Electric field in the equivalent cavity

For electrically thin antennas, i.e., those whose thickness h is significantly smaller than the wavelength λ in the substrate, the electric field in the equivalent cavity exhibits only the E_z component, which can be written using modal expansion as [Reference Richards, Yuen and Harrison22]

(1)\begin{equation}{E_z}\left( {x,y} \right) = \mathop \sum \limits_{m = 0}^\infty \mathop \sum \limits_{n = 0}^\infty {E_{mn}}{\Psi _{mn}}\left( {x,y} \right),\end{equation}

where ${E_{mn}}$ is the amplitude of each ${\text{TM}}_{mn}^z$ mode and ${\Psi _{mn}}\left( {x,y} \right) = {\text{cos}}\left( {m\pi x/{a_e}} \right){\text{cos}}\left( {n\pi y/{b_e}} \right)$.

The amplitude ${E_{mn}}$ is computed from the solution to the nonhomogeneous wave equation $({\nabla ^2} + {k^2}){\text{ }}{E_z} = j\omega {\mu _0}{\skew5\vec J}_f \cdot \skew3\hat{z}$:

(2)\begin{equation}{E_{mn}} = \frac{{j\omega {\mu _0}{\text{ }}{\mathop \iint \nolimits_S}{J_f}\Psi _{mn}^*\left( {x,y} \right){\text{ }}ds{\text{ }}}}{{\left( {{k^2} - k_{mn}^2} \right){\mathop \iint \nolimits_S}{{\left| {{\Psi _{mn}}\left( {x,y} \right)} \right|}^2}ds{\text{ }}}}{\text{,}}\end{equation}

where $S$ denotes the cavity surface located in the xy-plane, ${\skew5\vec J}_f$ is the total current density comprised of P sources representing the feed probe and metallic pins, ${k_{mn}} = \sqrt {{{\left( {m\pi /{a_e}} \right)}^2} + {{\left( {n\pi /{b_e}} \right)}^2}} $ is the resonant wavenumber, and $k = \omega \sqrt {{\mu _0}{\varepsilon _r}{\varepsilon _0}} $ is the wavenumber in a lossless cavity.

The pth source of current density amplitude J _0p ( $1 \leqslant p \leqslant {\text{ }}P$) is modeled as a strip with width L_p [Reference Garg and Bahl23] and height h, centered at the coordinates (x_p, y_p). Therefore, the total current density ${\skew5\vec J}_f$ is given by

(3)\begin{equation}{\skew5\vec J}_f\left( {x,y} \right) = \mathop \sum \limits_{p = 1}^P {J_{fp}}\left( {x,y} \right)\skew3\hat{z} = \mathop \sum \limits_{p = 1}^P {J_p}\left( y \right)\delta \left( {x - {x_p}} \right)\skew3\hat{z},\end{equation}

where

(4)\begin{equation}{J_p}\left( y \right) = \bigg\{ \begin{array}{*{20}{c}} {{J_{0p}}{\text{, if }}\left| {y - {y_p}} \right| \leqslant {L_p}/2} \\ {0{\text{, elsewhere }}} \end{array}.\end{equation}

After substituting (3) into (2) and evaluating the double integrals, the amplitude of each mode is written as

(5)\begin{equation}{E_{mn}} = \frac{{j\omega {\mu _0}}}{{{a_e}{b_e}}}\frac{{{\xi _m}{\xi _n}\mathop \sum \nolimits_{p = 1}^P{J_{0p}}{L_p}{\Psi _p}{\text{ }}}}{{{k^2} - k_{mn}^2}}{\text{,}}\end{equation}

where ${\Psi _p} = {\text{cos}}\left( {m\pi {x_p}/{a_e}} \right){\text{cos}}\left( {n\pi {y_p}/{b_e}} \right){\text{sinc}}\left( {n\pi {L_p}/2{b_e}} \right)$, with ${\xi _\tau } = 1$, if $\tau = 0$, and ${\xi _\tau } = 2$, otherwise, and ${\text{sinc}}\left( x \right) = {\text{sin}}\left( x \right)/x$.

Finally, applying (5) in (1), the electric field inside the cavity is established as a function of the current densities ${J_{fp}}$.

