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  1. 1. IESE TRANSACTIONS 120 Computer-Aided Signal KARL ON MfCROWAVE THEORY AND TSCHNIQUES, Determination Equivalent Microwave HARTllIANN, STUDENT of the IEEE, J. O. STRUTT, WILLI KOTYCZKA, FELLOW, CIRCUIT 2, FEBRUARY 1972 SmaU- MEMBER, IEEE, AND IEEE r-----+ % C7 c ‘(couector) II b’(base) 0 R2 %= !1 EQUIVALENT NO. Clj Abstract—The actual technology for the production of transistors is now suitable for bipolar transistors in the 1O-GHZ range. In order to obtain amplifier circuits in this microwave range, the knowledge of the exact equivalent circuit is essential. A computer method for the determination of the equivalent network is given. The final application and accuracy are shown for a particular microwave transistor. 1. THE ~-20, Network of a Bipolar Transistor MEMBER, MAX VOL. OF AN INTRINSIC u~e, by I’:Ya&.U~et 0 TRANSISTOR IN COMMON CONFIGURATION EXTRINSIC O N THE tion imate equivalent which BASIS of the only neglected RC line of direction the 2). This our transistor of the [1], capacitances CI take approx- we transistor current find Im into connection lows CT). ’21 e’ at Co and CT are the in the backward the resistance / . account the internal base Fig. 2. increasing frequency Approximation of the transadmittance VZ1,’ is complex. Im L n may Because 4-to-8 GHz), we a circle ‘21e’ (Fig. worse want to a defined [3] by becomes we only relative from Pritchard be approximated Y21.’ by zero current the circle the frequency approximate the (in trans- 3, crossing the real I Fig. (see Fig. 1) may Re equiv- range point. I’ w at increas- to find of Fig. Y21.’. zone, aDOr The (C6, 1) and consideration into and approximation network at its elements m A the (Fig. density junction RI takes Yzl,’ admittance axis and extrinsic slightly. circle case Winkel intrinsic emitter-base base frequency. alent Te The transadmittance that Intrinsic [10]. is doped shows 1. determina- and small C6 and resistance between ing the [2], The the at Fig. experimental elements recombination, elements. which the by of valid extrinsic The of equations e’(em[tter) ELEMENTS intrinsic circuit is EMITTER AND THE be expressed 3. as fol- Oxtmate fiiquency range Approximation of the transadmittance employed in our computer program. rue’ as : ~’ = Ul),e,” Y,I.I This equation can work of Fig. 4. (It = – & be represented is possible ‘b’”’ =_ 1/1 (1] i- jwL7 by the to employ additional net- additional net- Manuscript received January 18, 1971; revised April 12, 1971. The authors are with the Department of Advanced Electrical Engineering, Swiss Federal Institute of Technology, Zurich, Switzerland. Fig. 4. Additional network for the transadmittance Y21e/.
  2. 2. HAR’CMANN et al.: EQUNALF.!NT NSTWORK OF MICROWAVS 121 TRANSISTOR II 1, COMPUTER-AIDED EQUIVALENT L, L2 MEASURED + -+J)J= A. Relation DETERMINATION NETWORK BASED SCATTERING Between OF THE ON THE PARAMETERS the Scattering Parameters awd the Currents As 0 Fig. 5. Equivalent network of microwave transistor package. L1, and Lo—lead inductances between the reference plane (scattering parameters) and the package. L,, and L,—lead inductances between the package edge and the gold wire connection. -&, Lf, and L-inductances of the gold wires of the emitter, base, and collector lead-in wires to the transistor pill (L, = O, because of the actual construction). Cl, cZ, CS, and C4—capacltances of the package. C~—-capacitance between input and output. is well known parameters a pure the [6], resistance pedance, In of our source output terminals value Input 1) Z to B. General % + t-!! n 1 V and Additional 6. First the are I 1 I based, I are for Relationship between s parameters and on by by works the for more validity The complicated of described R8 and elements curves without impairing the method.) transistor, When L7 are also found by optimiza- the tion. the gold including this first complete the For initial this proce- values said element of the optimization needed only complete wires are not is values as initial network as, as long as in pill. optimization network con- optimization The are These under package, The of the shorted first is con- optimization the method pill) wires. range the The later. optimization example, this gold the of consecutive and collector short estimation. for for of the measured (without frequency which we find the and elements error the two base, 5 is used. in detail in alone obtaining which the currents. sev- range the with comat The until