Low-cost EM-Simulation-based Multi-objective Design Optimization

of Miniaturized Microwave Structures

Slawomir Koziel

, Adrian Bekasiewicz

, Piotr Kurgan

and Leifur Leifsson

Engineering Optimization & Modeling Center, Reykjavik University, Menntavegur 1, 101 Reykjavik, Iceland

Faculty of Electronics, Telecommunications and Informatics, Gdansk University of Technology, 80-233 Gdansk, Poland

Keywords: Miniaturized Microwave Structures, Design Optimization, Multi-Objective Optimization, Simulation-

Driven Design, Surrogate-based Optimization, Space Mapping.

Abstract: In this work, a simple yet reliable technique for fast multi-objective design optimization of miniaturized

microwave structures is discussed. The proposed methodology is based on point-by-point identification of a

Pareto-optimal set of designs representing the best possible trade-offs between conflicting objectives such as

electrical performance parameters as well as the size of the structure of interest. For the sake of

computational efficiency, most operations are performed on suitably corrected equivalent circuit model of

the structure under design. Model correction is implemented using a space mapping technique involving,

among others, frequency scaling. Our approach is demonstrated using a compact rat-race coupler. For this

specific example, a set of ten designs representing a Pareto set for two objectives (electrical performance

and the layout area) is identified at the cost corresponding to less than thirty high-fidelity EM simulations of

the structure.

1 INTRODUCTION

Design of miniaturized microwave structures for

contemporary wireless communication systems is a

challenging task. It involves, among others,

adjustment of designable (usually geometry)

parameters of the structure to satisfy multiple, often

conflicting objectives such as size, bandwidth, phase

response, etc. (Yeung and Man, 2011). Important

characteristics of compact structures, e.g., folded or

fractal-shaped couplers (Tseng and Chen, 2008;

Ghali and Moselhy, 2008; Liao, et al., 2005), are

densely packed layouts. Due to considerable

electromagnetic couplings between various parts of

such circuits, high-fidelity electromagnetic (EM)

analysis is the only way of accurate evaluation of

their electrical performance parameters.

Unfortunately, high-fidelity EM simulation is

computationally expensive, which turns out to be a

fundamental issue in simulated-driven design of

compact components. Conventional design strategies

such as repetitive parameter sweeps guided by

engineering experience or direct EM-driven

optimization—using, e.g., gradient-based or

derivative free methods (Nocedal and Wright, 2006;

Rios and Sahinidis, 2013)—require large number of

EM analyses, the total cost of which may be

unacceptable from practical point of view or even

prohibitive. On the other hand, alternative

techniques for performance evaluation (e.g.,

exploiting transmission line theory) are grossly

inaccurate. This is particularly true for highly

miniaturized circuits with coupled building blocks

(e.g. Bekasiewicz and Kurgan, 2014; Wincza and

Gruszczynski, 2013; Kurgan and Bekasiewicz, 2014;

Tsai, 2013; Koziel, et al., 2014).

These difficulties can be alleviated, to some

extent, by means of surrogate-based optimization

(SBO) techniques such as space mapping (SM),

which have proven their computational superiority

over traditional optimization algorithms applied to

the design of conventional microwave circuits. SBO

schemes benefit from low-cost surrogates that are

aligned with high-fidelity EM models through

adaptive corrections (Bandler et al., 2004b; Koziel et

al., 2006; Koziel et al., 2008). Because most of

operations are carried out on the corrected low-

fidelity model, and the high-fidelity EM simulation

is only launched occasionally (to verify the current

design and update the surrogate model), the overall

cost of the SBO process can be kept low.

As opposed to conventional designs, compact

767

Koziel S., Bekasiewicz A., Kurgan P. and Leifsson L..

Low-cost EM-Simulation-based Multi-objective Design Optimization of Miniaturized Microwave Structures.

