## 3 Electrically Small Planar Resonators: A Comparative Analysis and Circuit Models
### Topologies and Circuit Models of the Isolated Resonators  
Let us begin with the SRR already considered in the previous section [12]. As long as the distance between both rings is small (_i.e._, there is significant coupling between them), the particle can be considered to be electrically small and a quasi-static analysis to infer the SRR inductance and capacitance can be applied. Such an analysis was carried out in [7,15]. When the SRR is excited by an external time-varying magnetic field directed along the SRR axis (\(z\)-axis in Figure 3), the cuts on each ring force the electric current to flow from one ring to another across the slot between them, taking the form of a strong displacement current. The slot between the rings therefore behaves as a distributed capacitance, and the whole SRR has the equivalent circuit shown in Figure 3(b), where \(L_{s}\) is the SRR self-inductance and \(C_{o}\)/2 is the capacitance associated with each SRR half. This capacitance is \(C_{o}=2\pi r_{o}C_{pull}\), where \(r_{o}\) is the mean radius of the SRR, and \(C_{pull}\) is the per unit length capacitance along the slot between the rings. The total capacitance of this circuit is the series connection of the capacitance of both SRR halves, that is, \(C_{o}\)/4. Therefore, the resonance frequency \(\omega_{b}\) is given by:  
\[\omega_{o}=\frac{2}{\sqrt{L_{s}C_{o}}}=\sqrt{\frac{2}{\pi r_{o}L_{s}C_{pull}}} \tag{1}\]  
The inductance \(L_{s}\) can be modeled as the inductance of an average ring of radius \(r_{o}\), _i.e._, the mean radius of the ring, and width \(c\). Approximate expressions for \(L_{s}\), based on a variational calculation, are provided in [16]. Closed expressions for the per unit length capacitance are given in many microwave textbooks [17], and explicit expressions for \(L_{s}\) and \(C_{pull}\) are provided in [7]. The model of Figure 3 is valid as long as the perimeter of the ring can be considered small with regard to half wavelength, and the capacitance associated with the cuts on each ring can be neglected. Under such assumptions, the currents on each ring must vanish at the cuts, and the angular dependence of the currents on each ring can be assumed to be linear (so that the total current on both rings is constant). Such assumptions also imply that the voltage across the slots is constant in both SRR halves. Figure 2: (**a**) \(\lambda\)/2 ring resonator. (**b**) Split ring resonator (SRR). (**c**) First bulk SRR-based reported structure exhibiting backward wave propagation in a certain frequency band. Photograph courtesy of [PERSON]. [PERSON]. A more detailed circuit model, which takes into account the gap capacitance and includes a transmission line model for the slot between the rings has been reported in [18], but such a model converges to the model of Figure 3 when the capacitance of the cuts is neglected and the electrical length of the SRR is small [18]. As both approximations are usually fulfilled for any practical SRR design, the more simple model reported above is assumed valid in this paper. The closer the rings of the SRR of Figure 3 are the lower the resonance frequency and the smaller the electrical size of the particle result. However, in practice, the separation between rings cannot be smaller than a certain value that depends on the fabrication technology (typically 100 \(\,\mathrm{\SIUnitSymbolMicro m}\) for PCB technology). This means that the electrical size of edge coupled SRRs cannot be reduced below roughly \(\lambda/10\). To circumvent this limiting aspect, one possible solution is to print the rings face-to-face on opposite sides of a thin dielectric slab. Whilst this modification does not significantly affect the behavior of the resonator, the per unit length capacitance (_i.e._, the capacitance of the broadside coupled strips) is markedly increased, and hence the particles can be made as small as \(\lambda/30\)[15], or even smaller. Obviously, the penalty of these broadside coupled split ring resonators (BC-SRRs) is the need of two metallic levels for their implementation.