## 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 (*R*_{l,u}, *L*_{l,u}, *C*_{l,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.