# Some critical parameters for the statistical characterization of power density within a microwave reverberation chamber

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SOME CRITICAL PARAMETERS FOR THE STATISTICAL CHARACTERIZATION OF POWER DENSITY WITHIN A MICROWAVE REVERBERATION CHAMBER by ATINDRA KUMAR MITRA, B.S.E .E ., M.S.E.E. A DISSERTATION IN ELECTRICAL ENGINEERING Submitted to the Graduate Faculty o f Texas Tech U n iv e rs ity in P a rtia l F u lfillm e n t o f the Requirements fo r the Degree o f DOCTOR OF PHILOSOPHY Approved Chairperson o f the Committee £ ■ ____________________________ Accepted Dean o f the Graduate/School May, 1996 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9623845 Copyright 1996 by Mitra, Atindra Kumar All rights reserved. UMI Microform 9623845 Copyright 1996, by UMI Company. AH rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. ACK NOW LEDGM ENTS I would like to thank Professor Thom as T rost for his guidance on this project. I have found our frequent discussions with regard to the specifics of this project, as well as a num ber o f other discussions that were directed tow ards broader issues concerning EMC testing and microwave engineering, to be very insightful and enjoyable. I would like to thank Professor Thom as Krile, Professor H erm ann Krom pholtz, and Professor David Mehrl for their insights and suggestions. I would like to thank Professor Victor Shubov for serving as the Graduate Dean's Representative at my dissertation defense. I would like to express my gratitude and appreciation to m y parents for their support and encouragem ent throughout the course of my educational experiences. I would like to acknowledge Adriel Alvarado and Jam es Ledbetter for their contributions to the developm ent o f the m icrowave reverberation cham ber m easurem ent system that was used in this study. I would like to acknowledge Lonnie Stephenson for machining a num ber o f parts that were integrated into the m easurem ent system. I would like to thank Judy and Steve Patterson for their help in preparing this and related manuscripts and for their advice with regard to the com puter facilities at Texas Tech. ii Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. T A B L E OF CONTENTS ACKN O W LEDGM EN TS ....................................................................................................... ii A BSTRA CT ................................................................................................................................. v LIST O F TABLES ..................................................................................................................... vi LIST O F FIGURES ................................................................................................................... vii CHAPTER I. INTRO D U CTIO N .................................................................................................. 1 II. CH A M BER PARA M ETERS AND TH EO RETICA L FIELD M ODELS 7 Cham ber Quality Factor ................................................................................ 7 Transfer Function M odels ............................................................................ 10 Probability Density Functions ...................................................................... 13 Correlation Functions .................................................................................... 15 Sum m ary o f Critical Param eters .................................................................. 24 III. SIM U LA TIO N M ETH O D O LO G Y .................................................................. 27 IV. M EA SU REM EN T APPA RA TU S ..................................................................... 43 System Specifications .................................................................................... 43 Mechanical Design C onsiderations ............................................................. 50 RESU LTS A N D CO N C LU SIO N S .................................................................... 58 Power Transfer C haracteristics ................................................................... 58 V. iii Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. PDF o f Pow er Density .................................................................................. 66 SCF o f Pow er Density .................................................................................. 72 Summary and Discussion ............................................................................. 90 REFERENCES ........................................................................................................................... 95 A P P E N D K : M ATLAB SIM ULATION M-FELES .......................................................... 97 iv R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. ABSTRACT Statistical m odels for a com ponent of the power density inside a microwave reverberation cham ber are studied. These models include average chamber gain characteristics, an ideal probability density function (PDF), and an ideal spatial correlation function (SCF). The SCF model is extracted from a m ore general theory of complex cavities. Emphasis is placed on observing the applicability of these models as the conditions in the chamber are varied from non-ideal, multi-moded operation to ideal, "overmoded" operation in the high-frequency limit. The observations are made with m easurements and simulations. Deviations from the models are characterized by and related to a num ber of critical cham ber parameters such as the Q, the cham ber electrical size, and the frequency range o f chamber operation. The introduction o f a chamber param eter M, the number o f modes in a 3-dB bandwidth, provides useful information with regard to the behavior of the models in the low frequency limits of chamber operation. The introduction o f SCF measurements to microwave reverberation cham ber systems provides useful insights into the physical behavior o f the fields inside the chamber. v Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES 5.1 Tabulation o f Calulated M Values from M easured G ain Values .......................... 65 5.2 M easured M and a Values From the Q M easurem ent of Figure 5.3 Alongside M and a Values From the Corresponding Simulation Output .............................. 70 vi Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. LIST OF FIGURES 3.1 Plots o f the M inimum, M aximum, and Average o f the Pow er Density, in dB, from a Sample Cham ber Simulation ............................................................................. 35 3.2 Sam ple O utput File K l.O U T from Cham ber Sim ulation Program ....................... 37 3.3 Sample O utput File K 2 .0 U T from Cham ber Sim ulation Program ....................... 38 3.4 H istogram s o f Pow er D ensity Data for a Sim ulation Frequency o f 10 G H z and a C ham ber Size o f 1.0342 m X 0.8087 m X 0.5812 m ........................................ 39 H istogram s o f Pow er D ensity Data for a Sim ulation Frequency o f 10 G H z and a C ham ber Size o f 1.0342 m X 0.8087 m X 0.5812 m ........................................ 40 3.6 SC F Sim ulation Output for Two D ifferent Q V alues .............................................. 42 4.1 Block Diagram o f the Basic Reverberation Cham ber System that was Designed and C onstructed for this Study ..................................................................................... 44 D ata Sheet for the Prodyn A D -S 10 ( R ) W all-M ounted D -dot Sensors that are used to M easure the SC F in this Study ............................................................... 47 Reverberation Cham ber System that is used with HP8719A M icrow ave Netw ork A nalyzer to m ake SCF M easurem ents from 1 to 13.5 G H z ................ 49 4.4 Draw ing o f the Cham ber Door ..................................................................................... 54 4.5 D raw ing that Shows Cross-Section o f the Cham ber D oor G roove and E lastom er Seal .................................................................................................................. 55 4.6 Photograph o f Sensor Plate for C lose Spacing SCF M easurem ents .................... 57 5.1 C ham ber Gain versus Frequency in Configuration 1 ............................................... 59 5.2 Cham ber Gain versus Frequency in Configuration 2 ............................................... 60 5.3 C ham ber Q M easurem ent via Cham ber Gain M easurem ent with Two Receiving (Prodyn) D -dot Sensors (configuration 3) ............................................. 64 3.5 4.2 4.3 vii Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 5.4 Probability Density Function for EM Pow er Density in Configuration 1 ............ 67 5.5 Probability Density Function for EM Pow er Density in Configuration 2 ............ 68 5.6 M easured, Simulated, and Theoretical SCF o f the Pow er Density, versus Spacing, Inside the M icrow ave Reverberation Cham ber o f Figure 4.3 or Figure 4.1 ......................................................................................................................... 73 5.7 5.8 M easured, Simulated, and Theoretical SCF o f the Pow er Density, versus Frequency, Inside the M icrow ave Reverberation Cham ber o f Figure 4.3 or Figure 4.1 .......................................................................................................................... 82 Plots o f M easured SCF Response with 50 Paddle W heel Positions and 200 Paddle W heel Positions .................................................................................................. 89 viii Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER I INTRODUCTION The increasing popularity o f microwave reverberation chambers [1] for electromagnetic im munity testing applications has, in the recent past, motivated a number of investigations [2 - 8 ] with regard to obtaining accurate models for the electromagnetic fields within the chamber. These chambers are generally associated with a number of desirable features such as statistically uniform (or homogeneous) fields [ 1] as well as high field strengths in relation to the input power level. The statistical uniformity o f the field allows a very large cross-section o f the test object to be illuminated with a uniform (average) power level and is typically accomplished by varying the chamber boundary conditions with a rotating mechanical tuner (or paddle wheel). Standard microwave reverberation chambers are comprised o f a metallic rectangular enclosure (cavity) with a m otor-controlled paddle wheel installed in the vicinity of the top boundary. Statistical field samples are typically acquired at various points in the chamber for a large (usually about 200) number o f incrementally spaced paddle wheel positions. A primary chamber design objective is to obtain a spatial field uniformity such that the spatial variance o f the average field, calculated by averaging over all incremental paddle wheel positions, is minimized. As far as detailed information with regard to these chambers is concerned, NBS Technical Note 1092 [4], entitled “Design, Evaluation, and Use of a Reverberation 1 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Cham ber for Performing Electromagnetic Susceptibility/Vulnerability Measurements,” is currently perhaps the m ost commonly referenced guide to standard (m echanically stirred) microwave reverberation cham ber construction and evaluation. Three significant chamber quantities that are defined and described in this article are: (1) M ode Density: A form ula for the total num ber of m odes that can propagate “inside an unperturbed, lossless, rectangular cham ber” [4] at a particular frequency, f, is approximately given by Eq. 1.1. ( 1. 1) where a,b,d are rectangular chamber dimensions, c is the speed o f light, and f is the chamber source frequency under consideration. For an overmoded cavity, where the source wavelength is small in relation to the chamber dimensions, N(f) can be further approximated by the first term in Eq. 1.1. The derivative o f this term with respect to frequency is known as the mode density and can be expressed by Eq. 1.2. dN _ 8jtabd 2 d f~ ( 1.2) c3 This term can be evaluated at various frequencies within the range o f interest and considered with other significant cham ber quantities to determ ine the effectiveness o f a reverberation chamber. In other words, the lower frequency lim it o f a particular cham ber is typically associated with insufficient m ode density. (2) Composite Quality Factor: Eq. 1.3 is recom m ended for purposes of calculating the com posite (theoretical) Q o f the chamber. 2 C Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. where V is the cham ber volum e, S is the internal surface area o f the cham ber, 8 is the skin depth, X is the wavelength, and a,b,d are cham ber dim ensions. This com posite Q was originally developed in [3] and can be viewed as an average o f the Q ’s corresponding to the resonant modes about a particular frequency. Some o f the effects of lowering the Q by adding absorbing m aterial are reported to be (i) “decreases the effectiveness o f tuner” [4, p. 12], (ii) “increases the cham ber loss and hence increases the r f pow er required to obtain test fields o f the sam e level” [4, p. 12], (iii) “decreases the spatial statistical E-field uniformity” [4, p. 12], (3) Minimum Tuning R atio: “A reasonable guideline for proper operation o f the tuner is a m inim um tuning ratio o f 20dB ” [4, p. 7]. In other words, the ratio o f the maximum power to the m inim um pow er (at a selected test point and over all tuner positions) should be at least 20dB. Apparently, im plem entation of this criteria (though not theoretically derived) assures acceptable levels of field uniformity, or spatial hom ogeneity, for a large number o f applications. Careful consideration o f the discussion in this report seem s to indicate a trend tow ards applying a set o f very broad em pirically based guidelines towards the operation and analysis o f m icrowave reverberation chambers. In fact, the m ost concise set o f guidelines 3 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. available for the design o f these cham bers (to the author’s knowledge) is as follows ([3], also repeated in [4]): (1) “The volum e abd o f the cham ber should be as large as possible in order to have large values o f N(f) for stirring or tuning purposes.” [3, p. 20] (2) The ratios a2 : b2 : c 2 o f edge lengths squared should not be too rational in order to reduce fluctuations in N(f) (avoid m ode degeneracy) “ and, hence, to increase the uniformity in mode distribution.” [3, p. 20] While these relatively straightforward guidelines for choosing cham ber dim ensions [34] and evaluating proper cham ber operation (in the frequency range o f interest) are available, theoretical m odels for the fields inside the chamber are more difficult to obtain. These difficulties are prim arily due to the fact that, in order to achieve the “m ode stirring” required for field uniformity, a boundary is perturbed with an irregularly shaped m echanical tuner. Other m odeling problem s such as determ ining accurate wall conductivities and additional geom etrical constraints due to small excitation wavelength (or multi-mode excitation) make the possibility of obtaining a closed-form deterministic solution unrealistic. Thus, attempts to model these electromagnetic fields have lead to calculations o f [5-7] statistical models for field quantities under a variety o f idealized assumptions. Further investigations with regard to the validity o f these m odels might provide additional insight about the general operation of a m icrowave reverberation cham ber and could lead to the definition o f useful test parameters. 