2006 RISP International Workshop
on Nonlinear Circuits and Signal Processing (NCSP'06)
Waikiki Beach Marriott, Honolulu,Hawaii, USA, March 3-5, 2006

Frequency Domain Analysis of CMOS Amplifiers Containing Parasitic Capacitors
Yousuke Taniguchi† , Yoshihiro Yamagami†, Yoshifumi Nishio † and Akio Ushida‡†Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470/FAX:+81-88-656-7471
Email: yousuke@ee.tokushima-u.ac.jp‡Tokushima Bunri University
1314-1 Shido, Kagawa, Japan
Phone:+81-87-894-5111/FAX:+81-87-894-4201
Email: ushida@fe.bunri-u.ac.jp

Abstract
Distortion analysis of nonlinear circuits is very importantfor designing analog integrated circuits and/or communication systems. We consider here the frequency-domain analysis of CMOS operational amplifiers in the high frequency
domain. In this case, the parasitic capacitors among source,
drain, gate and substrate of MOSFETs give large effects tothe characteristics. In this study, we propose a harmonic balance method to solve the frequency response curves. It is
known that the determining equations are given by the analytical functions only when the nonlinear characteristics aregiven by polynomial functions. Unfortunately, many nonlinear devices are modeled by the use of many kinds of special
functions and piecewise continuous functions, so that we cannot apply analytical harmonic balance methods. Therefore,
we develop the Fourier Transfer Circuit Models (FTCM) using ABMs (Analog Behavior Models) of Spice which carries
out the Fourier transformation of MOSFETs containing nonlinear parasitic capacitors.
1. Introduction
Recently, owing to the rapid progress of integrated circuit
technologies, high frequency CMOS amplifiers have beenfrequently used in high performance RF communication systems [1-3]. On the other hand, the characteristics of MOSFETs have strong nonlinearity. Especially, the parasitic capacitors in the high frequency give large effects to the characteristics [2,4-6].
There are two techniques in the frequency domain analysis
of nonlinear circuits; namely, Volterra series [4-7] and harmonic balance methods [8-10]. The Volterra series methods
can be usually applied to weak nonlinear circuits. In this case,the characteristics of nonlinear elements should be describedby the power series expansions in the vicinity of the driving
point. The task is not easy for nonlinear electronic circuits
such as bipolar transistors and MOSFETs [6]. On the otherhand, the harmonic balance methods are usefully applied tothe circuits containing any kind of nonlinear devices. However, it is also not easy to derive the circuit equations and thedetermining equations for the harmonic balance method [8].

In this paper, we propose a new Spice-oriented harmonicbalance method solving the frequency response curves ofCMOS amplifiers [10-11]. Firstly, we develop the “packageddevice modules” executing the Fourier expansions of MOSFETs and bipolar transistors. Then, the determining equations are given in the form of coupled equivalent circuits. The
frequency response curves can be calculated by DC analysisof Spice. Thus, our simulators are really user-friendly.
We show the idea of the proposed FTCM in Section 2, andthe application to MOSFET in Section 3. The illustrative examples are introduced in Section 4.
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 exponential, square-root, piecewise continuous functions and so on [12]. For these devices,
the Fourier coefficients cannot be described in the analyticalforms. Therefore, using ABMs of Spice [13], we will realize the equivalent circuits executing the Fourier expansions
in the harmonic balance methods. Now, consider a nonlinear
function described byi = ĝ(v(t)).

(1)

Assume that the input and output waveforms are described asfollows;⎫K⎪⎪(V2k−1 cos kωt + V2k sin kωt) ⎪v(t) = V0 +⎪⎬k=1(2)K⎪⎪⎪i(t) = I0 +(I2k−1 cos kωt + I2k sin kωt) ⎪⎭k=1

where K shows the highest harmonic component in our analysis. Here, applying the discrete Fourier transformation to(2), the Fourier coefficients of the output current are describedas follows;- 389 -⎫T⎪
I0 = T1 0 ĝ(v(t))dt⎪⎪⎬
2 TI2k−1 = T 0 ĝ(v(t)) cos kωtdt.
T⎪
I2k = T2 0 ĝ(v(t)) sin kωtdt⎪⎪⎭
k = 1, 2, · · · , K(b) Saturation region(VGS > VT , VDS ≥ VGS − VT ):
(3)

