RL 768 THE TRANSMISSION LINE MODEL ANALYSIS OF RECTANGULAR PATCH ANTENNAS* Dipak L. Sengupta Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, Michigan 48109 ABSTRACT The transmission line model has been utilized to determine the input performance of a rectangular patch antenna excited by a coaxial probe. Approximate expressions have been developed for the resonant propagation constant (or the resonant frequency), quality factor and input resistance of such antennas which clearly indicate the effects of the various antenna parameters including the feed on these quantities under dynamic conditions. The results of computations based on the expressions developed are compared with available measured values. The present work establishes the transmission line model as a valid representation of rectangular patch antennas, and would facilitate the design of such antennas without elaborate numerical computation. The model has also been applied to determine the resonant parameters for a similar antenna tuned by passive metallic posts. Validity of an approximate expression derived for the resonant frequencies of such tunable antennas has also been established by comparison with available measured results. * The work was supported by the U. S. Army Research Office under Contract DAAG29-82-K-0076. RL-768 = RL-768

I. Introduction Early analyses of rectangular microstrip or patch antennas has been based on the TEM or parallel plate transmission line model [1,2,3]. Although this simple model requires less computation [3] and provides results with sufficient engineering accuracy, it cannot account for the effects of the patch aspect ratio, substrate thickness and the feed parameters [4]; moreover, due to the nature of the assumed open-circuit termination the predicted resonant frequency is generally larger than the actual one. Later, the cavity model [5,6] and its variations [4,7] have been developed for such antennas; these models provide more accurate results but involve detailed numerical computation. The present paper re-examines the transmission line model in greater detail and makes appropriate modifications to remove its abovementioned shortcomings, and to develop simple but accurate expressions so that the design parameters for rectangular microstrip antennas may be determined without elaborate numerical computation. The improved model is further modified to obtain the resonant parameters and the input performance of a tunable rectangular microstrip antenna [8,9,10] whose resonant or operating frequency is controlled by utilizing a number of passive conducting posts suitably located within the input region of the antenna. II. Equivalent Transmission Line Representation of a Rectangular Microstrip Antenna The transmission line model treats the rectangular patch radiator as a strip line resonator with no transverse field variation and assumes

-3 - the radiation to occur from the two transverse open edges [2,3]. It is well known [11] that the dominant mode of propagation in a strip line is the TEM or quasi TEM mode having negligible variation of fields in the transverse direction. It has been reported in [12,13] that it is the parallel plate type of TEM mode which is the dominant mode propagating along the longitudinal direction of a rectangular patch configuration, and that waves incident at the two open side edges suffer mostly total reflection thereby causing negligible radiation. Therefore, when properly excited only the two open edges normal to the direction of propagation take part in the radiation. It then seems natural to treat such an antenna as a section of parallel plate or strip transmission line terminated at both ends by admittances appropriate for the two radiating edges. This concept is utilized to obtain the equivalent transmission line model, and subsequently develop an equivalent lumped circuit representation for the antenna. Figure 1 shows a rectangular patch antenna of length i, width a, and excited from the back by a coaxial probe of radius r, the exact location of which is usually governed from desired impedance and polarized considerations. It is assumed that propagation takes place along the ~x-directions. The presence of two passive conducting posts, each of radius r0, in Fig. 1 is ignored for the present discussion; they are used to tune or control the resonant or operating frequency of the antennaand will be discussed later. Normally, the substrate of thickness d extends beyond the patch and may sustain surface waves for large d. However, as shown in [12], for sufficiently thin substrate,

-4 - the surface wave effects may be neglected and we therefore neglect the effects of the substrate beyond the patch. The ground plane is assumed to be of infinite extent which is justified from the fact that in a practical antenna it is much larger than the patch. The entire configuration of Fig. 1 with the exception of the two tuning posts may now be represented by an equivalent transmission line of length z, characteristic admittance Yc (impedance Z ), propagation constant S and terminated at each end by a normalized admittance Y (= YT/Yc, YT being the admittance of an open edge) as shown in Fig. 2 which also shows the source generator in series with a normalized (with respect to Z ) internal inductive reactance jXL located at a distance if from one edge of the patch. With a strip line excitation, the source in Fig. 2 should be replaced by an approximately located generator having a capacitive reactance. The characteristic admittance and the propagation constant of the transmission line are: acd Y =-1 e ="" 0 (1) c ZC nod ' 0 o with vp,s being the permeability and permittivity of free space, = c, c =, (2) c/ ~ 0 0 where e +1 + 2 ( lOd ) ' Ce 2 -'~ - ^1 +^ a (3)

