The Ellipsoidal Technique for Design Centering of Microwave Circuits Exploiting Space-Mapping Interpolating Surrogates

Abstract

A new technique for design centering of microwave circuits is introduced. This technique exploits the space-mapping interpolating surrogate (SMIS) with a modified ellipsoidal technique. The design centering solution for microwave circuits is obtained with a small number of fine model evaluations and, hence, the number of electromagnetic simulations is greatly reduced. Practical and demonstrative examples are included to show the efficiency of the new technique

Full text

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 10, OCTOBER 2006 3731
The Ellipsoidal Technique for Design
Centering of Microwave Circuits Exploiting
Space-Mapping Interpolating Surrogates
Hany L. Abdel-Malek, Member, IEEE, Abdel-karim S. O. Hassan, Ezzeldin A. Soliman, Associate Member, IEEE,
and Sameh A. Dakroury, Student Member, IEEE
Abstract—A new technique for design centering of microwave
circuits is introduced. This technique exploits the space-map-
ping interpolating surrogate (SMIS) with a modified ellipsoidal
technique. The design centering solution for microwave circuits
is obtained with a small number of fine model evaluations and,
hence, the number of electromagnetic simulations is greatly re-
duced. Practical and demonstrative examples are included to show
the efficiency of the new technique.
Index Terms—Computer-aided design (CAD) algorithms, design
centering, ellipsoidal technique, microwave circuits, space map-
ping (SM).
I. INTRODUCTION
DESIGN centering is an optimization process whose aim
is to find the nominal values of designable circuit param-
eters that maximize the yield. Design centering of nonlinear
microwave circuits is a great challenge [1]. The computational
overhead is one of the main difficulties in the optimization
process of these circuits as many electromagnetic (EM) simu-
lations would be required. This problem can be solved success-
fully by using the space-mapping (SM) technique [2], [3]–[6].
SM employs computationally fast coarse models to greatly
reduce the evaluation cost of the computationally expensive
full-wave EM fine models. The space-mapping interpolating
surrogate (SMIS) [2] aims to calibrate a space-mapped surro-
gate, via input and output mapping, to match the fine model
with high accuracy. In this study, the SM technique is integrated
with the modified ellipsoidal technique [7] to obtain the design
centering solution of microwave circuits with a small number
of EM simulations. In general, design centering methods are
either based on the Monte Carlo method [8]–[13] or on a geo-
metrical approach [7], [14]–[22]. Monte Carlo-based methods
are computationally expensive, as yield values are estimated by
using a large number of sample points. Attempts to reduce the
computational effort have been investigated. Methods based on
a geometrical approach aim to approximate the feasible region
Manuscript received March 16, 2006; revised June 4, 2006.
H. L. Abdel-Malek, A. S. O. Hassan, and S. A. Dakroury are with the Faculty
of Engineering, Department of Engineering Mathematics and Physics, Cairo
University, Giza 12211, Egypt (e-mail: sdakrory@hotmail.com).
E. A. Soliman is with the Faculty of Engineering, Department of Engi-
neering Mathematics and Physics, Cairo University, Giza 12211, Egypt, and
also with the College of Engineering, Department of Electrical and Computer
Engineering, King Abdul Aziz University, Jeddah 21589, Saudi Arabia.
Digital Object Identifier 10.1109/TMTT.2006.882881
by a geometrical body. The feasible region is a region in the
parameter space where design specifications are satisfied. The
ellipsoidal technique [22] approximates the feasible region
by a hyperellipsoid, which is the final hyperellipsoid of a
sequence of decreasing volume of different center and shape
hyperellipsoids. The center of this final hyperellipsoid is the
proposed design center. Reduction in the hyperellipsoid volume
is obtained through consecutive cuts. Each cut is a linearization
of the feasible region boundary at a selected boundary point.
The boundary points are located such that a great reduction
in the generated hyperellipsoids volume and, hence, a speedy
convergence, is achieved [7]. However, for microwave circuits,
finding the boundary points and linearization at these points
require a lot of computationally expensive EM simulations.
The proposed design centering technique employs the SMIS
surrogate [2] with the modified ellipsoidal technique to greatly
reduce this computational overhead. The Broyden formula
[23] offers a fast way to approximate the gradients required in
linearization. However, for some models, exact gradients can
be evaluated by the adjoint sensitivity technique [24], [25]. The
proposed technique is formulated in Section II. An algorithm
based on this technique is presented in Section III. Examples
of a two- and seven-section capacitively loaded impedance
transformer, a coupled line bandpass filter, and an electromag-
netically coupled Yagi antenna are given in Section IV.
II. ELLIPSOIDAL TECHNIQUE EXPLOITING SMISs
The design specifications of a microwave circuit define a re-
gion in the parameter space (feasible region), which can be de-
fined by
(1)
where is a vector of the design parameters,
, is a fine model response vector, is the number
of design parameters, is the number of constraints, and
is the constraint vector function.
The design center using (1) can be obtained by the ellipsoidal
technique. The ellipsoidal technique generates a sequence of
-dimensional hyperellipsoids. This sequence converges to a
final hyperellipsoid whose center is taken as the design center.
0018-9480/$20.00 © 2006 IEEE