4 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. This dissertation includes an evaluation o f the probability density function (PDF) m odel for the power density under certain limiting, non-ideal conditions as well as the derivation of a new transfer function model for the gain of a microwave reverberation chamber. The evaluation process is perform ed with an elaborate m easurem ent apparatus and via numerical simulations. The deviation o f the chamber measurements from the corresponding theoretical models is considered as a function o f some critical param eters such as chamber quality factor, electrical size, and m ode density. The primary focus of this study is the investigation o f a possible second-order statistical model for the fields inside the chamber. The mathematical form o f this model is extracted from a m ore general theory of “com plex cavities” and is termed the spatial correlation function (SCF) of the pow er density. The study is conducted by making actual SCF measurements and developing a computer SCF experim ent (simulation). The presentation of the m aterial is initiated in Chapter II with a detailed discussion of some relevant mathematical models. The first section o f this chapter outlines a more elaborate analysis o f cham ber Q calculations [9] that includes adjustm ents due to power liberated to any receiving sensors and antennas that may be included in a chamber m easurem ent configuration. The next section of Chapter II describes an approximate method for analyzing the gain versus frequency of a m icrowave reverberation chamber [10]. This method utilizes the overall Q models of the previous section. The third section o f this chapter describes an ideal first-order statistical model for the power density in a microwave reverberation, or mode-stirred, cham ber (MSC) while the fourth section 5 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. provides a concise sum m ary o f Lehm an’s [7] statistical theory o f “com plex cavities.” This fourth section includes a sketch o f the derivation of an ideal spatial correlation function in a “complex cavity.” As mentioned above, the applicability o f this type o f function to M SC ’s is the prim ary topic under consideration in this dissertation. The fifth section o f this chapter is a description and tabulation o f the various cham ber param eters that are to be considered in the subsequent analysis o f cham ber simulations and measurements. Chapter i n outlines the approach that is applied towards obtaining a statistical simulation o f an MSC. This approach, which uses a “m oving wall” [8 ] [12] to model the paddle wheel, incorporates a number of param eters such as chamber excitation frequency, size, and Q models. Chapter IV describes the entire m icrowave reverberation chamber m easurement system that is used in this study. Some details with regard to the significant mechanical design considerations along with the selected electrical properties o f the cham ber are included. Also, a block diagram o f the m easurement apparatus along with a description of the electronic control m echanisms are provided. The final chapter includes a set o f m easurements and simulation outputs. The measurements and sim ulations are analyzed in a variety of ways including via comparison with the appropriate theoretical model. Conclusions with regard to properties of the average cham ber pow er gain, the PD F of the pow er density, and the spatial correlation function o f the pow er density are presented. Emphasis is placed on observing trends in these functions due to variations, over preselected ranges, in a number o f critical chamber parameters. 6 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. CH A PTER II C HA M B ER PARAM ETERS A N D THEO RETICA L H E L D M ODELS Cham ber Quality Factor A significant elaboration of the theoretical Q analysis that is described in the introduction is contained in Phillips Laboratory Technical Report 91-1036 [9]. Here, an equation that accounts for the loading o f the receiving antennas and sensors is postulated based on an energy distribution argument. This overall Q is specified as a net Q and given by Eq. 2.1. net 1 1 ■+ N -------1 +N .. Q eqv a Q ^ a n t* (2 . 1) 1 b Q- where Q„,v is the com posite Q o f the cham ber (Eq. 1.3), Qant is the Q of a standard receiving antenna, Q g is the Q o f a standard B -dot probe, and Na ,Nb are the num ber o f standard antennas and B -dot probes, respectively. An expression for Qant is derived [9] from an expression for the average effective area of a receiving antenna with an incident signal that is randomly polarized [17]. This expression is given by Eq. 2.2. 7 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. where A is the average effective area, A, is the microwave wavelength, and D is the average directivity o f the antenna. Ideally, the average directivity o f an antenna in a m icrowave reverberation cham ber is unity [9] since a high degree o f statistical homogeneity o f the field can be assumed when the cham ber size is (ideally) much larger than the m icrowave source wavelength. W ith this assumption, the average power delivered to the antenna, P<j, can be calculated as follows: (2.3) where W is the average energy stored in the chamber, V is the volum e o f the chamber, c is the speed o f light and the quantity in the second parenthesis is the average pow er density in the chamber. This equation (Eq. 2.3) can be manipulated to yield an expression for Qant * (2.4) where co is the microwave radian frequency and the relation X = 2nc has been used. co A sim ilar approach is also applied in the report [9] to derive an expression for Q 6 . However, this derivation is not outlined here since the m easurem ents in this particular study are conducted with antennas and D-dot sensors. The final section of this report includes an analysis o f these Qnet models in comparison with a set o f m easured Q values. 8 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. The results indicate that the Qnct calculations match experimentally measured values significantly more closely than the QeqVmodel o f Eq. 1.3. Finally, an expression for the Q o f a wall-mounted D-dot sensor, denoted , can be derived by applying the above-mentioned approach to a first-order model for the operation o f a D -dot sensor [10]. This model [11] relates the voltage at the sensor output terminals to the electric field at the cham ber wall as follows: v . = R A e coE D o n (2.5) where v r) is the average o f the m agnitude o f the sensor output voltage, E n is the average of the m agnitude o f the normal electric field at the chamber wall, R is the sensor load resistance, A is the sensor equivalent area, and e 0 is the permittivity of free space. Also, a relationship between the average normal field magnitude at a cham ber wall and the average energy density in the cham ber is derived in [5] and is presented here as Eq. 2.6: £ E 2 =— o n 3 (2.6) where U is the average energy density in the chamber. These two expressions (Eq. 2.5 and Eq. 2.6) can be combined to obtain the following relationship for Q 6 : ooW coUVR Pd v j )2 3 V (2.7) 2 R A 2 e o co Eq. 2.1 can be modified to accom m odate D-dot sensors as follows: ( 2 .8 ) 9 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. where Nd is the num ber o f D -dot sensors. Transfer Function M odels A param eter that provides a considerable am ount o f insight with regard to the overall operation of a cham ber is the pow er gain, as a function o f frequency, between a transmitting antenna inside the cham ber and a receiving antenna/sensor inside the chamber. In this section, two first-order theoretical m odels for the chamber gain versus frequency are derived with the aide o f the Q„e,m odels o f the previous section. A calculation for the pow er transfer characteristic, or gain, of a chamber with a receiving antenna can be initiated from the definition o f the gain in Eq. 2.9. G P. =— ant p o (2.9) y ’ where Gant is the gain o f the cham ber with receiving antenna, P 0 is the power delivered to the cham ber from the transm itting antenna, and Pd is the power available to the receiving antenna from the chamber. Next, Eq. 2.9 can be manipulated as follows: (2.10) where co is the m icrowave radian frequency, W is the average energy stored in the chamber, Qnet is the overall Q o f the cham ber, and Q ^t is the contribution to the overall Q due to the receiving antenna. 10 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. The parallel loading effect o f the chamber walls and the antenna can be observed by evaluating Eq. 2.8 with Na= l, Nb=0, and Nd=0 (Eq. 2.11). 1 1 ^net ^ant (2 -ID ^eqv where Q ^v is the contribution to the overall Q due to the walls. Substitution of Eq. 2.11 into Eq. 2.10 yields the following simplified expression for the gain. f „ G ant i i r 1 1---------- , Q ant* Q eqv , 1 =-2-------------- -2 -^!— = ----o ant Q (2.12) * j_^___ a n t. Q ^eqv An ideal expression for QcqVcan be obtained by letting X « Q eqv =co W 3 V = --------P 2 li S5 eqv ^r a,b,d in Eq. 1.3. (2.13) where PeqVis the total pow er lost to the cham ber walls, V is the volume o f the chamber, |i r is the relative permeability o f the cham ber walls, 8 is the skin depth o f the cham ber walls, and S is the surface area of the cham ber walls. T his limiting case corresponds to the case of a highly “overmoded” cavity where the source wavelength is infinitesimally small in relation to the chamber dim ensions and is an approximation that is frequently applied in the analysis of microwave reverberation cham bers [4], 11 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Substitution o f the expression for QMt (Eq. 2.4) and Eq. 2.13 for QcqVinto Eq. 2.12 yields the desired expression for the gain o f the cham ber with a receiving antenna: °a n r <2-14> . lair1 ' „------- 1+i p i4 -a > 2-5 3 y c Jt where the relations ^ = a°d lt = g 0 ltr have been applied, with |i 0 the permeability o f free space and o the conductivity o f the cham ber walls. A calculation for the gain o f a cham ber with a receiving D-dot sensor can be perform ed in the same m anner as the previous gain calculation. The initial steps are identical, and Eq. 2.12 is modified as follows: Gc r — 5 7 i+ — y Q eqv (2 I5 ) where G ^ is the gain of the cham ber with receiving D -dot sensor. T he desired gain expression is obtained by substituting Eq. 2.7 for and Eq. 2.13 for Qcqv into Eq. 2.15 to yield: G ■ = ------ p = ^ _ —-----------------, 2^ r S -is 1 + J — ------- 9— ® (2.16) p 0° RA 2eo where the relations 5 = ,/—^ — and u ^ u ^ n have been applied. \c o p a 0 r 12 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. T he significance o f these relationships is discussed in detail in the results and conclusions chapter (Chapter V) where two sets o f gain m easurem ents are presented alongside plots o f the theoretical gain curves derived in this section. Probability Density Functions As m entioned in the introduction, investigations into possible statistical models for the fields in a reverberation chamber are motivated by the difficulties associated with finding closed-form determ inistic field solutions for chamber geometries. The statistical models that are presented in [ 6 ], entitled “ Statistical M odel for a M ode-Stirred Cham ber,” are representative o f current statistical treatments for the analysis o f reverberation chambers. Here, the PDF for the pow er density, at a point in the chamber, is given by the exponential distribution (chi square with two degrees o f freedom) in Eq. 2.17. f(p) = A e ' p/2(j2 2a (2.17) where p is the pow er density for one field com ponent (Ex, Ey, or E2) and is therefore proportional to the received pow er for m ost (linearly polarized) sensor/antenna configurations. In this treatment, the two possible polarization’s o f each field com ponent seem to be referred to as “in-phase and quadrature com ponents o f the field in each dim ension” [6 ]. This leads to the definition o f six “field com ponents” : components in three orthogonal directions each with an in-phase and quadrature com ponent cr, in Eq. 2.17, is defined as “the variance o f each of the six field com ponents” [6 ]. PD F’s for the 13 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. magnitude of the total field and for one dimension o f the total field are represented as chi distributions with six degrees of freedom and chi distributions with two degrees o f freedom, respectively. All three o f these PD F’s are “derived” based on an heuristic argument that is initiated by observing that, when a large num ber o f m odes are present in the chamber, each o f the six field components can be thought o f as a sum o f a large number of random variables, where the random nature o f the sum m ing elem ents (modes) is introduced by the paddle wheel. Then, by the central limit theorem , each o f these six components should be Gaussian. W hen points away from the wall are considered, a “reasonable assum ption” [6 ] is to consider these field com ponents to be independent and identically distributed. The forms of the PD F’s are specified by applying existing statistical theories for sums o f squared-Gaussian random variables. Histograms o f power density measurements from a highly overmoded (or electrically large) cavity along with goodness-of-fit-test results are also provided in [6 ]. The distributions o f this measured data are in close agreem ent with the theoretical exponential distribution of Eq. 2.17. Similar histograms are presented in chapter V for the case o f a chamber system with a range of excitation frequencies such that, in the low er end of the frequency spectrum, the chamber is electrically small. Also, histograms for measurements taken on a cham ber wall are presented. For these particular applications (i.e., PDF measurements), the pow er m easurem ents are processed in log m agnitude (or dB) and it is convenient, for comparison purposes, to perform the following change of variables on the PDF o f Eq. 2.17. 14 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. P = 101og(p) (2.18) This change of variables leads to the following distribution for the power (power density) in dB [7]. f p (P) = ( l/p /> e - z where z = e (2.19) , p ' = (10/ln 10)=4.343, and P is the average measured power in dB. The standard deviation for this distribution is always 5.57 dB [7]. Above the mean, the form o f this curve (Eq. 2.19) is such that the distribution falls towards zero rapidly, form ing a theoretical “c u t-o ff’ for the power m easurem ents. Correlation Functions The possibility o f extending the current statistical theory for the fields in the chamber to include a second-order statistical m odel is addressed in this section. As is the case in many practical statistical applications, the investigation of complicated, and perhaps unrealistic, joint PDF models is bypassed with the consideration o f more compact, and perhaps more easily interpretable, correlation function models. The theoretical fram ework for this study is based on Lehm an’s [7] “A Statistical Theory of Electromagnetic Fields in Complex Cavities.” In this eighty-page Phillips Laboratory Interaction Note, Lehm an initiates a lengthy discussion with regard to developing a general theory for the fields in irregularly shaped, or unsymmetrical, cavities. Here, Lehm an’s treatm ent o f the first-order statistics o f com plex cavities can be 15 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. inteipreted as a generalized and detailed m athem atical expression o f the heuristic treatm ent for reverberation cham bers in [6]. The first half o f this work [7] is dedicated to finding a determ inistic solution for the Efield in an arbitrarily-shaped cavity. This solution is deem ed unsuitable for computations and is used to find volum e averages that are shown to be equivalent to expectation operations in the second half o f the paper. T he form o f this solution, for the electric field, is known as an eigenvector, or m odal, expansion and is derived from an expansion solution to an inhom ogenous w ave equation for the m agnetic vector potential. The excitation is a filam entary source and a damping term (to m odel wall losses) is also incorporated into the wave equation. A m ajor portion o f this half o f the treatm ent is dedicated towards specifying the eigenvectors that are contained in the m odal expansion. Here, a seemingly novel approach is applied in the sense that an arbitrarily-shaped cavity is split into a large number, N, o f rectangular elem ents that are individually enlarged and defined as “virtual cavities.” The dim ensions o f these virtual rectangular cavities are chosen by extending each boundary of the corresponding cavity elem ent such that the extended boundary intersects a surface of the original arbitrarily shaped cavity. Thus, the dim ensions o f N virtual cavities that are derived from N cavity elements are specified. This formulation allows the real com plex cavity eigenvectors to be expressed as a superposition of known virtual cavity eigenvectors. These resultant eigenvectors can be used in the m odal expansion solution provided that an orthogonality condition is satisfied. This required orthogonality condition is, in turn, satisfied if Eq. 2.20 is satisfied. 