## 2 Materials and Methods
### Magnetoinductive Directional Filters  
A directional filter has four ports connected to lines of characteristic impedance \(Z_{0}\). A voltage source connected to port 1 then gives rise to incident and reflected waves \(A1_{1}\) and \(B1_{1}\) and transmitted waves \(B2_{2}\), \(B3_{2}\), \(B1_{4}\) from ports 2, 3, and 4. Port 2 has the response of a band-reject filter, and Port 4 that of a bandpass filter. In a matched and lossless device, \(S_{21}\) and \(S_{41}\) are complementary, \(S_{31}=0\) for all \(\omega\), and \(S_{11}=0\) at resonance. Equivalent properties apply no matter which port is used as input. We now develop the analysis for MI directional filters and consider their frequency response, bandwidth, and tuneability. Figure 2a shows the equivalent circuit of a filter based on a pair of infinite horizontal MI waveguides consisting of identical resonators, but with opposite coupling polarity so  
that the lower guide is right-handed (blue) and the upper guide left-handed (red). The mutual inductances are equal in magnitude, with \(M>0\), so the coupling coefficients are \(\kappa_{f}=\frac{2M}{L}\) and \(\kappa_{b}=-\frac{2M}{L}\). The waveguides are connected vertically via two resonators, which couple the guides via mutual inductances \(M_{C}\) and coupling coefficients \(\kappa_{c}=\frac{2M_{c}}{L}\), such that \(\kappa_{c}<\kappa_{f}\). The network has four ports, 1-4, formed by sections of an MI waveguide. The lower line supports currents \(I_{n}\), and the top line supports \(J_{n}\), while the two coupling resonators have currents \(K\) and \(P\). Note that guide terminations can also be represented by lumped impedances, as shown in Figure 2b. There are three main possibilities: an infinite MI waveguide, represented by \(Z_{0}\) (1); a resistive termination, represented by \(Z_{0M}\) (2); and a broadband termination to \(Z_{0M}\), achieved by halving the inductance and doubling the capacitance in the resonant element (3). Termination 2 provides an impedance match at \(\omega_{0}\), while termination 3 provides matching at two frequencies, \(\omega_{0}\) and \(\frac{\omega_{0}}{\sqrt{1-\kappa^{2}}}\), with low reflection in between [49]. For now, we assume type 1 terminations and that an MI wave is incident on port 1 from the left. Experience suggests that this wave will be partially reflected at \(n=0\) due to the discontinuity caused by the coupling resonator, and that the remaining power, less any loss, will appear as MI waves from ports 2 to 4. We write expressions for the currents \(I_{n}\) and \(J_{n}\) in terms of incident, reflected, and transmitted MI waves as:  
\[ \begin{array}{l}I_{n}=Ie^{-k_{f}n}+Re^{jk_{f}n},\ n\leq 0\\ I_{n}=T_{2}e^{-k_{f}n},\ n\geq 1\\ J_{n}=T_{3}e^{-k_{b}n},\ n\geq 1\\ J_{n}=T_{4}e^{jk_{b}n},\ n\leq 0.\end{array}  \tag{8}\]  
Here \(I\) and \(R\) are the amplitudes of the incident and reflected waves at port 1; \(T_{2}\), \(T_{3}\), \(T_{4}\) are the wave amplitudes from ports 2, 3, and 4; \(k_{f}=k_{f}^{\prime}-jk_{f}^{\prime\prime}\) is the propagation constant of the forward waveguide; and \(k_{b}=k_{b}^{\prime}-jk_{b}^{\prime\prime}\) is the propagation constant of the backward one. From Figure 1b and Equation (2), we can assume:  
\[k_{b}a=k_{f}a-\pi. \tag{9}\]  
Combining Equation (1) with circuit equations for the currents \(K\) and \(P\) in the coupling resonators, a set of coupled equations may be obtained at the six central resonators:  
\[ \begin{array}{l}ZI_{0}+Z_{m}(I_{-1}+I_{1})+Z_{mc}K=0,\quad ZI_{1}+Z_{m}(I_{ 0}+I_{2})+Z_{mc}P=0\\ ZK+Z_{mc}(I_{0}+J_{0})=0,\quad ZP+Z_{mc}(I_{1}+J_{1})=0\\ ZJ_{0}-Z_{m}(J_{-1}+J_{1})+Z_{mc}K=0,\quad ZJ_{1}-Z_{m}(J_{0}+J_{2})+Z_{mc}P=0. \end{array}  \tag{10}\]  
Figure 2: (**a**) MI directional filter equivalent circuit; (**b**) lumped element terminations: (1) magnetoinductive, (2) resistive, and (3) broadband resistive. Substituting the assumed solutions and using the dispersion equation, \(K\) and \(P\) may be eliminated, allowing the reflection and transmission coefficients to be calculated:  
\[