Circularly-Polarized Patch Antennas With Enhanced Bandwidth Based on Capacitively Coupled Orthogonal Patch Radiators

Circularly-polarized (CP) patch antennas with enhanced bandwidth based on capacitively coupled orthogonal patch radiators are proposed. Several patch radiators are alternately arranged one by one along the <inline-formula> <tex-math notation="LaTeX">${x}$ </tex-math></inline-formula>- and <inline-formula> <tex-math notation="LaTeX">${y}$ </tex-math></inline-formula>-direction, producing orthogonal currents and far fields. The adjacent patch radiators are coupled to each other and the coupling structure not only contributes to power distribution but also introduces 90° phase shift for circularly-polarized radiation. There is no need for an additional feeding circuit in the design. The far-field components are mapped to the excitation voltage of the equivalent circuit model and a quantitative design method based on the equivalent circuit model is proposed. The theoretical axial ratio response can be predicted and analyzed. Wideband CP patch antennas with 3 and 4 coupled radiators under a profile of <inline-formula> <tex-math notation="LaTeX">$0.035{\lambda }_{0}$ </tex-math></inline-formula> are designed, fabricated, and measured as examples. Full-wave simulation results agree well with the theoretical ones. Multiple minima in the axial ratio response are produced to form wide bandwidth. The measured overlap bandwidths of reflection coefficient, realized gain, and axial ratio for these two antennas are 4.9% and 6.3%, respectively.

Wideband CP patch antennas have been designed by including stacked patches [9], [10], parasitic elements [11], [12], [13], [14], pins [15], or metasurfaces [16] as well. Two more orthogonal modes are introduced by the stacked or gap-coupled patch to broaden the bandwidth [17]. Parasitic elements or metasurfaces are introduced to rearranged the current distribution of the antenna and achieve the equal magnitude and 90 • phase difference of two orthogonal far fields in a wide band. However, the coupling between the main patch and stacked patches or parasitic elements is often complicated in practice. Furthermore, it is difficult to calculate radiations from the parasitic elements or metasurfaces. As a result, there is currently no synthesis method available for the precise and effective design of these CP antennas. Despite the fact that the characteristic mode method can be used to extract the mode behavior of CP antennas [18], [19], [20], full-wave simulation for parameter sweep and optimization is still required.
Another class of wideband CP patch antennas is based on external feeding circuits. In [21], the impact of amplitude and phase imbalance on a dual differential fed patch antenna is analyzed. Beamwidth is increased by optimizing the phase imbalance of feeding circuits. A coupler [22] or a combination of Wilkinson power divider and quarter-wavelength delay line [23] is also commonly used in traditional feeding circuits. Since the resistor of coupler and Wilkinson power divider absorbs the unbalanced power, two ways of signals with the required phase and magnitude can be achieved in a wide band. Unfortunately, due to the power consumption of the resistor, this kind of CP patch antennas suffers from low radiation efficiency and low power capacity. The bandwidth of CP patch antennas can be further expanded by using the power divider cascaded with a wideband 90 • phase shifter [24], [25], [26], [27], [28]. Feeding circuits with metamaterial-line are also utilized to design wideband CP patch antennas [29], [30]. However, because these feeding circuits occupy a large non-radiating area, the entire antenna structure becomes more complicated and expensive. Linearly-polarized filtering antennas have been proposed in recent years. The feeding circuit and the patch antenna are considered as a whole bandpass filter network, where the patch antenna is modeled as the last resonator and the output port [31], [32]. This method has been developed to design wideband CP patch antennas. The bandpass feeding circuit is designed to provide equal excitation of two orthogonal modes of a patch antenna and then CP radiation is obtained by separating two orthogonal modes [33] or inserting a 90 • phase delay line [34], [35]. Later in [36], both the magnitude and phase responses are considered in the circuit model in order to achieve multiple AR minima and enhanced bandwidth in a co-design procedure. Furthermore, bandpass filtering circuits with modified resonators [37], [38], [39] are also demonstrated for wideband CP patch antennas. However, the feeding circuits of these antennas still occupy large area and the parasitic