Improved Analytical Model for a Proximity Coupled Microstrip Patch Antenna (PC-MSPA)

This article presents an accurate full analytical model to assess the impedance response of a proximity-coupled microstrip patch antenna (PC-MSPA). An improved formulation of the patch resonant frequency is used to calculate the quality factor, resonant resistance, and feeding circuit parameters of the antenna. The proposed model also assesses the PC-MSPA impedance response while considering the fabrication constraints. For the validation of the proposed model, at first, the antennas are simulated for <inline-formula> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula>-, <inline-formula> <tex-math notation="LaTeX">$C$ </tex-math></inline-formula>-, and <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula>-bands. Then, two prototypes are fabricated and measured. The proposed model predicts the antenna resonant frequency and impedance bandwidth with less than 1% error.

in the twentieth century [1], [2], it has been widely used due to its low profile, ease of fabrication, and easy integration with microwave circuits [3]. However, this antenna usually presents challenges when the bandwidth requirement is larger than 10%. Different strategies have been proposed in the literature to deal with these limitations [4], [5]. The feeding technique of an MSPA plays an important role in its frequency response. The probe-fed technique [6] produces a highly inductive impedance response limiting the bandwidth to a few percent. The aperture coupling feeding [7] produces low spurious radiation due to the isolation between the feed and the patch. The proximity coupling feeding [8] is more likely to behave with capacitive impedance and wider bandwidth.
Antenna modeling through analytical approaches allows accurate estimations of the antenna behavior, e.g., the bandwidth [9]. The first documented design guidelines and analytical models appeared in the early 1980s [6], [10], [11]. Diverse models for MSPAs with different feeding techniques were developed decades ago, beginning with the transmission line and the cavity models [7]. Rigorous studies and analyses for rectangular and circular MSPAs employing probe-and inset-fed techniques were performed in [6] and [12], providing an equivalent RLC circuit for the patch. In [13], an analysis of the aperture-coupled MSPA based on integral equations and Green's functions was presented, constituting an initial step in the development of circuit-based models. A detailed investigation of an electromagnetically coupled rectangular patch antenna was presented in [14], proposing an equivalent circuit based on the reciprocity theory. More recent models for the resonant frequency have been developed for a circular patch antenna [15] with less than 2% of error. For rectangular patches, existing models [16]- [18] predict the resonant frequency with errors between 1.5% and 5.5% for substrate thicknesses in the range between 0.003λ 0 and 0.05λ 0 .
Although there are models for microstrip patch antennas, limited models on the proximity-coupled microstrip patch antenna (PC-MSPA) are published. The accuracy of the PC-MSPA is compromised due to the complexity of the feeding mechanism and interaction with the patch voltage and current distribution. The impedance of the PC-MSPA cannot be directly determined under the end of the transmission line position unless the fringing fields around the edge and the variable stripline characteristic impedance are considered. In [8], a numerical circuit model for the PC-MSPA was obtained. The model consists of a capacitor in series with an RLC resonator, and the behavior of the feeding was predominantly capacitive when the overlapping section of the feed was low compared to the patch length. In [19], a circuit model for a circular PC-MSPA with a hybrid feed of an L-strip line was developed. In this model, the total capacitance resulted from the direct capacitances between the overlapping portions of the feed and the plates above and below it, as well as the fringing capacitances between the open ends of the feed and the patch and the ground plane. This is the nearest approach to the work proposed in this article, where the L-strip portion of the feeding in the original work can be suppressed to obtain the antenna geometry used in this work. However, this model is not accurate. Limitations in the accuracy of the resonant frequency and resonant resistance in microstrip patches significantly affect the accuracy of the impedance response model in PC-MSPAs. This article proposes an improved model based on an electric equivalent circuit to predict the impedance response of a two-layer single-material PC-MSPA.
This model includes the formulation of the patch resonant frequency, quality factor, and resonant resistance, as well as the feeding circuit elements. A mathematical and physical background on microstrip antennas is provided in Section II. Then, the proposed model of a PC-MSPA and its mathematical formulation are presented in Section III. In Section IV, four antenna prototypes are designed, fabricated, and tested to validate the proposed model and their results are discussed in Section V. Finally, the conclusion of this work is presented in Section VI.

