qingsha s. cheng, john w. bandler, and slawomir koziel · qingsha s. cheng ([email protected]),...

1.PHOTOCREDIT

Since the space mapping con-cept [1], [2] was first devel-oped, it has been successfullyapplied to microwave engi-neering design problems as

well as in other engineering fields (e.g.,[3], [4]). In the meantime, much re-search has been carried out on variousspace mapping and related approaches[5]–[8]. Space mapping has also beencombined with formulations such asneural networks [9]. Among the devel-opments in the art of space mapping,implicit space mapping [10] is probablythe simplest technique to implement.Here, no matrix calculation is needed.It can be generalized to include allother forms of space mapping in a gen-eralized implicit space mapping frame-work [5].

Qingsha S. Cheng,John W. Bandler, andSlawomir Koziel

February 2008 791527-3342/08/$25.00©2008 IEEE

Digital Object Identifier 10.1109/MMM.2007.910950

Qingsha S. Cheng ([email protected]), John W. Bandler, and Slawomir Koziel are with the Simulation Optimization Systems ResearchLaboratory, Department of Electrical and Computer Engineering, McMaster University, Hamilton, Ontario, Canada.

John W. Bandler is also with Bandler Corporation, Dundas, Ontario, Canada.

© IMAGESTATE

80 February 2008

In the microwave arena, circuit-theory based CADtools such as Agilent ADS [11] are fast and easy to usefor design optimization. On the other hand, electromag-netic (EM) simulators such as Agilent Momentum,Sonnet em [12], Ansoft HFSS [13], or FEKO [14] are usu-ally CPU-intensive but offer high-fidelity validations.Implicit space mapping is a simple approach to combinea circuit-theory based CAD model, or “coarse” model,and an EM simulator, or “fine” model, to achieve fastand accurate optimal design and modeling.

In both the fine model and corresponding coarsemodels, certain preassigned parameters exist; for exam-ple, dielectric constant and the height of the substrate.They are normally selected and their values fixed early inthe modeling and design process. Implicit space map-ping explores the flexibility of these parameters in thedesign optimization [10] and device modeling [15] tasks.The effects on the responses of microwave componentsof varying the values of these parameters may be as sig-nificant as those achieved by varying the optimizableparameters. In our space mapping technique, we applythis flexibility to the coarse model only.

Thus, we calibrate certain selected coarse model pre-assigned parameters in each space mapping iteration tomatch the response of the fine model. With these re-cal-ibrated preassigned parameters now fixed in value, theenhanced coarse model (surrogate) is then re-optimizedto obtain a new design. We apply this new design to thefine model. These steps are repeated until the design

specification is satisfied.Generally speaking, the

coarse model could be imple-mented using any circuit sim-ulator. In our examples, coarsemodels are realized in AgilentADS. We use Sonnet em andFEKO as the fine model evalu-ators. Agilent ADS schematicsorganize the ADS optimiza-tion engine and coarse andfine models to perform para-meter extraction (obtain amapping to re-match the mod-els) and surrogate optimiza-tion (obtain a new predictionof the desired optimal solu-tion). Figure 1 illustrates therelationship between the mod-els and simulators.

Space mapping belongs tothe category of surrogate-basedoptimization approaches. Thereis a rich literature concerningsurrogate-based optimization.In [16], the authors describe so-called approximation andmodel management optimiza-tion. This approach assumesthat the surrogate model satis-fies so-called zero- and first-order consistency conditionswith the high-fidelity model inquestion. A surrogate manage-ment framework and its appli-cations for engineering designare presented in [17]. Surrogateoptimization based on kriging

Figure 1. Space mapping implementation concept.

FineSpace

CoarseSpace

(Surrogate Optimization)Obtain New Prediction

Fine ModelCoarse Model

CircuitSimulator

(ADS, etc.)

MomentumSonnet em

Ansoft HFSSFEKO

DesignParameters

ResponsesDesign Parameters

Responses

PreassignedParameters

Optimization Engine(ADS, Matlab, etc.)