k W
(VGS − VT )2 (1 + λVDS ).
2L
The threshold voltage is given byID =(7)Next, let us apply the following trapezoidal integration forVT = VT 0 + γ( φ − VBS − φ).
(8)mula to (3); b
The parasitic capacitors are shown in Fig. 2. When the gatehf (x)dx = (f0 + fn ) + h(f1 + f2 + · · · + fn−1 ) (4) voltage is less than the threshold voltage, the channel does
2anot appear. However, in the linear region, the channel equally
where the step size of the integration is h = (a − b)/n. Then, spread between the source and drain terminals, so that C GD
the truncation error is given by f (2) h2 /12n, where (2) shows and CGS contain equal 12 Cox W L. On the other hand, in thethe second derivative. We show that (3) with (4) can be real- saturation region, the channel spread like as Fig. 2. C GD
ized by the equivalent circuit model.
contains 13 Cox W L and CGS does 23 Cox W L. They have also
The blocks in Fig. 1 are constructed by the ABMs of Spice, depletion capacitances given in (9) between substrates. Thus,
where the interval [0, 2π] of the integration is divided by 2K the characteristics of parasitic capacitors are given in Table 1
equal divisions. The value of θ k = 2π/2K is obtained by the [12]. The circuit model in high frequency is shown in Fig. 3.
node voltage at the Kth resistor in the resistive circuit.
AS Cj0, m  0.5.
(9)
CSB =
[1 + (VSB /φ)]m^0.5g(V,θ)cosNθ 2k2k^
0.5g(V,θ)sinNθ 2k2k

R

DGATE
C GD

C GS
C GBSOURCE^g(V,θ2k-1)cosNθ 2k-1
^g(V,θ2k-1)sinNθ 2k-1

R

Σ/2K

C GS

C SB

Bulk

G(n+)
CHANNELSUBSTRATE

C SB

C DB

C GB DRAIN(n+)

Coefficient of cosNθ

CGD

CGB

C DB2π^g(V,θ1)cosNθ 1^g(V,θ1)sinNθ 1R

Σ/2K

Figure 3: MOSFET in thefor nMOS.

high frequency.

Coefficient of sinNθ

Table 1: Terminal capacitances of a MOSFET.
Regions^0.5g(V,θ0 )cosNθ 0^
0.5g(V,θ0 )sinNθ 0RParasiticcapacitors
CGDCGS

V0 V2M

Figure 1: Fourier transfer circuit model.

To investigate the accuracy of our FTCM, we calculate thefollowing Fourier expansion:
ex cos θ= I0 (x) + I1 (x) cos θ + I2 (x) cos 2θ + · · · .

SFigure 2: Parasitic capacitors

Linear region
Cox W LD + 12 Cox W L
Cox W LD + 12 Cox W L

CBGCBD(5)Saturation region
Cox W LDCox W LD + 23 Cox W L
1 W LCox Cpn (VDB )3Cox +Cpn (VDB )0AD Cpn (VDB )+ 12 W LCpn (VDB )AS Cpn (VSB )+ 12 W LCpn (VSB )CBS

AD Cpn (VDB )AS Cpn (VSB )+ 23 W LCpn (VSB )The simulation result for h = 2π/20 in Fig. 1 is I 1 (10) =Note that the characteristics of the nonlinear parasitic ca2761 at N =1, x=10 which is exactly equal to the value fromthe table of Bessel function. Thus, we found that the FTCM pacitors in Table 1 are piecewise discontinuous functions.
can get the sufficiently exact solution even with 2K=10 to 20 The results obtained using the paramerter values given in Table 2 are shown in Figs. 4 and 5.
divisions of the interval 2π.
2.5

3. FTCM of MOSFETNow, consider FTCM of MOSFET. The drain current ofnMOS is given by the Schichman-Hodges model as follows[1-2]:(a) Linear region(VGS > VT , 0 < VDS < VGS − VT )k W VDS(VGS − VT ) −VDS (1 + λVDS ). (6)ID =L210nMOS drain current

Id[mA][fF]

2.0nMOS(paracitic capacitor)

8

VGS= 3[V]
1.5

6

1.0

C DG

4

VGS=2[V]0.5

CGSC DS2
VGS=1[V]

00

0.1

0.2

0.3

0.4

0.5

0.6

VDS[V]

Figure 4: Drain current of nMOS.

