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Improved formulae for the resonant frequencies of triangular microstrippatch antennasMelad M. Olaimata; Nihad I. Diba

a Department of Electrical Engineering, Jordan University of Science and Technology, Irbid, Jordan

Online publication date: 10 March 2011

To cite this Article Olaimat, Melad M. and Dib, Nihad I.(2011) 'Improved formulae for the resonant frequencies oftriangular microstrip patch antennas', International Journal of Electronics, 98: 3, 407 — 424To link to this Article: DOI: 10.1080/00207217.2010.547811URL: http://dx.doi.org/10.1080/00207217.2010.547811

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http://dx.doi.org/10.1080/00207217.2010.547811

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Improved formulae for the resonant frequencies of triangular microstrip

patch antennas

Melad M. Olaimat and Nihad I. Dib*

Department of Electrical Engineering, Jordan University of Science and Technology,P. O. Box 3030, Irbid, Jordan

(Received 10 June 2010; final version received 23 October 2010)

In this article, an improved formula for the effective side length (aeff) of theequilateral triangular patch antenna is presented. The computed resonantfrequencies for the first five modes, using this formula, are compared with theavailable experimental and theoretical results in the literature. It is shown that theproposed formula has better accuracy than other available expressions. More-over, the 308–608–908 and 308–308–1208 triangular patch antennas are studied.Specifically, improved formulae for the effective dielectric constant of thesepatches are derived.

Keywords: microstrip antennas; triangular microstrip antennas; resonantfrequency of patch antennas

1. Introduction

Microstrip antennas (MAs) consist of a patch of metallisation on a groundedsubstrate (Garg, Bhartia, Bahl and Ittipiboon 2001). It is well-known that MAs areof very narrow band; about 7% for a typical bandwidth. Therefore, it is important toaccurately determine the resonant frequency of a MA (Gang 1989). The cavity modelis one of the most widely used methods in the analysis of MAs. In this model, theregion between the conducting patch and the ground plane is assumed to be aresonant cavity bounded by electric walls on the top and the bottom and by magneticwalls on the sides (Garg et al. 2001). However, the theoretical resonant frequenciesobtained from the assumption of a perfect magnetic wall must be corrected if goodagreement between theoretical and experimental values is to be obtained.

Among the shapes that have attracted much attention lately is the triangularmicrostrip patch antenna (Helszajn and James 1978; Bahl and Bhartia 1980; Daheleand Lee 1987; Garg and Long 1988; Chen, Lee and Dahele 1992; Karaboga, Guney,Kaplan and Akdagli 1998; Guha and Siddiqui 2004; Nasimuddin and Verma 2005;Kimothi, Saxena, Saini and Bhatnagar 2008). This is due to their small sizecompared with other shapes like the rectangular and circular patch antennas. In thisarticle, improved formulae for the calculation of the resonant frequencies of threedifferent shapes of triangular patch antennas are presented. Specifically, theequilateral, the 308–608–908 and the 308–308–1208 triangular patch antennas are

*Corresponding author. Email: [email protected]

International Journal of Electronics

Vol. 98, No. 3, March 2011, 407–424

ISSN 0020-7217 print/ISSN 1362-3060 online

� 2011 Taylor & Francis

DOI: 10.1080/00207217.2010.547811

http://www.informaworld.com

Downloaded By: [Dib, Nihad] At: 08:51 13 March 2011

studied. This article is divided into three sections: section 2 is devoted to theequilateral triangular patch antenna (ETPA), in which an improved formula isproposed to calculate the effective side length (aeff) of this patch. Sections 3 and 4 aredevoted to the 308–608–908 and the 308–308–1208 triangular patch antennas,respectively, in which improved formulae for the effective dielectric constants ofthese patches are derived.

