2 Parallel-feedback network. F .] = / 1 2 ,a 2 /ll,a 1 1/21,a 1/22,a . where yn,a — 1 ( 1 ~ *S'll,a ) ( l + ‘S,2 2 ,a ) + <S'i21a ‘S'21)a A a 1 J —2 S 'i2 , a 1/12,a = Aa 1 “ 2 5 2 1 ,a 1/21,a = Aa L 1 1/22,a = (1 + •S'u a ) ( l - $ 2 2 ,a ) + 5 i 2 , a 5 2 l , a Aa £0 and [tt] = 1/ 11, 6 1/ 12, 6 . 1/ 2 1 , 6 1/ 22, 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 138 where Vn,b 2/ 22,6 2 /2 1 ,6 2 /l 2 ,6 _ 1 . _ J L Yv Z0( r + jx ) Z0 1 1„ / . • \ Z0(r+ jx ) Z0 ry ( „ ry ry Y Sn,a, S i 2 ,a, 521,a, and 522,a are the small signal S-parameters of the transistor Z a is the normalizing impedance (usually 50 Q, ) and measured at A a = (1 + 5 l l , a ) (1 + 5 2 2 ,0 ) ~ 5 l2 ,a * S 2 1 ,a The overall impedance m atrix of the composite network in Fig. 4b denoted by [Fc], is given by Y + y 'n ,a Y + y'l2,a Y + y'21>a Y + y'22,a. (A4.8) where 2 /jj,a = y i j , a Z 0, i , j = 1,2 Using the two port conversion relations from Y-parameters to S-parameters, it fol lows that Sxi,c is given by <j 11,C C1 ~ 2 / n X 1 + 2/ 2 2 ) + 2/ 122 /2 1 / , 4Qn (1+Vii)(l+Vi2)-Vi2vii ( * Expanding (A4.9) using the Y-parameters of the composite network gives q _ 11,C 1 + 2/ 2 2 ,c ~ 2/i 1 ,c ~ 2/n ,c2 /2 2 ,c + ! / i2 ,c 2 / 2 i,c f’/ U i n l 1 + l/2 2 ,c ^ + V u ,c + 2 /ll,c2 /2 2,c ~ 2/i2 ,c2/21,c j Substituting (A4.8) into (A4.10), we obtain Y ( - 2 /n ,o - ^ 2 2 ,a “ 2/ 2 1 ,0 ~ ^12,a ) + (* ~ ^11,a ~b ^22,a ~ ^ n ,a ^ 2 2 ,a ~b !/(2,a 2/21 ,a ) •S ll.c — b ( 2 + 2 /(1 |a + 2 /2 2 ,a "b 2/ 1 2 ,a 4" ^ 2 1 , a ) "b ( l + 2 / l l , a + 2/ 2 2 , 0 "b 2 / u , a 2 / 2 2 , a ^ 1 2 , 0 2 /2 1 ,a ) The above expression is in terms of Y = l / ( r + jx ). In order to obtain equivalent results to those of the series feedback network we must express S n )C in the form of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 139 (A-l.3). T hat is _Ya+ £ n 'c Yj +d (r + j x ) (r T jx ) _ ( r + jx ) /? + a (r + j x ) 9 + 7 To obtain an equation with the same constants as in the series case, we rename the above constants in terms of A, B, E, and D. Following the same procedure for 522,0 512,0 an(l 5>2i,c we arrive to the S-parameters of the composite network given below: (r + j x ) A Ptl + B p (r + j x ) E p + Dp (A 4 .ll) c (r T j x ) A P j 2 + Fp b22’c ~ (r + j x ) E p + Dp (44.12) lllC 1 (44.13) As Vl2,a ~ r + j x _ 1 1/21, a ~ r + jx S21-' - A l 5 'l2 ,c = (A4.14) where Ap, 1 = 1 —y ' \ \ >a ■^p,2 = 1 T + / ,a 2/l 1,a 2/ 2 2 ,a + 2/ l 2 ,al/2 1 ,a 122 ~ 2/ n , a — l/22,a —l/ll,al/22,a + l/l2,a2/21,a Bp = Fp = 2/ l l , a ~ 2/22,a ~ y'l2 ,a ~ 2/21,a (44.15) Dp = 2 + Bp I T 7/Jl a + 1/ 2 2 ,a+ 2 /ll ,a T 2/12, a + 1/ 2 1 , 0 2/22,a T 2/l 1 ,a2/22 ,a A 2 = ( 2 / l l ,a T l ) ( l / 2 2 , a T 1) l/ l2 ,a l /2 1 ,a — l/l2 ,a l /2 1 ,a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A P P E N D IX 5 F E E D B A C K M A P P IN G The equations for 5 u ,c and £ 22,0 for both the series and parallel feedback network are of the form _ (r + j x ) A + B Sa,c = ; !