Q4Microwave Theory And Techniques

Step 1 — select the lowpass prototype: for a Chebyshev filter with 0.5 dB equal-ripple passband response and N=3 reactive elements, the standard tabulated (Pozar/Matthaei) lowpass prototype g-values, normalized to g0=1 (source) and a cutoff Ωc=1 rad/s, are:

(the symmetric g1=g3 pattern is a general property of equal-ripple Chebyshev prototypes with odd N).

Step 2 — bandpass transformation parameters: the fractional bandwidth is FBW = 0.10 (10%) and the centre frequency f0 = 5 GHz, giving angular centre frequency ω0 = 2π(5×10⁹) = 3.1416×10¹⁰ rad/s.

Step 3 — lowpass-to-bandpass element transformation: each series lowpass inductor gk (odd-indexed elements, g1 and g3 here) becomes a series LC resonator, and each shunt lowpass element (g2 here) becomes a shunt LC resonator, using the standard transformation:

Step 4 — numerical evaluation for this design (Z0=75 Ω, ω0=3.1416×10¹⁰ rad/s, FBW=0.10):

For the first series resonator (g1=1.5963):

For the shunt resonator (g2=1.0967):

For the third series resonator (g3=1.5963, identical to g1 by the symmetric-prototype property): L3=L1≈38.11 nH, C3=C1≈0.0266 pF.

Result and practical realization: the design consists of three resonators — series L1C1, shunt L2C2, series L3C3 (a total of three series/shunt alternating sections) with the numerical values computed above. In practice, at 5 GHz these lumped LC values (tens of nH, fractions of a pF) are too small to realize with discrete lumped components reliably, so the filter would normally be implemented instead using equivalent distributed elements — coupled microstrip or stripline resonators, or coupled waveguide cavities — whose physical dimensions are derived from these same g-values via the standard coupled-resonator bandpass filter design equations (external Q and coupling-coefficient formulas), a distributed realization being the practical microwave engineering approach even though the underlying lumped-element g-value design procedure demonstrated above remains the conceptual starting point for any bandpass filter synthesis at this frequency.

Why 0.5 dB Chebyshev (equal-ripple) rather than Butterworth (maximally flat): the choice of a 0.5 dB equal-ripple Chebyshev response, rather than the alternative maximally-flat (Butterworth) response, is a deliberate design trade-off: for the same filter order N, a Chebyshev design achieves a sharper transition from passband to stopband (steeper roll-off just outside the passband edges) at the cost of a small, bounded ripple (here, 0.5 dB) within the passband itself, whereas Butterworth sacrifices this sharper roll-off in exchange for a perfectly flat (ripple-free) passband. For a bandpass filter with a narrow fractional bandwidth (10% here), the sharper Chebyshev roll-off is often essential to adequately reject nearby out-of-band signals with a practically realizable (low) filter order, which is why Chebyshev prototypes such as the one used in this design are the standard choice for narrowband microwave filters where selectivity, not perfectly flat passband response, is the primary design driver.

Physical interpretation of narrowband transformation validity: the lowpass-to-bandpass transformation formulas used above (Step 3) are strictly valid in the narrowband approximation, generally considered accurate for fractional bandwidths up to roughly 20-30%; since this design's FBW=10% falls comfortably within that regime, the direct g-value transformation approach used here gives accurate component values without needing the more complex exact (non-narrowband) bandpass transformation required for wider-bandwidth filter designs. This validity range is itself a further practical reason narrowband Chebyshev bandpass filters, of the type specified in this problem, are so commonly encountered in microwave system design — channel-select filters, image-reject filters and IF filters in communication receivers are almost always narrowband by this same 10-20% criterion, making the simple transformation procedure demonstrated in this solution directly applicable to the great majority of real bandpass-filter design problems encountered in practice.