424
IEEE
TRANSACTIONS ON
MICROWAVE
THEORY
AND
TECHNIQUES,
VOL.
36,
NO.
2,
FEBRUARY
1988
Circuit Optimization: The State
of
the Art
Abstract-This
paper reviews the current state of the art
in
circuit
optimization, emphasizing techniques suitable for modern microwave CAD.
It is directed at the solution of realistic design and modeling problems,
addressing such concepts as physical tolerances and model uncertainties. A
unified hierarchical treatment of circuit models forms the basis of the
presentation. It exposes tolerance phenomena at different parameter/
response levels. The concepts
of
design centering, tolerance assignment,
and postproduction tuning in relation to yield enhancement and cost
reduction suitable for integrated circuits are
discussed.
Suitable techniques
for optimization oriented worst-case and statistical design are reviewed. A
generalized
lp
centering algorithm is proposed and discussed. Multicircuit
optimization directed at both CAD and robust device modeling is formal-
ized.
Tuning is addressed in some detail, both at the design stage and for
production alignment. State-of-the-art gradient-based nonlinear optimiza-
tion methods are reviewed, with emphasis given to recent, but well-tested,
advances in minimax,
I,,
and
I,
optimization. lllustrative examples as well
as a comprehensive bibliography are provided.
I. INTRODUCTION
OMPUTER-AIDED circuit optimization is certainly
C
one of the most active areas of interest. Its advances
continue; hence the subject deserves regular review from
time to time. The classic paper by Temes and Calahan in
1967 [lo21
was one of the earliest to formally advocate the
use of iterative optimization in circuit design. Techniques
that were popular at the time, such as one-dimensional
(single-parameter) search, the Fletcher-Powell procedure
and the Remez method for Chebyshev approximation,
were described in detail and well illustrated by circuit
examples. Pioneering papers by Lasdon, Suchman, and
Waren
[73], [74], [lo81
demonstrated optimal design of
linear arrays and filters using the penalty function ap-
proach. Two papers in
1969
by Director and Rohrer
[48],
[49]
originated the adjoint network approach to sensitivity
calculations, greatly facilitating the use of powerful gradi-
ent-based optimization methods.
In
the same period, the
work by Bandler
[4],
[5]
systematically treated the formula-
tion of error functions, the least pth objective, nonlinear
constraints, optimization methods, and circuit sensitivity
analysis.
Manuscript received May 4, 1987; revised August
20,
1987.
This
work
was supported in part by the Natural Sciences and Engineering Research
Council
of
Canada under Grant A7239 and in part by Optimization
Systems Associates Inc.
J.
W.
Bandler is with the Simulation Optimization Systems Research
Laboratory and the Department
of
Electrical and Computer Engineering,
McMaster University, Hamilton, Canada L8S 4L7. He is also with
Optimization Systems Associates Inc., Dundas, Ontario, Canada L9H 6L1.
S.
H. Chen was with
t!!e
Department
of
Electrical and Computer
Engineering, McMaster University, Hamilton, Canada. He is now with
Optimization Systems Associates Inc., Dundas, Ontario, Canada L9H 6L1.
IEEE
Log
Number 8717974.
Since then, advances have been made in several major
directions. The development of large-scale network simula-
tion and optimization techniques have been motivated by
the requirements of the VLSI era. Approaches to realistic
circuit design where design parameter tolerances and yield
are taken into account have been pioneered by Elias
[52]
and Karafin
[68]
and furthered by many authors over the
ensuing years. Optimization methods have evolved from
simple, low-dimension-oriented algorithms into sophisti-
cated and powerful ones. Highly effective and efficient
solutions have been found for a large number of spe-
cialized applications. The surveys by Calahan
[
371,
Charalambous
[39],
Bandler and Rizk
[26],
Hachtel and
Sangiovanni-Vincentelli
[63],
and Brayton et
al.
[32]
are
especially relevant to circuit designers.
In the present paper, we concentrate
on
aspects that are
relevant to and necessary for the continuing move to
optimization of increasingly more complex microwave cir-
cuits, in particular to MMIC circuit modeling and design.
Consequently, we emphasize optimization-oriented ap-
proaches to deal more explicitly with process imprecision,
manufacturing tolerances, model uncertainties, measure-
ment errors, and
so
on.