Obtain a Mapping to Match the Models(Parameter Extraction)

Figure 2. Two illustrations of parameter extraction (PE) for matching the initial surro-gate (coarse model) to the fine model. The initial surrogate and fine model deviate fromeach other at the initial iteration. After the parameter extraction process, preassigned para-meter p0 becomes p1, and the new surrogate model is aligned with the fine model. Weshow two cases. In (a), the surrogate model is shifted to match the fine model as the preas-signed parameter vector changes from p0 to p1. In (b), the surrogate model is squeezed tomatch the fine model as the preassigned parameter vector changes from p0 to p1.

p1 p0 p1

p0

(a) (b)

InitialSurrogate

SurrogateAfter PE

FineModel Initial

SurrogateFineModel

SurrogateAfter PE

In both the fine model andcorresponding coarse models, certainpreassigned parameters exist; forexample, dielectric constant and theheight of the substrate.

February 2008 81

models are discussed in [18]. A survey of surrogate-based analysis and optimization methods is given in[19]. It should be emphasized that a characteristic fea-ture that differentiates space mapping from some othersurrogate-based optimization methods is that in ourspace mapping approach the surrogate model is con-structed using an available, low-fidelity (and physical-ly meaningful) model of the object response (the modelbeing a function of the actual design variables), ratherthan a pure interpolation or approximation.

There are many other approaches that aim atreducing the total time cost of fine model evaluations,such as the model order reduction (MOR) method.The MOR method [20] is a way of reducing unneces-sary complexity in a system in order to obtain a faster-to-compute model. It is a bottom-up approach, sinceit considers the details of the fine model and attemptsto reduce its order. However, space mapping workstop-down. It tries to reconcile two existing models, acoarse model with a fine model, even without theimplementation details of the fine model. MOR aimsat reducing the fine model’s complexity, while spacemapping aims at minimizing the number of calls tothe fine model.

Implicit Space MappingThe formulation of the implicit space mapping algo-rithm is presented in the sidebar (see “The ImplicitSpace Mapping Concept”) [10], [21]. Our goal is toobtain the fine model optimal design without goingto direct optimization of the fine model but insteadusing the surrogate model; i.e., the coarse modelwith updated values of the preassigned parameters.Parameter extraction and design optimization areperformed solely on the surrogate model. A predic-tion of the next fine model design is also obtainedthrough the surrogate.

We illustrate the space mapping process in Figure2. In this illustration, the solid lines are the finemodel representations and the dotted lines are the

Figure 3. Implicit space mapping algorithm flowchart[10], [21].

Test sensitivity using tuning andoptimization to decide on the design

and preassigned parameters.

Optimize the surrogate model w.r.t.available design parameters.

Simulate the fine model.

Specificationsatisfied? Yes

Exit

Extract the preassigned parametersto match the surrogate

with the fine model.

No

Update the surrogate.

Our goal is to design a fine model optimal solution

x∗ = arg minx

U(Rf (x)) (1)

Here, the fine-model response vector is denoted by Rf ,e.g., |S11| at selected frequency points. The fine-modeldesign parameters are denoted x. U is a suitableobjective function. In microwave engineering, U is typi-cally a minimax objective function with upper andlower specifications [22]; x∗ is the optimal design tobe determined. Implicit space mapping uses the fol-lowing iterative procedure to solve (1)

xk+1 = arg minx

U(Rc(x, pk)) (2)

where Rc(x, p) is a response vector of the coarsemodel with x and p being the design variables andpreassigned parameters, respectively. Rc(x, pk) is animplicit space mapping surrogate model with preas-signed parameters pk obtained at iteration k using theparameter extraction procedure

pk = arg minp

‖Rf (xk) − Rc(xk, p)‖ (3)

in which we try to match the surrogate to the finemodel. The initial surrogate model is Rc(x, p0), wherep0 represents the initial preassigned parameter val-ues. In other words, the surrogate model is thecoarse model with updated values of the preassignedparameters.

The Implicit Space Mapping Concept [10], [21]

82 February 2008

surrogates. In both cases, the selected preassignedparameters p1 are extracted such that the surrogate isclosely matched to the fine model. After the extraction,the new surrogate replaces the fine model, becomingthe function to be optimized. The solution is our newfine model solution. We can then simulate the finemodel for verification of its response. If the fine modeldesign specification is not satisfied, the steps can berepeated. Please refer to the sidebar for a formaldescription of the implicit space mapping technique.We summarize the implicit space mapping algorithm

in the flowchart of Figure 3.

Tapped-Line MicrostripFilter Illustration We now demonstrate theimplicit space mapping tech-nique on a simple optimiza-tion problem, the second-order tapped-line microstripfilter [23] shown in Figure 4.