DOI: 10.5220/0005127107670774

In Proceedings of the 4th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SDDOM-2014), pages

767-774

ISBN: 978-989-758-038-3

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

structures are typically developed based on novel

topologies and the influence of the structure size on

its performance capabilities cannot be foreseen

beforehand (Kurgan et al., 2012; Bekasiewicz et al.,

2012). To eliminate the risk of design failure in the

case of excessively stringent specifications that

cannot be met by a prototype circuit, multi-objective

optimization becomes a necessity. The goal here,

rather than a single optimum design, is to find the

entire set of designs (a so-called Pareto set)

representing the best possible trade-offs between

non-commensurable objectives. The most popular

solution approach is population-based metaheuristics

(Afshinmanesh et al., 2008; Deb, 2001; Jin and

Rahmat-Samii, 2010; Koulouridis et al., 2007;

Koziel and Ogurtsov, 2013; Yeung and Man, 2011).

While methods such as genetic algorithms or particle

swarm optimizers are capable of identifying the

entire Pareto set in one algorithm run, these methods

are of limited use for compact circuit design due to a

large number (from hundreds to tens of thousands)

of objective function evaluations involved (Koziel et

al., 2014; Koziel and Ogurtsov, 2013).

In this paper, we propose a computationally

efficient procedure for multi-objective simulation-

driven design of compact microwave passives. Our

methodology exploits surrogate-based optimization,

an equivalent circuit representation of the structure,

and space mapping correction techniques to perform

a point-by-point Pareto set identification. Our

approach is illustrated using a compact rat-race

coupler design.

2 CASE STUDY: COMPACT

RAT-RACE COUPLER

In this section, we provide a description of a specific

miniaturized microwave circuit to be used for

explaining and demonstrating the proposed multi-

objective design optimization methodology. We also

describe the design objectives that will be of interest.

2.1 Compact Rat-Race Coupler

Consider a novel structure of an equal-split

miniaturized rat-race coupler (RRC) shown in Fig. 1.

Figure 1: Geometry of a considered compact rat-race

coupler.

The structure miniaturization is achieved by folding

each 70.7 Ω section. The considered RRC is

designed on Taconic RF-35 substrate (ε

= 3.5, h =

0.762 mm, tanδ = 0.018). The input impedance is 50

Ω. The vector of coupler dimensions is: x = [l

, whereas w

= 1.7, l

= 15 remain fixed (all

dimensions in mm). The low- and high-fidelity

models of the structure are prepared with Agilent

ADS (Agilent ADS, 2011) and CST Microwave

Studio (CST, 2013) (~220,000 mesh cells and

simulation time ~15 minutes per design),

respectively. Lower/upper bounds l/u of the solution

space are represented by the following vectors: l = [2

10 17 0.2 0.5]

and u = [8 16 25 1.2 1.5]

. The initial

design is: x = [5 14 21 07 0.9]

2.2 Design Objectives

There are two objectives considered in the coupler

design: F

– maximization of bandwidth (defined as

intersection of |S

| and |S

| below –20 dB) centred

around the operating frequency of 1 GHz, and F

–

minimization of the RRC footprint (layout area).

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These two objectives are generally conflicting,

which means that reducing the layout area results in

reduction of the –20 dB bandwidth. The purpose of

multi-objective design in this case it to find out

possible trade-offs between the objectives.

Knowledge about these trade-offs is of fundamental

importance for the designer, especially when

selecting a structure for a particular application.

3 MULTI-OBJECTIVE DESIGN

METHODOLOGY

In this section, we formulate the multi-objective

design problem, discuss surrogate modelling using

space mapping, as well as describe the proposed

multi-objective optimization approach. The

numerical results obtained for the example structure

of Section 2 are presented in Section 4.

