1Analysis of Nonlinear Traveling Waves
by Frequency-Domain
Perturbation Method
Akio USHIDA, Yuichi TANJI and Yoshifumi NISHIO
Dept. of Electrical and Electronic Engineering, Tokushima tJniversity
2-1, Minami-Josanjima, Tokushima, 770 Japan
Tel. +81-886-56-7469
Fax. +81-886-56-7471
E-mail ushida@ee.tokushima-u. ac.jp

ABSTRACT
We discuss a numerical method for solving thenonlinear
transmission lines by the frequency-domain
perturbation method. To improve the convergence,
we introduce two new methods of the compensationand the homotopy techniques, which also help to makethe iteration stable. This kind of transmission lines
is widely used in the communication curcuits such as
GaAs integrated circuits, and varactor diode circuits.

1. INTRODUCTION
Nonlinear devices such as GaAs nonlinear transmission lines [1], varactor diodes and superconducting
transmission lines [2] have the nonlinear capacitors.
The velocity of traveling wave is given byv = l/~mso that if the capacitor is a decreasing function of thevoltages, it will have an interesting property such thatthe higher part of the waveform is the faster than thelower part of it. The property is used in many purposessuch as the picosecond pulse compression [1], picosecond shock-wave generations [3] and so on.
There are some papers discussing the phenomena fromthe physical points of view [1-5]. In reference [6], theshock-waves of nonlinear transmission line driven by asinusoidal generator are analyzed by the harmonic balance method, where the nonlinear transmission line isreplaced by the finite number of the discrete lumpedmodels. The reference [7] has analyzed the phenomenaby the application of the difference approximation tothe nonlinear partial differential equation. In reference[8], they have analyzed the nonlinear wave propagation
phenomena, where the 2-dimensional FFT is applied
to solve the circuit equation in the distance and time.
In this paper, we discuss an efficient numericalmethod for solving the nonlinear transmission lines.
The waveform is approximated by the Fourier expansion at each point on the line, and the coefficients arecalculated by a perturbation technique. Although, themethod is only applied to the weakly nonlinear systems, we have improved it with two techniques of acompensational and homotopy method.

2. FREQUENCY-DOMAINPERTURBATIONMETHOD2.1. Perturbation
MethodNow, consider a nonlinear transmission line terminated by linear subnetworks. The circuit equation is described by the following nonlinear partial differential equations:au–z6’@L= ~+?lR,

‘:

f?qc
=~+iG

(1)

where ~L and qc are the flux of a nonlinear inductor
and the charge of a nonlinear capacitor. We assume
the characteristics as follows:
iL =r@L+ ~~L(f#L), ~R = RiR + 6~R(iR)

uc = Sqc + 6~C(q~),

iG = GVG + 6;G(VG)

(2.1)
(2.2)

where e means a small constant, and ;L (#L), ~R(iR),tic(qc) and ;G(vG) are nonlinear terms. We have therelations of i s iL = i~ and v a vc = ‘VG.
To analyze the traveling-waves, consider the responses of nonlinear transmission lines driven by animpulse waveform as follows:
e(t)

21w’r
Em sin(—ql t)=:0+:&<T(3)

where T is a period of the impulse and k is an integer. Assume that the nonlinear transmission lineis terminated by a linear impedance ZL. Then, thecircuit equations (1) and (2) are mathematically reduced to the nonlinear partial equations constrainedby a linear boundary condition. In this paper, we apply the frequency-domain approach, and describe theinput impulse waveform by the Fourier expansion in
the complex domain as follows:
Me(t) =~l?~e~kwt,k=-M

~=—

27i
T

Copyright (C) 1996 IEEE. All Rights Reserved.

(4)

2

1/2

for a sufficiently large M. Now, let us assume the solutions at the mth iteration as follows:
Vm(x, t)=+5(rwk=– MMM

Vkmt?iktit,im(x, t) =~~k=-Mk.-hlf

MMI~eikWt
(5.1)

(5.2)

– qwl))’(8)

1
1

becomes sufficiently small for given a constant 6, However, we can not say anything whether the solution
waveform is sufficiently accurate or not.
Now, let us define the following residual error. The
solutions for the M frequency components be v~ (x, t),iM (x, t), Then, define the residual error as follows:

Then, we have the following perturbed equations:dVkrn—=dxdI~—dx=ITjkwL)Ip‘6(vR,k(im–l) - jk@,&#~-’))(6.1)) - jkLLL’VC,,(g:-’))(6.2)(iG,~(tJm-l

where L = I/I’,
C = 1/S and(6.3)=(~M(x, t) - ti(X,t))2

To+ (i~(x, t) - ~(x, t))’}(9)dt] 1’2-(G+jk@Vk”‘~QY[J{‘M=-(R+&c,k(&l))

c (Vkm–(6.4)

