angle does not vary with frequency. Butterworth lowpass filter design using FSS equivalent circuits Starting point: prototype lumped L, C Butterworth filter As an example of how to use FSS equivalent circuits for quick and efficient design of a practical filter, we can sketch out the process that would be followed in designing a 5-stage Butterworth filter (Hunter [2001], Matthaei [1964]) using a stack of 5 frequency selective surfaces, with 4 air spacers in between the FSS sheets. The lowpass prototype L,C ladder network is shown in Fig. 3.1.1-1 (Hunter [2001]). The cutoff frequency will be scaled to 7 GHz and the filter will be matched to 377 Ohms (the impedance of free space) on the input and output sides. The idea we'll follow is that the shunt capacitors will eventually be replaced by sub-resonant (capacitive) patch-type FSS sheets and the series inductors will be replaced by air spacers between the 5 FSS layers. Short transmission lines are approximately equivalent to series inductors. Transmission response of prototype lumped L, C filter The transmission magnitude and phase response of the scaled Butterworth L,C filter is shown in Fig. 3.1.2-1. Transmission magnitude is flat in the passband (below the 7 GHz cutoff frequency) and has a monotonically decreasing skirt on the high frequency side of the passband. The phase through the filter is linear throughout the 7 GHz passband, making this filter an ideal choice for a linear phase filter application, for example in the design of an ultra-wideband filter that approximates a true time delay transmission line. This is the baseline lumped L,C filter that will be the starting point for our 5-layer FSS Butterworth filter design. Now we begin the process of transforming the prototype Butterworth lumped L,C filter into an equivalent FSS Butterworth filter. Two modifications of the baseline lumped L,C filter will be necessary, in order to obtain the corresponding FSS filter. First, the series inductors will be replaced by their equivalent transmission line sections, and then the shunt capacitors will be replaced by capacitive frequency selective surfaces. First transformation: replace series inductors with transmission line spacers At this point in the development, the series inductors in the prototype L,C ladder network will now be replaced by sub-half-wavelength air spacers (which we will model as transmission lines) between the FSS layers. The thickness of the air spacers may be determined as shown in Fig. 3.1.3-1, in which we