sponse displayed in Figure 17 shows wider bandpasses due todiscontinuity effects. Moreover, one should note the introductionof tuning parameters in the inverters by means of step impedance.

The measured electrical response presented in Figure 18 showsa quite good agreement with the simulated one. The slight fre-quency shift is induced by technological dispersion.

This filter was achieved in a classical microstrip technology onan AR1000 substrate (�r � 10; h � 635 �m).

5. ADVANTAGES AND DRAWBACKS OF EACH STRUCTURE

Table 2 evidences the benefits and drawbacks of each structure interms of simplicity, flexibility, and rejection control.

The first solution, based on the integration of lowpass filters,gives low-spurious response. Unfortunately, it lacks flexibilitywhen the required frequency ratio differs greatly from three.

The dual-matching technique is very attractive because the twobandpass filters can be developed independently. However, thedual-band DBR is a very general solution and allows the designerto meet a very large range of frequency specifications. Flexibilityis the best term to qualify this topology.

6. CONCLUSION

This paper has reported on dual-band filters for multifunctional RFsystems. Three possible topologies were presented with their as-sociated synthesis techniques. The simulation results were vali-dated in microstrip technology, and the benefits and drawbacks ofeach structure were highlighted. This study showed that the dual-band DBR topology is a general solution, which appears as the bestone in terms of flexibility with respect to central frequencies andbandwidths ratios. Nevertheless, some additional work still needsto be done to develop dual-band inverters. The two other solutionswere easily designed, but their lack of flexibility reduces theirapplications. Moreover, a dual-band DBR constitutes the first stepfrom DBR to multiband DBR. Indeed, an n-band resonator will beconstituted of (n � 1) stopband structures.

REFERENCES

1. H.C. Bell, Zolotarev bandpass filters, IEEE MTT Dig, Phoenix, AZ,(2001).

2. R. Levy, Generalized rational function approximation in finite inter-vals using Zolotarev functions, IEEE Trans Microwave Theory TechMTT-18 (1970), 1052–1064.

3. H. Miyake, S. Kitazawa, T. Ishizaki, T. Yamada and Y. Nagatomi, Aminiaturized monolithic dual-band filter using ceramic laminationtechnique for dual mode portable telephones, IEEE MTT-S Int Micro-wave Symp Dig 2 (1997), 789–792.

4. L.-C. Tsai and C.-W. Hsue, Dual-band bandpass filters using equal-length coupled-serial-shunted lines and Z-transform technique, IEEETrans Microwave Theory Tech 52 (2004), 1111–1117.

5. C.-H. Chang, H.-S. Wu, J. Yang, and C.-K.C. Tzuang, Coalescedsingle-input single-output dual-band filter, IEEE MTT-S Int Micro-wave Symp Dig 1 (2003), 511–514.

6. C. Quendo, C. Person, E. Rius, and M. Ney, Integration of optimized

low-pass filters in band-pass filters for out-of-band improvement,Trans IEEE Microwave Theory Tech 49 (2001), 2376–2383.

7. C. Quendo, E. Rius, and C. Person, Narrow bandpass filters using dualbehavior resonators (DBRs), Trans IEEE Microwave Theory Tech 51(2003), 734–743.

8. C. Quendo, E. Rius, and C. Person, Narrow bandpass filters usingdual behavior resonators (DBRs) based on stepped impedance stubsand different-length stubs, Trans IEEE Microwave Theory Tech 52(2004),

9. E.G. Cristal, Tapped line coupled transmission lines with applicationto interdigital and combline filters, IEEE Trans Microwave TheoryTech MTT-23 (1975), 1007–1113.

10. S. Caspi and J. Adelman, Design of combline and interdigital filterswith tapped line input, IEEE Trans Microwave Theory Tech MTT-36(1988), 759–763.

11. C. Denig, Using microwave CAD programs to analyze microstripinterdigital filters, Microwave Journal 32 (1989), 147–152.