_ ■■■ (r + j x ) + D .J . (A5.1) where i = l or i=2, and A, B, E, and D are constants, [see (A4.7) and (A4.15)]. In what follows the analysis of how constant values of x and constant values of r map onto the 5 n )C plane is presented. The analysis is also valid for 5 2 2 ,c- From (A5.1), solving for (r + j x ) gives . D S n c —B z — r + j x = —---- =—---^ A — E S u tc Then, letting 5 n )C = a + jb, we obtain r I j x —D (a + i b) ~ B ( a D - B ) + jbD 3 A — E(a + jb) ( A - a E ) - j b E { Expressing A, B, E, and D in the form A = a1+ ja 2 B = Pi + j p 2 E = ei + j e 2 B = 61 + j S 2 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 141 (A5.2) can be expanded to . (aSi - /?i - b8 2 ) + j(aS 2 - f32 + b6 i) (ax - aex + be2) + j ( a 2 - ae2 - bei) ( . Using (A5.3) we can now find how constant values of r or x map onto the 5 n |C plane. Equating the real part of the right hand side of (A5.3) to r gives (qfii — fli — b S 2)(ai — a ei + b e 2 )+ (a82— P 2+b6\)(a2—a e 2— b ei) (ai - aei+b e 2 )2+(a2- a e 2- b ei)2 r , . Equation (A5.4) can be m anipulated into the form of the equation of a circle, that is (a - hr ) 2 + ( b - kr )2 = R l where 2rRc[AE*\ + Re[AD*] + Re[BE*] r~ 2 { r \ E \2 +Re[ED*]} 2rIm[AE*] + Im[AD*] + Im[BE*) 2 { r \ E \2 + Re[ED*]} y / \ A D \2 + \ B E \2 - 2Re[AB*]Re[ED*} - 2Im[AB*]Im[ED*] r~ 2 {r\E \2 + Re[ED*)} where hr, and kr are the center of the circle, and R r represents the radius of the circle. Furtherm ore, we can express the center in terms of polar coordinates, that is Cr = y / h 2 + k 2 (A5.6) Substituting (A5.5) into (A5.6) we obtain 2rAE* + AD* + BE* r ~ 2{r\E\+2Re(E*D)\ The radius of the circle in polar form is given by r |AD - BE\ 2[r\E]+2Rc(E‘ D)] 1 ' Following the same procedure, we now need to map constant values of x onto the S n jC plane. From (A5.3), we obtain (aS2 — f32 + b8 \)( ai — ae\ d~ be2) ~ ~ Pi ~ b6 2 )(a 2 — ae2 —bei) («i - aei + be2 ) 2 + (0:2 - ae2 - bei ) 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 142 which can be m anipulated to read (a - hx)2 + (b — k x ) 2 = R x2 where hx — 2xRe[AE*] + Im[A*D} - I m [B * E } 2 { x \ E \ 2 + Im[E*D]} 2xIm[AE*] + Re[A*D] - Re[B*E\ 2 { x \ E \2 -f- Irn[E*D}} (.45.8) yJ\A D \2 + \ B E \ 2 - 2Re[AB*)Re[ED*] - 2Im[AB*]Im[ED*} x ~~ 2 { x \ E \2 + Re[ED*]} The center and radius of the circle are given by x 2[x\E\+2Im(E*D')\ V and Rx = \AD - B E | 2 [r\E\+2Im(E*D)] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A P P E N D IX 6 O P T IM U M F E E D B A C K The equations for 5 n )C for both the series and parallel feedback network are of the form _ (r + j x ) A + B ( r*» (r +i j *x )\ Ern +, D — (A 6 .1 ) v ' where A, B, E, and D are constants and given by (A4.7) or (A4.15). By substituting r= 0 (maximum reflection coefficient) and |S'iijC|mai into (A6.1), we get jxA + B jxE + D \ S n tc Squaring both sides 2 l\Jll,c ^ll.clm ai |j x A + B \ 2 . . .2 \j x E + D\ but |j x A + B |2 = (j x A + B ) (j x A + B )* |j x E + D \2 = (j x E + D ) (j x E + D )* After expanding and using the property (AB* - A*B) = 2j I m ( A B * ) we get ,o |2 H.clmar |i?