Equal-element filter improves passband performance

Richard M Kurzrok, RMK Consultants, Queens Village, NY -- EDN, 3/15/2001

Designers originally conceived equal-element filters as all-pole microwave bandpass filters that provide minimum center-frequency insertion losses for specific values of resonator-unloaded Q (Reference 1). All resonators of the equal-element bandpass filter operate at the same loaded Q. For LC filters, the equal-element filter has another advantage. In the lowpass prototype, all inductors have the same value, and all capacitors have the same value. This minimum number of circuit elements provides design simplicity and reduces filter cost.

However, the equal-element filter's response shape has one severe shortcoming. Passband amplitude ripples, due to reflection, are unacceptable for some applications. In minimum-phase-shift filter circuits, group-delay ripples that preclude equalization accompany the amplitude ripples. At microwave frequencies, modifying the central resonator of a five-pole bandpass filter leads to improved performance (Reference 2).

Figure 1a shows the schematic of a nine-pole, equal-element, lowpass-filter prototype. Figure 1b shows a comparable schematic of a modified lowpass-filter prototype. By altering the filter input and output capacitors, the modified filter realizes substantial improvement in passband performance with some reduction in stopband selectivity. In the modified equal-element design, all filter inductors are still equal in value, and you need only two capacitor values in a convenient 2-to-1 ratio. Table 1 shows comparative theoretical amplitude responses for the nine-pole, equal-element lowpass filter and a nine-pole, modified, equal-element, lowpass filter with inductor-unloaded Qs of 100.

The reference frequency, at normalized frequency x=1.0, is not the 3-dB cutoff frequency for the filters in Figure 1. The 3-dB cutoff frequency occurs close to x=1.9. This feature differs from Butterworth and Chebyshev filters, for which x can equal 1.0 at 3-dB cutoff frequencies. You use this normalization for equal-element and modified equal-element designs to calculate the values of the circuit elements.

A nine-pole, modified, equal-element lowpass filter was designed at a reference frequency F_R of 4.681 MHz, for which x=1. For 50? input and output impedances Z₀, you calculate the normalizing inductance, L₀, and normalizing capacitance, C₀, as follows:

You then use these values of L₀ and C₀ to denormalize the filter to actual circuit-element values. Filter inductors L₁, L₂, L₃, and L₄ are equal to L₀=1.7 µH. Interior filter capacitors C, C₃, and C₄ are equal to C₀=680 pF. The filter input and output capacitors, C₁ and C₅, are equal to 0.5×C₀=340 pF. In the actual filter, the input and output capacitors are standard 330-pF values. The nine-pole, modified, equal-element, lowpass filter was constructed in a die-cast aluminum box with BNCs. The filter circuit was fabricated using vector board. All capacitors were 5%- tolerance polypropylene units. All inductors used 18 turns of number 26 magnet wire on Micro Metals' T37-2 toroids. Table 2 shows the measured amplitude-response data. The measured data provides reasonable correlation between theory and experiment and shows substantial improvement in amplitude response over most of the filter passband with some degradation in stopband performance.