2005 RISP International Workshop on Nonlinear
Circuit and Signal Processing (NCSP'05)
Hawaii, USA, Mar. 4-6, 2005.

Frequency Response Curves of Nonlinear Electronic
Circuits Using Fourier Transformer
Y.Yamagami†, Y.Nishio†, J.Kawata‡, A.Ushida‡†Department of Electrical and Electronic Engineering,
Tokushima University, Tokushima, 770-8506 JAPAN‡Department of Mechanical and Electronic Engineering
Tokushima Bunri University, Kagawa, 769-2193 JAPAN
Abstract

cient for getting the characteristics such as frequency response curves in wide frequency region.
In this paper, we present a new efficient Spice-oriented
harmonic balance method for calculating the frequency response curves at the fundamental and the higher harmoniccomponents in the distortion analysis. Integrated circuits
are usually composed of many nonlinear devices such as
diodes, bipolar transistors and/or MOSFETs, whose models are described by the different forms depending on the
demanded frequency domain. Namely, we need to take into
consider the parasitic elements in the higher frequency whichmake the models very complicate. However, the higher odermodels applicable in the wide frequency domain should be
used for calculating the frequency response curves. Firstly,
we have developed packaged modules of the nonlinear devices using the analog behavior models (ABMs) of Spice [12],which execute the Fourier transformations [11] of the devices. Thus, all the devices are replaced by the corresponding packaged modules. After then, the linear sub-circuit istransformed into the corresponding Cosine and Sine circuits
consisted of the resistive elements and controlled sources.
Now, we simply call the circuits harmonic circuits at corresponding order of the harmonic analysis.
Since we have developed the packaged modules of all
kinds of nonlinear devices such as diodes, bipolar transistors and MOSFETs as user’s library in my computer, it iseasy to get the frequency response curves with DC analysis
of Spice. Thus, we have developed the user-friendly simulator for calculating the frequency response curves of nonlinear
electronic circuits.
We show the Fourier transfer circuit in section 2, andthe packaged models of nonlinear devices in section 3. Theinteresting illustrative examples are shown in section 4.

Distortion analysis of nonlinear circuits is very importantfor designing analog integrated circuits. Especially, the frequency response curves are useful to know the global properties of the circuits. Of course, AC-analysis of Spice is used
for this purpose, but it can only get the response curves forthe linear incremental circuits at the DC operating point.
Thus, we need to develop a user-friendly simulator to calculate the response curves for the practical forced inputs.
The characteristic curves may be largely different from thoseof the incremental linear circuits. We propose here an efficient approach based on the Spice-oriented harmonic balancemethod, where the Fourier transformations of the nonlinear
devices such as bipolar transistors and MOSFETs are carried out with our Fourier transfer circuit model. Note thatthe circuit models are constructed by the analog behavior
models (ABMs) of Spice. The Fourier circuits corresponding to the determining equations in our harmonic balancemethod are schematically described by coupled resistive DC,
Cosine and Sine circuits. Therefore, the frequency response
curves can be easily obtained with a use of DC analysis of
Spice.

1. Introduction
The distortion analysis in the frequency domain is very
important for designing the nonlinear analog integrated circuits. Especially, the frequency response curves are very
useful to know the global behaviors of the circuits. The
Volterra series method has been widely used [1-5], because
the functions to solve the characteristic curves are obtainedin the analytical forms. The idea is based on the bilinear
theorem, which can be applied to the weakly nonlinear circuits. Although the algorithm is theoretically elegant, it is
not so easy to derive the higher order Volterra kernels needed
for the higher order distortion analysis [1]. Furthermore,
the nonlinear characteristics of the devices in the methodshould be described by the polynomial forms which can becalculated by the Taylor expansions in the vicinity of each
DC operating point of nonlinear devices [2]. This task is
not so easy, especially, for the complicated device models
such as the Gummel-Poon model of bipolar transistors andthe higher level models of MOSFETs in the high frequency
domain[6,7]. On the other hand, there have been proposed
many algorithms for calculating the exact steady-state waveforms of nonlinear circuits [8-10]. However, they are ineffi-