- 390 -0.7

000.1

0.2

0.3

0.4

0.5

0.6
0.7
VDS[V]

Figure 5: Parasitic capacitors of nMOS.

Observe that the parasitic capacitors give large effects in
the high frequency, though their order is 10 −15 F .

DLinearLinearG

subnetwork

circuitSmodulecircuit

B(b) kth cosine modulecircuit(c) kth sine module

VDD(Vt,2k−1 cos kωt + Vt,2k sin kωt) (10)(It,2k−1 cos kωt + It,2k sin kωt)module4. Fourier analysis of CMOS differential amplifiers
4.1 CMOS amplifier
Let us calculate the frequency response curves of differential amplifiers, whose MOSFET parameters are given byTable 2. In our simulations, we consider until the 3rd harmonic component.
First, we consider the amplifier circuit using the characteristics of a CMOS inverter circuit shown by Fig. 7.

M1Ct = D, G, S, B.
Here, D,G,S,B correspond to the terminal shown in Fig. 3.
First, we assume θ = ωt, and divide period 2π in equal stepsize [θ = 0, 2π/K, 4π/K, . . . , 2π] , and calculate the current
at each point. And we apply a trapezoidal integration formulaof equation (4) instead of equation (3) whose calculations arecarried out by Fig. 1 using Spice.
Thus, the Fourier coefficients are calculated as follows:it (t) = It,0 +kth sinecircuit SB

k=1KMOS
kth sine

Figure 6: Sine-Cosine circuits using MOS modules for kth harmoninccomponents.

Now, let us derive the FTCM of MOSFET. We assume thatthe terminal voltage is given by equation (10).
K

DGLinearB(a) Network

Other parameters：
AS = AD = 20 × 10−12 , tox = 30.4 × 10−9 , φ = 0.7
Cox = 1.135 × 10−3 .

vt (t) = Vt,0 +MOS
kth cosine

kth cosine

STable 2: MOS parameter [13] (MKS system).
MarkpMOSnMOS
L
1.2 × 10−6
1.2 × 10−6W
7.8 × 10−6
7.8 × 10−6
LD900.1 × 10−12900.1 × 10−12
pVT 0 , nVT 0-0.8311
0.6081
kp , K n
19.34 × 10−6
74.21 × 10−6λ0.10.2pγ, γ
0.3046
0.6166
Cj0259.97 × 10−6
280.65 × 10−6

DGvin-VDD = 3[V ], VB = 1.2[V ], C = 10[pF ], R = 100[kΩ].
Figure 7: CMOS amplifier circuit.
[V]
2.0
Vout

1.2
CMOS Amplifier[V]

t = D, G, S, B.
In the same way, the response of capacitor charges are also
expanded into the Fourier series as follows;

outM2

(11)

k=1+RvvoVBvout

0.8

CMOS inverter amplifier ω=109 [rad/sec](input-output characteristic)

1.5
fundamental component
1.00.40

3rd harmonic

0.5

vinqp (t) = Qp,0 +K(Qp,2k−1 cos kωt + Qp,2k sin kωt) (12)k=1ip (t) =

0.2

0

0.4

0.8

0.6

1.0

[µ s]

00

0.4

0.2

0.6

0.8

Vin

1.0 [V]

Figure 8: transient response of the Figure 9: Input and output characteristics

p = DS, GS, DG, DB, GB, SB.
We differentiate equation (12) with respect to time, the current are given as follows:K

2nd harmonic-0.4

CMOS amplifier.

of the CMOS inverter amplifier.

1.0
Vout[V]
0.8

CMOS inveter amplifier(frequency response)
fundamental component
6nω=10 x10 [rad/sec]
Vin=0.1[V]0.6kω(−Qp,2k−1 sin kωt + Qp,2k cos kωt). (13)0.4

k=10.2
3rd harmonic

Comparing each harmonic component, we can obtain equivalent circuits corresponding to determining equations. If weassume that the numbers of harmonic component is K, thenumbers of subcircuit are 2K + 1. They are combined with
controlled sources in each other [10].

