gradient-based optimization using parametric sensitivity ...lknockae/pdf/kfkd_2012.pdf ·...

INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDSInt. J. Numer. Model. 2012; 25:347–361Published online 20 January 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/jnm.839

Gradient-based optimization using parametric sensitivitymacromodels{

Krishnan Chemmangat*,†, Francesco Ferranti, Tom Dhaene and Luc Knockaert

Department of Information Technology, Ghent University–IBBT, Gaston Crommenlaan 8 Bus 201, B-9050 Ghent, Belgium

SUMMARY

A new method for gradient-based optimization of electromagnetic systems using parametric sensitivitymacromodels is presented. Parametric macromodels accurately describe the parameterized frequency behavior ofelectromagnetic systems and their corresponding parameterized sensitivity responses with respect to designparameters, such as layout and substrate parameters. A set of frequency-dependent rational models is builtat a set of design space points by using the vector fitting method and converted into a state-space form. Then,this set of state-space matrices is parameterized with a proper choice of interpolation schemes, such thatparametric sensitivity macromodels can be computed. These parametric macromodels, along with thecorresponding parametric sensitivity macromodels, can be used in a gradient-based design optimization process.The importance of parameterized sensitivity information for an efficient and accurate design optimization is shownin the two numerical microwave examples. Copyright © 2012 John Wiley & Sons, Ltd.

Received 19 May 2011; Revised 26 August 2011; Accepted 11 October 2011

KEYWORDS: gradient-based design optimization; parametric sensitivity; parametric macromodeling; interpolation

1. INTRODUCTION

When designing high-speed microwave systems, one aims at obtaining the optimal values of the designvariables for which the system responses satisfy the design specifications. This process is usuallycarried out through electromagnetic (EM) simulations. Optimal values of the design variables are oftendetermined using optimization algorithms (optimizers), which drive the EM simulator to obtain theresponses and their sensitivities in consecutive optimization iteration. Unfortunately, multiple consecutiveEM simulations are often computationally expensive. An alternative approach is to generate accurateparametric macromodels up first, which capture the parameterized frequency behavior of the EM systemsand their corresponding parameterized sensitivity responses with respect to design parameters, such aslayout and substrate parameters. Efficient and accurate parametric sensitivity information is required byoptimizers, which employ state-of-the-art gradient-based techniques to calculate the optimal designparameters. Parametric sensitivity macromodels are able to describe sensitivity responses not only in thevicinity of a single operating point (local sensitivity) but also over the entire design space of interest.

One of the most common approaches to calculate local sensitivities is the adjoint variable method.The main attractiveness of this approach is that sensitivity information can be obtained from at mosttwo systems analyses regardless of the number of designable parameters 1–3. However, these methodsinvolve the calculation of the system matrix derivatives with respect to each design parameters, whichis mostly carried out by finite difference approximations.

Recently, some interpolation-based parametric macromodeling techniques have been developed4–9, which interpolate a set of initial univariate macromodels, called root macromodels. In 4–6, a

*Correspondence to: Krishnan Chemmangat, Department of Information Technology, Ghent University–IBBT, GastonCrommenlaan 8 Bus 201, B-9050, Ghent, Belgium.†E-mail: [email protected]{This work was supported by the Research Foundation Flanders (FWO).

Copyright © 2012 John Wiley & Sons, Ltd.

348 K. CHEMMANGAT ET AL.

parametric macromodel is built by interpolating a set of root macromodels at an input–output level,whereas in 7–9, the interpolation process is applied to the internal state-space matrices of the rootmacromodels, therefore at a deeper level than in the transfer function-based interpolation approaches4–6. The methods 7–9 allow to parameterize both poles and residues; hence, their modelingcapability is increased with respect to 4–6, where only residues are parameterized.

A gradient-based minimax optimization process using parameterized sensitivity macromodels ispresented in this paper. As in 7–9, an interpolation process on the internal state-space matrices of the rootmacromodels is performed. However, in 7–9, the focus is on parametric macromodeling, which ensuresstability and passivity over the design space of interest. This is not strictly necessary for the calculation ofparametric sensitivities, which allows the use of more powerful interpolation schemes, which are at leastcontinuously differentiable. The parametric sensitivity macromodels avoid the use of finite differenceapproximations in the optimization process. Also, in 7–9, computationally expensive linear matrixinequalities are solved to guarantee preservation of passivity, which can be avoided in the present work.Parametric macromodels along with the corresponding parametric sensitivities are used in two pertinentnumerical examples, which confirm the applicability of the proposed technique to the optimizationprocess of microwave filters. The importance of parameterized sensitivity information to speedup the design optimization process is shown in the pertinent examples.

