formulas for the computation of the far-field radiation patterns of rectangular microstrip antenna...

Figure 14 Block diagram of the blind adaptive multisensor receiver

Ž .Figure 15 Eye diagrams of the received signal. a On sensor 1,Ž . Ž .SNR s 11.5 dB; b on sensor 3, SNR s 17.0 dB; c at the output of

Žthe blind adaptive four-sensor receiver, SNR s 20.0 dB SNR sth.20.4 dB

shown that the capability of the handset to reject a jammer isnot very good, but it is well suited for diversity reception.

REFERENCES

1. E. E. Altshuler, ‘‘A Monopole Antenna Loaded with a ModifiedFolded Dipole,’’ IEEE Trans. Antennas Propagat., Vol. AP-41, No.7, 1993, pp. 871]876.

2. A. B. Smolders and M. E. J. Jeuken, ‘‘Broadband MicrostripPhased-Array Antennas on a High-Permittivity Substrate,’’ JINA,Nice, France, 8-10 nov. 1994, pp. 650]653.

3. W. C. Y. Lee, Mobile Communications Engineering, MacGraw-Hill,New York, 1982.

4. H-C. Lin, ‘‘Spatial Correlations in Adaptive Arrays,’’ IEEE Trans.Antennas Propagat., Vol. AP-30, No. 2, 1982, pp. 212]223.

5. J. F. Diouris, J. Saillard, A. C. Tarot, C. Terret, and J. P. Blot,

‘‘Multisensor Receiver for Mobile Communications: An Experi-mental Study,’’ 28th Asilomar Conf. on Signals, Computers andSystems, 1994.

Q 1997 John Wiley & Sons, Inc.CCC 0895-2477r97

FORMULAS FOR THE COMPUTATIONOF THE FAR-FIELD RADIATIONPATTERNS OF RECTANGULARMICROSTRIP ANTENNA ELEMENTSWITH THICK SUBSTRATESMehmet Kara11 Weapons Systems DivisionDefence Science and Technology OrganisationPO Box 1500, Salisbury, SA 5108, Australia

Recei ed 13 Noëmber 1996

ABSTRACT: Formulas based on the two-slot, the caïty, and the surfacecurrent models to compute the far-field radiation patterns in both the Eand H planes for rectangular microstrip antenna elements are studiedand their älidity assessed. Three closed-form formulas are presented forthe calculation of the radiation patterns in the E plane for antennaelements with substrates thicker than those reported elsewhere. These arederi ed by modifying the two-slot model by the use of empirically deri edcorrection factors, taking into account the power radiated in space andsurface waës, the surface-wa e model coefficients, and the line exten-sion. The capability of the deëloped formulas is illustrated by compari-son between the computed results and experimental measurements. Thecomputed results are in good agreement with measurements. The H-planepatterns are computed from the two-slot model andror the caïtymodels. The half-power beamwidth in the E and H planes haë alsobeen determined for each of the antenna elements inëstigated in thiswork. Q 1997 John Wiley & Sons, Inc. Microwave Opt Technol Lett14: 360]367, 1997.

Key words: radiation patterns; far-field patterns; microstrip antennapatterns; rectangular parch antennas

1. INTRODUCTION

Radiation from a microstrip antenna occurs mainly from thefringing fields between the edge of the patch conductor andthe ground plane, as shown in Figure 1. The fields at theedges can be resolved into normal and tangential componentswith respect to the ground plane. The tangential componentsŽ .those parallel to the ground plane are in phase, and arecombined to give a maximum radiated field normal to theground plane.

Several methods have been proposed for the calculation ofthe far-field radiation patterns of microstrip antenna ele-

w xments 1]15, 17]19 . Many of these analysis techniques em-ploy approximations that are valid only for thin substrates.Other more general methods suffer from a lack of computa-tional efficiency, which in practice can restrict their useful-ness because of high computational time and costs.

