Insertion loss design of given LC ladder filter

Why the insertion loss method is used

The older image-parameter method gives a passband and stop-band but cannot set the exact response, so it needs many trial adjustments. The insertion loss method starts from a chosen power loss ratio:

PLR = 1 / (1 − |Γ(ω)|²) = 1 + M(ω²)/N(ω²)
Butterworth : PLR = 1 + k²(ω/ωc)^(2N)
Chebyshev   : PLR = 1 + k² T_N²(ω/ωc)

so it gives:

• full control of passband flatness or ripple, cut-off sharpness and phase,
• a systematic synthesis from tables (g-values), with a clear trade-off between order N and performance,
• a design that is easy to scale and transform to HP, BP or BS, and to convert to microstrip.

Design steps for the given 5-element ladder

The figure is a fifth-order low-pass ladder that starts with a shunt capacitor: C₁ (shunt), L₂ (series), C₃ (shunt), L₄ (series), C₅ (shunt). No values are given, so this example assumes a Butterworth response, N = 5, fc = 2 GHz and Z_s = Z_L = R₀ = 50 Ω.

Step 1: Specifications. fc = 2 GHz, maximally flat, 50 Ω terminations.

Step 2: Prototype g-values (g_k = 2 sin[(2k − 1)π/2N], N = 5):

Step 3: Impedance and frequency scaling (ωc = 2π × 2×10⁹ = 1.2566×10¹⁰ rad/s):

Ck = gk / (R0 ωc),  Lk = R0 gk / ωc
C1 = C5 = 0.618 / (50 × 1.2566e10) = 0.984 pF
L2 = L4 = 50 × 1.618 / 1.2566e10   = 6.438 nH
C3      = 2.000 / (50 × 1.2566e10) = 3.183 pF

Step 4: Convert to microstrip (stepped impedance). Choose Z_h = 120 Ω (narrow line) and Z_ℓ = 20 Ω (wide line):

inductor : βl = gk·R0 / Zh
capacitor: βl = gk·Zℓ / R0
C1, C5: βl = 0.618 × 20/50  = 14.16°
L2, L4: βl = 1.618 × 50/120 = 38.63°
C3    : βl = 2.000 × 20/50  = 45.84°

Each length is ℓ = (βl/360°)·λg at 2 GHz, where λg depends on the substrate (εr, h).

(Alternative: Richards' transformation turns each C into a λ/8 open stub of Z₀ = R₀/g (C₁: 80.9 Ω, C₃: 25 Ω) and each L into a λ/8 series short stub of Z₀ = R₀·g (L₂: 80.9 Ω). Kuroda identities then convert the series stubs into shunt stubs.)

Step 5: Simulate and tune the layout, adding the effects of step discontinuities, and adjust the lengths.

Microstrip implementation

 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)

Wide low-impedance pads make the shunt capacitors C₁, C₃ and C₅. Narrow high-impedance lines make the series inductors L₂ and L₄. All are printed on one substrate over a ground plane.