16 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 2 Jco s[K n ( r - r n )]dV = 0 (2.20) n=l V where N is the num ber o f cavity elements, in is the position vector to a selected com er o f the nth virtual cavity, K , is the wavevector for the nth virtual cavity, and V is the com plex cavity volume. This equation is Lehm an’s mathematical definition o f a com plex cavity and can be physically interpreted as follows: If the shape o f the cavity is sm ooth compared to a wavelength, the phase factor in Eq. 2.20 “will be slowly varying because the position vectors to the com ers o f the virtual cavities will all be highly correlated” [7, p. 20] and the left side o f the equation will be large. Similarly, “if the shape o f the cavity is very irregular compared to a wavelength” [7, p. 20], the left side o f Eq. 2.20 will be small. With regard to the general applicability of this criterion, Lehman states that “it will not be possible to a priori validate the complex cavity assumption for every cavity o f interest. The only practical way to proceed is to assume com plexity based on experience with other cavities and then use experim ental data to justify the assum ption” [7, p. 21]. However, with regard to the present development, this criterion is repeatedly applied during the calculation o f “the volum e averages o f all the powers of a single com ponent of the eigenvectors” [7, p. 23]. These calculations, along with the “ volume averages of products o f different eigenvector com ponents” [7, p. 28], are used in the developm ent o f the statistical models. The PD F’s for the field variables are calculated by treating r, the position vector, as a uniformly distributed random variable. This “postulate” leads to the equivalence of 17 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. volume averaging and statistical expectation. Also, though not directly stated, the distributions o f the eigenvector com ponents can be assum ed to be Gaussian since the large number of virtual cavity eigenvectors are a function o f the “random ” position vector and the actual eigenvector com ponents are a sum o f a large num ber o f virtual cavity eigenvector components. The developm ent proceeds in a relatively straightforward manner by building upon these basic results. The following ideal assumptions, in addition to the complex cavity requirement, are em bedded in the calculations: (1) Equal Energy Assumption: In a 3 dB bandwidth about a particular excitation frequency, the m odal Q ’s are approximately equal to the average Q o f the cavity and the modal frequencies are approxim ately equal to the excitation frequency. (2) The num ber o f m odes within a 3 dB bandwidth is large. The resulting PD F’s, from these com plex cavity calculations, are identical to the PD F’s derived in [6 ] for reverberation chambers. Clearly, the most significant simplifying assumption that is em bedded in the mathematical derivation of these P D F ’s is assumption (2) in the previous paragraph. This assumption is applied rather early in the treatment and is implemented by defining a variable, denoted as M, for the num ber o f m odes in a 3 dB bandwidth. An approximate expression for M can be derived from the m ode density form ula of Eq. 1.2 and is given by Eq. 2.21. M = -JtyVT q " (2'21) 18 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. where Q is the average o f the m odal Q ’s about a particular excitation frequency, to. The actual limiting process is applied during the derivation o f the PDF for the am plitude o f a com ponent of the “partial fields” where the term partial field is used to denote two m utually orthogonal electric field solutions each corresponding to a different polarization. This PDF calculation is a preliminary calculation that facilitates the straightforward derivation o f the rem ainder o f the PD F’s and correlation functions and allows the (deterministic) infinite modal expansions to be converted to the following statistical form: E us(r,t) = X us sin(cot) E ws(r,t) = X ws sin(cot) (2.22) where u and w denote polarizations and s represents a Cartesian com ponent (x,y, or z). Here, a fairly intricate m athematical procedure is applied to show that the characteristic function for all the X ’s is in the form o f Eq. 2.23. <i>x(u)~[l + | 3 V r M/2 (2.23) where |3 is the standard deviation o f a Gaussian eigenvector com ponent, u ks (r), and u is the “dum m y” variable in the transformed domain of the characteristic function. A form of the determ inistic solution (in terms o f eigenvector expansions) [7] for the partial fields is included here, in Eq. 2.24, for completeness. k(l+M /2) E us(r,t) = a 0 I u k,x( r 0 ) i k,s(r)sin(cot) (2.24) k'=k(l-M /2) where k=oVc is the wavenumber, r 0 is a position vector that specifies the location of a filamentary source oriented along the x-axis and a 0 is a constant that depends on param eters such as the length o f the filam entary source, the am plitude o f the source 19 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. current, the excitation (source) frequency, and the cavity Q. As m entioned above, u ^ r ) can also be expressed (in general) as the sum o f a large num ber of (virtual cavity) eigenvectors and is therefore assum ed Gaussian. (The solution for the w-polarization is identical to Eq. 2.24.) T he result of letting M go to infinity in Eq. 2.23 leads to the follow ing ideal form for the characteristic function o f the X ’s. --M 0V < M u )~ e 2 (2.25) The PDF for the X ’s is obtained by incorporating the constant, a 0 , into Eq. 2.25 and then taking the inverse Fourier transform to yield: f (x) = - = = L ----- - e 2Ma°p< xv (2.26) V2itMaop Thus, Lehm an has applied a rather elegant m athem atical procedure to show that the ideal PD F o f the partial field am plitudes are Gaussian. He then proceeds with a large sequence o f short, but detailed, calculations to derive expressions for the PD F’s and correlation functions for the field variables of interest. Two sets o f correlation functions are calculated for all the field variables. The first type o f correlation is denoted as spatial correlation and [7] “results because the fields are continuous functions o f position.” The second type o f correlation is denoted as temporal correlation and [7] “ will occur when the bandw idths o f two driving frequencies, say co and c o ', overlap.” These correlation functions are calculated for the [7, p. 1] “am plitude o f the com ponents o f the partial cavity fields, the magnitude o f a com ponent o f the tim e 20 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. averaged electric field, the square o f a com ponent o f the tim e average field, the time averaged com ponent o f the pow er density and the total energy density.” Inspection o f these functions indicates that the concept o f spatial correlation o f a com ponent of the power density is readily applicable to microwave reverberation chambers. A sketch o f the derivation of the spatial correlation function (SCF) for a com ponent of the pow er density can be initiated by noting the fact that the spatial correlation coefficient for the eigenvector com ponents is given by Eq. 2.27. r / x m sinfklrj - r , | ) K [u ks ( r ,), Uks ( r 2 )] = ^ ™ = K s( r , , r 2 ) (2.27) where k is the wavenum ber as a function o f frequency and ri , 12 are position vectors. This coefficient is derived by applying the standard definition o f a correlation coefficient, evaluating the expectation operators in this definition with the formulas for the volume averages that are derived in the first part o f the paper [7], and then sim plifying the result by applying the com plex cavity definition in Eq. 2.20. The next critical step towards obtaining the SCF o f the pow er density is the calculation o f the SCF for a com ponent of the partial fields. This is accomplished by once again using the definition o f the correlation coefficient, expanding the Xs(r,) and Xs(r2) terms with eigenvectors, applying the appropriate volum e average form ulas, and then substituting Eq. 2.27. The result turns out to be identical to the result in Eq. 2.27. 4 X u s ( l i ) X us( r 2) ] = K s (r i , r 2) k [ X ws ( t j ), X ws ( r 2 )] = K s ( r , , r 2 ) (2.28) 21 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. The desired result for the SCF of a com ponent o f the time averaged power density is obtained by observing the definition in Eq. 2.29 and further evaluating the individual expectation operators on the right-hand side of Eq. 2.30 by substituting Eq. 2.28 and the appropriate param eters (mean and variance) from the exponential distribution for a com ponent of the power density. {E . 2( l ,) E ! 2 ( r 2)> = i ( x „ ! (£ l) X „ ! (r 2 ) ) + i ( x „ 2 ( i 1)X „ . 2 (r! )) + (2.30) i(X „ .2(r,)X„2(r2)) + i { x wl2(r,)Xw.2(r2)) The final form o f this result is expressed in Eq. 2.31. (2.31) In the concluding section o f this paper, Lehm an’s com m ents with regard to the application of these results to reverberation cham bers are as follows [7, p. 65]: Although the statistical model developed herein is based on the assum ption that the position vector r in the cavity is a random variable, it is easy to dem onstrate that these statistical m odels also apply to the fields in mode-stirred chambers. The probability density functions and the correlation functions are shown to be independent of the shape of the cavity as long it satisfies the definition o f a complex cavity. Therefore, the statistical model is valid for the set o f all complex cavities with constant volume V and constant Q. All o f the cavities belonging to this set o f constant V and constant Q com plex shaped cavities is called an ensemble and the volum e averages are replaced by ensemble averages. For each stirrer position, a mode-stirred chamber represents a different shaped complex cavity having the same volume and Q as any other stirrer position. Therefore, m easurem ents performed at different stirrer positions correspond to measurem ents performed for 22 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. different members o f the ensem ble o f cavities. As a result, averages over stirrer positions are equivalent to ensemble averages which in turn are equivalent to volum e averages and the statistical m odels are applicable to the analysis o f mode-stirred cham ber test data. Closing discussions with regard to the com plex cavity, equal energy, and large m ode assum ptions are also included in the paper [7] with the general conclusion being that the relative validity of these assumptions is unresolved and requires further study. A m ajor segment o f chapter V o f this dissertation (Results and Conclusions) is dedicated to the analysis o f the applicability o f this SCF function to m icrowave reverberation chambers. Selected sets o f em pirical and com puter simulation outputs are included. Chapter ID provides a description of the simulation algorithm that is developed and implemented for the computer simulation o f an SCF experim ent in a m icrowave reverberation chamber. Chapter IV provides a detailed description of the measurement apparatus that is designed and implemented for purposes o f measuring the spatial correlation o f the average power density in an actual microwave reverberation chamber. 23 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Sum m ary of Critical Parameters A brief description o f seven critical param eters that effect the electrom agnetic response o f any standard m icrow ave reverberation cham ber is presented in this section. The objective is to introduce the reader to the significance o f these param eters im mediately following the presentation o f the theoretical m odels o f in tere st This type of approach should give the reader an increased awareness of the dom inant them e in this dissertation o f investigating the effects o f varying cham ber param eters on the applicability of a few relatively simple, idealized, m athematical (statistical) models. The param eters are described individually in the following numbered segments. (1) Excitation Frequency, f : The frequency o f the m icrow ave source is critical in the sense that large source w avelengths, in relation to the cham ber dim ensions, generate small mode densities and vice versa. In this study, the phrase “ideal cham ber operation” implies that the source w avelengths are small in relation to the cham ber dimensions. (2) Cham ber Size: The cham ber size, including the volum e and the surface area, is related to the quality factor, Q, and the Q is, in turn, directly related to the num ber of modes in the 3-dB bandw idth, M. (3) C ham ber Quality Factor. O : A t the present time, the Q is perhaps the m ost frequently discussed, m easured, and theoretically studied param eter o f a m icrowave reverberation chamber. H ow ever, w hile the Q net m odels are m ore accurate than the previous Qa,v models, there do not seem to be any sets o f docum ented m easurem ents available that do not deviate from the corresponding theoretical calculations by less than a factor o f two 24 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. [9]. M ost studies, for exam ple [4] and [6], report m easured Q ’s that are low er than values from the given theoretical m odels by a factor of 10 or more. (4) N um ber o f M odes in 3-dB Bandwidth. M : This param eter is one of the m ost significant param eters that is embedded in Lehm an’s theory o f complex cavities. As described in chapter V, a new approach for m easuring the average cham ber Q by applying the gain m odels that are derived in the first section o f this chapter is investigated in this study. These Q m easurem ents are used to determ ine M values, which are im portant with regard to the accuracy o f the PD F and SCF m odels that are presented in this chapter. (5) Paddle W heel Size: This param eter is not investigated in detail in this particular study. W u and Chang, in [18], have examined a hypothetical [18, p. 164] “2-D cavity with a 1-D perturbing body.” T he results from their sim ulations indicate that the dimension o f the perturbing body should be [18, p. 169] “ of the order o f two wavelengths or longer.” For the m easurem ents that are included in this dissertation, the effectiveness of the paddle wheel is evaluated from an analysis o f chamber gain measurements (Chapter V). ( 6) Number o f Paddle W heel Positions: As mentioned in the introduction, a typical microwave reverberation cham ber experim ent is usually designed to accommodate approximately 200 increm entally spaced paddle wheel positions. The effect of selectively excluding som e of these samples in a spatial correlation m easurem ent is documented in Chapter V. (7) Sensor and Antenna Critical Frequencies: Sensor and antenna critical frequencies occur due to electrom agnetic energy being stored in stray distributed capacitances (and 25 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. sometimes inductances) between structures within and in the vicinity o f the sensor or antenna. These frequencies affect the net, or overall, Q o f the cham ber and in certain extreme cases can effect PDF as well as SCF measurements. A discussion o f these effects is included in Chapter V. 26 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER HI SIM ULATION M ETHODOLOGY A com puter simulation o f a spatial correlation function (SCF) m easurement inside a microwave reverberation chamber is developed by modeling the cham ber as a rectangular cavity with one moving, or “perturbed,” wall and no paddle wheel. In this approach, field m easurements inside the chamber are simulated with values calculated from field equations for a set o f rectangular cavities with heights that vary in equally spaced increments from the height o f the paddle wheel to the height of the top o f the actual cham ber o f interest. Before proceeding with im plementing an algorithm based on the above-mentioned approach, a num ber o f authorities, including those cited in [14] and [15], have been consulted with regard to the possibility o f applying a standard numerical technique to this specific simulation problem. The general consensus is that the finite-difference timedomain (FD-TD) method [16], based on relatively straightforward second-order centraldifference approxim ations for the space and time derivatives in M axw ell’s equations, is m ore suited to this application than the m ore traditional ffequency-dom ain integral equation approaches. However, Y. Rahm at-Sam ii [15], a recognized authority on com putational electrom agnetics, pointed out, during a detailed discussion, that while the chamber under consideration may be considered electrically small for reverberation cham ber applications, it is considered electrically large for simulation applications. His com m ents referred to the fact that, while detailed FD-TD sim ulations have been 27 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. successfully conducted with m ultim ode cavities [13], a successfully completed detailed simulation (including paddle wheel and antennas) of a reverberation-chamber-sized apparatus has not been done. H e w as generally not optim istic about the possibility o f obtaining a accurate sim ulation (using FD-TD) o f the cham ber with reasonable execution times and reasonable com puter storage requirements. In another consultation [14], K.C. Chen , a authority on m icrowave reverberation chambers, expressed the view that sim ulation techniques such as FD-TD and the frequency domain m ethods would not lead to useful results in this application due to the fact that a successful simulation, after a lengthy developm ent process, w ould not yield a simulation that could be easily linked to cham ber parameters. After considering the com m ents provided by the consulted authorities and evaluating the results o f a literature search, a viable approach to the sim ulation is obtained by modifying the basic approach in [8 ], entitled “ An Investigation o f the Electromagnetic Field inside a M oving-W all M ode-Stirred Cham ber.” This technique, in [8] and [11], is based on obtaining the G reen’s function solution for a rectangular cavity and then repeatedly applying this solution to a cavity with a perturbed boundary. In other words, the location of an entire wall o f a rectangular cavity is perturbed in order to simulate a “m ode-stirred” cham ber response. T he direct application of this m ethod to the present simulation problem is not necessarily preferable since im portant cham ber parameters, such as the cham ber Q, are not directly em bedded in the G reen’s function approach to the solution for a rectangular cavity. This approach is modified by applying the “moving wall” 28 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. cham ber concept to a “m ode selection” procedure instead o f using the G reen’s function solution. This procedure leads to a simulation that is directly related to cham ber param eters such as Q and the num ber o f m odes in a bandwidth. A skeletal description o f this approach is as follows: —>A large num ber o f resonant frequencies for each rectangular cavity associated with a “moving wall” chamber is calculated. —>The frequency response of each resonant mode is approxim ated by a simple secondorder curve that is derived directly ffom Qnet. The shape of this curve is sim ilar to the frequency response characteristic of a high-Q narrow-band filter with the resonant frequency being analogous to the filter center frequency. —» The second-order curve is used to determ ine which m odes are significant at a frequency of interest —> A statistical field sample is obtained by summing the fields due to all significant modes at the frequency o f interest. This “ moving wall” algorithm is implemented with certain pre-processing steps in order to shorten overall execution times. The pre-processing steps involve the calculation, sorting, and archiving o f a large num ber o f resonant frequencies for all o f the rectangular cavities that are to be considered in the calculations. In this case, a simulation o f a chamber experim ent with 200 distinct paddle wheel positions is desired, and resonant frequency arrays are calculated, using Eq. 3.1, for 200 rectangular cavities. 