12. F. Mahe, G. Tanne, E. Rius, C. Person, S. Toutain, F. Biron, L.Billonnet, B. Jarry, and P. Guillon, Electronically switchable dual-band microstrip interdigital bandpass filter for multistandard com-munication applications, Proc 30th Euro Microwave Conf, Paris,2000.

13. E. Fourn, A. Pothier, C. Champeaux, P. Tristant, A. Catherinot, P.Blondy, G. Tanne, E. Rius, C. Person, F. Huret, MEMS switchableinterdigital coplanar filter, IEEE Microwave Theory Tech special issueon RF MEMS, 51 (2003), 320–324.

14. G.L. Matthaei, L. Young, and E.M.T. Jones, Microwave filters, im-pedance-matching networks and coupling structures, Artech House,Dedham, MA, 1980.

15. C. Quendo, E. Rius, and C. Person, An original topology of dual-bandfilter with transmission zeros, IEEE MTT Symp, Philadelphia, PA,2003.

© 2005 Wiley Periodicals, Inc.

OPTIMIZATION OF ARBITRARILYSHAPED H-PLANE PASSIVEMICROWAVE DEVICES THROUGHFINITE ELEMENTS AND THEANNEALING ALGORITHM

Javier Monge, Jose M. Gil, and Juan ZapataDepartamento de Electromagnetismo y Teorıa de CircuitosUniversidad Politecnica de MadridCiudad Universitaria s/n28040 Madrid, Spain

Received 16 February 2005

ABSTRACT: A global optimization method for the design of arbitrarilyshaped H-plane passive waveguide devices is presented. An adaptive simu-lated annealing algorithm (SA) is used to avoid local minima. No informa-tion on the initial shape is introduced. A hybrid method based on the seg-mentation technique, the finite-element method, and a matrix Lanczos–Pade

TABLE 2 Comparison of the Different Structures

Low DevelopmentComplexity

Bandwidth and Central-FrequencyPossibilities

Rejection Control(Low/High FrequencyBetween the Bands) Low Spurious Response

Bandpass filter with integratedlowpass filter

� � � ���

Dual matching ��� �� � ��Dual-band DBR � ��� ��� �

360 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 46, No. 4, August 20 2005

algorithm (SFELP) adapted to 2D space (2D SFELP) efficiently calculatesthe wideband response. Technological constraints and mechanical toler-ances are taken into account in the design process. © 2005 Wiley Periodi-cals, Inc. Microwave Opt Technol Lett 46: 360–367, 2005; Published on-line in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20987

Key words: finite-element methods; optimization methods; annealing;waveguide components; mesh generation

1. INTRODUCTION

The electromagnetic devices necessary for aerospace applicationsmust satisfy demanding constraints such as high-power handlingcapability, low losses, and robustness. Waveguide components areessential for these applications. Efficient and flexible electromag-netic simulators and global optimizers facilitate its design. Severalcomponents present translation symmetry along the y-axis (H-plane devices). Power splitters, diplexers, bends, and phase-shift-ers are examples of devices that can be implemented using thisgeometry. The only information needed to obtain the electricalproperties is the profile of the layout.

