_ - - l . . . . I . _ . . I . . . . i . . . . I . . . . I 0 100 200 300 400 500

Rdse (O)

Figure 2 w as a parameter

R,,,/KR,, as a function of K,, assuming the gate width

0 1000 2000 3000 4000 5000 K m

(a)

&./KR, 5

4.5

4

3.5

3

2.5

2

1.5

0 lo00 2000 3000 4000 5000 Kr.3

(b)

Figure 3 g , / K g n , as a function of Rdse assuming the gate width w as a parameter. (a) 50 p m < w < 200 pm; (b) 200 p m < w < 500 pm

single curves, in Figure 3(a) M.’ is considered in the range 50-200 p m and in Figure 3(b) w is considered in the range 200-500 pm.

Once the term K , is calculated for a given technological process and the FET gate width w is chosen, the resistor Rdse is derived from Figure 2 and the term g, is derived from Figure 3. Finally, the terms C,,,, CdJp and R,, are easily computed from Eqs. (191, (201, and (21).

It is clear that if the proposed set of equations and graphics is used, the circuit elements of the design-oriented FET model can be easily calculated, no matter the technolog- ical process adopted, for a FET of given dimensions.

CONCLUSIONS A new formulation of the design-oriented FET model was presented. The advantages of this formulation are the pro- cess-independent nature of the equations and the equations’ dependence from the gate width of the FET. Furthermore, the introduction of suitable graphics allows fast computation of the elements of the design-oriented FET model.

REFERENCES I .

2.

3.

4.

5.

6.

K. B. Niclas, W. T. Wilser, T. R. Kritzer, and R. R. Pereira, “On Theory and Performance of Solid-state Microwave Distributed Amplifiers,” IEEE Trans. Microwave Theory Tech., Vol. MlT-31, June 1983, p. 435. J. B. Beyer, S. N. Prasad, R. C. Becker, J. E. Nordman, and G. K. Hohenwarter, “MESFET Distributed Amplifier Guidelines,” IEEE Trans. Microwaue Theory Tech., Vol. MTT-32, March 1984, pp. 268-275. R. C. Becker and J. B. Beyer, “On Gain-Bandwidth Product for Distributed Amplifiers,” IEEE Trans. Microwacje Theory Tech., Vol. MTT-34, June 1986, pp. 736-738. M. Ross and R. G. Harrison, “Optimization of Distributed Mono- lithic GaAs Amplifiers Using an Analytical/Graphical Technique,” in 1988 IEEE MTT-S bit. Microwaue Symp. Digest, June 1989.

C. Paoloni and S. Kosslowsky, ”Application of Filter Theory in the Design of TWAs Based on FETs with Different Gate Widths,” MicrowaLie Opt. Techno]. Lett., Vol. 6, No. 4, 1993, pp. 261-266. C. Paoloni and S . D’Agostino, “An Approach to Distributed Amplifier Based on a Design-Oriented FET Model,” IEEE Trans. Microwaiie Theory Tech., Vol. MTT-43, No. 2, 1995, pp. 272-271.

pp. 379-382.

Received 6-6-96

Microwave and Optical Technology Letters, 13/4, 219-221 0 1996 John Wiley & Sons, Inc. CCC 0895-2477/96

CALCULATION OF THE RADIATION PATTERNS OF RECTANGULAR MICROSTRIP ANTENNA ELEMENTS WITH VARIOUS SUBSTRATE THICKNESSES Mehrnet Kara Weapons Systems Division Defence Science and Technology Organization P 0. Box 1500 Salisbury SA 51 08, Australia

KEY TERMS Radiation patterns, far-field patterns, patterns, microstrip antennas, patch antennas

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 13, No. 4, November 1996 221

ABSTRACT Formulas based on the two-slot, the ca l iy , and the electric sicrfnce current models to determine the far-field radiation patterns in both the E and H planes of rectangclar antenna elements h a i ~ been studied and their r~alldify has been assessed. Their c,ariutions were expentnentally r'erified by unuiyzing a set of newly designed rectangular microstrip antenna elements with r.anous substrate thicknesses. C 1446 Johri Wilqv & Sons. Inc.

1. INTRODUCTION

In the close proximity of an antenna, the radiation fields exhibit complex characteristics, because reactive components are present in addition to the radiated field. The reactive components vanish and only the radiated field remains in the far-field region when moving far enough away from the antenna. The distance R between the calibrated reference antenna and the antenna under test must satisfy the condi- tion R 2 2 D 2 / A o , where D is the largest dimension of the antenna and A,, is thc free-space wavelength. In addition, the reactive field may be considered to bc that region immedi- ately surrounding the antcnna, and the radiating field is that region, beyond the reactive field, in which the radiation pattern is dependent upon R [I]. All antenna elements con- sidered for this work are linearly polarized and the E- and H-plane radiation patterns are measured in the far-field region.

