ANALYSIS

OF MODULATORCIRCUITSMULTI-DIMENSIONAL

BASED

ON

FOURIER

TRANSFORMATION
Akio

USHIDA,

Yoshihiro

Department

YAMAGAMI,of Electrical

Tokushima

and Electronic

University,

NISHIOEngineering,

Tokushima,

ABSTRACT
There are many communication
tone signals such as modulator

Yoshifumi
770 JAPAN

A new SPICE oriented method is presented in this paFouriertransper which is based on both %dimensional
relaxation
methods. Asformationand ~requency-domain
sume that a given circuit is composed of nonlinear
resis-

circuits driven by multiand mixer.
If the input

tive sub-networks

and reactive elements

such as capacitors

frequency components
are largely dHferent in each other,the brute force numericalmethod will take an enormous
comput at ion time to get the steady-state
responses.
In

sources, theand inductors.
At first, using the substitution
circuit is partitioned
into two groups, namely, the nonlin-

this

ear resistive

subnetworks

and the reactive

substitution

waveforms

are described

paper,

we show a SPICE

on multi-dimensional

Fourierorientedalgorithm

transformation,

based

where all of

elements.

Theby 2-dimensional

the circuit analyses such as dc- and at-analysis
in the algorithm
are carried out with SPICE. On the other hand,a very simple sensitivity
analysis and 2-dimensional
FFT
are carried out by a Fortranprogram(or C program).
We found that the convergence ratio of our algorithm
isFourier expansions, and the coefficients are calculated by
the relaxation
method.
We have developed a very simple simulator
consistingof SPICE and a Fortran program, where all of the circuit
analysis are implemented
by SPICE. On the other hand,sufficiently
large, and can be appliedmunicant ion circuits.

very simple sensitivity
analysis and 2-dimensional
are carried out by the Fortran program.

to wide class of com-

FFT

1. INTRODUCTION
BASIC

2.
Many communication
circuits, such as modulators,
mixers and frequency
converters,
are driven by multi-tone
signals.
There are 2 basic approaches for the computation of the steady-state
responses: (1) frequency-domain
approach [1-2] and (2) time-domain
approach [3-4]. The
formercan be applied

only

to weakly

because the scale of determining
large for strongly
on numericalappliedables.

nonlinear

integration

to circuits

equationscircuits.

techniques,

havingnonlinearThen,a few number

the substitution

will be generallyof the state vari-sources at the partitioning=

VC,O + ~{VC,M...Icos vkt + VC,M sin v~t}{~L,2k..-lcos

i~(t)= ~L,o+ f

v = m~~u~resistive

exampleit contains

large

capacitances,for

period,andby thebrute

frequencies

is1000. If the step size is chosen h = T2/100 for
T2 = 1/$2, it will take 1000 x 100 numericalintegrations
for only one period (Tl = l/fI).
.f2/.f1=

Vkt+ 1~,2%sinVkt} (1.2)

k=ltional time, because the step size must be chosen sufEciently small depending on the carrier signal. Consider anof the two input

(1.1)

k=l

analysis) for getting thea brute force method (transient
steady-state
response, it will take an enormous computa-

a ratio

point

M
vc(t)

Generally,
modulatorsand mixers are driven by two
input signals, namely, the high frequency carrier and thelow frequency modulating
signal. In this case, if we use

such thatmethod,assumed of the form

is based

that can be efficiently

idea of our relaxation

Now, assume
consider a circuit as shown in Fig. 1 (a)l.
frethe two inputs e(t) and j(t) cent ain two independent
quency components WI and W2.

circuits,

becomes very

The latterTo focus on the main

APPROACH

TGener~lY, intewatedelements

a lsrgestesxiy-state
the circuit
resctive

into

such

circuits

two groups

elements

0-7803-4455-3/98/$10.00 (c) 1998 IEEEare composed

as transistors
it

the

and

long

force

method.

containingof CaP~itO~andIf in thiscnse,

diodes.

transienttakerespectively,(1.3)+ mz~uz

response

computation
Therefore,

nonlinearas shown

will

time

get

the

we partitioned

resistive

in Fig. l(b)continuetocircuitand+
taE33”E3‘
i
+ +o
b2

CI

e(t)

e(t)

v

Nonlinearresistive

‘cl

j(t)

%0

H

H

H220

‘L1

120

%?

t
co

+

VC i

L2

+

circuit

*

-

i L1

resistivei L

circuit

j(t)

? VC2

Nonlinear

i

t

210

AiL

‘L2

&L

-

1+

‘L El?

