NOLTA 2006
11-14 september
Bologna - ITALYOn Optimization Algorithm
for Attaining The Maximum DC Gain of CMOS Amplifiers
M. Oda† , Y. Yamagami† , Y. Nishio† , J. Kawata‡ , A. Ushida‡† Dept. of Electrical and Electronic Engineering, Tokushima University, Tokushima 770-8506, Japan‡ Dept. of Mechanical and Elecronic Engineering, Tokushima Bunri University, Kagawa 769-2193, Japan
Abstract—
CMOS operational amplifiers are most important building blocks of analog circuits. In this paper, we proposea Spice-oriented design algorithm of CMOS amplifies for
attaining the maximum DC gain. Our optimization algorithm is based on well-known steepest descent method,where the gradient direction is decided by the analysis ofsensitivity circuits for the designing parameters such as
MOS sizes and bias voltages. In order to develop theuser-friendly designing tool, we have firstly tried to replace
MOSFETs by the corresponding sensitivity modules. Afterthen, our steepest descent algorithm is realized by equivalent nonlinear RC circuits with ABMs (analog behavior
models), so that the solution can be found as an equilibrium point with transient analysis of Spice. We found fromsome numerical experiments that the optimum points canbe stably obtained by the transient analysis.

1. Introduction
CMOS operational amplifiers are widely used as thebuilding blocks in analog circuits [1-4]. For designing theamplifiers, there are several design criteria such as the maximum gain, the minimum power consumption and so on.
For example of attaining the maximum gain of an amplifier, the gain largely depends on the bias voltages and
MOS sizes, so that they are chosen as the design parameters. Traditionally, the optimum parameters were foundby trial and error process with Spice. It was really timeconsuming when the design parameters are increased [6].
On the other hand, the steepest descent method is widelyused for this kind of problems [5]. However, it is very efficient only when the objective functions are given in theanalytical form.
Hence, in this paper, we propose a practical steepest
descent algorithm based on Spice combining the sensitivity
analysis. Firstly, we need to define the objective function
as follow:
Φ(x, p), x ∈ Rn , p ∈ Rk
(1)
where
x: circuit variables such as voltages and currents.
p: optimization parameters such as bias voltages, resistor’svalues and the dimensions of MOSFETs W[width in µm]
and L [length µm] and so on [6,7]Now, let us discuss the design algorithm of amplifiers for
attaining the maximum DC gain based on Spice-oriented
steepest descent method [5,7]. Namely, the gradient direction is decided by the solutions of sensitivity circuits [8],and the steepest descent algorithm can be realized by theequivalent RC circuits combining with nonlinear controlledcurrent sources. In order to develop user-friendly simulators, we tried to transforms MOSFETs into the corresponding sensitivity modules. The optimization algorithm
to attain the maximum gain is given bydSv (p)dpi=−, i = 1, 2, . . . , k,
ds
dpi

(2)

where S(p) is the sensitivity gain and pi are optimization parameters. Unfortunately, it is impossible to evaluate dS(p)from the sensitivity analysis, so that we need to
dpi
introduce the numerical differentiation as follows:
Sv (p + ∆p) − Sv (p)dpi=−, i = 1, 2, . . . , k,ds∆piT∆p = (0, 0, . . . , ∆pi , . . . , 0)(3)

with sufficiently small ∆p. Then, replacing a variable “s”
by time “t”, our descent algorithm can be realized by theequivalent nonlinear RC circuits with the solutions of thesensitivity circuit and ABMs of Spice[9]. In this way, we
can find out the optimum parameters at the equilibrium
point from the transient analysis.
We show the sensitivity circuit using the tableau approach, and the above optimization algorithm in section 2.
The sensitivity modules of MOSFETs are shown in section
3. They are very useful to develop user-friendly simulators. We show interesting examples of inverter amplifier toattain the maximum gain and CMOS amplifier in section
4.

