## 3 RLC Circuit Impedance Boundary Formulation
Thus, after detecting a resonance in the admittance phase from its slope and zero crossings, and after sampling the associated peak in the admittance magnitude, the half-power bandwidth, \(\Delta\omega\), follows from Equation (5) (which may be estimated with simple finite differences). This approach is sufficient to obtain RLC parameters for each well-isolated resonance, and can be more robust than peak detection and half-power bandwidth estimation from the admittance magnitude alone. Nevertheless, this approach still has limitations for very-closely spaced resonances (examples can be seen in Figure 2) where the phase response at a resonance may not cross zero (and thus would not be detected with this approach so far). To deal with such issues, the _second derivative_ of the phase is additionally used to identify resonances, which can be written as:  
\[\left.\frac{\partial^{2}\angle Y(i\omega)}{\partial\omega^{2}}\right|_{\omega =\omega_{0}^{\prime}}=0\,,\quad\omega_{0}^{\prime}=\omega_{0}\sqrt{\sqrt{4-( \Delta\omega/\omega_{0})^{2}}-1}, \tag{6}\]where \(\omega_{0}^{\prime}\) is the angular frequency at which there is a zero crossing in the second-order derivative of the admittance phase. Furthermore, \(\omega_{0}^{\prime}\approx\omega_{0}\) provided that \((\Delta\omega/\omega_{0})^{2}\ll 1\), which means that \(\omega_{0}^{\prime}\) may be used as an initial estimate of \(\omega_{0}\) to seed the subsequent global optimisation. Identified resonances using this phase-derivative zero-crossing method (grey dotted lines) can be seen in Figure 2.  
Once a set of RLC triplet parameters has been identified given a set of resonant frequencies and associated bandwidths, a global optimisation using a Nelder-Mead method [39] as a non-linear minimisation algorithm is carried out to find the \(L^{(b)}\), \(R^{(b)}\) and \(C^{(b)}\) that minimise \(\epsilon=\|\gamma_{TMM}-\gamma_{RLC}\|\), where for a given slit, \(\gamma_{TMM}\) is the target admittance output from the TMM model of the QRM, and \(Y_{RLC}\) is the impedance output of the RLC circuit. Fit reflection coefficients of the two QRM slits previously illustrated (see Figure 2) can be seen in Figure 3. It can be observed that this methodology is successful in capturing the resonances within the QRM slits. The errors shown take into account magnitude and phase, and discrepancies can be seen at the upper limit of our range of frequencies (e.g., a resonance being ignored). Such discrepancies may be attributed to viscothermal losses in the TMM model which cause deviations from ideal second-order resonances. These could be mitigated using pairs of RLC triplets for each identified resonance to allow for better optimised results, but this was not pursued as a compromise of accuracy and model complexity. For later comparison purposes, a similar approach for modelling the equivalent surface impedance of the QRM is adopted in a Finite Element Method (FEM) study in COMSOL Multiphysics 5.3(tm), where the \(N=5\) impedance patches were modelled using built-in impedance boundary conditions with the complex impedance data obtained from the TMM model as input. The resulting scattering data between the QRM and the equivalent surface made of different RLC circuits is shown in the next section.