Insertion loss method and microstrip LPF

In the insertion loss method, the filter response is defined by the power loss ratio:

PLR = P_available / P_delivered = 1 / (1 − |Γ(ω)|²)
IL (dB) = 10 log PLR

A physically realisable PLR must be a ratio of even polynomials in ω, so the designer chooses a polynomial that gives the wanted response.

Steps of the method

1. Specify: fc, R0, ripple and minimum stopband attenuation.
2. Choose the response: Butterworth PLR = 1 + k²(ω/ωc)^(2N) for a flat passband; Chebyshev PLR = 1 + k²TN²(ω/ωc) for a sharper cut-off with ripple; linear-phase for flat group delay.
3. Find N from the stopband requirement.
4. Prototype values g0 … gN+1 from tables (R0 = 1 Ω, ωc = 1 rad/s).
5. Scale impedance and frequency: L = gR0/ωc, C = g/(R0ωc); transform to HP/BP/BS if needed.
6. Realise in transmission lines (Richards + Kuroda, stepped impedance), then simulate and tune.

Example: passive LPF using microstrip

Specs: fc = 2 GHz, R0 = 50 Ω, Chebyshev 0.5 dB ripple, N = 3 (π type: shunt C – series L – shunt C).

g1 = g3 = 1.5963,  g2 = 1.0967,  g4 = 1.0
C1 = C3 = 1.5963/(50×2π×2×10⁹) = 2.541 pF
L2      = 1.0967×50/(2π×2×10⁹) = 4.364 nH

Stepped-impedance microstrip (εr = 4.2, h = 1.58 mm, Zℓ = 20 Ω, Zh = 120 Ω):

C: βl = g·Zℓ/R0     L: βl = g·R0/Zh

 50Ω    ______        ______    50Ω
 ======|      |------|      |======
       |  C1  |  L2  |  C3  |
       |______| thin |______|
        wide          wide

All lines are shorter than λ/8 (45°), so the lumped approximation holds well. The 50 Ω feed lines are 3.13 mm wide.