Single-section series arm BPF and insertion loss

Single-section series-arm passive BPF in microstrip

A single-section band-pass filter has a series LC resonator in the series arm. It passes signals near its resonant frequency f₀ = 1/(2π√(LC)) and blocks others. In microstrip, the series resonator is made as a half-wavelength (λg/2) line coupled to the input and output lines through gaps. The gaps act as series capacitors, and the λ/2 line behaves like a series-resonant circuit near f₀.

 Lumped:   in --| |--UUU-- out
                C    L   (series arm)

 Microstrip (end-coupled):
 50 Ω            λg/2 resonator            50 Ω
 =========| |=========================| |=========
          gap1                         gap2

The gap widths set the coupling and so the bandwidth. A smaller gap gives more coupling and a wider band. Several such resonators in cascade give a higher-order end-coupled BPF. A parallel-coupled-line layout is a more compact alternative.

Filter design by the insertion loss method

The insertion loss method starts from the wanted power loss ratio and synthesises a network that gives it:

PLR = 1 / (1 − |Γ(ω)|²) = 1 + M(ω²)/N(ω²)
IL (dB) = 10 log10 PLR

Steps

1. Specifications: centre frequency f₀, fractional bandwidth Δ = (ω₂ − ω₁)/ω₀, passband ripple, stop-band attenuation, and R₀ = 50 Ω.
2. Response and order: Butterworth, PLR = 1 + k²(ω/ωc)²ᴺ (maximally flat), or Chebyshev, PLR = 1 + k²T_N²(ω/ωc) (equal ripple, sharper). Find N from attenuation charts.
3. Low-pass prototype: read the g-values from tables (R₀ = 1, ωc = 1). For example, Butterworth N = 3 gives 1, 2, 1.
4. Impedance scaling and LP → BP transformation, with ω → (1/Δ)(ω/ω₀ − ω₀/ω):
   • series L_k becomes a series LC: L' = R₀g_k/(ω₀Δ), C' = Δ/(ω₀R₀g_k)
   • shunt C_k becomes a parallel LC: L' = R₀Δ/(ω₀g_k), C' = g_k/(ω₀R₀Δ)
5. Microwave realisation: use admittance or impedance inverters with λ/2 resonators (end-coupled), λ/4 coupled lines (parallel-coupled), or stub resonators. Find the gap or coupling values from the inverter constants.
6. Layout and tuning: include gap, open-end and step effects, then simulate and tune the lengths and gaps.

The method gives a predictable, exactly specified response, which is why it is the standard method for microwave filters.