NUMERICAL MODELING OF ACTIVE DEVICES CHARACTERIZED BY MEASURED S-PARAMETERS IN FDTD

Transcription

1 Progress In Electromagnetcs Research, PIER 80, , 2008 NUMERICAL MODELING OF ACTIVE DEVICES CHARACTERIZED BY MEASURED S-PARAMETERS IN FDTD D. Y. Su, D. M. Fu, and Z. H. Chen Natonal Key Lab of Antennas and Mcrowave Technology Xdan Unversty X an, Chna Abstract A new FDTD modelng approach for actve devces characterzed by measured S-parameters s presented. Ths approach apples vector fttng technque and pecewse lnear recursve convoluton PLRC technque to complete modelng process, and does not need to know the equvalent crcuts of actve devces. It preserves the explct nature of the tradtonal FDTD method, and a general updated formula s derved. Furthermore, the man data-processng procedure s drectly handled over the frequency band of nterest, whch avods the tme-doman non-causal error n tradtonal technques. 1. INTRODUCTION Over the last decade, t s very popular to analyze hybrd mcrowave crcuts ncludng dstrbuted and lumped devces by usng electromagnetc smulators, such as ADS, Ansoft HFSS, IE3D and so on. However, not all these smulators ntegrate the lumped element and devce equatons wth the EM feld drectly, and the electromagnetc nteractons between ndvdual devces are neglected, so they do not accurately represent the physcal structure of the crcut. Thus these smulators are not sutable for analyzng hgh frequency ntegrated crcuts where the nteractons between the dstrbuted and lumped components play a vtal role. The fnte-dfference tme-doman FDTD method seems to be attractve for ths purpose [1]. Over recent ffteen years, much work has been reported for extendng the orgnal FDTD method to ncorporate lumped components [2 6]. Results ndcate that these extended FDTD approaches can successfully and accurately consder the electromagnetc nteractons between dstrbuted and lumped

2 382 Su, Fu, and Chen components. However, all these approaches are based on equvalent crcut models of lumped devces. Unfortunately, n the case of actve devces, only the measured S-parameters of the devce under certan basng condtons are provded by most manufacturers. Generally, we arenot easy to obtan equvalent crcut of the devce, replaced by It s very dffcult to obtan the equvalent crcuts of practcal complex actve devces, especally for the three-termnal devces. Moreover, some classes of devce are not even descrbed relably by the prevous technques. Therefore, the drect ncorporaton of actve devces characterzed by measured S-parameters nto FDTD formulatons s of great nterest. Lttle research has been reported n ths feld. In avalable papers [7, 8], the measured network parameters of actve devces n frequency-doman are frst transformed nto the tme-doman by drectly usng the nverse Fourer transform IFT technque. Then, the resultng tme-doman parameters are nputted nto the FDTD formulatons by convoluton products. In ths technque, the transformaton must ensure the system to satsfy the tme-doman causalty, but drect applcaton of IFT technque usually brngs noncausal tme doman data. In ths paper, a novel approach s developed to ncorporated actve devce characterzed by measured S-parameters nto FDTD algorthm by means of the vector fttng technque [9] and pecewse lnear recursve convoluton PLRC technque [10, 11]. It conssts of three man steps: Frst, the measured S-parameters are converted nto Y- parameters. Then each entry of whch s ftted nto a ratonal functon by usng vector fttng technque. Fnally, the coeffcents of the ratonal functons are drectly ncorporated nto FDTD equatons by means of PLRC technque. In ths proposed approach, the man data-processng procedure s drectly handled n frequency-doman, whch avods the tme-doman non-causalty. In addton, applyng the PLRC technque, we can obtan a general FDTD updated formula, whch preserves the explct nature of the tradtonal FDTD method [12 20] and mprove the computaton effcency. To show the valdty of ths proposed approach, t has been appled to calculate the S-parameters of a mcrowave FET amplfer. 2. THEORY ANALYSIS For the sake of concseness, the man focus s placed on the two-port actve devces. Assumng the two ports are assocated to the two electrc feld component E z1 and E z2. At nodes E z1 and E z2, the

