## 2 Materials and Methods
### Advanced Filters
#### 2.4.1 Infinite Rejection  
In the absence of loss, complete rejection occurs at \(\omega_{0}\) at port 2 when port 1 is used as an input. However, for finite Q-factors, the two signal paths from port 1 to port 2 in Figure 2a incur unequal losses, and some leakage into port 2 then occurs even at resonance. To equalise signal amplitudes along the two paths, the magnetic coupling strength between resonators 0 and 1 in the forward waveguide may be adjusted. We define a new positive coefficient, \(\kappa^{\prime}_{f}=\lambda\kappa_{f}\), between these elements and reconsider the port reflection and transmission coefficients. The circuit equations must be modified to:  
\[\begin{array}{c}ZI_{0}+Z_{m}(I_{-1}+\lambda I_{1})+Z_{mc}K=0,\ ZI_{1}+Z_{m}( \lambda I_{0}+I_{2})+Z_{mc}P=0\\ ZK+Z_{mc}(I_{0}+J_{0})=0,\ ZP+Z_{mc}(I_{1}+J_{1})=0\\ ZJ_{0}-Z_{m}(J_{-1}+J_{1})+Z_{mc}K=0,\ ZJ_{1}-Z_{m}(J_{0}+J_{2})+Z_{mc}P=0. \end{array} \tag{17}\]  
As before, we may substitute the assumed solutions and use the dispersion relation to obtain modified reflection and transmission coefficients. The transmission from port 2 may then be expressed as a rational polynomial in \(\lambda\) as follows:  
\[\begin{array}{c}\frac{T_{2}}{I}=\frac{p_{0}+\lambda P_{1}}{q_{0}+2\mu^{2} \rho^{2/\mu}+q_{0}\lambda^{2}}\\ p_{0}=\mu^{4}e^{2\
u/\mu}\left(1-e^{i2\
u/\mu}\right)\\ p_{1}=\left(e^{i4k_{f}/\mu}-1\right)^{2}+\mu^{2}e^{2\
u/\mu}\left(2-2e^{i4k_{f }/\mu}+\mu^{2}\left(e^{i2k_{f}/\mu}-1\right)\right)\\ q_{0}=e^{i2k_{f}/\mu}\left(e^{i6k_{f}/\mu}+e^{i4k_{f}/\mu}-e^{i2k_{f}/\mu}-1 \right)+\mu^{2}e^{2k_{f}/\mu}\left(2-2e^{i2k_{f}/\mu}-4e^{i4k_{f}/\mu}+4\mu^{2 }e^{i2k_{f}/\mu}-\mu^{2}\right)\\ q_{2}=1+e^{i2k_{f}/\mu}-e^{i4k_{f}/\mu}-e^{i6k_{f}/\mu}+\mu^{2}e^{i2k_{f}/\mu} \left(2+2e^{i2k_{f}/\mu}-\mu^{2}\right).\end{array} \tag{18}\]  
For infinite rejection at midband, we require the numerator of \(T_{2}/I\) to equal zero:  
\[\lambda=-\frac{p_{0}}{p_{1}}. \tag{19}\]  
At resonance, the propagation constant can be rewritten in terms of \(\kappa\) and \(Q\) alone:  
\[\left.k_{f}a\right|_{\omega=\omega_{0}}=\frac{\pi}{2}+jln\left(\sqrt{1+\frac {1}{\kappa^{2}Q^{2}}}-\frac{1}{\kappa Q}\right). \tag{20}\]  
Substituting into Equation (19), we obtain the condition for infinite rejection as:  
\[\lambda=\frac{\mu^{4}\rho^{4}}{\mu^{4}\rho^{4}-2\mu^{2}\rho^{4}+2\mu^{2}\rho ^{2}+\rho^{6}-\rho^{4}-\rho^{2}+1},\quad\rho=\sqrt{1+\frac{1}{\kappa^{2}Q^{2} }}-\frac{1}{\kappa Q}. \tag{21}\]  
Figure 4a shows the variation of \(\lambda\) with \(\mu\) for a range of Q-factors. Lower values of \(\lambda\) are required for higher losses, as well as for lower values of \(\mu\). In each case, \(\lambda\to 1\) as \(\mu\) increases, with sharper increases at lower losses. When \(\lambda\) is set as above, port 2 is isolated at resonance. However, there is an impact on filter performance; namely, port 3 is no longer fully isolated, the passband of port 2 reduces, and reflectance at port 1 is improved.  
Figure 4b shows the frequency dependence of scattering parameters for filters with lossy MI (1) and resistive narrowband (2) terminations for \(\kappa=0.6\), \(\kappa_{c}=0.3\), \(Q=200\), and \(\lambda=0.8746\) from Equation (21). Both cases demonstrate excellent directional filtering with reflectance mostly \(<\)\(-\)30 dB within the passband, as well as infinite rejection at resonance. If isolation is important, \(\lambda\) may be adjusted to balance rejection with leakage into port 3. The mutual inductance discontinuity generated by tuning \(\lambda\) implies that the complementary waveguide pair is no longer symmetrical, meaning the ports cannot be used interchangeably. Only the two ports immediately adjacent to the \(\lambda\)-tuned resonator pair would yield an infinite-rejection response. The remaining two ports would suffer from reduced performance if used as inputs.