radiation from the resonators could degrade the CP performance. In this paper, capacitively coupled orthogonal patch radiators are employed to design CP patch antennas with enhanced bandwidth. As shown in Fig. 1, n coupled patch radiators are placed one by one along the x-and y-axes, with one patch radiator adjacent to another one. A coupled-resonator filter network is formed by cascaded orthogonal patches, with the coupling between each two adjacent patches determining the power division and phase shift. In theory, n − 1 AR minima come up to achieve enhanced AR bandwidth, as shown in Fig. 1(c). The wide bandwidth of the proposed antenna is achieved by the multiple resonances of the multiple capacitively coupled orthogonal patch antennas. If n coupled patch antennas are employed, n orthogonal modes come up. n − 1 AR minima can be achieved by each two neighboring modes. At the frequencies of AR minima, the coupling coefficients of coupled patches are calculated to satisfy the requirement of equal magnitude and 90 • phase difference of orthogonal far fields. As a result, the wide bandwidth of the proposed antenna is achieved. In contrast to the conventional CP antenna with parasitic patches, the coupling effect between patches is clearly characterized and the response is accurately predicted with the circuit model in this work. Different from the reported filtering CP patch antennas, the proposed antennas in this work require no external feeding circuits and the patches themselves serve as radiators and feeding networks simultaneously. Fig. 2 shows the proposed CP antenna with 3 coupled patch radiators, which is implemented on a single-layer substrate. Three patch radiators are placed one by one along the x-and y-axes and the patch radiator is in proximity to another one. To minimize the effect of higher-order modes and ensure VOLUME 4, 2023 473 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. adequate coupling between patch radiators, the width of the patch radiator is smaller than its the length. TM 01 modes are excited in Patch 1 and 3, while TM 10 mode is excited in Patch 2. Thus, the radiations of Patch 1 and 3 are dominated by the x-direction surface currents and the radiation of Patch 2 is dominated by the y-direction surface currents. The couplings between patch radiators transmit power and introduce 90 • phase shift at the same time. As a result, two orthogonal far-field components with equal magnitude and 90 • phase difference will be produced for CP radiation. A coaxial probe is then placed along the center line of Patch 1 for excitation and impedance matching.

B. EQUIVALENT CIRCUIT MODEL AND QUANTITATIVE DESIGN
Next, as shown in Fig. 2(c), the proposed CP antenna with three radiator elements is efficiently designed based on the equivalent circuit model. To simplify the analysis, the dimensions of the three patch radiators are set to be identical and the resonant frequency of the patch is f 0 . As well known, the patch antenna can be equivalent as a half-wavelength transmission line with two parallel radiation conductances at two ends. The characteristic impedance and electrical length of the transmission line are represented by Z 01 and θ , respectively, and the radiation conductance is denoted as G A . The couplings between Patch 1 and 2 and between Patch 2 and 3 are modeled as the J inverters with characteristic admittances of J 12 and J 23 , respectively. The J inverter has a phase shift of 90 • . It should be noted that the patch radiator is usually equivalent to the parallel resonator with the lumped capacitor, inductor, and conductance in the design of traditional CP patch antennas. However, the phase shift of transmission cannot be accounted for with this lumped element circuit model. For example, the phase shift of the lumped parallel resonator is 0 • at the resonant frequency. But actually, when the signal passes through a single patch, there is a 180 • phase shift. Therefore, the transmission-line model is more accurate for the patch radiator in this design.
In the design of the conventional CP patch antennas, as discussed in [23], [40], [41], the voltages (V x and V y ) of the patch resonators in the equivalent circuit model correspond to the radiation fields. As a result, the magnitude ratio and phase difference of two orthogonal far-field components in this design can be expressed as |V y /V x | and ϕ(V y /V x ), respectively. Referring to the geometry and equivalent circuit model in Fig. 2, the complex amplitude ratio of the two orthogonal components is given by where V 1 , V 2 , and V 3 stand for the excitation voltages of Patches 1, 2, and 3, respectively. Next, V 1 and V 2 are tried to be expressed by V 3 .