Such realistic considerations arise
from design problems in which a large volume
of
produc-
tion is envisaged, e.g., integrated circuits. They also arise
from modeling problems in whch consistent and reliable
results are expected despite measurement errors, structural
limitations such as physically inaccessible nodes, and model
approximations and simplifications. The effort to for-
mulate and solve these problems represents one
of the
driving forces of theoretical study in the mathematics of
circuit CAD. Another important impetus is provided by
progress in computer hardware, resulting in drastic reduc-
tion in the cost of mass computation. Finally, the continu-
ing development
of
gradient-based optimization tech-
niques has provided
us
with powerful tools.
In this context, we review the following concepts: realis-
tic representations of a circuit design and modeling prob-
lem, nominal (single) circuit optimization, statistical circuit
design, and multicircuit modeling, as well as recent gradi-
ent-based optimization methods.
Nominal design and modeling are the conventional ap-
proaches used by microwave engineers. Here, we seek a
single point in the space of variables selected for optimiza-
tion which best meets a given set of performance specifica-
tions (in design) or best matches a given set of response
measurements (in modeling). A suitable scalar measure
0018-9480/88/0200-0424$01.00
01988
IEEE
BANDLER AND CHEN: CIRCUIT OPTIMIZATION: STATE
OF
THE ART
425
of the deviation between responses and specifications
which forms the objective function to be minimized
is the ubiquitous least squares measure (see, for example,
Morrison
[83]),
the more esoteric generalized
fp
objective
(Charalambous
[41])
or the minimax objective (Madsen
et
al. [80]).
We observe here that the performance-driven
(single-circuit) least squares approach that circuit design
engineers have traditionally chosen has proved unsuccess-
ful both in addressing design yield and in serious device
modeling.
Recognition that an actual realization of a nominal
design is subject to fluctuation or deviation led, in the
past, to the so-called sensitivity minimization approach
(see, for example, Schoeffler
[94]
and Laker
et
al.
[71]).
Employed by filter designers, the approach involves mea-
sures of performance sensitivity, typically first-order, that
are included in the objective function.
In reality, uncertainties which deteriorate performance
may be due to physical (manufacturing, operating) toler-
ances as well as to parasitic effects such as electromagnetic
coupling between elements, dissipation, and dispersion
(Bandler
[6],
Tromp
[107]).
In the design of substantially
untunable circuits these phenomena lead to two important
classes of problems: worst-case design and statistical de-
sign. The main objective is the reduction of cost or the
maximization of production yield.
Worst-case design (Bandler
et al.
[23], [24]),
in general,
requires that all units meet the design specifications under
all circumstances (i.e., a
100
percent yield), with or without
tuning, depending on what is practical. In statistical design
[l], [26], [30], [47], [97], [98], [loo], [loll
it is recognized
that a yield of less than
100
percent is likely; therefore,
with respect to an assumed probability distribution func-
tion, yield is estimated and enhanced by optimization.
Typically, we either attempt to center the design with fixed
assumed tolerances or we attempt to optimally assign
tolerances and/or design tunable elements to reduce pro-
duction cost.
What distinguishes all these problems from nominal
designs or sensitivity minimization
is
the fact that a single
design point is no longer of interest: a (tolerance) region of
multiple possible outcomes is to be optimally located with
respect to the acceptable (feasible, constraint) region.
Modeling, often unjustifiably treated as if it were a
special case of design, is particularly affected by uncertain-
ties and errors at many levels. Unavoidable measurement
errors, limited accessibility to measurement points, ap-
proximate equivalent circuits, etc., result in nonunique and
frequently inconsistent solutions. To overcome these frus-
trations, we advocate a properly constituted multicircuit
approach (Bandler
et
al.
[12]).
Our presentation is outlined as follows.
In Section 11, in relation to a physical engineering sys-
tem of interest, a typical hierarchy of simulation models
and corresponding response and performance functions
are introduced. Error functions arising from given specifi-
cations and a vector of optimization variables are defined.
Performance measures such as
lp
objective functions
(
lp
norms and generalized
1,
functions) are introduced and
their properties discussed.
We devote to Section I11 a brief review of the relatively
well-known and successful approach of nominal circuit
design optimization.
In Section IV, uncertainties that exist in the physical
system and at different levels of the model hierarchy are
discussed and illustrated by a practical example. Different
cases of multicircuit design, namely centering, tolerancing
(optimal tolerance assignment), and tuning at the design
stage, are identified.
A
multicircuit modeling approach and
several possible applications are described.
Some important and representative techniques in worst-
case and statistical design are reviewed in Section
V.
These
include the nonlinear programming approach to worst-case
design (Bandler
et
al.
[24],
Polak
[89]),
simplicial (Director
and Hachtel
[47])
and multidimensional (Bandler and
Abdel-Malek
[7])
approximations of the acceptable region,
the gravity method (Soin and Spence
[98]),
and the para-
metric sampling method (Singhal and Pine1
[97]).