29 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. where c = speed o f light, m, n, and p are integers, and a, b, and d are cavity dimensions. The heights o f the cavities are separated by increments o f (dmax - d,nin)/200, where dmin is the minimum cavity simulation height (paddle wheel height) and dmax is the maximum simulation height (height o f cham ber to be simulated). For a given height, d, a resonant frequency array is evaluated by varying (m,n,p) from ( 1, 1,0 ) to some “acceptably large” numbers. The “acceptably large” final values o f (m,n,p) are determined from an analysis of a mode density formula [2] combined with consideration of the desired simulation frequency range. (The number o f resonant modes in a particular frequency interval converges to a finite number as m, n, and p become sufficiently large.) The initial values of n and m are unity, instead o f zero, since in this particular study the wall-mounted Ddot sensors are designed to detect the z-component o f the field which, from an analysis of the field equations [3], is identically zero when n or m is zero. The simulation program processes the sorted resonant frequency data generated by the pre-processing program. It first determines a set of significant resonances, or modes, at a particular simulation frequency for a specific cavity height. These modes are selected by modeling the frequency response o f each cavity resonance, in the neighborhood of the desired simulation frequency, with a second-order curve, Eq. 3.2, that is analogous to the transfer function of an ideal RLC circuit 30 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. where cq, = a cavity m odal resonance in radians/s and Q net = overall Q o f the chamber.' The m agnitudes o f these second-order responses at the sim ulation frequency are used as a measure of the relative significance o f each mode. Specifically, the program scans the appropriate resonant frequency array for the resonance that is closest to the simulation frequency. The m agnitude o f the response associated with this nearest resonance is evaluated at the sim ulation frequency and stored as a variable called m ag i. T he responses associated with other neighboring resonances are evaluated successively, at the simulation frequency, in ascending and descending frequency order until resonances are encountered that generate a m agnitude that is less than . 1*mag 1 and the mode selection process is terminated. The results o f this mode selection process are sets o f m, n, and p indices and amplitudes that correspond to m odes that significantly contribute to the simulated chamber response at the chosen simulation frequency. Q nct, the overall theoretical Q of the cham ber with receiving D-dot sensors/sensor, is calculated from Eq. 2.8 and is given by Eq. 3.3 . (3.3) where, as defined in chapter 2 , QgqV , N = num ber o f D-dot sensors, V = volum e o f chamber, |ir = relative permeability o f chamber walls, 5 = skin depth o f chamber walls, S = surface area o f chamber walls, R = sensor load resistance, A 31 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. = sensor equivalent area, and £o = perm ittivity o f free space. The QcqVterm in this equation expresses the loading due to the cham ber walls while the term gives the loading due to a sensor. The sensor loading term is derived, in Chapter II, from an ideal first-order m odel for a D-dot sensor. One objective of this sim ulation is to study the simulated SC F response as Qnct is divided by various scale factors (i.e. Q = Qaet / 5 , Q = Qact / 10 ,...), corresponding to possible additional cham ber losses. The electric field, due to each m np mode, is calculated from the z-com ponent o f the Efield solution for an ideal rectangular cavity (Eq. 3.4) [3]. (3.4) where k k r = ^ k x2 + k y2 + k z2 , and Eo , Ej are am plitude constants in volts per m eter. Here, the first term represents the TE solution whereas the second term represents the TM solution. This equation is evaluated for all o f the selected mnp sets where each resulting term is m ultiplied by the corresponding m agnitude factor from the selection step. The total electric field, at a point on the bottom o f the chamber, is calculated by summing all o f the these individual terms. A pow er density sam ple is obtained by taking the magnitudesquared o f the total field. 32 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. In the sim ulation program , the aforementioned process o f calculating pow er density samples is em bedded inside tw o nested loops. The outer loop is the “ height perturbation loop” where a resonance frequency array corresponding to a particular cavity height is loaded into m em ory from disk storage. The inner loop is the “frequency loop” where a pre-determined initial simulation frequency is incremented by a specified amount. Two power density calculations, for two different spatial locations on the bottom o f the simulated cham ber, are perform ed within the inner loop. This procedure generates two power density samples for each simulation frequency and each rectangular cavity height The pow er density data is stored in two large m em ory segments. O ne segm ent contains all the data for one spatial location, while the other segm ent contains all the data for the other spatial location. A com puter-generated spatial correlation function output is obtained by correlating the data in the two memory segm ents using the usual productm om ent formula. In addition to calculating the SCF for a given spacing and frequency range, the sim ulation program generates a num ber o f outputs that allow for com parisons of this simulation approach to available theoretical and experim ental models for m icrowave reverberation chambers. The implementation of the “ moving w all” algorithm is a set of M atlab m-files that execute on a SUN10 SPA RC workstation. Two main m -file program s perform the actual calculations w hile a set of small m-files, that are called by the main programs, perform auxiliary calculations such as evaluation of the Qnet form ulas (Eq. 3.3), evaluation o f (second-order response) m agnitudes (Eq. 3.2), and generation o f mnp integers. The first 33 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. main program calculates, processes, and stores, on disk, approximately 150 megabytes of resonant frequency data. The second main program processes this data in segments and generates a simulated SCF o utput The most significant array sizes in this program, at any given time during the execution process, are three approximately 1,000,000 elem ent mnp integer arrays, one 50,000 - 75,000 elem ent integer array segm ent of resonant frequency data, one 50,000 - 75,000 elem ent double-precision array segm ent o f resonant frequency data, and two 51 x 200 elem ent double-precision arrays with calculated power density samples. In [8], the simulation output is evaluated by choosing 20 incrementally-spaced moving wall positions, over a wavelength, and then taking 20-sample E-field averages at several different points in the chamber. The algorithm is said to simulate a microwave reverberation chamber response since two of the observation points yield an E-field tuning ratio o f at least 40 dB and a homogeneity o f 2 dB. The results o f perform ing these tests on a sample output from the simulation program that is developed for this study yield sim ilar results: (power) tuning ratios o f well above 20 dB as well as a pow er density homogeneity o f better than 1 dB for the chosen cham ber size and at the highest simulation frequency o f 13.5 Ghz.. Plots o f the maximum, average, and minim um of the power density data from this sample simulation are shown in Figure 3.1. Inspection of this figure indicates that the average curve is generally 5-10 dB below the maximum curve. In an actual microwave reverberation chamber, shifts in the 5-10 dB range between the corresponding measured 34 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Simulated Chamber Responce 0 -20 S' -40 2 co -60 -80 -100 2 4 8 6 10 12 -20 m 2, CVJ -40 „ co -60 -80 -100 Frequency (GHz) Figure 3.1. Plots o f the M inimum, M aximum, and Average of the Power Density, in dB, from a Sample C ham ber Simulation. The two selected points, with power densities SI and S2, are spaced 17.5 cm apart and are located in the vicinity o f the center o f the bottom wall. The dimensions o f the simulated cham ber are 1.0342 m X 0.8087 m X 0.5812 m. T he Q values are from a 2 D-dot sensor Q net model that is divided by 10. 35 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. cham ber attenuation curves can be considered as another indicator o f efficient tuner operation [4]. Figures 3.2 and 3.3 are the two output text files that are generated by this sample run. In Figure 3.2, the colum ns from left to right represent the sim ulation frequency, the total num ber o f inodes processed by the program , the Q values from the chosen Q model, the average num ber o f m odes processed within the 3 dB bandwidth, the theoretical num ber o f modes in the 3 dB bandwidth, and the calculated spatial correlation coefficient. In Figure 3.3, the colum ns from left to right represent the simulation frequency, the minimum contribution to the overall cham ber response from a selected m ode, the average contribution to the overall chamber response from the selected modes, the minimum bandwidth o f the selected modes from the 200 height perturbations, the average bandwidth o f the selected modes from the 200 height perturbations, and the maximum bandwidth o f the selected modes from the 200 height perturbations. All these parameters are used for diagnostic purposes and are inspected periodically for unusual trends. M ost o f the param eters can be applied tow ards com paring simulations to m easurements and theoretical values. A third output data file contains all the calculated pow er density samples in compressed form for future analysis applications such as the plots in Figures 3.1, 3.4, and 3.5. An additional statistical test is perform ed to evaluate the quality o f the simulation output. This test involves tabulating the standard deviation o f the output pow er density samples and plotting them in histogram form. A sam ple test plot is shown in Figure 3.4. 36 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. f 1.000e+009 1.250e+009 1.500e+009 1.750e+009 2.000e+009 2.250e+009 2.500e+009 2.750e+009 3.000e+009 3.250e+009 3.500e+009 3.750e+009 4.000e+009 4.250e+009 4.500e+009 4.750e+009 5.000e+009 5.250e+009 5.500e+009 5.750e+009 6.000e+009 6.250c+009 6.500e+009 6.750e+009 7.000e+009 7.250e+009 7.500e+009 7.750e+009 8.000e+009 8.250e+009 8.500e+009 8.750e+009 9.000e+009 9.250e+009 9.500e+009 9.750e+009 1.000e+010 1.025e+010 1.050e+010 1.075e+010 1.100e+010 1.125e+010 1.150e+010 1.175e+010 1.200e+010 1.225e+010 1.250e+010 1.275e+010 1.300e+010 1.325e+010 1.350e+010 modes Q modes3 N3 K 2741 2092 790 1543 1963 2231 2373 2642 2474 2665 2845 3137 2491 3627 3846 3994 4591 5017 5342 6073 6542 7130 8038 8530 9243 10823 11276 11079 13587 14116 14762 16529 18048 19091 20909 21286 23462 25001 26514 28145 30223 31443 34035 36086 38498 40393 42580 44806 46933 50589 52792 5.759e+003 6.427e+003 7.027e+003 7.574e+003 8.079e+003 8.549e+003 8.989e+003 9.403e+003 9.794e+003 1.016e+004 1.052e+004 1.085e+004 1.117e+004 1.148e+004 1.177e+004 1.206e+004 1.233e+004 1.259e+004 1.283e+004 1.307e+004 1.331e+004 1.353e+004 1.374e+004 1.395e+004 1.415e+004 1.434e+004 1.452e+004 1.470e+004 1.487e+004 1.504e+004 1.520e+004 1.536e+004 1.551e+004 1.565e+004 1.579e+004 1.592e+004 1.605e+004 1.618e+004 1.630e+004 1.641e+004 1.652e+004 1.663e+004 1.674e+004 1.683e+004 1.693e+004 1.702e+004 1.711e+004 1.720e+004 1.728c+004 1.736e+004 1.744e+004 0.040 0.070 0.140 0.220 0.280 0.500 0.530 0.730 0.840 1.120 1.370 1.440 4.360 2.470 2.970 2.880 3.340 4.010 4.370 4.740 5.320 8.230 9.250 7.730 8.540 10.000 10.350 10.820 14.320 13.150 14.230 18.800 17.210 19.040 19.680 20.960 22.990 27.160 25.920 28.150 29.250 32.840 35.890 37.180 40.100 40.100 44.260 43.170 48.680 49.920 52.430 0.079 0.138 0.218 0.321 0.449 0.604 0.788 1.003 1.250 1.531 1.848 2.203 2.597 3.032 3.509 4.031 4.598 5.213 5.878 6.593 7.361 8.183 9.061 9.997 10.993 12.050 13.170 14.355 15.607 16.927 18.318 19.781 21.318 22.931 24.623 26.394 28.248 30.185 32.209 34.321 36.523 38.817 41.206 43.691 46.276 48.961 51.749 54.643 57.645 60.757 63.981 0.741 0.618 0.339 0.610 0.303 0.456 0.409 0.485 0.388 0.445 0.153 0.030 0.171 0.130 0.285 0.065 0.102 0.278 0.039 0.086 0.205 0.012 0.013 0.104 0.000 0.010 0.061 0.070 0.060 -0.021 -0.083 0.155 -0.007 0.090 0.011 0.060 0.039 0.112 0.125 0.036 -0.013 -0.083 -0.025 0.061 0.075 -0.060 -0.052 0.063 -0.029 -0.041 0.130 Figure 3.2. Sam ple Output File K 1 .0 U T from Chamber Simulation Program Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. f magmin 1.000e+09 1.250e+09 1.500e+09 t.750c+09 2.000e+09 2.250e+09 2.500e+09 2.750e+09 3.000e+09 3.250e+09 3.500e+09 3.750e+09 4.000e+09 4.250e+09 4.500e+09 4.750e+09 5.000e+09 5.250e+09 5.500e+09 5.750c+09 6.000e+09 6.250e+09 6.500e+09 6.750e+09 7.000e+09 7.250e+09 7.500e+09 7.750e+09 8.000e+09 8.250e+09 8.500e+09 8.750e+09 9.000e+09 9.250e+09 9.500e+09 9.750e+09 1.000e+10 1.025e+10 I.050e+10 1.075e+10 1.100e+10 1.125e+10 1.150e+10 1.175e+10 1.200e+10 1.225e+10 1.250e+10 1.275e+10 1.300e+10 1.325e+10 1.350e+10 0.008 0.023 0.239 0.130 0.038 0.040 0.088 0.085 0.174 0.108 0.152 0.128 0.997 0.462 0.262 0.239 0.292 0.278 0.382 0.439 0.452 0.917 0.997 0.665 0.624 0.646 0.703 0.716 0.995 0.714 0.872 1.000 0.898 0.961 0.872 0.894 0.896 0.898 0.843 0.935 0.953 0.984 0.972 0.996 0.993 0.972 0.941 0.968 0.995 0.978 0.979 magave minbw 0.098 0.133 0.313 0.293 0.338 0.418 0.456 0.500 0.568 0.617 0.670 0.695 0.997 0.804 0.835 0.835 0.861 0.892 0.900 0.902 0.924 0.963 0.997 0.952 0.960 0.968 0.972 0.977 0.996 0.984 0.985 1.000 0.989 0.992 0.988 0.990 0.994 0.994 0.993 0.995 0.997 0.997 0.998 0.998 0.998 0.998 0.998 0.998 0.999 0.998 0.999 0.000e+00 0.000e+00 1.319e+05 6.600e+05 0.000e+00 0.000e+00 5.945e+05 0.000e+00 8.393e+05 7.603e+05 7.720e+05 1.162e+06 1.751e+06 2.543e+06 2.118e+06 2.157e+06 2.600e+06 3.107e+06 2.646e+06 3.683e+06 3.393e+06 4.155e+06 3.554e+06 4.173e+06 3.879e+06 4.377e+06 4.522e+06 4.598e+06 4.890e+06 4.677e+06 4.977e+06 5.235e+06 5.351e+06 5.376e+06 5.603e+06 5.642e+06 5.736e+06 6.006e+06 6.093e+06 6.234e+06 6.316e+06 6.327e+06 6.466c+06 6.673e+06 6.788e+06 6.925e+06 7.021 e+06 7.121c+06 7.263e+06 7.248e+06 7.521 e+06 avebw 6.318e+07 3.232e+07 4.967e+06 9.519e+06 1.193e+07 1.062e+07 8.533e+06 8.046e+06 6.105e+06 5.989e+06 5.436e+06 5.420e+06 2.916e+06 4.390e+06 4.395e+06 4.498e+06 4.645e+06 4.514e+06 4.535e+06 4.766e+06 4.600e+06 4.523e+06 4.557e+06 4.864e+06 4.944e+06 5.025e+06 5.122e+06 5.165e+06 5.264e+06 5.385e+06 5.507e+06 5.536e+06 5.710e+06 5.809e+06 5.946e+06 6.040e+06 6.148e+06 6.247e+06 6.352e+06 6.457e+06 6.556e+06 6.665e+06 6.773e+06 6.871 e+06 6.999e+06 7.103e+06 7.215e+06 7.328e+06 7.425e+06 7.541e+06 7.653e+06 maxbw 2.179e+08 8.057e+07 8.409e+06 1.747e+07 6.266e+07 6.419e+07 3.105e+07 3.373e+07 1.717e+07 2.920e+07 2.150e+07 2.602e+07 3.540e+06 7.961e+06 1.437e+07 1.628e+07 1.341e+07 1.375e+07 1.094e+07 9.734e+06 9.669e+06 5.004e+06 4.719e+06 6.692e+06 7.876e+06 7.65 le+06 7.247e+06 7.317e+06 5.376e+06 7.383e+06 6.220e+06 5.662e+06 6.343e+06 6.090e+06 6.83 le+06 6.732e+06 6.838e+06 6.978e+06 7.320e+06 6.900e+06 6.927e+06 6.835e+06 6.993e+06 6.958e+06 7.092e+06 7.308e+06 7.680e+06 7.590c+06 7.520e+06 7.748e+06 7.775e+06 Figure 3.3. Sample O utput File K 2.0U T From Chamber Simulation Program 38 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. SIMULATED PDF FOR EM POWER DENSITY Relative Q xurence 0.2 i i i » ’T " i i--------------- r Location 1, Std. Dev. = 10.2 dB 0.15 0.1 0.05 . LL 0 Relative Ocarrence 10 20 ,ll 30 1 11III I lit 40 50 60 |S1| (dB) 70 80 70 80 90 Location 2, Std. Dev. = 10.3 dB 30 40 50 60 |S2| (dB) Figure 3.4. Histograms o f Power Density D ata for a Simulation Frequency of 10 GHz and a Cham ber Size o f 1.0342 m X 0.8087 m X 0.5812 m. The standard deviations are well above the theoretical value of 5.57 dB. 39 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. SIMULATED PDF FOR EM POWER DENSFTY 0.3 Location 1, Std. Dev. = 6.0 dB p O.2 -50 -40 -30 -20 |S1| (dB) -10 0.3 Location 2, Std. Dev. =5.8dB 1 0.2 g>0.1 1 4 0 -50 -40 -30 -20 |S2| (dB) Figure 3.5. Histogram s o f Power Density Data for a Simulation Frequency o f 10 GHz and a Cham ber Size o f 1.0342 m X 0.8087 m X 0.5812 m. T he standard deviations approach the theoretical value of 5.57 dB after adding a random phase term between the TE and TM com ponents o f the E-field. 