II. BACKGROUND AND THEORY FOR THE PC-MSPA
The geometry of a PC-MSPA is shown in Fig. 1(a) and (b), having two substrate layers, the patch, the ground plane, and a feeding transmission line between the layers. The bottom substrate (h 1 , ε 1 ) supports the feed at the top and the ground plane on the bottom, whereas the top substrate (h 2 , ε 2 ) supports the patch on the top. A substrate thickness ratio can be defined as r h = h 2 / h 1 . For this study, both substrates are of the same material and the same thickness, i.e., ε 1 = ε 2 = ε r and r h = 1. The patch length, which determines the resonant frequency ( f 0 p ), is set to L ≈ λ g /2. To make the impedance response stable, the ground plane size is L g = W g = 2λ 0 . The transmission line with length L f , used as a feed, has a section L 0 that overlaps the patch. The position x = 0 in Fig. 1(a) is the reference location where Z in is obtained.
From a circuit perspective, the MSPA can be modeled as an RLC parallel resonator in series with an equivalent circuit of the feed section. The equivalent circuit model of the PC-MSPA is shown in Fig. 1(c), where the RLC circuit and the LC circuit account for the patch and the proximity-coupled microstrip line, respectively [6]. The patch RLC circuit is fully defined by the resonant frequency f 0 p 15], [18], the quality factor Q p [6], [16], and the resonant resistance R p [16], [18]. The value of R p is related to the maximum resonant resistance R pM and the feed position L 0 . Using the antenna dimensions of the geometry shown Fig. 1(a) and (b), the parameters f 0 p , Q p , and R pM can be obtained.
Considering the patch as a microstrip transmission line of width W , over a substrate with a thickness of h T and a relative permittivity of ε r , the effective relative permittivity ε re can be calculated as [18] ε re = (ε r + 1)/(2) + (ε r − 1)/(2)(1 + 10h T /W ) −0.5 . (1) However, the more accurate effective relative permittivity ε rep used in the design procedure has been empirically determined to be as in (2). It is worth noting that (2) has been slightly modified from the original formulation of [17] to get more accurate resonant frequency On the other hand, the fringing fields, which are due to the finite dimensions of the patch, are counted through an small extension L [20]. A precise expression of L is listed in (3) where ζ 1 , ζ 2 , ζ 3 , ζ 4 , and ζ 5 are also defined in [20] Thus, the resonant frequency of the RLC resonator is where c 0 is the free-space speed of light and L e is the patch effective length L e = L + 2L. Hence, the wavelength (λ 0 p ) and the wavenumber (k 0 p ) at resonance are The resonator quality factor Q p [16] can be calculated considering the losses of the dielectric, the conductor, the radiation, and the surface waves through Q d , Q c , Q rad , and Q sw , respectively, and it is given by where δ d , μ, and σ are the substrate tangent loss, substrate permeability, and copper conductivity, respectively. The values of p, c 1 , and e hed r are defined in [16]. The resonant resistance R p [16] is related to its maximum value R pM and the effective feed position L 0e = L 0 + L along the patch effective length L e and R pM is expressed as The components of the parallel RLC circuit [16] for a microstrip patch are expressed as Then, the total impedance Z p of the parallel RLC circuit of the patch can be calculated using (13)

A. Problem and Proposed Solution
Although antenna models for the PC-MSPAs are available in the literature and some of them have become progressively more sophisticated, there are still limitations in predicting the impedance behavior for different lengths of the feed. To illustrate these limitations, Fig. 2 shows the impedance behavior of a PC-MSPA along different overlap ratios (l 0n = L 0 /L) (0 < l 0n < 1) using antenna Design 1 (see Table I). The limitations on the model can be observed when exciting the patch underneath its center (l 0n = 0.5), which produces an all-zero input impedance as R p = 0 according to (10), which is not true as observed in Fig. 2. The curves in Fig. 2(a) show a decaying resonant resistance as the feed becomes longer. This implies an alternative profile of the radiation resistance that does not necessarily correspond with the cosine-squared profile of (10). The patch resonant frequency and quality factor change slightly across the feed lengths, especially when 0.25 < l 0n < 0.75. Also, very high resistances and shifts in the patch resonant frequency are observed when l 0n > 0.90. Using lower overlap ratios, it is shown in Fig. 2(b) that the antenna response is predominantly capacitive, as already observed in [8]. On the other hand, for high overlap ratios, the response tends to be more inductive. No phase rotation is perceived in the response while l 0n < 0.75, indicating that the increase of the reactance is produced by larger capacitances and inductances. If only the capacitances are considered, then the response would be limited because the maximum reactance value would be half of the resonant resistance, which is not true especially when l 0n > 0.70.
In summary, the behavior of a PC-MSPA can be dependent on the overlap ratio as follows.
1) Short Overlaps (l 0n < 0.25): The impedance response has capacitive reactance, and the resonant resistance dramatically changes with slight variations of the feed length. An equivalent capacitor in series with the RLC resonator fits very well in this case.   equations for the patch resonator and the feeding inductance and capacitance, integrating them in a hybrid model structure of a transmission line in series with an LC-RLC circuit.