A
gener-
alized
lp
centering algorithm is proposed as a natural
extension to
1,
nominal design. It provides a unified
formulation of yield enhancement for both the worst case
and the case where yield is less than
100
percent.
Illustrations of statistical design are given in Section VI.
The studies in the last two decades on the theoretical
and algorithmic aspects of optimization techniques have
produced a great number of results. In particular,
gradient-based optimization methods have gained increas-
ing popularity in recent years for their effectiveness and
efficiency. The essence of gradient-based
lp
optimization
methods is reviewed in Section VII. Emphasis is given to
the trust region Gauss-Newton and the quasi-Newton
algorithms (Madsen
[78],
Mor6
1821,
Dennis and Mor6
The subject of gradient calculation and approximation is
[461).
briefly discussed in Section VIII.
11.
VARIABLES
AND
FUNCTIONS
In this section, we review some basic concepts of practi-
cal circuit optimization. In particular, we identify a physi-
cal system and its simulation models. We discuss a typical
hierarchy of models and the associated designable parame-
ters and response functions. We also define specifications,
error functions, optimization variables and objective func-
tions.
A.
The Physical System
The physical engineering system under consideration
can be a network, a device, a process, and
so
on, which has
both a fixed structure and given element types. We
manipulate the system through some adjustable parame-
ters contained in the column vector
GM.
The superscript
M
identifies concepts related to the physical system. Geomet-
rical dimensions such as the width of a strip and the length
of a waveguide section are examples of adjustable parame-
ters.
426
IEEE
TRANSACTIONS
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MICROWAVE
THEORY
AND
TECHNIQUES, VOL.
36,
NO.
2,
FEBRUARY
1988
-
-d
I
d-
4
In the production of integrated circuits,
+”’
may include
some fundamental variables which control, say, a doping
or photomasking process and, consequently, determine the
geometrical and electrical parameters of a chip. External
controls, such as the biasing voltages applied to an active
device, are also possible candidates for
+’.
The performance and characteristics of the system are
described in terms of some measurable quantities. The
usual frequency and transient responses are typical’ exam-
ples. These measured responses, or simply measurements,
are denoted by
FM(
+’).
B.
The
Simulation Models
In circuit optimization, some suitable models are used to
simulate the physical system. Actually, models can be
usefully defined at many levels. Tromp
[106], [lo71
has
considered an arbitrary number of levels (also
see
Bandler
et
al.
[19]).
Here, for simplicity, we consider a hierarchy of
models consisting of four typical levels as
FH=
FH(FL)
FL
=
FL(
+H)
v=
+”(+“)e
(1)
+L
is a set of low-level model parameters. It is supposed
to represent, as closely as possible, the adjustable parame-
ters in the actual system, i.e.,
+’.
+H
defines a higher-level
model, typically an equivalent circuit, with respect to a
fixed topology. Usually, we use an equivalent circuit for
the convenience of its analysis. The relationship between
+L
and
+H
is either derived from theory or given by a set
of empirical formulas.
Next on the hierarchy we define the model responses at
two possible levels. The low-level external representation,
denoted by
PL,
can be the frequency-dependent complex
scattering parameters, unterminated y-parameters, transfer
function coefficients, etc. Although these quantities may or
may not be directly measurable, they are very often used
to represent a subsystem. The high-level responses
FH
directly correspond to the actual measured responses,
namely
FM,
which may be, for example, frequency re-
sponses such as return loss, insertion loss, and group delay
of a suitably terminated circuit.
A realistic example of a one-section transformer on strip-
line was originally considered by Bandler
et al.
[25].
The
circuits and parameters, physical as well as model, are
shown in Fig.
1.
The physical parameters
+M
(and the
low-level model
+L)
include strip widths, section lengths,
dielectric constants, and strip and substrate thcknesses.
The equivalent circuit has
six
parameters, considered as
+H,
including the effective line widths, junction parasitic
inductances, and effective section length. The scattering
matrix of the circuit with respect to idealized (matched)
terminations is a candidate for a low-level external repre-
sentation
(
FL).
The reflection coefficient by taking into
account the actual complex terminations could be a high-
level response of interest
(
FH).
A
B
W1
w2
where
w
is
the strip width,
I
the length
of
the middle section,
E,
the
dielectric constant,
b
the substrate thickness, and
t,
the strip thickness.
OM
is represented in the simulation model by
&.