2. Equilateral triangular patch antenna

2.1. Introduction

Figure 1 shows the geometry of a probe-fed ETPA. Based on the cavity model, theresonant frequencies of an ETPA of side length a can be calculated using thefollowing formula (Helszajn and James 1978; Bahl and Bhartia 1980; Dahele andLee 1987; Garg and Long 1988; Chen et al. 1992; Karaboga et al. 1998; Guha andSiddiqui 2004; Nasimuddin and Verma 2005):

fm;n ¼2c

3affiffiffiffiffi2rp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 þmnþ n2

pð1Þ

where c is the speed of light in free space, and 2r is the dielectric constant of thesubstrate.

To account for the non-perfect magnetic wall assumption, two opinions appearedin the literature. One of them suggested replacing the side length a with an effectiveside length aeff as follows (Helszajn and James 1978):

aeff ¼ aþ hffiffiffiffiffi2rp ð2Þ

The second opinion suggested that besides using an effective side length, aneffective dielectric constant needs to be used which is given as follows (Bahl andBhartia 1980):

2eff ¼2r þ1ð Þ

2þ 2r �1ð Þ

41þ 12h

a

� ��12

ð3Þ

Figure 1. Geometry of an ETPA.

408 M.M. Olaimat and N.I. Dib


In an attempt to determine the proper correction factor for the ETPA,measurements were performed by Dahele and Lee (1987) and Chen et al. (1992),which showed that using Equation (2) gives better results compared to theexperimental ones. Equation (2), along with Equation (1), was used by Dahele andLee (1987) to calculate the resonant frequencies of the first five modes. It should benoted that Equation (2) gives the same effective side length for all modes ofoperation. However, since each mode has its own field distribution, it is believedthat different modes could have somewhat different effective side lengths. Thisnecessitates the derivation of a formula for the effective side length that is differentfrom Equation (2). In the next sub-section the mode numbers (m and n) areexploited and inserted in the calculation of the resonant frequency. As aconsequence, a better agreement with the experimental results (Dahele and Lee1987; Chen et al. 1992) is obtained.

2.2. Formulation

A number of methods (Helszajn and James 1978; Bahl and Bhartia 1980; Daheleand Lee 1987; Garg and Long 1988; Gang 1989; Chen et al. 1992; Karaboga et al.1998; Guha and Siddiqui 2004; Nasimuddin and Verma 2005) are available todetermine the resonant frequency of an ETPA. The theoretical resonant frequencyvalues obtained using these methods rely on the assumption that all modes havethe same effective side length. As indicated above, it is believed that each mode hasits own aeff. This is clear from the measured values reported by Dahele and Lee(1987) and Chen et al. (1992) and shown here in Tables 1–3 as fme. For example,using these measured values, the effective side lengths (in cm) for the first fivemodes (TM10, TM11, TM20, TM21 and TM30) are 10.26, 10.14, 10.23, 10.22 and10.3, respectively in Table 1; 8.82, 8.76, 8.85, 8.76 and 8.87, respectively in Table 2;and 4.06, 4.05, 4.12, 4.11 and 4.17, respectively in Table 3. These values arecalculated using Equation (1) by replacing the physical side length a with aneffective one aeff. This observation implies that the patch has somewhat differentvalues for the effective side length (aeff) according to the mode of operation. So,using curve fitting with the measured values, a new formula is proposed here forthe effective side length in which the side length varies with the mode indices (mand n) as follows:

aeff ¼ aþ 2ffiffiffiffimp hffiffiffiffiffi

2rp � 2

ffiffiffiffiffiffiffimnp hffiffiffiffiffi

2rp þ 197:1

nh2

a� 2rð4Þ

It should be noted that this formula is obtained using simple curve fitting of themeasured results (Dahele and Lee 1987; Chen et al. 1992) without the need to use anyoptimisation technique as in Karaboga et al. (1998).

2.3. Results and discussion

In this sub-section, a comparison between the computed values for the resonantfrequencies of several ETPAs using Equation (4), along with Equation (1), and otherresults reported in the literature is presented. Tables 1–3 show such a comparison, inwhich the measured values (Dahele and Lee 1987; Chen et al. 1992) are taken as the

International Journal of Electronics 409


Table

1.