|2 + ft2|2l|2 —2xI m( A B* ) \n \‘ + -ixImfED') ' J or (.46.3) + [ | S n , c | 2m „ | i 5 | 2 - | B | 2 ] = 0 143 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 144 It was pointed out in Chapter 5 that this equation can only have one solution at the tangential point of the r= 0 circle with the x opt circle; therefore we shall show th at (A6.3) has a double root. Equation (A6.3) is a quadratic expression in x of the form ax2 + bx + c = 0 The solutions of the quadratic expression are given by —b ± Vb 2 — 4ac Xi = 2a If b2 —4ac = 0 the quadratic equation will have a double root. From (A6.3) b2 = 4 [Im 2 ( A B ' ) + |5llie |J . „ W ( £ C * ) - 2 |S „ , . |L , I m ( A B ' ) I m { E D ' ) ] and 4a c = 4 [ | S „ X „ , | J 3D |2 + \AB\2 - \Su ,c\iax\AD\2 - | S „ , c | L J B E | 2] Letting A = 4ac —b2 = A i A 2 where Aa = [\S niC\2maxRe(ED*) - |5 „ fC|max|AJD - B E \ - Re(AB*)] A 2 = [|5lliC|2max M ^ ) + \Sn,c\max\ADBE\ - Re(AB*)] (A6.4) (A6.5) Recalling th at |5 'n iC|mai is given by [see (5.14)] \ A D - B E \ + \A D * + B E * \ |5 u ’c|mai " ---------- 2Re(E * D)----------- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A6>6) 145 it follows th at (A6.4) can be expressed, after some manipulations, in the form A [(AB* + A*B)(ED* + E*D) - 4Re(AB*)Re(ED*)] 1“ 4 Re(ED*) Since, (AB* + A*B) = 2Re(AB*) (ED* + E*D) = 2Re(ED*) then A i = 0 ==£• A = 0 or b2 = 4ac which implies th at at |5 n )C|max there is only one solution to the quadratic expres sion. Denoting the solution x = x opt, its value is X°r' b [2 I r n ( A B ’ ) - 2 I , n ( E D ' ) \ S n J L „ I } 2a 2 ||* m R « (, W - M P ) or |5 '1 i , c l 2m „ | E | 2 - |A |2 ( • , Since 522,c has the exact form as (A6.1) with the constant B replaced by the constant F, (A6.7) will yield the optimum x for a maximum \S 2 2 ,c\max where B is replaced by F and |5 n iC|max is replaced by |522,c|mau th at is \S 2 2 , A l . J m ( E D ‘ ) - I m ( A F ' ) X°>‘ = --------1W „ a I|£ P - W 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. U 6 '8) I A P P E N D IX 7 L IM IT S F O R T H E A R G U M E N T O F TIN We are interested in finding the intersection of the circles iT/wl = k with the 1IYI = 1. The |r/jv | := k (k > 1), represents a circle centered at zero; thus, it can be w ritten as x2 + y 2 = k 2 (A7.1) The |IV | = 1 relation represents a circle centered at C, = | C; | ZC,- with radius i?,\ This relation can be expressed in the form (x - a)2 + (y - b) 2 = R 2 (A7.2) where a — 1C, | cos LC{ (A7.3) b = jC, | sin