40 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Observation o f a num ber o f test plots indicates that the spread, or standard deviation, of this data is significantly above the theoretical value o f 5.57 dB (Chapter II). A careful analysis of the basic physical principles that are embedded in this original implementation indicate that the large standard deviations m ay be due to a relatively large number of occurrences o f the TE and TM term s in Eq. 3.4 adding in phase. The inclusion o f a random phase term between these TE and TM term s generates standard deviations that are in the neighborhood o f the theoretical standard deviations for the higher frequencies in the simulated frequency range. For the particular cham ber size that is considered in these sim ulations, these high frequencies correspond to the ideal case of large chamber electrical size. Figure 3.5 is a sam ple histogram that is generated by using data from the final form o f the simulation program . Figure 3.6 show s plots o f sim ulated SCF outputs for two different Q values with a spacing o f 2.5 cm. A selected set o f simulated correlation outputs will be discussed in detail in chapter V. The matlab M -file im plementation o f the simulation m ethodology that is docum ented in this chapter is included in the appendix. 41 with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Sinxiated SCF of EM Power Density 2 4 2 4 6 8 10 12 6 8 10 12 £ 0.5 Frequency (GHz) Figure 3.6. SCF Simulation O utput for Two Different Q Values. The spacing, r, is equal to 2.5 cm. The solid curves are theoretical values from Eq. 2.31 while the dotted curves are plots of the simulation output. 42 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. CHAPTER IV M EASUREM ENT APPARATUS System Specifications A block diagram o f the experim ental apparatus that was designed and constructed for this study is shown in Figure 4.1 [2]. The welded aluminum cham ber dimensions are 1.034 m by 0.809 m by 0.581 m while the microwave source frequency, from a network analyzer, varies from 1 G H z to 13.5 GHz. This frequency range corresponds to a wavelength range of 30 cm to 2.22 cm. Thus, the source wavelength is not negligible in relation to the cham ber dim ensions in the lower end o f the test spectrum. Here the apparatus allows for the accentuated observation o f non-ideal phenom ena to be contrasted with ideal theoretical calculations. A cham ber with these moderate dimensions is often referred to as an “electrically small” chamber. The aluminum alloy used to construct the cham ber is known as 6061 T6 and is found, in [19], to have a conductivity of o = 2.32 x 107 S/m. This param eter is em bedded in the theoretical QDCt formulas and is, in turn, also em bedded in the theoretical gain formulas o f chapter II as well as in the simulation program (appendix). Antenna 1 (Figure 4.1) is connected to the source o f a network analyzer and is denoted the transmitting antenna. This antenna is a log-periodic dipole-array with dipoles that allow for efficient transmission in the 1-18 GHz range. A ntenna 2 (Figure 4.1), which is also a (1-18 GHz) log-periodic dipole array, is denoted as a receiving antenna and is placed in the cham ber w hen it is desirable to 43 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. NETW ORK ANALYZER DIGITAL BUSES COM PUTER /\ MICROWAVE TRANSMISSION LINES MOTOR SENSORS PADDLE WHEEL ANTENNAS R E V E R B E R A T IO N C H A M B E R Figure 4.1. Block Diagram o f the Basic Reverberation C ham ber System that was Designed and Constructed for this Study. A ntenna 1 is the transmitting antenna and is connected to a netw ork analyzer source. T h e other antenna and the two D-dot sensors are selectively placed in the cham ber and connected to the receiving port/ports o f a netw ork analyzer in accordance with type o f m easurem ent that is desired. 44 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. evaluate the cham ber in a standard (documented) test configuration. A significant amount o f effort has been dedicated tow ards gathering and evaluating experim ental data in this (antenna-to-antenna) configuration in order to insure that this system is a functional reverberation cham ber apparatus and is acceptable for further experimental investigations. Some o f this prelim inary data is included in chapter V alongside plots of the theoretical model in Eq. 2.14. One noteworthy characteristic o f this antenna is that it cannot be m odeled as a point sensor since the locations o f the dipoles are at varying distances from the wall and the dimensions o f the larger dipoles are not infinitesimal in relation to the cham ber dim ensions. In other words, the power liberated to the receiving port o f the analyzer is not representative o f the pow er at a “point” in the chamber. In order to study in detail certain statistical characteristics o f the fields in the chamber (e.g. the spatial correlation o f the pow er density between two points), it is necessary to m odify the apparatus by replacing the receiving antenna with either one or two point sensors. This problem has been studied in detail with the decision being to replace the antenna with two ($2400/sensor) wall-m ounted D-dot sensors [4], These sensors measure the “tim e derivative o f the electric displacement, D, norm al to the w all” [2] and liberate an am ount o f pow er to an analyzer receiving port that is proportional to the pow er density present at the sensor tip. In C hapter V, the approxim ate theoretical model o f Eq. 2.16, developed for the cham ber response with receiving sensors, is com pared with a corresponding set of m easured data. Acceptable levels o f agreem ent have been observed and this basic experim ental apparatus is deemed suitable for further study o f the fields in 45 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. the chamber. The data sheet for these sensors, with all the critical dimensions, is shown in Figure 4.2. Alternatives to this wall-mounted D-dot sensor approach to making “point” measurements which em ploy a "free field" ( not wall mounted) probe are prohibitively expensive in term s o f the budgetary constraints for this project. For example, a system that is centered around an EM CO (The Electromechanics Company) M odel 7121 Broadband Probe could be used to measure the SCF in a cham ber if the following items were acquired: (1) One EM CO Broadband Probe - $5845 (2) Tw o or Three Hughes TW T (traveling wave tube) Am plifiers - $10,000 -$15,000 each (3) One Gigatronics Source - $27,450 Items (2) and (3) are necessary due to the sensitivity rating o f the probe which requires the presence o f higher field levels inside the cham ber than can be provided by the sources in the analyzer block o f Figure 4.1. The analyzer block in Figure 4.1 is implemented with either the HP 8753C or the HP 8719A m icrowave network analyzer. The 8753C has a source frequency range o f 300 kH z - 3 GHz and a m axim um output power level of 20 dBm, while the 8719A has a source frequency range of 130 M H z - 13.5 G H z and a m axim um output power level o f 10 dBm. In addition, the 8753C can be physically configured with two receiving ports to accept two D-dot sensor outputs while the 8719A has only one receiving port. Initially, the 8753C was used, along with a set of software control/data-acquisition 46 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. H 0 DEL C l i e t r s e i l I b ic A O -S IO (R ) AO -IO (A ) C qu I v . A r n l A i q l U IB -V F ro o . I b i m a i i tld b p o in t 1 K U e tiO O ( t r i e -1 0 1 > I1 0 H | H IG H , AO~Ct10(A ) Io l0 * * « * A O -1 1 0 (A ) > 2 .SON, L ll-V H .S Q H , A D -B 20(A ) AO-SO(A) A O -S O O (ft)t »IOHz >IOH, moow x (.) ) m < 1 .0 no Ix lO ^ a * IM IB -V ( .H n i < .l ) m t .I Q n o t.Ilm < .) S n i Mb i I m b O u tp u t (h ik t • 1S0V •ISOV • 1KV • IKV ikKV IkKV •IK V O u tp u t C o n n e c to r (F o n o lo l OSSN OSSH BMA SKA SNA BMA T ypo * F ti v c te o l t o a c (I n .) N .2 2 .22 .1 0 .0 1 t.tt 1.11 1 .1 2 0 1 .0 0 1 .0 0 i.a o 2 .1 0 - S . 10 S.SO 1 1 .1 2 L 1 ,1 0 T .M - .M A - .S I •0 0 - C .k i * .0 0 1 .0 0 1 2 .0 .0 0 • 01 .7 ) - .0 0 22.0 - .11 .72 - ■ .1 2 1 .0 2 AXIAL (A] Figure 4.2. D ata Sheet for the Prodyn A D -S 10 ( R ) W all-M ounted D-dot Sensors that are used to M easure the SCF in this Study. (Provided by Prodyn Technologies) 47 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. routines, to m easure the spatial correlation o f the pow er density within the chamber. The 8753C was chosen because o f its ability to accept both sensor outputs. It had the drawback, however, o f limiting the highest frequency of m easurem ent to 3.0 GHz. The values for the SCF were obtained by calculating a correlation coefficient for each analyzer frequency with a set sensor spacing. These calculations are perform ed by the control/data-acquisition software as a post-processing step by using the usual productm om ent estim ate [20] for the correlation in Eq. 4.1. where Pj = N 1 N X P jn n= 1 1 > *2 = ^ N ^ ^ 2n n= 1 ’ *>ln = Power d e n sity at location 1 for paddle wheel position n, P2n = power density at location 2 for paddle wheel position n, and N = total num ber o f paddle wheel positions. After carrying out these prelim inary 8753C measurements, a set of enhancem ents were m ade to the apparatus with the 8719A in the system. These enhancements included the integration o f a m icrowave switch (TA2F31 from DB Products) between the analyzer receiving port and the tw o cham ber output ports, shown in Figure 4.3, along with a com puter interface card designed for this switch. This modification allows for 1 -1 3 .5 G H z two-sensor SCF m easurem ents via the single receiving port on the 8719A. M ajor 48 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. MOTOR ANTENNA PADDLE WHEEL R E V E R B E R A T IO N CHAM BER SENSORS RF LINES SWITCH NETW ORK ANALYZER DIGITAL LINES COM PUTER Figure 4.3. Reverberation C ham ber System that is used with HP8719A M icrow ave Netw ork A nalyzer to m ake SCF M easurem ents from 1 to 13.5 GHz. Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. structural enhancem ents, that are discussed in the following section, included the integration o f a “low-backlash” gearbox that reduces paddle wheel vibrations. The basic m easurem ent param eters that these analyzers m easure are known as Sparam eters (or scattering param eters). Sn is the voltage reflection coefficient at analyzer p o rt 1, S22 is the voltage reflection coefficient at port 2 , S21 is the voltage transmission coefficient from port 1 to port 2 , and S i2 is the voltage transmission coefficient from port 2 to port 1. For reverberation cham ber applications, S 21 (or equivalently S 12) is of prim ary im portance since it gives the ratio o f the pow er at the receiving port to the power at the transm itting port and is therefore an indication o f the pow er level that is detected by an antenna/sensor. These S 21 values are proportional to the average pow er density in the neighborhood o f the antenna/sensor in the cham ber and are used directly as statistical sam ples for both the PDF and the SCF m easurements. Sn and S 22 m easurements can be used as correction factors for S2i measurem ents to obtain cham ber gain param eters between the transmitting antenna and the receiving sensor/antenna. These correction factors provide some compensation for antenna/sensor reflections and are discussed in m ore detail in C hapter V. M echanical Design Considerations Three significant structural upgrades that are implemented on the system o f Figure 4.3 are the placem ent of a “low-backlash” gearbox on the output shaft o f the stepper motor, a new door design for the cham ber enclosure with an elastomer seal, and a sensor plate 50 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. design that allows for close-spacing SCF measurements. These upgrades are discussed briefly in the following numbered segments. (1) Low-Backlash G earbox: T he gearbox (speed reducer) on the original m otor assembly was a standard worm-gear type. With this gearbox in place, a m axim um free-play o f approximately 0.9 degrees was m easured at the paddle wheel. For m ost of the measurements in this study, the paddle wheel is rotated in 200 1. 8-degree incremental steps. Therefore, the relatively large amount o f free-play could lead to a situation where the paddle wheel does not rotate to a unique position at each step. This inherent inaccuracy in the system was corrected by replacing the worm-type gearbox with a precision low-backlash planetary gearbox from Sterling Instruments. This particular gearbox (S9123A-PG010) has a m aximum backlash rating o f 6 arc m inutes (or 0.1 degrees). (2) D oor and Seal: The electromagnetic seal on the original cham ber system was a wire mesh gasket. The gasket was not attached to the door but was attached around the circumference of the chamber opening. This was accomplished by gluing a piece o f foam rubber weather stripping around the cham ber opening and then by gluing the wire mesh gasket to the edge o f the w eather stripping. The door w as fastened, with bolts, to the cham ber and the wire mesh gasket was compressed between the door and the chamber. Leakage tests were perform ed to evaluate the quality of this seal by connecting an external broadband log-periodic antenna to an analyzer source and placing it in the close vicinity o f the door. This allowed for an S 12 (transfer pow er ratio from the source to the 51 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. receiving port) analyzer m easurem ent to be used as an indicator o f the leakage levels into the cham ber when the analyzer receiving port was connected to an antenna in the chamber. The results o f perform ing several o f these leakage tests in the 1-10 GHz range indicated that leakage levels at a num ber o f frequencies were, on the average, about 10-15 dB above the average system noise level whereas m ost o f the leakage signals for the selected sweep frequencies were, on the average, only 0-5 dB above the system noise level. This problem, o f small segmented frequency ranges with leakage, was temporarily addressed by taping the edges o f the door to the cham ber with conductive tape. The wiremesh gasket and tape com bination yielded leakage levels that were only 0-5 dB above the system noise level for all sweep frequencies in the 1-10 GHz range. The S21 values in these tests were in the -90 to -80 dB range. This shielding system was used to conduct a small set of prelim inary SCF measurem ents but is not practical for obtaining a complete set o f SCF m easurem ents since adjusting the sensor spacing inside the cham ber would require repeatedly removing, cleaning, and applying strips of conductive tape. The developm ent o f a perm anent door and seal design was initiated by soliciting some m ajor vendors o f EM I products for technical literature with regard to state-of-the-art electromagnetic shielding design methodologies. In particular, handbooks from Chom erics [21] and Tecknit [22] provided detailed and fairly com prehensive material on this su b ject After careful consideration of the m aterial in these handbooks, a decision was made to design a door with a groove that can be fitted with a conductive elastom er gasket. The dim ensions o f the door along with the bolt specifications that w ere selected to m eet 52 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. certain m inim um torque requirem ents are shown in Figure 4.4. The dim ensions o f the groove and gasket are shown in Figure 4.5. The (nom inal) groove dim ensions were calculated from a set o f algebraic form ulas in [22 ] by substituting values for the m inim um and m axim um allowable gasket deflections along with the fabrication tolerance o f the gasket diam eter. Leakage tests perform ed on the im plementation of this design also yield leakage levels that are 0-5 dB above the system noise level and S 21 values that are in the 90 to -80 dB range. (3) Sensor P late: W hile it is possible to fasten the D-dot sensors to the bottom wall of the cham ber with conductive tape, it is preferable to have a thin, flat, rectangular plate with sm all threaded holes that allow for the sensors to be easily fastened to the plate with sm all screws. This type of plate was designed and fabricated for the Prodyn sensors with holes that allow for 6 sensor spacings ranging from 2.5 to 17.5 cm . SCF m easurem ents in this spacing range were conducted by fastening this plate to the bottom wall o f the chamber. Sensor spacings below 2.5 cm cannot be obtained with this sensor and plate m echanism due to the 1 inch diam eter o f the sensor ground disk (see Figure 4.2). Thus, a plate and sensor design that allows for close-spacing measurem ents was designed by fabricating two D-dot sensors from type 141 semi-rigid cables. The sensors were m ade by stripping about 1 inch of the outer conductor, along with the dielectric material, from the end of the cables and then bending the inner conductor 90 degrees. The resulting m onopoles w ere cut to a length o f 0.125 inches. These 141 cable-sensors are placed on a 53 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. C H A M B E R D O O R 1 .5 0 . e a c h s id e G ro o v e ra d iu s : 2 .0 0 in n e r. 2 .2 6 o u te r, each co rn er 2 1 .5 0 H o le fo r 1 /4 -2 8 b o lt. 5 p e r sid e ,G ro o v e f o r e la s to m e r s a s k e t /s e e d e ta il! 2 1 .5 0 M a te r ia l: a lu m in u m . 