3.1 Multi-Objective Design Problem

Formulation

Let F

(x)), where k = 1, …, N

obj

, be a kth design

objective. Typical objectives include electrical

performance parameters as well as the component

size (in particular the area occupied by the circuit

layout), the latter being critical for the design of

compact structures. In multi-objective scheme we

seek for a representation of a so-called Pareto-

optimal set X

, which is composed of non-dominated

designs such that for any x  X

, there is no other

design y for which the relation y  x is satisfied (y 

x, i.e., y dominates over x, if F

(y))  F

(x)) for

all k = 1, …, N

obj

, and F

(y)) < F

(x)) for at

least one k) (Deb, 2001).

3.2 Low-Fidelity Model. Surrogate

Modelling using Space Mapping

The most popular solution approaches for multi-

objective problems are undoubtedly population-

based metaheuristics (Venkatarayalu et al., 2005;

Guimaraes et al., 2006; Yang et al., 2008), including

genetic algorithms (Kuwahara, 2005) or particle

swarm optimizers (Jin and Rahmat-Samii, 2007).

The most important advantage of these techniques is

their ability to identify the entire Pareto set in a

single algorithm run. Unfortunately, the

computational cost of metaheuristic algorithms is

normally very high – typically, thousands or tens of

thousands of objective function evaluations

(Afshinmanesh et al., 2008; Chamaani et al., 2011;

Kuwahara, 2005). Consequently, metaheuristics are

only suitable for handling problems where

computational cost of objective evaluation is not of a

major concern. Here, EM-simulated high-fidelity

model is too expensive to be directly handled in a

multi-objective optimization setting. Therefore, we

use an auxiliary equivalent circuit (low-fidelity)

model R

, evaluated by means of a circuit simulator

(Bandler et al., 2001; Koziel et al., 2008; Bandler et

al., 2002), here, Agilent ADS (Agilent ADS, 2011).

Figure 2 shows the circuit model for the coupler

structure of Fig. 1.

The low-fidelity model is very fast, however, it is

not an accurate representation of R

. Its corrected

version, a surrogate model R

, will be utilized in the

optimization process (Bandler et al., 2004a; Cheng

et al., 2004). Based on initial inspection of the type

of misalignment between the low- and high-fidelity

models, implicit and frequency space mapping (SM)

seem to be the most suitable correction techniques.

More specifically, the surrogate model is defined as

() (; , )



xRxfp

scF

(1)

where R

c.F

is a frequency-scaled low-fidelity model,

whereas f and p are frequency SM and implicit SM

parameters, respectively.

Let R

(x) = [R

(x,



) R

(x,



) … R

(x,



)]

where R

(x,



) is evaluation of the circuit model at a

frequency



. Then, R

c.F

(x;f,p) = [R

(x,f



f

,p) … R

(x, f



f

,p)]

, with f

and f

being

frequency scaling parameters. Here, implicit SM

parameters p are dielectric permittivity as well as

thickness of the microstrip components of the circuit

corresponding to selected groups of components as

indicated in Fig. 2. SM parameters are extracted to

minimize misalignment between R

and R

follows:

[,]argmin|| () (;,)||

fp Rx R xfp

fcF

(2)

Figure 3 shows the responses of the high- and

low-fidelity model at certain design x, as well as the

response of the surrogate model R

at the same

design. It can be observed that the model alignment

is greatly improved, however, generalization

capability of the surrogate is limited (cf. Fig. 3(b)).

In particular, it is not possible to find a single set of

SM parameters that would ensure surrogate model

accuracy across the entire design space. As a

consequence, in order to lead towards a satisfactory

design, the surrogate has to be iteratively refined

during the optimization process.

Low-costEM-Simulation-basedMulti-objectiveDesignOptimizationofMiniaturizedMicrowaveStructures

769

3.3 Optimization Algorithm

Due to the limited generalization capability of the

SM surrogate model mentioned in Section 3.2, as

well as required reduction of the computational cost

of the multi-objective optimization process, our

design approach is based on point-by-point

identification of the Pareto set. In the first step, the

coupler structure is optimized taking into account

the first objective only (here, electrical parameters).