found parameters found values. very the values 7. Fig. is using elements, a digital measured frequency the emitter, of Fig. is described using are package within used (5) limit. transistor scattering – S,,). are varied circuit shorted are (4) calculated a defined values network dure (2) .s12 as compared [11 ], while package 50 Q, the (3) is found circuit the sideration the the equivalent measured of with is below nections for to .s21. of a specified parameters, measured network Z. equal parameters points equivalent steps. net’#Ork Complete equivalent circuit and additional the complex transadmittance I?z1,’. network compared The at the of Optimization scattering frequency ones, R3 Fig. equivalent calculated ug~ then (1 – s,,) = 0.01.(1 Procedure The of the a voltage and Excitation: 1;’ are “First are valid. 11” = – 0.01 eral by im- 13xcitaiion: Output The ports reference 7). 12’ = – 0.01 puter. the terminals 11’ = 0.01 2) scattering at both called input (see Fig. equations of the Z== 20 = 50 ~. at the Uol = U02 is equal If definition is terminated case, is applied following by transistor procedure is taken into is terminated, consideration (Fig. 6), 11. EQUIVALENT NETWORK COMPLETE In FOR THE the elements TRANSISTOR package For microwave influence of the transistors package parts of the leads bining Fig. 5 with equivalent network (gold we as shown wires) Figs. of the 1 and have may 4, we transistor to in Fig. consider 5. The be neglected. find the as shown following of the alone sections entire raises only the transistor optimization is described no special of the because the difficulties. the C. Determination ohmic Com- complete in Fiz. Many of the 6. OJ the Elements optimization error is based —on function. the methods The corresponding of the Transistor are based solution on the gradient of the method of present Fletcher paper and
  3. 3. 122 IEEE TRANSACTIONS Powell [4]. Because culated frequently, it use of computer as possible, their tance by has to between For four two culate the [5] We input we calculate quency the value. frequency the sum has sum values to of the errors be current with of the the squared minimum errors is 2n if n is the number –IR. @R ‘j@IL ‘+L juVc#Z V1~v = Zm. IICV +lCV — GICV. = Z.,. %lDV zii AC @rDT- and Un- Origina[ range. at each of the #R= R.CPB *L =juL . $L Q. =ju C. +~ cal- III-A). frequency number VR=R. IR VL =jcoL. IL Ic =jw C V, is we input (see Section over This First Component of AP Note; V and 1 are the voltage and current in the original network; ~ and @ are the voltage and current in the adjoint network, of the to be calculated. output integrate The only Sensitivity (component of ~) Branch Relation in Adjoint Resistance Inductance Capacitance Current-controlled voltage source (see Fig. 11) transadmit- 1972 I Branch Relation the However, are calculations. respectively not the on FEBRUARY plement Type Function value and do based elements TECHNIQUES, economical [5]. since AND TABLE be as simple is THEORY be calcu- limits. network excitation, to for Rohrer pill parameters with output like and The frequency scattering possible important calculation of the Error every has its determination be extended, defined Formation gradient Director Yzle’ is complex. varied D. that gradient paper theory said is very time Our excellent the ON MICROWAVE Network fre- 1 1 individual of the variable elements. 5o11 Sum E of the squared errors ‘Y[wl’(.iw) U1’(jh))c = -+ ~ ? Q k=l ] (~2’(joJk))c (~2’(j@k))m – [2 + W,’(jhc) + ~1’’(j%) I (~1’’(j~k)). – (~,’’(j%))m w,’’(jm) ] (l,’’(jcok))c – (l,’’(jok))m l’] Fig. 8. (Li)lOWer number ~2n; k’ individual m measured c of j~h) , ~,’( ~1” ( jtih), jo~) and This of the a further second error sum function network limit, sum current with would output rapidly comes near if its excitation. limit. the this elements process, attain as the But we the to E. physical limit. it is very essential that chosen variation upper exact limits program their the the this definition elements are only at values choose If useful limits corresponding may the subroutine of EL is not definition cannot yond zation (8) function. that be infinite. W used, EL is added function input ET is sum argue minimization L; be- limit of EL is in the varied the of .E~ optimi- between the Li. or lower E. Calculation ET respectively. Error with excitation. increases elements current excitation; input ju,) could because output with limit; W is the weight where functions; and network EY=E+EL. value; weight