A big effort is being made in improving computer-aideddesign (CAD) tools in order to avoid resorting to severalmanufacturing–measurement cycles until the final design isachieved. Both the flexibility and numerical efficiency of theanalysis method used, as well as the selection of the appropriateoptimizing algorithm— global, local, or mixed strategies—areimportant and interconnected issues related to CAD tools. Sev-eral combinations have been reported. Mode-matching methodshave been used for years by many researchers as analysis toolsin the CAD of waveguide components because of their accuracyand efficiency, although these methods have limitations in thegeometrical shapes that can be handled. A good review of thesetechniques can be found in [1]. In the past few years, hybridmethods based on neural models have been suggested for ob-taining fast simulations with sufficient accuracy for CAD pur-poses [2– 6]. In [7], an analysis method based on boundary-integral resonant-mode expansion and a gradient-basedoptimization algorithm was proposed for obtaining irregular-shaped waveguide H-/E-plane devices whose profile is definedby polygonal lines. The finite-element method (FEM) is apowerful analysis method which is also of significant interest.Although this method originally involved a high computationalcost when used in CAD tools, its latest improvements make ituseful for this task. In [8], a method based on a 3D FEM, anadjoining variable method, and the steepest descendent algo-rithm was used to design a waveguide-to-microstrip transition.A modified version of the adjoining variable method, whichdoes not require additional fields to be computed, was presentedin [9], where it was applied to the design of 2D waveguidedevices in combination with a quasi-Newton constrained opti-mizer. In [10], a gradient-based optimization technique in con-nection with the FEM was proposed for the CAD of 2Dwaveguide components. In spite of global optimization tech-niques such as evolution programs or simulated annealing (SA)being able to avoid local minima, these techniques have rarelybeen used as optimization algorithms associated with the FEM.The basic reason is that they require thousands of computationcycles before they arrive at the optimum design. This impliesthat the computation time necessary to calculate the response ofthe component, in a number of frequencies within the band,should be kept to seconds rather than minutes, which at presentseems out of the scope of available 3D-FEM implementations.However, this is not the case in 2D problems such as the

H-plane waveguide components. In [11], a fast 2D version ofthe FEM-segmentation method suggested in [12] was used on afrequency-by-frequency basis in combination with an evolutionprogram in order to optimize an H-plane broadband waveguide-matched load whose layout was defined by polygonal lines.

In this paper, an improvement of the previous method, outlinedin [13], is developed. An H-plane version of the mixed segmen-tation finite-element reduced-order model (SFELP) proposed in[14] has been developed. No assumptions with regard to the initialshapes are taken into account. Only the mechanical constraints areimposed. The electromagnetic simulator produces the generalizedadmittance matrix (GAM) that allows the evaluation of the scorefunction. The segmentation method splits the device into parts thatare analyzed separately using a 2D-FEM. A symmetric Pade viaLanczos algorithm (SyMPVL) efficiently obtains the widebandGAM of each segment. Finally, the generalized scattering matrices(GSMs) obtained from each GAM are interconnected, resulting inthe total GSM. With this technique, a drastic reduction in thematrices involved in the process and a frequency sweep areachieved at the same time. To accelerate the optimization process,the algorithm allows coarser mesh and the SA tolerates inaccuratesolutions in the early stages.

Flexibility in the definition of the profile is essential so thatthe optimizer can find the global minimum of the score func-tion. The finite elements used are mapped by means of 2nd-orderpolynomial functions so that 2D-SFELP gives good results fordevices whose profile is defined as a sequence of 2nd-orderpolynomials. The curved line that limits the shape satisfiesconditions that allows the implementation of the device in amilling machine, that is, curved segments with a radio greaterthan the drill radius (2 mm).

The developed SA algorithm finds, with a high probability ofsuccess, a global minimum by avoiding the local minimum resultsof the direct optimizer. The result obtained is then introduced intothe SA with the same score function multiplied by �1 and allow-ing variations in the geometrical variables equal to the tolerance ofthe milling machine. Thus, we know the worst possible result ofthe manufactured device.

2. FORMULATION

2.1 Two-Dimensional SFELPThe 2D SFELP formulation comes from the 3D SFELP ex-plained in [14]. The 3D SFELP uses a mixture of modalanalysis and 3D FEM to evaluate the GAM of one 3D region.In this case, we consider an arbitrarily shaped 3D region Vwhere we apply Galerkin’s method to the vectorial-wave equa-tion of the magnetic field. Then we use Green’s identities, applyessential and natural boundary conditions, and consider P portson the boundary surface S enclosing the region V. Finally, weuse modal expansion at ports to express tangential electric andmagnetic fields to obtain

�V

�� � W� � ��r�1� � H� � � k2W� � ��rH� ��dV

� j��0 �i�1

P �k�1

Vki � zi� � �

Si

W� � �zi�

� e� tki � xi, yi��dSi � W� � , H� � , (1)

