WIDEBAND IMPEDANCE MATCHING IN TRANSIENT REGIME OF ACTIVE CIRCUIT USING LOSSY NONUNI- FORM MULTICONDUCTOR TRANSMISSION LINES

Transcription

1 Progress In Electromagnetics Research C, ol. 8, 7 45, 01 WIDEBAND IMPEDANCE MACHING IN RANSIEN REGIME OF ACIE CIRCUI USING LOSSY NONUNI- FORM MULICONDUCOR RANSMISSION LINES A. Amharech * and H. Kabbaj Laboratoire LSSC-Equipe de recherche CEM, Faculté des Sciences et echniques-fès, Université Sidi Mohamed Ben Abdellah, B.P. 0- Route d Imouzzer, FES, Maroc Abstract his paper focuses on the electromagnetic compatibility domain, coupling in microwave circuits and wideband (WB) impedance matching in time domain using a purely temporal method, such as the centered-points Finite Difference ime Domain (FDD). he paper here presents a new approach of WB impedance matching in transient regime and coupling context, of active circuits such as multiple complex nonlinear components (represented here by metal semiconductor fieldeffect transistors (MESFEs)), using Nonuniform Multiconductor ransmission Lines (NML) with frequency dependent losses and FDD as modeling method. he FDD method has several positive aspects such as the ease to introduce nonlinear components in the algorithm, the ease to use NML and the gain in simulation time and memory space. Also the FDD method allows the study of WB impedance matching in time domain without recourse to the frequency domain. Systematic comparisons of the results of this method with those obtained by PSpice are done to validate this study. hese comparisons show a good agreement between the method presented here and PSpice. he technique presented in this paper shows higher efficiency and ease to implement when compared to PSpice in regard to the treatment of frequency dependent losses, or shapes of transmission lines. Received 16 December 011, Accepted 1 February 01, Scheduled 16 March 01 * Corresponding author: Amine Amharech (amineetudes@yahoo.fr).

2 8 Amharech and Kabbaj 1. INRODUCION Impedance matching is a very important concept in telecommunications and microwave domain. o obtain the best performance of the circuits, it cannot be ignored. here are many techniques used to match loads to sources, such as the use of single stub, multi-stubs, uniform transmission line, sections of quarter-wave uniform transmission lines, tapered transmission line, or nonuniform and arbitrary nonuniform transmission line for matching impedance in narrow or wide band frequency [1 6]. hese techniques are usually used in the frequency domain and in non coupling context, just for single load as in [5, 7]. his is no longer the case today where the trend is towards integration of circuits and the context in which the coupling between the circuits can no longer be ignored. With the development of numerical data communications, the demand of high speed and wide-band circuit increases rapidly [7 11]. For these reasons the study of the transient domain of components is becoming more important, specifically for the microwave components lie MESFEs and their matching circuits taing into consideration frequency dependent losses and the nonlinear operation of the transistors. his paper presents a technique that uses regular NML with frequency dependent losses to achieve a WB impedance matching of multi-mesfes inputs in transient domain. he study is based on FDD method and the multi-mesfes are modeled by their large signal scheme.. ANALYSIS OF NML USING HE FDD MEHOD Under the assumption of a EM field structure and in the absence of external electromagnetic field, the ML with frequency dependent losses can be described, in time domain, by a system of partial differential equations as: z (z, t) + Z int (t) I (z, t) + L I (z, t) = 0, (1) t z I (z, t) + G (t) (z, t) + C (z, t) = 0, () t where is the convolution product. In the case of NML, the line is divided into multiple segments, of different sections (Figure 1). Each segment has its own distributed parameters (they can change from one segment to another).

3 Progress In Electromagnetics Research C, ol. 8, 01 9 Nonuniform ransmission line segments (Zint i, G i, L i, C i ) (Zint i+1, G i +1, L i +1, C i+1 ) 0 l position along the line z Figure 1. Model of a nonuniform transmission line. his leads to the new system of equations, where the parameters Z int, L, C and R depend on the position z along the line: z (z, t) + Z int (z, t) I (z, t) + L (z) I (z, t) = 0, (3) t z I (z, t) + G (z, t) (z, t) + C (z) (z, t) = 0. (4) t he N 1 vectors and I contain the N lines voltages as [(z, t)] i = i (z, t) and the N lines currents as [I(z, t)] i = I i (z, t), where the entry in the ith row of a vector is denoted as [] i, z is the position along the line, t denotes time. he N N matrices Z int, L, C and G are the per-unit length, conductor internal impedance, inductance, capacitance, and conductance describing the ML. In the case of a uniform line, the internal impedance contains both resistance and internal inductance (due to the magnetic flux in the conductors) as Z int (s) = R + sl int (s). In general, the impedance is a nonlinear function of frequency. hus, the description of the frequency dependent loss is introduced by using the widely used Nahman approximation of the internal impedance Z int (s) = A + B s as in [1] and Qingjian formulation of conductance G(s) = G dc + G df s (s+1/τ p) as in [13], where s is the Laplace transform variable.