3732 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 10, OCTOBER 2006
However, working with (1) involves a lot of computationally ex-
pensive fine model evaluations. Instead, the SMIS is employed
and the feasible region is approximated by
(2)
where is the SMIS surrogate. The SMIS is
obtained by satisfying the following matching conditions:
(3)
(4)
where and are the Jacobians of the surrogate and fine
model, respectively. The SMIS is initially constructed based on
the coarse model and updated through SM iterations. In each
SM iteration, the ellipsoidal technique is invoked and a design
centering point is obtained. This new center is validated by the
fine model and is used to update the next SMIS surrogate. En-
hanced improvement of the SMIS is achieved by satisfying (3)
at all preceding design centering points (global matching). The
ellipsoidal technique is then restarted with the updated SMIS
and the next design center is obtained.
A. Modified Ellipsoidal Technique
Assume that is the design center obtained in the
SM iteration. In the th iteration, the feasible region is approx-
imated by . The ellipsoidal
technique starts with a sufficiently large hyperellipsoid [22] con-
taining the feasible region with initial center . An itera-
tion of the ellipsoidal technique involves the following.
Assume that the current hyperellipsoid is
with center and hyperellipsoid
matrix , which is symmetric and positive definite. A
hyperplane is obtained by linearizing the most
promising constraint at the best boundary point that results
in the greatest volume reduction (as shown in Section II-B).
This hyperplane is used to generate the next hyperellipsoid
where and
are given by [22]
(5)
and
(6)
where
and (7)
The new hyperellipsoid is smaller in volume than the previous
one with a reduction ratio in hyperellipsoid volume given by
[22]
(8)
where
(9)
The iterations of the hyperellipsoid technique continue until
no significant reduction in the hyperellipsoid volume can be
achieved. The center of the final hyperellipsoid, denoted by
, is considered the next design center and is fed into
the next SM iteration.
B. Selection of Boundary Points
The reduction ratio in hyperellipsoid volume is monoton-
ically decreasing function in [22]. Thus, the best boundary
point is the one that maximizes the value of . If the current hy-
perellipsoid center is not feasible, then a boundary point can
be located by carrying out a line search starting from a feasible
point in the direction
(10)
until hitting a boundary constraint , say, at . Hence, using
(7), the value of is given by
(11)
which is a positive value resulting in a significant reduction
ratio.
If the point is feasible, then the best boundary point is lo-
cated by using the following steps.
First, the most promising constraint is selected. All con-
straints are linearized around the hyperellipsoid center . This
linearization is given by
(12)
where is the gradient of at ,
. The constraint boundary can be approxi-
mated by a hyperplane given by
(13)
The corresponding value of for this hyperplane is evaluated
by using (7) as
(14)
The constraints are then renumbered according to the de-
scending values of . The most promising constraint , say, is
selected according to this numbering order such that the max-
imum is obtained. Second, the boundary point on constraint
is found by carrying out a line search starting from and taking
small steps in the direction
(15)