Radiation from a microstrip antenna occurs mainly from the fringing fields between the edge of the patch conductor and the ground plane. The fields at the edges can be resolved into normal and tangential components with respect to the ground plane. The tangential componcnts (those parallel to the ground plane) are in phasc. and are combined to give a maximum radiated field normal to the ground plane.

Several analysis techniques havc been proposed for the determination of the far-field radiation patterns of microstrip antenna elements [ I - 181. Many of these analysis techniques employ approximations that are valid only for thin substrates. Other more general approaches suffer from a lack of compu- tational efficiency, which in practice can restrict their useful- ness because of high computational time and costs.

This article investigates formulas based on the two-slot model, the cavity model, and the electric surface current model methods for calculating the far-field radiation patterns in both the E and H planes. This is because the formulas are applicable to the pattern analysis of rectangular microstrip antenna elements and they arc procedurally simplc for com- puting the radiation patterns. Their respective regions of validity in theory and applicability for a given antenna ele- ment have also becn established by comparing the measured results with those obtained from formulas based on these models.

Note that in this work the ( 1 , 0) mode has been used. because in this mode the magnetic currents on the two

opposite radiating edges are constant. The pattern this mode produces is linearly polarized and has a broadside maximum. The magnetic currents on the remaining two edges suffer a phase reversal. and hence the fields radiated by them largely cancel.

2. EXPERIMENTAL PROCEDURES

The far-field radiation patterns in the E and the H planes were measured for rectangular microstrip antenna elements fabricated on PTFE (polytetrafluoroethylene) woven glass laminate material ( E , = 2.50 and 2.55 and loss tangent is 0.002) and Duroid microwave substrate. which is made of a glass microfiber-reinforced PTFE composite ( E , = 2.22 and loss tangent is 0.001; see Table 1). The substrate thicknesses were varied from 0.0071Ad to 0.1039Ad (0.17 mm I h 5 3 mm). The configuration of a probe-fed rectangular mi- crostrip antenna element with dimensional parameters is shown in Figure 1. The radiation-pattcrn measurement re- quires two antennas: one to transmit thc signal, the other to receive it. The antenna under test acts as a receiving antenna and is mounted on a large 183-cm-diameter circular alu- minum ground plane in a microwave anechoic chamber.

All of the antenna elements considered for this work are fcd by a standard 5 0 4 coaxial line. The APC-7 connector was attached to the back side of the printed circuit board, with its central conductor passing through the ground plane and the dielectric substrate and connecting to a suitable position on the patch.

The test path of the automatic microwave network ana- lyzer system was then used to measure both the E- and the H-plane radiation patterns at the polar coordinates of the microstrip test antenna element. This was carried out by determining the magnitude of the scattering transfer coeffi- cient S,, for the free-space path loss between the radiating test antenna clement and the receiving horn. In addition, the dimensions of the substrate material on which the patch was printed. are 100 X 100 mm. Absorbing material is placed around the perimeter of the ground plane to reduce diffrac-

2 t, P(R.e.0)

GROUND PLANE

Figure 1 radiating slots with dimensional parameters

Rectangular antenna element represented as two slot-

TABLE 1 Properties of the Antenna Elements

f, 0, Patch W (mm) a (mm) h (mrn) Er (GHz) (deg.) 1 7 / 4 NO. L (mm)

I 11.85 7.90 4.10 0.17 2.22 8.450 99 0.0071 2 11 3 3 10.63 3.90 0.79 2.55 7.730 105 0.0326 3 13.m 12.70 4.25 1.63 2.55 6.560 100 0.0569 4 14.85 14.03 4.60 2.52 7.55 5.800 107 0.0778 5 12.80 11.70 3.40 3.00 2.50 6.570 83 0.103')

222 MICROWAVE AND OPTICAL TECHNOLOGY LE-ERS / Vol 13, No 4. November 1996

tion from the edge. This inhibits measurements at angles greater than approximately 80" off the z axis, but it extends the angular range over which measurements can be ade- quately made with a finite ground plane size.

The accurate measurement of far-field radiation patterns in the E and H planes involves a number of factors, such as the physical and effective dimensions of the antenna element, its feed-point location, its operational frequency, and the environment in which it is operated (e.g., ground plane). In addition, the feeding and construction discontinuities can cause asymmetry in the radiation patterns.

3. ANALYSIS

The two-slot model, the cavity model, and the electric surface current model have been used for calculating the far-field radiation patterns in the E and H planes, and their variations are compared with measurements.

The basic formulas for computing these patterns are given in the following sections.