-5 - = 1 + 1.393 ( + 0.667 (d) in (d+ 1.444), (4) and r', W(= 2rf), c are the relative permittivity of the substrate, the radian frequency and the velocity of light in free space, respectively. The above relations are well known [3,11,14] and have been found to provide an accurate representation for a strip line. It should be noted that for sufficiently small d/a (or d/a -> 0) the equivalent permittivity ce of the substrate and the form factor a given by (3) and (4), respectively, reduce to ce r er and a - 1, respectively; in generalce and a are slowly varying functions of (d/a) and for typical d/a used for patch antennas, ce and a are slightly smaller and larger than cr and 1, respectively. The normalized terminating admittance in Fig. 2 is Y = G + jB where G and B represent the (normalized) radiation conductance and capacitive susceptance appropriate for the reradiating aperture. Exact determination of G and B is a difficult boundary value problem. Although formal expressions for G and B have been derived in [12,13] using the Wiener-Hopf technique, they are too complicated, and meaningful results can only be obtained through detailed numerical computation. Various empirical approximations to G and B have also been found useful [2-4] in the previous analyses of such antennas. For the present purpose we shall obtain approximations to G and B from the known results for a semi-infinite parallel plate waveguide radiating into free space [15,16]. Using the image principle and the aperture admittance of the open-end of a parallel plate waveguide radiating into free space [17] it can be shown that

-6 - G - 2d (5) 2ace,. ~/7/T 2'ire B = 6de in ( yd ) (6) where, y = 1.78107, e = 2.71828. It should be noted that (5) is the same as that obtained in [12] under thin substrate and no surface wave propagation. The nature of approximations involved to obtain (5) and (6) are such that the parameter a be sufficiently large. The normalized inductive reactance of the probe may be obtained from considerations similar to the probe excitation of a parallel plate waveguide propagating only its dominant mode [17,18], and is given by X = a n 2. (7) L 27v y3r( This completes the description of the transmission line circuit equivalent to the coaxial probe fed untuned rectangular patch antenna. For a similarly excited rectangular patch antenna tuned by passive conducting posts (Fig. 1) the equivalent transmission line circuit is similar to Fig. 2. except that the susceptances appropriate for the posts would be inserted across the line at the corresponding locations of the posts. For example, with posts of radii r0 in Fig. 1 the required susceptance of each post to be inserted in Fig. 2 would be the reciprocal of that given by (7).

-7 - III. Resonance Parameters of an Untuned Rectangular Patch Antenna In this section we at first obtain the general resonance conditions for a patch antenna in the absence of the exciting probe inductance. Later, the effects of the probe inductance are taken into account. Finally, various resonant parameters of the antenna are obtained and compared with available measured results. A. Resonance Conditions The input admittance (Y in) of a section of transmission line of length x, characteristic admittance Yc' propagation constant B and terminated by (normalized) Y is given by the following well known expression Y. in - Y + j tan Bx Y 1 + jY tan x (8) It is found from the equivalent circuit of Fig. 2 that in the absence of the exciting probe inductance the admittance seen by the generator is the same as the total admittance Yin at the terminals A-A. By using (8) it can be shown that AA = (G + G ) + j(B + B ), (9) C 1 2 1 2 with G = --— (10) 1,2 cos2 BI 2+ (B2 + G2)sin2 St - B sin 2Bz 1,2 1'e, 1, - I JL, I