The obtained value F

(1)

)) at the optimum

design x

(1)

determines, together with the

corresponding value of the second objective (here,

layout area), F

(1)

= F

(1)

)), the extreme points

of the Pareto set.

Figure 2: Equivalent circuit model of the coupler structure

of Fig. 1 with highlighted regions with different implicit

SM parameters p (model implemented in Agilent ADS).

(a)

(b)

Figure 3: Responses of the high- and low-fidelity coupler

models as well as the SM surrogate (a) at certain design x

(at which the surrogate is extracted), and (b) at some other

design. Plot (b) indicates limited generalization capability

of the surrogate.

In the subsequent steps, we set the threshold

values for the second objective F

(j)

, and optimize

the structure with respect to the first objective so that

the above threshold value is preserved:



()

,( ())

arg min ( )





xRx

(3)

Here, x

(j)

is the jth element of the Pareto set. The

process is continued until F

(j)

)) is still

satisfactory from the point of view is given design

specifications.

Problem (3) is solved using the SM surrogate

model (cf. Section 2.2) and it is itself realized as an

iterative process



()

(.)

(.) (.)

,( ())

arg min ( )





xR x

xRx

jk jk

(4)

0.7 0.8 0.9 1 1.1 1.2 1.3

-50

-40

-30

-20

-10

Frequency [GHz]

S-parameters [dB]

0.7 0.8 0.9 1 1.1 1.2 1.3

-50

-40

-30

-20

-10

Frequency [GHz]

S-parameters [dB]

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where

(.) (.) (.)

() (; , )RxRxf p

kjkjk

scF

(5)

and

(.) (.)

[,]

arg min || ( ) ( ; , ) ||





xRx

jk jk

fcF

(6)

The starting point for the algorithm (4) is x

(j–1)

(the previously obtained Pareto set point). Normally,

two iterations of (4) are sufficient to obtain x

(j)

which is because the starting point is already a good

approximation of the optimum. In practice, the

thresholds F

(j)

can be obtained as F

(j)



 F

(j–1)

with



< 1 (e.g.,



= 0.95), or F

(j)

= F

(j–1)

–



with



> 0 (e.g.,



= 0.05



(1)

The computational cost of the entire multi-

objective design process using the proposed

methodology can be estimated (in terms of the

number of EM simulations of the structure) as NK,

where N is the number of point in the Pareto set, and

K is the average number of iterations (4) necessary

to obtain the next point. In practice, K  3.

It should also be mentioned that another

important design goal, i.e., |S

| = |S

| at the

operating frequency (here, 1 GHz), ensuring equal

division of the signal power between ports 2 and 3

of the circuit, is handled implicitly. More

specifically, it is enforced for each design obtained

in the optimization process by applying an additional

penalty function to F

in (4) that penalizes designs

for which | |S

| – |S

| | > ds at the operating

frequency (here, we use ds = 0.1 dB, as an

acceptable inaccuracy level).

4 NUMERICAL RESULTS

The coupler structure of Section 2 has been designed

using the multi-objective design methodology

described in Section 3. The first design obtained by

using (4) without any area constraints resulted in

-20 dB bandwidth of 281 MHz and the layout area

of 570 mm

. Nine other designs have been obtained

by setting up F

(j)

to 540, 500, 475, 450, 425, 400,

375, 350, and 325 mm

, respectively. Figure 4

shows the obtained representation of the Pareto

front.

For the layout area of 300 mm

, it was

impossible to obtain a design with positive value of

–20 dB bandwidth, which essentially means that 300

is a lower limit (in terms of layout area) for

practically useful designs for this particular coupler

topology. This is—from the designer standpoint—an

important information regarding miniaturization

limitations, which may be utilized, e.g., to

discriminate structures suitable for a given (in

particular, space-limited) application.

Table 1 and Figure 5 show the numerical data

and frequency characteristics for the selected

designs. It can be observed that the coupler size can

be reduced by over 40 percent with respect to its

original size (corresponding to the best possible

electrical performance), while maintaining

acceptable performance.

