Characterization of group delay in linearly and circularly polarized lossless stacked patch antennas

Authors


Abstract

[1] In this study, we investigate the group delay characteristics of linearly and circularly polarized lossless stacked patch antennas. A new closed-form analytic expression for group delay was derived for both polarizations to obtain physical insight and have an efficient technique for conducting parametric studies of the antenna group delay characteristics. The analytical model was constructed by employing an equivalent circuit, consisting of resonant RLC sections that model the lossless patches. The group delay is calculated via the frequency derivative of the input phase function (disregarding the far-field phase contribution, a second-order effect), which is obtained in closed form. The group delay is a function of the circuit parameters (Rl,u, Ll,u, Cl,u) , the values of which must be determined for a particular antenna. We employ a reduced-order Padé approximation to extract the parameter values by making use of sampled data obtained from either a full-wave simulation or alternatively via measured data. A circuit synthesis methodology is then employed to map the extracted Padé expansion coefficients onto a known impedance function. Comparison of the analytical results with the full-wave numerical calculations of the impedance, phase, and group delay quantities show very good agreement. The reduced-order approximation is a balanced compromise between the fidelity of the equivalent circuit model and the ability to obtain the group delay in an analytically exact fashion. Results shown herein establish that the group delay of a stacked patch antenna is proportional to the patch Q and the patch capacitance. The peak group delay of the antenna approaches an asymptotic value as the input resistance increases without bound, and the rate at which this occurs is inversely proportional to the patch Q. Results also illustrate that the peak group delay of the stacked patch approaches infinity as the inductance decreases, and approaches an asymptotic value as the inductance increases. Interestingly, all of these characteristics are independent of antenna polarization.

1. Introduction

[2] Microstrip patch antennas have become ubiquitous in terms of the extensive variety of applications they have been associated with [e.g., Abdelaziz and Nashaat, 2007; Sabban, 2007; Heckler et al., 2004; Yoharaaj et al., 2006; Deffendol and Furse, 1999]. In recent years, patch antennas have been employed with large success in Global Positioning System (GPS) user and ground equipment [Kaplan, 1996; Counselman, 1999; Boseley and Waid, 2005; Kim et al., 2004a, 2004b; Johnson et al., 2001]. A number of commercial GPS receiver vendors make use of patch antennas in their equipment. Although patch antennas have been analyzed quite extensively over the years [Richards, 1988; Jackson et al., 1997; Waterhouse, 1999; Pozar, 1992; Li et al., 2004; Munson, 1974; Agrawal and Bailey, 1977; Bahl and Bhartia, 1980; Garg et al., 2000; Itoh and Menzel, 1981], little attention has been paid to the group delay characteristics of these antennas [van Graas et al., 2004; Otoshi, 1993]. With their use in GPS navigational equipment, however, the group delay behavior has become a parameter of increasing importance. This is related to the operational demands of the application. In obtaining the most accurate navigational solution, and in turn the most accurate positioning information, the GPS system must have the ability to calculate the signal propagation time from the satellite to the receiver accurately. In propagating through the earth's atmosphere, signals experience delays introduced by the ionospheric layer, which can deleteriously affect the evaluation of the total propagation time [Kaplan, 1996]. Advanced techniques, such as a differential frequency mode, can be employed to quantify and compensate for the ionospheric delay, and in turn maintain the accuracy of the navigational solution [Kaplan, 1996]. However, the receiving antennas themselves introduce an additional delay due to their dispersive nature. The antenna group delay, which is defined here as the frequency derivative of the far-field phase, can add to the ionospheric delay and various other error terms to reduce the overall accuracy of the positioning information. Thus, a careful characterization of the group delay of antennas intended for GPS applications is quite important.

[3] In this study, we will investigate the group delay characteristics of linearly and circularly polarized lossless stacked patch antennas. To this end, a new closed-form analytic expression for group delay is derived for both linear and circular polarization. The development of the analytic expression is motivated by two key requirements: the desire for an accurate representation of the group delay for a lossless stacked patch configuration with linear or circular polarization, and the need for an efficient means of computing the group delay, which would in turn lead to a simple methodology for conducting parametric/trade studies of the antenna group delay characteristics.