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 46, No. 4, August 20 2005 361

where is the set of trial W� and test H� functions, �r and �r arethe relative permeability and permittivity scalars, respectively,k is the wave number at the angular frequency �, and zi is anormal vector pointing inwards in the region on the surface Si

of each port.Now we focus on the rectangular-waveguide H-plane

devices. Our goal is to reduce the problem to two dimensionsby applying the y�-axis translation symmetry. In each access,we can describe the field as an infinite sum of TE and TMmodes. The magnetic field inside the H-plane devices is as-sumed as

H� � �n�0

Hx� x, z�cos�kyy�x� � Hy�x, z�sin�kyy�y�

� Hz�x, z�cos�kyy�z�, (2)

where ky � (n/b) and b is the height of the waveguide.Using Eq. (2) and the modes at the ports and introducing them

into Eq. (1), we obtain an expression where the test and trialfunctions do not depend on the variable y as follows.

For the first term, we obtain

�V

� � W� � �r�1� � H� dV � �

b

�r�

S

�Wz

x

Hz

x�

Wz

x

Hx

z�

Wx

z

Hz

x�

Wx

z

Hx

z �dV

for n � 0;

b

2�r�

S

�ky2�WzHz � WxHx� � ky�Wz

Hy

z� Hz

Wy

z� Hx

Wy

x� Wx

Hy

x ��

Wy

z

Hy

z�

Wz

x

Hz

x�

Wz

x

Hx

z�

Wx

z

Hz

x�

Wx

z

Hx

z�

Wy

x

Hy

x �dS

for n � 0;

for the second term, we obtain

�V

k2W� � ���r�H� �dV � �bk2�r �

S

�WxHx � WzHz�dS

for n � 0;

bk2�r

2 �S

�WxHx � WyHy � WzHz�dS

for n � 0;

the excitations are given by

j��0 �i�1

P �k�1

Vki � zi� �

Si

W� � �zi� � e� tki � xi, yi��dSi

� �bj��0 �

i�1

P �k�1

Vki � zi�

� � �Wx� x, z� x�� � �zi� � �eytki y���dxi, for n � 0;

b

2j��0 �

i�1

P �k�1

Vki �zi�

��Si

W� �xi, zi� � �zi� � e�tk

i �xi��dxi for n � 0.

After this operation, we discretize using test and trial functions,whose domain belongs to ℜ2, and taking a finite number of modesmi in each port gives

G�Hc� � j��0�BN��V�; �Hc� � CN,

where G is the system matrix and BN is related to the modes at theoutput ports.

Proceeding thusly as in [12], we obtain the GAM for theH-plane devices as follows:

Y�k� � jk

�0BN

T�k��G � k2M��1BN�k�.

As in the general 3D case, the SFELP method includes twomore steps:

1. the application of the SyMPVL algorithm, which gives usthe wideband response efficiently;

2. the connection of the different GSM matrices of each re-gion, in which a passive microwave circuit can be seg-mented.

2.2 The Optimization AlgorithmThe optimization algorithm used is adapted for our problem basedon the annealing algorithm described in [15]. This is an adaptiveversion that modifies the value of the control parameter and thenumber of iterations of the metropolis loop, and has a stop crite-rion. All these values are stabilized dynamically by the contribu-tion of the aggregate functions.

The inner loop of an annealing algorithm can be understood asa finite homogeneous Markov chain. The following terminologyand concepts that are used in this paper are defined as follows.

● The state space S with states s: this set consists of all possiblecombinations of values of the optimization variables. Eachcombination forms an array whose terms define a validconfiguration of the device under development.

● The set � whose terms are pairs of states. Only jumpsbetween the states in pairs in this set are allowed. This set hasa great influence on the efficiency of the algorithm. It must

362 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 46, No. 4, August 20 2005

satisfy:

space connectivity: �k�1

�k � S � S;

symmetric: � s � S � s � S��s, s � � � 3 �s , s� � ��;

andreflexive: � s � S��s, s� � ��.