Based on this observation, this article investigates formu-Ž .las based on the two-slot model TSM , the cavity model

Ž . Ž .CM , and the electric surface current model ESCM meth-ods for calculating the far-field radiation patterns in both theE and the H planes, respectively. This is because of theirapplicability to the pattern analysis of rectangular microstrip

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 14, No. 6, April 20 1997360

Figure 1 Rectangular microstrip antenna element represented astwo radiating slots with dimensional parameters

antenna elements and their procedural simplicity for comput-ing the radiation patterns. Their respective regions of validityin theory and applicability for a given antenna element havealso been established by comparing the measured results withthose obtained from these methods. For antenna elementswith substrates in the range of h F 0.0815l , where h is the0substrate thickness and l is the free-space wavelength, the0TSM and CM have been verified and successfully used tocalculate the far-field radiation patterns in the E and H

w xplanes 13 . However, the author knows of no method avail-able in the literature that calculates the E-plane radiationpattern for antenna elements with thicker substrates. Thereis also no experimental work that has been carried out onantenna elements with substrate ranges considered in thisarticle.

To compute the far-field radiation patterns of rectangularmicrostrip antenna elements, expressions based on the TSMand used for thin antennas were modified with the use of newcorrection factors. These correction factors were derived em-pirically by the use of a curve-fitting method. The correctionfactor expressions explicitly show the dependence of radia-tion patterns on both the space- and surface-wave powers,the surface-wave modal coefficients, and the line extension topatch resonant length. This is because the lines of the fring-ing field do not stop abruptly at the edges of the patch. Thereis a stray field extending beyond the patch edges that can beinterpreted as an electrical lengthening of the patch, whichimplies an amount of stored energy.

The new expressions were used to calculate the radiationpatterns of antenna elements with thick substrates, and ob-tained results were then compared with the patterns obtainedfrom the TSM, the ESCM, and measured results.

Ž .Note that in this work the 1, 0 mode was used, becausein this mode the magnetic currents on the two oppositeradiating edges are constant. The magnetic currents on theremaining two edges suffer a phase reversal, and hence thefields radiated by them largely cancel. The pattern this modeproduces is linearly polarized and has a broadside maximum.

The main aims of this work are to

v Carry out a study on far-field radiation patterns in boththe E and the H planes of rectangular microstrip an-tenna elements with various substrate thicknesses

v Discuss formulas based on the TSM, the CM, and theESCM method for computing these patterns

v Determine the threshold of the applicability and suit-ability of these formulas

v Develop formulas that can determine the patterns forantenna elements with substrates thicker than thosereported elsewhere

v Conduct experiments for radiation patterns in the E andH planes

v Verify the calculated results by comparison with experi-mental measurements

v Study the 3-dB beamwidths of these antenna elementsv Study the effect of the edges of the ground plane to the

radiation patterns

The formulas used to calculate the radiation patterns andtheir modification are discussed briefly in the following sec-tions.

2. ANALYSIS

For antenna elements with substrate thicknesses smaller than0.0815l , the TSM, the CM, and the ESCM methods have0been successfully used to calculate the far-field radiation

w xpatterns in the E and H planes 13 . For thicker substrates,however, modifications to the TSM method have been carriedout.

2.1. Formulas Based on the Two-Slot Model Method. Gener-ally, the fringing fields that give rise to the slot model forpatch antenna radiation occur around the four sides of thepatch. The fringing fields at the top and bottom of a patchŽ .patch-length walls are depicted in Figure 1. These fieldsrepresent the preferred sources of radiation for the antenna,which is vertically polarized. The fields along the side wallsŽ .patch width walls radiate horizontally polarized power andcannot be avoided. Based on this observation, the antennaelement can be regarded as an antenna system equivalent tothe two-slot model shown in Figure 1. The E-plane patternŽ .x-z plane is determined by multiplying the radiated field ofa single slot by an array factor corresponding to the arrange-ment of a two-slot array. The total radiation field in the Eplane of a rectangular microstrip antenna element operating

w xin the TM mode is given by 14 as10

eyjk 0 R k h0Ž .E u N s yjV Wk sin sin uu fs0 0 0 ž /ž /4p R 2

y1k h k L0 0 Ž .= sin u cos sin u , 1ž / ž /2 2

where V is the voltage at either of the two ends of the patch,0k is the free-space wave number, L is the patch length, R is0the distance from the antenna, and u and f are the sphericalcoordinates defined in Figure 1.