2. Fourier transfer circuit model
Analog integrated circuits are usually composed of manykinds of nonlinear devices such as diodes, bipolar transistors and MOSFETs, whose Spice models are described by
the special functions such as the exponential, square-root,
piecewise continuous functions and so on [7]. For these devices, it was impossible to execute the Fourier expansions in
the analytical forms. Therefore, we have tried to obtain the
Fourier expansions in the circuit models with ABMs of Spice[11]. We first show a simulator carrying out the Fourier expansion of nonlinear one port resistive element driven by a1

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periodic input. Simplicity, assume that the nonlinear resistor is a voltage-controlled element as follows:

0.5^g(V,  2K)cos N 2K
0.5^g(V, ) sin N R

2K

i = ĝ(v)(1)

Let us assume the input and the output waveforms as follows;Xi(t) = I0 +(I2k−1 cos kωt + I2k sin kωt)







2

(2)

R

^g(V , 1 )cosN  1^g(V , ) sin N1
1R

0.5^g(V,  0)cosN 0^ , 0) sin N 
0.5g(V0

Fig.1 Fourier transfer circuit model.
(3)

To investigate the accuracy of our Fourier transfer circuit model,
we first calculate the following function:
ex cos θ = I0 (x) + I1 (x) cos θ + I2 (x) cos 2θ + · · ·Note that the Fourier coefficients Ik can be given by the explicit
functions of (V0 , V1 , . . . , V2M ) only if the nonlinear function ĝ(v)is described by the polynomial function. Therefore, we will propose here a Fourier transfer circuit model which can be applied
to any kind of circuit elements as shown in Fig.1. Each Fourier
coefficient for ĝ(v) can be obtained by the discrete Fourier transformation as follows:

Z 2π
Z0 2π

1
1
ĝ(v) cos kωtdt, I2k =π 0
πk = 1, 2, . . . , M

Z 2π

IN (x) =

0




(4)
Now, let us apply a trapezoidal integration formula to (4) as
follows;

Z bĝ(v)dt =ah(ĝ0 + ĝn ) + h(ĝ1 + ĝ2 + · · · + ĝn−1 )
2

(5)

ID =

ex cos θ cos N θdθ,

N = 0, 1, 2, . . .

(9)

−π

KW
L

h

(VGS − VT ) −

VDS
2

i

VDS (1 + λVDS ) (10.1)

2. Saturation region: (VGS > VT , VDS > VGS − VT )ID =KW(VGS − VT )2 (1 + λVDS )L(10.2)The result of Fourier expansions for the input VGS cos ωt isshown in Fig.2.. Thus, the Fourier transfer circuit model Fig.1
will be applied to any kind of circuit elements contained in analog
integrated circuits.

M
X(V2k−1 cos kθ + V2k sin kθ)Z π

1. Linear region: (VGS > VT , VGS − VT ≥ VDS > 0)

where the step size of the integration is h = (a − b)/n. Then,the truncation error is given by ĝ (2) h2 /12n, where (2) shows thesecond derivative. Replacing the integrations in (4) by (5), we
can realize the equivalent circuit model for the relations (4) with
ABMs of Spice. To understand the circuit model, we assume theinputv(θ) = V0 +1
2π

The simulation result for h = 2π/20 with Fig.1 is I1 (10) = 2761
at N = 1, x = 10 which is exactly equal to the value from thetable of Bessel function [13]. Thus, we found that the Fouriertransfer circuit model can get the sufficiently exact solution evenwith n = 10 to 20 divisions of the interval 2π.
Next, we apply it to the Fourier transformation of MOSFET,
whose first level in Spice model is described by a piecewise continuous function [6] as follows:ĝ(v) sin kωtdt I2k1 =(8)

whose Fourier coefficients are given by modified Bessel functions[13] as follows:







ĝ(v)dt

Coefficient of SinN

V2M

V0




1
2π

  where M denotes the highest harmonic component to take account in the analysis. Thus, the output Fourier coefficients aredescribed by the function of the input V as follows:

I0 =



Coefficient of CosN

k=1

I0 = f0 (V0 , V1 , · · · , V2M )I1 = f1 (V0 , V1 , · · · , V2M )...............................