III. PROPOSED MODEL FOR THE PC-MSPA
In this section, a new model for a PC-MSPA is presented. The model is divided into three parts: a patch RLC resonator, a feeding circuit, and a feeding equivalent transmission line. Notice that for moderate overlaps, on which this work focuses, the equivalent transmission line can be ignored. An equivalent circuit of the proposed model is presented in Fig. 1(c). The input impedance of the PC-MSPA Z in is given by (14) This means that the real part of Z in only depends on the patch resonator, whereas the imaginary part depends on both the patch resonator and the feeding.

A. Patch RLC Resonator
The parameters f 0 p , Q p , and R pM can be calculated considering the average relative permittivity of the two substrates ε r , the total thickness h T = h 1 + h 2 , and the length of the patch L. For R p , the length of overlapping section of the feed L 0 is also required for the calculation. Because the coupling mechanism in the PC-MSPA depends on the substrates thickness ratio, the expressions (4), (7), (8), and (9) of f 0 p , Q p , R pM , and R p of Section II need to be reformulated.
The RLC resonant frequency in PC-MSPAs ( f 0 p ) has been observed to shift upward with respect to the RLC resonant frequency of the same patch but with probe feeding. To consider this shift and use (4), a factor F f 0 is applied as expressed in (15). In order to avoid any confusion, the RLC resonant frequency calculated in (4) is renamed to f 0r , and thus, the corresponding wavelength (5) is also renamed to λ 0r . Then, f 0 p is expressed as where F 0 = 1.02 − 0.045/(ε r ) 1/2 and F 1 = (0.7376/r h + 0.4754)/(ε r ) 1/2 . The result of (15), obtained by curve fitting techniques [21], has an error less than 1% with simulated data of square PC-MSPA designs and over the range ε r ∈ [1.7; 3.66], r h ∈ [0.75; 1.25], and h T ≤ 0.1λ 0r /(ε r ) 1/2 . After f 0 p is calculated, (5)-(6) can be used to obtain λ 0 p and k 0 p , which will be used to get Q p and R p .
Following the patch cavity model, its effective length L e can be calculated using (16), where f 0 p has been calculated in (15). Hence, the value of L for PC-MSPAs can be rewritten as in (17) The RLC quality factor in a patch is a function of the resonant frequency, the relative permittivity, and the patch effective dimensions (L e and W e ). By comparison with simulated data, it was observed that a value of W e as in (18) enables prediction of the quality factor of the square PC-MSPA with errors less than 10%. It is important to indicate that this value of W e only represents a mathematical estimate Consequently, the RLC quality factor of PC-MSPAs can be calculated using (7) considering the resonant frequency f 0 p of (15) and the effective dimensions of (16) and (18).
The RLC resonant resistance R p of PC-MSPAs can be rewritten as in (20). The value of R pM expressed in (19) was obtained from (9) using L 0 = 0 and a square patch, and it can be calculated using The shape curve F Rp is expressed in (21) for short and moderate overlaps assuming that h 1 = h 2 F Rp = 32.38 1 − l 0.005 0n + 0.14.
A more accurate expression of F Rp that includes the influence of the thickness ratio r h = h 2 / h 1 is These terms, obtained by curve fitting techniques with nonlinear regression [21], provide a fast-decay exponential shape dictated by p 1 and a slow-decay shape, dictated by p 2 . Typical values of p 1 are around several tens, while p 2 is a value between 0.5 and 1.5. Therefore, for moderate overlap ratios, the first term of (22) can be ignored to calculate F Rp .