2. GENERATION OF ROOT MACROMODELS

Starting from a set of data samples s; →gð Þk;H s; →gð Þk� �Ktot

k¼1, which depend on the complex frequency

s= jo and several design variables →g ¼ g nð Þ� �Nn¼1 , such as layout features or substrate parameters, a

set of frequency-dependent rational macromodels is built for some design space points by means ofthe vector fitting (VF) technique 10–12. Each root macromodel has the following form:

R→gk

sð Þ ¼XNP

n¼1

c→gkn

s� a→gkn

þ d→gk : (1)

The terms in the rational model (1), a→gkn ; c

→gkn ; and d

→gk , represent poles, residues, and feed forward

terms, respectively, at the design point →gk ¼ g 1ð Þk1; . . . ; g Nð Þ

kN

� �. The problem of finding the unknown

coefficients in (1) is nonlinear, as the poles a→gkn appear in the denominator. The idea of the VF technique

is to recast this nonlinear identification problem into a linear least-square problem that is solved iteratively[10]. A pole-flipping scheme is used to enforce stability [10], whereas passivity assessment and enforce-ment can be accomplished using the standard techniques 13,14. This initial step of the proposed methodresults in a set of stable and passive frequency-dependent rational models, called root macromodels.

Two design space grids are used in the modeling process: an estimation grid and a validation grid.The first grid is utilized to build the parametric macromodels. The second grid is utilized to assess thecapability of parametric macromodels of describing the system under study in a set of points of thedesign space previously not used for their construction. To clarify the use of these two design spacegrids, we show in Figure 1 a possible estimation and validation design space grid in the case of twodesign parameters →g ¼ g 1ð Þ; g 2ð Þ� �

. A root macromodel is built for each estimation grid point in thedesign space. This set of root macromodels is interpolated, as explained in the following section, tobuild a parametric macromodel that is evaluated and compared with original data related to the validationdesign space points.

3. PARAMETRIC MACROMODELING

Each root macromodelR→gk

sð Þ, corresponding to a specific design space point →gk ¼ g 1ð Þk1; . . . ; g Nð Þ

kN

� �, is

converted from the pole-residue form (1) into a state-space form:

R→gk

sð Þ ¼ C→gk

sI� A→gk

� ��1B→gkþ D→

gk: (2)

Copyright © 2012 John Wiley & Sons, Ltd. Int. J. Numer. Model. 2012; 25:347–361DOI: 10.1002/jnm

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

g(1)

g(2)

Estimation GridValidation Grid

Figure 1. Estimation and validation grids for a general two-parameter design space.

GRADIENT-BASED OPTIMIZATION 349

Then, this set of state-space matrices A→gk;B→

gk;C→

gk;D→

gkis interpolated entry by entry, and the

multivariate models A →gð Þ;B →gð Þ;C →gð Þ;D →gð Þ are built using multivariate interpolation schemes togenerate a parametric macromodel R s; →gð Þ for the entire design space 7,8:

R s; →gð Þ ¼ C →gð Þ sI� A →gð Þð Þ�1B →gð Þ þ D →gð Þ: (3)

The computationally cheap piecewise linear interpolation cannot be used to generate parametricsensitivity macromodels, as it is not continuously differentiable. A proper choice of interpolationschemes, which are at least continuously differentiable, is necessary. In this work, three interpolationmethods are investigated, namely the cubic spline (CS) interpolation, the piecewise cubic Hermiteinterpolation (PCHIP), and the shape preserving C2 cubic spline interpolation (SPC2). They arebriefly described in what follows.

3.1. Cubic spline interpolation

Given some data samples xi; yið Þni¼1, the CS interpolation method builds a cubic polynomial for eachinterval of the data set xi≤ x≤ xi+ 1, i = 1, . . ., n.

si xð Þ ¼ ai x� xið Þ3 þ bi x� xið Þ2 þ ci x� xið Þ þ di: (4)

The coefficients of the cubic polynomials are obtained by imposing the first-order and second-orderderivative continuity at each data point along with a not-a-knot end condition and then solving atridiagonal linear system [15]. Once these coefficients are computed, the derivatives of the overallspline interpolation function can be analytically calculated in terms of its coefficients ai, biand ci for xi≤ x< xi+ 1, i= 1, 2, . . ., n� 1. If the data under interpolation are in a matrix form, eachentry of the matrices is independently interpolated.

The univariate CS interpolation can be extended to higher dimensions by means of a tensor productimplementation [15].