A VQ

Lo

I

I

(c) Sensitivity circuit for calculation(b) Circuit partition(a) Nonlinear circuitof the variational

value

Fig.1 Schematic diagram of our relaxation method

where ml~,m2~ are integers

Msatisfying
AZL(t)

s B,

[mlkl

lm2kl

<

AIL,O + ~

=

(1.4)

B

{AIL,2k-1

cos

Vkt

i-

sin u~t}AIL,2~

k=l(7.2)
for some sufficiently
Assuming
uniquelarge B.

that

the

steady-state

original

solutionand iL (t)satisfies

circuitscribedhas athen WC(t)(2.1)‘uLz(t)= o(2.2)

capacitor

the higher-order
expansionflc2(~c2),#L2

and inductor

=

&2

=

@L~
——
&L2

(2),

and ne-of Auc (t) and AiL (t)inof nonlinear

F1 (vi+terms,

AVC, i; + AZL) = ZC1(UCj+’

are de+ZC2(V:)

=

terms

(6) into

we obtain

equation:+ icz(t) = o= ‘v~~ (t) –glecting

the Taylor

by
qc2

Then,

by (l),

determining

F~ (WC, ZL) = icl(t)Now, assume the nonlinear

in Flg.l(a)described

the following

F2(WC,i~)

V2+1 (t),i~+l(t)from

Substituting

(3)

(~L2)

F2(’zT$+ AVC,“~+1), IL

i34c2(vc)

dAvc

the

dt

+

~~ + AZL)=

VLl(V~+l,+

ZC2(4+1)

= ~

i~+l)

(8.1)

– VL2(i~+1)

we haveS~4c2 dvC2——‘C2 =Let us calculatetion technique.

&C2

dt

‘

‘L2

diL2

VLl(’f&Z~)

avLj:’iL)Lhc

+

(4)

dt

‘vL2

the steady-state
response using an iteraAssume the solution at the jith iteration(i;)

avL’&~’’L)AiL@L1(ZL)

dAiL

= ~

6’iL

dt

–

Now, define the residualis given by+sources as follows:

E~(t)

=

Zcl(V’&jZ~) + iC2(vA)

(9.1)

c~(t)

= ‘L1(VL;‘~) – vL2(@(9.2)

Since the relation(8) for calculating

linearsystem,time-varying

V~+l(t)the solution= v~(t)+ Avc(t),at the (j + l)thi~+l(t)iteration,= z~(t)put+ AiL(t)Auc (t)and AiL (t)are described

where the variations

(6)

H120

H21O

H220

for uc = VCO,
{AVc,2k-1)(=&I(lJc,iL~
avC
zh~l(uc,i~)
avC

by

M

Auc (t)= AVc,o+~

(

~llo

cos Ukt + AVc,2k sin Vkt}

k=l
(7.1)

i~

=

iHI,

and

a~c2
co=—
avc2 ~Go‘

0-7803-4455-3/98/$10.00 (c) 1998 IEEE

Aucand AiLis a

it is not easy to solve even if

it is linear.
Therefore, we approximate
systems as follows:
the time-invariant
To evaluate(8.2)

the equations

aicI(Vc,tL)
84L
av~l(wc,i~
6’i~

by

J

(10.1)

where Vco and iLo are dc solutions
Thus,

the relation

at the operating(8) can be described

Next,

points.

choose sufficiently

6 and e inas follows:

we draw

(

Hllo

H121J
H2113 H22(J)(2)+(: -9(%)
()E;

(~)

—
——

SPICE

the

small
(13),

dc circuit

stopping

voltagesourceconditions

respectively.

diagram,andAt

first,

solve itis replacedwithand each inductor

source with

(11)

and

2. Thus, each capacitor

stituting
V&.,

(12)

by

by a sub-a de-voltage

by a substituting

sourcecurrent

I~o.