4 30 Amharech and Kabbaj Where, for NML: 1 A(z)=, σ m w (z) t m B(z)= µ σ (w (z)+t m ), Gdc (z)= σ di C (z), (5) ε where σ m is the conductors conductivity, t m is the conductors thicness, w(z) is the conductors width along the line, σ di is the dielectric conductivity, µ is the permeability, ε is the permittivity and τ p is the time constant characteristic of the dielectric substrate. In the time domain, the dc portion of the impedance (A(z)) affects the late-time response whereas the high-frequency portion (B(z)) affects the early-time response. In order to adequately characterize the NML, both portions should be included. In time domain and according to the expression of Z int in [1], the convolution product Z int (z, t) I (z, t) becomes: Z int (z, t) I (z, t) = Lap 1 (Z int (z, s) I (z, s)) = Lap (A 1 (z) I (z, s)+b (z) 1 ) si (z, s). (6) s Knowing that Lap 1 ( 1 s ) = 1 πt, Z int (z, t) I(z, t) of Equation (6) becomes: Z int (z, t) I (z, t) = A (z) I (z, t) + B (z) t 1 I (z, t τ) dτ. (7) π τ (t τ) Also, in time domain and according to the expression of G in [13], the convolution product G(z, t) (z, t) becomes: G (z, t) (z, t)=lap 1 (G (z, s) (z, s)) =Lap 1 G dc (z)(z, s)+g df 1 (z)( )s(z, s). (8) s+ 1 τ p herefore, G (z, t) (z, t)=g dc (z)(z, t)+g df (z) t 0 0 τ/τp (z, t τ) e dτ. (9) (t τ) he FDD method is widely used in solving various inds of electromagnetic problems wherein lossy, nonlinear, and/or non homogeneous media may be considered. In order to evaluate voltages and currents along the transmission line, NML Equations (3) and (4) are translated into finite-difference expressions. he line axis z is divided into ndz cells, each of equal length z and the time variable t in ndt segments of equal length t.

5 Progress In Electromagnetics Research C, ol. 8, i m e t (n+3/) t () t (/) t n t t / I -1 z n (-3/) z (-1) z (-1/) z z Position (a) I n+3/ I / +1 z 1 I 1 ndz I ndz ndz +1 z z z Z=0 z/ z (ndz-1) z Z=l=ndz z (ndz-1/) z (b) z Figure. Illustration of (a) the FDD discretization of the voltages and currents along the ML and (b) the locations of the solution variables on the line. In order to ensure stability in the FDD solution of the ML equations, the discrete voltage and current solution points are not located at the same point; they are interlaced. Each voltage and adjacent current solution point are separated by z/. In addition, the discrete voltages and currents must be similarly interlaced in time with the time points for the voltages and for the currents being spaced by t/ [14] as illustrated in Figure. he use of the second order central differences according to the scheme in Figure (a) to discretize the derivatives in the NML, and according to Nahman approximation of Z int and Qingjian formulation of G and to Equations (7) and (9), Equations (3) and (4) leads to the recursive formula for interior point: [ ] 1 [ ] 1 z +1 + t L I n+ 3 I n [ ] A I n+ 3 +I n B [ ] n I n+ 3 m π t m=0 I n+ 1 (m+1) m dτ = 0, (10) τ m