-8 - B - (B2 + G2 - 1)tan a,1 2 - B tan2, 1,2 B: (11a) 1'2 1 + (B2 + G2) tan2 z,2 - 2B tan pB1 1,2 1,2 r sin (5 - 2gl2 ) (11lb) (1 + B2 + G2) - r cos(E - 2 1,2) where r2 = 4B2 + (B2 + G2 - 1)2 (12) tan = 2B(13) B2 + G2 - 1 and the subscripts 1,2 on the right-hand sides of (10) - (11) correspond to those on the left-hand sides. It is now demanded that resonance is obtained when the input admittance of the antenna is real; i.e., Im(YAA/Yc) = 0 in (9). Using this condition along with (11) - (13) and after considerable algebra it can be shown that at the dominant (or lowest) resonance C = SQ which yields the following transcendental equation determining the values of B at the dominant resonance: tan = 2B (14) B2 + G2 - 1 which has been frequently quoted in the literature [2,13,19]. It can now be shown that under the assumption XL = 0 the resonant conductance of an antenna fed at a distance = 9f from one of the radiating edges 2 f is given by

-9 - r 2G (15) (15) c cos2.Jf + (B2 + G2)sin2 Sf - B sin 2Bf from which we obtain the following expression for the resonant resistance Rr of an antenna excited from one edge (Zf = 0): z R - c (16) r 2G B. Resonant Frequency Solution of (14) yields the propagation constant P at resonance from which the resonant frequency is obtained by using (2). Substitution of (5) and (6) in (14) indicates that the latter is a complicated transcendental equation which is generally solved by numerical means [2,3,19]. Also, it is not obvious from (14) how the resonant frequency of an antenna varies with its characteristic parameters. In the following we develop an explicit expression for the resonant frequency from an approximate solution of (14). Based on the observation B,G << 1 appropriate for most antennas of practical interest, we simplify (14) to tan I = -2B = - 2d ln e, (17) c e yd Tr' (1 where we have used (6) to obtain the last form. To solve (17) we make use of the exoerimentallv observed fact that at the fundamental

-10 - resonance the propagation constant:w/2, and assume that the solution Br of (17) is such that = 7(1 - 6/7T) (18) where 6/1 << 1. Under the assumption of (18), (17) can be linearized in 6 and we obtain 6 e in yd 2d - -F e 2 1 + 2d I (19) Using (2), (18) and (19) we obtain the following expression for the resonant or operating frequency fr fo (2d 2 Ll + y2d (In — 2) 1 + 7ra O' yd / I (20) where f = (21) 2R Xce is the resonant frequency under the assumption B = 0. Equation (20) has been reported earlier [20,21]. Previously [1,3] the resonant

-11 - frequency of a patch antenna as obtained from the transmission line model considerations has been given by f0, with ce = r, which is independent of the patch width and the substrate thickness [5,7]. The present expression (20) contains the correction factor to be applied to f, and it clearly indicates the dependence of the resonant frequency on all the basic parameters of the antenna including the feed. Before discussing the accuracy of (20) we shall examine the validity of (17) to approximate (14). To this end, we have used the Newton-Raphson method to solve (14) numerically for rectangular patch antennas having dimensions reported in [4]; only two or three iterations were sufficient to obtain convergence solutions. The 6/'r values obtained numerically from (14) and from (19) are shown in Table 1 where we have also indicated the relevant parameters of the antennas used. The results indicate that (17) is a good approximation to (14), and that (19) can be used to calculate the desired values of 6/r. The resonant frequencies of the antennas considered in Table I have been calculated by using (20) and the results along with the values of f0 are shown in Table II where, for comparison, we also show the corresponding measured values obtained from [4]. For the antenna parameters used, the results of Table II indicate that (20) overestimates the measured resonant frequency by about 0.2 to 1.6 percent whereas the overestimation by fo (i.e., by (21) ranges from three to six percent. C. Effects of the Probe Inductance From resonant circuit considerations it is expected that an inductive probe (Fig. 2) would slightly increase the resonant frequency