where Y 01 = 1/Z 01 . The patch can be considered as a half-wavelength resonator and its susceptance slope is b r = π Y 01 /2. According to the definitions of the quality factor and J inverter, the quality factor of the patch and the coupling coefficients between the adjacent patch radiators are given by where Q is the quality factor of the single patch radiator. J 12 is the characteristic admittance of the inverter between Patch 1 and 2, while J 23 is the characteristic admittance of the inverter between Patch 2 and 3. k 12 and k 23 are the coupling coefficients between the patch radiators. The ABCD matrix of the circuit between V 1 and V 3 can be obtained by multiplying the ABCD matrix of each element. ] are the ABCD matrixes of two inverters. According to the definition of ABCD matrix, the relationship between V 1 and V 3 can be obtained. where I 3 is the current that flows into Patch 3 in the equivalent circuit model, as denoted in Fig. 2(c). Similarly, the ABCD matrix of the circuit between V 2 and V 3 is calculated as Then the relationship between V 2 and V 3 can be obtained as Thus far, the complex amplitude ratio of the two orthogonal components can be obtained by substituting (8) and (12) into (1). It can be found that the complex amplitude ratio is only determined by the quality factor of the single patch radiator and the coupling coefficients between the adjacent patch radiators. It should be noted that when the patch is at the resonant frequency, the electrical length θ of the patch equals π . Consequently, θ can be transformed to the frequency with the following equation.
The resonance frequency of the single patch radiator is initially specified in the design. Then the dimension of the single patch radiator is determined for a selected substrate and the quality factor of the single patch radiator can be obtained as well. As a result, there are only two unknown parameters, k 12 and k 23 , in the expression of the complex amplitude ratio. Assuming that perfect CP radiation is achieved at the frequency of f 1 , the complex amplitude ratio should meet the conditions of equal magnitude and 90 • phase difference.  Two equations are created by expanding the real and imaginary parts of (14), which can be used to calculate two unknown parameters, i.e., k 12 and k 23 , can thus be calculated. With all the parameters determined, the AR response can be predicted with the complex amplitude ratio [41,Ch. 2].
where AR and ε are negative in the case of left-hand circular polarization and positive in the case of right-hand circular polarization.

C. THEORETICAL ANALYSIS
The theoretical results obtained using the aforementioned design method are discussed in order to highlight a few attractive features of the proposed CP antenna. Firstly, the AR responses under different frequencies of the AR minimum are investigated. The frequency is normalized with the resonant frequency of the single patch and f 0 is equal to 1.
Assuming that the quality factor of the single patch radiator is 23.5. Different values of f 1 /f 0 are chosen as shown in Table 1, and the coupling coefficients between coupled radiators can be calculated using the proposed design method. The theoretical AR responses are plotted in Fig. 3(a). It is observed that two minima are generated in the AR response, resulting in a widened bandwidth. Because the frequencies of the two minima are symmetric about the center frequency, only the frequency f 1 is required in the proposed design method. The ripple at the center frequency decreases as two minima in the AR response move inwards. The coupling coefficient k 12 almost keeps unchanged while k 23 gradually decreases in these cases. In other words, the frequencies of two minima are mainly affected by the coupling coefficient k 23 when the quality factor of the single patch is given. Secondly, the AR responses under different values of quality factor of the single patch are discussed. The frequency of the minimum in the AR response is set as f 1 /f 0 =0.985. Different values of Q are chosen as shown in Table 2, and the coupling coefficients between coupled radiators can be calculated with the proposed design method. The theoretical AR responses are plotted in Fig. 3(b). Two minima are observed in the AR response and their positions of minima remain unchanged as expected. When the quality factor of the single patch increases, the ripple at the center frequency rises, and the 3-dB bandwidth decreases. The required coupling coefficient k 12 decreases significantly while k 23 changes only slightly. As a result, the coupling coefficient k 12 is primarily determined by the quality factor of the patch radiator.