Observe that the relations (6.1) and (6.2) are ordinary differential equation, and the second terms at theright hand side correspond to the forced terms calculated at (m – l)th iteration. The solutions of (6.1) and(6.2) are written as follows:

where O(X, t),;(z,t) are the exact solutions, and‘u~ (x, t), iM (x, t) are the approximate solutions at
rnth iteration with M frequency components. Since
it is impossible to get the exact solutions in (9), we assume the solutions obtained with the larger frequency
components &l’as the exact solutions.
The residual error (9) between the solutions with M
and M’ frequency components are given by&J/f =(Vfyl) - w-w’~[ lkl=M

1/2+S(Ip(l:-r(o)’Ikl=kf1

If the residual error EM is not so small, we need tochoose much larger M for the waveform approximation, and again repeat our perturbation method.
where the propagation constant ~k and characteristic
impedance i?k,o are given by& = I/(jkwL+R)(jkJC

jkwL + R+ G), Zk,o =/jkwCi-GThe first two terms of (7) correspond to the zero-input
responses, and ftikm(x) and @(x)to the zero-state
responses calculated by a numerical integration technique. The constant parameters A~ and B~ in (7) areevaluated by the boundary conditions of the transmission line. Thus, all of the Fourier coefficients Vjm andIF are calculated,and ~~ and Q~ are decided by(6.3) and (6.4). The iteration is continued until thevariationam= [ ~ (vkm(z)-v~-’(z))’
Lk=-M

2.2+ Modified
Perturbation
Method It is said
that the perturbation method is only applied to theweakly nonlinear circuits. In this section, we will modify the method such that it can be applied to the muchstronger nonlinear circuits. Assume that the nonlinear characteristics given in (2) are monotone increasing
functions, and for simplicity, describe it as follows:
U1 = HU2 + e~(u’),

H = diag(r, R, S, G)

where U1 = (iL, v~, VG,iG)~,and ~fi(~’)=(lo)

u’ = (@L, iR, qc, ~G)~~ (;L(#L),i~(i~),

ti~(q&G(@))T.

Since the convergence of our perturbation method
largely depens on the magnitude of perturbed terms~~(u2), we need to decrease it compared with the linear term. Thus, we introduce small constant
AH E diag(AI’, AR, AS, AG)

Copyright (C) 1996 IEEE. All Rights Reserved.

Now, describe the zero-state response of (7) in the integration form.

parameter in the following form:

as a compensation(11)

ul = (H + AH)u2 + (et(u~) - AHu~)

The second term is the nonlinear function corresponding to the perturbed term which can be be largely
decreased by choosing a suitable AH. We found fromour numerical examples that the convergence ratio islargely improved by introducing of the compensation
parameter,
Homotopy method is sometimes used when the iteration method does not guarantee the convergency.
We will apply it to our perturbation method. Let us
introduce a parameter p for the nonlinear elements (2).
U1 =(H+AH)uz + ~(6k(UZ)-+c2,k(z)xl,kxyJi)
xy~(z) = cl,k(x)Ekx—ce-AJJZ-S)&k(qn-l,s)dS(?

(13.1)/oxflh(~)=

c“xfl~h(~) – d“xz,h(lo~-l,$)(13.2)

where Cl ,h(a), Cz,,$(a) are decided from the boundary conditions ancl (7).
Let us adopt the following norm

II w II=(12)It is clear that the relation (12) for p = O corresponds
to linear elements, and the nonlinearity is gradually
increased by choosing {p :01}. At p = 1, it reduces to the original function. If we choose a small
variation Ap per iteration such ass)dsz-@kAHuz),

p:o+l

w~-l,–,’ik(Z-s)jl,k(/o1/1?0~w2(t)dt(14)

Then, we have the following convergence conditions.
Theorem:Assume that, for an approximate solution(w?, w~)T, there are constants D1 ,D2, Ko, L1 and Lz
satisfying as follows:(i)

Define the solution domain by:

Pm = Pm-l + AP
our perturbation can also get the solution stablely.
2.3. Convergence
Condition
Now, let us consider
the convergence condition of our algorithm. For simplicity, let us introduce the following symbols for variables in (6) and (7):[1w,-w:

II<D2}(15.1)(ii) Define norms of the coefficient matrices by:(a). For the solutions, we setly(z)QT(x))(b). For the perturbed terms, we set

Xz,k(tup, Z) =({L,k(#~-l)vc,k(#rl)))

where WI = (v, i)~, W2 = (d~, qc)T.
Set the coefficient matrices of (6) as follows:
Ah =o( G + jkwC

R+ jkwLO(iii) Assume the perturbed terms satisfy the following Lipschitz conditions:

W; II(15.3)
for all O ~ x ~ 1,where L1 and Lz are Lipschitz
constants.
(1&(W;,X) -XZ(W;,X) It< L2 IIW; -(iv) Set the maximum values of the variables inz = [0, 1] as follows:)
Lo )Copyright (C) 1996 IEEE. All Rights Reserved.

Furthermore, if it satisfies the condition,

WI0,4(15.5)

KR,[IFII<l(a)nonlineartransmission

Iina

0.2

A-1
A-(jKl(forP =Kz
K&!(K2 + Lz) )our perturbation method (13) will converge to theunique solution (wY, W2M)T.
We found from the theorem that our perturbational
iteration will be converged to an approximate solution
XmI (/) if the nonlinear term e is small.
3. ILLUSTRATIVE

0.0(b) linear

linetrensmlsslon

0.2

0.0

0.2

EXAMPLES
0.0

As an example of the nonlinear transmission line,
let us consider to analyze the shock-wave to an impulse
response. Assume the transmission line is terminatedby a resistor RL = 10 and0.0

0.20.4w~ = o,o15iR + o.li~vc = lollqc+ 0.5 X 1034q:,

iG=

Figure 2: Impulse responses(a) The responseof nonlinear

transmission

REFERENCES[1] M .J.W. Rodwell, “Nonlnear

4. CONCLUSIONS

AND

1.2COnverganeOrazlos
1.0 -—A =0.0CX3.-A =0.030—A -0.0$0.....A=o.Ioo

0.6,-0.4 -line for pi-and broadbandphase

Electron. Lett.,vol.23, no.3, pp.109-110,1987[2] R.K.

Arora and J.L. Thaker, “Electromagnetic
pulse transmission on superconducting interconnects”, Int. J. Electronics, VO1.71,no.1, pp.117-125,
1991[3] C.J.
D. M.

Madden,R.A.

Bloomand Y.C,

Marsland,Pao,

M.J.W.

Rodwell,“Hype rabrupt-doped

GaAs nonlinear transmission line for picosecond
shock-wave generation”, Appl. Phys. Lett., VO1.54,
no.11, pp.1019-1021,[4] R. Landauer,

1989
waves in nonlinear

transmis-sion lines and their effect on parametricamplifica-tion”,“ShockIBM Journal, October, pp.391-401, 1960

[5] D. Jager and F-J. Tegude,“Nonlinear wave propagation along periodic-loaded transmission line”,Appl. Phys., VO1.15,pp.393-397, 1978

C. Camacho-Penalosaand I. Molina-Fernandez,“Harmonic balance analysis of nonlinear transmission lines”, Electron. Lett,, VO1.24,no.9, pp.12351236, 1988

[7] R.H. Freeman and A.E.

Karbowiak, “An investigation of nonlinear transmission lines and shockwaves”, J. Phys. D: Appl. Phyg,, VO1.1O,pp.633643, 1977

0.2 0.0o5Fig. 2(c)1: Convergencecompensational&transmission

pulse compression

modulation,”[6]0.8 -Figure

cosecond

REMARKSWe have presented a frequency-domain perturbation algorithm for calculating transient responses of anonlinear transmission lines. We can largely improvethe convergence by introducing two methods of thecompensation
method and homotopy method.

line (b) The

line where nonlinear terms

are set to be zeros (c) Input impulse

0.015~G + O.lv~

For the input impulse, we choose Em = 0.25, T = 1
and k = 16 in (3). We found from the example thatour perturbation method is largely improved by thecombinations of compensation method and homotopy
method,0.8

Fig. 2(d)response of linear transmissioni~ = 350 x @l@L,

0.6[nzec]=1115

10lzerdons

ratios for various

parameter, where variation

Xm(l)- Xm-’(l) II[8] I. Molina-Fernandez,

C. Camacho-Penalosa and J.
Ramos, “Application of the two-dimensional Fouriertransform to nonlinear wave propagation phenomTheory Tech.,ena”, IEEE Trans. on Microwave

VO1.42, no.6, pp.1079-1085,

1994

Copyright (C) 1996 IEEE. All Rights Reserved.