-12 - Table I 6/7r Values for Rectangular Patch Antennas (d = 0.1524 cm, cr = 2.5; a and 2 variable) (cm) (cm) e Eq. (14) Eq. (19) 4.100 4.140 2.390 1.135 0.0392 0.0404 6.858 4.140 2.428 1.088 0.0408 0.0417 10.800 4.140 2.452 1.060 0.0422 0.0422 11.049 6.909 2.453 1.059 0.0280 0.0283.,... Table II Calculated and Measured [4] Values of Resonant Frequency for Rectangular Patch Antennas (d = 0.1524 cm, 6r= 2.5; a and z variable) a f o (MHz) fr (MHz) (cm) (cm) Eq. (21) Eq. 20 Measured 4.100 4.140 2343 2248 2228 6.858 4.140 2325 2228 2200 10.800 4.140 2314 2216 2181 11.049 6.907 1386 1347 1344

-13 - given above. It is important to know this upward shift in the frequency because the measured resonant frequency is associated with the probe inductance. To simplify the analysis we assume that the antenna is fed from one edge (i.e., f= = 0 = O, Z = in Fig. 2) and observe that near resonance 2 1 SP -a. Under these conditions and using the approximation B,G << 1, it can be shown from (9) and (1la) that the admittance (without the probe) at the terminals A-A located at the edge is YAA y A 2G + j(2B + tan X). (22) C Combining (22) with the probe reactance (Fig. 2) it is found that the total impedance seen by the generator of an edge-fed antenna is: Zin 2G 2B + tan = + (2B tan - - )2 CL 4G2 + (2B + tan f,)2 L 4G2 + (2B + tan g,)2' (23) Applying the resonance condition (Im Zin = 0) to (23) we now obtain the following XL tan2 Sz - (1 - 4BYL)tan Bk + [4(G2 + B2)XL - 2B] = 0. (24) The appropriate solution of (24) which would yield the resonant B under the present conditions is

-14 - tan,z = -2B + - (1 - 16G2X2)1/2 (25) 2XL 2XL which should be compared with (17). We make further approximation 16G2XL < 1 valid for practical antennas to obtain the following from (25) tan SB = -2B + 4G2XL, (26) with G, B and XL given by (5), (6) and (7), respectively. Equation (26) can be solved in a manner similar to the case with XL = 0 (i.e., solution of (17) described earlier. It can now be shown from (26) that the relative upward shift in the resonant frequency caused by the probe inductance is Afr 4G2X L 4G2X r L L f r -2B ( rr where fr is given by (20) and in the last form of (27) we have used the approximation w >> 2B. After introducing (5), (7) in (27) we obtain the following explicit expression for the frequency shift: rf - 2 (d ) (a) 1 In 29) (28) e With a stripline excitation, a similar expression for the downward shift of the resonant frequency may be obtained after replacing XL in (27) by the equivalent series reactance of the capactive junction.

-15 - For an edge fed antenna having d = 0.1524 cm, r = 0.064 cm, 0 a = 4.10 cm, z = 4.140 cm and r = 2.5, the calculated resonant frequency fr = 2248 MHz with XL = 0 (Table 2). The frequency shift due to the probe inductance as calculated from (28) is Af r = 1.02 x 10-3 (~ 0.1 percent) f r D. Resonant Resistance For matching purposes it is of practical interest to know the (input) resistance of the antenna at resonance as seen by the generator. This resistance consists of the antenna radiation resistance and the resistances associated with the copper and dielectric losses. The latter two are usually very small [4] and will be neglected. For an edge-fed antenna the desired resonant resistance Rr can be obtained easily by using (26) and (23) and is given by Z c 1 R. (29) r 2G 1 + 4G2X2 (29) Note that with XL = 0, (29) reduces to the value given earlier by (16). Neglecting the effects of the probe the input resistance of the antenna at resonance may be obtained from r 20- 1 (e-) ohms. (30) r 2G=r (1 - 6:r) a