Comparisonbetweenthemeasuredandcalculatedresonantfrequencies

(inMHz)

ofanETPA

witha¼

10cm

,h¼

0.159cm

,and2 r¼

2.32.

Mode

f me

(Daheleand

Lee

1987)

f pre

f ga

(Gang

1989)

f hj

(Helszajn

and

James

1978)

f gu

(Karaboga

etal.1998)

f bb

(Bahland

Bhartia

1980)

f MoM

(Chen

etal.

1992)

f gl

(Garg

and

Long1988)

f na

(Nasimuddin

and

Verma2005)

f guh

(Guhaand

Siddiqui2004)

TM1,0

1280

1286

1340

1299

1281

1413

1288

1273

1281

1285

TM1,1

2242

2226

2320

2251

2218

2447

2259

2206

2219

2226

TM2,0

2550

2551

2679

2599

2562

2826

2610

2547

2562

2570

TM2,1

3400

3400

3544

3438

3389

3738

3454

3369

3390

3400

TM3,0

3824

3802

4019

3898

3842

4239

3875

3820

3844

3855

Error

!45

606

189

66

1367

190

81

66

72

Note:Totalabsolute

errorisincluded.



Table

2.

Comparisonbetweenmeasuredandcalculatedresonantfrequencies

(inMHz)

ofanETPA

witha¼

8.7

cm,h¼

0.078cm

,and2 r¼

2.32.

Mode

f me

(Chen

etal.

1992)

f pre

f ga

(Gang

1989)

f hj

(Helszajn

and

James

1978)

f gu

(Karaboga

etal.1998)

f bb

(Bahland

Bhartia

1980)

f MoM

(Chen

etal.

1992)

f gl

(Garg

and

Long1988)

f na

(Nasimuddin

and

Verma2005)

TM1,0

1489

1492

1532

1500

1488

1627

1498

1480

1490

TM1,1

2596

2596

2654

2599

2577

2818

2608

2564

2581

TM2,0

2969

2969

3065

3001

2976

3254

2990

2961

2980

TM2,1

3968

3967

4054

3970

3937

4304

3977

3917

3942

TM3,0

4443

4437

4597

4501

4464

4880

4480

4441

4469

Error

!10

437

106

79

1418

88

102

79

Note:Totalabsolute

errorisincluded.



Table

3.

Comparisonbetweenmeasuredandcalculatedresonantfrequencies

(inMHz)

ofanETPA

witha¼

4.1

cm,h¼

0.07cm

,and2 r¼

10.5.

Mode

f me

(Chen

etal.

1992)

f pre

f ga

(Gang

1989)

f hj

(Helszajn

and

James

1978)

f gu

(Karaboga

etal.

1998)

f bb

(Bahland

Bhartia

1980)

f MoM

(Chen

etal.

1992)

f gl

(Garg

and

Long1988)

f na

(Nasimuddin

and

Verma2005)

f guh

(Guhaand

Siddiqui2004)

TM1,0

1519

1490

1577

1498

1501

1725

1522

1494

1519

1516

TM1,1

2637

2593

2731

2594

2600

2988

2654

2588

2630

2626

TM2,0

2995

2967

3153

2995

3002

3450

3025

2989

3037

3032

TM2,1

3973

3961

4172

3962

3971

4564

4038

3954

4018

4011

TM3,0

4439

4435

4730

4493

4503

5175

4518

4483

4556

4548

Error

!117

800

129

128

2339

194

143

211

198

Note:Totalabsolute

errorisincluded.



reference values to compute the total absolute error. The entries of fme, fpre, fga, fhj,fgu, fbb, fMoM, fgl, fna, and fguh represent, respectively, the values measured (Daheleand Lee 1987; Chen et al. 1992), calculated by the present method, calculated byGang (Gang 1989), calculated by Helszajn and James (1978), calculated byKaraboga et al. (1998), calculated by Bahl and Bhartia (1980), calculated by usingmethod of moments (Chen et al. 1992), calculated by Garg and Long (1988),calculated by Nasimuddin and Verma (2005) and calculated by Guha and Siddiqui(2004).