Thirdly, the 3-dB AR bandwidth of the proposed CP antenna is investigated. As shown in Fig. 3(a), for a given Q, the ripple at the center frequency will reach 3 dB by moving two minima in the AR response. In this case, the theoretical maximum 3-dB AR bandwidth is obtained for the given Q. The AR bandwidth of the proposed CP antenna is compared with the one of the traditional CP patch antenna. As discussed in [41], the AR bandwidth of the traditional CP patch antenna based on two degenerate modes can be calculated as follows.
where ARmax dB is the maximum allowable axial ratio in [dB]. The 3-dB AR bandwidths of the proposed and traditional CP patch antennas under different values of Q are shown in Fig. 4. When Q increases from 16 to 30, the fractional bandwidth (FBW) of the proposed antenna is reduced from 8.4% to 4.5%, while the FBW of the traditional CP patch antenna decreases from 2.2% to 1.2%. It can be found that the FBW of the proposed antenna is almost 3.5 times that of the traditional CP patch antenna. Furthermore, the required coupling coefficients can be calculated using the proposed design method at the same time, as shown in Fig. 4. When Q increases from 16 to 30, the coupling coefficient, k 12 and k 23 , gradually decrease from 0.13 to 0.07 and from 0.043 to 0.023, respectively. It can be found that k 12 is always larger than k 23 . From the point-view of physical insight, the values of coupling coefficients determine the ratio of power divided to each patch. The configuration of k 12 and k 23 in our design ensures that the two orthogonal farfield components can satisfy the magnitude condition for CP radiation.

D. SIMULATION AND EXPERIMENTAL VERIFICATION
Next, the proposed antenna structure and design method are verified by full-wave simulation and experiment. The whole antenna structure is implemented on a single-layer substrate with a thickness of 3 mm, relative permittivity of 2.2, and loss tangent of 0.0012. The resonant frequency is set to 3.48 GHz and then the length of each patch can be determined when the width of the patch is set to 12 mm. Then the quality factor of a single patch can be extracted by full-wave simulation, which is 23.5 in this design. The frequency of the AR minimum is chosen as f 1 /f 0 = 0.985. The required coupling coefficients can be calculated with the proposed design method, which are k 12 = 0.086 and k 23 = 0.025. As seen from the geometry of the proposed antenna in Fig. 1, it can be understood that k 12 and k 23 vary as a function of the coupling gaps (g 1 and g 2 ). The values of coupling coefficient (k) under specified physical dimensions can be numerically extracted by two natural resonant frequencies (f 01 and f 02 ) of two coupled patch radiators via full-wave simulation [43].
Thus, the initial physical dimensions can be determined. After minor fine tuning, the final physical dimensions can all be determined as tabulated in Table 3. It should be noted that the lengths of the three patch radiators are slightly changed due to the effects of coupling and the feeding probe.
As shown in Fig. 5, the simulated magnitude ratio (MR) and phase difference (PD) of two orthogonal far-field components are compared with the theoretical ones, which agree well with each other. In the operating band, small ripples around 1 and −90 • appear in the magnitude ratio and phase difference responses, respectively. Two minima appear in the magnitude ratio response. The phase difference response is almost antisymmetric around the center frequency. One peak is slightly larger than −90 • below the center frequency and one minimum is slightly smaller than −90 • above the center frequency. Fig. 6 plots the simulated surface current distributions of the proposed CP antenna with 3 coupled patches at 3.425 GHz over a single period. At t = 0, the surface currents on Patch 1, 2, and 3 are along −y, +x, and +y axis directions, respectively. The currents on Patch 1 are stronger than those on Patch 3. Thus, the total equivalent surface current is along the lower left direction. Similarly, at t = 1/4T, 2/4T, and 3/4T, the total equivalent surface current directions are indicated in Fig. 7(b), (c), and (d), respectively. During one period, the current direction rotates anticlockwise, producing right-handed circularly polarized radiation wave. Fig. 7 shows the simulated |S 11 | and AR responses of the proposed CP antenna with 3 coupled patches under different values of gaps. When g 12 increases, the magnitude and the center frequency of the reflection coefficient decrease, resulting in better impedance matching. When g 12 increases from 0.2 to 0.3 and 0.4 mm, the frequency of the left minimum in the AR response decreases from 3.435 to 3.426 and 3.416 GHz while the right one remains nearly unchanged. The resulting AR bandwidth is extended as the ripple in the center frequency rises. When g 23 changes from 2.2 to 2.4 and 2.6 mm, both the two minima in the AR response move inwards, resulting in decreased bandwidth and ripple variation.