For the sake of simplicitywe only use two designparameters, L1 and g asdefined in Figure 4. The finemodel is simulated in FEKO[14]. The design specifica-tions are |S21| ≥ −3 dB for4.75 GHz ≤ ω ≤ 5.25 GHz,and |S21| ≤ −20 dB for 3.0Figure 5. Coarse model of the second-order tapped-line microstrip filter (Agilent ADS).

MLINTL1W=1 mmL=L1–4 mm

MLINTL2W=1 mmL=L1–4 mm

MTEETee 2W1=1 mmW2=1 mmW3=1 mm

MLOCTL 6W=1 mmL=3 mm

MLINTL4W=1 mmL=3 mm

MLINTL3W=1 mmL=3 mm

Term 1Z=50 Ω

MLOCTL5W=1 mmL=3 mm

Term 2Z=50 Ω

MCLIN CLin 1W=1 mmS=g mmL=10–L1 mm

MTEE Tee 1W1=1 mmW2=1 mmW3=1 mm

Figure 6. Second-order tapped-line microstrip filter responses: fine model (—) versus coarse model (---). (a) Initial design.(b) A good match is obtained after extracting the preassigned parameters. (c) A good fine model response is obtained in justtwo iterations.

3 4 5 6 7−40

−30

−20

−10

0

Frequency [GHz]

(a)

3 4 5 6 7Frequency [GHz]

(b)

3 4 5 6 7Frequency [GHz]

(c)

|S21

|

−40

−30

−20

−10

0

|S21

|

−40

−30

−20

−10

0

|S21

|

Figure 4. Geometry of the second-order tapped-linemicrostrip filter [23].

In/Out

3 mm

1 mm

L110 mm

In/Out

g

February 2008 83

GHz ≤ ω ≤ 4.0 GHz and 6.0 GHz ≤ ω ≤ 7.0 GHz. Thecoarse model shown in Figure 5 is the circuit equiva-lent of the structure in Figure 4 and is implemented inAgilent ADS [11].

We want to optimize our filter using implicit spacemapping with the dielectric constant εr and the heightH of the substrate as tuning parameters. Initial valuesof the parameters are 9.9 and 100 mils, respectively, forboth fine and coarse models. These parameters remainfixed in the fine model; however, we are going to tunethem in the coarse model, according to the implicitspace mapping methodology.

The initial design, L1 = 6.977 mm and g = 0.060mm, is the optimal solution of the coarse model withrespect to our specifications. Figure 6(a) shows the fineand coarse model responses at the initial design. Notethat neither the coarse nor fine models satisfy thedesign specifications. Also,there is quite a significant mis-alignment between the fineand coarse models both withrespect to center frequency andbandwidth. We now performthe parameter extraction proce-dure. Figure 6(b) shows thefine model and surrogatemodel (i.e., the tuned coarsemodel) response at the initialdesign. We (re)optimize the de-sign parameters in our surro-gate with the newly obtained

preassigned parameter values. A new set of designparameter values is found to supply to fine model. Inthis manner, we obtain a good fine model design in justtwo iterations [Figure 6(c)].

For comparison purposes we also perform thedirect optimization of the fine model using Matlab[24]. It turns out that the implicit space mapping

Figure 8. A microstrip hairpin filter is “half” implemented, since the structure is symmetric.

MSOBND_MDSBend5Subst="MSub1"W=55 mil

MSUBMSub2H=H1 milEr=Er1Mur=1Cond=1.0E+50Hu=3.9e+034 milT=0.7 milTanD=0.002Rough=0 mil

MSub

MSUBMSub3H=H2 milEr=Er2Mur=1Cond=1.0E+50Hu=3.9e+034 milT=0.7 milTanD=0.002Rough=0 mil

MSub

MSUBMSub1H=H milEr=ErMur=1Cond=1.0E+50Hu=3.9e+034 milT=0.7 milTanD=0.002Rough=0 mil

MSub MLINTLcSubst="MSub1"W=55 milL=Lc mil

MLINTLbSubst="MSub1"W=55 milL=Lb mil


MCLINCLin2Subst="MSub3"W=55 milS=S2 milL=L4 mil

MCLINCLin1Subst="MSub2"W=50 milS=S1 milL=L3 mil

MLINTLaSubst="MSub1"W=40 milL=La mil

MLINTL2Subst="MSub1"W=40 milL=L2 mil

TL1Subst="MSub1"W=40 milL=L1 mil

MLINTL0Subst="MSub1"W=65 milL=100.0 mil

PortP2Num=2

CC4C=.01 pF

CC1C=.01 pF

PortP1Num=1

CC2C=.01 pF

CC3C=.01 pF




MTEETee1Subst="MSub1"W1=40 milW2=40 milW3=65 milMLEF

Figure 7. The microstrip hairpin filter [25].