[4] The analytical model is constructed by employing a simple equivalent circuit for the lossless stacked patch configuration. The circuit consists of resonant RLC (tank) sections that model each patch. The total number of tank sections is two for the linearly polarized case and four for the circularly polarized case. The group delay of the antenna is calculated approximately by taking the frequency derivative of the input impedance phase function. In actuality, the group delay of the antenna is composed of the contribution from two constituents – the input impedance and the far-field phase. Since the far-field phase is typically a weak function of frequency, we choose to disregard it for the purposes of the present study and approximate the group delay solely as the frequency variation of the antenna input impedance. The accuracy of this approximation is proportional to the Q of the patches. This approximation renders the relevant group delay expressions in closed form, since the input impedance and in turn the input phase functions can be expressed analytically for either polarization. Both expressions are functions of the lower and upper patch circuit parameters (Rl,u, Ll,u, Cl,u), the values of which must be determined for a particular antenna. To arrive at these values, we employ a reduced-order Padé approximation to extract the parameter values by making use of sampled data obtained from either a full-wave simulation (using Ansoft's HFSS, for example) or alternatively via measured data [Miller, 1998a, 1998b, 1998c; D'Amore and Sarto, 1994]. A circuit synthesis methodology is employed, which conveniently maps the extracted rational-polynomial coefficients from the Padé expansion onto an impedance function that is derived from the equivalent circuit [Van Valkenburg, 1955; Yu et al., 2005; Mangold and Russer, 1999; Schaubert, 1979].

[5] In practice, we could use a full-wave field solver (e.g. Ansoft HFSS) to compute the complex input impedance of a candidate antenna, and employ the procedure outlined herein to extract the relevant equivalent circuit parameters. Once these are known, the group delay of this antenna is obtained quite readily from the closed-form expressions derived here. This procedure can be employed in an iterative fashion to design a particular patch that meets some given maximum group delay specification. Alternatively, we could conceivably employ a more computationally efficient field solver (e.g. Method-of-Moments) and combine it with the approach proposed herein and an appropriate optimization methodology to yield a self-contained antenna synthesis tool.

[6] The results will illustrate that the parameter extraction routine does a very good job of capturing the frequency variation of the impedance and phase functions. Comparison of the analytical results with the HFSS numerical calculations of the impedance, phase, and group delay quantities will show fairly good agreement, especially in light of the fact that our Padé expansion consists of a first-order numerator polynomial and a second-order denominator polynomial. This reduced-order approximation is a necessary compromise between the fidelity (and complexity) of the equivalent circuit model and the ability to obtain the impedance component of the group delay in an analytically exact fashion.

2. Formulation

2.1. Group Delay of a Lossless Linearly Polarized Stacked Patch Antenna

[7] A cross-sectional illustration of the stacked patch topology addressed here is shown in Figure 1, where h1,2 are the thicknesses of the lower and upper substrate layers, respectively, εr1,2 are the layer relative permittivities, and dp is the input probe diameter. The equivalent circuit of this linearly polarized lossless stacked patch antenna is shown in Figure 2.

Figure 1.

Cross-sectional view of the stacked patch topology. Note the coaxial input probe feeding the lower patch.

Figure 2.

Transmission line equivalent circuit for a lossless linearly polarized stacked patch antenna.

[8] The individual resonant RLC circuits modeling the lower and upper patch have equivalent resonant impedances given by [Deshpande and Bailey, 1982; Pozar, 1982; Lumini and Lacava, 2001]

equation image

and

equation image

where Ql = 2π flRlCl and Qu = 2π fuRuCu are the lossless quality factors for the lower and upper patch, respectively, and fl,u represent the corresponding patch resonance frequencies for the fundamental TM10 mode. In particular, fl = 1/equation image = c/(2L1effequation image) and fu = 1/equation image = c/(2L2effequation image), where L1,2eff denote the effective electrical lengths of the lower and upper patch, respectively [Hammerstad, 1975]. In (1) and (2) we have also defined the frequency ratios frl = f/fl and fru = f/fu. The total input impedance of the stacked patch arrangement is thus Zin = Zl + jXp + n2Zu, which includes the coax input probe inductance,