As shown in Table 1, the present formula gives better results, and its valuesare the closest to the measured values, with a total absolute difference of 45 MHz.Tables 2 and 3 show that the absolute differences between the resultsobtained using the present formula and the measured values reported byChen et al. (1992) are only 10 MHz and 117 MHz, respectively. Table 4 showsthe total absolute error (that is the sum of the absolute errors in Tables 1–3)between the measured and the calculated resonant frequencies. The resultsobtained by the proposed formula are in very good agreement with themeasured values, and are better than the results obtained by other methodsreported in literature. About 36.3% error reduction is obtained using the presentformula for the effective side length, compared with the best results that wereobtained by Karaboga et al. (1998). Regarding the results taken from (Guha andSiddiqui 2004), only the errors obtained from the designs in Tables 1 and 3 areincluded in Table 4 since the results for the design in Table 2 were notincluded in Guha and Siddiqui (2004). It is anticipated that the effective sidelength, aeff, expression proposed in this study, has good accuracy in the range of2.32 5 2r 5 10.6 and 0.005 5(h/ld) 50.034, where ld is the wavelength in thesubstrate.

3. Triangular patch with 308–608–908 angles

Figure 2 shows the geometry of the 308–608–908 triangular patch antenna. Recently,this triangular patch antenna was briefly studied by Kimothi et al. (2008). The 308–608–908 triangular waveguide and cavity have been theoretically treated by Huang(1999), Overfelt and White (1986), Zhang and Fu (1991), in which a formula tocalculate the cutoff wave-number for these structures has been presented. Thisformula is used here to calculate the resonant frequencies of the 308–608–908triangular patch antenna.

The formula for the cutoff wave-number for the 308–608–908 triangularwaveguide is given as follows (Overfelt and White 1986; Zhang and Fu 1991;Huang 1999):

kc ¼2p

affiffiffi3p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 þmnþ n2ð Þ

qð5Þ

Thus, the resonant frequencies of the corresponding triangular patch can becalculated by the following formula:

fr ¼ckc

2pffiffiffiffiffi2rp ¼ c

affiffiffiffiffiffiffiffiffi3 2rp


q: ð6Þ



Table

4.

Totalabsolute

errors

betweenthemeasuredandcalculatedresonantfrequencies.

Mode

f me

(Daheleand

Lee

1987)

f pre

f ga

(Gang

1989)

f hj

(Helszajn

and

James

1978)

f gu

(Karaboga

etal.1998)

f bb

(Bahland

Bhartia

1980)

f MoM

(Chen

etal.

1992)

f gl

(Garg

and

Long1988)

f na

(Nasimuddin

and

Verma2005)

f guh

(Guhaand

Siddiqui2004)

Error

!172

1843

424

273

5124

472

326

356

270



The product (fr.a) as a function of 2r is shown in Figure 3 for various values of mand n. Such curves can be used for initial design purposes. The lowest order resonantfrequency corresponding to the dominant TM10 mode is:

f1;0 ¼c

affiffiffiffiffiffiffiffiffi3 2rp ð7Þ

Tables 5–9 show a comparison between the resonant frequencies obtained usingEquation (6) and those obtained by the full-wave IE3D simulator (Version 14.0,www.zeland.com) for several 308–608–908 triangular patch antennas with differentdimensions and substrates. In these tables, IE3D results are taken as the reference tocompute the relative error of the computed resonant frequencies. For illustrationpurposes, Figures 4 and 5 show the simulated reflection coefficient (in dB) of the 308–608–908 triangular patch antennas with the dimensions mentioned in Tables 5 and 6,respectively. As shown in these figures, the reflection coefficient has different valuesfor different modes. This depends on the feeding position which can be chosen suchthat a specific mode has better reflection coefficient (i.e. better matching) than othermodes. For example, the feeding position used in Figure 4 gives a return loss betterthan 10 dB for both TM20 and TM30 modes.