ABDEL-MALEK et al.: ELLIPSOIDAL TECHNIQUE FOR DESIGN CENTERING OF MICROWAVE CIRCUITS EXPLOITING SMISs 3733
where is the gradient of constraint . The Jacobian in (15)
is evaluated either by updating the Broyden formula or by an
adjoint sensitivity technique after each step and the direction
is updated. This process permits the rotation of the search to find
the promising boundary point [7]. If the value of drops during
the line search process below a certain prespecified value , the
constraint is discarded and the next promising constraint is tried
until the best boundary point is found. If the search hits the
boundary at and (preset value), then final
refinement is carried out by using a boundary search technique
[7] to reach the best boundary point resulting in the greatest
possible reduction in hyperellipsoid volume.
C. Boundary Search Technique [7]
Starting from , a sequence of boundary points converging
to the best boundary point can be generated as follows:
The value of at the boundary point is given by
(16)
Find a nonfeasible point given by
(17)
where is initially taken as and , .A
new boundary point is obtained by a line search starting from
in the direction
(18)
The process is repeated and (17) and (18) are used itera-
tively until a fixed point is reached. This fixed point is the best
boundary point that maximizes the value of [7].
D. SMIS [2]
The SMIS is constructed by applying linear input and output
mapping to the coarse model. The SMIS response at the th
iteration is given by
(19)
where
(20)
, are the output mapping parameters,
is the coarse model response vector, and
is the input mapping. The input mapping is given
by
(21)
where and are the input mapping
parameters . The mapping parameter
is taken as [2]. The other map-
ping parameters , , and , are obtained
by parameter extraction (PE) with the objective of satisfying (3)
at the design centers , , and (4) at . The PE
aims at solving the optimization problem given by
(22)
where is a residual representing the deviation
of the surrogate from the fine model and is given by
.
.
.elements
elements
(23)
where and are the th columns of and , re-
spectively. Hence, the surrogate is updated by the new mapping
parameters and the ellipsoidal technique is restarted with the
updated surrogate. The process continues until the final design
center point is reached.
III. ALGORITHM
The proposed technique proceeds in iterations. In each iter-
ation, an SMIS surrogate is used in the approximation of the
feasible region then the ellipsoidal technique is applied to find a
design center point. The technique can be summarized by the
following algorithm. The coarse model is taken as the initial
surrogate. A feasible point of the current SMIS surro-
gate is assumed to be known. An initial hyperellipsoid ,
containing a feasible region is selected such that its volume is
greater than times any estimate of the feasible region volume.
Step 1) Set , , and initialize , ,
, .
Step 2) If the center is feasible go to Step 4).
Step 3) Starting from , do a line search in the direc-
tion given by (10) to find the boundary point
then go to Step 12).
Step 4) Find hyperplane approximations for the constraints
by using (13).
Step 5) Find for each hyperplane by using (7), renumber
the constraints according to the descending values
of .
Step 6) Set the current point to .
Step 7) Select the available most promising constraint (i.e.,
the one with the highest value of ). Take a small
step in the direction given by (15), recalculate
and at the new point and then evaluate . If the
constraint is hit, then go to Step 11).
Step 8) If and all constraints are exhausted, then go
to Step 15).
Step 9) If and there are available constraints, dis-
card the current constraint, select the next most
promising constraint, go to Step 6).
Step 10) If , set the current point to the new point and
go to Step 7).
Step 11) If , apply the boundary search technique by
using (17) and (18) to get the best boundary point
.