2. Sensitivity analysis
and Spice-oriented optimization algorithm
Sensitivity analysis
The steepest descent method is the most basic optimization
approach, where the gradient direction is decided by solutions of the sensitivity circuit. Let us derive the sensitivity
circuit via tableau approach [8]. The tableau equation is
given by:

"

Ki
0
A

Kv
1
0

0
−AT
0

#"

i
v
vn

#

"
−

g(v, i)
E
AJ

#

"
=

0
0
0

#

(4)
The first row is coming from the Ohm’s low, second andthird are Kirchhoff’s voltage and current lows, respectively.
A is the incidence matrix and Ki , Kv are (b × b) squarematrix composed of “1” and “0” for b elements. E and J

999

show the voltage and current sources. Now, we define thesensitivity as follow:
Sensitivity: Let us define the sensitivity by
Sk,i ≡ lim

∆xk

(5)

∆pi →0 ∆pi

Example: Now, consider a simple example for deriving
the sensitivity circuit. We assume that c3 in nonlinear
resistori = c1 v + c3 v 3
is chosen as an optimization parameter. Then, the sensitivity circuit is given by Fig.2(b).

Now, let us calculate the variational equation from (3). For
parameter variations of the nonlinear resistors, we have¯¯g(v0 + ∆v, i0 + ∆i) + ∂g(v,i)∆p∂p
v0 ,i0¯∂g(v,i) ¯' g(v0 , i0 ) + ∂v v ,i ∆v
0 0 ¯¯+ ∂g(v,i) ¯∆i + ∂g(v,i) ¯
∆p
∂i

∂p

v0 ,i0

R1

E

Ki
0
A


−

Kv
1
0

¯
¯

0
−AT
0

#"

∂g(v,i)Si,pi∂i
v0 ,i0

δ(i)
A‹(i)



Si,pi
Sv,pi
Svn ,pi

v0 ,i0



=



#

−

¯
¯

∂g(v,i)Sv,pi∂v
v0 ,i0¯¯∂g(v,i)δ(i)∂pv0 ,i0

0
0

0
0

2
0,

...
...,

i
1

...
...,

k
0

G(v0)

v03

Sv

(a) nonlinear resistive circuit.
(b) Sensitivity circuit.
Fig.2. Sensitivity circuit.

 Variational method: We perturb c3⇒ c3 +∆c3 , then, we have the following circuit equation and the variationalc1 (v0 + ∆v) + (c3 + ∆c3 )(v0 + ∆v)3 +,

where ‹(i) means a delta function satisfying
1
0,

v

Si

equation:(7)‹(i) = [R1(6)

Thus, we have the following circuit equation for calculating
the sensitivities:"i(v0 + ∆v) − E
=0
R1

∆v
=0
R1
Therefore, the sensitivity of the resistor voltage is given by⇒ c1 ∆v + ∆c3 v03 + 3c3 v02 ∆v +

∆v
v03
=−
, for G(v0 ) = c1 + 3c3 v02∆c3 →0 ∆c3
G(v0 ) + R1

]T .

Sv = lim

1

Thus, from (6), we can derive the sensitivity circuit forith pi parameter. Note that the circuit configuration isequal to the original one except for the nonlinear elements
being replaced by the linear incremental resistors at theoperating points V0 , I0 . Now, we summarize the algorithm
for deriving the sensitivity circuit.
1. When a voltage source is chosen as an optimization
parameter, it is set to “1[V]”. Other sources are removed as the short-circuits.
2. When a current source is chosen as an optimization
parameter, it is set to “1[A]”. Other sources are removed as the open-circuits.
3. When a resistor chosen as an optimization parameter,
it is replaced by the linear incremental
resistor with
¯
¯
‹(i).
Other resisthe controlled source ∂g(v,i)∂p
v0 ,i0tors are only replaced by the incremental resistors.
These transformations are shown in Fig.1.
Resistive elements

Sensitivity elementsRi0
j(v 0) = dg(v )
dp
v=v

0

Voltage-controlled resistor^
i=g(v)

Current-controlled resistor^
v=r(i)

G(v0 )e(i 0) = dr(i)dp i=i0

R(i0 )

Fig.1. Sensitivity elements for resistors.