3 Progress In Electromagnetcs Research, PIER 80, FDTD updated equaton s expressed n followng dscrete form as ε En+1 z1 Ez1 n [ ] = H n+1/2 J n+1/2 1 ε En+1 z2 t Ez2 n = t [ H z1 ] n+1/2 z2 J n+1/2 2 where s the permttvty of substrate, s the tme ncrement, and denote the devce current through the two ports, respectvely. The devce can be descrbed by ts Y-parameters matrx n the Laplace doman as follows [ ] [ ][ ] I1 s Y11 s Y 12 s V1 s = 2 I 2 s Y 21 s Y 22 s V 2 s where V p and I p p =1, 2 denote the voltage and current at the two ports Fttng Data wth Vector Fttng Technque In ths secton, the data-fttng procedure s dscussed. Accordng to the vector fttng technque, each entry of Y-matrx can be approxmated by the followng ratonal functon. Y pq s = N =1 1 c s a + g + sh, p, q =1, 2 3 The resdues c and poles a are ether real quanttes or come n complex conjugate pars, whle g and h are real. In general, c, a, g and h are dfferent for each Y pq s. N s the order of ratonal functon. We are to estmate all coeffcents n 3 so that a least squares approxmaton of Y pq s s obtaned over a gven frequency nterval. To smplfy notaton, the ndexes p, q wll be gnored n the followng dscusson [21]. We assume that the S-parameters of the actve devce have been measured at M frequency ponts s k = j2πf k k =1,...,M over the frequency band of nterest [f mn,f max ]. By use of standard networktheory expressons, these measured S-parameters are transformed nto Y-parameters Y s k. Now our goal s to get the coeffcents of the ratonal functon n 3, and a ratonal fttng technque must be selected. In the paper [14], a comparson of dfferent fttng technques s made, and vector fttng technque shows to be robust, accurate and quckly, so ths technque s adopted n ths paper.

4 384 Su, Fu, and Chen In ths fttng process, t s worth notng that to avod leadng to an ll-condtoned set of equatons, we carry out the fttng procedure over the normalzed frequency band [ fmn, f max ], where the normalzed frequency s defned as s k = j2π f k, fk = f k /f ave, wth f ave = 1 2 f mn + f max. The man fttng procedures are descrbed as follows: Step 1: Specfy a set of startng poles a, whch are dstrbuted over the frequency range of nterest, and ntroduce a weght functon σ s. Multply Y s wth the weght functon and ntroduce a ratonal approxmaton. σ s =1+ σ s Y s N =1 N =1 c s ā 4 c s ā + g + s h 5 where ā = a /f ave, c = c /f ave and h = h f ave. Furthermore, t s noted that the two equatons have the same poles, and Equaton 4 shows that the poles of Y s become equal to the zeros of σ s. Step 2: Substtutng 4 nto 5 yelds the followng equatons: 1+ N c Y s s ā =1 N =1 c s ā + g + s h 6 A set of measured Y s k for a gven frequency pont s k k =1, 2,...,M s replaced nto 6, whch leads to an over-determned lnear problem n the unknowns c, g, h and c : where [ A k = 1 s k ā 1 A k x = b k k =1, 2,...,M 7 1, 1, s k, Y s k Y s ] k s N ā N s k ā 1 s k ā N x = [ c 1 c N,g, h, c 1 c N ] T, bk = Y s k Step 3: Solve 7 va lnear least squares for the unknown { c }, and calculate the zeros of 4, then replace new poles ā by the zeros found. Repeatng the above procedure, we can obtan the approxmate poles a of the orgn problem 4 n few teratons. Step 4: Fnally, the rest unknowns c g and h can be solved accordng to the poles found and Y s k.

5 Progress In Electromagnetcs Research, PIER 80, To perform a stable FDTD smulaton, t s worth notng that none of the poles of Y pq s n 3 should be placed on the rght-hand sde of the s-plane. Hence, a verfcaton and correcton procedure f necessary must be appled at step 3 to turn the poles ā on the rght-hand sde of the s-plane f any nto the left-hand sde. Furthermore, vector fttng technque can also be appled drectly to vector functons wth the assumpton that all elements n the vector have dentcal poles. Hence, consderng the Y-matrx of a two-port devce, we can express t as a vector functon: Y s =[Y 11 s,y 12 s,y 21 s,y 22 s] T 8 Then, applyng an dentcal weght functon σs, we can obtan four ratonal functons of Y-parameters after runnng only one tme Dervng FDTD Formula wth PLRC Technque In ths secton, we dscuss the ncorporaton of the resultng ratonal functons nto FDTD algorthm by usng the PLRC technque n detal. Assume a convoluton operaton I n = n t 0 V n t τ Y τdτ 9 Accordng to the PLRC technque, we defne two varables as: χ m = ξ m = m+1 t m t 1 t Y τdτ 10 m+1 t m t If the varables meet the followng relatonshp ρ = χ m χ m 1 = τ m ty τ dτ 11 ξ m ξ m 1 12 the tme-doman current I n z can be updated usng recursve equaton: Iz n+1 =χ 0 ξ 0 Vz n+1 + ξ 0 Vz n + ρiz n 13