V2M = f2M (V0 , V1 , · · · , V2M )  ^g(V,  2K-1 )cosN 
2K-1^g(V,
2K-1 ) sin N  2K-1

R


M
X


v(t) = V0 +(V2k−1 cos kωt + V2k sin kωt) 


k=1
M

2K

(6)

k=1
30

The Fourier transfer circuit model for calculating the N th higher
harmonic component is shown in Fig.1. Applying integral formula(5) to (4), we have

Z 2π
I2N −1 =
0

ID[µ]
20


π
π

ĝ(v) cos N θdθ =(ĝ0 + ĝ2K ) +(ĝ1 cos N θ1 


2K
K



+ĝ2 cos N θ2 + · · · + ĝ2K−1 cos N θ2K−1 )Z 2ππĝ(v) sin N θdθ =(ĝ1 sin N θ1 + ĝ2 sin N θ2
I2N =
K
0
+ · · · + ĝ2K−1 sin N θ2K−1 )

10








(7)
The blocks in Fig.1 are constructed by the ABMs of Spice whichcalculate the each term in the relation (7). In the calculations, theinterval [0, 2π] of the integration is divided by 2K equal divisions,so that the value of θk = 2π/2K is obtained as a node voltage at
the kth resistor in the resistive circuit.

Fourier Expansion of MOSFET

N=1
N=2
N=3
N=4
N=5

0
0.0

2.0

4.0

6.0

8.0

VGS[V]

10.0

Fig.2 Fourier transformation for MOSFET.

3. Fourier transformation of nonlinear
circuits
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Now, we consider the distortion analysis of nonlinear electronic
circuits. Analog ICs contain many kind of nonlinear devices suchas diodes, bipolar transistors and MOSFETs. In this section,
we show a technique to replace the devices by the corresponding modules using the Fourier transfer circuit model in section 2.
These customizations are very useful to develop user-friendly simulators of our harmonic balance method. In this case, we need totake account of the parasitic capacitors in the modelings at higher
frequency domain [6-7]. Assume that the currents and charges ofthe devices have voltage controlled nonlinear characteristics. The
Spice models are given as follows:(a) Diodes:
In the modelings of diodes at higher frequency domain, the currents iD are composed of the DC current id and charge qD given
as follows:³

id = IS

Now, let us consider Fourier circuit models of the above devices. In oder to develop the used-friendly simulators, we makethe Fourier transfer circuit models for all of the devices in theforms of modules. At first, we consider the Fourier transfer modelof a bipolar transistor given by (12) and (13). Assume they are
driven from 3 terminal voltage as follows:
vC = VC,0 +(VC,2k−1 cos kθ + VC,2k sin kθ) 




k=1


M

X

M
X

vB = VB,0 +vdkT) − 1 , Vt =
nVt
q

(11)

θ = ωt

The charge characteristics are described by the piecewise continuous functions as follows:
Cj0 Anτ IS An+exp
qD =[1 − (vd /φB )]m
nVt
qD =

³

Cj0 An³vd
nVt

We calculate the currents (iC (θi ), iB (θi ), | θi = (2π/K)i, i =0, 12, . . . , K) and the charges (qBC (θi ), qBE (θi ) by (13) and (12)with θi = i × (2π/K). Thus, using Fourier transfer circuit shownin Fig.1, we can get the Fourier coefficients described by´, for (vd < Fc φB )mvd
1 − Fc (1 + m) +φB

(12.1)´MX[1 − Fc ](1+m)³´τ IS Anvd+
exp, for (vd ≥ Fc φB )nVt
nVt

iC = IC,0 +(12.2)

iD

where the examples of the default values [6] are given by
Is = 10−14 [A], n = 1, τ =10−10 [s], φB = 0.8[V ], m =0.5, Cj0 = 2. × 10−12 [F ], Fc =0.5, An = 1. k = 1.381 ×
10−23 [J/K], q = 1.602 ×
10−19 [C], and T is the device
temperature in degrees Kelvin.
Thus, Vt ' 0.026[V ].