Next, the proposed CP antenna is fabricated and measured for further verification. The antenna is measured

FIGURE 8. Simulated and measured reflection coefficients, realized gains, and AR responses of the proposed CP antenna with 3 coupled patch radiators.
using Keysight E5071C Vector Network Analyzer (VNA) for reflection coefficients and SUNYIELD SY-16M antenna measurement system for radiation characteristics, respectively. The simulated and measured results of reflection coefficients, realized gains and AR responses are plotted in Fig. 8. The results show good agreement between the simulated and measured values, except for a small frequency shift. The measured 10-dB return-loss bandwidth is from 3.38 to 3.61 GHz, with only one minimum in the reflection coefficient response. It should be noted that the couplings between patch radiators affect the performances of AR and S 11 at the same time. In our design, the coupling coefficients are specified to obtain multiple minima in the AR response. However, under these coupling coefficients between patch radiators, the response of S 11 may not be optimal, and multiple reflection zeros can not be achieved at the same time. Despite this, the 10-dB return-loss bandwidth is still sufficient. The peak realized gain is 7.56 dBic, and the realized gain response sharply rolls down outside the operating band. The measured 3-dB gain bandwidth is from 3.35 to 3.59 GHz. The simulated AR response matches well with the measured one, except for a small frequency discrepancy. It is found that the frequency shift of S 11 is slightly smaller than that of AR. This is because the AR response is more sensitive to the antenna configuration, and the discrepancy may be caused by the assembly error of the feeding part and tolerance of the antenna dimension. Two minima come up at around 3.45 GHz and 3.55 GHz, which efficiently broadens the AR bandwidth. The fractional bandwidth with AR<3 dB is 4.7% and 4.9% in the simulation and measurement, respectively. The measured 3-dB AR bandwidth ranges from 3.42 to 3.59 GHz. Consequently, the overlap bandwidth of reflection coefficient, realized gain, and AR is from 3.42 to 3.59 GHz and the corresponding fractional bandwidth is 4.9%.
The simulated and measured efficiencies agree well with each other, as shown in Fig. 9. The efficiency in the operating band is approximately 90%. The normalized lefthand circularly polarized (LHCP) and right-hand circularly polarized (RHCP) far-field radiation patterns in both x-z and y-z planes are measured at 3.5 GHz and the results are presented in Fig. 10. The simulated radiation patterns agree well with the measured ones, and boresight radiation is obtained as expected. The RHCP field intensity is approximately 17 dB higher than that of the LCHP counterpart in the boresight direction. The measured half-power beamwidths in the x-z and y-z planes are found to be 94.5 • and 63.5 • , respectively. These two values are different because the geometry of the proposed antenna is not symmetrical.

A. GEOMETRY AND DESIGN
In this section, the radiator number of the proposed CP antenna is increased to 4 to achieve wider bandwidth, and the antenna's geometry is depicted in Fig. 11. Similarly, 4 patch radiators are placed one by one along x-and y-axes, and each patch radiators are in proximity to the next one. TM 01 modes are excited in Patches 1 and 3, and TM 10 modes are excited in Patches 2 and 4. The corresponding equivalent circuit model is shown in Fig. 12, which is similar to the one of the CP antenna with three coupled patch radiators. The patch antenna can be equivalent as a half-wavelength transmission line with two parallel radiation conductances at two ends, and the coupling between the patch radiators is modeled as a J inverter. The voltages (V x and V y ) of the patch resonators in the equivalent circuit model correspond to the radiation fields [17], [34], [35]. Therefore, the complex amplitude ratio of the two orthogonal far-field components is given by where V 1 , V 2 , V 3 , and V 4 are the excitation voltage of Patches 1, 2, 3, and 4, respectively. Following that, V 1 , V 2 , and V 3 are tried to be expressed with V 4 , which can be calculated with the ABCD matrix as discussed in the previous section. It can be derived that the complex amplitude ratio in (17) is only determined by  the quality factor of the single patch (Q) and the coupling coefficients (k 12 , k 23 , and k 34 ). k 34 is related to the inverter J 34 , which is expressed as In the design, the quality factor of the single patch is given at first. Assuming that perfect CP radiation is achieved at the frequency of f 0 and f 1 , the complex amplitude ratio should meet the conditions of equal magnitude and 90 • phase difference.
It should be noted that the complex amplitude ratio is a pure imaginary quantity at f 0 . So only one equation is obtained from (19a). However, by expanding the real and imaginary parts of (19a) and (19b), three equations are obtained with three unknown parameters, and so k 12 , k 23 , and k 34 , can be calculated.