6 3 Amharech and Kabbaj and hus and 1 [ z +G df I n+3/ I / n m=0 ] I / t C [ m = D E I / + D z n m [ n ] 1 + Gdc ] (m+1) m [ + n ] e t τp τ dτ = 0. (11) ( ) z +1 D B n, π t where = 1,,..., ndz. (1) ( L D = t + A + B ) 1, π t (13) E = L t A + B, (14) π t ] (m+1) n = n m=1 [ I n+ 3 m I n+ 1 m m dτ τ, (15) = M N n + M ( ) I n+ 1 z 1 1 In+ M G df n m=1 [ ] m f(m), where =1,,..., ndz+1. (16) ( 1 M = n m t (C +C 1 )+ 1 4 N = 1 t (C +C 1 ) 1 4 f (m)= f (0)= (m+1) m 1 0 ( ) G dc +Gdc ( ) 1 G df +Gdf 1 f(0)), (17) ( ) G dc +Gdc ( ) G df +Gdf 1 f(0), (18) e t τp τ dτ (19) e t τp τ dτ (0) Incorporating boundary conditions for = 1 (in the source) and = ndz + 1 (in the load), leads to: 1 = P 1 Q 1 1 n + ( z P 1 R 1 s + n ) s s I / 1 P 1 G df n [ 1 m 1 1 n m ] f (m), (1) m=1

7 Progress In Electromagnetics Research C, ol. 8, where and ndz+1 = P ndz+1q ndz+1 ndz+1 n +P ndz+1 z P ndz+1 G df P 1 = Q 1 = P ndz+1 = Q ndz+1 = C ndz+1 t n ndz+1 m=1 [ m ndz+1 ( I / ndz I L +In L ) n m ndz+1] f (m), () ( ) 1 C1 t + Gdc 1 + Gdf 1 f (0) + R 1 s, (3) ( ) C1 t Gdc 1 + Gdf 1 f (0) R 1 s, (4) ( 1 C ndz+1 + Gdc ndz+1 + G df t ndz+1 f (0)), (5) n I n Gdc ndz+1 + G df ndz+1f (0), (6) [( 1) z, n t], (7) I [( 1/) z, n t]. (8) For the stability of the FDD relations, the time step t must respect the Courant condition [15]. herefore t must be equal to or less than the v p max (Equation (9)). Where v p max is the largest phase velocity found in the propagation along the line. t z. (9) v p max In the case of linear loads (resistance), ndz+1 = L = R L I L. hus the Equation () becomes: where ndz+1 O ndz+1 = = O ndz+1x ndz+1 ndz+1 n + O ndz+1 z n O ndz+1 G df [ m ndz+1 ( C ndz+1 t X ndz+1 = C ndz+1 t ndz+1 m=1 I / ndz n m ndz+1] f (m), (30) ) 1 + Gdc ndz+1 + G df ndz+1 f (0)+ 1 z R 1 L, (31) Gdc ndz+1 + G df ndz+1 f (0) 1 z R 1 L. (3)

8 34 Amharech and Kabbaj When the transmission lines are connected by nonlinear loads, the above method may be similarly implemented and there is no change in the topological equations since the KL and KCL equations are independent of the branch relations. he current load (I L ) in Equation () is replaced by its nonlinear expression according to the nonlinear load. he solution of these equations starts with an initially relaxed line having zero voltages and currents values. First voltages along the line are solved for a fixed time from Equation (16) in terms of the previous solutions, and then currents are solved from Equation (1) in terms of these and previous values. 3. EQUAIONS A HE ENDPOINS OF NML FOR NONLINEAR LOADS Here the NML treated are loaded by multi-mesfes, in a commonsource configuration. he interest is focused on the nonlinear behavior of the transistor MESFE. Hence, as in [16] for single transistor, but without taing into account the gate resistance, or as in [17] for the case of multi-transistors, the intrinsic large signal equivalent scheme of the MESFE is adopted (Figure 3). It is taen from [18]. Where C gs ( g (t)) ( ) C gs0 ( b = g (t), for g < (F C. b ) C gs0 ( 1 3 F C b g (t) ), for g (F C. b ) (1 F C ) 3 (33) I L ndz +1 Gate g Cgs R i C gd Ids =f(g,ds) Ids G d C ds Drain i R L = 50 Ω Source Figure 3. Configuration of the end of NML loaded by a MESFE (the intrinsic large-signal model of a MESFE is in the dashed square).