3 /8 th ic k /A ll d im e n s io n s in in ch es') Figure 4.4. Drawing of the C ham ber D oor 54 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. C R O S S -S E C T IO N D E T A IL O F G R O O V E A N D G A SK ET Gasket: 1/4 diameter metal-filled \ silicone elastomer 3/8 Groove: 0.205 deep. 0.260 wide ("All dimensions in inches) Figure 4.5. Drawing that Show s Cross-Section of the Cham ber Door Groove and Elastomer Seal. 55 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. plate with a semicircular slit in the middle. After the cables are placed in the slit, clamps are placed on the cables to insure that the outer conductor is in contact with the plate and the spacing between the monopoles rem ains fixed. Figure 4.6 shows a photograph of this plate with the 141 cable-sensors attached. The sensor spacing is 0.5 cm. A m ajor feature o f this structure is the conductive tape, with two holes, that is placed over the ends o f the cables. The tape covers the region between the sensors and provides a m ore sym m etric current flow around the m onopoles. M easurem ents, at 2.5 cm , without the conductive tape yield lower m easured correlation coefficients than m easurements taken with the Prodyn sensors with the same spacing. M easurem ents with the tape yield correlations that agree, on the average, with those from the Prodyn sensor at a spacing of 2.5 cm. The results are not identical, however, since random fluctuations are observed in all the SCF measurements in this study. This effect is discussed in m ore detail in the following chapter along with SCF m easurements for close sensor spacings o f 0.5 cm , 1.0 cm, and 1.5 cm and SCF measurem ents for six larger spacings ranging from 2.5 cm to 17.5 cm. 56 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Figure 4.6. Photograph o f Sensor Plate for C lose Spacing SCF M easurem ents. The sensors in this design are m ade from 141 semi-rigid cables. 57 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER V RESULTS AND CONCLUSIONS Pow er Transfer Characteristics T he apparatus shown schematically in Figure 4.1 was used to m easure cham ber gain for com parison with the theoretical m odels of Eq. 2.14 and Eq. 2.16. T he H P 8719A was selected as the network analyzer since it allows for com parisons over a larger frequency range. This system is referred to as configuration 1 when the receiving antenna is in place and as configuration 2 when the receiving antenna is replaced with a receiving D -dot sensor. As mentioned in chapter IV, the antennas are linearly polarized log-periodic dipole arrays (W atkins-Johnson W J-48195, 1.0 to 18.0 G H z) and are m ounted well apart and with perpendicular polarizations in order to minim ize direct coupling between them. The D-dot sensor is a surface-m ounted asymptotic conical dipole from Prodyn Technologies (Figure 4.2) and is mounted to a cham ber wall where the electric field is perpendicular to the polarization o f the transmitting antenna to m inim ize coupling. The sensor is used at frequencies up to its 3-dB point of 10 G H z, where its response has fallen to 3 dB below the first-order m odel o f Eq. 2.16, by correcting the sensor output values for this fall-off during data analysis. Plots o f m easured cham ber gain for configuration 1 and configuration 2 are shown in Figure 5.1 and Figure 5.2, respectively [10]. The corresponding theoretical m odels are also plotted in these Figures, Eq. 2.14 in Figure 5.1 and Eq. 2.16 in Figure 5.2. The measurements presented in these Figures were conducted with a 21-point frequency sweep 58 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. (transmitting antenna, receiving antenna) theoretical -10 exptl. max. G21 (dB) exptl. avg. -20 -30 -40 -50 -60 FREQUENCY (GHz) Figure 5.1. Chamber Gain versus Frequency in Configuration 1. 59 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. (transmitting antenna, receiving sen sor) -10 theoretical -20 . exptl. max. exptl. avg. CM 0 -4 0 exptl. min. -50 -60 -70 FREQUENCY (GHz) Figure 5.2. Cham ber Gain versus Frequency in Configuration 2. 60 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. from 1 to 10 GHz. A total o f 153 separate gain m easurem ents w ere accum ulated for each frequency point. Each o f these m easurem ents was taken with the paddle wheel adjusted to a unique angular position controlled by the stepper m otor, which was program m ed to turn the paddle wheel one com plete revolution in 153 equal angular increm ents. The gain values in Figures 5.1 and 5.2 w ere obtained from S21 and Sn measurements as follows: 2 S21 G 21 ~ 1- r (5-1) S11 where G 21 is the m easured pow er gain from the transm itting port (port 1 in Figure 4.1) to the receiving port (port 2 in Figure 4.1), |S2 i| is the m agnitude o f the voltage transmission coefficient from port 1 to port 2, and |Sj j| is the m agnitude of the voltage reflection coefficient at port 1. The 1—|Si j |2 term is included to account for pow er returned to the source from the transmitting antenna. An im portant characteristic o f the experim ental curves in both Figure 5.1 and Figure 5.2 is the large, greater than 20 dB, difference between the m axim um and m inim um values o f G 21 over the entire frequency interval. This large difference is generally desirable with regard to proper reverberation chamber operation and is an indication o f a properly functioning paddle wheel. As m ight be expected, the theoretical curves in both Figure 5.1 and Figure 5.2 are only approxim ate representations o f the actual response o f the chamber. T he slopes of the theoretical curves match the general trends o f the slopes o f the experim ental m axim um and 61 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. average curves, but the theoretical values tend to be larger than the experim ental. This discrepancy can perhaps be attributed to factors such as losses in the paddle wheel blades, antenna internal and m ismatch losses, and losses through the access panel (door) gasket, which were not modeled and included in the calculations. Regarding cham ber wall losses, which were m odeled, the value used for the wall conductivity, c , w as 2.32107 S/m. Although this is the handbook value [19] for our particular alum inum alloy (6061T6), it may in fact be too high [23]; and this would add to the discrepancy. Som e insight into the physical phenom ena that determ ine the shape o f the cham ber response can be obtained from an examination o f equations used in the models. For example, for configuration 1 the GT5dependence in the second term o f the denom inator of Eq. 2.14 can be attributed to the loading o f the receiving antenna decreasing with ci/ combined with the chamber wall loss increasing with co1/2. Specifically, the ratio o f Q ’s in Eq. 2.12 can be rewritten as a pow er ratio as follows: (5.2) Here, Pd co2 from Eq 2.3 and P«,v °= (01/2 from Eq. 2.13 since 8 to 1/2. A t low frequencies, however, the first term in the denom inator o f Eq. 2.14 dom inates. Thus there are two distinct frequency regions. The transition point between these tw o regions can be calculated by setting the second term in the denom inator equal to unity and solving for the corresponding value o f ox T he result is a transition point for our cham ber at 3.34 G Hz. B elow this transition point, Pd > Pcqv in Eq. 5.2, so that pow er 62 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. extraction from the chamber by the receiving antenna dominates over pow er loss to the chamber walls. Similarly, for configuration 2 the cd3/z dependence in the second term o f the denom inator o f Eq. 2.16 is due to the cd2variation in the D-dot sensor loading combined with the col/2 variation in the cham ber wall loss. (The frequency dependence o f the sensor loading can be observed by solving for Pd in Eq. 2.7.) The first term of the denom inator o f Eq. 2.16 becomes im portant only for high frequencies- well above 10 G H z for our particular cham ber and sensor. This is the range where the pow er extracted by the sensor is dominant. Over the interval o f our measurements, 1 - 1 0 GHz, the cham ber wall loss dominates. An immediate application o f the theoretical gain models of Eq. 2.14 and Eq. 2.16 can be developed from observation of Eq. 2.10. This equation expresses the chamber gain as a ratio of the cham ber Q , Qnet, over the antenna or sensor Q. Therefore, a frequency domain approach to measuring the Q o f a microwave reverberation chamber, in a given configuration, is to measure the cham ber gain and then calculate Q nct values by applying the appropriate m odel for the sensor/antenna Q. This procedure w as carried out for the chamber system o f Figure 4.3, referred to as configuration 3, with the two Prodyn D-dot sensors installed as the receiving sensors. Eq. 2.7 was used as the sensor model with R = 50 ohms and A = 10^ m2. A plot o f the resulting Q measurements versus frequency is shown in Figure 5.3 where the smooth curve is Qnet, given by Eq. 2.8, divided by seven since this gives the best fit to the measurements. Table 5.1 is the tabulation o f a series of 63 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 6 Measured Chamber Q Versus Frequency x 10 5 4 O 3 2 1 0 2 4 6 8 10 12 Frequency (GHz) Figure 5.3. Cham ber Q M easurement via Cham ber Gain M easurem ent with Two Receiving (Prodyn) D-dot Sensors (configuration 3). The jagged curve represents the m easured values whereas the smooth curve is QDCt, from Eq. 2.8, divided by seven. The gain m easurem ent was taken with a 21-point frequency sweep and 200 incremental paddle wheel positions. 64 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Table 5.1. Tabulation o f Calculated M Values from M easured G ain Values. Each colum n, from left to right (not including f), represents a step in the calculation process. f M easured Gain M easured q d 1.0000e+009 1.6250e+009 2.2500e+009 2.8750e+009 3.5000e+009 4.1250e+009 4.7500e+009 5.3750e+009 6.0000e+009 6.6250e+009 7.2500e+009 7.8750e+009 8.5000e+009 9.1250e+009 9.7500e+009 1.0375e+010 1. 1000e +010 1.1625e+010 1.2250e+010 1.2875e+010 1.3500e+010 -45.56 -35.19 -33.20 -27.88 -28.29 -28.96 -24.60 -26.33 -24.30 -21.35 -19.89 -22.46 -19.53 -19.45 -21.62 -19.96 -23.64 -16.42 -17.52 -20.57 -19.77 2.62e+007 1.61e+007 1.17e+007 9.12e+006 7.49e+006 6.35e+006 5.52e+006 4.88e+006 4.37e+006 3.96e+006 3.62e+006 3.33e+006 3.08e+006 2.87e+006 2.69e+006 2.53e+006 2.38e+006 2.25e+006 2.14e+006 2.04e+006 1.94e+006 Q 3-dB BW 7.28e+002 4.88e+003 5.58e+003 1.48e+004 1.1 le+ 004 8.08e+003 1.9 le+ 0 0 4 1.13e+004 1.62e+004 2.90e+004 3.7 le+ 0 0 4 1.89e+004 3.43e+004 3.26e+004 1.85e+004 2.55e+004 1.03e+004 5.14e+004 3.79e+004 1.79e+004 2.05e+004 1.37e+006 3.33e+005 4.03e+005 1.94e+005 3.15e+005 5.10e+005 2.48e+005 4.74e+005 3.70e+005 2.29e+005 1.96e+005 4.17e+005 2.48e+005 2.80e+005 5.27e+005 4.07e+005 1.07e+006 2.26e+005 3.24e+005 7.21e+005 6.59e+005 M 0.62 0.40 0.93 0.73 1.75 3.94 2.54 6.21 6.03 4.55 4.66 11.72 8.11 10.57 22.72 19.84 58.58 13.86 22.02 54.21 54.43 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. calculations that were perform ed, including the calculations for the Q m easurem ent, that lead to a set o f values for the num ber o f m odes in a 3-dB bandwidth, M. These calculations were perform ed sequentially from left to right by processing the gain m easurem ents (second colum n) first and arriving at the values for M by substituting the results from the 3-dB bandwidth calculation (fifth column) for d f in the mode density formula o f Eq. 1.2. The resulting dN values are the M values. The 3-dB bandw idth was obtained simply by observing the relationship BW = f/Q where the Q's are the m easured Q values of column four. PDF o f Power Density A study of the probability density function for a com ponent o f the pow er density in the special (small, non-ideal) cham ber o f Figure 4.1 was initiated by exam ining the sample distributions, at each frequency, for the gain measurem ents of Figures 5.1 and 5.2. This involved plotting and analyzing the form o f the histograms for the 153 m easured samples at each frequency. Figure 5.4 shows these histograms [2] at the low est and the highest m easurem ent frequencies, 1 and 10 GHz, for the receiving antenna configuration (configuration 1) and Figure 5.5 show s the histogram s [2] at 1 and 10 G H z for the receiving D -dot sensor configuration (configuration 2). Inspection o f these figures indicates that there is good agreem ent at 10 GHz between the m easured distribution and the theoretical PDF (Eq. 2.19) both aw ay and on the wall. At 1 GHz, poor levels of agreem ent are observed since the distributions o f the m easured values are much "flatter" 66 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. CD O c £ w 3 O 0.2 O O CD > 0.1 05 (1 GHz, aw ay from wall) curve = theoretical bars = experim ental CD CE L ■Win 0 -50 -40 -30 -20 |S211 (dB) CD O c (10 GHz, aw ay from wall) CD w w 3 O 0.2 O O > '■g 05 - CD cc -30 -20 |S211 (dB) -10 Figure 5.4. Probability Density Function for EM Power Density in Configuration 1. The bin sizes for the histogram s are 0.25 dB. T he standard deviation for the 1 GHz data is 6.61 dB and the standard deviation for the 10 GHz data is 5.35 dB. 67 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CD O c f£ o 0.2 ■ CJ O .1 0.1 - (1 GHz, on the wall) curve = theoretical bars = experim ental 0) DC -80 |S211 (dB) CD o c 2t _> 13 0.2 O o (10 GHz, on the wall) o .1 0.1 ja CD DC 0 -60 -5 0 -40 -30 |S211 (dB) -20 -10 Figure 5.5. Probability Density Function for EM Power Density in Configuration 2. The bin sizes for the histograms are 0.25 dB. The standard deviation for the 1 G H z data is 8.57 dB and the standard deviation for the 10 G H z data is 5.36 dB. 68 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. than the shapes o f the corresponding theoretical curves. In configuration 1, the standard deviation o f the m easured data varies from 6.61 dB, at 1 GHz, to 5.35 dB at 10 GHz. In configuration 2, the standard deviation o f the m easured data varies from 8.57 dB , at 1 GHz, to 5.36 dB at 10 GHz. As m entioned in chapter II, the theoretical value for this standard deviation is 5.57 dB. The upward shifts in the m easured standard deviations, as a function of decreasing frequency, indicate that the distribution w idens as the num ber of m odes in a 3-dB bandwidth, M, decreases. Inspection of the sam ple standard deviations of the m easurem ents used to obtain the Q o f Figure 5.3 can provide m ore insight into the relationship between the transition of the non-ideal m easured PDF's at 1 GHz to the m ore ideal forms