3.2. Piecewise cubic Hermite interpolation

The PCHIP method is a monotonic shape preserving interpolation scheme. As in the CS interpolation,each data interval is modeled by a cubic polynomial similar to (4):

pi xð Þ ¼ fiH1 xð Þ þ fiþ1H2 xð Þ þ diH3 xð Þ þ diþ1H4 xð Þ; (5)

where dj ¼ dp xjð Þdx ; j ¼ i; iþ 1; and the Hk xð Þ are the usual cubic Hermite basis functions for the

interval xi≤ x< xi+ 1, i= 1, 2, . . ., n� 1 : H1(x) =f((xi+ 1� x)/hi), H2(x) =f((x� xi)/hi), H3(x) =� hic((xi+ 1� x)/hi), H4(x) = hic((x� xi)/hi), where hi= xi+ 1� xi, f(t) = 3t2� 2t3, c(t) = t3� t2. The first-order derivative di at each data point xi is calculated such that the local monotonicity is



preserved [16]. An extension to higher dimension can be performed by a tensor product imple-mentation [15]. The calculation of derivatives is performed in the same way as in the CS inter-polation case. This interpolation scheme works better for non-smooth data sets, wherein the CSscheme could result in overshoots or oscillatory behavior of the derivatives. However, thePCHIP method is only continuous in first derivatives, which affects the smoothness of thederivatives [16].

3.3. Shape preserving C2 cubic spline interpolation

The SPC2 interpolation is a monotonicity preserving interpolation scheme similar to PCHIP.However, in contrast to the PCHIP method, which is only continuous in first derivative, theSPC2 method is a second-order derivative continuous interpolation scheme. The idea here isto add two extra break points in each subinterval of the data, such that enough degrees offreedom are generated to construct a CS interpolant, which is globally second-order derivativecontinuous [17]. As the monotonicity of the data is preserved, this scheme works better withrespect to the CS method for non-smooth data sets. Similar to the CS and the PCHIP inter-polation schemes, a multivariate SPC2 interpolation is performed using a tensor productimplementation [15].

4. PARAMETRIC SENSITIVITY MACROMODELS

Once the parametric macromodel R s; →gð Þ is built, the corresponding sensitivities can be computed bydifferentiating (3) with respect to the design parameters →g

@

@→gR s; →gð Þ ¼ @C →gð Þ

@→gsI� A →gð Þð Þ�1B →gð Þ þ C →gð Þ sI� A →gð Þð Þ�1 @A vec→gð Þ

@→gsI� A →gð Þð Þ�1B →gð Þþ

C →gð Þ sI� A →gð Þð Þ�1 @B→gð Þ

@→gþ @D →gð Þ

@→g: (6)

In (6), @@

→gR s; →gð Þ is based on the parameterized state-space matrices A →gð Þ;B →gð Þ;C →gð Þ;D →gð Þ and the

corresponding derivatives, which are computed efficiently and analytically using the interpolationmethods described in Section 3.

5. GRADIENT-BASED MINIMAX OPTIMIZATION

Parametric sensitivity macromodels can be used in the optimization process of EM systems.Considering microwave filters, a typical optimization process begins by defining passband andstopband specifications in terms of the frequency responses, which are reformulated in the formof a cost function Fi

→gð Þ, at optimization frequency samples si, i = 1, 2, . . .Ns to be minimized:

Fi→gð Þ ¼ Ri

L � R si;→gð Þ or R si;

→gð Þ � RiU; i ¼ 1; 2; . . . ;NS: (7)

In (7), RiL and Ri

U represent lower and upper frequency response thresholds, respectively, atfrequency samples si spread over the frequency range of interest. A negative cost indicates thatthe corresponding specification is satisfied, whereas a positive cost denotes that the specificationis violated. The minimization (7) can be performed by several state-of-the-art optimization algo-rithms. In this paper, we use a minimax optimization algorithm [18], which uses the cost func-tion (7) and its gradients with respect to design parameter →g giving the optimum designparameters g* as

g� ¼→g

argmin maxi

Fi→gð Þ½ �

� : (8)

The complete optimization process starting from the proposed parametric macromodeling techniqueis depicted in Figure 2.


Figure 2. Complete optimization process flowchart.


6. NUMERICAL EXAMPLES

6.1. Double folded stub microwave filter

A double folded stub (DFS) microwave filter on a substrate with relative permitivity Er = 9.9 and athickness of 0.127mm is modeled in this example. The layout of this DFS filter is shown in Figure 3.The spacing S between the stubs and the length L of the stub are chosen as design variables in additionto frequency. Their corresponding ranges are shown in Table I. The design specifications of thisband-stop filter are given in terms of the scattering parameter, similar to [19]

S21j j≥� 3 dB for freq≤ 9 GHz or freq≥ 17 GHz (9)

S21j j≤� 30 dB for 12 GHz≤ freq≤ 14 GHz: (10)

From the design specifications (9, 10), a cost function (7) is formulated in terms of S21 and→g ¼ S; Lð Þ.

Figure 3. Layout of the DFS band-stop filter.

Table I. Design parameters of the DFS band-stop filter.

