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Table 2 compares the resonator values of this first filter example to the example produced in ref. 3. The results of the comparison are also depicted in Fig. 4. As can be seen from the comparison of the two example filters, while the closed-form equations presented in the current report are not quite as accurate as the relations used in ref. 3, they are easier to implement and deliver reasonable results.

4. These graphs represent the iterative method developed by Howard and Lin3 (blue line) and the results of using the closed-form equations from the current report (red line).

For a second comparison of example filters, the example of ref. 2 will be used. It is a five-section Chebyshev filter with 0.01-dB ripple and 1% bandwidth at 4 GHz. The lowpass prototype parameters for this filter design are g0 = 1; g1 = 0.7563; g2 = 1.3049; g3 = 1.5773; g4 = 1.3049; g5 = 0.7563; and g6 = 1. Using Eq. 17, Δ is found to be 0.7716. Using Eq. 18 to find the inner resonators result in the following: l1 = l4 = 3.9964 cm and l2 = l3 = 4.3303 cm.

Equations 13 and 15 are used to find the outer resonator lengths. Since these are arbitrary, the largest g values are chosen to calculate the lengths for the outer resonators. The outer lengths then are l0 = l5 = 4.6432 cm.

As Fig. 5 shows, the results of the interactive method of ref. 2 and the closed-form method of the present report are much closer, as the bandwidth for these example filters is much more narrow (1%). The bandwidth plotted in Fig. 4 is much wider (about 7.5%) and the results between those two methods (the present approach and ref. 3) are more diverse.

5. These graphs offer the iterative method of Craven and Mok2 (blue line) and the results of using the closed-form equations from the current report (red line).

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