Figure 2. A probe-fed 308–608–908 triangular patch antenna.

Figure 3. Variation of (fr.a) with 2r for various modes of the 308–608–908 triangular patch.



http://www.zeland.com)

In Tables 5 and 6, where the relative permittivity (2r) is low (54), all calculatedresults are larger than those obtained by simulation. On the other hand, in Tables 7–9, where the relative permittivity is high (44), all calculated results are less thanthose obtained by simulation. Since the effective permittivity (2eff) is always less thanthe relative permittivity, and the resonant frequency is inversely proportional to the

Table 5. Calculated and simulated resonant frequencies in (MHz) of a 308–608–908triangular patch antenna with a ¼ 100 mm, h ¼ 1.59 mm, and 2r ¼ 2.32, fed at (60, 10) mm.

Mode IE3D Calculated Relative error

TM1,0 1127 1137 þ10TM1,1 1937 1969 þ32TM2,0 2237 2274 þ37TM2,1 2995 3008 þ13TM3,0 3339 3411 þ72Error ! þ164



TM1,0 1298 1307 þ9TM1,1 2241 2264 þ23TM2,0 2580 2614 þ34TM2,1 3442 3458 þ16TM3,0 3880 3921 þ41Error ! þ123

Table 7. Calculated and simulated resonant frequencies in (MHz) of a 308–608–908triangular patch antenna with a ¼ 70 mm, h ¼ 1 mm, and 2r ¼ 6, fed at (42, 8) mm.


TM1,0 1017 1010 77TM1,1 1774 1750 724TM2,0 2038 2020 718TM2,1 2698 2673 725TM3,0 3040 3030 710Error ! 784






relative permittivity, as shown in Equation (6), replacing relative permittivity (2r)with an effective relative permittivity (2eff) for the designs with high relativepermittivity could enhance the calculated results. To the best of our knowledge, sucha formula for the effective relative permittivity for the 308–608–908 triangular patchantenna is not available in the literature. A formula is derived below.

Using the approach presented by Mythili and Das (1998), the effectivepermittivity (2eff) for any microstrip patch antenna can be written in terms of

Table 10. Calculated and simulated resonant frequencies in (MHz) of a 308–608–908triangular patch antenna with a ¼ 70 mm, h ¼ 1 mm, and 2r ¼ 6, fed at (42, 8) mm.

Mode IE3D cn ¼ 0 cn ¼ 0.2 cn ¼ 0.223 cn ¼ 0.25 cn ¼ 0.279 cn ¼ 0.3

TM1,0 1017 1010 1017 1018 1019 1020 1021TM1,1 1774 1750 1761 1763 1765 1767 1768TM2,0 2038 2020 2034 2036 2038 2040 2042TM2,1 2698 2673 2691 2693 2696 2699 2701TM3,0 3040 3030 3051 2054 3057 3060 3063Error ! 84 35 33 30 33 40

Note: Total absolute error is included.




Figure 4. Simulated reflection coefficient (in dB) of a 308–608–908 triangular patch antennawith a ¼ 100 mm, h ¼ 1.59 mm, 2r ¼ 2.32, fed at (60, 10) mm.



a function F, which depends on the relative permittivity and the ratio of the aperturedimensions to substrate thickness of the MA as follows:

2eff ¼2r þ 1

2þ 2r � 1

2F 0 < F < 1 ð8Þ

where

F ¼ 1� 2r cn2r � 1� Fringing area on the plane of the patch

area of the microstrip patchð9Þ

and (cn 4 0) is a coefficient to be determined to find F.The fringing area for the 308–608–908 triangular patch antenna is shown as a

shaded region in Figure 6 where it is assumed that the effective area extends by adistance of h (substrate thickness) over the physical plane of the patch (Mythili andDas 1998). Using simple geometry, the following function can be derived for thefunction F:

F ¼ 1� 2r cn2r � 1�h2 6:4641þ 2:732 a

h

� �a2

2ffiffi3p

� � ð10Þ

Figure 5. Simulated reflection coefficient (in dB) of a 308–608–908 triangular patch antennawith a ¼ 87 mm, h ¼ 0.78 mm, 2r ¼ 2.32, fed at (50, 10) mm.

