# Wideband Lorenz Resonance Reconfigurable Metasurface for 5G+ Communications
## 2 Unit Cell Modeling and Design
### Equivalent Circuit Model  
According to the propagation path of the incident electromagnetic wave, the equivalent circuit (EC) model of the MSRRA can be modeled by means of using equivalent inductors, capacitors, resistors, and transmission lines, which represent the metal patch, dielectric mediums, the ground plane, and varactor diodes in the MSRRA system. A sketch of the MSRRA and its equivalent circuit are illustrated in Figure 1. The second substrate's medium layer with permittivity \(\epsilon_{2}\) and propagation constant has no contribution to the MSRRA and is neglected, since it is under the ground plane. The total EC model is a parallel RLC resonant circuit. The analysis of the equivalent circuit model is conducted through the following steps:  
(1) Analysis of metal patch with a varactor diode: The metal patch is modeled using lumped R, L, and C components so that a resonant equivalent circuit can be obtained which, together with the varactor, provides equivalent capacitance for the entire EC model. When the MSRRA is illuminated, the induced electric field appears across the gap between the two microstrip patches. On average, this effect can be equivalent to a sheet capacitance \(C_{p}\). The high-frequency currents flow back and forth between the capacitive gap, so the patches can be equivalented as serials of effective inductive impedance \(L_{p}\) and a resistance \(R_{p}\) as shown in Figure 2a,b. A MACOM varactor diode (MAVR-011020-1411) [30], Ref. [31] is used in this MSRRA, and can provide extra capacitance. With the spice model provided by the manufacturer, we can easily obtain an EC model of the metal patch with a varactor on it, as shown in Figure 2c. Finally, we can set up the whole model, as shown in Figure 2d; the MSRRA can be equivalented to a parallel RLC resonant circuit. The \(C_{p}\) in Figure 2d is described in Equation (1) [32] as:  
\[C_{p}=\frac{L(\epsilon_{1}+\epsilon_{2})}{\pi}Cosh^{-1}\left(\frac{a}{g}\right), \tag{1}\]  
where \(L\) is the length of the metal patches, \(g\) is the gap width, and \(a=2\times w+g\) is the total length of the metal patch from edge to edge. The structure is surrounded by \(\epsilon_{1}\) on one side, and \(\epsilon_{2}\) on the other. Additionally, closed-form expressions for \(L_{p}\) and \(R_{p}\) in the EC model are not readily available. They must be de-embedded using electromagnetic simulations  
Figure 1: (**a**) Side view of the multi-substrate layer MSRRA. Each substrate medium is characterized by its thickness \(l_{x}\), permittivity \(\epsilon_{x}\), and propagation constant \(\gamma_{x}\). (**b**) Transmission line-based EC model of a multi-substrate layer MSRRA. The metal patch is represented by a shunt admittance. of the discontinuity due to their unique geometry. A microstrip patch without a varactor is simulated using full-wave electromagnetic simulation software CST to extract the S-parameters. \(L_{p}\) is calculated using the geometric structure's inherent resonant frequency. Finally, the computation of \(R_{p}\) needs to simulate the reflection coefficient of the patch. Using the measured power in the radiated field, the resistance of the patch can be extracted if the magnitude of the voltage established in the patch slots is known. (2) Analysis of substrate medium and ground plane: Both the substrate medium and ground plane are modeled as transmission lines and are mainly handled as the lumped L component of the entire EC model. As shown in Figure 1b, \(Z_{\text{eq}}\) is the input impedance of the substrate with the ground plane and is calculated by:  
\[Z_{eq}=j\times Z_{TL}\tan(\beta l), \tag{2}\]  
where \(l\) is the thickness of the substrate medium, \(\beta=2\pi/\lambda\) is the propagation constant in the medium, \(\lambda\) is the wavelength, and \(Z_{TL}=\sqrt{\mu/\varepsilon}\) is the impedance introduced by the substrate medium. Figure 2: Equivalent circuit model of the MSRRA unit cell. (**a**) Metal patch and the equivalent lumped elements. (**b**) EC model of the metal patch.