TIME DOMAIN ANALYSIS OF ACTIVE TRANSMISSION LINE USING FDTD TECHNIQUE (APPLICATION TO MICROWAVE/MM- WAVE TRANSISTORS)

Progress In Electromagnetics Research, PIER 77, 309–328, 2007

TIME DOMAIN ANALYSIS OF ACTIVETRANSMISSION LINE USING FDTD TECHNIQUE(APPLICATION TO MICROWAVE/MM-WAVETRANSISTORS)

K. Afrooz, A. Abdipour, A. Tavakoli, and M. Movahhedi

Microwave/mm-Wave & Wireless Communication Research LabRadio Communication Center of ExcellenceElectrical Engineering DepartmentAmirkabir University of Technology424 Hafez Ave., Tehran, Iran

Abstract—In this paper, an accurate modeling procedure for GaAsMESFET as active coupled transmission line is presented. This modelcan consider the effect of wave propagation along the device electrodes.In this modeling technique the active multiconductor transmissionline (AMTL) equations are obtained, which satisfy the TEM wavepropagation along the GaAs MESFET electrodes. This modelingprocedure is applied to a GaAs MESFETs by solving the AMTLequations using Finite-Difference Time-Domain (FDTD) technique.The scattering parameters are computed from time domain results overa frequency range of 20–220 GHz. This model investigates the effectof wave propagation along the transistor more accurate than the slicemodel, especially at high frequencies.

1. INTRODUCTION

Due to the required increasing performance and lower cost, monolithicmicrowave integrated circuits (MMICs) are evolving with a largenumber of closely packed passive and active structures, severallevels of transmission lines and discontinuities on the same chip [1].With the increasing flow of data in telecommunication world, thehigh-performance electronics that are based on MMIC technologiesoperating at high speeds, frequencies, and sometime over very broadbandwidths are needed. Also by increasing the operating frequency,devices and circuits need to more and more accurate techniquesfor modeling and simulation [2]. For accurate device modeling an

310 Afrooz et al.

electromagnetic interaction must be taken into account, especiallywhen the gate width is on the order of the wavelength [2]. Whenthe device dimension become comparable to the wavelength the inputactive transmission line, the gate electrode, has a different reactancefrom the output transmission line, the drain electrode. Therefore, theyexhibit different phase velocities for the input and output signals. Soby increasing the frequency or device dimension the phase cancellationdue to the phase velocity mismatching will effect the performance ofthe device [3, 4]. In such cases, wave propagation effect influences theelectrical performance of the device, so this phenomena needs to beconsidered accurately in device modeling. The full wave analysis andglobal modeling approach can be used to consider the wave propagationeffect along the device structure, accurately. But, this type of analysisis time consuming and needs a huge CPU time. Although, someefficient numerical methods have been recently proposed for simulationtime reduction [5–9], but it sounds that this analysis approach needsmore attention for implementing in simulation software. On theother hand, device behavior in high frequencies can be well describedusing semi-distributed model which can be easily implemented inCAD routines of simulators [9–12]. But the semi-distributed modelcannot consider wave propagation effect and phase cancellation on theelectrical performance of the device [2]. In the paper, we propose anew modeling procedure for GaAs MESFET which can consider theeffect of wave propagation along the device electrodes. In the proposedapproach, the transverse electromagnetic (TEM) wave propagation isinvestigated on the electrodes of the device. In this model, the devicewidth is divided into infinity segments. Each segment is consideredas a combination of three coupled lines and a conventional equivalentcircuit of a GaAs MESFET. Its parameters in the MESFET modelcircuit are obtained from DC and low frequency measurements (Fig. 2).The transmission line theory is applied to a segment of transistor toobtain the wave equation in a GaAs MESFET structure. Now thissystem of differential equations (active multiconductor transmissionline (AMTL) equation) must be solved. Since a time domain analyticalsolution doesn’t exist for this system, this problem is needed tosolve using a numerical technique. The Finite Difference TimeDomain (FDTD) method is widely used in solving various kinds ofelectromagnetic problems, wherein lossy, nonlinear, inhomogeneousmedia and transient problem can be considered [13]. This techniqueis used to solve the obtained equations. By applying this modelingtechnique, the scattering parameters of a sub micrometer-gate GaAstransistor are calculated over a frequency range of 20–220 GHz. Theresults achieved from this model (Fully distributed) are compared with

Progress In Electromagnetics Research, PIER 77, 2007 311

slice model. It is shown, at the low frequencies, the results of semi-distributed and fully distributed models are the same. By increasingthe frequency, the results of two models are not in a good agreement.Because the fully distributed model is a modified version of the semi-distributed one when the number of slices increases to the infinity andconsider the wave propagation, we expect the proposed method becomemore accurate. This is confirmed by comparing the S-parameterscalculated using both models over a wide frequency band.