LaLbS1

S2S2

S1 LaLb2Lc

L2

L1

εr

L3L4

L4

L3

L2

L1

H

Implicit space mapping is a simpleapproach to combine a circuit-theorybased CAD model and an EMsimulator model to achieve fastand accurate optimal designand modeling.

84 February 2008

optimization of our filter, which requires only threeevaluations of the fine model (plus a small overheadrelated to parameter extraction and optimization ofthe surrogate model), takes only about 17 minutes.This is substantially faster than the direct optimiza-tion, which requires 54 fine model evaluations andtakes about 4.5 hours.

Microstrip Hairpin Filter DesignWe now describe in detail the design process of themicrostrip hairpin filter [25] example of Figure 7,where the implicit space mapping algorithm, includ-ing the coarse/surrogate model evaluation, is imple-mented in Agilent ADS. The fine model responses areloaded from the result of an external EM simulatorusing the ADS SnP file. In three iterations, we have a

Sonnet em design with much better performance thanreported in [25].

We use the original design specifications of[25]: |S11| < −16 dB for 3.7 GHz ≤ ω ≤ 4.2 GHz; |S21| < −28 dB for ω ≤ 3.2 GHz and ω ≥ 4.7 GHz.

Our coarse model is the same ADS circuit model asin [25]. A “half filter” is implemented in ADS, as shownin Figure 8. The two half-filters are connected in a“back-to-back” fashion [25]. We use L1, L2, L3, L4, S1,and S2, as design parameters. An ADS test optimizationand tuning process show that if we add Lc to the set ofdesign parameters, the flexibility will improve the per-formance of the filter. The test also shows that L2 shouldbe set to a small value. So we fix L2 to zero. The designvariable set now becomes L1, L3, L4, S1, S2, and Lc. Weimplement a coarse model optimization schematic inFigure 9. The coarse model is capable of obtaining a bet-ter passband performance than the original designspecification. We then tighten the specification to|S11| < −40 dB for the passband and |S21| < −30 dB forthe stopband.

In Figure 10, the ADS test shows that the specifica-tion is easily satisfied. To compensate for the designparameter value changes, we use a set of preassignedparameters. The uniform dielectric constant εr and

Figure 9. The coarse (surrogate) model optimization process implemented in Agilent ADS.

OptimOptim1OptimType=GradientMaxIters=100DesiredError=−10StatusLevel=4FinalAnalysis="None"NormalizeGoals=noSetBestValues=yesSaveSolns=yesSaveGoals=yesSaveOptimVars=noUpdateDataset=yesSaveNominal=yesSaveAllIterations=noUseAllOptVars=noUseAllGoals=yes

OPTIM

S_ParamSP1Start=3 GHzStop=5 GHzStep=.01 GHz

S-Parameters

Goalq2Expr="db(S(2,1))"SimInstanceName="SP1"Min=Max=−30Weight=RangeVar[1]="Freq"RangeMin[1]=4.7 GHzRangeMax[1]=5 GHz

Goal

Goalq1Expr="db(S(2,1))"SimInstanceName="SP1"Min=Max=−30Weight=RangeVar[1]="Freq"RangeMin[1]=3 GHzRangeMax[1]=3.2 GHz

Goal

VARVAR3H=31 {t} {-o}Er=3.38 {t} {-o}

VARVAR17H2=31 {t} {-o}Er2=3.38 {t} {-o}

VARVAR16H1=31 {t} {-o}Er1=3.38 {t} {-o}

VarEqn

VARVAR14S1=9.78091 {t} {o}S2=14.9824 {t} {o}L1=272.621 {t} {o}L2=0 {-t} {-o}L3=382.047 {t} {o}L4=327.149 {t} {o}La=80 {t} {-o}Lb=100 {-t} {-o}Lc=70.0426 {t} {o}

GoalqExpr="db(S(1,1))"SimInstanceName="SP1"Min=Max=−40Weight=RangeVar[1]="Freq"RangeMin[1]=3.7 GHzRangeMax[1]=4.2 GHz

Goal

TermTerm2Num=2Z=50 Ω

half_hairpin_brady_detach_H_ErX2

half_hairpin_brady_detach_H_ErTerm Term1Num=1Z=50 Ω

VarEqn

VarEqn

VarEqn

1 2 2 1 +

+ X1

The implicit space mappingoptimization of our filter, whichtakes only 17 minutes, issubstantially faster than thedirect optimization, which requires4.5 hours.