equation image

where γe is the Euler constant. In addition, the coupling between the lower and upper patches is modeled via a coupling transformer with turns ratio n:1, as shown in Figure 2. We chose this representation because the complicated coupling physics is reduced to a single parameter representation (the turns ratio). However, we could have equivalently chosen any other canonical two-port network to model the coupling phenomena, such as a ‘T’ or ‘Π’ circuit. For example, a ‘T’ representation with capacitors in the series arms of the circuit and an inductor in the shunt arm would effectively model the quasi-static electric and high-frequency magnetic coupling effects, respectively. Now, we define Xl = 2Ql (frl − 1) and Xu = 2Qu (fru − 1), so that

equation image

The real and imaginary parts of (4) can be extracted readily after some simple algebra,

equation image

The phase of Zin can then be written as

equation image

The input impedance component of the group delay for the stacked patch antenna is given by the frequency derivative of the phase response, i.e.,

equation image

To compute (7) analytically, we first define A(f) to be the argument of the arctan function in Figure 6. As a result, we can express (7) as

equation image

The frequency derivative in (8) can be obtained readily, and as such (8) represents an analytic closed-form expression for the group delay of a linearly polarized lossless stacked patch antenna.

2.2. Group Delay of a Lossless Circularly Polarized Stacked Patch Antenna

[9] The treatment for a circularly polarized lossless stacked patch follows a similar approach. The corresponding equivalent circuit model for the CP stacked patch is shown in Figure 3.

Figure 3.

Transmission line equivalent circuit for a lossless circularly polarized stacked patch antenna.

[10] Note that in this transmission line model, each patch is modeled by two resonant RLC circuits consisting of Zequation image (Requation image, Lequation image, Cequation image) for the lower patch, and Zequation image (Requation image, Lequation image, Cequation image) for the upper. This allows for the slightly different resonance frequencies along the two patch axes. This is of course the mechanism by which we obtain circular polarization [Jackson and Alexopoulos, 1991]. To compute the input impedance of this model, we first make the assumption that the patch geometries are almost square, i.e., LW (in fact, W/L = (2Q + 1)/(2Q − 1) [Dong et al., 2006]), and the probe is located along the diagonal of the fed patch. With this assumption, it can be readily shown that the excitation coefficients for the TM10 and TM01 modes are equal. As a result, we can simplify the model somewhat by letting Requation imageRequation imageRl, Requation imageRequation imageRu, Qequation imageQequation imageQl and Qequation imageQequation imageQu.

[11] Based on this circuit model, the equivalent impedances for the lower and upper patches are given by

equation image

and

equation image

where fequation imagef/fx, fequation imagef/fy, fx = fl,u (1 + 1/2Ql,u) and fy = fl,u (1 − 1/2Ql,u) with the appropriate subscript chosen to denote either the lower or upper patch. With these definitions in place, we can express the input impedance of the lossless CP stacked patch arrangement as

equation image

where Xp is the probe inductance given by (3), and Xl = 2Ql (frl − 1) and Xu = 2Qu (fru − 1) are defined analogously to the LP case. The input impedance can be expressed as

equation image

At this stage, we introduce the following definitions to simplify the notation. Let fequation image ≡ 1 + (1 − Xl)2, fequation image ≡ 1 + (1 + Xl)2, fequation image ≡ 1 + (1 − Xu)2, fequation image ≡ 1 + (1 + Xu)2, gequation image ≡ (1 − Xl), gequation image ≡ (1 + Xl), gequation image ≡ (1 − Xu), and gequation image ≡ (1 + Xu). The phase response of the CP stacked patch configuration can then be expressed as

equation image

Once again, we let A(f) denote the rational argument of the arctan function in Figure 13. The input impedance component of the group delay for the lossless CP stacked patch is once again given by (8).