Figure 1. The schematic of GaAs MESFET.

2. MODEL IDENTIFICATION

A typical millimeter-wave field effect transistor is shown in Fig. 1. Thedevice consists of three coupled electrodes fabricated on a thin layerof GaAs supported by a semi-insulating GaAs substrate. As operatingfrequency of the microwave GaAs MESFETs increase to the millimeterwave range, the dimensions of the electrodes become comparable tothe wavelength. In this situation the transmission line properties ofthe electrodes need to be considered [22–24]. Fully distributed modelis one of the accurate models which is applied to calculate the effect ofwave propagation along the electrodes of a GaAs MESFET.

For low enough frequencies, the longitudinal EM field is verysmall in magnitude as compared to the transverse field [2, 3].Therefore we can consider quasi-TEM modes and the generalizedactive multiconductor transmission line (AMTL)’ equation. Thisequation can be used to describe the instantaneous voltage and current

312 Afrooz et al.

dL

gL

sL sR

dR

gR

gpC

spC

dpCdgpC

iR

gsC

gdC

+

-gsmvg ' dsC dsG

gsv'

gspC

Passive part

Active part

dgM

gsM

Figure 2. An equivalent circuit model of a differential length of atransistor (AMTL).

relationship in the transistor. Consider an element portion of lengthof a three-Active transmission line. We intend to find an equivalentcircuit for this line and derive the transistor equations. An equivalentcircuit of a portion of the transistor is shown in Fig. 2. Eachsegment is represented by a 6-ports equivalent circuit which combinesa conventional MESFET small signal circuit model and another circuitelement to account the coupled transmission line effect of the electrodestructure where the all parameters are per unit length. By applyingkirchhoff’s current law to the left loop of the circuit in Fig. 2 and inthe limit as ∆z → 0, we obtain the following three equations:

∂

∂zId(z, t) + C1

∂

∂tVd(z, t) − C12

∂

∂tVg(z, t) − C13

∂

∂tVs(z, t)

+Gds(Vd(z, t) − Vs(z, t)) + GmVg(z, t) = 0 (1)

∂

∂zIg(z, t) − C12

∂

∂tVd(z, t) + C2

∂

∂tVg(z, t) − C23

∂

∂tVs(z, t)

+Cgs∂

∂tVg(z, t) = 0 (2)


∂

∂zIs(z, t)−C13

∂

∂tVd(z, t)−C23

∂

∂tVg(z, t)+C3

∂

∂tVs(z, t)−Cgs

∂

∂tVg(z, t)

−Gds(Vd(z, t) − Vs(z, t)) − GmVg(z, t) = 0 (3)

Also, we can write an extra equation:

´Vg(z, t) + Vs(z, t) + RiCgs∂

∂tVg(z, t) − Vg(z, t) = 0 (4)

whereC1 = Cdp + Cds + Cdsp + Cdg + Cdgp

C2 = Cgp + Cgsp + Cdg + Cdgp

C3 = Csp + Cds + Cdsp + Cgsp

C12 = Cdg + Cdgp

C13 = Cds + Cdsp

C23 = Cgsp

similarly, applying the kirchhoff’s voltage law to the main node of thecircuit and in the limit as ∆z → 0 in Fig. 2 gives:

∂

∂zVd(z, t) + RdId(z, t) + Ld

∂

∂tVd(z, t) + Mdg

∂

∂tVg(z, t)

+Mds∂

∂tVs(z, t) = 0 (5)

∂

∂zVg(z, t) + RgIg(z, t) + Lg

∂

∂tVg(z, t) + Mdg

∂

∂tVd(z, t)

+Mgs∂

∂tVs(z, t) = 0 (6)