vC1

M
X

iC1

qBC = QBC,0 +

iB1

M
X

qBE = QBE,0 +Network

³

³

vBN

³

´³

vBE
Vt

³

´

D

D
+

+

vC1,2kvC1,2k-1
kth vB1,2k-1
Cos-circuit

vE1,2k-1´

i B1,2k-1

iC1,2k-1
Q1

iBN,2k-1

i CN,2k-1
QN

vCN,2k-1
vBN,2k-1´

vEN,2k-1

1
vBC
1
−1
(13.1)
exp
+
QB
BR
V
´ ´
´ ´
³ ³
³ t³
IS
IS
vBE
vBC
iB =
−1 +
− 1 (13.2)
exp
exp
QF
Vt
QR
Vt
where
1
vBC
vBE
'1−
−
QB
VAF
VAF
The two capacitor charges are the same as (12). An examples ofthe default values other than diode are shown in section 4 [6].
(c) MOSFETs
Nowadays, the sizes of MOSFETs are becoming smaller andsmaller, so that there are many types of the models [6,7]. The simplest Shichman-Hodges model (level 1) is given in (10). We alsoneed to consider four parasitic capacitor charges qDS drain-gate,
qGS source-gate, qBS substrate-source and qDB drain-substrateat the higher frequency domain. Their charge characteristics are
given by (12) in the simplest model [6,7].

i D,2k-1

vD,2k-1-

i D,2k-1

vD,2k-1-−1´(15.2)

QN

vEN

exp




(QBE,2k−1 cos kθ + QBE,2k sin kθ) 



iBN

(b) Bipolar transistors:
The collector and base currents of bipolar transistors at higher
frequency domain also consist of the DC currents and the chargesof the base-collector, and the base-emitter. The DC currents are
given by Gummel-Poon model as follows:
IS
QB




(QBC,2k−1 cos kθ + QBC,2k sin kθ) 



k=1

iCN

Fig.3 Nonlinear electronic
circuit.

iC =(15.1)

k=1

Q1

vE1
vCN




(IB,2k−1 cos kθ + IB,2k sin kθ) 



k=1

D

vB1

Linear(IC,2k−1 cos kθ + VI,2k sin kθ) 



k=1

M
X

iB = IB,0 +
+
vD
-

(14)





X


vE = VE,0 +(VE,2k−1 cos kθ + VE,2k sin kθ) 




k=1

´

exp(

(VB,2k−1 cos kθ + VB,2k sin kθ)

k=1
M

kth
vB1,2kSin-Circuit

v E1,2ki B1,2k

i C1,2k
Q1

iBN,2kiCN,2kQN

vCN,2kvBN,2kvEN,2k

−IS

(a)

(b)

Fig.4 Fourier circuit of Fig. 3 at kth higher harmoniccomponent.
Next, the charges (15.2) need to be transformed into the corresponding currents as follows:
M
X

iBC =




(−kQBC,2k−1 sin kθ + kQBC,2k cos kθ) 



k=1

M
X

iBE =



(−kQBE,2k−1 sin kθ + kQBE,2k cos kθ) 



(16)

k=1

Thus, the Fourier coefficients of the collector and base currents3- 501 -

are given byX
iC = IC,0 +(IC,2k−1 − kQBC,2k ) cos kθ
k=1+(IC,2k + kQBC,2k−1 ) sin kθ]M
X

iB = IB0 +5. Conclusions and remarks











M

The distortion analysis is very important for designing the highfrequency analog integrated circuits. Especially, it is useful toinvestigate the properties of the frequency response curves in thewide domain.
In this paper, we have proposed an efficient technique for calculating the frequency response curves of nonlinear electronic circuits. At first, the nonlinear devices such as bipolar transistors
and MOSFETs are transformed into the modules which execute
the Fourier transformations. Using the modules, the nonlinear
circuit is transformed into the Fourier transfer circuit which iscorresponding to the determining equations of the harmonic balance method. Once, we have developed the modules for all kindsof nonlinear devices, it can be easily applied any kind of circuits.
The Fourier circuits are solved with the DC analysis of Spice.
Thus, our simulator is quite user-friendly.
As future works, we are going to develop frequency analyzers
with two independent input frequencies where we adopt two dimensional Fourier transformation technique. It will be applied toanalysis of mixers and modulators.







[(IB,2k−1 + kQBC,2k + kQBE,2k ) cos kθ 



k=1

+(IB,2k − kQBC,2k−1 − kQBE,2k−1 ) sin kθ](17)Note that the transistor modules are only constructed by the
AMBs of Spice, so that it does not contain any dynamic element.
This technique is easily applied to the diodes and MOSFETs.
On the other hand, the linear sub-network in Fig.3 is transformto the Cosine and Sine circuits [11], whose the Fourier circuit forkth higher harmonic component is shown in Fig 4. Thus, theFourier circuit corresponding to the determining equation of theharmonic balance method can be easily calculated by the use of
DC analysis of Spice.