Far field

Using the magnetic surface currents on the magnetic walls of the equivalent cavity, the far-zone radiated field by the microstrip antenna can be computed by applying the Equivalence Principle and the electric vector potential [Reference Lo, Solomon and Richards24]. The expressions for the far-field components ${E_\theta }_{mn}\left( {\theta ,\phi } \right)$ and ${E_\phi }_{mn}\left( {\theta ,\phi } \right)$ for the ${\text{TM}}_{mn}^z$ modes are well-established in the literature [Reference Lumini, Cividanes and da S. Lacava25, Reference James and Wood26] and are not detailed here.

The radiation pattern of a canonical rectangular microstrip antenna is often well characterized by the ${\text{TM}}_{01}^z$ or ${\text{TM}}_{10}^z$ modes in LP designs. In contrast, for CP radiators, the superposition of these two modes adequately represents the far field. However, in this work, due to the multiple sources within the cavity, a strong dependence on several modes to describe the ${E_z}$ field, as given by (1), is observed. As a result, the far field can be accurately described only if the radiated fields by each ${\text{TM}}_{mn}^z$ mode are superimposed. Thus,

(6)\begin{equation}{E_\theta }\left( {\theta ,\phi } \right) = \mathop \sum \limits_{m = 0}^\infty \mathop \sum \limits_{n = 0}^\infty {E_\theta }_{mn}\left( {\theta ,\phi } \right),\end{equation}

(7)\begin{equation}{E_\phi }\left( {\theta ,\phi } \right) = \mathop \sum \limits_{m = 0}^\infty \mathop \sum \limits_{n = 0}^\infty {E_\phi }_{mn}\left( {\theta ,\phi } \right),\end{equation}

The dependency on the number of modes to describe the radiated fields (6) and (7), as well as the input impedance will be further explored in Section III–A.

Input impedance

The complex power delivered by the pth source to the cavity can be calculated using the complex Poynting’s theorem as ${P_{in}}_p = - \frac{1}{2}\int\int\int\limits_V {E_z} J_{fp}^*dv$. Hence, the total complex power delivered to the cavity is obtained by summing the powers delivered by all sources. According to the circuit theory, the power supplied by the pth source is ${P_{in}}_p = \frac{1}{2}{Z_{in}}_p{\left| {{I_{in}}_p} \right|^2}$, leading to the input impedance of the pth port:

(8)\begin{equation}{Z_{in}}_p = - \frac{1}{{{{\left| {{I_{in}}_p} \right|}^2}}} \int\int\int\limits_V {E_z}J_{fp}^*dv,\end{equation}

where ${I_{in}}_p = {J_{0p}}{L_p}$ is the complex current on the pth source.

Substituting (1) and (5) into (8) and evaluating the triple integral yields the input impedance of the pth port:

(9)\begin{equation}{Z_{in}}_p = \frac{{ - j\omega {\mu _0}h}}{{{a_e}{b_e}}}\mathop \sum \limits_{q = 1}^P \frac{{{I_{in}}_q}}{{{I_{in}}_p}}\mathop \sum \limits_{m = 0}^\infty \mathop \sum \limits_{n = 0}^\infty \frac{{{\xi _m}{\xi _n}\Psi _p^2}}{{{k^2} - k_{mn}^2}}{\text{.}}\end{equation}

Modeling the cavity as a P-port linear circuit, represented by a $\left[ Z \right]$ matrix of elements ${Z_{pq}}$, the voltage on the pth port is given by ${V_{in}}_p = \mathop \sum \limits_{q = 1}^P {Z_{pq}}{I_{in}}_q$. Then, the input impedance of the pth port is ${Z_{in}}_p = \mathop \sum \limits_{q = 1}^P {Z_{pq}}{I_{in}}_q/{I_{in}}_p$. This leads to the following expression for the element ${Z_{pq}}$:

(10)\begin{equation}{Z_{pq}} = \frac{{ - j\omega {\mu _0}h}}{{{a_e}{b_e}}}\mathop \sum \limits_{m = 0}^\infty \mathop \sum \limits_{n = 0}^\infty \frac{{{\xi _m}{\xi _n}\Psi _p^2}}{{{k^2} - k_{mn}^2}}{\text{.}}\end{equation}

Up to this point, we have assumed that the antenna has P ports. We impose that port 1 corresponds to the feed probe, while the others are terminated with loads ${Z_L}_q$, and without loss of generality, assuming ${I_{in}}_1 = 1{\text{ A}}$, the following linear system is found:

(11)\begin{align} &\left[ {\begin{array}{*{20}{c}} {{Z_{22}} + {Z_L}_2}&{{Z_{23}}}& \ldots &{{Z_{2P}}} \\ {{Z_{32}}}&{{Z_{33}} + {Z_L}_3}& \ldots &{{Z_{3P}}} \\ \vdots & \vdots & \ddots & \vdots \\ {{Z_{P2}}}&{{Z_{P3}}}& \ldots &{{Z_{PP}} + {Z_L}_P} \end{array}} \right].\left[ {\begin{array}{*{20}{c}} {{I_{in}}_2} \\ {{I_{in}}_3} \\ \vdots \\ {{I_{in}}_P} \end{array}} \right] \nonumber\\ &\quad = \left[ {\begin{array}{*{20}{c}} { - {\text{ }}{Z_{21}}} \\ { - {\text{ }}{Z_{31}}} \\ \vdots \\ { - {\text{ }}{Z_{P1}}} \end{array}} \right].\end{align}

The currents ${I_{in}}_q$ are determined by solving (11), and consequently, the antenna input impedance is calculated as ${Z_{in}}_1 = \mathop \sum_{q = 1}^P {Z_{1q}}{I_{in}}_q$. These currents play a crucial role in computing (5) and subsequently in determining the far field described in (6) and (7).

Effective loss tangent

The effects of radiated power ${P_i}$, metallic loss ${P_c}$, and dielectric loss ${P_d}$ are computed and incorporated into the model using the effective loss tangent concept [Reference Richards, Yuen and Harrison22]. This parameter is calculated for each ${\text{TM}}_{mn}^z$ mode as ${\delta _{ef}}_{mn} = \left( {{P_i} + {P_c} + {P_d}} \right)/\left( {2{\omega _{mn}}{\text{ }}{W_e}_{mn}} \right)$, where ${\omega _{mn}}$ is the angular resonant frequency given by ${\omega _{mn}} = {c_0}{k_{mn}}/\sqrt {{\varepsilon _r}} $, and ${\text{ }}{W_e}_{mn}$ represents the stored electric energy in the cavity for each mode. Therefore, the effective wavenumber of each mode is computed as ${k_{ef}}_{mn} = k{\left( {1 - j{\text{ }}{\delta _{ef}}_{mn}} \right)^{1/2}}$, which replaces $k$ in (2), (5), (9), and (10), properly incorporating the losses into the model. The surface wave loss is ignored in ${\delta _{ef}}_{mn}$ due to its negligible effect on electrically thin antennas [Reference Lo, Solomon and Richards24].

The challenge of computing ${\delta _{ef}}_{00}$ of the ${\text{TM}}_{00}^z$ mode is a known issue in the literature [Reference Garg and Bahl23], which seems not clearly solved yet. In this study, we address this challenge by employing the following rule: ${\delta _{ef}}_{00} = {\delta _{ef}}_{10}$, if a_e > b_e, or ${\delta _{ef}}_{00} = {\delta _{ef}}_{01}$, if a_e < b_e. The accuracy of this choice is demonstrated in the next section. Furthermore, the computation of ${\delta _{ef}}_{mn}$ is conducted for the unloaded antenna, similar to a classical rectangular microstrip patch [Reference Lo, Solomon and Richards24].

Circularly polarized antenna

Initially, we analyze the CP antenna illustrated in fig. 2(a). It consists of a square patch with a side length a = 37.19 mm (a_e = 40.25 mm) and a 50 Ω probe positioned at x ₁ = 11.3 mm and y ₁ = a/2. Four metallic pins with a radius of 0.5 mm are placed at a distance d = 2.0 mm from the antenna corners, connecting grounded capacitors to the patch. The antenna is printed on a laminate with a thickness of 3.048 mm, characterized by a relative permittivity ε_r = 2.55 and loss tangent tan δ = 0.002. For proper CP operation at 1.575 GHz, the capacitances, acting as perturbations in the cavity, are tuned as follows: capacitances C ₁ = 1.346 pF are connected in vias 2 and 4, resulting in Z_{L 2} = Z_{L 4} = —j75 Ω. Similarly, capacitances C ₂ = 1.296 pF are connected in vias 3 and 5, leading to Z_{L 3} = Z_{L 5} = —j78 Ω.