2.3. Synthesis of Equivalent Circuit Parameters

[12] The analytical expression for the stacked patch group delay shown in (8) (based on the equivalent circuits in Figures 2 and 3) are functions of the (Rl,u, Ll,u, Cl,u) parameter values, which are yet unknown. To compute these circuit element values, we employ a conventional network synthesis procedure [Van Valkenburg, 1955; Yu et al., 2005; Mangold and Russer, 1999; Schaubert, 1979]. The complex input impedance as a function of frequency is first computed using an accurate full-wave solver, such as Ansoft's HFSS (Copyright, Ansoft Corp.). Alternatively, measured data can also be utilized, depending on the model's intended application(s). A Padé approximation is then employed to generate a reduced-order model of the input impedance for the stacked patch arrangement using model-based parameter estimation [Miller, 1998a, 1998b, 1998c; D'Amore and Sarto, 1994].

[13] To this end, we first note that the input impedance of a single patch antenna can be written as

equation image

where s = = j(2π f) is the complex (Laplace domain) frequency. We note that (14) above is cast in terms of a rational polynomial representation,

equation image

where N0 = 0, N1 = 1/C, D0 = 1/(LC), D1 = 1/(RC), and D2 = 1 are readily determined by direct comparison with the algebraic form in (14). The circuit element values for each patch can thus be obtained via the simple synthesis procedure outlined herein, where the numerical values of the unknown coefficients in (15) are computed by sampling the data derived from the full-wave simulation (or measurements) at two distinct frequencies. Since the impedance data is complex, this leads to four linear equations that can be solved for the four unknown coefficients, N0, N1, D0, and D1 [Van Valkenburg, 1955]. This procedure is followed twice, once to extract (Rl, Ll, Cl) for the lower patch, and once again to extract (Ru, Lu, Cu) for the upper. This is analogous to the ‘windowing’ approach described by Miller [1998a, 1998b, 1998c]. We should note that a much more accurate Padé representation of the impedance function can be obtained by employing higher-order polynomials for the numerator and denominator of the rational expansion in (15). This, however, would disrupt our ability to arrive at an analytically exact form for the input impedance component of the antenna group delay. Alternatively, we could also employ oversampling of the full-wave or measured data and use a least-squares algorithm to solve the overdetermined linear system.

3. Results

3.1. Linearly Polarized Lossless Stacked Patch Antenna Group Delay

[14] As an initial validation test of the parameter extraction routine, we compute the input impedance of a stacked patch using a high-order Padé approximation. The order of the numerator and denominator polynomials was chosen as ten. A comparison of this result with the HFSS full-wave simulation of the patch is shown in Figure 4. The results of the MBPE model are almost indiscernible from those of HFSS. This result validates the extraction routine and is an indication that we have the capacity to model the input impedance characteristics of the patch to a high degree of fidelity. The tradeoff in using such a high-order approximation is that we no longer have the ability to formulate a closed-form expression for the antenna group delay.

Figure 4.

Comparison of stacked patch input impedance from HFSS and a high-order Padé approximation.

[15] Figure 5 illustrates the results of using a low-order rational approximation for the linearly polarized stacked patch input impedance. In particular, we employ a linear function for the numerator, and a quadratic polynomial for the denominator, thus matching the orders required by our circuit synthesis model shown in (15). The circuit model results deviate slightly from the HFSS results. In light of the result shown in Figure 4, we can attribute this entirely to the decrease in model order. As already mentioned, the entire dual-band frequency characteristics of the antenna are captured using only four distinct sample points, two for the lower band (centered at fo = 1.227 GHz) and two for the upper band (centered at fo = 1.5 GHz).

Figure 5.

Comparison of linearly polarized stacked patch input impedance from HFSS and an equivalent circuit model based on a low-order MBPE scheme.

[16] The input phase for the linearly polarized stacked patch is shown in Figure 6. The HFSS and circuit model results are both shown for comparison purposes. There is some discrepancy above 1.5 GHz, but this is unavoidable because the four sample points must be judiciously distributed around the band centers. The group delay variation versus frequency for the linearly polarized patch is shown in Figure 7. A comparison of the HFSS result with the closed-form analytical expression derived from the equivalent circuit shows excellent agreement between the two calculations. We should note, however, that the HFSS calculation takes hours to calculate, while the analytical expression takes only seconds. As expected for a linearly polarized patch, the group delay is inversely proportional to the resonance frequency. In fact, for a single patch, it can be shown that the peak group delay is given by τmax = Q/(π fo). This behavior is quite evident in Figure 7. The lower patch has a delay of ∼15.5 ns, while the upper has a delay of ∼11.5 ns, according to the equivalent circuit model. The HFSS results do not exhibit the predicted frequency dependence. This may be due to the coarser sampling of the impedance data as well as the use of a low-order finite differences scheme to numerically compute the frequency derivative of the phase function – two issues that are circumvented with the use of the closed-form expression derived herein.

