Insertion loss method for filter design

The insertion loss method designs a filter from a chosen power loss ratio (frequency response), then builds a lumped low-pass prototype that gives exactly that response, and finally scales and converts it to the required filter and to microwave form.

Power loss ratio

PLR = P_available / P_load = 1 / (1 − |Γ(ω)|²)
IL  = 10 log10(PLR)  dB
|Γ(ω)|² = M(ω²) / [M(ω²) + N(ω²)]
=> PLR = 1 + M(ω²)/N(ω²)

Common responses

• Maximally flat (Butterworth): PLR = 1 + k²(ω/ωc)²ᴺ. Flattest passband, slower roll-off.
• Equal ripple (Chebyshev): PLR = 1 + k² T_N²(ω/ωc). Ripple in the passband, sharper cut-off.
• Elliptic: ripple in both bands, sharpest cut-off.
• Linear phase (Bessel): good group delay, poor roll-off.

Design steps

 Filter specs (fc, ripple, attenuation)
            |
 Choose response + order N
            |
 Low-pass prototype (g-values, R0=1, ωc=1)
            |
 Impedance & frequency scaling,
 LP -> HP / BP / BS transformation
            |
 Microwave realisation (Richards, Kuroda,
 stepped-impedance, coupled lines)

1. Write the specifications: cut-off frequency, passband ripple, stop-band attenuation and impedance.
2. Choose the response type and find the order N from attenuation charts.
3. Take element values g₁…g_N from prototype tables (for example Butterworth N = 3: g = 1, 2, 1).
4. Scale impedance (L' = R₀L, C' = C/R₀) and frequency (divide by ωc), and transform to HP, BP or BS if needed.
5. Convert lumped L and C into distributed microstrip elements and check the response by simulation.

Advantages: full control over passband and stop-band response, a systematic and repeatable synthesis, and easy trade-off between order, flatness and roll-off. It has replaced the older image-parameter method.