Parameter Min Max

Frequency (freq), GHz 5 20Spacing (S), mm 0.1 0.25Length (L), mm 2.0 3.0


2 2.2 2.4 2.6 2.8 30.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

Length parameter [mm]

Spac

ing

Para

met

er [

mm

]

Estimation GridValidation Grid

Figure 4. Estimation and validation grids for the parametric macromodeling of the DFS filter.


The scattering matrix S s; S; Lð Þ has been computed using the ADS MOMENTUM1 software. The

number of frequency samples was chosen to be equal to 31. The estimation and validation grid pointsfor the design variables are shown in Figure 4. The average simulation time for each design space point(S, L) has been found to be equal to, TSimAvg = 32.87 s. A set of stable and passive root macromodelshas been built for all design space points in the estimation grid of Figure 4 by means of VF with a fixednumber of poles NP = 18, selected using an error-based bottom-up approach. Each root macromodelhas been converted into a state-space form, and the set of state-space matrices has been interpolatedusing the CS, PCHIP, and SPC2 methods. Let us define the absolute error

Err →gð Þ ¼i; j; kmaxðj Ri; j sk;

→gð Þ � Hi; j sk;→gð Þ� Þ (11)

i ¼ 1; . . . ;Pin; j ¼ 1; . . . ;Pout; k ¼ 1; . . . ;Ns;

where Pin and Pout are the number of inputs and outputs of the system, respectively, and Ns is equal tothe number of frequency samples. The worst-case absolute error over the validation grid is chosen toassess the accuracy and the quality of parametric macromodels

→gmax ¼ →gargmaxErr →gð Þ; →g 2 validation grid (12)

Errmax ¼ Err →gmaxð Þ: (13)

The maximum absolute error (13) for the parametric macromodel over the validation grid of Figure 4is � 58.45, � 50.23, and � 50.23 dB, respectively, using the CS, PCHIP, and SPC2 interpolationschemes. The CS interpolation scheme gives the minimum worst-case error (13) for this specificexample, and hence, it has been used in the optimization process to generate the parametric macro-model and corresponding sensitivities. Figure 5 shows the parametric behavior of the magnitude ofS21 as a function of frequency for five different values of S and L = 2.5mm. In Figure 5, the darkestcurve of S21 corresponds to the largest value of S. Similarly, Figure 6 shows the magnitude of S21for five different values of L, with S = 0.175mm.

The cost function (7) and its gradients calculated using (6) have been supplied to the minimaxoptimization algorithm (8), resulting in the optimum design parameter values S* and L*. To showthe advantage of supplying parametric sensitivity information to the optimizer to speed up theoptimization process, two cases have been considered:

• Case I: No sensitivity information is supplied to the minimax algorithm and the algorithm estimateswith the help of finite difference approximation computed using the parametric macromodel.

• Case II: The sensitivity information is calculated from (6) and supplied to the minimax algorithm.

1MOMENTUM EEsof EDA, Agilent Technologies, Santa Rosa, CA.


5 10 15 20−70

−60

−50

−40

−30

−20

−10

0

Frequency [GHz]

|S 21

| [dB

]

S increasing

Figure 5. DFS filter: magnitude of S21 as a function of frequency for five values of S and L= 2.5mm.

5 10 15 20−70

−60

−50

−40

−30

−20

−10

0

Frequency [GHz]

|S 21

| [dB

] L increasing

Figure 6. DFS filter: magnitude of S21 as a function of frequency for five values of L and S= 0.175mm.


In addition to that, in order to show the advantage of using a parametric macromodel, we performedthe same optimization problem by using the gradient-based minimax optimization routine in the ADSMOMENTUM software. Table II compares these three optimization approaches in terms of optimizationtime for a particular optimization case. Table II shows the relevant speed up in the optimization processobtained using the parametric macromodel. We note that the generation of the parametric macromodelrequires some initial ADS MOMENTUM simulations and therefore an initial computational effort of3714.31 s (for the estimation and validation design space points in Figure 4). However, once theparametric macromodel is generated and validated, it acts as an accurate and efficient surrogate ofthe original system and can be used for multiple design optimization scenarios, for instance, changingfilter specifications. Figure 7 shows the magnitude of S21 for the optimization case of Table II. Theactual data generated by the ADS MOMENTUM software at the optimum design space point (S*, L*)obtained using the parametric macromodel are shown by asterisks in Figure 7. As seen, this is in good

Table II. DFS filter: optimization using parametric macromodel and ADS MOMENTUM software.

Method (S0, L0) (mm) (S*, L*) (mm)Optimization

time (s)Cost function at(S*, L*) (dB) [1]

Parametricmacromodel

Case I (0.1500, 2.6364) (0.2408, 2.1802) 19.88 �1.67Case II (0.1500, 2.6364) (0.2408, 2.1802) 1.23 �1.67

ADS MOMENTUM (0.1500, 2.6364) (0.2303, 2.1580) 5115.00 0.00

[1]The gradient-based minimax optimization routine of ADS MOMENTUM software uses the minimax L1 error function, whichcannot take a value less than zero.