Figure 6.

Comparison of linearly polarized stacked patch input phase from HFSS and the Padé approximation.

Figure 7.

Comparison of linearly polarized stacked patch group delay from HFSS and the closed-form analytical expression derived from the equivalent circuit.

[17] Due to the analytical nature of our closed-form expression for group delay, we can now investigate the effect of various parameters on the group delay of a stacked patch antenna. For example, Figure 8 illustrates the relationship between individual patch Ql,u and the peak (resonance) group delay of the antenna. Note that this result is for a stacked configuration, consisting of the coupled pair of patches. Thus, although we are varying the Ql,u of either the lower or upper patch, the group delay characteristic shown is that of the stacked arrangement and thus accounts for the coupling phenomena. Also, we should note that Cl,u and Ll,u are varying in order to maintain the constant resonance frequency and input resistance for the respective patch in this result. It is apparent from Figure 8 that the group delay of a linearly polarized stacked patch is directly proportional to the patch Ql,u. This behavior is again consistent with that of a single patch, where τmax = Q/(π fo). The results in Figure 8 also indicate that for a given patch Ql,u, the lower patch has a slightly higher group delay than the upper patch. This is due to the lower resonance frequency of the lower patch.

Figure 8.

Variation in peak (resonance) group delay versus Q for a linearly polarized stacked patch.

[18] The variation in peak group delay with individual patch input resistance is shown in Figure 9. Again, we are varying the resistance of either the lower or upper patch, and computing the group delay of the stacked arrangement. We are holding Ql,u constant for this result, and as a result Cl,u and Ll,u are varying in order to maintain the constant Ql,u and respective resonance frequencies. Two interesting trends are noticeable from these results. First, the antenna group delay approaches an asymptotic value as the input resistance increases (∼16 ns for the lower patch and ∼11 ns for the upper). Second, the rate at which this asymptote is approached is inversely proportional to the patch Ql,u.

Figure 9.

Peak group delay variation versus patch resistance for a linearly polarized stacked patch arrangement.

[19] The variation in peak antenna group delay with individual patch inductance Ll,u for the linearly polarized case is shown in Figure 10. Again, we are varying either the lower (Ll) or upper (Lu) patch inductance, and computing the group delay of the stacked configuration. For this result, the patch resonance frequencies fequation image and input resistances Rl,u are held constant. The patch capacitance Cl,u (and in turn the Ql,u) is allowed to change in order to maintain the desired resonance; the index is chosen consistent with the choice of lower or upper patch parameters. The results illustrate that the group delay asymptotically approaches infinity as Ll,u → 0. In addition, the antenna group delay changes very little as Ll,u increases towards larger values. We should note that for the practical cases considered in this study, the patch inductance is typically ∼0.3 nH. For relatively small changes about this value, the antenna group delay is not very sensitive.

Figure 10.

Peak group delay variation versus patch inductance for a linearly polarized stacked patch arrangement.

[20] Finally, Figure 11 illustrates the relationship between peak antenna group delay and patch capacitance Cl,u. For this result, Rl,u is held constant, and Ll,u is allowed to vary to maintain the desired resonance frequencies fequation image again. The patch Ql,u is obviously changing proportionally with the capacitance. The results indicate that antenna group delay is strongly dependent on the patch capacitances. This is consistent with results shown earlier because the patch Ql,u is directly proportional to the input capacitance, and the antenna group delay is directly proportional to Ql,u, as we have already seen. For the practical cases considered here, the capacitances are on the order of 30–50 pF. The results indicate that in this neighborhood, a 10 pF change in capacitance can give rise to a 7 ns change in antenna group delay, which is obviously significant.

