and lower 3 dB points are given as the solution to the equation: where, So, if the center frequency and the width of the resonance are determined from first principles codes, the L,C of the equivalent circuit may be readily obtained by fitting the reflection response of the equivalent resonant circuit to the reflection response of the actual FSS, and in this way, the circuit parameters L,C are readily extracted. Once that is done, then we can use the equivalent circuit model for multi-layer FSS design. Any nearby dielectrics should be included in the equivalent circuit. For small values of ω, the impedance of the inductor, jωL, is smaller than the impedance of the capacitor, 1/jωC, therefore the capacitor dominates the shunt impedance and so the patch-type bandstop FSS is capacitive below resonance. We'll use this fact in section 2.3.1 to design a lowpass FSS filter using equivalent circuits. Bandpass FSS filter equivalent circuit and transmission response The equivalent circuit for a mesh-type bandpass FSS is shown in Fg. 2.4.2-1. The admittance of the parallel connection of inductor and capacitor is (Desoer, Kuh [1984]): and this admittance is zero (open-circuit condition) when When the parallel combination of inductor and capacitor produces an open circuit, all energy is transmitted. In the same way, the magnitude of the transmission coefficient of the bandpass filter is: Below resonance, the admittance of the inductor, 1/jωL is greater than the admittance of the capacitor jωC, therefore the mesh-type bandpass FSS is inductive below resonance. Comparison of equivalent circuit response and actual FSS response Fig. 2.4.3-1 shows the comparison in reflection between a single-layer crossed dipole FSS and its fitted equivalent circuit. The equivalent circuit is a series connection of a capacitor and inductor placed in parallel across the transmission line, as in Fig. 2.4.1-2. This resonator produces a short circuit condition at resonance. The fit is very good below the resonance though not nearly as good above. The real FSS has a reflection null at 18.7 GHz (the frequency at which the wavelength equals the unit cell dimension of .630"), which is not accounted for in the equivalent circuit model. The null is known as a Wood's anomaly and is caused by the inverse square root singularity in the spectral domain Green's function () going to infinity. Physically, this represents a uniform plane wave propagating in the plane of the FSS. In the spatial domain,