# Dual-Frequency Linear-to-Circular Polarization Converter for Ka-Band Applications
## 2 Design of the Unit Cell
### Equivalent Circuit Model  
In the literature, simplified equivalent models are largely used for the analysis of radiating structures Some of them use sophisticated analysis methods based on characteristic modes [30; 31], whereas others resort to a more intuitive and experience-based construction of the circuit [32; 33]. In the following, we adopt the second approach, leaving the first one to future developments.  
Figures 2-4 show the elements used in the elementary cell along with their equivalent circuit model. The ranges of physically realizable values of capacitances and inductances shown in the figures were established using Equations (5) and (6) of [34]. These closed-form equations have been used to estimate the feasible ranges from the minimum line/gap widths provided by the fabrication process. In our work, we used a cell periodicity of 7.15 mm. If a minimum realizable width of 50 \(\upmu\)m for the conducting strips and the gaps is used in the equations, the feasible ranges for inductances and capacitances are 0.01-6.5 nH and 0.1-328 fF respectively.  
Figure 2: Equivalent circuit model of the Jerusalem cross for an x- polarized impinging wave. (**a**) Equivalent circuit; (**b**) transmission coefficient with Cx1 = 6.4 fF, Lx1 = 6.1 nH, Cx2 = 2.3 fF, Lx2 = 4.8 nH.  
The proposed configuration is regarded as a two-mode microwave circuit because the polarizer modifies the orthogonal and parallel incident fields differently. For an x-polarized wave incidence, the inductances and capacitances related to the horizontal branch of the cross are selected to be zero at a frequency between the two bands (Figure 2). In this manner, two passbands are generated, separated by the resonant stopband. For y-polarized waves, the vertical branch of the cross provides a zero at a high out-of-band frequency (Figure 3), offering a wide passband able to cover both bands. An 'I'-type strip is added in a separate level to optimize magnitudes and phases of the y-polarized wave with minimum perturbation on the x-polarized wave. The 'I'-type strip is sized to provide a pole at a frequency between the two bands (Figure 4).  
The three components described in Figures 2-4 are then combined together to realize the configuration of the elementary cell shown in Figure 1. The equivalent circuit of the whole cell is shown for the x- and y-polarized waves in Figure 5a,b, respectively. In this model, each foam spacer between two layers is modelled with a \(h_{\mathit{foam}}\) long transmission line with characteristic impedance \(Z_{foam}=Z_{0}=377\)\(\Omega\) and propagation constant \(\beta_{0}=\omega/c\), where \(c\) is the speed of light. Each dielectric substrate is modelled with a \(h_{\mathit{sub}}\) long transmission line with characteristic impedance \(Z_{\mathit{sub}}=Z_{0}/\sqrt{\epsilon_{r}}\) and \(\beta_{\mathit{sub}}=\beta_{0}\sqrt{\epsilon_{r}}\).  
Figure 4: Equivalent circuit model of the ‘I’-type strip for a y- polarized impinging wave. (**a**) Equivalent circuit; (**b**) transmission coefficient with Cy = 72 fF, Ly = 6.3 nH.  
Figure 3: Equivalent circuit model of the Jerusalem cross for a y-polarized impinging wave. (**a**) Equivalent circuit; (**b**) transmission coefficient with Cy1 = 2.3 fF, Ly1 = 3.8 nH, Cy2 = 3.4 fF, Ly2 = 4.3 nH.  
The circuits shown in Figure 5 can be used for a further optimization of the parameters using Keysight Advanced Design System (ADS) to satisfy the objective of maximizing the \(-3\) dB AR bandwidths at Rx and Tx band respectively.