∂

∂zVs(z, t) + RsIs(z, t) + Ls

∂

∂tVs(z, t) + Mds

∂

∂tVd(z, t)

+Mgs∂

∂tVg(z, t) = 0 (7)

The above equations could be simplify in two matrix equations asfollows:

∂

∂z

Id

Ig

Is

0

+

∂

∂t

C1 −C12 −C13 0−C12 C2 −C23 Cgs

−C13 −C23 C3 −Cgs

0 0 0 RiCgs

Vd

Vg

Vs

Vg

+

Gds 0 −Gds Gm

0 0 0 0−Gds 0 Gds −Gm

0 −1 1 1

Vd

Vg

Vs

Vg

= 0 (8)

314 Afrooz et al.

∂

∂z

(Vd

Vg

Vs

)+

∂

∂t

(Ld Mdg Mds

Mdg Lg Mgs

Mds Mgs Ls

) (Id

Ig

Is

)

+

(Rd 0 00 Rg 00 0 Rs

) (Id

Ig

Is

)= 0 (9)

where Id, Vd, Ig, Vg and Is, Vs are the drain, gate and source currentsand voltages, respectively and Vg is gate-source capacitance voltage.

Position

Tim

e

Z = 0 Z = k Z = Nz+1

1nI +

o

nIo

2/1n1I +

2/1n1kI +

+2/1n

kI +

2/3nkI +

2/1nNzI -

1n1NzI +

+

n1NzI

+

nkV

1n1kV +

+

1nkV +

2

Z2

Z

2t

D

∆

∆

2

Z∆

Figure 3. The relation between the spatial and temporaldiscretization to achieve second-order accuracy in the discretizationof the derivatives.

3. THE FDTD SOLUTION OF THE AMTL EQUATION

The fully distributed model of a GaAs MESFET is embodied in theAMTL equations

∂

∂zI(z, t) + C

∂

∂tV(z, t) + GV(z, t) = 0 (10)

∂

∂zV(z, t) + L

∂

∂tI(z, t) + RI(z, t) = 0 (11)


where

V(z, t) = [Vd(z, t), Vg(z, t), Vs(z, t)]T (12)

I(z, t) = [Id(z, t), Ig(z, t), Is(z, t)]T (13)´V(z, t) = [Vd(z, t), Vg(z, t), Vs(z, t), Vg(z, t)]T (14)

I(z, t) = [Id(z, t), Ig(z, t), Is(z, t), 0]T (15)

Now a suitable technique should be selected to solve the AMTLequations. One of the best methods which can be used to find thesolution of AMTL equation is FDTD technique. The FDTD techniqueseeks to approximate the derivatives with regard to discrete solutionpoints defined by the spatial and temporal cells [13–15]. The AMTLequations are coupled, first-order partial differentional equations likeMaxwell’s equations. Applications of the FDTD method to the full-wave solution of Maxwell’s equations have shown that accuracy andstability of the solution is achieved if we choose the electric andmagnetic field solution points to alternate in space and be separatedby one-half the position discretization, e.g., ∆z/2 [14]. We choose thesolution times for these two equations to also be interlaced in time andseparated by ∆t/2. In order to insure stability of the discretization andto insure second-order accuracy we interlace the Nz +1 voltage points,V1, V2, . . . , VNz , VNz+1, and the Nz current points, I1, I2, . . . , INz , asshown in Fig. 3 [16–19]. Each voltage and adjacent current solutionpoint is separated by ∆z/2. In addition, the time points are alsointerlaced, and each voltage time point and adjacent current time pointare separated by ∆t/2 as illustrated in Fig. 3. With applying the finitedifference approximation to (10) and (11) gives:

V n+1k+1 − V n+1

k

∆z+ L

In+ 3

2k − I

n+ 12

k

∆t+ R

In+ 3

2k + I

n+ 12

k

2= 0 (16)

Ikn+ 1

2 − ´Ik−1n+ 1

2

∆z+ C

V n+1k − V n

k

∆t+ G

V n+1k + V n

k

2= 0 (17)

where we denote

Vji ≡ V ((i − 1)∆z, j∆t) (18)

Vji ≡ V((i − 1)∆z, j∆t) (19)

Iji ≡ I((i − 1

2)∆z, j∆t) (20)