3734 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 10, OCTOBER 2006
Step 12) Find the hyperplane approximation for the active
constraint around .
Step 13) Use (5) and (6) to find the next hyperellipsoid ,
,find from (8).
Step 14) If , set and go to Step 2).
Step 15) Set , .
Step 16) If stop the algorithm with the final
solution , .
Step 17) Evaluate , .
Step 18) Apply PE (22) and (23) to find the mapping parame-
ters , , , . and up-
date the surrogate.
Step 19) Set , , and go to Step 2).
Remarks: computational saving can be achieved by consid-
ering the following.
•In the early SM iterations and if the design center of the
last SMIS is infeasible with respect to the fine model, the
ellipsoidal technique can be stopped at a feasible point be-
fore reaching the accurate solution.
•During the search towards the constraint boundary [see
Step 7)], Broyden updates can be applied to approximate
only the gradient of the constraint considered.
IV. EXAMPLES
A. Two-Section Capacitively Loaded Impedance Transformer
A good example to demonstrate the design centering
problem is the two-section TL impedance transformer [7].
Here, , where at
the frequency points , GHz. The
coarse model is an ideal two-section TL, whereas the fine
model is a capacitively loaded TL with pF. The
characteristic impedances are taken as the design parameters
, while the normalized lengths with respect
to the quarter-wave length at the center frequency of 1 GHz
are taken as . The initial
surrogate is taken as the coarse model giving an initial de-
sign of the fine model as . The final design
center and the final hyperellipsoid ma-
trix are reached after five SM
iterations.
The feasible region can be approximated by a hyperellipsoid
using the final hyperellipsoid scaled by [22]. It is a smaller
region and, hence, is expected to give pessimistic yield estima-
tion. The final solution and the scaled final hyperellipsoid are
shown in Fig. 1. The initial and final yield are evaluated via
the Monte Carlo method with 1000 sample points by using the
actual region and the approximate region assuming normally
distributed parameters. The results assuming independent pa-
rameters are shown in Table I. Table II shows the results for cor-
related parameters with the covariance matrix given by ,
, and . The final yield and the number of fine
model evaluations are shown in Table III. The ellipsoidal tech-
nique is applied directly to the fine model (direct optimization)
and the results are also shown in Table III. The proposed tech-
nique shows high efficiency compared with the direct optimiza-
tion of the fine model.
Fig. 1. Final solution and final hyperellipsoid for the two-section TL example.
TABLE I
RESULTS FOR THE TWO-SECTION TL EXAMPLE
ASSUMING INDEPENDENT PARAMETERS
TABLE II
RESULTS FOR THE TWO-SECTION TL EXAMPLE
ASSUMING CORRELATED PARAMETERS
B. Seven-Section Capacitively Loaded
Impedance Transformer
Another example is the seven-section TL impedance
transformer [3], where
and at the frequency points ,
GHz. The coarse model is an
ideal seven-section TL, whereas the fine model
is a capacitively loaded TL with pF. The
characteristic impedances are fixed at
,
whereas the design parameters are the normalized lengths

ABDEL-MALEK et al.: ELLIPSOIDAL TECHNIQUE FOR DESIGN CENTERING OF MICROWAVE CIRCUITS EXPLOITING SMISs 3735
TABLE III
COMPARISON OF THE PROPOSED TECHNIQUE AND DIRECT OPTIMIZATION
TABLE IV
RESULTS FOR SEVEN-SECTION TL EXAMPLE
ASSUMING INDEPENDENT PARAMETERS
with respect to the quarter-wave length at the center
frequency 4.35 GHz. The initial surrogate is taken as
the coarse model giving initial design of the fine model
as .
The final design center
and
the final hyperellipsoid matrix are obtained after four
SM iterations, as shown in (24) at the bottom of this page.
The initial and final yield are evaluated via the Monte Carlo
method with 1000 sample points by using the actual region as-
suming normally distributed parameters. The results assuming
independent parameters are shown in Table IV. Table V shows
the results for correlated parameters. The final yield and the
number of fine model evaluations are shown in Table VI.
C. Coupled-Line Bandpass Filter
A third example is the design of a coupled-line bandpass filter
[27] shown in Fig. 2. The design constraint functions are given
by GHz GHz
GHz GHz
GHz GHz
(25)
TABLE V
RESULTS FOR THE SEVEN-SECTION TL EXAMPLE
ASSUMING CORRELATED PARAMETERS
TABLE VI
COMPARISON OF THE PROPOSED TECHNIQUE AND DIRECT OPTIMIZATION
Fig. 2. Coupled-line bandpass filter [27].
where dB at frequency . The substrate
thickness is taken as 1.272 mm and . The design pa-
rameters are in millimeters, as shown
in Fig. 2. The simulation of this example is performed using an
in-house planar solver based on the method of moments (MoM)
[28], [29]. The fine model is meshed with three width and 15
(24)