LCi From (A7.1) y2 = k2 - x 2 (A7.4) Substituting (A7.4) into (A7.2) we obtain —2ax + a2 + k 2 + b2 — Rjf s 2 = k2 - x 2 2 b 146 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A7.5) 147 Since a2 + b2 = Cf it follows th at after some manipulations (A7.5) can be written in the form a ( ad\ ( d2 1+^J +{f) x+{i *2 < ? - k '‘ ) = 0 ^ where d = k 2 + Cl - R] (A7.5) Substituting (A7.3), (A7.6) into (A7.5) we get i + ^ W b2 ) i + ^ ? 44^ W \ s in 2 L C i ) i + \ a d \ _ {\Cj\coslCj) (k 2 + C l - R l ) b2 ) \C i^s in2l C i # 462 A ) 1 tan2LCi k 2 + Cl - R 2 \Ci\tanlCi s in l C i (k* + C j f - t i p 4\Ci\ 2 sin 2£Ci which corresponds to (5.22). Finally, once Xj are found from the quadratic equation (5.22), yi can be found substituting X{ into (A7.4). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A P P E N D IX 8 IN S T A B IL IT Y CIRCLES We are interested in mapping onto the IV plane the iT/wl = k circle, namely |r/N l = S 1 2 S 2 1 Tt + 1 —S 2 2 T t = k (A8.1) From (A8.1) we can write 1-Si i - S u S 2 2 T t + 5,i25 2 i r r | = k |1 - S ^ IV l (A8.2) | 5 „ - A r T | = f c | i - 5 22r T | (A8.3) A — S n S 22 — 5 i 2 5 2i (A8.4) or where Squaring (A8.3) ( S u - A F t ) ( S n - A T t )* = k 2 (1 - S 22r T ) (1 - S 22r T )* (A8.5) Expanding (A8.5) and grouping terms gives | r r |2 ( | A | 2 - k 2 \ s 22 12 ) + r T {k 2s 22 - a 5 ? a) + ( k 2s ; 2 - s n A * ) (AS.6) = k 2 - \S n \2 148 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 149 Multiplying both sides of (A8.6) by (|A |2 — fc2|S2212) gives \Tr \2 (|A |2 - Ar2|5 22|2) 2 + T t {k 2 S 2 2 ~ A S*n ) (|A |2 - fc2|S22|2) + (.48.7) (*2S2*2 - S n A*) (|A |2 - *2|S22|2) = {k2 - \ S n ?) (|A|3 - * 2|S22|2) M anipulating the right hand side of (A8.7) and regrouping (fc2 — |S „ |2) (|A |2 - f c 2|S22|2) (48.8) = |fcS2,S12|2 - (V.S22 - AS,',) ( i 2S2*2 - A *S„) Substituting (A8.8) into (A8.7) gives irvi2 ( t 2|s 22|2 - |A|2) 2 - r T ( k 2s 22 - a s ; , ) ( t 2|s 22|2 - |A |2) - r ^ ( t 2s 2*2 - s „ A*) (fc2|S22|2 - |A|2) (48.9) + |*2|S22|2 - | A | 2|2 = |M ,2S2,|2 a careful examination shows th a t the above expression can be factored into the product of two sums, namely Tt {k 2 \S2 2 \2 - |A|2) - (A:2|S22|2 - AS?*)* [Tt (k2\S22\2 ~ |A |2) - (fc2|522|2 - AS*,)] (A8.10) = \kSi 2 S 2 1 \2 The right-hand product term is the complex conjugate of the first product term. Thus, (A8.10) can be expressed in the form T t (k 2 \S2 2 \2 - |A|2) - (fc2|S22|2 - A ^ ) * ] 2 = |fcS12S21|2 (48.11) or (k 2 \S2 2 \2 - |A|2) we obtain Tt ~ (i-2|s 22|2 - a s ; ,) ‘ (*2|S22|2 - | A | 2) k S i 2 S21 ( P | 5 2 2 |2 - | A | 2 ) (A9.12) Equation (A8.12) represents the equation of a circle whose center is given by Ck = (k2\s22\2 - a s ^ y (fc2|S22|2 - | A | 2) and its is Rk = k S \ 2 S2i ____ (fc2!S’22|2 - | A | 2) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A P P E