Iji ≡ I((i − 1

2)∆z, j∆t) (21)

316 Afrooz et al.

Solving these equations give the required recursion relations:

Vn+1k =

(C∆t

+G2

)−1

(C∆t

− G2

)Vn

k −In+ 1

2k − I

n+ 12

k−1

∆z

(22)

In+ 3

2k =

(L∆t

+R2

)−1(

L∆t

− R2

)In+ 1

2k −

Vn+1k+1 − Vn+1

k

∆z

(23)

The leap-frog method is used to solving the AMTL equationsbecause of its simplicity and accuracy. First, the solutions start withan initially relaxed line having zero voltage and current values. Then,voltages along the electrode of transistor are solved for a fixed timefrom (22) in terms of the previous solutions and then currents aresolved for from (23) in terms of these and previous values.

3.1. Boundary Condition

The Equation (22) for k = 0 and k = Nz + 1 become

Vn+11 =

(C∆t

+G2

)−1

(C∆t

− G2

)Vn

1 − In+ 1

21 − I

n+ 12

0∆z2

(24)

Vn+1Nz+1=

(C∆t

+G2

)−1

(C∆t

− G2

)Vn

Nz+1 −In+ 1

2Nz+1 − I

n+ 12

Nz∆z2

(25)

By considering Fig. 3 this equation requires that we replace ∆z with∆z2 only for k = 1 and k = Nz + 1. Referring Fig. 4 we will denote

the currents at the source point (z = 0) as I0 and at the load point(z = L) as INz+1 . By substituting this notation in to (24) we obtain:

I =Vn

in + Vn+1in − Vn

1 − Vn+11

2Rs(26)

where

Vin =

(Vind

Ving

Vins

), Gs =

1Rs

=

(Gsd 0 00 Gsg 00 0 Gss

)

I =Vn

in + Vn+1in − Vn

1 − Vn+11

2Rs

(27)


where

Vin =

Vind

Ving

Vins

0

, Gs =

1Rs

=

Gsd 0 0 00 Gsg 0 00 0 Gss 00 0 0 0

Similarly, we impose the terminal constraint at z = L with substitutingINz+1 into (25) as follow:

INz+1 =Vn

Nz+1 + Vn+1Nz+1

2RL(28)

where

GL =1

RL=

(GLd 0 00 GLg 00 0 GLs

)

INz+1 =Vn

Nz+1 + Vn+1Nz+1

2RL

(29)

where

GL =1

RL

=

GLd 0 0 00 GLg 0 00 0 GLs 00 0 0 0

The finite difference approximation of Equation (17) can be written asfollows:for k = 1

Vn+11 =

(C∆t

+G2

)−1

(C∆t

− G2

)Vn

1 − In+ 1

21 − I

n+ 12

0

∆z

2

=(

C∆t

+G2

+1

Rs∆z

)−1 (C∆t

− G2

− 1Rs∆z

)Vn

1

− 2∆z

(In+ 1

21 − Vn

in + Vn+1in

2Rs

)(30)

for k = 2, 3, . . . , Nz

Vn+1k =

(C∆t

+G2

)−1

(C∆t

− G2

)Vn

k −In+ 1

2k − I

n+ 12

k−1

∆z

(31)

318 Afrooz et al.

for k = Nz + 1

Vn+1Nz+1 =

(C∆t

+G2

)−1

(C∆t

− G2

)Vn

Nz+1 −In+ 1

2Nz+1 − I

n+ 12

Nz

∆z

2

=

(C∆t

+G2

+1

RL∆z

)−1 (C∆t

− G2

− 1RL∆z

)Vn

Nz+1+2

∆zINz

(32)

and for Equation (16):for k = 2, 3, . . . , Nz

In+ 3

2k =

(L∆t

+R2

)−1(

L∆t

− R2

)In+ 1

2k −

Vn+1k+1 − Vn+1

k

∆z

(33)

The voltages and currents are solved by iterating k for a fixed timeand then iterating time.

inV

sRLR

oI1I1V 1NzV

+

1NzI+

NzI

Figure 4. Discretization of the terminal voltages and currents.

