## 2 Materials and Methods
### Magnetoinductive Directional Filters
\begin{split}\frac{R}{I}=\frac{-\left(e^{j2k_{f}a}+1\right)\mu^{2}}{ 1+2\mu^{2}e^{j2k_{f}a}-e^{i4k_{f}a}},\quad\frac{T_{4}}{I}=\frac{\left(1-e^{j2 k_{f}a}\right)\mu^{2}}{1+2\mu^{2}e^{j2k_{f}a}-e^{i4k_{f}a}}\\ \frac{T_{2}}{I}=\frac{1-e^{j4k_{f}a}}{1+2\mu^{2}e^{j2k_{f}a}-e^{i4 k_{f}a}},\quad\quad\frac{T_{3}}{I}=0.\end{split}  \tag{11}\]  
Here \(\mu=\frac{M_{c}}{M}\) is the mutual inductance ratio. Note that the coefficients above are complex valued and frequency dependent. They are expressed only in terms of \(\mu\) and \(k_{f}\), which may be found from Equation (2). However, care must be taken to ensure that \(k_{f}^{\prime}\geq 0\) and \(0\leq k_{f}^{\prime}\leq\pi\). The results imply that no signal leaves port 2 at resonance, provided the guides are lossless. Similarly, the transmission from port 4 is maximum at resonance, while the reflectance from port 1 is zero. Port 3 is isolated at all frequencies, both within and outside the MI passbands. This behaviour is inherent in the way signals are combined. The phase shift at resonance along path P1 in Figure 2a is \(\frac{\pi}{2}\), while the phase shift along path P2 is \(\frac{3\pi}{2}\). If the device is lossless, the signal from port 2 is then zeroed by cancellation. If Q-factors are finite, losses are different along P1 and P2 and cancellation is partial. Two signal paths, P3 and P4, to port 3 are in antiphase for all frequencies. Although the results above provide valid amplitude reflection and transmission coefficients, they offer no insight on power flow in devices that are either lossy or operating out of band. We therefore define three sets of discrete power wave pairs, \(\{A1_{1},B1_{1}\}\), \(\{A2_{2},B2_{2}\}\), and \(\{A1_{4},B1_{4}\}\), for ports 1, 2, and 4, as shown in Figure 2a, as follows:  
\[ \begin{split} A1_{1}=\frac{-Z_{m}I_{-1}+Z_{0}I_{0}}{2\sqrt{k_{e} (Z_{0f})}},\;\;A2_{2}=\frac{Z_{m}R_{2}-Z_{0}I_{1}}{2\sqrt{k_{e}(Z_{0f})}},\;\; A1_{4}=\frac{Z_{m}I_{-1}+Z_{0}I_{0}}{2\sqrt{k_{e}(Z_{0b})}},\\ B1_{1}=\frac{-Z_{m}I_{-1}-Z_{0}^{\prime}I_{0}}{2\sqrt{k_{e}(Z_{0f})}}, \;\;B2_{2}=\frac{Z_{m}R_{2}+Z_{0}^{\prime}I_{1}}{2\sqrt{k_{e}(Z_{0f})}},\;\; B1_{4}=\frac{Z_{m}I_{-1}-Z_{0}^{\prime}I_{0}}{2\sqrt{k_{e}(Z_{0b})}}.\end{split}  \tag{12}\]  
Here \(Z_{0f}=j\omega Me^{-jk_{f}a}\) and \(Z_{0b}=-j\omega Me^{-jk_{b}a}\) are the characteristic impedances of the forward and backward guides. In fact, \(Z_{0f}=Z_{0b}\) whenever Equation (9) applies. Power wave scattering parameters with respect to port 1 may then be found as:  
\[S_{11}=\frac{B1_{1}}{A1_{1}},\;S_{21}=\frac{B2_{2}}{A1_{1}},\;S_{41}=\frac{B1 _{4}}{A1_{1}}. \tag{13}\]  
Using the transmission and reflection coefficients, we may obtain analytic expressions for the scattering parameters of a directional filter with MI line terminations (1) as:  
\[ \begin{split} S_{11}=\left(\left(1-e^{2k_{f}^{\prime\prime}a} \right)\left\{e^{2k_{f}^{\prime}a}+e^{2(k_{f}^{\prime\prime}+j2k_{f}^{\prime})a }\right\}+\mu^{2}\left(1+2e^{2(k_{f}^{\prime\prime}+jk_{f}^{\prime})a}-e^{2k _{f}^{\prime}a}\right)\right)/d\\ S_{21}=\left(e^{(k_{f}^{\prime\prime}+jk_{f}^{\prime})a}-e^{(k_{f }^{\prime\prime}+j3k_{f}^{\prime})a}+e^{-(k_{f}^{\prime\prime}+jk_{f}^{\prime}) a}-e^{(-k_{f}^{\prime\prime}+jk_{f}^{\prime})a}\right)/d\\ S_{41}=\mu^{2}\left(e^{2k_{f}^{\prime\prime}a}-1\right)/d,\\ d=1+2\mu^{2}e^{2(k_{f}^{\prime\prime}+jk_{f}^{\prime\prime})a}-e^{4 (k_{f}^{\prime\prime}+jk_{f}^{\prime\prime})a}.\end{split}  \tag{14}\]  
Moduli squared of Equations (11) and (14) are equal whenever \(k_{f}^{\prime\prime}=k_{b}^{\prime\prime}=0\).