The radiation pattern in the E plane is derived from theabove equation as

y1k h k h k L0 0 0Ž .F u s sin sin u sin u cos sin u .E ž / ž / ž /2 2 2Ž .2

Ž .The total radiation field in the H plane y-z plane of therectangular microstrip antenna element operating in the TM10

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 14, No. 6, April 20 1997 361

w xmode is given by 14 as

eyjk 0 R k W0Ž .E u N s yjV Wk sin sin uf usp r2 0 0 ž /ž /4p R 2

y1k W0 Ž .= sin u cos u , 3ž /2

where W is the width of the patch.The expression for the far-field radiation pattern in the H

Ž .plane is derived from 3 as

y1k W k W0 0Ž . Ž .F u s sin sin u sin u cos u . 4H ž / ž /2 2

Ž . Ž .It is of interest that Eqs. 2 and 4 correspond to equa-Ž .tions derived by the transmission line model TLM , when the

w xh is assumed to be zero 15 .

2.2. Formulas Based on the Ca¨ity Model. This model treatsthe region between two parallel conductor planes, consistingof a patch conductor and ground plane, as a cavity boundedby the electric walls and a magnetic wall along the peripheryof the patch. The fields in the antenna element are assumed

w xto be those of this cavity 16 .This cavity will support quasidiscrete TM modes trans-mn

verse to z, where m and n are the mode numbers associatedwith the y and x directions, respectively. The radiating mode

w xof interest is the TM mode, obtained for L s l r2 14 ,10 dwhere l is the wavelength in the dielectric substrate. Fordthe E plane

k h k L0 0 eŽ . Ž .F u , 908 s cos cos u cos sin u , 5f ž / ž /2 2

and for the H plane

k W0 esin sin už /k h 20Ž . Ž .F u , 08 s cos cos u cos u , 6u ž / k W2 0 e

sin už /2

where W and L are the effective dimensions, taking intoe eaccount the fringing fields at the edge of the patch. In thecase of the TM mode, W and L may be approximated as10 e e

follows:

« «' eW eLw Ž .x Ž .W s W 1 q d W 7ae «r

and

Ž .L s L 7be

with

Ž .h 0.164 « y 1rŽ .d W s 0.882 q 2½W «r

« q 1 Wr Ž .q 0.758 q ln q 1.88 . 8ž / 5p« hr

In the case of the TM mode,01

« «' eL eww Ž .x Ž .L s L 1 q d L 9ae «r

and

Ž .W s W , 9be

where « is the relative permittivity of the substrate materialrand « is the effective relative permittivity as a function ofew

w xthe patch width, which can be calculated from 17

Ž .1r210hŽ . Ž . Ž .« s 0.5 « q 1 q « y 1 1 q . 10eW r r ž /W

« is the effective relative permittivity as a function of theeLŽ .patch resonant length, and can be calculated from 10 by

Ž .replacing W with L. The term d L can be calculated fromŽ .8 by replacing W with L.

2.3. Formulas Based on the Electric Surface Current Model.The antenna element is replaced by an assumed surfacecurrent distribution, and the fields are solved by taking intoaccount the presence of the dielectric layer. Calculation takesplace in the Fourier domain. The far field, calculated asymp-totically from this solution, is used to get the radiationpatterns of the antenna element.

The radiation pattern in the E plane is functionally ex-

TABLE 1 Properties of the Antenna Elements with Thick Substrates

Patch L W a h f u ur E HŽ . Ž . Ž . Ž . Ž . Ž . Ž .No. mm mm mm mm GHz « hrl deg. deg.r d

1 15.20 10.00 3.45 4.76 5.820 2.55 0.1475 74 772 19.70 12.00 2.55 6.26 4.660 2.55 0.1553 70 873 27.56 12.56 3.20 9.52 3.580 2.55 0.1814 68 914 34.00 10.80 3.70 12.81 3.150 2.55 0.2148 65 100


w xpressed as 18, 19

p sin u2cos ž /2 «' eW2Ž .F u , f s 0 s « 1 q « cot k h «'Ž .E eW r 0 r 22Ž .sin u y «eW

2 Ž 2 .cos « y sin ur Ž .? , 112 2 2 2 2Ž . '« y sin u q « cos u cot k h « y sin ur r 0 rž /

and for the H plane

pF u , fsH ž /2

Wk sin u sin f02 2s 1 q « cot k h « sin c'Ž .r 0 r ž /2

cos2 u? ,

2 2 2 2Ž . '« y sin u cot k h « y sin u q cos ur 0 rž /Ž .12

where « is the effective relative permittivity as a functioneWŽ .of the patch width, and can be calculated from 10 .

