## 2 Theory and Principle
### Lines Loaded with a Single Symmetric Resonator (Resonance-Based Structures)  
In general, if a planar resonant element is etched in close proximity to a transmission line, coupling between the line and the resonator arises, and the line prevents the transmission of signals at the fundamental resonance frequency and at higher-order resonance frequencies. Depending on the line and resonator topology, orientation, and position, the line-to-resonator coupling may be electric, magnetic, magneto-electric (or mixed), or it may be cancelled (in this case providing total transmission at the resonance frequency). From now on, we will focus the attention only on frequencies in the vicinity of the fundamental resonance frequency of the resonant element. In lines loaded with a single and symmetric resonant element, and assuming that the resonator is under the influence of the field generated by the line, a necessary condition to prevent line-to-resonator coupling is a perfect alignment between the line axis and the symmetry plane of the resonator. Under these circumstances, if the symmetry planes of the line and resonator are of different electromagnetic nature (one an electric wall and the other one a magnetic wall), then net coupling is cancelled, the resonator is not excited, and the line is transparent [19; 27]. Conversely, if the symmetry planes are of the same electromagnetic nature (either an electric wall or a magnetic wall) the resonator is driven and the transmission coefficient exhibits a resonance (notch). Note that certain lines support more than one fundamental mode of propagation (e.g., differential microstrip lines may propagate even- and odd-modes), and hence the electromagnetic nature of the symmetry plane depends on the considered propagating mode. This means that in such lines, selective mode suppression is possible [27], as will be discussed later. If the symmetry planes of the line and resonator are misaligned, then, in general, line-to-resonator coupling arises. Note that in the context of this paper an electric/magnetic wall is a virtual plane with an anti-symmetric/symmetric charge distribution at both sides. Besides, even- and odd-modes are symmetric and anti-symmetric modes, respectively, that propagate in symmetric structures, such as in a symmetric pair of coupled lines. In summary, the line-to-resonator coupling may be controlled or modulated, which in turn modulates the associated resonance. According to this feature, a possible designation for such structures may be coupling-modulated resonance-based structures or simply resonance-based structures. To clarify the previous assertions related to the coupling (resonance) prevention, consider as an illustrative example a CPW transmission line (see Figure 2). The fundamental CPW mode is an even mode and hence the transmission line exhibits a magnetic wall at its symmetry plane. If we align with such a line an SRR (it exhibits an electric wall at its symmetry plane at the fundamental resonance), the resonator will not be excited (Figure 2a). In terms of fields, for this configuration there is a perfect cancellation of magnetic fields in the inner region of the SRR, and the resonator is not excited because there is no net magnetic field penetrating the SRR [19]. However, for the slot (odd) mode of the CPW, the resonator will be excited, since the symmetry plane of the line is an electric wall for that mode (Figure 2b), and a net magnetic field in the SRR, able to induce electric currents in the particle, arises. Thus, selective mode suppression in the vicinity of the SRR fundamental resonance is possible [28]. It will be shown later (Section 4) that selective mode suppression in balanced microstrip lines is possible by using CSRRs, and particularly useful for common-mode suppressed balanced lines.