3736 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 10, OCTOBER 2006
TABLE VII
YIELD RESULTS FOR THE COUPLED-LINE BANDPASS FILTER
ASSUMING INDEPENDENT PARAMETERS
TABLE VIII
YIELD RESULTS FOR THE COUPLED-LINE BANDPASS FILTER
ASSUMING CORRELATED PARAMETERS
length segments of the microstrip lines, while the coarse model
is meshed with one width and five length segments.
The constraints functions are evaluated at all frequency points
GHz.
The initial surrogate is taken as the coarse model giving
as an ini-
tial fine model design. The final design center solution
is reached after
six SM iterations. The initial and the final yields are evaluated
via the Monte Carlo method with 100 sample points assuming
normally distributed parameters. The results assuming inde-
pendent parameters are shown in Table VII.
Results for correlated parameters are shown in Table VIII.
Much higher yield is achieved with the obtained de-
sign center in comparison with the minimax center
.
D. Electromagnetically Coupled Microstrip Yagi Antenna
One last example is the electromagnetically coupled Yagi
antenna array [27] (see Fig. 3). The array consists of a driven
element, a reflector, and two directors. The driven element
is electromagnetically coupled to the feeding microstrip
line, which runs below the array. The design parameters are
in millimeters,
as shown in Fig. 3. The design constraint functions are given by
Fig. 3. Electromagnetically coupled microstrip Yagi antenna [27].
TABLE IX
YIELD RESULTS FOR THE ELECTROMAGNETICALLY COUPLED MICROSTRIP
YAGI ANTENNA ASSUMING INDEPENDENT PARAMETERS
for GHz GHz,
at 26 frequency points ,
where dB . The simulation of this example is
performed using an in-house planar solver based on the MoM
[28], [29]. The fine model is meshed with three width and ten
length segments of the microstrip lines, while the coarse model
is meshed with one width and three length segments.
The initial surrogate is taken as the
coarse model giving an initial design of
. The final design center
is reached after
four SM iterations. The initial and the final yields are
evaluated via the Monte Carlo method with 100 sample
points assuming normally distributed parameters. The
results assuming independent parameters are shown in
Table IX. Results for correlated parameters are shown
in Table X. A much higher yield is achieved with the
obtained center in comparison with the minimax center
.

ABDEL-MALEK et al.: ELLIPSOIDAL TECHNIQUE FOR DESIGN CENTERING OF MICROWAVE CIRCUITS EXPLOITING SMISs 3737
TABLE X
YIELD RESULTS FOR THE ELECTROMAGNETICALLY COUPLED MICROSTRIP
YAGI ANTENNA ASSUMING CORRELATED PARAMETERS
V. C ONCLUSION
In this paper, a new technique for design centering of mi-
crowave circuits is introduced. The proposed technique has been
successfully applied to solve the design centering problem for
different microwave circuits. The integration of SM with the
modified ellipsoidal technique has reduced the number of fine
model evaluations required to obtain a good design center and,
hence, the computational effort is reduced. Yield values have
been dramatically increased for all examples considered. The
technique shows a significant increase in yield values for the
design center obtained in comparison with the minimax solu-
tion as a center.
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Hany L. Abdel-Malek (S’75–M’78) was born in
Cairo, Egypt, in 1949. He received the B.Sc. degrees
in electronics and communications engineering and
mathematics (with honors) from Cairo University,
Giza, Egypt, in 1970 and 1972, respectively, and
the Ph.D. degree in electrical engineering from
McMaster University, Hamilton, ON, Canada, in
1977.
From 1970 to 1974 he was an Instructor with
the Department of Engineering Mathematics and
Physics, Cairo University. From 1974 to 1978, he
was with the Department of Electrical Engineering, McMaster University. In
1978, he returned to the Faculty of Engineering, Department of Engineering
Mathematics and Physics, Cairo University, where he is currently a Professor.
His main research interests are in the area of circuit theory, computer-aided
circuit and system design, numerical methods, and optimization techniques.
Dr. Abdel-Malek was the recipient of a 1977 Post-Doctoral Fellowship pre-
sented by the National Research Council of Canada.