3. FORMULAS BASED ON MODIFICATIONS OF THE TSM

Ž . Ž . Ž .To check the accuracy of Eqs. 2 , 5 , and 11 , in Figures4]7 the E-plane radiation patterns of all antenna elementsgiven in Table 1 have been calculated and plotted, togetherwith those obtained from measurements. As can be seen,neither the TSM nor the ESCM yield an explicit formula forthe patterns for the specific range of substrate thickness, so

Ž .correction factors for Eq. 2 were established empirically.

3.1. Formulas Based on the TSM and Empirically Deri edCorrection Factors. The existence of a dielectric substrateover a conducting ground plane in a microstrip antennaelement can cause the excitation of surface waves along theair-dielectric interface. Such waves propagate without attenu-ation in a direction parallel to the interface, whereas in thenormal direction, they decay exponentially, without propaga-tion. These waves radiate some of their energy as they reachthe microstrip discontinuity, and the electromagnetic fieldsfrom this radiation add to the space-wave fields to contributeto the total radiation fields from the microstrip structure. Asthe substrate thickness is increased, a large number of dis-crete surface-wave modes are also excited in the air-dielectricboundary of the microstrip antenna beyond the TM . The0TM surface mode has a zero cut-off frequency and is0therefore always present. The strength of the excitation dueto the surface wave provides an estimate of the losses in thesubstrate, and hence the efficiency of the antenna, as can beseen in Figures 2 and 3. In this section the effect of surfacewaves on radiation patterns is considered empirically.

3.1.1. Correction factor based on the surface-wave modalcoefficients. The radiation field for the antenna can be deter-

Ž .mined from the field due to radiation, F u , and a newEcorrection factor Ka

Ž . Ž .F s K F u . 13E a E

Figure 2 Variation of P rP , fractions of surface-wave power tos rincident power, as a function of substrate electrical thicknesses of allantenna elements given in Tables 1 and 2

Figure 3 Efficiencies versus substrate electrical thicknesses of allantenna elements given in Tables 1 and 2

Ž . Ž .F u can be calculated from 2 . The correction-factor ex-Epression of best fit for the radiation pattern was derivedempirically as a function of the surface-wave modal coeffi-cients, patch length, and substrate thickness, and is given by

w Ž . Ž . x Ž .K s cos sin k L cos k h sin u , 14a 1 2

where k is the cutoff wave number and k is the wave1 2number in the air region.

For the continuity of the tangential dielectric and mag-netic fields, the propagation constants in both the air and

w xdielectric regions are identical, and can be represented as 12

k2Ž . Ž .tan k h s « 151 r k1

and

2 2 2 Ž . Ž .k q k s k « y 1 . 161 2 0 r


When hrl < 1, k is found by0 1

1r212 2 2 2 Ž . Ž .'k s y« q « « q 4h k « y 1 , 171 r r r 0 rž /'h 2

which is accurate provided that

Ž . Ž .tan k h f k h , 181 1

or for more accurate results the Maclaurin series expansionŽ .can be used for tan hk :1

2 173 5 71Ž . Ž . Ž . Ž .tan k h f k h q k h q k h q k h q ??? .1 1 1 1 13 15 315Ž .19

Ž . Ž .Substituting Eq. 15 into 19 yields for k2

k 2 171 3 5 71 Ž . Ž . Ž . Ž .k s k h q k h q k h q hk . 202 1 1 1 13« 15 315r

3.1.2. Correction factor based on the space- and surface-wavepowers. The radiation field for the antenna can be deter-mined from the field due to radiation and an empiricallyderived correction factor due to surface waves as follows:

Ž . Ž .F s K F u , 21E b E

where K is the correction factor:b

1 Pr Ž .K s cos sin u , 22b (ž /2 Ps

Žwhere P is the power radiated in the space waves directr.main beam power and P is the power radiated in surfaces

waves.w xThe power radiated in the space waves is given by 20

22 Ž .R k k h 1 20 0 0 Ž .P f 1 y q , 23r 2ž /3p « 5«r r

and the power radiated in surface waves is given by

3 3Ž . Ž .k h « y 10 r2 Ž .P f R k , 24s 0 0 34«r

where R s 120p V is the vacuum-medium resistance.0The TM surface mode becomes stronger as substrate0

thickness increases, which reduces the radiation efficiency, ascan be seen in Figures 2 and 3. In Figure 2 efficiencies areplotted as a function of substrate electrical thicknesses for allantenna elements listed in Tables 1 and 2. The ratios of thesubstrate surface-wave power, P , to the incident power as asfunction of substrate electrical thicknesses of all antenna

elements given in Tables 1 and 2 are shown in Figure 3. Ascan be seen, surface-wave power generated by a single ele-ment increases with substrate thickness. Note that this workonly considers the propagation of the dominant lowest-orderTM mode, which is excited for usual values of the nor-0malized substrate thickness hrl and has a zero cutoffdfrequency.