N D IX 9 P H A S E A N G L E M A P P IN G We are interested in finding the relation between the angle a in the T i n plane and the angle <f>in the IV pl~ne, where IV and T i n are relate by T j n — S 11 + S 1 2 S21 1 —.S22IV solving for IV we get rv = A - pn £ 2 2 1 (-49.1) IN where A = S 1 1 S 2 2 ~ S 1 2 S 21 Expanding (A9.1) in terms or real and imaginary we get ip I / ' _ -fo ifti] --R e[r J i y ] + i ( ^ [ ^ n ] - I m [ T IN]) 1 T| 9 i2e[A]-fle[S'22r /Jv ] + j ( J m [ A ] - / m [ 1S22r/iV]) . V . Therefore, f , _! (Im[S n } - I m [ V i N] ) \ j Z ( /m [ A ] - Im [522r /N ] ) \ * = taU { JtalS„l-Jfc[T ,w] I - * ™ { iJe[A]-H e[S22r /N] ) { M '2) Equation (A9.2) can be further simplified by invoking the trigonometric identity tan X(A) —tan X(B) = tan 1 A B ^) 150 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 151 Hence where A= ~ Jm jrjjv]) R e [ S n ] ~ i2 c [r/jv ] and B = ( - M A ] - I m [S22^IN}) i2 e [A ] — i? e [ 5 ,2 2 r / ^ ] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. VITA Branko Avanic was born in Cochabamba, Bolivia on April 30, 1960. His parents are Branko Avanic and Desanka Avanic. He received his prim ary and secondary education from “Colegio Anglo Americano” in Cochabamba, graduating with a bilingual diploma on October 1978. In January 1979 Branko enrolled in the College of Engineering at the University of Miami to pursue a career in Electrical Engineer ing. He graduated with a Bachelor of Science in Electrical Engineering, “Magna Cum Laude” , in December 1981. In January 1982 he was adm itted to the Graduate School of the University of M iam i, and in May 1983 was conferred the degree of M aster of Science in Electrical Engineering. Upon graduation in June of 1988 he accepted a job with Motorola, Inc. as an RF engineer, and was also employed by the University of Miami as an Adjunct Faculty member where he taught undergraduate classes in the evenings. In May of 1986, with the encouragement of faculty members, family and friends, he decided to resign his position at Motorola in order to pursue the deg. -e of Doctor of Philosophy in Electrical Engineering at the University of Miami. While enrolled as a full time graduate student, Branko was also a full time Lecturer at the University of Miami, where he taught several undergraduate classes in the fields of electronics, communication and circuit theory. In December 1988 Branko received the “Eliahu I. and Joyce Jury Award”, an award given to the top graduate student in the Department of Electrical and Computer Engineering at the University of Miami. He was granted the degree of Doctor of Philosophy by the University of Miami, Coral Gables, Florida on May 1990. Permanent Address: P.O. Box 8433, Coral Gables, Florida 33124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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