Figure 11.

Peak group delay variation versus patch capacitance for a linearly polarized stacked patch arrangement.

3.2. Circularly Polarized Lossless Stacked Patch Antenna Group Delay

[21] We now shift our attention to the circularly polarized stacked patch configuration. The input impedance for the circularly polarized stacked patch configuration is shown in Figure 12. A comparison of the HFSS results with those of our reduced-order model reveals relatively good agreement between the two models. Once again, it is important to point out that the Padé approximation is making use of only four distinct sample points to extract the dual-frequency characteristic shown in Figure 12. So it is quite remarkable that we are able to reproduce such relatively complex behavior with such a low-order model. The resistance curves in the plot exhibit the characteristic dips at the resonance frequencies (∼1.26 GHz and 1.6 GHz), while the reactance curves exhibit a small inflection, or ‘blip,’ in their response. These features are consistent with and due to the circular polarization of the antenna.

Figure 12.

Comparison of circularly polarized stacked patch input impedance from HFSS and an equivalent circuit model based on a low-order MBPE scheme.

[22] The input phase as a function of frequency for the circularly polarized stacked patch is shown in Figure 13. Again, the agreement between the reduced-order model and HFSS is fairly good. The phase function exhibits the small inflection ‘blips’ in the response at the resonances, which is a manifestation of the behavior evident in the resistance and reactance characteristics of Figure 12.

Figure 13.

Comparison of circularly polarized stacked patch input phase from HFSS and a high-order Padé approximation.

[23] The frequency variation of group delay for the circularly polarized stacked patch configuration is shown in Figure 14. The HFSS and equivalent circuit model results are shown for comparison purposes. We note that in contrast to the response shown in Figure 7, the circularly polarized case exhibits a dual-peak behavior over each frequency sub-band. In addition, the group delay at the resonance frequencies themselves (∼1.26 GHz and 1.6 GHz) is in fact slightly negative. The delay is also negative in between bands as well. The HFSS result also indicates the existence of another higher-order patch mode resonance at 1.8 GHz, which we are not concerned with and can safely disregard for the purposes of the present study.

Figure 14.

Comparison of circularly polarized stacked patch group delay from HFSS and the closed-form analytical expression derived from the equivalent circuit.

[24] The dependence of peak group delay on individual patch Ql,u for the circularly polarized case is shown in Figure 15. Due to the dip in the resistance at the resonance frequencies for the circularly polarized case, we choose to compute the antenna group delay at one of the actual peaks of the response. The two peaks associated with a particular resonance are located approximately at f(τgmax) ≈ fl,u (1 ± equation image) [Dong et al., 2006]. The results illustrate that the group delay of the antenna is once again proportional to Ql,u, indicating that this phenomenon is independent of antenna polarization.

Figure 15.

Variation in peak group delay versus Q for a circularly polarized stacked patch. The peak values here are located approximately at f(equation image) ≈ fl,u (1 ± equation image). The lower patch is resonant at fl ≈ 1.26 GHz, and the upper patch is resonant at fu ≈ 1.6 GHz.

[25] The peak group delay versus patch input resistance for the circularly polarized patch is shown in Figure 16. We vary the resistance of a single patch, and compute the group delay of the stacked configuration. We hold Ql,u constant, and as a result Cl,u and Ll,u vary in order to maintain a constant Ql,u and resonance frequencies. Again, the antenna group delay approaches an asymptotic value as the input resistance increases (∼8 ns for the lower patch and ∼5 ns for the upper). The rate at which this asymptote is approached is inversely proportional to the patch Ql,u. This behavior is also very similar to that encountered for the linearly polarized case, so it is apparent that the phenomenon is independent of polarization.

Figure 16.

Peak group delay variation versus patch resistance for a circularly polarized stacked patch arrangement.