B. Feeding Circuit
The feeding structure can be modeled as an in-series LC circuit, as shown in Fig. 1(c). The variation of the feeding inductance and capacitance (L T , C T ) is shown in Fig. 3(b) and (c) as a function of the overlapping ratio l 0n . These values were obtained by selecting the combination (L T , C T ) that produced the least impedance error between the model and simulated results of Z in . Hence, L T and C T can be expressed as in (30) and (31) for moderate overlap ratios as demonstrated in the following.
First, for ε r1 = 2.2, μ r1 = 1, and f 0 p1 = 2.945 GHz where A 1 = 0.1587, B 1 = −11, and B 3 = 1.797. Since the patch is designed to have a length of L = λ g /2 and the overlapping length of the feeding L 0 depends on L to have l 0n constant, then (26) and (27) can be generalized as Thus, replacing (28) and (29) in (26) and (27) where A 1s = 0.4674, B 1s = −48.05, B 3s = 7.85, f 0 p is in GHz, L T is in nH, and C T is in pF. As observed in (30) and (31), the feeding equivalent inductance is negligible at short overlaps, but it becomes significant as the overlap ratio increases. Also, the equivalent capacitance increases up to a maximum around l 0n = 0.45 and decreases as the overlap ratio increases.

C. Feeding Equivalent Transmission Line
For short and moderate overlaps, the transmission line of Fig. 1(c) can be ignored since the response does not get shifted and the input reactance increases almost linearly with the frequency, i.e., having an inductance. However, for large overlaps, an equivalent transmission line of length L 0 needs to be added to the LC feed circuit model, where L 0 ≤ L 0 . Physically, this line may be due to the proximity between the open ends and the nonplanar feature of the structure, which is more noticeable at large overlaps. Because the overlapping portion of the feeding is between the patch and the ground plane, the equivalent transmission line takes the form of a stripline (with impedance Z 0s ) [22] instead of an embedded microstrip line (with impedance Z 0u ) [23], [24]. Notice that the equivalent resonant resistance may be less than that plotted in Fig. 3(a) when including this equivalent transmission line. The feeding inductance and capacitance may not continue following the pattern pictured in Fig. 3(b) and (c), as l 0n → 1, Q p may decrease and R p may get close to zero.

IV. VALIDATION OF THE PROPOSED MODEL
The proposed model is validated with simulated and measured results of the impedance of PC-MSPAs designed at 3, 3.5, 5.4, and 9.4 GHz. The proposed antenna design is shown in Fig. 1(a) and (b) and the dimensions of the proposed designs are listed in Table I. In order to show the generalized capability of the proposed model, the proposed designs are different from the ones used to get the model equations. Different overlap ratios l 0n are used to show the model assessment, regardless of the maximum return loss that the antenna gets at f 0 . An assessment in both Z in and S 11 parameters is presented. Furthermore, the radiation patterns are evaluated and discussed at the end of this section.
The model performance using the deembedded wave port in HFSS (ideal condition) is shown in Fig. 4 through a comparison between the modeled and simulated responses of the input impedance and the S 11 parameter. The results were obtained considering an ideal wave port in the simulated designs and deembedding the port from the position x = −(L f − L 0 ) to x = 0. In addition, the values of the patch RLC resonator parameters are compared in Table II between the modeled and simulated values.
The assessment of the model performance using a coaxial probe feed (real condition) is performed through the comparison between the modeling, simulations, and measurements of antenna designs 2 and 3 of Table I. Even though the antenna fabrication was intended to replicate the same specifications during the fabrication process, an air gap h a and a slight movement of the patch p m occurred. The effect of fabrication imperfections is shown in Fig. 5. This effect needs to be considered in the model to have more accurate results.
The air gap h a is included in the model by reformulating the effective thickness used for the patch substrate h 2 (32) instead of the physical thickness h 2 . Thus, the thickness ratio and the total thickness are also affected by h 2 in (33) and (34) Moreover, the average relative permittivity [25] ε r is calculated as The patch movement along the x-axis is also considered in the model, especially to determine the overlap ratio, which is a meaningful parameter needed to accurately determine the resonant resistance R p . In Fig. 5(c), this shift is indicated as p m , which is positive if the patch is moved as indicated in the arrows and negative if moved in the opposite direction. The inclusion of p m in the model is done with an effective overlap ratio of l 0n calculated from the expected overlap ratio l 0n as The described dimensions are listed in Table III. The specifications not mentioned were already listed in Table I.
The modeled impedance Z in from the reference position at x = 0 is transformed to the measurement position as Z in0 , as shown in Fig. 5(b), located under the ground plane and connected through a 50 coaxial cable and a probe.
Hence, Fig. 6 shows a comparison between modeled, simulated, and measured values of the impedance and S 11 parameter when the antennas include a coaxial connector under the specific location (x = −(L f − L 0 )). This 50 connector has an inner and outer radius of 0.65 and 2.10 mm, respectively, and the dielectric is Teflon-based material (ε r = 2.0). Table IV lists a comparison between measured, simulated, and modeled resonant frequencies f 0 , in-bandwidth minimum f l and maximum f h frequencies, and the impedance bandwidth of the antenna designs from the S 11 parameter data.