The high-level
parameters
of
the equivalent circuit are
where
D
is
the effective linewidth,
L
the junction parasitic inductance,
and
I,
the effective section length. Suitable empirical formulas that
relate
OL
to
OH
can be found in
[25].
For a particular case, we may choose a certain section of
this hierarchy to form a design problem. We can choose
either
+L
or
+H
as the designable parameters. Either
FL
or
FH
or a suitable combination of both may be selected as
the response functions. Bearing this in mind, we simplify
the notation by using
+
for the designable parameters and
F
for the response functions.
C.
Specifications
and
Error
Functions
The following discussion on specifications and error
functions is based on presentations by Bandler
[5],
and
Bandler and
Rizk
[26],
where more exhaustive illustrations
can be found.
We express the desirable performance of the system by a
set of specifications which are usually functions of certain
independent variable(s) such as frequency, time, and tem-
perature. In practice, we have to consider a discrete set of
samples of the independent variable(s) such that satisfying
the specifications at these points implies satisfying them
BANDLER AND CHEN: CIRCUIT OPTIMIZATION: STATE
OF
THE ART
0,
421
11.
I
I
I
I
b
I
ep
parameter
space
unacceptable
,
ptable
*b
I
(b)
Fig.
2.
Illustrations of (a) upper specifications, lower specifications, and
the responses of circuits
a
and
b,
(b) error functions corresponding to
circuits
a
and
b,
(c) the acceptable region, and (d) generalized
/p
objective functions defined in
(13).
almost everywhere. Also, we may consider simultaneously
more than one lund of response. Thus, without loss of
generality, we denote a set of sampled specifications and
the corresponding set of calculated response functions by,
respectively,
j=1,2,-.
*,
m
SJ
,
q(
+),
j
=
1,2;
-
,
m.
(2)
Error functions arise from the difference between the
given specifications and the calculated responses. In order
to formulate the error functions properly, we may wish to
distinguish between having upper and lower specifications
(windows) and having single specifications, as illustrated in
Figs. 2(a) and 3(a). Sometimes the one-sidedness of upper
and lower specifications is quite obvious, as in the case of
designing a bandpass filter. On other occasions the distinc-
tion is more subtle, since a single specification may as well
be interpreted as a window having zero width.
In the case of having single specifications, we define the
error functions by
.
.
,
m (3)
where wJ is a nonnegative weighting factor.
We may also have an upper specification
SuJ
and a
lower specification
S/,.
In this case we define the error
eJ
(
+)
=
w,
IF/
(+
)
-
SJI,
j
=
parameter space
(empty acceptable region)
aF
0'
xb
b
Oa
,,L
i,
le
b
I1
ii
Fig.
3.
Illustrations of (a) a discretized single specification and two
discrete single specifications (e.g., expected parameter values to be
matched),
as
well
as
the responses of circuits
a
and
b,
(b) error
functions related
to
circuits
a
and
b,
(c) the (empty) acceptable region
(i.e., a perfect match is not possible) and (d) the corresponding
lp
norms.
functions
as
.U,(+>
=
wuJ(5(+)-sUJ)7
jEJu
e,(+>
=w/,(q(+)-s/,)y
JEJ/
(4)
where
wuJ
and wIJ are nonnegative weighting factors. The
index sets as defined by
J,
=
{
jl,
j*,*
.)
jk}
J/
=
{
&+I7
jk+2,.
*
*
9
jm
1
(5)
are not necessarily disjoint (i.e., we may have simultaneous
specifications). In order to have a set of uniformly indexed
error functions, we let
..
e, =e,,(+),
J=
J,,
i=1,2;..,k
e,=-e,J(+),
j=j,,
i=k+l,k+2,...,m.
(6)
The responses corresponding to the single specifications
can be real or complex, whereas upper and lower specifica-
tions are applicable to real responses only. Notice that, in
either case, the error functions are real. Clearly, a positive
(nonpositive) error function indicates a violation (satisfac-
tion) of the corresponding specification. Figs. 2(b) and
3(b) depict the concept of error functions.
D.
Optimization Variables and Objective Functions
lem by the following statement:
Mathematically, we abstract a circuit optimization prob-
(7)
minimize
U(
x)
where
x
is a set of optimization variables and
U(x)
a
scalar objective function.
Optimization variables and model parameters are two
separate concepts.
As
will be elaborated on later in this
paper,
x
may contain a subset of
+
which may have been
normalized or transformed, it may include some statistical
variables of interest, several parameters in
+
may be tied
to one variable in
x,
and so on.
X
428
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TRANSACTIONS
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VOL.
36,
NO.