E; (t)

Thus, we have the equivalentsensitivity
circuit shown by
Fig.l(c).
It can be easily solved by the phasor technique.
Observe that, although
the convergence ratio may be decreased for the strongly nonlinear
circuits, the algorithm
is very simple
vergence.

and produces

The iterations

will

the exact

continue

Avc (t) and AiL (t) satisfy

until

solutionafter

the variational

the following

con-valuesstopping

1. Set v~ (t) = V~o and i!(t)

AVC

II+

condi-

wl-components

11<8

AIL

= [A.l~,o, . . . . A1~,2A4][AVC,O, . . . . AVC,2M],=

for a sufficiently
small 6. Furthermore,
if the residual
rent does not satisfy the following
condition:II‘(e~(t))z+(e;(t))’dt <// 70J(t) 11= 1

cur-capacitors.

If they cannotof the algorithm

the numberof nonlinear

ples, we recommend

will

be decreased

capacitors.

to partition

accordinga circuit

SPICE

are

tion

SPICE

purposes

analysis

is widelyused for manysuch as de-analysis,of SPICE,

circuit

at-analysis,transientis implemented

by ac-

and a very simple Fortranare linear,

we need not apply

11<6,

go to 5.
set v&+l (t) = v~(t)andj

5. Estimate

II e~(t)Otherwise,

=j

-t-l.

II in (13).

+ Avc

and i~+l (t) =Go to Step 1.
If II d (t)It< e, stop.

increase B in (1.4) and go to Step O.

We have carried out the algorithm
PSPICE of MicroSim co.

with

IBM

PC loading

AN ILLUSTRATIVE

Considerinputsa mixer

circuitshown by Fig.2(a).

It has two

of

pro-

algorithme2(t)O. A given circuit is partitioned
into nonlinear
resistive circuits and reactive elements with substitution
sources.
Considering
the amplitudesof the signal
inputs

EXAMPLEel (t) = 0.01 sin 2m x 50 x 106t[V]Implementationand carrierofsimula-(or C-program).

in (l).

[7].

expansions

IMPLEMENTATION

and so on. Our simulator

and de-analysis
gram

FFT,

Otherwise,

4.
Nowaday,

elements

2-dimensional

i~(t)+AiL,

into subcircuits

at only coupling capacitors, whose capacitors voltages
considered as substation voltage sources in Fig.l(b).

3.

to

exam-

the reactiveAI;

be ne-

In our many

separately

Fouriers&(t) and g;(t) given by (9). Thus, the4. Estimate
relation (11 ) can be easily solved by t he phasor technique, and get Avc(t)and AZL(t).
If II AV& II + 1]

glected at the high frequency, we need to take account of
the nonlinear capacitors in Fig. 1(a). Thus, the computer
efficiency

and w2-components,

we have 2-dimensionaland v~l (t).

the responses of nonlinear
reactive el3. Calculate
ements, and describe them by the 2-dimensional
Fourier expansions of i&2(t) and v~2(t). Note that if

(13)

e

for a large T and a small c, then we need to increase theFourier terms M in (l).
Note that the nonlinear
resistive network in Fig.l(a)may have small parasitic

Thus,i-&(t)(12)

II AILac-

2. Solve the, nonlinear resistive circuit with (e(t), j(t),v~ (t),
ii(t))
by de-analysis
of SPICE.
In thiscase, 2-dimensional
FFT can be carried out by
the applicationof the onedmensional
FFT to thetion[/ AVC= l~o. Applying

sweep of SPICE, determine H1lO, . . . . H220, Co and
Lo at zero frequency.
Set j = O[7], set the highest

harmonic

M=

0.01sin27r

X 51 x 10Gt[V]We partition
the circuit at the two capacitors Cl and C2,
because they are considered as sufficiently
large compared
2Introduce
the compensational
linearity
is strong [8].

0-7803-4455-3/98/$10.00 (c) 1998 IEEE

resistors

RC and—Rc if the non-

5. Conclusions

and Remarks

El

In

-+

this

paper,

Fourier

4late the steady-stateo-signals such as modulators
not depend

I

Trl

Tr2

be easily

Tr3

inputsE2

modified

sitivity

1
E3

Ii

the Fortranclto the analysis

input

The efficiency

does

Furthermore,it canof multiple-frequency

than two, and to the noise analysis.

e2(t) -

=R2

=1000,

1

RG = 200fl,

Fig.2(a)Mixer
R3=R4=10K2,
EI

= 5V,

the computer

efficiency.