input error value; output Now, total value; We jak) 1’”( upper The input j+,), lower ith and adjoint values J%+), ~~”( 11’ ( jcw) , 1“( frequency frequency calculated ~1’( ith (LJ.PPe, kl Original (6) where 1,”( Network I IZ + I Adjoint Y’;k 1’ – U,’(jw))m of the Error of the Gradient Sum Function EL (Li)lower + 1) Gradient (-w..er —— . z {x, – (L,),ower (L,)..,., (7) tion } -xi is based network where This Xi normalized ‘n number + WI’’(joJJ network of variable ~(~l’’*(jq))C element network (see III-E); With elements; – (~1’’”(jtih))~} lowing with method [5] and of the Error on the its mutually makes the Sum pairing use definition Function of adjoint of E: The originally network a theorem calculaspecified (see Fig. of Tellegen of E one can derive 13). [7]. the fol- equations: ~(~,’’(j~k))c wl’’(jow))c + 13p the Wz’’(jbk) [ (~2’’*(j@k))c – (~2’’*(j@k))m] ap 1} (9)
  4. 4. et al.: HARTMANN ---B ----Eir_3 EQUIVALENT NETWORK oF MIC3Z01VAVFi TRANSISTOR c, ----- RL T I’ I q Original Fig. 12, network Relation Adjoint between original and current-contl-oiled voltage L4 network adjoint sour,:e. network for a ----- ---- Fig. 9. r———— I_’_’_- ‘—l ic..??.2u!2??_o (?2 C* 123 Transformed network corresponding to Fig. 10. CT —--- RL C* R2 C3 I I’ LA ----- -.-— Fig. 10. Part I’ of the original network of a broken line. Fig. 6 as delimited by C7 L 1 AdJolnt Additmr@{ netwc,rk Fig. 13. Complete addltiona( adjoint [network network and adjoint additional network, ------------Fig. 11. Transformed network corresponding to Fig. 9. where Pth P network Because element; for = ap Re ~ (lo) [s’+s”]; first-order sensitivity with input s“ Re first-order sensitivity with output real [5] the Now [5], directions we must Output voltages lated. to input and of Fig. change the currents the measured values, 8; value, the UO1= 1 V, U02 = O necessary branch Fig. original network output currents have to is not work and we line network of the input analyses, excitation, (see Fig. of the network insert are calcumust the have very 8 are sign essential of the rela- of the been may Adjoint In~at sources of Fig. 8. part deterorig- by an interrupted use of an additional Y210’. The to the ad joint original 9, its current a further The of the source net- network, source having transformation 11. source voltage voltage to Fig. UW., as a current-controlled Ul, U2 as current-controlled consider voltage of the is equal With Fig. is completely network: of the is marked because part transplaced. We following network 6 which 10 is equivalent w-e obtain procedure transadmittance other ~’ is one. of the adjoint simple sensitivity I. This previous ad joint the The to Table in Fig. for +0.01.&. excitation: to the inal network 12’= c) Determination derived: two output frequency Because I are perform excitation every and of Table have Input At analogous excitation; i.e., according mination relations with a) excitation; part. we namely, error current of 111-A; b) s’ the tions calculated k=l V.): the theory k=n dE In this and sources, 1’ depends I) (by (by l’) on 1: Network Brarwh : According *elk’ = ((~1’’(jw))m – (Id’’),) k “ w’,’(jc.-%), = frequency . point. . are (11) . . treated In the to [5] the as shown adjoint current-controlled in Fig. network voltage sources 12. we must insert the voltage
  5. 5. IEEE TRANSACTIONS 124 Q ON MICROWAVE THEORY AND TSCHNIQUJ=, FEBRUARY 1972 d, Read: parameter 3) Initia( values b) Excitation-Response situation at thespeclfied Apply analysis Cait Uo, lvolt = store - program and the branch voltages Form error excitation and currents necessary for gradierrt calculation u~~=ovo(t frequency points {k )m ,(l;k )m,(l;k)m,(l;k)m k= r the input excitation L ~~1, ‘t’~2 1,.. ....n !I Calculate and store part ial gradient Form theerror E; relative to input excitation components excitation the Call analysis program to calculate the branch — for input (grad’) voltages Appty error excitation to ad joint and currents V& ,$$2 network !I Cal[ anatysis program and store the branch vo(tage$ App[y the output excitation u~, = o Vo(t U02 = 1 volt and currents for gradient Apply error excitation %$,.’i’~2 Form error necessary excitation ‘i& ,4’~2 to ad joint network calculation 1 , r Ca(culate Calcu(ate the total gradient Form the error E; relative to output excitation partial (grad’ + grad”) and store the Cal 1 ana(ysis partial gradient components for output excitation (grad”) — - program to calcu(ate the branch voltages and currents t Yes END Adjust network parameters