at higher frequencies and M. For example, inspection o f the standard deviation values in the seventh column o f Table 5.2 indicates that the first sample standard deviation that is between 5 and 6 dB occurs at about 4.5 or 5 GHz. Observing the corresponding M values in the sixth column o f this table indicates that the m easured distribution could converge to an ideal one for M values as low as 3 or 4. This m eans that Lehm an's assumption o f an infinite M may not restrict the applicability o f his theory as much as one might have supposed. A sim ilar type o f analysis can be perform ed on the sim ulation output. Here, standard deviation values, from a post-processing program that calculates a set o f statistics (and also generates histograms) for the sim ulation samples, are tabulated and observed in the fourth colum n o f Table 5.2. These particular sample standard deviations result from a simulation run with Q = Qnct / 10 and a distance of 8 cm between calculated field points (or samples) and correspond to the sim ulation run, from the overall set o f simulations, that 69 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Table 5.2. Measured M and o Values From the Q Measurement of Figure 5.3 Alongside M and a Values From the Corresponding Simulation Output Simulation f 1.0000e+009 1.2500e+009 1.5000e+009 1.7500e+009 2.0000e+009 2.2500e+009 2.5000e+009 2.7500c+009 3.0000e+009 3.2500e+009 3.5000e+009 3.7500e+009 4.0000e+009 4.2500e+009 4.5000e+009 4.7500e+009 5.0000e+009 5.2500e+009 5.5000e+009 5.7500e+009 6.0000e+009 6.2500e+009 6.5000e+009 6.7500e+009 7.0000e+009 7.2500e+009 7.5000e+009 7.7500c+009 8.0000e+009 8.2500e+009 8.5000e+009 8.7500e+009 9.0000e+009 9.2500e+009 9.5000e+009 9.7500e+009 1.0000e+010 1.0250e+010 1.0500e+010 1.0750e+010 l.lOOOe+OlO 1.1250e+010 1.1500e+010 1.1750e+010 1.2000e+010 1.2250e+010 Mave 0.04 0.07 0.14 0.22 0.28 0.50 0.53 0.73 0.84 1.12 1.37 1.44 4.36 2.47 2.97 2.88 3.34 4.01 4.37 4.74 5.32 8.23 9.25 7.73 8.54 10.00 10.35 10.82 14.32 13.15 14.23 18.80 17.21 19.04 19.68 20.96 22.99 27.16 25.92 28.15 29.25 32.84 35.89 37.18 40.10 40.10 Measurement M* 0.08 0.14 0.22 0.32 0.45 0.60 0.79 1.00 1.25 1.53 1.85 2.20 2.60 3.03 3.51 4.03 4.60 5.21 5.88 6.59 7.36 8.18 9.06 10.00 10.99 12.05 13.17 14.36 15.61 16.93 18.32 19.78 21.32 22.93 24.62 26.39 28.25 30.18 32.21 34.32 36.52 38.82 41.21 43.69 46.28 48.96 a 9.15 9.49 6.74 7.27 8.72 9.26 8.49 7.84 7.98 7.15 6.96 6.98 6.88 6.82 5.88 7.02 6.87 5.64 6.90 6.36 6.29 6.27 6.67 6.14 6.34 6.09 6.10 6.01 5.77 5.96 6.31 6.44 5.62 5.25 4.85 5.18 5.42 5.46 5.44 5.96 6.18 6.03 6.48 6.54 5.17 5.93 f M a 1.0000e+009 0.62 6.99 1.6250e+009 0.40 8.37 2.2500e+009 0.93 6.72 2.8750e+009 0.73 7.46 3.5000e+009 1.75 6.25 4.1250e+009 3.94 6.42 4.7500e+009 2.54 5.25 5.3750e+009 6.21 5.77 6.0000e+009 6.03 5.51 6.6250e+009 4.55 5.24 7.2500e+009 4.66 5.63 7.8750e+009 11.72 6.66 8.5000e+009 8.11 6.28 9.1250e+009 10.57 5.69 9.7500e+009 22.72 5.25 1.0375e+010 19.84 6.05 1.1000e+010 58.58 5.62 1.1625c+010 13.86 5.95 1.2250e+010 22.02 5.37 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Table 5.2. Continued. Simulation f 1.2500e+010 1.2750e+010 1.3000e+010 1.3250e+010 1.3500e+010 Mave 44.26 43.17 48.68 49.92 52.43 Measurement M* 51.75 54.64 57.64 60.76 63.98 a f 5.54 5.56 6.24 5.73 5.60 M a 1.2875e+010 54.21 5.04 1.3500e+010 54.43 5.75 most closely matches the conditions in the chamber for the m easurem ent o f Figure 5.3. Inspection o f these standard deviation values, which are for one o f the two field points, also indicates that the first sample standard deviation that is between 5 and 6 dB occurs at about 4.5 or 5 GHz. One o f the output files (in the form of Figure 3.2) from this simulation run also shows that, in the 4.5 - 5 GHz range, the average num ber o f selected modes that fall within the 3-dB bandwidth is approximately 3 or 4. These average M values are shown in the second colum n o f Table 5.2. Also, the theoretical values o f M (from the output file) are included in the third column of Table 5.2. These values range from about 3.5 to 4.5 in the 4.5 - 5.0 G H z range. 71 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. SCF o f Power Density Figure 5.6 is a set o f plots that show m easured and simulated spatial correlations versus spacing w hile Figure 5.7 is a set o f plots that show m easured and simulated spatial correlations versus frequency. Lehm an's theoretical curve, Eq. 2.31, is plotted alongside all o f these results. All of the measurements were taken with the m easurem ent system of o f Figure 4.3 (denoted as configuration 3). M easurem ents with sensor spacings o f 2.5, 5.5, 8.0, 11.5, 13.5, and 17.5 cm were taken with the Prodyn sensors (Figure 4.2) in place whereas m easurem ents with sensor spacings of 0 .5 ,1 .0 , and 1.5 cm were taken with the 141 cable-sensor plate (Figure 4.6) in place. Fifty-one discrete frequency points between 1 and 13.5 GHz, with 0.25 G H z increments, were used for both the m easurem ents and the simulations. The com plete set o f simulation runs include three runs, with Q = Qn(:t , Q = Qnct /10 , and Q = Qnct / 100 , for each o f the sensor spacings o f the above-mentioned measurements. (Qnci is given by Eq. 2.8 for the case of two receiving D -dot sensors.) The SCF values from the Q = Qn<:t /10 runs were chosen for the plots o f Figures 5.6 and 5.7 since Q„ct /10 is o f the same order o f m agnitude as the measured cham ber Q's o f Figure 5.3. The SCF outputs from the sim ulations with Q = QDet/100 were found to be in close agrrem ent with Lehman's theoretical curve in the upper half o f the selected frequency range, where large values of M occur. Slightly low er levels o f agreem ent with the theoretical curve are observed in the S C F outputs from the Q = Qnet /10 runs in the upper half o f the frequency range. In contrast, the SCF outputs from the Q = Qnet runs are significantly higher than 72 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. SCF of EM Power Density f = 1.00 GHz f= 1.25 GHz ® 0.5 ® 0.5 0 5 10 15 20 0 5 f = 1.50 GHz 10 15 20 f = 1.75 GHz ® 0.5 ® 0.5 0 5 10 15 20 0 5 f = 2.00 GHz 10 15 20 f = 2.25 GHz ® 0.5 0 5 10 15 20 0 Spacing (cm) 5 10 15 20 Spacing (cm) Theo. = Solid M eas. = Dashed Sim. = D otted Figure 5.6. M easured, Simulated, and Theoretical SC F o f the Pow er Density, versus Spacing, Inside the M icrow ave Reverberation Cham ber o f Figure 4.3 or Figure 4.1. The measurem ents were taken with the m easurem ent system of Figure 4.3. (a) Plots for frequencies o f 1, 1.25, 1.50, 1.75, 2.00, and 2.25 GHz. 73 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. SCF of EM Power Density f = 2.50 GHz f = 2.75 GHz <u 0.5 a) 0.5 -4/- - 0 5 10 15 20 0 5 1 f = 3.00 GHz ® 0.5 10 15 20 f = 3.25 GHz a> 0.5 0 0 5 10 15 20 0 1 f = 3.50 GHz 10 5 15 20 f = 3.75 GHz o 0.5 V ' o 0 5 10 15 20 0 Spacing (cm) 5 10 15 20 Spacing (cm) Theo. = Solid M eas. = Dashed Sim. = Dotted Figure 5.6. Continued, (b) Plots for frequencies o f 2 .5 ,2 .7 5 , 3.00, 3.25, 3.50, and 3.75 GHz. 74 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. SCF of EM Power Density 1 f = 4.00 GHz f = 4.25 GHz cu 0.5 a> 0.5 0 0 5 10 15 20 0 5 1 f = 4.50 GHz ® 0.5 cd 10 15 20 f = 4.75 GHz 0.5 0 0 5 10 15 20 0 1 f = 5.00 GHz cd 5 0.5 cd 10 15 20 f = 5.25 GHz 0.5 0 0 5 10 15 0 20 Spacing (cm) 5 10 15 20 Spacing (cm) Theo. = Solid M eas. = Dashed Sim. = Dotted Figure 5.6. Continued, (c) Plots for frequencies o f 4 .0 0 ,4 .2 5 ,4 .5 0 ,4 .7 5 , 5.00, and 5.25 GHz. 75 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Correlation SCF of EM Power Density 1 0.5 g> 0.5 0 0 0 5 Correlation 1 10 15 0 a> 0.5 0 0 1 10 15 5 1 0.5 5 f = 5.75 GHz 20 f = 6.00 GHz 0 Correlation 1 f = 5.50 GHz 0 20 15 20 f = 6.25 GHz 5 1 f = 6.50 GHz 10 10 15 20 f= 6.75 GHz 0.5 0 0 0 5 10 20 0 Spacing (cm) 5 10 15 20 Spacing (cm) Theo. = Solid Meas. = Dashed Sim. = Dotted Figure 5.6. Continued, (d) Plots for frequencies o f 5.50, 5 .7 5 ,6 .0 0 ,6 .2 5 , 6.50 and 6.75 GHz. 76 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Correlation SCF of EM Power Density f = 7.00 GHz 0.5 a> 0.5 Correlation 0 5 10 15 0 20 5 f = 7.50 GHz 10 15 20 f = 7.75 GHz 0.5 a> 0.5 0 Correlation f = 7.25 GHz 5 10 15 20 0 5 f = 8.00 GHz 10 15 20 f = 8.25 GHz 0.5 0 5 10 15 0 20 Spacing (cm) 5 10 15 20 Spacing (cm) Theo. = Solid Meas. = Dashed Sim. = Dotted Figure 5.6. Continued, (e) Plots for frequencies o f 7.0 0 ,7 .2 5 , 7 .5 0 ,7 .7 5 , 8.00, and 8.25 GHz. 77 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Correlation SCF of EM Power Density f = 8.50 GHz 0.5 ® 0.5 Correlation 0 5 10 15 20 0 5 f = 9.00 GHz 10 15 20 f= 9.25 GHz 0.5 ® 0.5 0 Correlation f= 8.75 GHz 5 10 15 0 20 5 f = 9.50 GHz 10 15 20 f= 9.75 GHz 0.5 ® 0.5 0 5 10 15 20 0 Spacing (cm) 5 10 15 20 Spacing (cm) Theo. = Solid M eas. = D ashed Sim. = D otted Figure 5.6. Continued, (f) Plots for frequencies o f 8.50, 8.75, 9.00, 9.25, 9.50, and 9.75 GHz. 78 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Correlation SCF of EM Power Density f= 10.00 GHz 0.5 <d 0.5 Correlation 0 5 10 15 20 0 5 f = 10.50 GHz 10 15 20 f= 10.75 GHz 0.5 a) 0.5 0 5 1 Correlation f= 10.25 GHz 10 15 20 0 5 f= 11.00 GHz 10 15 20 f= 11.25 GHz 0.5 0 0 5 10 15 20 0 Spacing (cm) 5 10 15 Spacing (cm) 20 Theo. = Solid M eas. = Dashed Sim. = Dotted Figure 5.6. Continued, (g) Plots for frequencies o f 10.00, 10.25, 10.50, 10.75, 11.00, and 11.25 GHz. 79 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. SCF of EM Power Density 1 f= 11.50 GHz f= 11.75 GHz ® 0.5 0 0 5 1 10 15 20 0 5 f= 12.00 GHz 10 15 20 f = 12.25 GHz a> 0.5 0 0 5 1 10 15 0 20 5 f = 12.50 GHz 10 15 20 f = 12.75 GHz a> 0.5 0.5 0 0 5 15 10 Spacing (cm) 0 20 5 10 15 Spacing (cm) 20 Theo. = Solid Meas. = Dashed Sim. = Dotted Figure 5.6. Continued, (h) Plots for frequencies o f 11.50,11.75, 12.00, 12.25, 12.50, and 12.75 GHz. 80 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. SCF of EM Power Density 1 f= 13.00 GHz f= 13.25 GHz P 0.5 __I 0 0 5 10 15 20 f= 13.50 GHz c o f = 1.00 GHz s 1 0.5 p 0.5 o O 0L 15 10 1 0 20 5 10 15 20 {=1.50 GHz f = 1.25 GHz A' p 0.5 0 0 5 10 15 Spacing (cm) 0 20 5 10 15 Spacing (cm) 20 Theo. = Solid Meas. = Dashed Sim. = Dotted Figure 5.6. Continued, (i) Plots for frequencies o f 13.00, 13.25, and 13.50 GHz. Plots for frequencies of 1.00,1.25, and 1.50 G H z are repeated for convenience. 81 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. SCF of EM Power Density r = 0.5 cm c 0.8 v \ O 5 0.6 CD g 0.4 0.2 ° - 0.2 r = 1.0 cm c 0.8 o 5 0.6 CD g 0.4 \ \ 0.2 ° v - 0.2 Frequency (GHz) Theo. = Solid Meas. = D ashed Sim. = Dotted Figure 5.7. M easured, Sim ulated, and Theoretical SC F o f the Pow er Density, versus Frequency, Inside the M icrow ave Reverberation Cham ber o f Figure 4.3 or Figure 4.1. The m easurem ents were taken with the m easurem ent system o f Figure 4.3. (a) Plots for spacings o f .5 and 1.0 cm. 82 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. SCF of EM Power Density r= 1.5 cm Correlation 0.8 0.6 0.4 0.2 - > \ , ' ' 't — • rV + V 0.2 1 r = 2.5 cm Correlation 0.8 0.6 0.4 0.2 0 - 0.2 2 4 6 8 Frequency (GHz) 10 12 Theo. = Solid Meas. = Dashed Sim. = Dotted Figure 5.7. Continued, (b) Plots for spacings o f 1.5 and 2.5 cm. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. SCF of EM Power Density r = 5.5 cm C 0.8 o 5 0-6 0) fc 0.4 0.2 ° - 0.2 r = 8.0cm c 0 .8 o '•S 0 .6 0.2 - 0.2 Frequency (GHz) Theo. = Solid Meas. = Dashed Sim. = Dotted Figure 5.7. Continued, (c) Plots for spacings o f 5.5 and 8.0 cm. 84 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. SCF of EM Power Density Correlation 0.8 0.6 -A’, 0.4 / 0.2 - ■ ^ 0.2 T T T T T r = 13.5 cm Correlation 0.8 - 0 6 -. 0.4 0.2 - 0.2 Frequency (GHz) Theo. = Solid M eas. = Dashed Sim. = Dotted Figure 5.7. Continued, (d) Plots for spacings o f 11.5 and 13.5 cm. Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. SCF of EM Power Density r= 17.5 cm C 0.8 o 5 0-6 CD * 0.4 0.2 - \ / - 0.2 r = 0.5cm c 0.8 o V \ 5 0.6 a> g 0.4 0.2 ° - 0.2 Frequency (GHz) Theo. = Solid Meas. = Dashed Sim. = Dotted Figure 5.7. Continued, (e) Plots for spacing o f 17.5 cm. Plots for spacing o f 0.5 cm are repeated for convenience. 86 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. the corresponding theoretical values in the upper half of the frequency range. Here, inspection of the output files shows that even at the highest sim ulation frequency o f 13.5 GHz, the (average) num ber o f selected m odes that are within the 3-dB bandwidth is only about 5 whereas in the Q = Qnet /10 and Q = Q„et/100 cases the values for this param eter are about 50 and 500, respectively. Inspection o f the forms o f the m easured SCF response in Figures 5.6 and 5.7 reveals the following three trends. (1) High Spatial Correlations at Large Spacings and Low Frequencies: This effect can be observed in Figure 5.6(a)-(c) and Figure 5.7(c)-(d). The measurem ents seem to "converge," on the average, to Lehm an's theoretical curve at frequencies above the 4.5 to 5.0 G H z range. The frequency range below 4.5 G H z w as cited, in the previous section, as a region that can be associated with insufficient m ode densities for the chamber in this study. (2) Random (Jagged) Fluctuations in the ('Locally') M onotonic N ature of the R esponse: This lack of "smoothness" can be observed in all the o f measurements as well as in all o f the simulation outputs. An effort to analyze this effect by taking the Fourier transform (FFT) o f a num ber o f m easured SCF sam ple outputs did not reveal a "signature" in the transform dom ain. As shown in Figure 5.8, this effect can be related to the num ber of data acquisition samples, or paddle w heel positions, that are program m ed into the m otor control algorithm o f the m easurem ent system. All o f the SCF measurements in this study were taken with 200 paddle wheel positions. Selectively removing every fourth sample 87 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. from a selected m easurem ent set (Figure 5.8) indicates that the peak-to-peak levels of these fluctuations decrease as the number o f samples increases. (31 Low Correlation at Close Sensor Spacings and at High Frequencies: This effect can be observed in Figure 5.7(a)-(b). Since the presence o f undesirable signals is a relatively common problem in microwave circuits, a number of possibilities were carefully investigated to determ ine the physical nature o f these low correlations. Some o f the possibilities that were eliminated by inspection o f some additional sets of measurements and additional simulations along with some rough hand calculations were: (i) decorrelation due to electrom agnetic scattering at close sensor spacings, (ii) the existence o f slowly propagating or heavily attenuated surface waves, (iii) large sensor size in relation to the signal wavelengths, and (iv) the existence o f non-standard waveguide modes that do not fall into the realm of standard theories for the operation o f an overmoded rectangular cavity. Further consideration o f this effect led to the hypothesis that the decorrelation may be due to the unsymmetric nature of the tape in the vicinity of the 141-cable sensors. In other words, the "bump" in the tape between the tw o sensors may be causing the sensors to operate in a non-ideal manner. As mentioned in chapter IV, this was not considered a problem atic structural flaw prior to conducting the SCF measurements since trial m easurements at a spacing of 2.5 cm yielded acceptable levels of agreem ent with trial measurements taken with the Prodyn sensors in place. However, since a 2.5 cm SCF m easurem ent with the Prodyn sensors requires placing them in direct contact with each 88 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. SCF of EM Power Density Correlation 0.8 0.6 0.4 N = 50 0.2 - 0.2 0.8 Correlation 0.6 0.4 N = 200 0.2 - 0.2 Frequency (GHz) Figure 5.8. Plots o f M easured SCF Response with 50 Paddle W heel Positions and 200 Paddle W heel Positions. The data for the N = 50 plot was obtained from the N = 200 data by selecting every forth sam ple from the 200 sample measurement. The sensor spacing for this m easurem ent was 0.5 cm. Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. other, it is conceivable that the 2.5 cm m easurem ents with these sensors also are associated with structural dissymm etries. This hypothesis o f dissym m etries in the 141 cable-sensor plate was verified by m aking a final SC F m easurem ent with a modified sensor structure. This structure was im plemented by drilling two sm all holes in the bottom of the cham ber that are spaced 1 cm a p a rt Then, two 141 cable-sensors w ere press-fitted into the holes from the outside o f the cham ber such that only the inner conductor o f the cables was protruding into the cham ber with the flat cham ber wall between the outer conductors of the cables. Significandy higher values o f spatial correlation, at the higher frequencies, were m easured with this structure in relation to the 1 cm spacing m easurem ent with the 141 cable-sensor plate. Summary and Discussion The follow ing outline serves as a sum m ary o f the m ajor developm ents and results that are docum ented in this dissertation project. —> Design and Construction o f M icrow ave Reverberation C ham ber M easurem ent Apparatus with Source W avelengths such that the Cham ber is Electrically Small at the Low er End o f the Source Frequency Spectrum. —» Development o f Algorithm that Allows for Com puter Simulation o f SCF Experim ent by Generating Suitable Statistical Pow er Density Samples. —» M atlab Im plem entation o f SCF Algorithm. -» Derivation o f Average Cham ber Gain Models. 