● The selection function , given by S � S 3 [0, 1]: thisfunction gives the probability of jumping from a current states to the next state s . Like the previous item, this one has agreat influence in the efficiency of the algorithm as well. Itmust satisfy:

� �s,s ��� � �s, s � � 0�,

� �s,s ��� � �s, s � � 0�,

� s�S � �s �S�

�s, s � � 1� ,

� s�S � s �S � �s, s � � �s , s��.

● The score function �, given by S 3 ℜ� is the function thatevaluates the device configuration defined by the state s.

2.2.1 The Move Set. The first step in the configuration of thealgorithm is the discretization. For example, if the state space ismade up of variables that represent the physical lengths, thediscretization step should be similar to the precision of the tech-nology used in the implementation.

The second step is to choose the movement strategy. In eachstep, we obtain a random array of n independent integers that arecyclically added to the array s. We use this cyclical sum becausethe elements of the array s have upper and lower limits. In ourcase:

● s � S � �n, with n the number of optimization variables;● @s�S @s �S m � �n[(s, s ) � � N s � s Q m].

With this strategy, a selection function and a move set � aregenerated that satisfy the properties defined above.

2.2.2 The Selection Function. The move set and the selectionfunction have a considerable effect on the efficiency of the anneal-ing algorithm. As has been proven in [15], the convergence state isindependent of the selection function. In this implementation anadaptive is used.

In the first loops in the annealing algorithm, a small diameter isdesirable to make all states easily reachable. A smooth change inthe scores are desirable in the last steps for achieving the equilib-rium densities quickly. In this implementation, all of this is doneby selecting the value m according to the aggregate functions:

m�last steps� � m�in the beginning� � 0.3.

2.2.3 The Score Function. The score function also has a greatinfluence in the efficiency of the process. A smooth variation of thescore between states is desirable for obtaining equilibrium andbeing able to measure the aggregate functions. But a quick changein the score is better for guiding the process to the optimum state.In our experience, a linear variation of the score function worksbetter than a quadratic one. A different score function must be usedin each optimization problem.

2.2.4 The State Space (Constraints). The profile of the devicemust fit the constraints in order to be physically implementablewith the technology used. In this optimization method, admissiblestates are the only ones included in the space S so that thealgorithm converges towards a solution that fits the constraints.The examples presented here satisfy the following criteria.

● Precision milling machine (�50 �m): The solution obtainedby the algorithm is sufficiently robust to mechanical errors.This solution does not provoke a big change in the scorefunction as a result of variations in its dimensions which aresmaller than the precision of the milling machine.

● Minimum drill radius (2 mm): The device profile must havea curvature greater than the minimum usable drill radius inthe implementation process. Solutions with curvaturessmaller than the drill radius are not implementable.

2.2.5 Arbitrarily Shaped Mesh. In the optimization process, a setof variables are defined to conform to the arbitrary shape of theH-plane device in order to obtain the desired electrical response.Additionally, some electrical parameters such as the permittivity�r could be the object of optimization as well.

The profile is made up of straight lines and a sequence ofcurved segments. These curved segments are defined by 2nd-orderpolynomials. We impose the continuity of the first derivate be-tween adjacent curved segments. Typically, the profile is param-eterized by defining the position of the junctions of the segments.Then we obtain the coefficient of the 2nd-order polynomial of eachsegment.

The mesh is generated automatically from the profile. Internalnodes are calculated for obtaining a regular mesh limited by thecurved segments.

In our implementation of the algorithm, the 2nd-order curvedsegment is the best way to define the shape because the elementsof the mesh are mapped with 2nd-order transformations.

2.2.6 Local Optimization. The output of the annealing algorithmis introduced as the starting point of a local optimizer. The 2DSFELP for H-plane devices does not give us any information aboutits gradient. Its numerical evaluation involves a large computa-tional cost, so a simplex method is employed. The output of thesimplex method is then the final design.