Table 1: Multi-objective design optimization of rat-race

coupler: selected results.

Design Variables [mm] Objectives

d W

–20 dB

Bandwidth

[MHz]

Layout

Area

[mm

]

4.18 13.20 20.68 0.994 0.865 281 570

3.83 11.76 20.44 0.825 0.877 270 500

4.10 13.78 21.14 0.581 0.887 260 450

4.25 12.17 22.12 0.400 0.923 202 400

3.95 10.87 21.71 0.350 0.936 174 375

4.37 12.33 22.52 0.350 0.820 151 350

Figure 4: Pareto set obtained using the proposed multi-

objective design optimization methodology.

Both Figure 4 and Table 1 indicate the

conflicting nature of the considered objectives:

reduction of the layout area of the circuit inevitably

results in degrading its electrical performance, here,

–20 dB bandwidth.

200 250 300 350 400 450 500 550 600

100

150

200

250

300

Coupler layout area [mm

]

-20 dB bandwidth [MHz]

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771

(a)

(b)

(c)

Figure 5: Frequency characteristics for selected coupler

designs, corresponding to the layout area 570 mm

(a),

448 mm

(b), and 375 mm

(c).

All the designs along the Pareto front are

nominally satisfying the basic design goal (i.e., both

| and |S

| are lower than –20 dB and centred

around the operating frequency of 1 GHz). However,

the designs with wider –20 dB bandwidth (such as

the one shown in Fig. 5(a) versus that in Fig. 5(c))

are electrically better because of higher chance of

satisfying design specifications in case of

unavoidable manufacturing tolerances (a

consequence of which will be a deviation of actual

characteristics of the fabricated circuit with respect

to the nominal ones).

The total cost of the design process corresponds

to less than 30 high-fidelity model evaluations (~7.5

hours), including the overhead related to multiple

evaluations of the circuit model R

(the latter does

not exceed 20 percent of the overall EM simulation

cost). It should be noted that direct multi-objective

optimization of the high-fidelity EM antenna

model R

would not be possible (the expected cost of

a few thousand of model evaluations is practically

prohibitive).

5 CONCLUSIONS

In this paper, a technique for low-cost multi-

objective design optimization of miniaturized

microwave structures has been proposed. The design

speedup has been obtained through the usage of

appropriately corrected, fast equivalent circuit model

of the structure under design. Another important

component was point-by-point Pareto set

identification through constrained single-objective

optimizations. As a result, the number of high-

fidelity EM simulations of the structure was greatly

reduced (to less than three per identified Pareto set

point). As demonstrated using a compact RRC

coupler, a set of designs corresponding to best

possible trade-offs between conflicting objectives

(here, electrical performance and the layout area of

the structure) has been obtained at a low

computational cost, corresponding to less than thirty

EM simulations of the coupler. According to our

knowledge, this is a first successful attempt to solve

the low-cost multi-objective design problem of

compact structures exploiting surrogate-based

optimization.

The future work will aim at extending the

presented methodology to cases with larger number

of conflicting design objectives, as well as applying

it to other classes of structures, especially antennas,

where fast equivalent circuit models are normally

unavailable.

0.5 1 1.5

-40

-35

-30

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-20

-15

-10

-5

Frequency [GHz]

S-parameters [dB]

0.5 1 1.5

-40

-35

-30

-25

-20

-15

-10

-5

Frequency [GHz]

S-parameters [dB]

0.5 1 1.5

-40

-35

-30

-25

-20

-15

-10

-5

Frequency [GHz]

S-parameters [dB]

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ACKNOWLEDGEMENTS

The authors thank Computer Simulation Technology

AG, Darmstadt, Germany, for making CST

Microwave Studio available. This work was

supported in part by the Icelandic Centre for Research

(RANNIS) Grant 13045051.

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