3.1. Formulas Based on the Two-Slot Model. Generally, the fringing fields that give rise to the slot model for patch antenna radiation occur around the four edges (side walls) of the rectangular patch. The fringing fields along the patch- length edges are depicted in Figure 1. These fields represent the preferred sources of radiation for the antenna, which is vertically polarized. The fields along the patch-width edges radiate horizontally polarized power and cannot be avoided. Based on this observation, the antenna element can be re- garded as an antenna system equivalent to the two-slot model shown in Figure 1. The E-plane pattern ( x - I plane) is deter- mined by multiplying the radiated field of a single slot by an array factor corresponding to the arrangement of a two-slot array. The total radiation field in the E plane of a rectangular microstrip antenna element operating in the TM1, mode is given by [13] as

where

For the H plane (y-z plane),

where

where k , is the propagation constant in free space, W and L are the patch width and length, V, is the voltage across the slot, and 8 and 4 are the spherical coordinates defined in Figure 1.

3.2. Formulas Based on the Cavity. This model treats the region between two parallel conductor planes, consisting of a patch conductor and ground plane, as a cavity bounded by the electric walls and a magnetic wall along the periphery of the patch. The fields in the antenna element are assumed to be those of this cavity [15]. This cavity will support quasidis- Crete TM,, modes transverse to z , where rn and n are the mode numbers associated with the y and x directions, re- spectively. The radiating mode of interest is the TM,, mode, obtained for L = A d / 2 .

For the E plane

F,(8,90") = cos( cos 8 ) cos( 2 knLt sin . 8 ) . ( 5 )

and for the H plane

sin( knW, sin . 8 )

cos 6 , (6)

where We and L, are the effective dimensions, taking into account the fringing fields at the patch edges. In the case of the TM,, mode, We and L , may be approximated as follows [ 131:

and

w, = w, with

0.164(&, - 1)

&, 2

+c[0.7%3+ln(g + 1.88)]}, (9) r & r

where ceW is the substrate effective permittivity as a function of patch width W, and can be calculated from [16]

Few = 0.5 ( E , + 1) + (6, - 1) ( 1 + - l;) -''.i]. (10) [ and eer. is the effective relative permittivity as a function of the patch length L , and can be calculated from (10) by replacing W with L.

3.3. Formulas Based on the Electric Surjace Current Model. The antenna element is replaced by an assumed surface current distribution, and the fields are solved taking into account the presence of the dielectric layer. Calculation takes place in the Fourier domain. The far field, calculated asymp- totically from this solution, is used to get the radiation patterns of the antenna element.

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 13, No. 4, November 1996 223

90' 90'

I

'270'

Figure 2 Measured far-field radiation pattern in the E plane at 8.450 GHz for antenna element No. 1 in Table I compared with results obtained from Eqs. ( 5 ) and ( 1 1 )

The radiation pattern in the E plane is functionally ex- pressed as [17, 181

T sin 6, cos2 -

(sin' o - E?,, 1- F , ( 0 , 4 = 0 ) = &,,[l + ErCOt:(k,,h&)] (2J;;i

C O S ' ( E , - sin? 0 )

( c , - sin? 0 1 + c; cosz Hcot'( ko/iJ=) X , (11)

and for the H plane

= [ l + & r c o t y k ( , h G ) ]

(12)

The radiation pattern in the E plane of rectangular mi- crostrip antenna elements with substrates in the range of h s 0.0815h0 can be calculated by Eq. (2) or (51, because both equations yield similar results. The results obtained from these formulas agree well with measured results at desired frequency range values. as can be seen in Figures 2-6. However, the agreement between the measured results and those calculated using Eq. (11) has been found not to be promising. As is evident from the close agreement between the measured and calculated patterns in the H plane shown in Figures 7-9, the formulas based on the TSM and the CM calculate the patterns for antenna elements with substrates in the range of h I 0.0815h,, with good accuracy. Similar to the

Figure 3 Measured far-field radiation pattern in the E plane at 7.730 GHz for antenna element No. 2 in Table 1 compared with results obtained from Eqs. (5) and (11)

90'

- Measured

Eauation (1 1) I

. . '270'

Figure 4 Measured far-field radiation pattern in the E plane at 6.560 GHz for antenna element No. 3 in Table I compared with results obtained from Eqs. ( 5 ) and (11)

E-plane case, the agreement between the measured radiation pattern in the H-plane results and those obtained from Eq. (12) is once again not promising.

Note that the results for the radiation pattern in the E and H planes obtained from Eqs. (2) and (4) are similar to the results obtained from (5) and (61, respectively.