5 10 15 20

−80

−70

−60

−50

−40

−30

−20

−10

0

Frequency [GHz]

|S21

| [dB

]

Before optimizationSolution: No derivatives suppliedSolution: Derivatives suppliedADS data at the solution

Figure 7. DFS filter: magnitude of S21 before and after optimization.

5 10 15 20

−80

−70

−60

−50

−40

−30

−20

−10

0

Frequency [GHz]

|S12

| [dB

]

Before optimization

Solution: Using ADS Momentum Optimization

Figure 8. DFS filter: magnitude of S21 before and after optimization.


agreement with the parametric macromodel response. Figure 8 shows the solution obtained using thegradient-based minimax optimization routine of ADS MOMENTUM software. As clearly seen in Figures 7and 8, the optimal solutions satisfy all the requirements (9, 10), which are shown by thin solid black lines.

Figure 9 shows the value of cost function with respect to the number of cost function evaluations incases I and II, which confirms the improved convergence of the optimization when parametricsensitivity information are provided. The convergence time taken by cases I and II are 13.76 s and

0 200 400 600 800 1000−5

0

5

10

15

20

25

Number of cost function evaluations

Cos

t fun

ctio

n va

lue

No derivative suppliedDerivative supplied

Figure 9. DFS filter: cost function during optimization.



1.73 s, respectively. The trajectory of the optimum design space point (S*, L*) during optimizationfor case I is shown in Figure 10 with different points showing the output of some particulariterations along with the elapsed time. A similar curve is plotted in Figure 11 for case II. ComparingFigures 10 and 11, we can see that the time for convergence of case II is considerably less comparedwith case I.

Table III shows the comparison of cases I and II for some important optimization measures, whichare related to 200 optimization trials, starting from different initial design points spread over the designspace. Table III confirms that there is a considerable reduction in the number of cost function evaluationsand the optimization time if derivatives are supplied (case II).

6.2. Hairpin bandpass microwave filter

In this example, a hairpin bandpass filter with the layout shown in Figure 12 is modeled [19] . The relativepermittivity of the substrate is Er = 9.9, whereas its thickness is equal to 0.635mm. The specifications forthe bandpass filter are given in terms of the scattering parameters S21 and S11:

S21j j > �2:5 dB for 2:4 GHz < freq < 2:5 GHz (14)

S11j j < �7 dB for 2:4 GHz < freq < 2:5 GHz (15)

S21j j < �40 dB for freq < 1:7 GHz (16)

S21j j < �25 dB for 3:0 GHz < freq: (17)

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.00 sec

1.02 sec

1.22 sec

1.79 sec

1.85 sec

16.31 sec

19.88 sec


Spac

ing

Para

met

er [

mm

]

(S*,L*)

(S0,L0)

Figure 10. DFS filter: the trajectory of the optimal design space point (S*, L*) during optimization for case I.

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.00 sec

0.25 sec

0.31 sec

0.42 sec

0.85 sec

1.23 sec


Spac

ing

Para

met

er [

mm

]

0.51 sec

(S*,L*)

(S0,L0)

Figure 11. DFS filter: the trajectory of the optimal design space point (S*, L*) during optimization for case II.


Table III. DFS filter: comparison between cases I and II.

Method

Number of cost function evaluations Optimization time (s)

Max Mean STD Max Mean STD

Case I 20 004 861.33 2360.10 135.30 6.05 16.31Case II 1601 50.56 138.82 41.10 1.43 3.62

Figure 12. Layout of the hairpin bandpass filter.


As shown in Figure 12, three design parameters have been chosen for the design process, namely,the spacing between the port lines and the filter lines S1, the spacing between the two filter lines S2,and the overlap length L in addition to frequency. The ranges of the different design variables areshown in Table IV.

The scattering matrix S s; S; Lð Þ has been computed using the ADS MOMENTUM software. Thenumber of frequency samples was chosen to be equal to 41. As in the first example, two design spacegrids are used in the modeling process. The average simulation time for each design space point(L, S1, S2) has been found to be equal to TSimAvg = 34.30 s. A set of stable and passive root macromodelshas been built for the estimation grid of 6� 4� 4 (L� S1� S2) samples by means of VF with a fixednumber of poles, NP = 18, selected using an error-based bottom-up approach. Each root macromodelhas been converted into a state-space form, and the set of state-space matrices has been interpolatedby the CS, PCHIP, and SPC2 methods. The maximum absolute error (13) of the models over thevalidation grid of 5� 3� 3 (L� S1� S2) samples is equal to � 42.57, � 38.27, and � 38.27 dB forthe CS, PCHIP, and SPC2 methods, respectively. The CS technique has been used in the optimiza-tion process of this example, as it gives the best accuracy. Figure 13 shows the parametric behaviorof the magnitude of S21 as a function of frequency for five different values of L and S1 = 0.295mm,S2 = 0.695mm. Figure 14 shows the magnitude of S21 when the parameter S1 changes.