6 386 Su, Fu, and Chen We ntroduce four auxlary current ntenstes I pq sp, q =1, 2, and 2 can be expressed as I p s = I pq s p =1, 2 14 where q=1,2 I pq s =Y pq sv q s 15 Then, dvde the known ratonal functon of each Y-parameter nto two parts: Y pq s = N =1 c s a + g + sh = N =1 Y pq, s+y pq,o s 16 Thus, the tme-doman current I n+1/2 pq can be expressed as: N Ipq n+1/2 = =0 I n+1/2 pq, + I n+1/2 pq,o 17 We dscuss the two rght terms of the above formulaton n detal. 1. If the resdues c and poles a are real quanttes, Y pq, s = c s converted to the tme-doman by nverse Fourer s a transform technque Y pq, t =c e a t ut 18 ut s the unt step functon. By ntroducng 18 nto 10 and 11, we can get: χ m, = c 1 e a t e ma t a c [ ξ m, = 1 a 2 t a t e a t 1 ] e ma t Obvously, 12 can be met ρ = χ m, χ m 1, = ξ m, ξ m 1, = e a t 21

7 Progress In Electromagnetcs Research, PIER 80, In ths case, the tme-doman current Ipq, n can be updated usng recursve equaton: Ipq, n+1 = χ 0, ξ 0, Vq n+1 + ξ 0, Vq n + ρ Ipq, n If c and c +1, a and a +1 are complex conjugate pars, the Equatons are yet satsfed, but they can be smplfed. From the defnton of 10 12, we can prove that χ m, and χ m,+1, and ξ m,+1, ρ and ρ +1 are all complex conjugate pars, so ξ m, I n+1 pq, and I n+1 real quantty. pq,+1 I n+1 pq, + I n+1 pq,+1 = 2Re are also complex conjugate pars and ther sum s a +Re Ipq, n+1 ξ 0, { =2 Re χ 0, ξ 0, Vq n+1 V n q + Re ρ I n pq, } 23 So only one complex of a complex conjugate par s needed. In other words, f a and a contan 2N g conjugate complex, only N g varables need to be saved. 3. Substtutng Y pq,o s = g + sh nto 15, we can obtan I pq,o s = g + sh V q s. Appled the transformaton of s / t, the followng central-dfference formulaton can be acheved Ipq,o n+1/2 = g /2+h / t Vq n+1 + g /2 h / t Vq n 24 Fnally, substtutng nto 17 and assumng a are composed of N r complex total number N = N total current as: where real quanttes and N g r +2N g and a pars of conjugate, we can wrte the Ipq n+1/2 = A pq Vq n+1 + B pq Vq n + Ipq,t n 25 A pq =χ 0,t ξ 0,t + g /2+h / t 26 B pq =ξ 0,t + g /2 h / t 27 χ 0,t = 1 N r χ 0, + 2 =1 N r +N g =N r +1 Re χ 0, 28

8 388 Su, Fu, and Chen ξ 0,t = 1 N r ξ 0, + 2 =1 Ipq,t n = 1 N r ρ +1 2 I n+1 pq, = =1 χ 0, ξ 0, N r +N g =N r +1 Re ξ 0, N r +N g Ipq,+ n =N r +1 [ Re ρ +1 I n pq, ] Vq n+1 + ξ 0, Vq n + ρ Ipq, n 31 The relatonshps of the voltage wth the electrc feld and the current ntensty wth the current densty are approxmately expressed as Vz n = Ez n dz Ez n z 32 Iz n+1/2 = Jz n+1/2 dxdy Jz n+1/2 x y 33 From 14, the tme-doman current at each port can be expressed as Ip n+1/2 = Ipq n+1/2 p =1, 2 34 q=1,2 Fnally, substtutng and 25 nto 1, we can obtan the FDTD formulaton of the electrc feld at each port [ ] Ez1 n+1 Ez2 n+1 = 1 [ ] A11 + ε/α t A 1 [ 12 T n] 1 α A 21 A 22 + ε/α t T2 n 35 where α = z x y, Tp n = ε [ ] t En zp + H n+1/2 αb pq Ezq n + 1 zp x y In pq,t 36 q=1,2 From 35, we can fnd that ths approach preserves the full explct nature of the standard FDTD method. 3. NUMERICAL EXAMPLE To llustrate the valdty and accuracy of the proposed approach descrbed above, we apply ths approach to analyze a mcrowave