9 Progress In Electromagnetics Research C, ol. 8, I ds ( g (t), ds (t)) ( = A 0 δ+a 1 g (t)+a g (t) +A 3 g (t) 3) tanh (α ds (t)), (34) where b is the built-in potential of the Schotty gate, F C is forward bias depletion capacitance coefficient and δ is a (N 1) vector of 1. he expressions taen from [18, 19] are scalars and are written to describe a single transistor. In this study several transistors are simultaneously treated. So for that, these expressions were developed to describe multi-transistors. Scalars are now (N 1) vectors for voltages and currents, and (N N) matrices for the transistors parameters. Where N is the number of transistors treated. From Figure 3 (as in [17]) and in terms of finite differences, the expression of I, and are extracted and are: L I ( L = C gs g ndz+1 = ndz+1 ( g + R i C gs g [ ] [ ] = R L (C n ndz+1 ds C n ndz+1 gd n t t t ( ) +G d + A 0 δ+a 1 g +A g + A 3 g 3 tanh ( α )) (35) ) [ ] g g n + C gd [ n t t ndz+1 + n ] [ + C gd ( ) [ ]] g + R i C gs g g n g, (36) t t ) [ ] g n g t Replacing I L of Equation (36) and ndz+1. (37) of Equation (37) in Equation () leads to a new equation relating g and. With the combination of the new equation relating g and and the Equation (35), a new system of equations is obtained, from which the values of g and are deduced. Once g and are determined, the value of is deduced. ndz+1 o solve this new obtained system of equations, where the only unnown term is g, the use of the Newton-Raphson method as shown below is required: H g + f ( g ) = 0, (38) where f is a term of g and where the nonlinear components are collected.

10 36 Amharech and Kabbaj he matrix g is found by solving the following equation: ( ) m+1 ( ) g = m g (JH ) 1 H, (39) where J H = dh g = ( g ) m. (40) d g Once g found, can be determined and thereafter ndz+1. he expressions of, I L and ndz+1 in Equations (35), (36) and (37) are originally scalars according to Figure 3 and to the expressions of nonlinear elements taen from [18, 19]. hey are developed to (N 1) vectors (where N is the number of conductors or transistors in the line) to describe a system of multiconductor transmission line loaded by multi MESFEs. 4. NUMERICAL EXAMPLES he study is focused on the wide band impedance matching of the multi MESFEs in time domain using NML. It is based on the use of tapered lines. For that and as an application of NML, the NML (Figure 4) whose section and characteristic impedance change continuously, is chosen. his structure of NML, which consists of triangular NML, allows to go, gradually, from one impedance to another while avoiding reflections. Figure 4. riangular NML used to implement the wideband impedance matching of the multi-mesfes inputs. For the numerical examples two cases of loads are chosen: linear loads as resistances and nonlinear loads as MESFE transistors First Example For the first example, the system as shown in Figure 5 is considered. his example is considered to validate our study in the linear case. he input source is a ramp of 0.5 in amplitude, with a rise time of 100 ps. he length of the line is cm. W st, W end, S st and S end are the dimensions of the conductors line. hey are chosen to obtain the required characteristic impedance in the

11 Progress In Electromagnetics Research C, ol. 8, beginning and in the end of the line. In this example the beginning characteristic impedance of the line should be 50 Ω. Gradually, an impedance of 100 Ω is reached (line s end characteristic impedance). he results of this configuration are compared to those of a mismatched case, where the line is uniform. he length of the line is cm and is loaded by a resistance of 100 Ω. he parameters of this line are C = e 1 F/m and L =.10418e 6 H/m, which gives a characteristic impedance of 04 Ω. he comparison between Figure 6 and Figure 7 shows that the settling time of mismatched circuit (1. ns) is bigger than the matched Figure 5. Configuration of a WB impedance matching in transient domain of a resistance using NML. (a) Figure 6. Input and output voltage for mismatched circuit (uniform transmission line). (a) FDD. (b) PSpice. (b)

12 38 Amharech and Kabbaj one (0.4 ns). Also, the matcher circuit eliminates overshoot and the distortions, due to the elimination of reflections. 4.. Second Example In the second example, the system as shown in Figure 8 is considered. he transistor is in common-source configuration. he input source is a ramp of 0.5 in amplitude, with a rise time of 100 ps. he length of the line is. cm. he parameters of the transistor are shown below: C gs0 = pf, C ds = 0.6 pf, C gd = 1.5 pf, R i = 1 Ω, G d =0. ms, b = 0.7, α = 1.6, A 0 = , A 1 = , A = (a) (b) Figure 7. Input and output voltage for WB matched circuit in transient regime. (a) FDD. (b) PSpice. Figure 8. Configuration of a WB impedance matching in transient domain of a non linear MESFE using NML.