-16 - where 6/71 is given by (19). For the edge-fed patch antennas discussed earlier, the resonant resistance has been computed by using (30), and the results are shown in Table III where we also show the corresponding measured values obtained from [4]. The results indicate that except for the lowest value of a, the agreement between the calculated and measured values of resistance may be considered to be fair. E. Q of the Patch Antenna The bandwidth of the patch antenna is intimately related to its quality or Q-factor. Hence, it is of considerable interest to obtain an explicit expression for the Q-factor. Again, we consider an edgefed antenna and ignore the effects of the probe and the dielectric and copper losses associated with the antenna, i.e., we shall obtain an expression for the radiation Q or Q r of the antenna. To derive the desired expression for Qr we at first develop an equivalent lumped circuit parameter for the rectangular patch antenna near resonance which may be found useful for other applications as well. Referring to Fig. 2 and assuming YL = 0, we rewrite (22) for the input admittance near resonance as: YA yAA= 2G(S) + j[2B(5) + tan z], (31) y c where we have explicitly shown the dependence of various parameters on 3. Note that at resonance tan rz + 2B(r3 ) = 0, (32a)

-17 - Table III Resonant Resistance of Edge-Fed Rectangular Patch Antennas (d = 0.1524 cm, er = 2.5, r0 = 0.064 cm; a and z variable) a Rr (ohms) (cm) (cm) e 6/7T Eq. (30) Measured [4] 4.100 4.140 2.390 0.0404 195 280 6.858 4.140 2.428 0.0417 118 115 10.800 4.140 2.452 0.0422 75 65 11.049 6.909 2.453 0.0283 120 102

-18 - and near resonance 6 = B + A Using (32) we obtain from (31) (33b) YAA(Br + A) Y C 2G(r) j [ ea ln eI ) AS + ZA, (33c) where it has been assumed that G(Br + Ag) = G( ). Let the inductance and capacitance per unit length of the strip line comprising the patch be denoted by Le and Ce, respectively. Then from the considerations of the TEM mode characteristics it can be shown that: c ee 'e o LeCe = Ce/Y Ag = CeA,/Yc,. (34) Ce = CoSe"a/d e 0 e.I Using (34) we obtain the following from (33) CY 2d + 27 j YAA(r + A) 2YG() + j -e In rrd _+ Ce Am L- -] (35)

-19 - Comparing (35) with the admittance near resonance of a parallel combination of R,L,C having a resonant frequency wr = 1/L-C, it can be shown that the equivalent R and C of the patch antenna are: Z O~ R AA R = G (36a) r r C = e In (ed ) ~C C aa ad /0F1 27r\ o, e 0 - + -- - In (36b) 2d v ~d ' It is interesting to note that the first term in (36b) is one half the equivalent static capacitance of the patch and the second term is the dynamic contribution to the capacitance due to the two radiating apertures. Using the basic definition of Qr as Qr = WrCR, (37) we obtain the following expression for the radiation Q of the patch antenna _ _ 1 / _-_ 2_ _ 4G + In e yd ) r r r + -^- 1n e. (37) 2d +1n In \ yd yd '

-20 - It is interesting to observe that if the dynamic contribution to C is ignored, _ r_ __ T-2B (38) r 4G - 4G (38) ^r the last form of which is exactly the same as given in [12]. Equation (37) shows explicitly the dependence of Qr on the various parameters of the antenna. As an application of (37), consider the patch antenna having a = 6.858 cm, Z = 4.14 cm, d = 0.1524 cm and cr = 2.50. From Table I we have ce = 2.428 cm and a = 1.088. This, from (37) it is found that Qr = 35.88 + 1.23 = 37, which indicates that the contribution to the dynamic term (i.e., second tern in (37) is negligible. The resonant frequency of the antenna is from Table 2: fr = 2228 MHz. Thus, the theoretical bandwidth of the antenna is f Af = 60 MHz. Qr No experimental results are available for comparison. However, corresponding theoretical values obtained by detailed numerical computation using the cavity model [4] for the same antenna are Q = 30 and Af = 73 MHz.