Figure 6. Fringing area of the 308–608–908 triangular patch antenna.



Using Equations (8)–(10), we obtain:

2eff ¼ 2r � p � cn ð11Þ

where

p ¼ 2r �h2 6:4641þ 2:732 a

h

� �a2ffiffi3p� � ð12Þ

For the sake of comparison, these formulae are applied to the designs with highrelative permittivity (i.e. the designs in Tables 7 and 9), and the obtained results areshown in Tables 10 and 11. In these tables, the given error is the total absolutedifference between the calculated frequencies and the simulated ones. For the designin Table 10, p ¼ 0.42. It can be seen that all the results obtained using 2eff are betterthan those calculated without using 2eff (i.e. cn ¼ 0). The best results are obtainedwith cn ¼ 0.25, which gives an effective relative permittivity of 2eff ¼ 5.9. For thedesign in Table 11, p ¼ 0.88. Again, all obtained results using 2eff are better thanthose calculated without using 2eff. The best results are obtained with cn around 0.33.With this value, the effective permittivity becomes 2eff ¼ 10.21, and about 74%absolute error reduction is obtained compared with the results obtained without theuse of an effective permittivity (cn ¼ 0).

4. Triangular patch antenna with 308–308–1208 angles

To the best of our knowledge, this is the first time that this patch shape has beenstudied. Figure 7 shows the geometry of the 308–308–1208 triangular patch antenna.The 308–308–1208 triangular waveguide has been treated by Overfelt and White


Mode IE3D cn ¼ 0 cn ¼ 0.25 cn ¼ 0.3 cn ¼ 0.33 cn ¼ 0.35 cn ¼ 0.4

TM1,0 1332 1304 1318 1320 1322 1323 1326TM1,1 2297 2258 2283 2286 2290 2292 2297TM2,0 2644 2607 2636 2640 2644 2646 2652TM2,1 3504 3449 3487 3492 3498 3500 3508TM3,0 3940 3911 3954 3960 3966 3969 3978Error ! 188 67 59 49 49 56


Table 12. Calculated and simulated resonant frequencies in (MHz) of a 308–308–1208triangular patch antenna with a ¼ 100 mm, h ¼ 1.59 mm, and 2r ¼ 2.32, fed at (0, 0.6) mm.


TM1,0 2263 2274 þ11TM1,1 3846 3939 þ93TM2,0 4600 4549 751TM2,1 5949 6017 þ68Error ! þ121



(1986) in which a formula to calculate the cutoff wave-number for such a waveguidewas presented. This formula is used here to calculate the resonant frequencies of the308–308–1208 triangular microstrip patch antenna.

The formula for the cutoff wave-number for the 308–308–1208 triangularwaveguide is given as follows (Overfelt and White 1986; Zhang and Fu 1991):

kc ¼4pffiffiffiffiffi3ap


qð13Þ

Thus, the resonant frequencies can be calculated using the following formula:

fr ¼ckc

2pffiffiffiffiffi2rp ¼ 2c

affiffiffiffiffiffiffiffiffi3 2rp


qð14Þ

As expected, the resonant frequency of the 308–308–1208 triangular patch antenna isdouble the resonant frequency of the 308–608–908 triangular patch antenna. Figure 8shows a plot of the product (fr � a) as a function of 2r for the first four modes. Thelowest order resonant frequency is

f1:0 ¼2c

affiffiffiffiffiffiffiffiffi3 2rp ð15Þ

Tables 12–15 show the resonant frequencies calculated using Equation (14) ascompared to the simulated ones for several 308–308–1208 triangular patch antennas

Figure 8. Variation of (fr.a) with 2r for various modes of the 308–308–1208 triangular patchantenna.

