LPF prototype conversion and second order HPF

Low-pass prototype (Butterworth / Chebyshev)

A low-pass prototype is a normalised LC ladder with source resistance g₀ = 1 Ω and cut-off ωc = 1 rad/s. Its element values g₁…g_N are chosen so that its power loss ratio follows the chosen approximation.

 g0=1    g2 (L)       g4 (L)
 -/\/\-+--UUU--+--UUU--+--  g(N+1)
       |       |        |    (load)
     g1 (C)   g3 (C)   ...
       |       |        |
 ------+-------+--------+--

• Butterworth: g_k = 2 sin[(2k − 1)π/(2N)], with k = 1 (3 dB at ωc). For N = 3: 1, 2, 1.
• Chebyshev: g-values depend on the ripple and are taken from tables. For 0.5 dB ripple, N = 3: 1.5963, 1.0967, 1.5963, with g₄ = 1.0.
• The order N is found from attenuation-versus-|ω/ωc| − 1 charts for the required stop-band attenuation.

Conversion to real values and other filter types

Impedance scaling: L' = R₀L, C' = C/R₀, R' = R₀.

Frequency scaling and transformations (ω₀ = √(ω₁ω₂), Δ = (ω₂ − ω₁)/ω₀):

• LP → HP: ω → −ωc/ω, so inductors become capacitors and capacitors become inductors.
• LP → BP: ω → (1/Δ)(ω/ω₀ − ω₀/ω). A series L becomes a series LC (L' = R₀g/(ω₀Δ), C' = Δ/(ω₀R₀g)). A shunt C becomes a parallel LC.
• LP → BS: the inverse of the BP transformation.

The lumped elements are then realised with transmission lines using Richards' transformation and Kuroda identities, or stepped-impedance sections.

Second-order high-pass π-section in microstrip

Assumed: fc = 2 GHz, R₀ = 50 Ω, so ωc = 1.2566×10¹⁰ rad/s. A π-section has a shunt L, a series C and a second shunt L. A strictly second-order filter has only two reactive elements, so the second shunt arm is left out (half-π). A full symmetric π with both shunt arms is third order. Both sets of values are given:

2nd order (Butterworth N = 2, g1 = g2 = 1.4142):
shunt L1  = R0 / (ωc g1)   = 50 / (1.2566e10 × 1.4142)
          = 2.813 nH
series C2 = 1 / (R0 ωc g2) = 1 / (50 × 1.2566e10 × 1.4142)
          = 1.125 pF
Full π (Butterworth N = 3, g = 1, 2, 1):
L1 = L3 = R0 / ωc = 3.979 nH,  C2 = 1 / (2 R0 ωc) = 0.796 pF

 Lumped:  o----+---| |---+----o
               |   C2    |
              L1        L3  (L3 omitted for N = 2)
               |         |
          o----+---------+----o

 Microstrip (top view):
 50 Ω ======+====| |====+====== 50 Ω
            |   gap or  |
          short  inter-  short
          stub  digital  stub
          (L1)   C2     (L3)
            |           |
          [via]       [via]  -> ground

• Series C2: a gap or interdigital capacitor in the main line (1.125 pF for N = 2, or 0.796 pF for the full π).
• Shunt L1 (and L3): a short-circuited stub ending in a via. By Richards' transformation, a λ/8 shorted stub at fc with Z₀ = ωcL gives the required reactance: Z₀ = 50/1.4142 = 35.4 Ω for N = 2, or Z₀ = 50 Ω for the full π. Alternatively, a shorter high-impedance stub can be used, for example Z₀ = 100 Ω with tan βl = 0.354 (βl ≈ 19.5°) for the 2.813 nH inductor.