Sv = −v03
G(v0 ) + R11

which is exactly equal to the solution from the variational
method.
Spice-oriented optimization techniqueNow, we consider optimization problems of DC circuits,
and define the objective function as follows:Φ(v, i, p) = φ̂(v1 , v2 , . . . , vn , i1 , i2 , . . . , in , p1 , p2 , . . . , pk )(8)
where v, i are circuit variables, and p is the optimization
parameters. Applying the steepest descent method to (7),
we have
∂p1
∂s
∂p2
∂s

Linear resistor
v=Ri

Sensitivity analysis:
From the sensitivity circuit
Fig.2(b) with ∂i/∂c3 |v0 = v03 , we have the voltage dropat the resistor G(v0 ) is given= −φ̂(Sv1 ,p1 , . . . , Svn ,p1 , Si1 ,p1 , . . . , Sin ,p1 , 1, 0, . . . , 0) 
= −φ̂(Sv1 ,p2 , . . . , Svn ,p2 , Si1 ,p2 , . . . , Sin ,p2 , 0, 1, . . . , 0)···············



∂pk
,
0,
0,
.
.
.
,
1)
,
.
.
.
,
S
,
S
,
.
.
.
,
S
=
−
φ̂(S
i
,p
i
,p
v
,p
v
,p
n
n
1
1
k
k
k
k
∂s
(9)
where {Svi ,pj , Sii ,pj , i = 1, 2, . . . , n, j = 1, 2, . . . , k} are solutions from the sensitivity analysis. Note that we needk times sensitivity analysis to execute the steepest descentmethod. Changing variable “s” with time “t”, the equation(8) is realized by a coupled RC circuits using controlledcurrent sources as shown in Fig.3, where the value capacitance Cs are related to the convergence speed, and R0 s
are sufficient large dummy resistances. On the other hand,

1000

+
^ (S

v p, ... ,S v p,S
1 1

n 1

, ... ,S i p, 1,0, ... ,0)

i1 p1

R0 p

C

n 1

2. Saturation region
kn W
∂iD
=
(vGS0 − vT 0 )(1 + λvDS0 ) (16)
GGS ≡
∂vGS
L
kn W
∂iD
GDS ≡
=
(vGS0 − vT 0 )2 λ
(17)
∂vDS
2L
∂iD
γkn W (vGS0 − vT 0 )(1 + λvDS0 )
GBS ≡
=
(18)
p
∂vBS
2L
φ − vBS0

1

+
^ (S

v p, ... ,S v p,S i p, ... ,S i p, 0,1, ... ,0)
1 2

n 2

1 2

p

C

n 2

R0 2

+
^

(Sv p, ... ,S v p,S
1 k

n k

, ... ,S i p, 0,0, ... ,1)

i1 pk

R0pk

C

n k

For the optimization parameter (W ,L), we have the following current sensitivities:-

Fig.3. Circuit configuration of the steepest decent method.

1. Linear region

h

i

kn
vDS0
(vGS0 − vT 0 ) −vDS0 (1+λvDS0 ), for L,
L
2
(19)
h
i
vDS0
kn W
(vGS0 − vT 0 ) −vDS0 (1+λvDS0 ), for W,
Sin = − 2L2(20)
2. Saturation regionkn
Sin =(vGS0 − vT 0 )2 (1 + λvDS0 ), for L,
(21)
2L
kn W
(vGS0 − vT 0 )2 (1 + λvDS0 ), for L, (22)Sin = −
2L2
Sin =the optimization attaining the maximum gain is realized by
the use of numerical differentiations as given in (2). The
relation (2) is also realized by the circuit shown in Fig.4
with small ∆pi . where S(p + δ(∆pi )) and S(p) are so

-



+



pi ) ) - S (p )

S (p+

p

C
i

R0 p

i=1,2, ... , ki-where “0” means voltage at the operating point. These
modules are realized with ABMs of Spice, whose nMOS
module is shown by Fig.5 (b). We can also realize pMOS

Fig.4. Circuit configuration of attaining the maximum gain.