according to F(etcher - PoweN method Fig. source (+) into source of the obtain the Except I. In ad joint for gradient the controlling controlled network case must “n diagrams The be zero. voltage Thus we elements, calculated gradient Go to r for computer the according component partial to Table numerical trast would Re . [S’+ Re [ variable have verified been 2) Gradient be determined 3) Normalization For the Elements: OR1)’ – (14) (~Rl”@RI)”]O gram it (in the our sidered. sensitivity case Thus of the I&), the the controlling controlled relationship branch current of (14) has has elements to be con- we to unity need The the total { – (~RiI[@R2 (1R2[@R2 + + method leads relationships @R2’])’ @m’1)”}. (15) of this EL: Function by a simple analytical of the Gradient proper that (initial normalized normalized axi Re by the con- number to an chapter numerically. dET — gradient (n being the network value gradient X~O) at 8E —“t’”: dxi This gradient calculation. and optimization of the Network subroutine elements the probe nor- start. Hence components. gradient of the dE —+— = ax{ ilEL to be changed as follows: . the this The the Error is essential malized For Thus saving. Of of 2n calculations elements). time becomes S”] –(lR1o determination require important can dlz —. 8R1 optimization. The of 13. branch are of RI the Flow of Fig. the controlling components the 14. branch. branch 1 6’X; error function is
  6. 6. HARTMANN et Q1.: EQUIVALENT NETwORK OF MICROWAVE TRANSISTOR 125 1 — 78 -90” Fig. 16. Scattering parameter s,, (transistor AT- IO IA: Ic, 3 mA). A measured; u calculated. -1 Fig. 15. AT-101A: Scattering parameters su and siz (transistor Avantek UcE, 10V; lc, 3 mA). A measured; U calculated. Ld where S,2 p’ initial value of the P actual value of the This computer wave transistor Without in the have The flow in Fig. 14. the Control Data Type of Technology bias to much by more computer used for which of the are Collector—Base Current 0165 0.023 .4 0.005 0.158 0,006 5 Im(SIZ) laborious 17. used 0 1s6 -00!5 6 -0026 0166 -0037 7 -0035 0162 -0050 8 — Scattering in summarized these programs is one of the computers Swiss Federal Institute computed AT-101 BY values A) is This final model C,=o. oll C,=o. ool C,= O.013 C,= O.005 C,= O.152 C,= O.316 C,=4.011 in the frequency AT-101A: range from As a comparison, and those shown the calculated in measured with Figs. the scattering final parameters, computed model, 15–17. ACKNOWLEDGMENT of a microwave at the following The authors Laboratory, wish Swiss to Federal of the above thank Institute U. Gysel, Microwave of Technology, for work. REFERENCES 10 V 3 mA C,= O.085 is valid su (transistor 3 mA). 4 to 8 GHz. his discussions L,=o. zl L2=0.20 parameter 10V; Ic, APPLICATION given: Voltage -0013 METHOD Capacitances (pF) . 0032 UCE, been OBTAINED Inductances (nH) L,= O.09 L,=O.31 L,3=0.36 L,=O.42 (S12) OIL6 are Avantek L,=O.16 R? 0 IL5 computer Fig. programs have PRESENTED the (Type conditions Im(s12) 0146 8 the PROGRAMS RESULTS THE an example, Reverse Collector 4 of Zurich. OF As 6500, Center ACTUAL transistor from calculation network procedure computer Computer SOME of the above The of the T7. range been COMPUTER diagrams out [GHZ] 0136 a micro. [9]). lV. carrying to gradient equivalent would and applied frequency described f R@ (5,2) element. been CALCULATED 0.162 of optimization [8] the the determination has ~,2 — element; network program GHz. (see network MEAsuRED [1] Resistors (Q) [2] R,= 14.8 R,=279.8 R,= 7.7 R,= 0.3 [3] [4] [5] [6] J. Te Winkel, “Drift transistor simplified electrical characterization, ” Electron. Radio Eng., vol. 36, 1959, pp. 28@-288. W. Baechtold, “The small-signal and noise properties of bipolar transistors in the frequency range 0.6 to 4 GHz/s” (in German), thesis 4207, Swiss Fed. Inst. Tech., Zurich, Switzerland. R. L. Pritchard, Electrical Characteristics qf Transistors. New York: McGraw-Hill, 1967. R. Fletcher and M. J. D. Powell, “A rapidly convergent descent method for minimization,” Cow@.d. ~., vol. 6, no. 2, 1963, pp. . . 163-168. S. IV, Director and R, A. Rohrer, “Automated network design— The frequency-domain case, ” IEEE Trans. Givcwit Theory, vol. CT-16, Aug. 1969, pp. 330--337. Hewlett-Packard, “S-parameters-Circuit al]alysis and de-