3.2. Formulas Based on the TSM and the Line Extension. Forthis case the antenna element is assumed to be an open-circuit microstrip line. The electric fields at the open end of amicrostrip line are distorted by an abrupt termination at theedges, resulting in fringing electric fields that leak out intothe dielectric surrounding the conductor pattern and also outinto the air region and around the substrate. Because thepatch is modeled as a transmission line of length L, anelectrical lengthening of D L will occur, due to this open-endeffect, at both patch ends. This forms the slots, as shown in

Ž .Figure 1. To improve the accuracy of 2 the line extensionhas been taken into account by replacing the physical patch

Ž .length by L q 2 D L . The resulting formula for the radia-Ž .tion pattern in the E plane x-z plane of rectangular an-

tenna elements with thick substrates can be given as

k h0sin sin u Ž .ž / k L q 2 D L2 0Ž . Ž .F u s cos sin u , 25E k h 20

sin u2

where D L is the edge extension as a function of patch length,w xand can be calculated from 21 by replacing W with L as

follows:

y1Ž .« q 0.300 L LeLD L s 0.412 h q 0.264 q 0.813 .ž / ž /Ž .« y 0.258 h heL

Ž .26

It has been found that formulas based on the TSM andthe CM for calculating the H-plane patterns are valid forrectangular microstrip antenna elements with substrates ofany thickness, as will be shown in the following section.

4. RESULTS

Far-field radiation patterns were measured for angles fromy908 to 908 in both the E and H planes for all antennaelements given in Tables 1 and 2. These antenna elementswere fabricated on PTFE substrate materials of 2.33 F « Fr2.55. Results obtained from the methods described above arecompared with these pattern measurements. All radiationpatterns for both the E and H planes were measured andcalculated at resonant frequency f . Furthermore, in thesertables a denotes the feed point locations of the antennaelements.

The radiation patterns of rectangular microstrip antennaelements are hemispherical, but both the E- and H-plane

TABLE 2 Properties of the Antenna Elements with Thin Substrates

Patch L W a h frŽ . Ž . Ž . Ž . Ž .No. mm mm mm mm GHz « hrlr d

1 19.60 18.10 6.27 1.57 4.805 2.33 0.03842 12.80 11.70 3.40 3.00 6.570 2.50 0.0991


Figure 4 Measured far-field radiation pattern in the E plane at5.820 GHz for antenna element number 1 in Table 1 compared withcalculated results


patterns have a null in the horizontal plane "908. Evidentlyany E-plane radiation pattern in this plane occurs in the formof surface waves.

Figures 4]7 show the measured E-plane radiation pat-terns of thick antenna elements given in Table 1 or of thickantenna elements 4, 7, 9, and 17 of tables published inw x Ž . Ž . Ž .22]25 compared with results obtained from 2 or 5 , 11 ,Ž . Ž . Ž . Ž . Ž .13 , or 21 , and 25 . The patterns obtained by 13 , 21 ,

Ž .and 25 coincide very well with measured values at desiredangular regions below "608 on the E plane. But the results

Ž . Ž . Ž .obtained from 2 or 5 and 11 are not valid for antennaelements with thick substrates. As is shown in these figuresthe E-plane radiation patterns vary significantly with sub-



strate thickness. They are also affected by the surface wavesand the edges of the ground plane.

Figures 8]10 show the selected measured H-plane radia-tion patterns of thick antenna element numbers 4, 7, and 17

w xof tables published in 22]25 compared with results obtainedŽ . Ž . Ž .from 6 and 12 . The patterns obtained by 6 coincide very

Ž .well with measured values, whereas 12 fails to calculatethese pattern accurately.