Figure 2. CP antenna loaded by capacitors. (a) Geometry. (b) Axial ratio. (c) Input impedance. (d) Comparisons with HFSS.

These values were obtained through the following systematic procedure. First, all four capacitances were initially set to 1.3 pF, providing an intermediate capacitive reactance within the range offered by the chosen varicap BB833 from Infineon Technologies. Next, the equivalent side length a_e was adjusted to achieve a resonance close to the target operating frequency of 1.575 GHz, resulting in a LP antenna at this stage. Finally, a parametric study was conducted to determine the values of C ₁ and C ₂. This analysis involved slight variations to the initial capacitances, allowing for the optimization of the antenna’s circular polarization performance.

The choice of the upper limit (UL) for the indexes m and n in the summations is crucial for the performance of the proposed model. This aspect is examined by computing the input impedance of the proposed antenna. Assuming the same UL for both m and n, we calculate the broadside axial ratio (AR) [Reference Balanis27] and input impedance for different values of UL, as shown in fig. 2(b) and (c), respectively. Initially, with UL = 1, a resonance at approximately 2.3 GHz would be expected for the unloaded cavity. However, due to the capacitive loading of the vias and the coupling of the modes, the antenna resonates at nearly 1.71 GHz (blue curves in fig. 2(c)). When UL tends to 50, both the input impedance and AR converge (black curves) and exhibit the performance of a CP antenna. This analysis suggests that the two resonant modes depicted by the black curves in fig. 2(c) are strongly influenced by the modes of the unloaded cavity that resonate away from the operating frequency.

To validate our analysis, we simulated the antenna using HFSS. The comparison, illustrated in fig. 2(d), demonstrates a close agreement for both input impedance and broadside axial ratio. Next, tuning the capacitances and the probe position in HFSS (C ₁ = 1.405 pF, C ₂ = 1.325 pF, and x ₁ = 9.5 mm), the results closely matched those obtained from the cavity model. The updated load impedances are Z_{L 2} = Z_{L 4} = —j72 Ω and Z_{L 3} = Z_{L 5} = —j76 Ω. Note that only minor adjustments were necessary to achieve the final geometry, highlighting the potential of the model as a predesign tool.

Antenna operating in the TM₀₀ mode

The antenna operating in the ${\text{TM}}_{00}^z$ mode is a well-known radiator based on shorting pins, renowned for its monopole-like radiation pattern [Reference Nunes, Moleiro, Rosa and Peixeiro9, Reference Fong and Chair10]. To evaluate the effectiveness of our formulation, we analyzed the antenna depicted in fig. 3(a) and assumed UL = 50 in the sums. Following the methodology described in [Reference Liu, Zhu and Liu11], the radiator is designed for dual-band resonance. The lower band is governed by the ${\text{TM}}_{00}^z$ mode, with a resonant frequency close to 1.4 GHz, while the upper band is controlled by the ${\text{TM}}_{01}^z$ mode, close to 1.8 GHz. The design is conducted using the microwave laminate mentioned earlier, and the final dimensions are listed in fig. 3(a). Comparisons with HFSS for input impedance versus frequency and the radiation pattern in both bands are presented in fig. 3(b), (c), and (d), demonstrating good agreement. The slight deviation in the resonant mode at 1.4 GHz may be attributed to challenges in characterizing the loss tangent of the ${\text{TM}}_{00}^z$ mode, as mentioned above.

Figure 3. Dual mode antenna loaded by metallic vias. (a) Geometry. (b) Input impedance. (c) Radiation pattern ${\text{TM}}_{00}^z$. (d) Radiation pattern ${\text{TM}}_{01}^z$.

According to our formulation, the resonance at 1.4 GHz is strongly influenced by neighboring resonant modes, suggesting a perturbed mode with an ${E_z}$ field distribution similar to ${\Psi _{00}}$. Hence, this resonance was classified as ${\text{TM}}_{00}^z$. It is important to note that there is no resonant frequency associated with ${k_{00}}$. Moreover, as the pins are positioned along the region of the null of ${\Psi _{01}}$, the resonance at 1.8 GHz remains unaffected by the presence of the pins and is controlled by the unloaded ${\text{TM}}_{01}^z$ mode.