LC Canonical (Ladder) Filters

Overview

Classical lumped-element filter topology using inductors and capacitors arranged in a ladder network. Supports all four filter classes: lowpass, highpass, bandpass, and bandstop.

Principle

Normalized lowpass prototype coefficients are transformed into actual component values based on the desired filter type. The network can begin with either a shunt or series element, yielding first-series or first shunt topologies.

Topology Options

CLC (1st shunt)

Begins with a shunt capacitor (to ground).

Lowpass:

Input ──┬──[L]──┬──[L]──┬── Output
        │       │       │
       [C]     [C]     [C]
        │       │       │
       GND     GND     GND

LCL (1st series)

Begins with a series inductor.

Lowpass:

Input ──[L]──┬──[L]──┬──[L]── Output
             │       │
            [C]     [C]
             │       │
            GND     GND

Design Equations

Lowpass

Element values are scaled from the normalized prototype:

Shunt capacitor:

\[C = gₖ / (2π × fc × Z₀)\]

Series inductor:

\[L = gₖ × Z₀ / (2π × fc)\]

Highpass

Inductors and capacitors swap roles (dual network):

Shunt inductor:

\[L = Z₀ / (2π × fc × gₖ)\]

Series capacitor:

\[C = 1 / (2π × fc × gₖ × Z₀)\]

Bandpass

Each lowpass element is replaced by a series or parallel resonant circuit:

Series inductor → series LC resonator:

\[Lₛ = gₖ × Z₀ / (2π × BW) Cₛ = BW / (gₖ × Z₀ × (2π × fc)²)\]

Shunt capacitor → parallel LC resonator:

\[Cₚ = gₖ / (2π × BW × Z₀) Lₚ = BW × Z₀ / (gₖ × (2π × fc)²)\]

Bandstop

Each lowpass element is replaced by its dual resonator:

Series inductor → parallel LC resonator (series arm):

\[Lₚ = gₖ × Z₀ × BW / ((2π × fc)²) Cₚ = 1 / (gₖ × Z₀ × 2π × BW)\]

Shunt capacitor → series LC resonator (shunt arm):

\[Lₛ = Z₀ / (gₖ × 2π × BW) Cₛ = gₖ × BW / (Z₀ × (2π × fc)²)\]

Where:

Parameter	Description
fc	Cutoff frequency (LP/HP) or center frequency (BP/BS)
BW	Bandwidth (BP/BS)
Z₀	System impedance
gₖ	Normalized lowpass prototype coefficients

Input Parameters

Parameter	Range	Default	Description
Cutoff freq (fc)	—	1 GHz	LP/HP cutoff
Center freq (fc)	—	2 GHz	BP/BS center
Bandwidth (BW)	—	400 MHz	BP/BS bandwidth
Order (N)	2 – 10	3	Number of elements
Topology	CLC / LCL	CLC	Shunt-first or series-first

Semilumped Option

For lowpass and highpass filters, shunt capacitors and/or series inductors can be replaced by transmission line sections to create a hybrid lumped/distributed implementation.

Shunt capacitor replacement:

\[l_C = (λ₀ / 2π) × arcsin(2π × fc × Z_min × C)\]

Series inductor replacement:

\[l_L = (λ₀ / 2π) × arcsin(2π × fc × L / Z_max)\]