The calculations were carried out for all antenna elementslisted in Tables 1 and 2. It is therefore understood that theformulas are also applicable to antenna elements on thinsubstrates. Figures 11 and 12 show the measured E-plane


Figure 8 Measured far-field radiation pattern in the H-plane at5.820 GHz for antenna element number 1 in Table 1 compared withcalculated results

Figure 9 Measured far-field radiation pattern in the H plane at4.660 GHz for antenna element number 2 in Table 1 compared withcalculated results

radiation patterns of thin antenna elements given in Table 2Ž . Ž . Ž . Ž .compared with results obtained from 2 or 5 , 21 , and 25 .

Ž . Ž .Note that Eqs. 13 and 21 yield approximately the sameresults.

The half-power beamwidths in the E and H planes of allantenna elements considered for this article have beenmeasured and are listed in the eight and ninth columns ofTable 1.

In summary,

v The agreement between the calculated and measuredresults for patterns in the E plane is very good, exceptfor the noticeable effects of the diffraction at largeangles from the z axis due to the imperfect ground planeused in the measurements.

Figure 10 Measured far-field radiation pattern in the H plane at3.150 GHz for antenna element number 4 in Table 1 compared withcalculated results


v The E-plane patterns of thick antenna elements exhibitvariations below "608 that could be caused by surfacewave and diffraction at the ground plane edges.

v The agreement between the calculated and measuredresults for patterns in the H plane is also good.

v Despite the use of a very thick substrate, the microstripantenna element exhibited satisfactory and nearly sym-metrical E- and H-plane radiation patterns.

5. CONCLUSION

This article has verified the applicability of the TSM, the CM,and the ESCM methods for computing the far-field radiationpatterns in both the E and the H planes of rectangular



microstrip antenna elements. It has also demonstrated thatthe formulas based on the TSM and CM methods are foundto yield the radiation patterns in the E plane of antennaelements with substrates on the order of h F 0.0815l with0

w xgood accuracy 13 , but they become increasingly inaccurateas the substrate thickness is increased. It has also beendemonstrated that the applicability of the formulas based ontwo models for predicting the radiation patterns in the Hplane are not restricted to certain material and substratethicknesses. They are valid for all antenna elements intro-duced in this article.

The inadequate performance of these formulas in predict-ing the radiation patterns in the E plane of antenna elementswith thicker substrates makes it necessary to modify theseformulas to fit the measured results. This has been done byconsidering the TSM method by taking into account the

Ž .surface-wave modal coefficients 13 , the power radiated inŽ .the space wave, the surface wave 21 , and the line extension

Ž .25 . The modified formulas give the radiation patterns in theE plane for antenna elements with thin and thick substrateswith good accuracy, as is evident from the close agreementbetween the calculated and measured results. Measurementsconfirmed the calculated results for all antenna elementslisted in Tables 1 and 2.

The effect of substrate thickness on the far-field radiationpatterns for rectangular microstrip antenna elements hasbeen shown to be quite significant. The characteristic of thepattern is changed for different substrate thicknesses.

The relationship between the half-power beamwidth inthe E and H planes and the substrate thicknesses has alsobeen given.

REFERENCES

1. K. Chang, Handbook of Microwa e and Optical Components, JohnWiley and Sons, New York, 1989, Vol. 1.

Ž .2. J. R. James and P. S. Hall Eds. , Handbook of Microstrip Anten-Ž .nas, Peter Peregrinus IEE , London, 1989, Vols. 1 and 2.

3. P. Bhartia, K. Rao, and R. Tomar, Millimeter-Wa e Microstrip andPrinted Circuit Antennas, Artech House, Dedham, MA, 1991.

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15. I. J. Bahl and P. Bhartia, Microstrip Antennas, Artech House,Dedham, MA, 1980.

16. K. R. Carver, ‘‘Practical Analytical Technique for the MicrowaveAntenna,’’ in Proceedings of the Workshop on Printed CircuitAntenna Technology, New Mexico State University, Las Cruces,October 1979, pp. 7.1]7.20.

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19. J. Ashkenazy, S. Shtrikman, and D. Treves, ‘‘Electric SurfaceCurrent Model for the Analysis of Microstrip Antennas onCylindrical Bodies,’’ IEEE Trans. Antennas Propagat., Vol. AP-33,March 1985, pp. 295]300.

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Q 1997 John Wiley & Sons, Inc.CCC 0895-2477r97


formulas for the computation of the far-field radiation patterns of rectangular microstrip antenna...

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