Dlutions from the sensitivity analysis. Thus, the optimum
point can be found by the transient analysis of Spice.

D
vD

vG

3. Sensitivity modules of MOSFETs

iD
vB

G

B

Sin

G
GGSSV,GS

iS

vS

B

GDSSV,DS G BSSV,BSS
S

Now, let us derive the sensitivity modules of MOSFETs
in DC. Once these modules are stored in our computer as
the packages, our optimization algorithms can be easily
carried out with Spice. We apply piecewise continuous
Shichman-Hodges model [1-4] for nMOS as follows:
1. Linear region (vGS > vT , 0 < vDS < vGS − vT )
iD =

kn W
L

h

vDS
2

(vGS − vT ) −(10)

2. Saturation region(vGS > vT , vDS ≥ vGS − vT )kn W
(vGS − vT )2 (1 + λvDS )2L
where the threshold voltage is given byiD =vt = vT 0 + γ³p

φ − vBS −Fog.5. (a) nMOS, (b) nMOS module.

sensitivity modules in the same way.

4. Illustrative examples.

ivDS (1 + λvDS )(b) nMOS module(a) nMOSCMOS inverter amplifier.
Most of CMOS operational amplifiers contain some kindsof inverters Fig.6(a) as the building blocks, whose transfer
characteristic is given by Fig.6(b), where MOS parameters
are given by Table 1.

(11)

[V]

E2

3
Vout
2.5

M1

Inverter characteristic

2

p ´

+

φ

(12)

E1
M2

vout
RL -

1.5
1
0.5
0

Thus, we have the following sensitivity elements at thecorresponding operating points:

0

0.2

0.4

0.6

0.8

E 1 [V]1(a) CMOS inverter.
(b) Input-output characteristic.
Fig.6. CMOS inverter.

1. Linear region
GGS ≡
GDS ≡

h

kn W
∂iD
vDS0 (1 + λvDS0 )
=
∂vGS
L

∂iD
kn W
×
=
∂vDS
L

3 2
− λvDS0− vDS0 + (1 + 2λ)(vGS0 − vT 0 )
2
GBS ≡

∂iD
γkn W vDS0 (1 + λvDS0 )
=
p
∂vBS
2L
φ − vBS0

i

(13)

(14)

(15)

1001

Table.1 MOS parameter[9] (MKS system).
Mark
pMOS
nMOS
L
1.2 × 10−6
1.2 × 10−6W
7.8 × 10−6
7.8 × 10−6LD900.1 × 10−12900.1 × 10−12
pVT 0 , nVT 0-0.83110.6081
kp , Kn19.34 × 10−6
74.21 × 10−6
λ
0.1
0.2
pγ, nγ0.3046
0.6166Cj0
259.97 × 10−6
280.65 × 10−6

VDD

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

M5
M6
M1

M4

M 2 M3- Vin

The gain is given by the following relation+Vin
M 8 M9M7

dVout
Gout =
.
dE1

M10

vout

(23)

VB
M11

M12
M 14

If the input voltage is set at series to E1 . It seems to havethe maximum around E1 = 0.37[V ]. Now, let us calculate the maximum gain with the steepest descent method
shown is section 2.2. Our Spice-oriented optimization algorithm is realized the equivalent circuit shown by Fig.7,
where the above 3 circuits are inverters, and the below 3
are corresponding sensitivity circuits, where they are set
to E1 = 1[V ] in the sensitivity analysis because of the gainofE1 . The optimum point is obtained by the following

M13

VSS

Fig.9(a) CMOS AB amplifier.