  7. 7. IEEE TRANSACTIONS 126 ON MICROWAVE sign, ” Application Note 95, Sept. 1968. [7] C. A. Desoer and E. S. Kuh, Basic Ciwait Theory, vol. 2. New York: McGraw-Hill, 1967. [8] W. Thommen and M. J. 0. Strutt, “Noise figure of UHF Transisters, ” IEEE T?am. Electron Devices, vol. ED-12, Sept. 1965, pp. 499–500. [9] W. Thommen, “Contribution to the signal- and noise equivalent circuit of UHF bipolar transistors” (in German), thesis 3658, Cavity Perturbation of the ISMAIL MEMBER, IEEE, AND Swiss Fed. Inst. Tech., 1965, Zurich, Switzerland. W. Beachtcld and M. J. O. Strutt, “Noise in microwave tranTrans. -kfic~owave Theo~y Tech., vol. MTT-16, sisters, ” IEEE Sept. 1968, pp. 578-585. K. Hartmann, W. Kotyczka, and M. J. 0. Strutt, “~quivalent networks for three different microwave bipolar transwtor packages in the 2-10 GHz range, ” E1.ctYon. Lett., no. 18, Sept. 1971, pp. 51@511. [10] [11] for C(IM PLEX GEORGE semiconductor T two ductor slab been or fills to reported by recently achieved with the this that is the for used Nag is very section lems. First, enough problem between the in attenuators sample should the due and re- ple at other phase has has precise, since the 20 dB) hand, and when the ac- commercial shifters. the The fact transverse [10]. quency shift in the interaction sample surement of placed Manuscript received January 28, 1971; revised May 28, 1971. This work was supported by the National Aeronautics and Space Administration under Grant NGL 23-005-183. I. I. Eldumiati is with Sensors, Inc., Ann Arbor, Mich. G. I. Haddad is with the Electron Physics Laboratory, Department of Electrical Engineering, university of Michigan, Ann Arbor, Mich. 48104. under a high that Q-factor, the er than that of the the energy the and energy perturbing the the If the sample in the material can the meaas sample cavity is chosen is much it can parameters be related the properties sample cavity, samstrong and the of a reentrant the defre- the The for size of the sample within by cavity material stored in in in the post con- resonant field, suitable the to techniques inserting perturbations. stored change by very central sam- difficult microwave the electric fields changes external guide are measured and cavity method of small a result the change the this of the of the perturbation of maximum between Imakes mode be is sam- InSb. of a resonant region may parameters Q-factor to the surface like cavity material the idea problem TEIO measuring for mode of a circular scheme material with erroneous a is normal mode a gap ” Helm’s contact to the This air even to give the a TEIO using a small suggested the converting into this is effect. field is tangential by The termining ple by method is that large sample walls, “gap electric a brittle second ductivity [7]- fact boundary. with The been accuracy mode, fill and field the realize the guide electric very to available completely surface, rectangular the The (nearly a transmission when method [4]. of the prob- get the shown [6] this to section; is always Helm serious cross when was IEEE hard waveguide effect overcome advantage ple’s serious Recently, to two is very waveguide and this MEMBER, poses it there sample [5]. complex the is not high using more and important the Second, fitting, results cases to fill crystal. SENIOR waveguide many becomes single tight Material of the in samples and materials, On a a semiconsection This method small, further one [1 ]– [3 ] and and reflection is very is degraded standards authors Datta to be measured method determine of measured first a waveguide high-conductivity phase-angle curacy been the coefficients. many by the with VSWR made transmission reviewed especially In are conductivity has methods. completely measurements flection material different Dielectric I. HAD DAD , cross takes microwave Measurement and transducer INTIIODUCTION MTT-20, NO. 2, FEBRUARY 1972 VOL. Semiconductor Abstract—Cavity perturbation techniques offer a very sensitive and highly versatile means for studying the complex microwave conductivity of a bulk material. A knowledge of the cavity coupling factor in the absence of perturbation, together with the change in the reflected power and the cavity resonance frequency shift, are adequate for the determination of the material properties. Thk eliminates the need to determine the @factor change with perturbation which may lead to appreciable error, especially in the presence of mismatch loss. The measurement accuracy can also be improved by a proper choice of the cavity coupling factor prior to the perturbation. HE TECHNIQUES, Conductivity of a Bulk 1, ELDUMIATI, I. AND Techniques Microwave Constant THEORY is with such small- be shown as a result to the cavity