4. NUMERICAL RESULTS

The proposed approach is used for modeling a sub micrometer-gateGaAs transistor. The device had a 0.3 × 560 µm gate. The input andoutput nodes were connected to the beginning of the gate electrode andthe end of drain electrode. The transistor was biased at Vds = 3 v andIds = 10 mA. The source and load resistance are 50 Ω. Moreover, thebeginning and the end of source electrodes are grounded. A schematicof the considered transistor is shown in Fig. 5. The element valuesused in the distributed model are given in Tables 1 and 2 [10]. Thescattering parameters of this transistor are calculated over a frequencyrange of 20–220 GHz using fully distributed and slice models from thetime domain analysis and are drawn in Fig. 6. It is clearly shownthat both results of the fully distributed and the slice models are the


Table 1. Numerical values of distributed model elements for passivepart.

The distributed model elements Numerical values (per unit length)

Ld 780 nH/m

Ls 780 nH/m

Lg 161 nH/m

Mgd 360 nH/m

Mgs 360 nH/m

Mds 240 nH/m

Cgp 0.6 pF/m

Cdp 87 pF/m

Csp 148 pF/m

Cgdp 29 pF/m

Cgsp 29 pF/m

Cdsp 61 pF/m

Rd 900Ω/m

Rs 900Ω/m

Rg 34300Ω/m

Table 2. Numerical values of distributed model elements for activepart at Vds = 3 v and Ids = 10 mA.

The distributed model elements Numerical values (per unit length)

Cgs 0.771 nF/m

Cds 0.0178 nF/m

Cgd 0.1178 nF/m

Gm 146.42 S/m

Ri 0.002Ω/m

Gds 15.46mho/m

same at the low frequency (in this problem under 40 GHz) but byincreasing the frequency, the results of two models would be different.In order to test actual predictive capabilities of the propose approach,the model was adopted to predict the electrical behavior of two devicestructures (a 0.3 × 840 µm and a 0.3 × 1120 µm). The parameters of

320 Afrooz et al.

Figure 5. Schematic of the simulated transistor.

50 100 150 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Frequency (GHz)

|S11

|

50 100 150 2000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency (GHz)

|S12

|

50 100 150 2000

0.2

0.4

0.6

0.8

1

Frequency (GHz)

|S21

|

50 100 150 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Frequency (GHz)

|S22

|

(a) (b)

(c) (d)

Figure 6. The scattering parameters of transistor with 560µm gatewidth. “—” fully distributed model, “ × × × ” slice model. (a)magnitude of S11, (b) magnitude of S21, (c) magnitude of S21, (d)magnitude of S22.

these transistors is obtained using scaling method [21]. In Figs. 7 and8 the scattering parameters obtained for two device structures usingthe new approach and slice model are shown, respectively. It is obviousthat by increasing the device dimension and the frequency, differencebetween fully distributed and slice model also increases. The mostobvious difference between two models are appeared for the magnitudeof S11.


50 100 150 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Frequency (GHz)

|S11

|

50 100 150 2000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency (GHz)

|S12

|

50 100 150 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

Frequency (GHz)

|S21

|

50 100 150 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Frequency (GHz)

|S22

|

(a) (b)

(c) (d)


As the equations show, the fully distributed model is a modifyversion of slice model when the number of slices has been increasedto infinity. Therefore, the results of fully distributed model ismore accurate than the slice model, specially at the high frequencyapplications and devices which their dimensions are comparable withwavelength. It is due to this fact that fully distributed model isbased on solving the wave equation in the transistor structure whilethe slice model is based on circuit modeling. Therefore, the fullydistributed model can consider the effect of wave propagation along thedevice electrode more accurate than the slice model, especially whenthe device dimension is comparable with the wavelength. Becausethe gate electrode has a different reactance from the drain electrode,Therefore they exhibit different phase velocities for the input andoutput signals. By increasing the frequency or device dimension, thephase cancellation phenomena due to the phase velocity mismatchingcannot be neglected. This effect can be consider in fully distributed

322 Afrooz et al.

50 100 150 2000

0.2

0.4

0.6

0.8

Frequency (GHz)

|S11

|

50 100 150 2000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency (GHz)

|S12

|

50 100 150 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

Frequency (GHz)

|S21

|

50 100 150 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Frequency (GHz)

|S22

|

(a) (b)