13 Progress In Electromagnetics Research C, ol. 8, Figure 9. Input and output voltages in the case of mismatch circuit loaded by a MESFE (FDD). (a) (b) Figure 10. Input and output voltages in the case of WB matched circuit loaded by MESFE. (a) FDD and (b) PSpice , A 3 = 0.001, F C = 0.5. he input resistance of the MESFE was measured by a ime Domain Reflectometer (DR) and is equal to 15 Ω. In this example the beginning characteristic impedance of the line should be 50 Ω. Gradually, an impedance of 15 Ω is reached (line s end characteristic impedance). his impedance corresponds to the MESFE input impedance, as measured by the DR. he results of this configuration are compared to those of a mismatched case, where the line is uniform with a characteristic impedance of 04 Ω. he length of the line is. cm and is also loaded by a transistor with the same parameters. he comparison between Figure 9, and Figure 10 on the one hand

14 40 Amharech and Kabbaj Figure 11. ransistor output voltage in the case of mismatched circuit loaded by MESFE (FDD). (a) Figure 1. ransistor output voltage in the case of matched circuit loaded by MESFE. (a) FDD and (b) PSpice. (b) and Figure 11 and Figure 1 on the other, shows that the settling time of mismatched circuit ( >5 ns) is bigger than the matched one (1.5 ns). Also, the matcher circuit eliminates overshoot and the distortions, due to the elimination of reflections and increased the output voltage of the matched transistor from approximately 55 m to about 85 m hird Example In the third example, the system as shown in Figure 13 is considered. he transistors are in common-source configuration. he input source is a ramp of 0.5 in amplitude, with a rise time of 100 ps. he line is composed of five conductors, plus the reference conductor. he length of the line is cm.

15 Progress In Electromagnetics Research C, ol. 8, wo cases are treated: lossless line and lossy one. For the case of line with frequency dependent losses: σ di =0.0 S/m, and G df = 0 ms m, where = is the identity matrix. he parameters of the transistors are shown below: C gs0 = pf, G d = 0. ms, C ds = 0.6 pf, C gd = 1.5 pf, R i = 1 Ω, b = 0.7, α = 1.6, A 0 = , A 1 = , A = , A 3 = 0.001, F C = 0.5. he MESFEs used here are identical to the one used in the second example. herefore, each one has 15 Ω in the input impedance. In the case of lossless NML, results are shown in Figure 14. For the case of NML with frequency dependent losses, obtained results are shown in the Figure 15. As in the first example and in the second one, Figure 14 and Figure 15 show that the WB impedance transformer circuit eliminates overshoot and the distortions, due to the elimination of reflections. With this matched circuit, a short settling time (1.5 ns) is obtained, compared to the one of mismatched circuit (> 5 ns), the cancellation of the distortions and the overshoot. he dimensions of the triangular NML in case of one conductor Figure 13. Configuration of a WB impedance matching in transient regime of a nonlinear multi-mesfe using NML.

16 4 Amharech and Kabbaj (a) (b) (c) Figure 14. Inputs, outputs and transistors voltage in the case of WB matched multi-mesfes for the lossless NML. (a) FDD, (b) PSpice, (c) FDD, and (d) PSpice. (d) (a) Figure 15. (a) Inputs, outputs voltages and (b) transistor outputs voltages in the case of WB matched multi-mesfes for the NML with frequency dependent loss. (b)

17 Progress In Electromagnetics Research C, ol. 8, loaded with one MESFE are different from the dimensions of the triangular NML in the case of multi-conductors loaded with several MESFEs. It is due to the coupling between conductors. 5. CONCLUSION he wideband impedance matching of multi-nonlinear load using NML, whose section and characteristic impedance change continuously, were described and simulated in transient domain and coupling context. his type of impedance matching, using NML, is based on the use of tapered lines. A computational approach based upon the centered-points finite-difference time-domain (FDD) technique for evaluating voltages and currents along the NML with frequency dependent losses (metallic losses and dielectric losses) was developed and used. his technique allows to introduce, easily, multi-nonlinear complex components in the algorithm, as the GaAs MESFEs studied here. A new system of equations describing nonlinear elements in MESFEs, based on matrices was developed. his system of equations was solved by using the Newton-Raphson s method. Several examples were treated. In all the examples, the WB impedance transformer eliminates the overshoot and the distortions, due to the elimination of reflections. As a result, it reduces the settling time of the circuit. he results of the simulation with FDD method have been compared to those obtained in PSpice, in the case of one conductor plus the reference conductor, connected to linear element (resistance), in the case of one conductor and a reference conductor, connected to strongly nonlinear element (one GaAs MESFE) and in the case of six (including the reference conductor), frequency dependent, lossless and lossy nonuniform conductors connected to strongly nonlinear elements (five GaAs MESFEs). A good level of agreement has been achieved. In addition to being very flexible in treating nonuniform lines (can deal with arbitrary NML) and frequency dependent losses, the proposed method has no limitation in the number of conductors (compared to some software). Moreover, the CPU time of this approach is less than the conventional time-domain to frequencydomain (DFD) solution technique. REFERENCES 1. Collin, R. E., Foundations for Microwave Engineering, McGraw- Hill, 1996.