4. An illustrative example
Let us calculate the frequency response curves of a tunedamplifier circuit [13] shown in Fig. 5, where we take accountthe DC, and the fundamental to the third harmonic components.

References[1] M.Schetzen, The Volterra and Wiener Theorems of Nonlinear Systems, John Wiley and Sons, 1978.

Q
C0

R0

RL

Vin

C

[2] D.D.Weiner and J.E.Spina, Sinusoidal Analysis and Modeling of Weakly Nonlinear Circuits with Application to Nonlinear Interference Effects, Van Nostrand Reinhold, 1980.

L

E1[3] P.Wambacq and W.Sansen, Distortion Analysis of Analog
Integrated Circuits, Kluwer Academic Pub., 1998.

E2[4] B.J.Leon and D.J.Schaefer, “Volterra series and Picard iteration for nonlinear circuits and systems,” IEEE Trans. on
Circuits and Systems, vol.CAS-25, no.9, pp.789-793, 1078.

Fig.5 Tuned- amplifier.
vin (t) = 0.05 cos ωt, E1 = E2 = 10[V ], C0 = 1[µF ],C = 50[pf ], L = 20[µH], R0 = 2[kΩ], RL = 1.5[kΩ][5] L.O.Chua and C-Y.Ng, “Frequency-domain analysis of nonlinear systems: formulation of transfer functions,” IEE
Trans. Electronic Circuits and Systems, Vol.3, no.6, pp.257269, 1979.

The transistor is modeled by a Gummel-Poon model, whose parameters are given by
IS = 10−14 [A]BF = 470
BR = 1

τ = 10−10 [sec]Cj0 = 0.1[pF ]VC0Vt = 0.026[V ]VAF = 150[V ]

VC1

VB0

C0

I 01

I C1


VE1

VD2

I B0

R0

02

VB1
I B1

EM

R0

VD1

02

I
VE2

C2

VB2

I L1

+
RL

C0

[7] M.Miura, T.Shiino and K.Mori, Circuit-Simulation Technique and MOSFET modeling, Riaraizu Rikou Senta(Japanese), 2003

VC2



I 01

I C0
VE0

CVS1

LI L2

I L2

+

I B2

Vc1
-

RL

R0

VS1

-

-

LI L1

[9] K.S.Kundert, J.K.White and A.Sangiovanni-Vincentelli,
Steady-State Methods for Simulating Analog and Microwave
Circuits, Kluwer Academic Pub., 1990.

Fig.6 Fourier circuit of the tuned-amplitude.

[10] A.Ushida, T.Adachi and L.O.Chua, “Steady-state analysis of
Nonlinear Circuits based on hybrid methods,” IEEE Trans.
on Circuits and Systems-I: Fundamental Theory and Applications, vol.39, no.9, pp.649-661, 1992.

The Fourier circuits are shown in Fig.6, where the angular frequency is changed in the following manner:
ω = 104 × 10t
To get the DC solutions stably, we set two DC sources as follows:[11] A.Ushida, Y.Yamagami and Y.Nishio, “Frequency responses
of nonlinear networks using curve tracing algorithm,”ISCAS
2002, vol.I, pp.641-644, 2002.

VD1 = VD2 = 10t, for 0 < t < 1, VD1 = VD2 = 10[V ], for t > t[12] MicroSim: PSpice A/D Circuit Analysis User’s Guides, MicroSim Co. 1995.

3

6
[V]
5

[mV]

[13] K.K.Clarke and D.T.Hess, Communication Circuits: Analysis and Design, Addison-Wesley Pub. Co., 1971.

2

4

Third higher harmonic3[8] T.J.Aprille and T.N.Trick, “Steady-state analysis of nonlinear circuits with periodic input,” Proc. IEEE, vol.60, pp.108114, 1972.

CVc1(c) Sine-circuit(b) Cosine-circuit(a) DC[6] R.L.Geiger, P.E.Allen and N.R.Strader, VLSI: Design Techniques for Analog and Digital Circuits, McGraw-Hill, 1990.

φB = 0.75
Fc = 0.74

Fundamental

1

2Second higher harmonic

1
0
105

6

10

107

 rad/sec] 108

0

5

10

6

10

107

 rad/sec] 108

Fig.7 The frequency response curves.

4- 502 -