-21 - IV. Tunable Rectangular Patch Antenna It is known [8,9,10] that the addition of passive metallic posts within the input region of a rectangular patch antenna increases its resonant or operating frequency without significantly altering its input impedance or radiation pattern and without increasing the complexity of the external feed network. A variety of such tuning post arrangements have been developed; for example, it has been reported [9], that the range of operating frequency can be increased to 20 percent with a pair of posts placed along the center line of the antenna and in excess of 50 percent with more posts located along and offset from the center line. The present section utilizes the transmission line model discussed in the previous section to determine the resonant characteristics of tunable rectangular patch antennas. A. Numerical Evaluation of the Resonant Parameters We consider a rectangular patch antenna fed by a coaxial line at one edge and tuned by a single post located (y = a/2 in Fig. 1) at a distance i from the feed edge. Under these conditions the equivalent transmission line circuit is as shown in Fig. 3 where -jBL represents the inductive susceptance of the post, and the other parameters are as explained before. For simplicity, it is assumed that the radius of the post is the same as that of the probe; therefore, BL = l/XL where XL is given by (7). For more than one post located along the transverse direction at ~, BL should be changed appropriately. With more set or sets of 2 L similar posts placed at various distances, the appropriate susceptance be placed at the corresponding locations in Fig. 3. The resonant

-22 - frequency or propagation constant of a tunable antenna may now be found by demanding the vanishing of the imaginary part of the admittance seen by the generator obtained by multiple application of (8) in conjunction with Fig. 3. From a knowledge of 6r all other resonant parameters of the antenna can be obtained in a manner described in the previous section. In general, this is best accomplished by numerical computation. In the following we shall obtain an approximate expression for the resonant frequency which may be found useful for the design of such antennas. B. Approximate Expression for the Resonant Frequency For simplicity we neglect the effects of the probe inductance, and obtain the following for the total admittance at the AA terminals in Fig. 3: YAA Y = (G + G ) + j(B + B - BL) (39) C 1 2 1 2 where all the notations are as explained before. Applying the condition Im(YAA/Yc) = 0 to (39), and after considerable algebra using (10) through (13) it can be shown that the resonant B satisfies the following equation [l+B2+G2-r cos{(;-j+)+ B(zl- 2)}][l+B2+G2-r cos{ (E-8)-B(1 -2)}] sin(-6R,) = BL 2r(l+B2+G2)cos 3(k -k )-2r2 cos(C-Bk) 1 2 + + =. (40) 1 2

-23 - Since BL << 1, close examination of (40) indicates that at the desired solution C - z =2 O. It is also known that B,G << 1, B2 + G2 << 1 which imply that r = 1, C = X - 2B. Under these approximations it can be shown that (40) simplifies to sin( - 2B - St) -BL cos2 (). (41) Using the small argument approximation on the left side of (41) it can be shown that - (r - 2B) + BL cos2( ). (41) Finally, identifying ~ra - r - 2B (Eqs. (17) and (18)), where 8r being the resonant propagation constant in the absence of the post we obtain the following expression for the resonant propagation constant from (41) ~ = Br + BL cos2-) (42):=T/ L Introducing the value of BL (= 1/XL: Eq. (7)) we obtain the following expression for the operating frequency of a rectangular patch antenna tuned by a single post - f I 1 cos2 (43) fr 1"a 2 In Y7rr0

-24 - where fr is given by (20). Note that (43) indicates that f = f when i = k/2,a fact confirmed by experiment [9]. For N similar posts 2 located at k = kn, and ignoring the mutual effects between the posts (43) can be generalized and it can be shown that the operating frequency of rectangular patch antenna tuned by N similar posts located at Z is 2n f - f l + 1: sin[-! 1 cos2( )1 op r j ff/ au 2k k, U - In -y-rr (44) C. Results Figure 4 shows the measured [9] operating frequency and input VSWR vs the tuning post location for a rectangular patch antenna fed at one edge by a coaxial probe and tuned with a single post (having the same radius as that of the probe) located along the central line of the antenna. Corresponding theoretical values shown in Fig. 4 were obtained numerically by using (8) in conjunction with the equivalent transmission line circuit shown in Fig. 3. We have also calculated the operating frequencies using (43),and the results were almost identical with the theoretical values shown in Fig. 4 indicating thereby that (43) can be used with advantage for such purposes. The theoretical fop vs i. / curve is symmetrical about z /k = 0.5 where the operating frequency 2 2 is unaffected by the post (i.e., fop = f ); the latter fact is also borne out by the measured results. The slight asymmetry in the measured frequency variation is attributed to that caused by the feed.