18 44 Amharech and Kabbaj. Pozar, D. M., Microwave Engineering, John Wiley & Sons, Liao, S. Y., Microwave Circuit Analysis and Amplifier Design, Prentice-Hall, Ha,.., Solid-State Microwave Amplifier Design, John Wiley & Sons, Khalaj-Amirhosseini, M., Wideband or multiband complex impedance matching using microstrip nonuniform transmission lines, Progress In Electromagnetics Research, ol. 66, 15 5, Huang, X.-D., X.-H. Jin, and C.-H. Cheng, Novel impedance matching scheme for patch antennas, Progress In Electromagnetics Research Letters, ol. 14, , Zhang, B, A D-band power amplifier with 30-GHz bandwidth and 4.5-DBM P sat for high-speed communication system, Progress In Electromagnetics Research, ol. 107, , Reynolds, S. K., B. A. Floyd, U. R. Pfeiffer,. Beuema, J. Grzyb, C. Haymes, B. Gaucher, and M. Soyuer, A silicon 60-GHz receiver and transmitter chipset for broadband communications, IEEE J. Solid-State Circuits, ol. 41, No. 1, , Dec Powell, J., H. Kim, and C. G. Sodini, SiGe receiver front ends for millimeter-wave passive imaging, IEEE rans. Microw. heory ech., ol. 56, No. 11, , Nov Nicolson, S.., A. omins, K. W. ang, A. Cathelin, D. Belot, and S. P. oinigescu, A 1., 140 GHz receiver with on-die antenna in 65 nm CMOS, IEEE Radio Frequency Integrated Circuits Symposium, 9 3, Jun Lin, Y.-J., S. S. H. Hsu, J.-D. Jin, and C. Y. Chan, A GHz ultra-wideband CMOS low noise amplifier with currentreused technique, IEEE Microwave and Wireless Components Letters, ol. 17, No. 3, 3 34, Mar Nahman, N. and D. Holt, ransient analysis of coaxial cables using the sin effect approximation A + B s, IEEE ran. on Circuit heory, ol. 19, No. 5, , Sept Yu, Q. and O. Wing, Computational models of transmission lines with sin effect and dielectric loss, IEEE rans. on Circuits And Systems I: Fundamental heory and Applications, ol. 41, No., , Feb Orlando, A. and C. R. Paul, FDD analysis of lossy, multiconductor transmission lines terminated in arbitrary loads, IEEE rans. Electromagnetic Comp., ol. 38, No. 3, ,

19 Progress In Electromagnetics Research C, ol. 8, Aug Li, K., M. A. assoudji, R.. Shin, and J. A. Kong, Simulation of electromagnetic radiation and scattering using a finite difference time domain technique, Comput. Appl. in Eng. Education, ol. 1, No. 1, 45 6, Sept./Oct Kabbaj, H. and J. Zimmermann, ime-domain study of lossy nonuniform multiconductor transmission lines with complex nonlinear loads, Microwave and Optical echnology Letters, ol. 9, No. 5, , June Amharech, A. and H. Kabbaj, Analysis of multiconductor transmission line loaded by multi MESFE transistors modeled by their large signal scheme: An FDD approach, International Journal on Communications Antenna and Propagation, ol. 1, No. 3, Kuo, C. N., B. Houshmand, and. Itoh, Full-wave analysis of pacaged microwave circuits with active and nonlinear devices: An FDD approach, IEEE rans. Microwave heory ech., ol. 45, No. 5, , May Su, H.-H., C.-W. Kuo, and. Kitazawa, A novel approach for modeling diodes into FDD method, PIERS Proceedings, Marraesh, Morocco, Mar. 0 3, 011.