2nd harmonic

01

2

34

frequency

5nFigure 10: Frequency responses of CMOS inverter amplifier.

The output voltage waveform v out (t) to the input vin (t) =0.1 sin 107 t is shown by Fig. 8. The output waveform is- 391 -largely distorted. Next, changing the amplitude of the input
voltage from V in = 0 to 1, the responses of the output waveforms and the higher harmonic components are shown by Fig.
9. Next, we calculated the frequency response curves, where
the frequency is changed by ω = 10 6+n for a variable n, andassume the input by V in = 0.1[V ]. Thus, we can obtain thefrequency response curves shown by Fig. 10.
4.2 Differential amplifier [2]Next, we consider an n-channel differential amplifier givenby Fig. 11. The results in this case are shown in Figs. 12 and13.

the output characteristics at ω = 10 9 [rad/sec]. The results
are shown by Fig. 13.
5. Conclusions and remarks
In this study, the nonlinear devices such as bipolar transistors and MOSFETs were transformed into the modules executing the Fourier transformations. Using these device modules, the nonlinear circuit was transformed into the Fouriercircuit corresponding to the determining equations of the harmonic balance method. They were solved with the DC analysis of Spice.
References

VDD[1] S.M.Kang, and Y.Leblebici, CMOSW Digital Integrated Circuits:
Analysis and Design, McGraw-Hill Co.Inc., 1996.
M4

M5[2] G.Roubik, Introduction to CMOS OP-AMPs and Comparators,Wiley-Interscience Pub., 1999.

C+Rv+vin

M2

M1[3] F.Maloberti, Analog Design for CMOS VLSI Systems, KLuwer Academic Pub., 2001.

out-v in-[4] P.Wambacq and W.Sansen, Distortion Analysis of Analog IntegratedCircuits, Kluwer Academic Pub., 1998.
M3
V1[5] D.Leenaerts, J.V.Tang, and C.Vaucher, Circuit Design for RF
Transceivers, KLuwer Academic Pub., 2001.

VB[6] A.Bauer and W.Schwarz, “Algorithms for the efficient computation ofVolterra Kernels in the behavior analysis of periodically excited nonlinear dynamical circuits and systems,” Proc. of NOLTA’05, pp.509512, 2005.

VDD = VB = 3[V ], V1 = 2[V ], C = 10[pF ], R = 100[kΩ].
Figure 11: An n-channel differential amplifier.
Vout
0.3
[V]

[7] M.Schetzen, The Volterra and Wiener Theorems of Nonlinear Systems,
John Wiley and Sons, 1987.

Differential Amplifier (frequency response)
0.2

fundamental component[8] K.S.Kundert, J.K.White and A.Sangiovanni-Vincentelli, Steady-State
Methods for Simulating Analog and Microwave Circuits, Kluwer Academic Pub., 1990.

0.1

2nd harmonic
3rd harmonic

700.5

1.0

1.5

2.0

2.5

[9] 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.

nω=10 x10 [rad/sec]
3.0

3.5
4.0 n 4.5
frequency

Figure 12: Frequency response curves of the differential amplifier.
Vin = 0.1.

[10] A.Ushida, Y.Yamagami and Y.Nishio, “Frequency responses of nonlinear networks using curve tracing algorithm,”Proc. of ISCAS 2002,
vol.I, pp.641-644, 2002.

Vout
2.5
[V]

[11] A.Ushida, J.Kawata, Y.Yamagami and Y.Nishio, “Intermodulationanalysis of nonlinear circuits using two-dimensional Fourier transformation,” Proc. of NOLTA’05, pp.282-285, 2005.

Diffrential amplifieri(input-output characteristic)

2.0
1.5

fundamental component[12] MicroSim: PSpice A/D Circuit Analysis User’s Guides, MicroSimCo., 1995.

1.0
0.5

[13] J.E.Meyer, “MOS models and circuit simulation,” RCA Review, Vol.
32, 1971.

3rd harmonic 2nd harmonic

000.5

1.0

1.5 Vin[V]2.0Figure 13: Input and output characteristics of the differential amplifier.
ω = 109 [rad/sec].

The results in Fig. 12 are the frequency response curves forthe input [Vin = 0.1[V ]]. Next, we calculated the input and- 392 -