-25 - The disagreement between the measured and calculated frequencies appears to be maximum when the post is located at the radiating edges. This may be due to the fact that edge effects have been ignored in obtaining the effective susceptance of the post. Similar results are shown in Fig. 5 for the same antenna tuned by a pair of similar posts located symmetrically along the center line of the antenna. Here again (44) has been found to reproduce the numerically computed values without significant error. Resonant parameters have been determined for the same antenna with four similar tuning posts symmetrically located along the two radiating edges each pair being located at a distance of 2.5 cm from its nearest side edge, and with five pairs of similar posts spaced symmetrically 1.5 cm apart along the two radiating edges. The results are shown in Table IV where some of the available measured [9] results are also given for comparison. Further calculations indicated that (44) can be used with advantage to determine the operating frequencies of antennas using up to six tuning posts. V. Conclusions The input performance of a rectangular patch antenna fed by a coaxial probe has been investigated by using the transmission line model. Approximate expressions have been developed for the resonant frequency (or propagation constant), quality factor and input resistance of the antenna which indicates clearly the influence of various antenna parameters including the feed under dynamic conditions. The validity

-26 - Table IV Resonant Parameters for a Rectangular Patch Antenna Tuned by Four and Ten Posts z = 6.2 cm, a = 9.0 ct, 4 = 0.16 cm, r0 = 0.064 cm, Er = 2.55 Operating Frequency f (GHz) No. of Posts OP VSWR Numerical Eq. (44) Measured (Numerical) 4 2.187 2.013 1.995 1.95 10 2.409 3.238 2.240 3.9

-27 - of theoretical values obtained has been confirmed by comparison with available experimental results. The present work establishes the transmission line model as a valid representation of rectangular patch antennas, and the expressions derived would facilitate the design of such antennas without elaborate numerical computation. The model has also been used to determine the resonant parameters of a similar antenna tuned by a number of passive metallic posts. The results based on the theory developed were compared with available experimental results for antennas using one, two, four and ten tuning posts. Aprroximate expression given can be used to obtain the operating frequencies of such tunable antennas using up to six tuning posts; numerical method is preferable for antennas using more than six similar tuning posts. VI. Acknowledgements The author is grateful to Dr. A. Sindoris and Mr. F. G. Farrar of the Harry Diamond Laboratories (HDL) for many discussions, and for the measured results for tunable rectangular patch antennas which were developed at the HDL.

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Figure Captions Figure 1 Rectangular microstrip antenna excited by a coaxial probe and tuned by two metallic posts. 2 Equivalent transmission line circuit for a rectangular patch antenna. 3 Equivalent transmission line circuit for an edge-fed rectangular patch antenna tuned by a single post. 4 Operating frequency (upper curves) and VSWR (lower curves) vs. post locations for a single post tuned rectangular patch antenna. Calculated;- Measured. 5 Operating frequency (upper curves) and VSWR (lower curves) vs., post location for a double post tuned rectangular patch antenna. Calculated; ----— Measured.

Tuning Posts s// -- Sa /< — Substrate.. ri 7r // I / Figure 1. Rectangular microstrip antenna excited by a coaxial probe and tuned by two metallic posts.

A Y=G+jB Xj Y=G+jB I r ------ ------------- n ---------------------- A 1 / Generator Figure 2. Equivalent transmission line circuit for a rectangular patch antenna.

A JXL — - t jxL I I 1 Y=G+jB Figure 3. Equivalent transmission line circuit for an edge-fed rectangular patch antenna tuned by a single post.

post post a lY = 62 cm a=9.0 cm d=0.16cm r =0.064cm dr=2.55 1700.1600 1500 1400 Feed N ' 00 - - / N N I-I U ca) C7' Q) Su ai%, / - VSWR - 3 2 I 0.0 0.2 04 0.6 0.8 1.0 Figure 4. Operating frequency (upper curves) and VSWR (lower curves) vs. post locations for a single post tuned rectangular patch antenna. Calculated; ---— Measured.

1700.1600 N -' ~ 1500 G) U1400 1300 c00 Figure 5. /Q9 -2 Operating frequency (upper curves) and VSWR (lower curves) vs. post location for a double post tuned rectangular patch antenna. Calculated; ----- Measured.