90 with perm ission of the copyright owner. Further reproduction prohibited without perm ission. —> M easurements o f Cham ber Gains. —> Developm ent o f Frequency-Dom ain M ethod to M easure Cham ber Q U sing Chamber Gain Models. —> Calculation o f M Values for Chamber. —» Evaluation o f M easured PDF's Under Non-Ideal (Electrically Small) Cham ber Conditions Using Histogram s and Sample Standard Deviations. -» Evaluation o f M easured PDF's Under Non-Ideal Conditions Via Consideration of Cham ber M V alues and O utput from SCF Simulation Program. -» M easurem ent o f the SCF in a M icrow ave Reverberation Chamber. —» Comparison o f SCF M easurem ents and Simulations to Lehm an's Theoretical SCF Curve for Complex Cavities. T he introduction and investigation o f M as a microwave reverberation cham ber param eter is perhaps the m ost significant o f these developments. Further study o f this param eter could lead to a sim ple criterion that provides an accurate low-frequency bound for the operation o f a microwave reverberation chamber. Form ulas that are presently under consideration [25] by other investigators include a 60-modes criterion and a 6 x f0 criterion, where fo is the fundamental resonant frequency of the cham ber cavity. T he 60-modes criterion can be applied by either finding the 60th mode o f the chamber cavity by com puter counting or by setting N equal to 60 in Eq. 1.1 and solving for f. A com puter counting calculation for the cham ber in this study (Figures 4.1 and 4.3) yields a low frequency cut-off o f 844 M Hz. This criterion seems to have originated from NBS 91 with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Technical Note 1092 [4, p. 21] where the authors state that the "practical lower frequency limit for using the NBS enclosure as a reverberation chamber is approximately 200 MHz. This lower limit is due to a num ber of factors including insufficient mode density, limited tuner effectiveness, and ability to uniformly excite all modes in the chamber." The dimensions o f this NBS cham ber are such that a frequency o f 200 M Hz corresponds to the existence o f approxim ately 60 distinct modes from 0 to 200 MHz. However, the 60modes criterion does not seem to have been specified or generally recom m ended within the body o f this paper. The fundamental frequency for the cham ber cavity (Figures 4.1 and 4.3) in this study is 235 MHz. Six times this value yields a chamber low frequency cut-off of 1.41 GHz. This criterion originated from the analysis of a sample set o f chamber data [25]. In this case, plots o f the cham ber VSW R versus frequency were analyzed, and the 6 x fo term was found to correspond to the low-frequency edge of a region on the plot where the VSWR response is approximately flat. Literature with regard to the underlying physical principles behind this criterion has not been cited at the present time. The M equals 3 or 4 criterion that was proposed in the second section of this chapter as a possible criterion for the low frequency limit o f the chamber in this study corresponds to a low frequency cut-off o f 4.5 - 5.0 GHz. W hether or not this criterion develops into an acceptable bound for the low-frequency operation of a m icrowave reverberation chamber is an open question since this criterion, like the other two that are described 92 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. above, was not developed via a formal m athem atical procedure but is proposed based on the observation o f a lim ited set of data. Additional analytical methods that are currently under investigation [25] and could lead to criteria for the low-frequency cut-off of a m icrowave reverberation cham ber are m ethods based on applying Kolm ogorov-Sm irnov goodness-of-fit tests to the m easured PDF's and Lehm an's "unstirred energy" calculation for a m icrowave reverberation chamber. Published literature with regard to these investigations are not presently available. Possible applications for the SCF inside a m icrowave reverberation cham ber include the consideration of "correlation lengths" as part o f the overall testing procedures [24]. This concept is particularly well-suited to applications that use a m icrowave reverberation cham ber test environm ent to simulate the actual EM I environm ent of interest. For example, inspection o f Lehm an's theoretical SCF curve (Eq. 2.31) for null points yields that the first null occurs at: |l .- l 2 | = R «r = V 2 . (5.3) This X/2 quantity can be used to investigate the concept of correlation length in a m icrowave reverberation cham ber in the sense that, under ideal conditions, the correlation between the power density at two points becomes negligible at and beyond this spacing. However, interesting cases such as the m easured SCF response at 2 GHz in Figure 5.6a exist that do not conform to this straightforward formulation. Accurate characterization o f the correlation length could lead to situations w here the com prehensive testing of a 93 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. particular device in a cham ber will not also require the placem ent o f all the surrounding equipm ent from the device's intented operational environment. Only equipm ent that falls within a particular correlation length would need to be placed in the chamber. 94 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. REFERENCES [ 1] D. A. Hill, “Electronic M ode Stirring for Reverberation Cham bers,” IE E E Trans. Electromagn. Compat., vol. EM C-36, no. 4 , pp. 294-299, Nov. 1994. [2] T.F. Trost, A.K. Mitra, and A.M. Alvarado, “Characterization o f a Sm all M icrow ave Reverberation Cham ber,” Proceedings o f the 11th International Z urich Sym posium and Technical Exhibition on EMC, pp. 583-586, M arch 1995. [3] B.H. Liu and D.C. Chang, “Eigenmodes and the Com posite Quality Factor o f a Reverberating Cham ber,” N B S Tech. N ote 1066, Aug. 1983. [4] M.L. Crawford and G.H. Koepke, “Design, Evaluation, and Use of a Reverberation Chamber for Performing Electromagnetic Susceptibility/Vulnerability M easurements,” N B S Tech. N o t e 1092, Apr. 1986. [5] J.M. Dunn, “L o c a l, High-Frequency Analysis o f the Fields in a M ode-Stirred Chamber,” IE E E Traits. Electromagn. Compat., vol. EM C-32, no. 1, pp. 53-58, Feb. 1990. [6] J.G. Kostas and B. Boverie, “ Statistical M odel for a M ode-Stirred Cham ber,” IE E E Trans. Electromagn. Compat., vol. EM C-33, no. 4, pp. 366-370, Nov. 1991. [7] T.H. Lehman, “A Statistical Theory o f Electromagnetic Fields in Com plex Cavities,” Phillips Laboratory Interaction N ote 494, M ay 1993. [8] Y. Huang and D.J. Edwards, “An Investigation o f the Electrom agnetic Field inside a M oving-W all M ode-Stirred Chamber,” The 8th IE E Int. Conf. on EM C, Edinburgh, UK, pp. 115-119, Sept. 1992. [9] T.A. Loughry, “Frequency Stirring: An Alternate Approach To M echanical M odeStirring For The Conduct O f Electromagnetic Susceptibility Testing,” Phillips Laboratory Tech. R eport 91-1036, Nov. 1991. [10] A.K. M itra, T.F. Trost, “ Pow er Transfer Characteristics o f a M icrow ave Reverberation Cham ber,” to appear in IE E E Transactions on EM C, May, 1996. [11] C.E. Baum et al., “ Sensors for Electromagnetic Pulse M easurem ents Both Inside and Away from N uclear Source Regions,” IE E E Trans. Electrom agn. Compat., vol. EM C-20, no. 1, pp. 22-35, Feb. 1978. 95 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. [12] Y. Huang, “The Investigation of Chambers for Electromagnetic Systems,” Ph.D Dissertation, University o f Oxford, 1994. [13] M. Iskander et al., “FDTD Simulation o f M icrowave Sintering o f Ceramics in M ultimode Cavities,” IE E E Trans. M icrow ave, vol. M TT-42, no. 5, pp. 793-800, M ay 1994. [14] K.C. Chen, private communication, Sandia N ational Laboratory, Sept. 1994. [15] Y. Rahmat-Samii, private com munication, UCLA, Sept. 1994. [16] A. Taflove and K.R. Umashankar, “The finite-difference time-domain method for numerical modeling of electromagnetic wave interactions with arbitrary structures,” in M.A. M organ, Ed., Finite-Elem ent and Finite-D ifference M ethods in Electrom agnetic Scatter. Amsterdam: Elsevier, 1990. [17] C.T. Tai, “On the Definition o f the Effective Aperture of Antennas,” IEEE Trans. Antennas Propagat., vol. AP-9, pp.224-225, M arch 1961. [18] D. I. Wu and D.C. Chang, “The Effect o f an Electrically Large Stirrer in a ModeStirred Cham ber,” IE E E Trans. Electromagn. Compat., vol. EM C-31, no. 2, pp. 164-169, M ay 1989. [19] J. H. Potter, H andbook o f the Engineering Sciences, Vol. II, Van Nostrand, 1967. [20] R. N. Rodriguez, “Correlation,” Encyclopedia o f Statistical Sciences, vol. 2, pp. 193-204, W iley, 1982-. [21] EM I Sheilding Engineering Handbook, Chom erics Inc., W oburn, M A, 1989. [22] Sheilding D esign Guide, Tecknit EMI Sheilding Products, Cranford, NJ, 1991. [23] D.A. Hill et al., “Aperture Excitation o f Electrically Large, Lossy Cavities,” IEEE Trans. Electromagn. Compat., vol. EM C-36, pp. 169-178, Aug. 1994. [24] T.F. Trost and A.K. M itra, "Eectromagnetic Compatibility Testing Studies," Final Technical R eport on G rant NAG-1-1510, NASA Langley Research Center, January 15,1996. [25] G. Freyer, Consultant, M onument, CO, private com m unication, M arch 1996. 96 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. APPENDIX M ATLAB SIM ULATION M-FDLES (1) B W R E S1.M (Used to permanently store resonant frequency data for simulation) %***** Calculate, Sort, and Select Cavity Resonances for Reverberation Chamber %***** Simulation clkl=clock; c=2.99792458E8; a= 1.0342; b=.8087; d=.5812; lpnum=95; simres lc=zeros(l,51); m g l(l)= l; mg l(2:52)=zeros( 1,51); bwhl(l)=300e6; bwhl(2)=75e6; bwh 1(3:9)=50e6*ones( 1,7); bwh 1(10:21 )=25e6*ones( 1,12); bwhl(22:5 l)=50e6*ones(l,30); ddmax=.2; ddl=ddmax/200; dl=d-ddmax-ddl; df=.25e9; for h= 1:200, %Perturbation Loop lenl=0; save stat.out h -ascii dl= dl+ ddl; fres = (c/2) * ( (fresl + fres2 + ((pl/dl).A2)) .A .5 ); [fres,k]=sort(fres); ff=.75e9; for i= 1:51, %Frequency Loop ff=ff+df; f(i)=ff; [fpt0,10] = min( abs( fres - ( (f(i)-bwhl(i)) * ones(size(fres))) ) ) ; [fpt 1,11] = min( abs( fres - ( (f(i)+bwhl(i)) * ones(size(fres))) ) ) ; Ien2=len 1+11-10+1; 97 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. fbw 1(len 1+1 :len2)=fres(10:l 1); kbw 1(len 1+1 :len2)=k(10:l 1); [fptc,lc(i)] = min( abs( fbwl - ( f(i) * ones(size(fbwl))) ) ) ; lenl=len2; m gl(i+l)=lenl; end datak = ['data' int2str(h)]; fill = [datak ’.mat']; eval(['save' f i l l ' fbwl kbwl m gl lc']); end clk2=clock; save et.mat clkl clk2 (2) SIM RES.M (Used by BW RES1.M ) %*** Integers For Cavity Resonances Starting *** %*** With m=l n=l p=0 ************************** lp=lpnum; len=(lp*2)*(lp+1); <^5|cs|c:jcsf:sfej|csfc:jcijej}cj(c}(cji<:}c:fcjjc5|ejj<5j<s{eijc!}ca|csie:ic:$cs|cjj<sjcjfc ml=zeros(l,len); mm=l; il= l; i2=(lp)*(lp+l); del=(lp)*Gp+l); fori 100= l:(lp), m 1(i 1:i2)=mm*ones( 1,del); mm=mm+l; il=il+del; i2=i2+del; end nl=zeros(l,2*len); nn=l; il= l; i2=(lp+l); del=0p+l); fori 100= I:(lp). nl (i 1:i2)=nn*ones( 1,del); nn=nn+l; il=il+del; i2=i2+del; 98 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. end i2=i2-del; while i2 < le n , il= i2+ l; i2=2*i2; n l(il:i2 )= n l(l:il-l); end nl=nl(l:len); %*********%******************** pl=zeros(l,2*len); pl(l:lp+l)=0:l:lp; i2=lp+l; while i2 < le n , il=i2+l; i2=2*i2; p l(il:i2 )= p l(l:il-l); end pl=pl(l:len); fresl = (m l/a).A2; fres2 = (nl/b).A2; (3) SIM 57.M (This is the actual simulation program) %***** Mode Selection in Microwave Reverberation ************* %***** Chamber By Perturbing Height ************************** %***** AND Correlation Calculation *************************** %***** From Loughry's Field Equations *********************** %***** Resonant Frequency Arrays Generated By BWRES1.M ******* r=.175; rot=(21/180)*pi; x0=.5; y0=.325; x l=x0+(r*cos(rot)/2); x2=x0-(r*cos(rot)/2); y l=y0-(r*sin(rot)/2); y2=y0+(r*sin(rot)/2); j= (-l)A5; c=2.99792458E8; sig=2.32E7; u=4*pi*lE-7; e=8.8542E-12; 99 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. a= 1.0342; b=.8087; d=.5812; S=2*a*b + 2*b*d + 2*a*d; V=a*b*d; AA=lE-4; R=50; eta=377; rand(’seed',suni( 100*clock)); ^********** clkl=clock; f=le9:.25e9:13.5e9; eqv50 BD3 = (f./Q3); N3 = ((8*pi*V)/(cA3)) * ((f.A2).*BD3); fll=f-(.5*BD3); ful=f+(.5*BD3); % ********** lpnum=95; simresl modes=zeros(l,51); modes3=zeros( 1,51); mag 1=zeros(51,200); fll0=zeros(51,200); fu 10=zeros(51,200); bw 10=zeros(51,200); fid 10='e l.out'; fopen(fidlO,'w'); % ********** ddmax=.2; ddl=ddm ax/200; dl=d-ddmax-ddl; for h= 1:200, %Perturbation Loop save stat.out h -ascii dl=dl+ddl; datak = ['data' int2str(h)]; fhl = [datak'.mat']; eval(['load' fill]); fres = fbwl; k = kbwl; clear fbwl kbwl for i= 1:51, %Frequency Loop %Model Excitation Resonances inc=0; l=lc(i); ii=k(l); 100 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. evml inc=inc+l; modes(i)=modes(i)+1; if (ffes(l)>fll(i)) & (fres(l)<ful(i)), modes3(i)=modes3(i)+l;, end data(inc,l)=ml(ii); data(inc,2)=nl(ii); data(inc,3)=pl(ii); data(inc,4)=mag; magl(i,h)=mag; 10=1; 1= 1+ 1 ; evml while mag > (.l*magl(i,h)) ii=k(l); inc=inc+l; modes(i)=modes(i)+1; if (fres(l)>fl 1(i)) & (fres(l)<ful(i)), modes3(i)=modes3(i)+l;, end data(inc,l)=ml(ii); data(inc,2)=nl(ii); data(inc,3)=pl(ii); data(inc,4)=mag; 1=1+ 1 ; if l==mg 1(i+1) fprintf(fid 10,'Insufficient Modes At f=% 8.3g h = %4.0f\n',f(i),h) break end evml end fulO(i,h)=fres(l-l); 1= 10; 1= 1 - 1 ; evml wliile mag > (.l*magl(i,h)) ii=k(l); inc=inc+l; modes(i)=modes(i)+1; if (fres(l)>fll(i)) & (fres(l)<ful(i)), modes3(i)=modes3(i)+l;, end data(inc,l)=ml(ii); data(inc,2)=nl(ii); data(inc,3)=pl(ii); data(inc,4)=mag; 1= 1- 1 ; if l==mgl(i) fprintf(fid 10,'Insufficient Modes At f=% 8.3g h = %4.0f\n',f(i),h) break end 101 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. evml end fllO(i,h)=fres(l+l); ^Calculate Field and Power Density At Two Points kx=(pi/a)*data(l :inc,l); ky=(pi/b)*data(l :inc,2); %kz=(pi/dl)*data(l :inc,3); %kxy=((kx.A2)+(ky.A2)).A(.5); %kr=((kx.A2)+(ky.A2)+(kz.A2)).A(.5); ph 1=exp(j*2*pi*rand(inc, 1)); erl = data( 1:inc,4) .* sin(kx*xl) .* sin(ky*yl); eil = data(l:inc,4) .* sin(kx*xl) .* sin(ky*yl) .* phi; er2 = data(l:inc,4) .* sin(kx*x2) .* sin(ky*y2); ei2 = data(l:inc,4) .* sin(kx*x2) .* sin(ky*y2) .* phi; emagl=abs(sum(erl)+sum(eil)); emag2=abs(sum(er2)+sum(ei2)); s 1(i,h)=(emag 1A2)/(2*eta); s2(i,h)=(emag2A2)/(2*eta); d ea r erl ei 1 er2 ei2 data phi end end %*************************** save pden.m atfsl s2 <^s|<5|es|csJcsjcsfc:Jcsfcaie:Jc5jcsjcsJesjcjjc:fc}jc5jes|«:fc:jes|{}fcs|cj}c:}cs|c fidl='kl.out'; fid2='k2.out'; fopen(fidl,'w’); fopen(fid2,'w'); modes3=modes3/200; bwlO=fulO-fllO; for i=l:51 cc=corrcoef(s l(i, 1:200),s2(i, 1;200)); ccc(i)=cc(l,2); fprintf(fidl,'%8.3e %10.0f %14.3e %10.3f %10.3f %10.3Nt',f(i),modes(i),Q3(i),modes3(i),N3(i),ccc(i)) magmin=min(mag 1(i,:)); magave=mean(mag 1(i,:)); maxbw=max(bw 10(i,:)); minbw=min(bw 10(i avebw=mean(bwlO(i,:)); fprintf(fid2,'%8.3e %10.3f %10.3f %10.3e %10.3e %10.3eVi',f(i),magmin,magave,minbw,avebw,maxbw) end fclose('alT); clk2=dock; 102 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. save etsim.mat clkl clk2 (4) EQV50.M (Used by SIM 57.M ) %**calc qfac=10; sd = (pi*u*sig*f) .A (-.5); Q1 = ( (3/2)*V*ones(size(sd))) . / (S*sd); Q I 1 = 2*Q1; %** 2-sensor model Q2 = (3/2) * ( (V*ones(size(f)))./ (R*(AAA2)*e*2*pi*f)); Q3 = (1/qfac) * ( (Q1.*Q2)./(Q11+Q2) ); (5) S1MRES1.M (Used by SIM 57.M ) %*** Integers For Cavity Resonances Starting *** %*** With m=l n=l lp=lpnum; Ien=(lpA2)*(lp+l); ml=zeros(l,len); mm=l; il= l; i2=(lp)*(lp+l); del=(Ip)*0p+l); fo ri 100= 1:(lp), m 1(i 1:i2)=mm*ones( 1,del); mm=mm+l; il=il+del; i2=i2+del; end nl=zeros(l,2*len); nn=l; il= l; i2=(lp+l); del=(lp+l); fo ri 100= l:(lp), n 1(i 1:i2)=nn*ones( 1,del); nn=nn+I; il=il+del; i2=i2+del; end 103 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. i2=i2-del; while i2 < le n , il=i2+l; i2=2*i2; n l(il:i2 )= n l(l:il-l); end nl=nl(l:len); pl=zeros(l,2*len); pl(l:lp+l)=0:l:lp; i2=lpf 1; wliile i2 < le n , il=i2+l; i2=2*i2; p l(il:i2 )= p l(l:il-l); end pl=pl(l:len); (6) EVM 1.M (Used by SIM57.M) for SIM57.M to calc, cavity response************ %**calc. magnitude of second order response******** fO=fres(l); num=fOA2; den l=-(f(i)A2)+(fOA2); den2=f(i)*fO/Q3(i); den=(denlA2 + den2A2 ) A .5; mag=( l/Q3(i))*num/den; 104 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.

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