9 Progress In Electromagnetcs Research, PIER 80, Fgure 1. Structure of mcrowave FET amplfer. FET amplfer crcut, whch structure s shown n Fgure 1. The S- parameters of ths amplfer are calculated. The parameters of the mcrostrp lne used for our computatons are thckness of the substrate H = mm, wdth of the metal strp W = 2.4 mm, delectrc constant of the substrate ε = 2.2, and thckness of the metal strp zero, whch corresponds to the characterstc mpedance of the mcrostrp lne 50 Ω. In FDTD smulaton, the space steps are x = 0.4 mm, y = 0.4 mm and z =0.265 mm. The total mesh dmensons are 44 x 102 y 20 z. The dmenson of all metal strps can be referred to lterature [6]. The frst-order Mur s absorbng boundary condton s adopted to truncate the computng regon, except the boundary of z = 0. A Gaussan pulse source wth an nternal resstor 50 Ω s used to excte the crcut at the 1st port, and a matched load 50 Ω s connected to the 2nd port. The tme step s t =0.53 ps. A JS8851-AS FET s used n ths amplfer crcut, as shown n Fgure 1. The S-parameters of FET wth 41 frequency ponts lnearly dstrbuted over a frequency-band 4 GHz 8 GHz are obtaned, whch are llustrated n Fgure 2. Accordng to the proposed a b Fgure 2. S-parameters of FET a real part b magnary part.

10 390 Su, Fu, and Chen approach, the S-parameters over a fnte-band are frst converted nto the Y-parameters, whch can be expressed as a vector form,.e., Y s = [Y 11 s,y 12 s,y 21 s,y 22 s]. Applyng the teratve procedure of vector fttng technque and defnng the startng poles of 4e9, 5.33e9, 6.67e9, 8e9, then we obtan the fourth order ratonal functon approxmatons of Y-parameters wth an dentcal set of poles. Convergence s reached n two teratons. Fnally, the resultng coeffcents of the ratonal functons are drectly ncorporated nto the FDTD updated equatons by usng the PLRC technque. The smulaton s performed for 1600 tme steps. We calculate the S- parameters of ths amplfer crcut, whch are compared wth those n reference [6]. The compared results are shown n Fgure 3. It can be seen from Fgure 3 that good agreement s acheved over the frequency range of nterest. It demonstrates the accuracy and valdty of the proposed approaches. Fgure 3. S-parameters of the mcrowave FET amplfer crcut. 4. CONCLUSION Most of the extended FDTD methods are based on equvalent crcut models of lumped devces, but equvalent crcuts of most practcal actve devces are not easy to obtan. To overcome ths problem, n ths paper, we propose a novel approach for ncorporatng lumped devce nto the FDTD smulator. The vector fttng technque and PLRC technque are adopted to complete the modelng task. Some detals n the fttng process are dscussed to ensure the stable FDTD smulaton, and a full-explct FDTD updated formulaton s derved. Fnally, a numercal example s analyzed to llustrate the valdty of

11 Progress In Electromagnetcs Research, PIER 80, ths proposed method. It s useful for desgner to accurately analyze hybrd mcrowave crcuts ncludng practcal actve devces. ACKNOWLEDGMENT Ths work s supported by the Scence Fund of Chna No and No REFERENCES 1. Taflove, A., Computatonal Electrodynamcs-The Fnte- Dfference Tme-Doman Method, 2nd edton, Artech House, MA, Su, W., D. A. Chrstensen, and C. H. Durney, Extendng the two-dmensonal FDTD method to hybrd electromagnetc systems wth actve and passve lumped elements, IEEE Trans. Mcrowave Theory Tech., Vol. 40, No. 4, , Pket-May, M., A. Taflove, and J. Baron, FDTD modelng of dgtal sgnal propagaton n 3-D crcuts wth passve and actve lumped loads, IEEE Trans. Mcrowave Theory Tech., Vol. 42, No. 8, , Kuo, C. N., R. B. Wu, B. Houshmand, and T. Itoh, Modelng of mcrowave actve devces usng the voltage-source approach, IEEE Mcrowave Guded Wave Lett., Vol. 6, No. 5, , Chu, Q. X., X. J. Hu, and K. T. Chan, Models of small mcrowave devces n FDTD smulaton, IEICE Trans. Electron., Vol. E86- C, No. 2, , Reddy, V. S. and R. Garg, An mproved extended FDTD formulaton for actve mcrostrp crcuts, IEEE Trans. Mcrowave Theory Tech., Vol. 47, No. 9, , Zhang, J. Z. and Y. Y. Wang, FDTD analyss of actve crcuts based on the S-parameters, Asa Pacfc Mcrowave Conference, , Ye, X. N. and J. L. Drewnak, Incorporatng two-port networks wth S-parameters nto FDTD, IEEE Mcrowave Wreless Comp. Lett., Vol. 11, No. 2, 77 79, Gustavsen, B. and A. Semlyen, Ratonal approxmaton of frequency responses by vector fttng, IEEE Trans. Power Delvery, Vol. 14, No. 3, , 1999.

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