### I. Introduction

### II. Operation Principle of an FPC Antenna

*Γ*

*is the reflection magnitude of the PRS,*

_{PRS}*φ*

*is the reflection phase of the ground plane,*

_{GND}*φ*

*is the reflection phase of the PRS,*

_{PRS}*l*is the distance between the PRS and the ground, and

*f*

*is the radiation pattern of the source. If the FPC antenna has obtained its maximum gain, the denominator in the modification must be the minimum value, and*

^{2}### III. Gain Equation of an FPC Antenna with a Finite Dimension

*w*×

*w*). The source antenna operating at 5 GHz is located on the center of the ground. The utilized substrate for the antenna has the relative permittivity of 10.2 and a thickness of 1.6 mm. Three cases with different PRS reflection magnitudes of 0.53, 0.78, and 0.9 have been simulated. In addition, to obtain the PRS’s reflection magnitudes, we have used substrates, which have a fixed thickness at 3.2 mm and permittivity values of 6.15, 10.2, and 20. In addition, the height of a PRS from the ground (

*l*) can be calculated using Eq. (2). Fig. 3 shows the simulated peak gains and Eq. (1) versus the dimension of an FPC antenna using various reflection magnitudes of a PRS. The simulated results were obtained using the ANSYS HFSS simulator (ANSYS Inc., Canonsburg, PA, USA). As shown in Fig. 3, the peak gain calculated by Eq. (1) with an infinite dimension is independent of the dimension of a PRS and is different from the full-wave simulated peak gain. All gain values versus the dimension of the FPC antenna go up and down because of the edge diffraction effect due to a finite ground condition. The full-wave simulated peak gain of an FPC antenna also increases and becomes saturated due to a specific dimension. The specific dimension where the gain starts to saturate increases as the reflection magnitude of a PRS becomes closer to one. To guarantee the number of multiple reflections to reach the maximum gain, the sufficient dimension of a PRS is required. Otherwise, the reflected wave would leak from the FPC antenna, resulting in a lower gain. Even though the reflection magnitude of the PRS is close to one in the case of a finite dimension with about 1.5λ

_{0}or less, the gain of the FPC antenna cannot reach its maximum value (as shown in Fig. 3). The reason for this is because the larger the reflection magnitude of a PRS is, the bigger the dimension of a maximum gain should be. It has also been noted that, when the dimension of the PRS and the ground is less than 1.5λ

_{0}, the edge diffraction is dominant compared to the effect of multiple reflections. Eq. (1), which is proportional to the reflection magnitude of a PRS, is reasonable and valid in the case of a finite dimension with about 1.5λ

_{0}or more. Therefore, by using the curve fitting from the simulated results where the dimension is 1.5λ

_{0}or more, the equation for the peak gain required in order to include the dimension of an FPC antenna can be obtained with Eqs. (3) and (4).

*Γ*

*is the reflection magnitude of the PRS and*

_{PRS}*n*(=

*w*/λ

_{0}) is the normalized dimension of an FPC antenna by the free space wavelength. Fig. 4 presents the peak gains with Eq. (4), the simulation, and the measurement versus the dimension of an FPC antenna for various reflection magnitudes of a PRS. The peak gain was measured in the anechoic chamber system to confirm the accuracy of the proposed gain equation. The anechoic chamber is composed of a shield enclosure (4 m × 2.5 m × 2.5 m), a pyramidal absorber, a network analyzer, a positioner, a turn table, and a dual polarized transmitting antenna. The peak gains calculated from Eq. (4) are in good agreement with the simulated and measured results. Since Eq. (4) is a function of both a dimension and a reflection magnitude in the resonance condition, the exact gain of an FPC antenna with a dimension of 1.5λ

_{0}or more can be obtained through the equation.

### IV. Conclusion

_{0}. It will be helpful to be able to predict the gain of an FPC antenna.