The cost function (7) and its gradients calculated using (6) have been supplied to the minimaxoptimization algorithm (8), resulting in the optimum design parameter values L*, S�1, and S

�2. To show

the improved convergence of the optimization when derivatives are supplied, two cases areconsidered as in the previous example. In addition to that, in order to show the advantage of usinga parametric macromodel, the same optimization problem has been performed using the gradient-based minimax optimization routine in the ADS MOMENTUM software. Table V compares thesethree optimization approaches in terms of optimization time for a particular optimization case.Table V shows the relevant speed up in the optimization process obtained using the parametricmacromodel. As mentioned in the previous example, the generation of the parametric macromodel

Table IV. Design parameters of the hairpin bandpass filter.

Parameter Min Max

Frequency (freq), GHz 1.5 3.5Length (L), mm 12.0 12.5Spacing (S1), mm 0.27 0.32Spacing (S2), mm 0.67 0.72


1.5 2 2.5 3 3.5−45

−40

−35

−30

−25

−20

−15

−10

−5

0

Frequency [GHz]

|S 21

| [dB

]

L increasing

igure 13. Hairpin filter: magnitude of S21 as a function of frequency for five values of L with(S1, S2) = (0.295, 0.695) mm.


F

1.5 2 2.5 3 3.5−45

−40

−35

−30

−25

−20

−15

−10

−5

0

Frequency [GHz]

|S 21

| [dB

]

S1 increasing

Figure 14. Hairpin filter: magnitude of S21 as a function of frequency for five values of S1 with(L, S2) = (12.25, 0.695) mm.

Table V. Hairpin filter: optimization using parametric macromodel and ADS MOMENTUM software.

Method L0; S01; S02

� �(mm) L�; S�1; S

�2

� �(mm)

Optimizationtime (s)

Cost function atL�; S�1; S

�2

� �(dB)

Parametricmacromodel

Case I (12.2778, 0.2700, 0.6867) (12.0568, 0.2984, 0.7200) 13.46 �0.76Case II (12.2778, 0.2700, 0.6867) (12.0568, 0.2984, 0.7200) 0.79 �0.76

ADSMOMENTUM

(12.2778, 0.2700, 0.6867) (12.0036, 0.2700, 0.6826) 1251.00 0.00

requires an initial ADS MOMENTUM simulation cost of 4836.30 s. However, once the parametricmacromodel is generated and validated, it acts as an accurate and efficient surrogate of the originalsystem and can be used for multiple design optimization scenarios, for instance, changing filterspecifications. Figure 15 shows the magnitude of S21 for the optimization case of Table V. Theactual data generated by the ADS MOMENTUM software at the optimum design space point L�; S�1; S

�2

� �obtained using the parametric macromodel is shown by asterisks in Figure 15. As seen, this is in agreementwith the parametric macromodel response. The requirements (14–17) are shown by the thin black solidlines. A magnified view of the passband is shown in Figure 16 for clarity. Similar results are given forthe magnitude of S11 in Figure 17. As clearly seen, all the requirements are met for the optimal design


1.5 2 2.5 3 3.5−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

0

Frequency [GHz]

|S 21

| [dB

]


Figure 15. Hairpin filter: magnitude of S21 before and after optimization.

2.25 2.3 2.35 2.4 2.45 2.5 2.55 2.6 2.65−18

−16

−14

−12

−10

−8

−6

−4

−2

0

Frequency [GHz]

|S 21

| [dB

]


Figure 16. Hairpin filter: a zoomed-in view of Figure 15.

2.25 2.3 2.35 2.4 2.45 2.5 2.55 2.6 2.65−45

−40

−35

−30

−25

−20

−15

−10

−5

0

Frequency [GHz]

|S 11

| [dB

]





point. Figures 18 and 19 show similar results for the solution obtained using the gradient-basedminimax optimization routine of ADS MOMENTUM software. Here also, all the design specifications(14–17) are met. Some important measures of the optimization process related to 1000 trial runs areshown in Table VI, which clearly shows the better convergence properties of case II.

1.5 2 2.5 3 3.5−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

0

Frequency [GHz]

|S 21

| [dB

]

Before optimizationSolution: Using ADS MomentumOptimization


2.25 2.3 2.35 2.4 2.45 2.5 2.55 2.6 2.65−45

−40

−35

−30

−25

−20

−15

−10

−5

0

Frequency [GHz]

|S11

| [dB

]

Before optimization

Solution: Using ADS Momentum Optimization


Table VI. Hairpin filter: comparison between cases I and II.