3. RESULTS

For the following examples, we have used the optimization tool(2D SFELP � simulated annealing) in a Pentium III (Coppermine)at 1 GHz with a random access memory of 1 Gbyte, typicallyobtaining the response for 50 points of frequency over the band fora given geometry in about 3.5–10 s, depending on the mesh and thecircuit size.

In each iteration, a fast frequency-sweep analysis is performedusing SyMPVL. We also make three frequency-by-frequency anal-yses at different points in order to verify the accuracy of thefrequency-sweep analysis. When the error of the sweep analysis isgreater than a given tolerance, we reanalyze the device on afrequency-by-frequency basis.

3.1 90° BendThe first example we present is a two-port device, which isinteresting due to its simplicity. It only has one propagating mediawithout losses (the vacuum �0) and its electrical response iscompletely described by the S11 parameter. This circuit has beenconsidered by many authors (see [16] where a matching septum

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 46, No. 4, August 20 2005 363

technique is presented). We want to find a 90° bend with the lowestwideband reflection coefficient for a WR-90 waveguide. The de-vice is symmetric and its profile has been parameterized with onlyfour variables. These variables represent the length of the radiuswith fixed inclinations. These points define four second ordercurves that are used to generate the internal nodes of the meshautomatically (see Fig. 1).

3.1.1 The Score Function. Because the reflection coefficient is theonly parameter needed to determine everything, it is evaluated atseveral frequencies throughout the band. The score of the state s is�(s) � 100 � maxf�band(�S11( f )�dB). The optimization algo-rithm minimizes this function.

3.1.2 Results. Figure 2 shows this device. Only four geometricaloptimization variables are necessary to achieve the best result.Although this has been attempted with eight variables, we obtainedsimilar results and the former approach took much longer as aresult of the increase in state space.

In Figure 3, the return loss is shown. A wideband reflectioncoefficient less than �35 dB is achieved.

The response measured is located between the optimized curveand the “worst-case” curve. We observe a higher reflection thanexpected only at the beginning of the band. The “worst-case”response represents the combinations of errors less than �50 �min each radio that produces the highest reflection coefficient.

3.1.3 Mechanical Tolerances. Random perturbations in each ra-dius are introduced which simulate the implementation errors ofthe milling machine. The worst case is obtained optimizing with�(s) � �maxf�band(�S11( f )�dB). The perturbations are equal toor less than �50 �m. This gives us an idea of the tolerance of ourdesign.

In order to establish the minimum reflection coefficient that canbe modeled using our analysis tool, the return loss versus thefrequency of several straight waveguides have been computed, asshown in Table 1. The waveguide electrical lengths are taken in thesame range as the electrical length of the bend.

From this table it can be stated that computed return lossesunder 61 dB are not reliable.

3.2 DiplexerThe relationships between the scattering parameters involve acomplex but powerful cost function. A symmetric three-port junc-tion is designed for the implementation of a diplexer to be usedwith two preassigned branching filters. One band, b1, is from 9.8to 10.05 GHz and the other, b2, is from 10.55 to 10.8 GHz.According to [17], the conditions that the scattering parametersmust verify are given by

�S11� � �S22� �S33�,

Figure 1 Optimized H-plane bend in a WR-90 with R1, R2, R3, and R4as variables

Figure 2 View of the 90° bend

Figure 3 Return loss of the bend of the configuration shown in Fig. 1

TABLE 1 Return Loss in dB of an Empty Straight WaveguideAnalyzed Using the 2D SFELP

GHz/cm 1.5 2.5 3.5 4.5

8.2 �96.6 �121.4 �108.0 �95.010 �102 �97.0 �61.0 �89.011 �79.8 �106.5 �104.5 �102.612.4 �95.1 �102.2 �84.6 �86.7

364 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 46, No. 4, August 20 2005

�S33� �1

3f � f1, f � f2,

�S11

�f

�S12

�f

�S22

�f 0, (3)

where f1 is the central frequency of the band b1 and f2 is the centralfrequency of the band b2.

