平成19年度電気関係学会四国支部連合大会

390

17-60

A Spice-Oriented Designing Algorithmof Two-Stage CMOS Op-Amps
T. Ishikawa1
Y. Yamagami1
Y. Nishio1
A. Ushida2(Tokushima University1 Tokushima Bunri University2 )1. Introduction
In this study, we propose a Spice-oriented design algorithmof CMOS operational ampliﬁers, which is based on the steepest
descent method. For attaining both the maximum gain and minimum power consumption, we need to set the suitable channel
width of two-stage CMOS op-amps and the reference current as
low as possible. The proposed algorithm whose circuit is realized
by ABMs (Analog Behavior Models) of Spice. Firstly, we need todeﬁne the objective function as follow:
Φ(x, p), x ∈ Rn , p ∈ Rk(1)wherex: circuit variables such as voltages and currents.
p: optimization parameters such as channel width of CMOS andreference current and so on. The gradient direction is decided
by the solutions of sensitivity circuits, and the steepest descentalgorithm can be realized by the equivalent RC circuits combining
with nonlinear controlled current sources.

2.2.2 a current source of nMOS sensitivity circuit with parametervariationSin,L , Sin,W :(a) linear region(VGS > VT , 0 < VDS < VGS − VT )a current source with variation of width W：Sin,W =GDSVDSGBS VBSB
GDSVDS

S inG

B
GGSVGS

VBS

VGS
S

GBS VBS

VGS

VBS

S

(a)

(b)

Fig. 1 Sensitivity modules of nMOS.
2.2.1 nMOS（the case element parameter is not change）(a) linear region(VGS > VT , 0 < VDS < VGS − VT )GGS ≡∂IDkn W
=VDS (1 + λVDS )
∂VGSL

∂ID
GDS ≡∂VDS{}
3kn W
2− λVDS− VDS + (1 + 2λ)(VGS − VT )=L
2GBS ≡∂ID
γkn W VDS (1 + λVDS )
=
√∂VBS
2L
φ − VBS(2)

(3)

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(b) saturated region(VGS > VT , VDS ≥ VGS − VT )
∂ID
∂VGS
∂ID
GDS ≡
∂VDS∂ID
GBS ≡
∂VBS
GGS ≡

=
==VDS2}

VDS (1 + λVDS ) (8)kn(VGS − VT )2 (1 + λVDS )(9)2L
It is enable to consider pMOS similarly if you attend to directionof current.
Sin,W =

3. Steepest descent method for attaining the maximum
gain+VDD

Q8Q8v

Q1

Q9+v

Q1C0J
Q3Q3CL
VoutQ6RL

VSS

Now, we consider Spice-oriented optimization technique for attaining the maximum gain of two-stage CMOS op-amps. We
need to introduce the numerical diﬀerentiation technique for theoptimization of the gain, because the gain is obtained by the sensitivity circuit. The gradient for attaining the maximum gain isgiven bydSv (p)dpi=−, i = 1, 2, …, k,(10)dsdpi
where S (p) is the gain, and p is optimization parameter. UndS(p)fortunately, it is impossible directly to evaluate dp from theisensitivity analysis, so that we introduce the numerical diﬀerentiation as follows;
Sv,i (p + Δ p) − Sv (p)dpi=−, i = 1, 2, …, k,(11)dsΔ piVDS

G(VGS − VT ) −Fig. 2 Two-stage CMOS op-amps conﬁguration.

DVDS

GGSVGS{(b) saturated region(VGS > VT , VDS ≥ VGS − VT )a current source with variation of width W：

2. Sensitivity modules of MOS
2.1 Sensitivity analysis: The steepest descent method is themost basic optimization approach, where the gradient direction
is decided by the solutions of sensitivity circuit. The sensitivity
circuit conﬁguration is equal to the original one except for thenonlinear elements being replaced by controlled source.
2.2 Sensitivity modules of MOS transistor: For simplicity,
we consider nMOS transistor modeled by Ebers-Moll model asshown in Fig.1. Fig.1(a) is sensitivity circuit module and Fig.1(b)is sensitivity circuit module with parameter variation.
DKnLkn W(VGS − VT )(1 + λVDS )Lkn W(VGS − VT )2 λL
γkn W (VGS − VT )(1 + λVDS )√2L
φ − VBS(5)

Δ p = (0, 0, …, Δ pi , …, 0)Twith suﬃciently small Δ p. Replacing the auxiliary variable “s”
by time “t”, our descent algorithm can be realized by the equivalent nonlinear RC circuits, and the optimum parameters can befound by equilibrium point of the transient analysis. We get thefollowing result; when the minimum current is set to 50 [µA], we
attained the maximum gain = −20.4.

4. Conclusions and remarks
In this study, we proposed a Spice-oriented designing algorithmof two-stage CMOS op-amps, the optimization technique is basedon the steepest descent method, whose gradient is calculated by
the sensitivity circuits. The equivalent steepest descent circuit
consists of ABMs of Spice, and the optimum point can be foundby the solution of the transient analysis. This conclusion provide
us to use CMOS more practically.

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References(7)[1] A. Ushida and M. Tanaka, Electronic Circuit’s Simulations, Korona Pub. Co., 2002 (Japanese).