February 2008 85

substrate height H are not sufficient to compensate forthe changes in the design parameters. We introduce dif-ferent dielectric constants εr1 and εr2, and substrateheight H1 and H2 for the coupled lines “CLin1” and“CLin2,” respectively, as annotated in Figure 8 (sub-strate components MSub2 and MSub3 are assigned toCLin1 and CLin2, respectively; MSub1 is assigned tothe rest of the microstrip components).

In the fine model, we use Sonnet em with a very finecell size of 1 mil × 1 mil (see Figure 11 and Figure 12).Each frequency point takes about 50 minutes to com-plete on a Pentium 4 with 3.4 GHz CPU and 2 GB RAM.Total simulation time for a 25-frequency-point sweep isaround 17 hours.

One of the most important steps, parameter extrac-tion, is implemented entirely in ADS (Figure 13). Thefine model solution is imported from a Touchstone filegenerated by Sonnet em. The surrogate optimization isimplemented by ADS in exactly the same way as inFigure 9 but with different preassigned parameter val-ues for each iteration.

In the initial iteration, we use an adaptive frequen-cy sweep, since it is shifted away from the specifica-tion as seen in Figure 14. (The passband details arenot important at this stage.) The first iteration shiftsthe responses back to the desired frequency bands

(Figure 15). We conducted a fine frequency sweep inthe passband region. The second iteration finds agood solution without much difficulty (Figure 16).The third iteration fine tunes the responses. A returnloss of −34 dB in the passband is obtained, as shownin Figure 17. In Table 1 we list the parameter values ofthe initial and the final designs.

The solution reported in Table 1 can be furtherimproved by performing one iteration of a spacemapping algorithm using the so-called output space

Figure 12. The fine model in Sonnet em: a three-dimen-sional view.

1

2

Figure 11. The fine model implementation in Sonnet em.

65

5555

65

50

40

505555 55

L4

40

La Lb Lc Lb LaS1 S2 S2 S1

40

100

260L1

4050

5055

L4

L3L3L1

1 2

100260

55

Figure 10. Coarse model optimal responses |S11| (---) and|S21| (—); the desired specification [25] (—) and the tight-ened specification (. . . ).

3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0

−50

−40

−30

−20

−10

−60

0

Frequency (GHz)

|S11

| and

|S21

| in

dB

In the fine model, we use Sonnetem with a very fine cell size of1 mil x 1 mil.

86 February 2008

mapping surrogate model, which allows us to ensurea perfect match with the fine model at the currentiteration point [5]. The improved solution is L1 = 280,L3 = 448, L4 = 342, S1 = 9, S2 = 12, Lc = 86 (dimen-sions in mil), and the corresponding fine modelresponse is shown in Figure 18.

DiscussionOur design starts from Brady’s back-to-back ADSdesign [25]. In his paper [25], the author probablyexpected a narrower passband from the Sonnet emsimulation and the test fabrication, so he applied awider bandwidth for the passband in the ADS circuit

Figure 13. The parameter extraction process implemented in Agilent ADS.

OptimOptim1OptimType=GradientMaxIters=100DesiredError=−10StatusLevel=4FinalAnalysis="None"NormalizeGoals=noSetBestValues=yesSaveSolns=yesSaveGoals=yesSaveOptimVars=noUpdateDataset=yesSaveNominal=yesSaveAllIterations=noUseAllOptVars=noUseAllGoals=yes

SaveCurrentEF=yes

OPTIM

S_ParamSP1Start=3 GHzStop=5 GHzStep=.01 GHz

S-Parameters

GoalqExpr="abs(db(S(1,1))–db(S(3,3)))"SimInstanceName="SP1"Min=Max=0Weight=RangeVar[1]=RangeMin[1]=RangeMax[1]=

Goal

Goalq1Expr="abs(db(S(2,1))–db(S(4,3)))"SimInstanceName="SP1"Min=Max=0Weight=RangeVar[1]=RangeMin[1]=RangeMax[1]=

