S-parameters and Butterworth/Chebyshev filter responses

Usefulness of S-parameters in microwave network analysis

• They relate incident and reflected waves, which are measurable at microwave frequencies, while voltage and current are not (and are not unique in waveguides).
• They are measured with matched loads, avoiding open/short terminations that are hard to realise and that make active devices unstable.
• |S11| and |S21| directly give return loss, VSWR, insertion loss and gain; reciprocity and losslessness show as symmetry and unitarity of [S].
• They are measured accurately with a vector network analyzer and are the natural language of filter and amplifier design.

S-parameters of a two-port network

 b1 = S11·a1 + S12·a2        a = incident wave
 b2 = S21·a1 + S22·a2        b = reflected wave

• S11 = b1/a1 (a2 = 0): input reflection coefficient.
• S21 = b2/a1 (a2 = 0): forward transmission coefficient.
• S22 = b2/a2 (a1 = 0): output reflection coefficient.
• S12 = b1/a2 (a1 = 0): reverse transmission coefficient.

Butterworth and Chebyshev responses in the insertion loss method

In the insertion loss method, a filter is specified by its power loss ratio:

 PLR = Power available from source / Power delivered to load
     = 1/|S21(ω)|² = 1/(1 - |Γ(ω)|²)

For a lossless filter, |S21|² + |S11|² = 1.

Why these forms are used:

1. |Γ(ω)|² of a passive, lossless, reciprocal LC network is an even function of ω, so it can be written as a ratio of polynomials in ω²:

 |Γ(ω)|² = M(ω²)/(M(ω²) + N(ω²))
 PLR = 1 + M(ω²)/N(ω²)

2. Any PLR of this form (PLR ≥ 1, even, rational) is physically realisable as a ladder of L and C elements. Taking N = 1 gives an all-pole low-pass prototype, the simplest ladder.
3. The designer then chooses the polynomial M to give the desired passband shape:

Butterworth (maximally flat):

 PLR = 1 + k²(ω/ωc)^(2N)

All derivatives of PLR are zero at ω = 0, so the passband is as flat as possible; the stopband falls at 20N dB/decade. k = 1 gives the 3 dB point at ωc.

Chebyshev (equal ripple):

 PLR = 1 + k²·T_N²(ω/ωc)
 T_N(x) = cos(N·cos⁻¹x),    |x| ≤ 1
 T_N(x) = cosh(N·cosh⁻¹x),  |x| > 1

Passband ripple = 1 + k²; the cutoff is much sharper than Butterworth for the same order N (stopband ≈ k²(2ω/ωc)^(2N)/4).

4. Both are polynomial (all-pole) responses, so the same ladder prototype structure (g₁, g₂, … gₙ tables) serves both. Through impedance and frequency scaling, low-pass to high-pass/band-pass/band-stop transforms, and Richards' transformation and Kuroda identities, the same prototype becomes a microwave filter of stubs or coupled lines.

Hence the Butterworth and Chebyshev responses are the standard (common) responses of prototype microwave two-port filters: they are the simplest realisable PLR functions, one optimizing flatness and the other optimizing selectivity, and both map directly onto S21 = 1/√PLR of the two-port.