[1]

E3 = 12V

K. S. Kundert,G. B. Sorkin“Applyingharmonic
Trans.

[2]

10.0
[3]

0.0
2.0

A. Ushidaandcircuits

mme@lo6 eeo)

[4]

waveform

signalnoise

capacit antes in the transistors.

the waveforms

as follows:

We assume[5]77v~,o+ ~ ~ {v&@ Cos(kwl + nwz)-t
k=l n=][6]+Vi,.kn,2.

We found thatsin(kwl+V&,IS– nwz)t},sin(kwl+ znwz)t2 =1,2[7]

response can be obtained in 3
Note that if we apply a brute

it will take an enormous computationtime
because the relative frequency difference of two inputs isvery small, and the response contains a very low frequency
method,circuit,”Tech.,

VO1.MTT-36,

two

and

pp.766-779,

1984.

H. Tanimoto,“An

met hod

excitations,”

VO1.CAD-9,

of

IEEE

signals,)’

analysisfrequency

Design,analysis

multi-tone

VO1.CAS-31,

H. Tanimoto,~oi~e analysis

effi-

of

nonlin-IEEE

Trans.

no.3,

signalexcitation

sources,”IEEE

Trans.

Theorypp.225-235,

590,

1993.

A. Ushida,

T. Adachi

andVO1.CAS-40,

L. O. Chua,

circuits

basedTrans.

Circuits

Syst.-I:

Fundamental

cations,

VO1.CAS-39,

R.

Telichevesky

andDesignpp.581-“Steady-state

analy-

on hybrid

TheorysanJose,

A. Ushida,

L. O.Chua

andT. Sugawara

algorithm

forfrequency

components,”

A. Ushidalinear
“

Int.

and

Jour

pp.249-269,

of lMHz.

0-7803-4455-3/98/$10.00 (c) 1998 IEEE

L. O.ChuaAoj

Primer,
July,

“A

Ca-

1996.

substitution

circuits

with

Int.

of Circuit

TheoryJour

Theorymultiand1987.

“Steady-state

frequency-domain

Circuit1989.

California,

non-linear

VOI.15, pp.327-355,

circuits:

Appli-

and

1992.

System,solvingIEEE

method,”SpectreRF

K. Kundert,

Fundamen-

no.9,

pp,649-661,

acyclostationary

Circuits Syst. -I:

sis of nonlinear

no.8,

and
T. Sugawara,
circuitswithincludingand Applications,Applications,[8]T. Itakura
for nonlinear

large

tal

dence

VO
ICO –
— VO
2C0 = 2.388 [V] by the de-analysis

of SPICE. The steady-state
iterationsof our algorithm.

byfrequency

with

M. Okumura,C(Numericalperiodic

to the parasitic

almost-periodic

1990.

I

-mo L

7w2)t

the

A. Sangjovanni-Vincentelli,toT. Sugawara

smallComputer-aided–

be decreased

research problem.

“Frequency-domain

drivenSyst.,

M. Okumura,ear circuitscos(kw

will

TheoryL. O. Chua,

Circuits

cient

-10.0

+Vi,kn,C2

sen-

out by1988.

nonlinearTrans.

andbalance

Microwave

pp.366-378,=

simpleare carried

REFERENCES
c1 =CZ=O.OlpF

outputeficiecy

This is a future

circuit

EZ = 2.5V,

Steady-state

Another

FFT

program.

IEEE

Fig.2(b)by SPICE.

Note that, for very high frequency, we need to take into
account parasitic capacitors of transistors.
If the numbera’computerforce

to calcu-

by 2-frequency

values.

analysis and 2-dimensional

is increased,vJt)

2-dimensional

appliedand mixers.

analyses are implemented

R3

RI

response driven

on the frequency

greaterthat

We have developed a very simple simulator
cosisting of
SPICE and a Fortran program,where all of the circuit

J

E3

presented

can be efficientlyv ...
;: !32

RI: :we havetransformation

response
relaxation

and Applications,of nonmethod,
VO1.17,