2,
FEBRUARY
1988
Typically, the objective function
U(
x)
is closely related
to an
lp
norm or a generalized
lp
function of
e(+).
We
shall review the definitions of such
lp
functions and dis-
cuss their appropriate use in different contexts.
E. The
lp
Norms
The
lp
norm (Temes and
Zai
[103])
of
e
is defined as
l/P
II~II~=
C
IejIp
*
(8)
[;rl
I
It provides a scalar measure of the deviations of the
model responses from the specifications. Least-squares
(I,)
is perhaps the most well-known and widely used norm
(Morrison
[83]),
which is
llell2
=
[
~l~ej12]1’2.
(9)
The
1,
objective function is differentiable and its gradi-
ent can be easily obtained from the partial derivatives of
e.
Partly due to this property, a large variety of
I,
opti-
mization techniques have been developed and popularly
implemented. For example, the earlier versions of the
commercial CAD packages TOUCHSTONE
[
1041
and
SUPER-COMPACT
[99]
have provided designers solely
the least-squares objective.
The parameter
p
has an important implication. By
choosing a large (small) value for
p,
we in effect place
more emphasis on those error functions
(ej’s)
that have
larger (smaller) values. By letting
p
=
CO
we have the
minimax norm
llellm
=
maxlejl
(10)
J
which directs all the attention to the worst case and the
other errors are in effect ignored. Minimax optimization is
extensively employed in circuit design where we wish to
satisfy the specifications in an optimal equal-ripple manner
PI,
[13l, [14],
On the other hand, the use of the
I,
norm, as defined by
[4Ol, [42], [65], [67], [go], [85].
m
IIeIIl=
C
IejI
(11)
;=1
implies attaching more importance to the error functions
that are closer to zero. This property has led to the
application of
I,
to data-fitting in the presence of gross
errors
[22], [29], [66], [86]
and, more recently, to fault
location
[8], [9], [27]
and robust device modeling
[12].
Notice that neither
llellm
nor
llelll
is differentiable in
the ordinary sense. Therefore, their minimization requires
algorithms that are much more sophisticated than those
for the
1,
optimization.
F.
The One-sided and Generalized
lp
Functions
By using an
lp
norm, we try to minimize the errors
towards a zero value. In cases where we have upper and
lower specifications, a negative value of
ej
simply indicates
that the specification is exceeded at that point which, in a
sense, is better than having
ej
=
0.
Ths fact leads to the
one-sided
lp
function defined by
l/P
~p’(e>
=
C
IejIp
(12)
[JEJ
1
where
J
=
{
jle,
2
O}.
Actually, if we define
max
{
ej,O},
then
H,’(e)
=
[le+
[Ip.
use of a generalized
lp
function defined by
e,’
=
Bandler and Charalambous
[lo], [41]
have proposed the
Hl (e)
if the set
J
is not empty
(13)
Hp(e)
=
H;
(e)
otherwise
i
where
Hp-(e)
=
-
[;l
(
-ej)-’
(14)
In other words, when at least one of the
ej
is nonnegative
we use
Hi,
and
Hp-
is defined if all the error functions
have become negative.
Compared to
(12),
the generalized
lp
function has an
advantage in the fact that it is meaningfully defined for the
case where all the
e,
are negative.
This
permits its
minimi-
zation to proceed even after all the specifications have
been met,
so
that the specifications may be further ex-
ceeded.
A classical example is the design of Chebyshev-type
bandpass filters, where we have to minimize the gener-
alized minimax function
Hm
(
e
)
=
max
{
ej
}
.
(15)
J
The current Version
1.5
of TOUCHSTONE
[lo51
offers
the generalized
lp
optimization techniques, including
minimax.
G. The Acceptable Region
We use
H(e)
as a generic notation for
Ilellp,
H,+(e),
and
Hp(e).
The sign of
H(e(+))
indicates whether or not
all the specifications are satisfied by
+.
An acceptable
region is defined as
R,
=
{
+IH(e(+))
01
(16)
Figs.
2(c),
2(d), 3(c), and 3(d) depict the
lp
functions and
the acceptable regions.
111.
NOMINAL CIRCUIT OPTIMIZATION
In a nominal design, without considering tolerances (i.e.,
assuming that modeling and manufacturing can be done
with absolute accuracy), we seek a single set of parameters,
called a nominal point and denoted by
+O,
which satisfies
the specifications. Furthermore, if we consider the func-
tional relationship of
+H=
+”(+“)
to be precise, then it
does not really matter at which level the design is con-
ceived. In fact, traditionally it is often oriented to an
equivalent circuit.
A
classical case is network synthesis
’