Method

Number of cost function evaluations Optimization time (s)

Max Mean STD Max Mean STD

Case I 53 183 114.39 1689.71 565.14 1.29 17.95Case II 99 13.73 5.79 4.69 0.76 0.28


Another important thing to be noted here is that with the increase in the number of designparameters, the initial number of EM simulations increases considerably because of the curse ofdimensionality. Adaptive sampling schemes, which take into account influence of the designparameters on the system behavior, could be used to properly sample the design space prior tothe parametric macromodeling and help resolve this issue. For instance, in the second example,from Figures 13 and 14, it is seen that the overlap length L of the hairpin filter is more influentialthan the spacing S1, which allows one to sample along S1 more sparsely using a wise adaptivesampling scheme, thereby reducing the number of initial EM simulations needed for the construc-tion of parametric macromodels and the corresponding computational effort.

7. CONCLUSIONS

Gradient-based design optimization of microwave systems using parametric sensitivity macromodelshas been presented in this paper. Parameterized frequency-domain data samples are used to build a



set of root macromodels in a state-space form. Then, this set of state-space matrices is parameterizedusing suitable interpolation schemes, which are continuously differentiable. This allows to accuratelyand efficiently calculate parametric sensitivities over the entire design space of interest. Parametricmacromodels and corresponding sensitivities are used for the gradient-based design optimization intwo proposed numerical examples, which confirm the applicability of the proposed technique tothe optimization process of microwave filters. Also, the importance of parameterized sensitivityinformation to speed up the design optimization process has been shown in the examples.

REFERENCES

1. Li D, Zhu J, Nikolova NK, BakrMH, Bandler J. Electromagneticoptimisation using sensitivity analy-sis in the frequency domain. IETMicrowaves, Antennas Propagation2007; 1(4):852–859. doi:10.1049/iet-map:20060303.

2. Dounavis A, Achar R, Nakhla M.Efficient sensitivity analysis of lossymulticonductor transmission lineswith nonlinear terminations. IEEETransactions on Microwave Theoryand Techniques 2001; 49(12):2292–2299. doi:10.1109/22.971612.

3. Nikolova NK, Bandler JW, BakrMH. Adjoint techniques for sensitiv-ity analysis in high-frequency struc-ture CAD. IEEE Transactions onMicrowave Theory and Techniques2004; 52(1):403–419. doi:10.1109/TMTT.2003.820905.

4. Deschrijver D, Dhaene T. Stabil-ity and passivity enforcement ofparametric macromodels in timeand frequency domain. IEEETransactions on Microwave Theoryand Techniques 2008; 56(11):2435–2441. doi:10.1109/TMTT.2008.2005868.

5. Ferranti F, Knockaert L, Dhaene T.Parameterized S-parameter basedmacromodeling with guaranteedpassivity. Microwave and WirelessComponents Letters, IEEE Oct2009; 19(10):608–610. doi:10.1109/LMWC.2009.2029731.

6. Ferranti F, Knockaert L, Dhaene T.Guaranteed passive parameterizedadmittance-based macromodeling.IEEE Transactions on Advanced

Copyright © 2012 John Wiley & Sons, Ltd

Packaging 2010; 33(3):623–629.doi:10.1109/TADVP.2009.2029242.

7. Ferranti F, Knockaert L, Dhaene T,Antonini G. Passivity-preservingparametric macromodeling for highlydynamic tabulated data based onLur’e equations. Microwave Theoryand Techniques, IEEE Transactionson 2010; 58(12):3688–3696. doi:10.1109/TMTT.2010.2083910.

8. Triverio P, Nakhla M, Grivet-TalociaS. Passive parametric macromodel-ing from sampled frequency data.2010 IEEE 14th Workshop on SignalPropagation on Interconnects (SPI)2010; 117–120.

9. Triverio P, Nakhla M, Grivet-TalociaS. Passive parametric modeling ofinterconnects and packaging com-ponents from sampled impedance,admittance or scattering data. 3rdElectronic System-Integration Tech-nology Conference (ESTC). 2010;1–6, doi: 10.1109/ESTC.2010.5642961.

10. Gustavsen B, Semlyen A. Rationalapproximation of frequency domainresponses by vector fitting. IEEETransactions on Power Delivery1999; 14(3):1052–1061. doi:10.1109/61.772353.

11. Gustavsen B. Improving the polerelocating properties of vector fit-ting. IEEE Transactions on PowerDelivery 2006; 21(3):1587–1592.doi:10.1109/TPWRD.2005.860281.

12. Deschrijver D, Mrozowski M,Dhaene T, De Zutter D. Macro-modeling of multiport systemsusing a fast implementation ofthe vector fitting method. IEEE

.

Microwave and Wireless Com-ponents Letters 2008; 18(6):1587–1592. doi:10.1109/LMWC.2008.922585.