Assuming that the quality factor of the single patch is 23.5, different values of f 1 /f 0 are chosen, as shown in Table 4. The required coupling coefficients can be calculated with the proposed design method, and AR responses are also obtained, as plotted in Fig. 13. Three minima appear in the AR response to form a wide bandwidth. As two minima outside the center frequency move inwards, the in-band ripples gradually decline and the minimum in the center frequency remains unchanged. As a result, the AR bandwidth gradually decreases. As shown in Table 4, the coupling coefficient k 12 is almost kept unchanged, while k 23 and k 34 gradually decrease. Therefore, for a given Q, the AR minima are mainly determined by k 23 and k 34 .

B. SIMULATION AND EXPERIMENTAL VERIFICATION
The proposed CP antenna in this section is also implemented on a single-layer substrate with a thickness of 3 mm, relative permittivity of 2.2, and loss tangent of 0.0012. The resonant frequency and width of the single patch are set as 3.48 GHz and 12 mm, respectively. The extracted quality factor of the single patch is 23.5. The frequency of the AR minimum is given by f 1 /f 0 = 0.97. The required coupling coefficients can be calculated with the proposed design method, which are figured out with k 12 = 0.1318, k 23 = 0.048, and k 34 = 0.025. k 12 , k 23 , and k 34 will vary with the coupling gap widths g 1 , g 2 , and g 3 , respectively. The initial physical dimensions can be determined by numerical extraction of coupling coefficients in full-wave simulation. After minor fine tuning, the final physical parameters are determined and tabulated in Table 5.
In Fig. 14, the simulated magnitude ratio, phase difference, and AR responses of the antenna are compared with the theoretical ones. The simulated results match well with the theoretical ones, verifying the proposed design method. Small ripples of the magnitude ratio and phase difference responses appear in the operating band. One more ripple appears in both the magnitude ratio and phase difference responses when compared to the responses of the CP antenna with three coupled patch radiators in Fig. 5. In the theoretical magnitude ratio response, the values of the local minimum at the center frequency reaches 1. In the theoretical response of phase difference, the local maximum at low frequency and the local minimum at high frequency are both exactly −90 • .
The simulated surface current distributions of the proposed CP antenna with 4 coupled patches at 3.465 GHz over a single period are plotted in Fig. 15. The surface currents on Patch 1 and 3 are along −y axis direction, while those on Patch 2 and 4 are along +x axis direction, as shown in Fig. 15(a). The total equivalent current is along the lower VOLUME 4, 2023 479 left direction. Similarly, it can be found that the total equivalent surface current direction rotates anticlockwise during one period as shown in Fig. 15. The simulated |S 11 | and AR responses of the proposed CP antenna with 4 coupled patches under different values of gaps are plotted in Fig. 16.
As g 12 increases, both the magnitude and frequency of the minimum of the reflection coefficient decrease. When g 12 increases from 0.1 to 0.2 and 0.3 mm, the ripples of the AR response increase and the two left minima move towards low frequency. When g 23 increases from 1.5 to 1.9 and 2.3 mm, the center frequency of reflection coefficient nearly remains unchanged, but the magnitude increases. The value of g 23 mainly affects the frequency of the right minimum in the AR response. When g 23 increases, the right minimum moves towards low frequency. There is little effect of g 34 on the |S 11 | response. The variation of g 34 mainly affects the symmetrical property of the AR response. As g 34 increases, the left ripple goes down while the right one rises. The proposed CP antenna with 4 coupled patch radiators is fabricated and measured. Fig. 17 depicts the simulated and measured reflection coefficients, realized gains, and AR responses. Good agreement is observed between simulation and measurement. The measured 10-dB return-loss bandwidth is from 3.36 to 3.69 GHz. The peak realized gain is 7.75 dBic and the measured 3-dB realized gain bandwidth covers the frequency range of 3.34 to 3.60 GHz. There is a small frequency discrepancy between the simulated and measured AR responses, with the center frequency shifting from 3.48 to 3.50 GHz. Three minima appear in the AR response to construct enhanced bandwidth. The simulated and measured fractional bandwidths with AR<3 dB are 7.3% and 7.1%, respectively. The measured 3-dB AR bandwidth ranges from 3.38 to 3.63 GHz. As a result, the overlap bandwidth of reflection coefficient, realized gain, and AR is from 3.38 to 3.60 GHz, and the corresponding fractional bandwidth is 6.3%.