(c) (d)


model. Fig. 9 depicts the voltage gain as a function of the gate widthat 80 GHz. It is obvious that by increasing the gate width, voltage gaindecrease periodically due to the phase cancellation. The time domainload voltage obtained from the fully distributed and slice models fora transistor with 560µm when excited by sinusoidal source (80 GHz)are shown in Fig. 10. The magnitude and the phase of the solutionfor the fully distributed model is different from slice model. The mostimportant reason for this difference is the phase cancellation that istaken account in to by the fully distributed model. Figs. 11 and12 show the time domain variation of voltage in the beginning andthe end of drain and gate electrodes at 100 GHz frequency using fullydistributed model, respectively. Fig. 12 depicts the gate voltage atthe beginning and the end of electrode. The magnitude of voltage atthe end of the gate electrode is less than the voltage at the beginningthe gate electrode. This is mainly due to the gate ohmic resistance.Fig. 11 shows the transistor has amplification at the end of the drain


0.2 0.4 0.6 0.8 1 121.2 1.4 1.60

0.05

0.1

0.15

0.2

0.25

Width ( mm )

Vo

ltag

e g

ain

Figure 9. The transistor voltage gain versus gate width at 80 GHzfrequency.

0 20 40 60 80 100 120 140 160 180 200-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Time (ps)

Lo

ad V

olt

age

( m

V )

Slice modelFully distributed mode l

Figure 10. Time domain load voltage from slice and fully distributedmodels at 100 GHz frequency.

324 Afrooz et al.

0 10 20 30 40 50 60 70 80 90 10 0

-0.3

-0.2

-0.1

0

0.1

0.2

-0.3

-0.2

-0.1

0

0.1

Time (ps)

Vo

ltag

e (m

v)

End of drain electrodeBeginning of drain electrode

Figure 11. The voltage of transistor with 560µm gate width at thebeginning and end of the drain electrode.

0 10 20 30 40 50 60 70 80 90 100-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Time (ps)

Vo

ltag

e (m

v)

end of gate electrodebeginning of gate electrode

Figure 12. The output voltage of transistor with 560µm gate widthat the beginning and the end of the gate electrode.


electrode. The Fig. 13. shows the distribution of the maximum valueof voltage on the drain, gate and source electrodes for a transistor with560 micro meter gate length when the device is excited by a sinusoidalvoltage source (80 GHz).

0 100 200 300 400 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Length ( micro meter)

Vol

tage

(V

)

Gate electrodeDrain electrodeSource electrode

Figure 13. Distribution of the maximum value of the voltage on thedrain, gate and source electrodes of a transistor with 560 micro metergate length.

5. CONCLUSION

A new approach for analysis of microwave/mm-wave transistor hasbeen presented. This new method can consider the effect of wavepropagation along the device electrodes, accurately. The derivedequation (AMTL) is solved using FDTD technique. The scatteringparameters are obtained from the time domain results for 20–220 GHzfrequency band. The simulation results show that the proposedmethod and the slice model have the same results at the low frequency.But, by increasing the frequency to the mm-wave range where thedevice dimensions are comparable with the wavelength, a differenceappears between the results of two models. This is due to theconsideration of the wave propagation and the phase cancellation bythe proposed model. Moreover, another advantage of this model is theeasy of its integration in to CAD optimizers.

326 Afrooz et al.

ACKNOWLEDGMENT

The authors would like to thank Iran Telecommunication ResearchCenter (ITRC) staffs for supporting this work. They also appreciateDr. G. Moradi from Amirkabir University of Technology (TehranPolytechnic) for his good comments.

REFERENCES

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2. Alsunaidi, M. A., S. M. S. Imtiaz, and S. M. Ghazaly,“Electromagnetic wave effects on microwave transistors using afull-wave time-domain model,” IEEE Trans. Microwave TheoryTech., Vol. 44, 799–808, 1996.

3. Ghazaly, S. M. and T. Itoh, “Traveling-wave inverted-gate field-effect transistor: concept, analysis, and potential,” IEEE Trans.Microwave Theory Tech., Vol. 37, 1027–1032, 1989.

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Documents

TIME DOMAIN ANALYSIS OF ACTIVE TRANSMISSION LINE USING FDTD TECHNIQUE (APPLICATION TO MICROWAVE/MM- WAVE TRANSISTORS)