[26] The dependence of peak group delay on patch inductance is illustrated in Figure 17. Again, we vary the inductance of a single patch, and compute the group delay of the stacked topology. The resonance frequencies fequation image and input resistances Rl,u are held constant. The patch capacitance Cl,u (and Ql,u) is allowed to vary to maintain the desired resonance of either the lower or upper patch. The group delay asymptotically approaches infinity as Ll,u → 0. In addition, the antenna group delay changes very little as Ll,u increases towards larger values. For the practical cases considered in this study, the patch inductance is typically ∼0.1 nH. The antenna group delay is relatively insensitive to small changes in inductance about this nominal value range. This behavior is similar to that encountered for the linearly polarized case, so it is apparent that this phenomenon is also independent of polarization.

Figure 17.

Peak group delay variation versus patch inductance for a circularly polarized stacked patch arrangement.

[27] The variation in peak antenna group delay with patch capacitance is shown in Figure 18 for the circularly polarized case. In this calculation, Rl,u is held constant, and Ll,u is allowed to vary to maintain the fixed resonance frequencies fequation image. The patch Ql,u varies proportionally to the capacitance. The results indicate that the antenna group delay is directly proportional to Cl,u. For the practical cases considered, the capacitances are on the order of 0.1–0.3 nF. The results indicate that in this neighborhood, a 10 pF change in capacitance can give rise to a 7 ns change in antenna group delay, which is again significant.

Figure 18.

Peak group delay variation versus patch capacitance for a circularly polarized stacked patch arrangement.

4. Conclusions

[28] In this study, we have investigated the group delay characteristics of linearly and circularly polarized lossless stacked patch antennas. To this end, a new closed-form analytic expression for group delay was derived for both linear and circular polarization. The development of the analytic expression was motivated by two requirements: the desire for an accurate representation of the group delay for a lossless stacked patch configuration with linear or circular polarization, and the need for an efficient means of computing the group delay, which would in turn lead to a simple methodology for conducting parametric/trade studies of the antenna group delay characteristics.

[29] The analytical model was constructed by employing a simple equivalent circuit for the stacked patch configuration. The circuit consists of resonant RLC (tank) sections that model each patch. The total number of tank sections is two for the linearly polarized case and four for the circularly polarized case. The group delay of the antenna is calculated approximately by taking the frequency derivative of the input impedance phase function. In actuality, the group delay of the antenna is composed of the contribution from two constituents, the input impedance and the far-field phase. Since the far-field phase is typically a weak function of frequency, we chose to disregard it for the purposes of the present study and approximate the group delay solely as the frequency variation of the antenna input impedance. The accuracy of this approximation is proportional to the Q of the patches. This approximation renders the relevant group delay expressions in closed form, since the input impedance and in turn the input phase functions can be expressed analytically for either polarization. Both expressions are functions of the circuit parameters (Rl,u, Ll,u, Cl,u) , the values of which must be determined for a particular antenna under investigation. To arrive at these values, we employed a reduced-order Padé approximation to extract the parameter values by making use of sampled data obtained from either a full-wave simulation (using Ansoft's HFSS, for example) or alternatively via measured data. A circuit synthesis methodology is employed, which conveniently maps the extracted rational polynomial coefficients from the Padé expansion onto an impedance function that is derived from the equivalent circuit. The results illustrate that the parameter extraction routine does a very good job of capturing the frequency variation of the impedance and phase functions. Comparison of the analytical results with the HFSS numerical calculations of the impedance, phase, and group delay quantities show fairly good agreement, especially in light of the fact that our Padé expansion consists of a first-order numerator polynomial and a second-order denominator polynomial. This reduced-order approximation was a necessary compromise between the fidelity of the equivalent circuit model and the ability to obtain the input impedance component of the group delay in an analytically exact fashion.

[30] With the analytical model in hand, we then proceeded to conduct an investigation of the relationship between the antenna group delay and various relevant parameters. Results shown herein establish that the group delay of a stacked patch antenna is proportional to the patch Q and the patch capacitance. We also found that the peak group delay of the antenna approaches an asymptotic value as the input resistance increases without bound, and the rate at which this occurs is inversely proportional to the patch Q. Results shown also illustrate that the peak group delay of the stacked patch approaches infinity as the inductance decreases without bound, and approaches an asymptotic value as the inductance increases. Interestingly, we found that all of these characteristics are independent of antenna polarization.

Ancillary