A. PC-MSPA Radiation Patterns
The expressions of the far-field radiation patterns of a PC-MSPA over a theoretical infinite ground plane are given in (37) and (38) using the patch dimensions and substrate where X = k L/2 sin θ cos φ, Y = kW/2 sin θ sin φ, Z = k H T cos θ , sinc α = (sin α)/α, and tanc α = (tan α)/α are defined in [26].  The fields produced by the PC-MSPA can be obtained using the equivalent radiating slots using the patch cavity model [18]. The length of each slot is represented by the width of the patch (W ). The separation between slots is dictated by the patch length (L). In the previous expressions, the term sinc Z can be ignored since sin(Z ) can be approximated to Z when very thin substrates are used. The modified 2nd Ludwig's definition of the cross polarization is used [27], [28]. According to [28], there are two variants of this definition, depending on the feeding orientation. The variant 2-I is used for patches fed along the x-axis, as shown in Fig. 5. The simulated and measured radiation patterns of antenna designs 2 and 3 are shown in Fig. 7, where a good agreement between them is found for the E-, H-, and D-planes.

V. DISCUSSION
The plots in Fig. 4 show great agreement between modeled and simulated impedance responses when the model is validated using the deembedded wave port in HFSS (ideal condition). A frequency shift is more perceptible in D2 as it has the lowest thickness. A resonant resistance shift is observed in D1, which is the electrically thickest model. Comparing the modeled and simulated real parts of the design responses, very low errors are observed. Errors are slightly more perceptible in D2, primarily as a consequence of the frequency shift. The S 11 plots of the second row of Fig. 4 show that the −10 dB bandwidth is predicted very well, as is the resonant frequency of the four antenna designs, although some difference in the coupling level due to calculation errors in R p is observed in D1. It is important to note that the inductance L T has a critical role in the accuracy of the feed reactance, especially in D3, which has a reactance response above j 0 . This would not have been possible if the model considered only a capacitor, as seen in the third column of Fig. 4.
The comparison of patch RLC parameters listed in Table II shows that the resonant frequency f 0 p presents the errors less than 0.5%. This high accuracy is necessary because it is used as part of the required variables to calculate the other two parameters (Q p and R p ) and the feeding parameters (L T and C T ). The quality factor presented errors less than 3.5%, not showing dependence of these errors on substrate thickness. However, the resonant resistance can be more accurately determined for antennas with thinner substrates.
A model performance using a coaxial probe (real condition) also presents good agreement, even though these cases require additional transmission line transformations. As shown in Fig. 6, the impedance response and the reflection coefficient are very well predicted. The model required impedance transformations to convert the impedance response at x = 0 to the actual port location. A slight difference is noticed, but a good match in the resonant frequency and the bandwidth is obtained.
The model is able to predict the resonant frequency with errors less than 0.2%. It is important to indicate that the measurement uncertainty was also counted in the model computation, as the patch length directly affects f 0 p . Considering the frequency interval where |S 11 | < −10 dB, the model is able to predict the impedance bandwidth with error less than 1%, as listed in the last rows of Table IV. VI. CONCLUSION An accurate mathematical model of the PC-MSPA has been proposed and validated. The relative position of the feed along the patch was defined by the ratio l 0n . This ratio was used to develop an exponential-based equation for the resonant resistance, as well as a frequency-based equation for the feeding capacitance and inductance. The proposed equivalent circuit and equations allowed a simple but accurate model to be built for the PC-MSPA. The model was validated with antenna designs on the S-, C-, and X-bands. In all cases, the feed and patch had a moderate overlap ratio. Thus, the calculated −10 dB frequency interval and bandwidth presented errors less than 1% when comparing simulated and measured results, despite the differences found in the coupling level at frequencies near the resonance. The differences in the coupling level may be decreased by adjusting the equation of the patch resonant resistance. Also, the far-field radiation patterns presented high symmetry and moderately high cross-polarization isolation levels, showing that the PC-MSPA model can be useful to build a design with a highly accurate predicted impedance response. He joined the National Institute of Research and Training in Telecommunications, Peru, working in electronics engineering with a focus on electromagnetism. He worked in the X-and P-band SAR image processing to estimate forest changes in the Amazon, having a six-month stay at the Department of Telecommunications, State University of Campinas, Campinas, Brazil.