The expressions employed in the design process with the an-nealing algorithm are given by

��s� � a � b � c,

a � maxf��b1�b2�� �S11�dB � �S33�dB�,

b � maxf��b1�b2� �S33�dB � 20 � log�1

3�

c � maxf��b1�b2��S12� fi� � S12� fi�1��.

The minimization of this score function produces a device that fitsEq. (3).

Figure 4, shows the profile obtained from the optimizationparameters used. This profile is implementable using a millingmachine with a drill radius of 2 mm.

In Figure 5, the magnitude of the scattering parameters of thethree-port junction designed for use as a diplexer is presented.Figure 5 shows the proximity of S11 and S33 in the bands ofinterest. In addition, their values are around �9.5 dB, as expected.The S12 parameter has a slow variation in both bands.

3.3 Phase ShiftersIn the prior example, we designed an arbitrarily shaped imped-ance-matched dielectric-slab-filled waveguide phase shifters forWR-90. In [18, 19], many 90° phase shifters were designed byusing a similar configuration of the dielectric. The dielectric ma-terial chosen is Rexolite, a styrol polymerizate compound with�r � 2.54 and tan � � 6.6 � 10�4 (25°C, 10 GHz), because it isrelatively easy to handle mechanically. The design implements adifferential phase shifter of 90° between 9.25 and 10.75 GHz with

Figure 4 Optimized H-plane three-port junction in a WR-90 configuredas a diplexer with V1 � 7.401E-02 cm, V2 � 2.996E-02 cm, V3 ��0.168 cm, V4 � 0.108 cm, H � 0.366 cm, and R � 0.198 cm

Figure 5 Magnitudes of the configuration shown in Fig. 4

Figure 6 Optimized H-plane 90° dielectric-slab-filled waveguide phaseshifter in a WR-90 with V1 � 0.114 cm, V2 � 0.160 cm, V3 � 0.250cm, V4 � 0.812 cm, and D � 2.504 cm

Figure 7 Return loss of the 90° double dielectric-slab-filled waveguidephase shifter of the configuration shown in Fig. 6

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 46, No. 4, August 20 2005 365

the smallest length and phase error. In Figure 6, the optimizationparameters are presented. The experience teaches us that 4 param-eters are a good compromise between appropriate electrical re-sponse and computational efforts.

3.3.1 The Score Function. In this case, we use an expression thathighlights the electrical constraints to the geometrical ones. This isa valid strategy when constraints of a different nature come to-gether.

��s� � �200 � Dif �PhaseError � 2� � �ReturnLoss � 30�;

100 �PhaseError

2� 100 �

30

ReturnLoss� D

if �PhaseError � 2� � �ReturnLoss � 30�;

100 �PhaseError

2� 100 � D

if �PhaseError � 2� � �ReturnLoss � 30�;

100 �30

ReturnLoss� 100 � D

if �PhaseError � 2� � �ReturnLoss � 30�;

PhaseError � 90 � ��TE10 � d � �S21�,

ReturnLoss � ��S11�dB.

3.3.2 Results. The phase shifter with the dielectric-slab in themiddle of the waveguide achieves a good performance over a shortlength because of the highest intensity of the electric field in thatspace, i.e., most of the electric field is inside the dielectric. It hasa length of 25.04 mm. Figure 7 shows the reflection coefficient,which is better than �30 dB in all the band of interest, and Figure8 shows the differential phase error, which is better than �5°.

4. CONCLUSION

A powerful global-optimization tool for the design of H-planecomponents has been presented in this paper. No assumption onthe initial geometry was made. The SA algorithm found a globalsolution, with a low probability of falling in a local minimum. Avery flexible way of determining the profile was used. Second-order polynomial curved segments limited the mesh that is filled

with triangular elements mapped with 2nd-order polynomials. The2D SFELP efficiently evaluated the response by applying segmen-tation methods, the FEM, and the Pade via Lanczos algorithm. Amethod for measuring the sensitivity as a result of tolerance errorshas been described.