13. Gustavsen B, Semlyen A. Fast pas-sivity assessment for S-parameterrational models via a half-size testmatrix. IEEE Transactions onMicrowave Theory and Techniques2008; 56(12):2701–2708. doi:10.1109/TMTT.2008.2007319.

14. Gustavsen B. Fast passivity en-forcement for S-parameter modelsby perturbation of residue matrixeigenvalues. IEEE Transactions onAdvanced Packaging 2010; 33(1):257–265 doi:10.1109/TADVP.2008.2010508.

15. de Boor C. A Practical Guideto Splines. Springer-Verlag NewYork, Inc.: New York, USA,2001.

16. Fritsch FN, Carlson RE. Monotonepiecewise cubic interpolation. SIAMJournal on Numerical Analysis1980; 17(2):238–246.

17. Pruess S. Shape preserving C2cubic spline interpolation. IMAJournal of Numerical Analysis1993; 13(4):493–507.

18. Du DZ, Pardalos PM.Minimax andApplications. Kluwer Academic Pub-lishers: Dordrecht, The Netherlands,1995.

19. Ureel J, De Zutter D. Gradient-based minimax optimization ofplanar microstrip structures withthe use of electromagnetic simula-tions. International Journal ofMicrowave and Millimeter-WaveComputer-Aided Engineering 1997;7(1):29–36.

AUTHORS’ BIOGRAPHIES

Krishnan Chemmangat was born in Kerala, India, on December 28, 1984. Krishnanreceived his Bachelor of Technology (B.Tech) degree in Electrical and Electronics Engi-neering from the University of Calicut, Kerala, India in 2006 and his Master of Technology(M.Tech) degree in Control and Automation from the Indian Institute of Technology Delhi,New Delhi, India in 2008. Since April 2010, he is active as a PhD student in the researchgroup Internet Based Communication Networks and Services (IBCN) at Ghent University.His research interests include parametric macromodeling, parameterized model orderreduction, and system identification.

Int. J. Numer. Model. 2012; 25:347–361DOI: 10.1002/jnm


Copyright © 2012 John W

Francesco Ferranti received his BS degree (summa cum laude) in Electronic Engineeringfrom the Università degli Studi di Palermo, Palermo, Italy, in 2005; his MS degree (summacum laude and honors) in Electronic Engineering from the Università degli Studidell’Aquila, L’Aquila, Italy, in 2007; and his PhD degree in Electrical Engineering fromthe University of Ghent, Ghent, Belgium, in 2011. He is currently a Post-DoctoralResearch Fellowwith the Department of Information Technology (INTEC), Ghent University,Ghent, Belgium. His research interests include parametric macromodeling, parameterizedmodel order reduction, electromagnetic compatibility numerical modeling, and systemidentification.

Tom Dhaene was born in Deinze, Belgium, on June 25, 1966. He received his PhD degreein Electrotechnical Engineering from the University of Ghent, Ghent, Belgium, in 1993.From 1989 to 1993, he was Research Assistant at the University of Ghent, in the Departmentof Information Technology, where his research focused on different aspects of full-wave elec-tromagnetic circuit modeling, transient simulation, and time-domain characterization of high-frequency and high-speed interconnections. In 1993, he joined the EDA company Alphabit(now part of Agilent). He was one of the key developers of the planar EM simulator ADSMOMENTUM. Since September 2000, he has been a Professor in theDepartment ofMathematicsand Computer Science at the University of Antwerp, Antwerp, Belgium. Since October 2007,he is a Full Professor in the Department of Information Technology (INTEC) at Ghent Univer-sity, Ghent, Belgium. As author or co-author, he has contributed to more than 150 peer-reviewed papers and abstracts in international conference proceedings, journals, and books.He is the holder of three US patents.

Luc Knockaert received his MSc degree in Physical Engineering, MSc degree inTelecommunications Engineering, and PhD degree in Electrical Engineering from GhentUniversity, Belgium, in 1974, 1977, and 1987, respectively. From 1979 to 1984 and from1988 to 1995, he worked in North–South cooperation and development projects at theUniversities of the Democratic Republic of the Congo and Burundi. He is presently affil-iated with the Interdisciplinary Institute for BroadBand Technologies (www.ibbt.be) anda Professor at the Dept. of Information Technology, Ghent University (www.intec.ugent.be). His current interests are in the application of linear algebra and adaptive methods insignal estimation, model order reduction, and computational electromagnetics. As authoror co-author, he has contributed to more than 100 international journal and conferencepublications. He is a member of MAA and SIAM and a senior member of IEEE.

iley & Sons, Ltd. Int. J. Numer. Model. 2012; 25:347–361DOI: 10.1002/jnm

gradient-based optimization using parametric sensitivity ...lknockae/pdf/kfkd_2012.pdf ·...

Documents