The simulated and measured efficiencies in Fig. 18 match well with each other, although the measured efficiency is slightly lower than the simulated one. Specifically, the simulated and measured efficiencies at 3.48 GHz are 93% and 88.6%, respectively. Fig. 19 shows the simulated and measured results of normalized radiation patterns at 3.5 GHz  and good agreement is observed. The measured half-power beamwidth in the x-z and y-z planes are 96 • and 58.5 • , respectively. The maximum cross-polarization levels in the x-z and y-z planes are −17.6 dB and −15.7 dB, respectively. Therefore, the proposed CP antenna can radiate satisfactory RHCP waves with low cross-polarization.

IV. CP ANTENNA WITH MORE COUPLED RADIATORS
In the previous two sections, wideband CP antennas with three and four coupled radiators are designed, and two and   three minima in the AR response are achieved. Similarly, the CP antenna with more coupled orthogonal patch radiators can be further designed. Multiple patch radiators are placed one by one along the x-and y-axes and the adjacent patch radiators are coupled to each other. Then the proposed design method can be employed to calculate the required coupling coefficients between adjacent coupled radiators and predict the AR response. In theory, for the CP antenna with n coupled radiators, n − 1 minima in the AR response can be produced to form a wide bandwidth. If more than 4 coupled radiators are employed, wider bandwidth is achieved but a larger coupling coefficient is required. The theoretical procedure of the proposed design method may become more complicated. To address this, the multiple patches can be placed on the multiple layers to obtain strong coupling and more compact size. Furthermore, based on the same design method, the patch radiators can be replaced with other kinds of antenna radiators, such as slot antenna.
After that, a comprehensive comparison among different wideband CP patch antennas is summarized in Table 6. The size of the proposed antenna is 0.32*0.95λ 2 0 and the profile is 0.035λ 0 . Our work exhibits medium size and low profile, when compared to other wideband CP patch antennas. It can be seen that the works in [11], [12], [25], [26], [27], [28], [34], and [36] employ multi-layer structures or sequentially rotated patches, suffering from complex configuration. Extra feeding circuits consisting of Wilkinson power divider and wideband phase shifter are utilized, which occupy large area. The unbalanced signals are absorbed by the resistor of the power divider to ensure equal magnitude and quadrature phase for wide AR bandwidth. But the antenna efficiency will be degraded. Among the single-layer structures [13], [14], [15], [16], our work exhibits medium bandwidth and gain under a low profile. In [13] and [14], parasitic elements are employed but no synthesis design method is reported. The antenna design mainly replies on time-consuming fullwave simulation. The thick substrate is used in [18], resulting in the high profile and increased cost. Finally and importantly, if compared with these reported works, a clear design mechanism and a quantitative design method based on the equivalent circuit model are proposed in this paper. Multiple coupled patches are employed to introduce multiple minima in the AR response to form wide bandwidth. The coupled patches serve as both radiators and feeding networks at the same time. There is no need for an additional feeding circuit in the design, resulting in a compact structure. The required coupling coefficients can be calculated to determine the physical dimensions. The magnitude ratio and phase difference of the orthogonal far-field components, as well as the AR response, can be accurately predicted, improving design efficiency significantly.

V. CONCLUSION
CP patch antennas with enhanced bandwidth based on capacitively coupled orthogonal patch radiators have been proposed in this paper. A quantitative design method based on the equivalent circuit model and closed-form solution has been proposed to efficiently design and analyze the proposed antenna. Wideband CP patch antennas with three and four coupled radiators are designed, fabricated, and measured. Multiple minima appear in the AR response to form wide bandwidth. The theoretical and full-wave simulation results are found to be in good agreement. The proposed antenna and design method are furtherly verified by the measurement. The wideband CP radiation characteristic and simple structure make the proposed antenna a good candidate for a variety of applications. Furthermore, the proposed antenna can be employed as a subarray element to construct a large-scale antenna array with enhanced radiation gain.