All of these elements configure a very flexible and efficientCAD tool that is able to obtain arbitrarily shaped H-plane passivewaveguide components. The method can be easily extended toE-plane waveguide components.

This tool has been successfully used in several examples. A 90°bend was implemented and good agreement between the theoret-ical and measured results was obtained.

ACKNOWLEDGMENT

This work was supported by CICYT, Spain, under contract no.TIC2001-2739.

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Figure 8 Differential phase error of the 90° dielectric-slab-filledwaveguide phase shifter of the configuration shown in Fig. 6

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© 2005 Wiley Periodicals, Inc.

DUAL FREQUENCY PLANARINVERTED-L BOW-TIE PATCHANTENNA

Wenhui Shen, Xilang Zhou, and Zhiyong ShanDepartment of Electronic EngineeringShanghai Jiaotong UniversityShanghai 200030, P. R. China

Received 3 February 2005

ABSTRACT: The characteristics of a planar inverted-L (PIL) bow-tie patch antenna in the 1–3.5-GHz band are investigated. The pro-posed design is based on a modified PIL patch antenna in a finite-sized ground with 50 microstrip feed line. The patch antenna canachieve a more flexible frequency ratio and has the advantage ofeasy fabrication. The experimental results are presented and dis-cussed. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett46: 367–369, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20988

Key words: PIL bow-tie patch antenna; dual frequency

1. INTRODUCTION

The increasing demand for wireless communication has attractedsignificant interest in dual-band or multiband antenna designs thatare easily integrated with the rest of the system. These systemsoften required compact antennas. To meet these requirements,several techniques to achieve size reduction and dual frequency

have been proposed. The planar inverted-L (PIL) microstrip patchantennas are suitable for dual-frequency and compact operation[1–3]. The geometry of the antenna in general has two portions: thehorizontal portion and the vertical portion. The change of either thehorizontal or vertical portion leads to the change of frequencyratio. In [1], two identical slits were inserted at the vertical portionin order to change the resonant frequency so as to obtain variousfrequency ratios. In [2], the horizontal portion is an isoscelestrapezium. The change of frequency ratio is dependent on thechange of the tapering angle. In [3], the change of frequency ratioalso relies upon the change of tapering angle, while the side of thehorizontal portion is circular instead of linearly tapered (as in [2]).

In this paper, a novel PIL patch antenna suitable for dual-frequency as well as compact operation is proposed. The horizon-tal portion of the patch is a bow-tie plate. The experimental resultsfor the return loss, radiation pattern, and frequency ratio aredisplayed.

2. ANTENNA DESIGN

Figure 1 illustrates the geometry of the modified PIL patch an-tenna. It has a total length of (L � h), with horizontal length L andvertical length h. The horizontal portion is selected to be a bow-tieplate and suspended at a height h above a finite-sized groundsubstrate. The ground substrate has a thickness of h1 and relativepermittivity �r, on which a 50 microstrip feed line is printed. ThePIL patch is centered above the microstrip feed line, with the edgeof the patch’s vertical portion directly connected to the feed line.The portion of the feed line below the patch’s horizontal portion

Figure 1 Configuration of the proposed PIL bow-tie patch antenna

Figure 2 Measured return loss vs. frequency for the proposed PILbow-tie patch antenna (�r � 4.4, L � 35 mm, h � 10 mm, h1 � 1 mm,d � 17 mm, and ground-plane size � 130 � 130 mm2)

TABLE 1 Dual-Frequency Performance and Bandwidth of thePIL Bow-Tie Patch Antenna

l [mm] d [mm]

f1

[GHz],BW [%]

f2

[GHz],BW [%] f2/f1

Antenna 1 10 17 1.628, 4.15 2.775, 2.70 1.70Antenna 2 20 17 1.675, 4.46 2.835, 3.17 1.69Antenna 3 30 17 1.680, 4.91 2.963, 2.78 1.76Antenna 4 40 17 1.690, 6.21 2.995, 2.00 1.77

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 46, No. 4, August 20 2005 367