Microstrip LPF implementation and prototyping

Low-pass filter in microstrip

A microwave LPF is built in microstrip by replacing the lumped L and C of a ladder prototype with short transmission-line sections. Two common forms are used.

1. Stepped-impedance (hi-Z / low-Z) filter

A short line (βl < π/4) of high impedance Z_h behaves like a series inductor. A short line of low impedance Z_ℓ behaves like a shunt capacitor:

βl = L·R0 / Zh   (inductor, L normalised)
βl = C·Zℓ / R0   (capacitor, C normalised)

Z_h is made as high as can be etched (narrow strip, about 100–150 Ω), and Z_ℓ as low as practical (wide strip, about 10–20 Ω).

 Stepped-impedance microstrip LPF (top view)
       ___        _____        ___
 =====|   |______|     |______|   |=====
 50 Ω | C1| L2   | C3  |  L4  | C5| 50 Ω
      |___|      |_____|      |___|
 wide = low Z0 (shunt C)
 narrow = high Z0 (series L)

2. Stub filter (Richards' transformation + Kuroda identities)

• Richards' transformation maps a shunt capacitor to an open-circuited stub and a series inductor to a short-circuited series stub, each λ/8 long at fc.
• Kuroda identities change the series stubs into shunt open stubs separated by λ/8 unit elements, so all stubs are easy to make in microstrip.

       open stubs (λ/8 at fc)
        | |      | |      | |
 50 Ω ==+=+======+=+======+=+== 50 Ω
          unit elements (λ/8)

Prototyping of a low-pass filter

A low-pass prototype is a normalised LC ladder with source resistance g₀ = 1 Ω and cut-off ωc = 1 rad/s.

  g0     g2 (L)
 --/\/--+--UUU--+-- ... --+
         |       |         g(N+1) load
       g1 (C)   g3 (C)
         |       |
 --------+-------+---------

1. Choose the response: Butterworth (maximally flat), PLR = 1 + (ω/ωc)²ᴺ, or Chebyshev (equal ripple), PLR = 1 + k²T_N²(ω/ωc).
2. Find the order N from the required attenuation at a given stop-band frequency, using attenuation-versus-normalised-frequency charts.
3. Read the g-values from tables. For Butterworth, g_k = 2 sin[(2k − 1)π/(2N)]. For example, N = 3 gives g₁ = 1, g₂ = 2, g₃ = 1.
4. Scale to the real impedance R₀ and cut-off ωc: L_k = R₀ g_k / ωc and C_k = g_k / (R₀ ωc).
5. Realise the elements in microstrip as above and check the response with a simulator.