## 3 Circuit Models
### CPWs Loaded with ELC Resonators  
The ELC resonator (Figure 9) is a bisymmetric resonant particle [39]. At the fundamental resonance frequency, the currents in the two arms flow in opposite directions (one is clockwise and the other one  
Figure 8: Simulated common-mode and differential-mode insertion loss \(|S_{cc21}|\) and \(|S_{dd21}|\) for the order-three structure that results by cascading three-unit cells like the one shown in Figure 7. Inter-resonator distance is 0.15 mm. From [36]; reprinted with permission from IEEE 2011. Figure 7: Simulated common-mode insertion loss \(|S_{cc21}|\) for the structure shown in the inset. CSRR dimensions are: \(c=0.2\) mm, \(d=0.2\) mm, \(W=1\) mm, \(S=2.5\) mm, and side length of 7.6 mm. The considered substrate is _Rogers RO3010_ with thickness \(h=1.27\) mm and dielectric constant \(\varepsilon_{r}=10.2\). Extracted parameters are \(L_{e}=9.86\) nH, \(C=1.06\) pF, \(C_{c}=2.68\) pF and \(L_{c}=3.36\) nH. From [36]; reprinted with permission from IEEE 2011.  
is counterclockwise, as Figure 9 illustrates). The particle can be excited by means of a time-varying uniform electric field applied to the direction orthogonal to the electric wall (\(y\)-direction), but it cannot be excited by means of a uniform magnetic field polarized in the \(z\)-direction, since there is not a net magnetic dipole in that direction at the fundamental resonance. However, the particle can be excited by means of counter-magnetic fields applied to the two loops, as Figure 9 shows. From these assertions, it follows that if the particle is etched on the back side of a CPW transmission line (or on top of it), it can be excited depending on its relative orientation with regard to the line [40, 41]. If the magnetic wall of the particle is aligned with the line axis (also a magnetic wall for the fundamental CPW mode), the ELC will be excited and a notch in the transmission coefficient is expected. Conversely, the CPW will be transparent if the ELC is oriented with the electric wall aligned with the line axis. Obviously for intermediate angles, the particle will be coupled to the line and the coupling strength can be controlled by the angle. These ELC-loaded lines are therefore useful as rotation sensors for alignment, displacement and velocity measurements, as will be shortly discussed. The proposed equivalent circuit model of an ELC-loaded CPW is depicted in Figure 10a [41]. This circuit is valid as long as the cell length is electrically small (losses are not considered). The ELC resonator is represented by the inductances \(L_{e}\) and the capacitance \(C_{e}\), modeling the inductive loops and the capacitive gap, respectively. The CPW is modeled by its inductance and capacitance, \(L\) and \(C\), respectively, and it is divided into two identical halves for convenience. Finally, each half is magnetically coupled to each loop through the mutual inductances \(M_{\theta}\) and \(M^{\phi}\), both being dependent on the angle \(\theta\) (different dots are used to distinguish the magnetic coupling sign associated to each of the halves). Therefore, the frequency response of the circuit of Figure 10a directly depends on the angle \(\theta\). When \(\theta=0^{\circ}\), due to symmetry, \(M_{\theta}=M^{\phi}\
eq 0\), which means that the currents flowing on the CPW induce a pair of equal and anti-phase voltages in the loops; no net voltage is induced in the resonator due to an absolute cancellation. In other words, there is not a net magnetic coupling, the resonator cannot be magnetically driven, and the resulting model is that of a conventional transmission line. On the other hand, as \(\theta\) increases, the magnetic coupling is complementarily distributed; \(M^{\phi}\) increases at the expense of a decrease in \(M_{0}\). Hence for \(\theta>0^{\circ}\), a net induced voltage arises and the line is indeed capable of magnetically exciting the resonator. The larger the angle, the higher the induced voltage.