25

50

Gain Optimization (MOS size)
Max gain

15

20

nW,pW[µm]nW,pW(µm)
1010nL,pL(µm)
0

5

INV0

E2

RL

INV1

INV2

RL

RL

(b)

S (E ,E )

1[V]

1

1[V]

2



S1

RL

[sec]

1.0

0.0

0.2

0.4

0.6

0.8

[sec]

1.0



Inverter
S2

RL

0.8

S(E ,E + E )
1 2
2

1[V]

InverterInverter
S0

0.6−1.206[V ], M ax Gain = 40.12(c)
S (E 1+ E ,E )
1 2

0.4

pL = nL = 1µm, pW = nW = 10µm, VB =

E1
(a)

0.2

Fig.9(b) Optimization for sizes, Max Gain =21.21,(c) Optimization for sizes and bias VB .

InverterInverterInverter
E1

-20

0.0

E2 + E2

VB (x10)-10nL,pL[µm]E2

Max gain30

0

E 1+ E 1

Gain Optimization

40

20

RL

5. Conclusions and remarks
(Sb)

(Sa)

(Sc)

Fig.7. Equivalent circuit for getting the maximum gain.

steepest descent method shown Fig.4.
,E2 )−Sv (E1 ,E2 )1
C dE= − Sv (E1 +∆E1∆E
dt
1
Sv (E1 ,E2 +∆E2 ,)−Sv (E1 ,E2 )dE2
C dt = −
∆E2
5

Vout

E1

0
Gain
-5

-5
-10

(24)

5

E2

0

¾

E1- gain characteristic
Gain

-10

Gain optimization-15

In this paper, we propose a Spice-oriented design algorithm of CMOS amplifies for attaining the maximum DC
gain. The optimization algorithm is based on the steepest
descent method such that the gradient direction is decided
by the analysis of the sensitivity circuit. Since our optimization method is realized by the nonlinear RC circuitswith ABMs, the solution is found by the transient analysis
of Spice. From our numerical experiments, we can get thesolution, stably. For future work, we will apply our method
to bipolar transistor circuits and the others.

References

-15

-20

-20

Gain

-25

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

-25

-30
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35
0.4
Time

Fig.8 (a) Gain optimization

-30
0

0.3

0.6

0.9

1.2

E 1 [V]

1.5[2] F.Maloberti, Analog Design for CMOS VLSI Systems,
KLuwer Academic Pub., 2001.

(b) Gain sensitivity.

E1 = 0.74696[V ], E2 = 1.9467[V ], Gout,max = −28.444, C = 0.01,
RL = 1000[kΩ].

Thus, the optimum point corresponds to the equilibrium
point, which is found by the transient analysis of Spice. To
investigate the result, we calculate the gain characteristic
for the variable E1 with E2 = 1.946 as shown in Fig.8(b).
The result is exactly equal to the result from Fig.8(a).
AB amplifier
Now, we consider AB amplifier shown in Fig.9(a) [1]. It isa class of differential amplifier and consisted of 14 MOSFETs. At first, we optimize the gain by changing the sizes(nL, nW, pL, pW ) between 1µm < L, W < 10µm, where
we classify the transistors into 3 groups according to pMOS
mirror circuits (1,2,3,4,5,6), nMOS differential inputs andmirror circuit (1,3,4,5,6,8) and nMOS current sources (2,7).
We also consider the bias voltage VB so that we optimized
seven parameters and have the result shown by Fig.9 (c).

1002

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Univ. Press, 2004.
[5] J.A.Snyman Practical Mathematical Optimization: An Introduction to Basic Optimization Theory and Classical and
New Gradient-Based Algorithm, Springer, 2005.
[6] K.Ono, K.Saeki,Y.Sekine,“A design of an operational amplifier with high DC gain,” The Papers of Tech. meeting on
Electronic Circuits, IEE Japan, ECT-04-99, pp.1-5, 2004.
[7] M.Oda, Y.Yamagami, Y.Nishio, J.Kawata and A.Ushida, “A
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[8] L.O.Chua and P-M Lin, Computer-Aided Analysis of Electronic Circuits: Algorithms and Computational Techniques,Prentice-Hall, 1975.
[9] MicroSim: PSpice A/D circuit Analysis User’s Guides, MicroSim Co. 1995.