## 3 Circuit Models
### CPWs Loaded with Pairs of Coupled SRRs
Figure 13: Typical topology of a CPW loaded with a pair of SRRs (**a**), and lumped element equivalent circuit model, considering magnetic coupling between SRRs (**b**). The model considers the general case of different SRRs, or identical SRRs with different dielectric loading (in the latter case, \(L_{1}=L_{2}\)). If \(M\)' \(\
eq\) 0 and the structure is symmetric (\(M_{1}=M_{2}=M\), \(\omega_{01}=\omega_{02}=\omega_{00}\), \(L_{1}=L_{2}=L_{r}\), and \(C_{1}=C_{2}=C_{r}\)), the two solutions of (2) are (\(k_{M}\) is the magnetic coupling coefficient):  
\[\omega_{z}=\frac{\omega_{o}}{\sqrt{1\mp\frac{M^{\prime}}{L_{r}}}}=\frac{\omega _{o}}{\sqrt{1\mp k_{M}}} \tag{3}\]  
However, one of the solutions, the one with the (-) sign in the radicand of Expression (3) (\(\omega_{+}\)), is not actually a transmission zero frequency. The reason is that this frequency also nulls the numerator of the last term in Expression (1). By applying l'Hopital's rule, it follows that the series impedance is finite at the frequency \(\omega_{+}\); hence, we can conclude that only one transmission zero, to the left of \(\omega_{00}\) (since \(M\)' \(>\) 0), appears in the transmission coefficient for symmetric structures with magnetically coupled SRRs [26]. The transmission zero frequency for the symmetric case, \(\omega_{0}\)-, can alternatively be obtained by applying the magnetic wall concept to the circuit of Figure 13b. Before that, it is necessary to transform the magnetically coupled inductors \(L_{r}\)'s of the circuit of Figure 13b to the equivalent T-circuit model (see Figure 14) [44]. Notice that the inductance of each resonator at both sides of the symmetry plane (magnetic wall) is given by \(L_{r}\)+\(M\)', from which Expression (3) with the (+) sign results. It should be mentioned that for the symmetric case, the model of [42] provides the same macroscopic results. However, the equivalent inductance of the resonators in the model presented here includes the effects of \(M\)', _i.e._, in [42]\(M\)' is inherently included in the SRR inductance. For the asymmetric structure (different coupled SRRs), but considering that their fundamental resonance frequencies are identical (\(L_{1}\
eq L_{2}\) and \(C_{1}\
eq C_{2}\) where \(\omega_{1}=\omega_{2}=\omega_{00}\)), Expression (2) gives:  
Figure 11.0  
Note that Expression (4) is formally identical to Expression (3), but replacing the inductance \(L_{r}\) with the geometrical mean of the two inductances \(L_{1}\) and \(L_{2}\) (both expressions can be written in terms of the magnetic coupling coefficient, \(k_{M}\)). The two frequencies given by Expression (4) null the denominator of the last term in (1), but, in general, none of them nulls the numerator, thus providing two transmission zeros. However, there is a specific condition:  
Figure 11.1 **(a)** Circuit model of the structure of Figure 13 for the symmetric case, with the inductive coupling between resonators transformed to the equivalent T-model (indicated on the left); **(b)** circuit that results by applying the magnetic wall concept. Figure 11.2  
that gives only one transmission zero (at \(\omega_{\text{-}}\)), since the numerator of the last term in (1) is also null at \(\omega_{\text{+}}\), and the impedance of the series branch is finite at this frequency (as the application of l'Hopital's rule reveals). The last analysis corresponds to the asymmetric case with arbitrary resonator frequencies (\(L_{1}\
eq L2\) and \(C_{1}\
eq C_{2}\) with \(\omega_{1}\
eq\omega_{2}\)).