3738 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 10, OCTOBER 2006
Abdel-karim S. O. Hassan was born in Cairo,
Egypt, in 1956. He received the B.Sc. degree (with
honors) in electronics and communications engi-
neering from Cairo University, Giza, Egypt, in 1979,
the B.Sc. degree in mathematics from Ain Shams
University, Cairo, Egypt in 1981, and the M.Sc.
and Ph.D. degrees in engineering mathematics from
Cairo University, in 1984 and 1989, respectively.
Since 1979, he has been with the Physics, Faculty
of Engineering, Department of Engineering Math-
ematics, Cairo University. From 1994 to 2002, he
was with the Department of Mathematics, Teacher’s College, Dammam, Saudi
Arabia. In 2002, he returned to the Department of Engineering Mathematics
and Physics, Cairo University, where he is currently a Professor. His research is
involved with optimization theory and techniques and computer-aided design.
His interests are in engineering optimization, statistical circuit design, and
optimal design of microwave circuits.
Ezzeldin A. Soliman (S’97–A’99) was born in
Cairo, Egypt, on May 18, 1970. He received the
B.Sc. degree (honors) in electronics and communi-
cations engineering and M.Sc. degree in engineering
physics from Cairo University, Giza, Egypt, in
1992 and 1995, respectively, and the Ph.D. degree
(summa cum laude) in electrical engineering from
the University of Leuven, Leuven, Belgium, in 2000.
In 1992, he joined the Faculty of Engineering,
Cairo University, where he was initially a Demon-
strator and then an Associate Professor in 2005.
From 1996 to 2000, he was a Graduate Researcher with both the Interuniversity
Microelectronics Center (IMEC), Leuven, Belgium, and the Department of
Electrical Engineering (ESAT), University of Leuven. In 2002, he was a Vis-
iting Professor with IMEC. From 2002 to 2003, he was a Post-Doctoral Fellow
with the Department of Electrical and Computer Engineering, McMaster Uni-
versity, Hamilton, ON, Canada. He is currently an Associate Professor with the
Department of Electrical and Computer Engineering, College of Engineering,
King Abdul Aziz University, Jeddah, Saudi Arabia. His research interests
include computational electromagnetics, development and characterization of
planar antennas in multilayer technology, neural-network modeling of EM
problems, and EM-based optimization techniques.
Sameh A. Dakroury (S’02) was born in Cairo,
Egypt, in 1973. He received the B.Sc. degree (with
honors) in electronics and communications engi-
neering and M.Sc. degree from Cairo University,
Giza, Egypt, in 1995 and 2001, respectively.
In 1996, he joined the Faculty of Engineering,
Cairo University, where he was a Research and
Teaching Assistant with the Department of Engi-
neering Mathematics and Physics. From January
2002 to August 2003, he was a Research Assistant
with the Simulation Optimization Systems Research
Laboratory, McMaster University, Hamilton, ON, Canada. In September 2003,
he rejoined the Engineering Mathematics and Physics Department, Cairo
University, where he is currently a Research and Teaching Assistant. His
doctoral research concerns design centering of microwave circuits using recent
advances in SM.