4. RESULTS

Far-field radiation patterns were measured for angles from -90" to 90" in both the E and H planes. The radiation patterns of rectangular rnicrostrip antenna elements are hemispherical, but both the E- and H-plane patterns have a null in the horizontal plane f90". Evidently, any E-plane

224 MICROWAVE AND OPTICAL TECHNOLOGY LEnERS / Vol 13, No 4, November 1996

90' 90'

I '270'

Figure 5 Measured far-field radiation pattern in the E plane at 5.800 GHz for antenna element No. 4 in Table 1 compared with results obtained from Eqs. (5) and (11)

Figure 7 Measured far-field radiation pattern in the H plane at 8.450 GHz for antenna element No. 1 in Table 1 compared with results obtained from Eqs. ( 6 ) and (12)

90'

90'

- - - Equation (11) - - - - Equation (5) - Measured

I I 270'

I ' 270' Figure 8 Measured far-field radiation pattern in the H plane at 7.730 GHz for antenna element No. 2 in Table 1 comoared with results obtained from Eqs. (6) and (12)

Figure 6 Measured far-field radiation pattern in the E plane at 6.570 GHz for antenna element No. 5 in Table 1 compared with results obtained from Eqs. (5) and (11)

radiation pattern in this plane occurs in the form of surface waves.

All radiation patterns for both the E and H planes were measured and calculated at resonant frequency.

Figures 2-6 show the selected measured E-plane radiation patterns of thin antenna element numbers 2, 4, 8, 12, and 14 of the tables published in [19-211 compared with results obtained from Eqs. (5) and (11). The patterns obtained by Eq. (5) agree very well with measured values at desired angular

regions below +60" on the E plane, whereas Eq. (11) fails to predict these patterns satisfactorily.

Note that diffraction from the edges of the ground plane has a profound effect on these antenna patterns above k 60". In the main beam region, the ideal patch pattern is modu- lated by diffracted fields.

Figures 7-9 show the selected measured H-plane radia- tion patterns of thin antenna element numbers l, 2, and 5 of Table 1 compared with results obtained from Eqs. (6) and (12). The patterns obtained by Eq. (6) coincide well with measured values, whereas Eq. (12) fails to accurately calcu- late them.

MICROWAVE AND OPTICAL TECHNOLOGY LElTERS / Vol. 13, No. 4, November 1996 225

90” radiation patterns in the H plane are not restricted t o certain material and substrate thicknesses.

T h e effect of substrate thickness on the far-field radiation patterns for rectangular microstrip antenna elements has been shown to be quite significant.

- - - - Equation (6) Equation (12)

L I U

Figure 9 Measured far-field radiation pattern in the H plane at 0.570 GHz for antenna element No. 5 in Table 1 compared with results obtained from Eqb. (6) and (12)

Note that the €-plane beamwidth decreases with increas- ing substrate thicknesses. and the H-plane slightly increases.

In summary,

T h e agreement between the calculated and measured result for patterns in the E plane is very good, except for the noticeable effects of the diffraction at large angles from the z axis due to the imperfect ground plane used in the measurements.

60” that could be caused by surface waves and diffraction at thc ground plane edges. The agreement between the calculated and measured result for patterns in the H plane is also good. The microstrip antenna element exhibited satisfactory and nearly symmetrical E- and H-plane radiation pat- terns.

The E-plane patterns exhibit variations below

5. CONCLUSION

A study of the far-field radiation patterns in both the E and the H planes for antenna elements with substrates ranging in electrical thickness, defined as h / A , , from 0.0071 to 0.1039 and physical thickness from 0.17 to 3.00 mm has been carried out.

This article has verified the applicability of the TSM, the CM, and the electric surface current methods for computing the far-field radiation patterns in both the E and H planes of rectangular microstrip antenna elements. I t has also demon- strated that the formulas based o n the TSM and the C M methods yield radiation patterns in the E plane of antenna elements with substrates on the order of h I 0.0815A0 (or 0.1 3A,) with good accuracy, but they become increasingly inaccurate as the substrate thickness is increased. I t has also been demonstrated that the applicability of the formulas based on the TSM and the CM methods for predicting the

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12. D. M. Pozar, Analyses and Design of Considerations for Printed Phased Array Antennas, Handbook of Microstrip Antennas, Vol. 1, Peter Peregrinus, London, 1989, Chap. 12.

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18. J . Ashkenazy. S. Shtrikman, and D. Treves, “Electric Surface Current Model for the Analysis of Microstrip Antennas on Cylindrical Bodies,” IEEE Trans. Antennas Propagat. ~ Vol. AP-33, March 1985, pp. 295-300.

19. M. Kara, “Formulas for the Computation of the Physical Proper- ties o f Rectangular Microstrip Antenna Elements with Various Substrate Thicknesses.” Microwaue Opt. Tedinol. Lett., to be published.

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pp. 7.1-7.20.

Receiced 5-21-94

Microwave and Optical Technology Letters, 13/4. 221-226 D 1996 John Wiley & Sons, Inc. CCC 0895-2477/96

226 MICROWAVE AND OPTICAL TECHNOLOGY L E l T E R S / Vol 13 No 4 , November 1996