## 3 Circuit Models
### CPWs Loaded with Pairs of Coupled SRRs  
CPW transmission lines loaded with pairs of SRRs can be used as sensors based on frequency splitting. Typically, the coupling between resonators cannot be neglected due to their close proximity. Thus, the modeling of these loaded lines must include not only the coupling between the line and the \begin{table}
\begin{tabular}{c c c c c c} \hline
**0 (Degrees)** & \(C\) (pF)** & \(L\) (nH)** & \(C_{e}\) (pF)** & \(L_{e}\) (nH)** & \(M\) (nH)** \\ \hline
30 & 5.86 & 5.95 & 3.09 & 25.6 & 0.94 \\
60 & 5.73 & 6.22 & 3.09 & 25.6 & 1.91 \\
90 & 5.55 & 6.57 & 3.06 & 25.6 & 2.76 \\ \hline \end{tabular}
\end{table} 
Table 1: Extracted lumped element values of the circuit of Figure 10b for the structure of Figure 11.  
Figure 11: Layout of the circular-shaped ELC-loaded CPW structure (for a specific angle of \(30^{\circ}\)). The considered substrate is _Rogers RO3010_ with \(h=1.27\) mm and \(\varepsilon_{r}=11.2\). The dimensions are: for the line, \(W\) and \(G\) are tapered such that the characteristic impedance is \(50\)\(\Omega\); for the ELC resonator: mean radius \(r_{0}=8.05\) mm, capacitor outer radius \(r_{1}=11\) mm, \(w_{2}=s=0.2\) mm, and \(w_{3}=0.5\) mm. Figure 12: **(a)** Magnitude and **(b)** phase of the reflection and transmission coefficients given by the lossless electromagnetic simulation for the structure of Figure 11 and by the circuit simulation for the models of Figure 10. The circuit parameters are indicated in Table 1. From [41]; reprinted with permission from IEEE 2013.  
resonant elements, but also the inter-resonator coupling [26]. A typical CPW loaded with a pair of square-shaped SRRs and its lumped element equivalent circuit are depicted in Figure 13[26]. Line-to-resonator coupling is accounted for through \(M_{1}\) and \(M_{2}\), and inter-resonator coupling through _M'_. The per-unit-cell inductance and capacitance of the line are \(L\) and \(C\), respectively, and the SRRs are described by the resonant tanks \(L_{1}\)-\(C_{1}\) and \(L_{2}\)-\(C_{2}\). The model considers the general case of an asymmetric structure. The circuit of Figure 13b is a \(\pi\)-circuit where the shunt branches are purely capacitive. Thus, the transmission zeros (or resonance frequencies of the coupled resonators) are given by the poles of the series reactance. From [PERSON]'s equations, including the different mutual couplings, applied to the series branch of the circuit of Figure 13b, the impedance of this branch is found to be:  
\[Z_{s}(\omega)=j\omega\Delta+j\omega^{3}\times\left\{\frac{C_{1}M_{1}^{\ 2}\left(1-\frac{\omega^{2}}{\omega_{2}^{2}}\right)+C_{2}M_{2}^{\ 2}\left(1-\frac{\omega^{2}}{\omega_{2}^{2}}\right)+2\omega^{2}M_{1}M_{2}M^{ \prime}C_{1}C_{2}}{\left(1-\frac{\omega^{2}}{\omega_{1}^{2}}\right)\left(1- \frac{\omega^{2}}{\omega_{2}^{2}}\right)-\omega^{4}M^{\prime\ 2}C_{1}C_{2}}\right\} \tag{1}\]  
where \(\omega_{1}=(L_{1}C_{1})^{-1/2}\) and \(\omega_{2}=(L_{2}C_{2})^{-1/2}\) are the resonance frequencies of the isolated resonators. By forcing the denominator of the last term to be zero, the notch (or transmission zero) frequencies can be derived, namely:  
\[\omega_{z}^{2}=\frac{\omega_{1}^{2}+\omega_{z}^{2}\pm\sqrt{\left(\omega_{1}^{2 }-\omega_{2}^{2}\right)^{2}+4M^{\prime\ 2}\ \omega_{1}^{4}\omega_{2}^{4}C_{1}C_{2}}}{2[1-M^{ \prime\ 2}\ \omega_{1}^{2}\omega_{2}^{2}C_{1}C_{2}]} \tag{2}\]  
Note that these frequencies do not depend on the mutual couplings, \(M_{I}\) and \(M_{2}\), between the line and the SRRs, but depend on the mutual coupling, \(M\)', between the SRRs. If inter-resonator coupling is negligible (\(M\)' = 0), which is reasonable if the SRRs are separated enough, the solutions of Expression (2) are simply \(\omega_{I}\) and \(\omega_{2}\). In this case, the equivalent circuit is formally identical to the one reported in [42], but with two different resonators in the series branch, due to the asymmetry of the unit cell with regard to the line axis.