Goal

TermTerm4Num=4Z=50 ΩS2P

SNP1File="C:\shasha\Research\Sonnet_hairpin\hairpin_sym_tight_spec_0.s2p"

21RefTerm

Term3

Z=50 ΩNum=3

VARVAR3H=24.4742 {t} {o}Er=2.9103 {t} {o}

VARVAR16H1=25.8823 {t} {o}Er1=2.76591 {t} {o}

VARVAR17H2=26.7509 {t} {o}Er2=2.9517 {t} {o}

VARVAR14S1=10 {t} {-o}S2=15 {t} {-o}L1=273 {t} {-o}L2=0 {-t} {-o}L3=382 {t} {-o}L4=327 {t} {-o}La=80 {t} {-o}Lb=100 {-t} {-o}Lc=70 {t} {-o}





VarEqn

VarEqn

VarEqn

VarEqn

2 121+ +

++

−

February 2008 87

simulator (coarse model) optimization specification;i.e., the passband is changed to 3.55 GHz ≤ ω ≤ 4.4GHz from 3.7 GHz ≤ ω ≤ 4.2 GHz.

However, the trade-off is that the increased band-width deteriorates the passband performance. Whenthe ADS design is applied to Sonnet em, the specifica-

tion is barely satisfied, while the bandwidth is muchwider than necessary. Space mapping uses a parameterextraction procedure in each iteration. The parameterextraction procedure “absorbs” the difference betweenthe Sonnet em model and the ADS model. Therefore,we can keep the desired specification as our coarse

Figure 17. Fine model responses |S11| (---) and |S21| (—)after the third iteration, the desired specification [25] (—),and the tightened specification (. . . ).

3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0

−50

−40

−30

−20

−10

−60

0

Frequency (GHz)

|S11

| and

|S21

| in

dB

Figure 18. Fine model responses |S11| (---) and |S21| (—)at the final solution improved by one space mapping iteration with an output space mapping surrogate model,the desired specification [25] (—), and the tightened specification (. . . ).

3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0

−50

−40

−30

−20

−10

−60

0

Frequency (GHz)

|S11

| and

|S21

| in

dB

Figure 14. Fine model responses |S11| (---) and |S21| (—)at the initial solution, the desired specification [25] (—),and the tightened specification (. . . ).

3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0

−50

−40

−30

−20

−10

−60

0

Frequency (GHz)

|S11

| and

|S21

| in

dB

Figure 15. Fine model responses |S11| (---) and |S21| (—)after the first iteration, the desired specification [25] (—),and the tightened specification (. . . ).

3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0

−50

−40

−30

−20

−10

−60

0

Frequency (GHz)

|S11

| and

|S21

| in

dB

Figure 16. Fine model responses |S11| (---) and |S21| (—)after the second iteration, the desired specification [25] (—),and the tightened specification (. . . ).

3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0

−50

−40

−30

−20

−10

−60

0

Frequency (GHz)

|S11

| and

|S21

| in

dB

TABLE 1. Microstrip hairpin filter designparameter values.

Initial Design (mil) Design (Coarse Model Final On-Grid Variables Optimal Design) Design (mil)

L1 273 280

L3 382 448

L4 327 341

S1 10 9

S2 15 12

Lc 70 87

88 February 2008

model or surrogate optimization specification. With aproper choice of design parameters, we can even usethe tightened design specifications and thus obtain abetter Sonnet em design.

SummaryIn this article we reviewed the implicit space map-ping concept. We illustrated it using a simple tapped-line microstrip filter example. We demonstrated therobustness of our approach by performing an accu-rate design of a microstrip hairpin filter. A detailedand easy-to-follow design optimization procedurewas provided and the Agilent ADS implementation ofthe algorithm was described. A good electromagneti-cally validated design in Sonnet em was obtained in afew fine model simulations.

AcknowledgmentsThis work was supported in part by the NaturalSciences and Engineering Research Council ofCanada under Grants RGPIN7239-06, STPGP336760-06 and by Bandler Corporation. We thank SonnetSoftware, Inc., for em, EM Software & Systems-S.A.(Pty) Ltd. for FEKO, and Agilent Technologies forADS. We also thank the reviewers for their construc-tive suggestions.

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Space mapping uses a parameterextraction procedure in eachiteration, which “absorbs” thedifference between the Sonnet emmodel and the ADS model.

qingsha s. cheng, john w. bandler, and slawomir koziel · qingsha s. cheng ([email protected]),...

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