Figure 7. A probe-fed 308–308–1208 triangular patch antenna.



with different dimensions and substrates. As shown in Tables 12 and 13, where2r ¼ 2.32, the total relative error is positive which means that the calculated valuesare, on average, greater than the values obtained by simulation. In contrast, the totalrelative error is negative in Tables 14 (2r ¼ 4.6) and 15 (2r ¼ 10.5), which indicatesthat the calculated values are, on average, less than the values obtained bysimulation. As was done with the 308–608–908 triangular patch, it will be shown that



TM1,0 2609 2614 þ5TM1,1 4480 4528 þ48TM2,0 5259 5228 731TM2,1 6906 6916 þ10Error ! þ32



TM1,0 2337 2307 730TM1,1 3935 3994 þ59TM2,0 4700 4615 785TM2,1 6098 6105 þ7Error ! 749



TM1,0 2698 2607 791TM1,1 4523 4516 77TM2,0 5273 5215 758TM2,1 7048 6899 7149Error ! 7305


Mode IE3D cn ¼ 0 cn ¼ 0.2 cn ¼ 0.23 cn ¼ 0.28 cn ¼ 0.29

TM1,0 2698 2607 2644 2649 2659 2660TM1,1 4523 4516 4580 4588 4606 4608TM2,0 5273 5215 5288 5298 5318 5321TM2,1 7048 6899 6995 7009 7036 7039Error ! 305 182 178 179 180




if the relative permittivity is replaced by an effective one, in the designs with highrelative permittivity the difference between the calculated results and the simulatedones gets smaller. For illustration purposes, Figure 9 shows the frequency responsefor the patch in Table 12.

For the 308–308–1208 triangular patch antenna, the fringing area is shown as ashaded region in Figure 10. Using simple geometry, the following function can bederived for the function F:

F ¼ 1� 2r cn2r�1�h2 2:155 a

hþ 9:1962� �

a2

4ffiffi3p

� � ð16Þ

Using Equations (8), (9), and (16), we obtain:

2eff ¼ 2r �pcn: ð17Þ

where

p ¼ 2r �h2 2:155 a

hþ 9:1962� �

a2

2ffiffi3p

� � ð18Þ

Figure 9. Simulated reflection coefficient (in dB) of a 308–308–1208 triangular patch antennawith a ¼ 100 mm, h ¼ 1.59 mm, and 2r ¼ 2.32, fed at (0, 0.6) mm.

Figure 10. Fringing area of the 308–308–1208 triangular patch antenna.



These formulae are applied to the design in Table 15, and the obtained results areshown in Table 16. For this design, p ¼ 1.436. All obtained results using 2eff arebetter than those calculated without using 2eff. The best results are obtained withcn ¼ 0.23 which gives an effective relative permittivity of 2eff ¼ 10.21. As shown inthe table, the error obtained at this value is the minimum and equals 178 MHz.Using an effective dielectric constant; a 41.6% absolute error reduction is obtained.

5. Conclusions

In this article, several shapes of triangular patch antennas were studied. A newexpression for the effective side length was proposed for the resonant frequency of anETPA. The mode indices were taken into account to calculate the effective sidelength of the patch. The very good agreement between the measured values(available in the literature) and our computed resonant frequencies supports thevalidity of the derived formula for this shape. In addition, the resonant frequenciesfor the 308–608–908 and 308–308–1208 triangular patch antennas were studied. Newformulae for the effective dielectric constant of both antennas were presented.Comparison between the calculated and full-wave simulated results was performed,and a very good agreement was obtained which supports the validity of the derivedformulae for these patches.

References

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Nasimuddin, K.E., and Verma, A.K. (2005), ‘Resonant Frequency of an EquilateralTriangular Microstrip Antenna’, Microwave and Optical Technology Letters, 47, 485–489.

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