## 3 Minimum Nonlinearity PL Design Guidelines
### PL Design Trade-Offs  
From the expressions obtained above, it can be observed that the nonlinear perturbation comes from the waves reflected by the loads, given by the term \(\mathbf{S_{AB}}\left(\mathbf{\Gamma_{B}}^{-1}-\mathbf{S_{BB}}\right)^{-1} \mathbf{S_{BA}}\,\mathbf{a_{A}}\). Accordingly, if the nonlinear effects are to be minimized there are two approaches:  
1. Reducing the \(\mathbf{S_{AB}}\), \(\mathbf{S_{BA}}\) terms, that is, the coupling between external and internal ports. This is the underlying philosophy of the PL as there is a physical separation between the feed ports and the switches. Note that this coupling can be controlled by the PL height above the basic antenna, adding one degree of freedom to the design. However, changing the PL height over the basic antenna also impacts the radiated field and port match, requiring a delicate balance. 2. Reducing the \(\left(\mathbf{\Gamma_{B}}^{-1}-\mathbf{S_{BB}}\right)^{-1}\) harmonic terms. This could be attained by a careful antenna design that provides an adequate matching between the terms in the subtraction, however, given that normally the minuend will be dominant, this is mostly technology dependent, reinforcing the need for accurately characterizing the switch nonlinearities and the benefits of using these models as a detailed design tool. To examine the effects of the PL height '\(h\)' over one basic antenna and its effects on the reconfigurability of parameters like port matching, radiated field, and nonlinearities, the parasitic-layer-based reconfigurable-antenna illustrated in Figure 1 is taken as a reference. A parametric variation of '\(h\)' is performed on this reference antenna, aiming to determine a feasible trade-off region which can be used as design guideline, even for other PL pixellations and number of switches, as well as other antenna configurations. Therefore, the interest is in average or expected behavior rather than in particular switch states behavior. Aggregated port match bandwidth, maximum steering angle, and average power margin are used as metrics of comparison. The PL based RA is composed of a basic 2-ports aperture-coupled microstrip patch radiator matched at \(f_{0}=3.5\) GHz, on a \(th=3\) mm thick Rogers RO4003 substrate \(\epsilon_{r}=3.55\). The PL is a \(3\times 3\) pixellation supported on a \(th=0.51\) mm thick Rogers RO4003 substrate, placed at a distance \(h\) from the basic antenna, with size and switches numbering as illustrated in Figure 11. This design is based on the single-port presented in [28], which was fabricated and measured with good agreement between simulations and measurements, working from 2.4 to 2.5 GHz, providing steering in nine directions \(\theta_{0}\in\{-30^{\circ},0^{\circ},30^{\circ}\}\), \(\phi_{0}\in\{0^{\circ},45^{\circ},90^{\circ},135^{\circ}\}\) with an average gain of 6.5 dB. The parametric sweep is performed using the frequency-domain solver of CST studio, for \(h=\lambda_{0}/\{40,20,15,12,10,9,8,6,5,4\}\), retaining for each case the 'unloaded' S-parameters file and the radiated fields generated by each 'port'. In a later step, the IMPM is applied to determine the metrics of interest. As the number of switch configurations is relatively low (4096), exhaustive search is feasible in a short time for each height value. The sweep results for antenna port 1 are summarized in the set of plots of Figure 12 which represent the probability distribution of configurations for a given figure of merit in terms of the variations of the parameter \(h\). Results for port 2 are almost identical and are omitted. The first that can be observed in Figure 12a is that, as expected, the PL loading alters the port impedance in a way that when the separation is smaller, the obtainable bandwidth is higher, and when separation grows, convergence to the bandwidth of the simple antenna is achieved. This is in a good agreement with the experimental results in [40], for a higher complexity PL where a statistical sweep is performed for practical reasons.