LC Canonical (Ladder) Filters | Qucs-S S-parameter Viewer Documentation

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# 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:

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

Where:

Parameter | Description
--- | ---
Z_min | Minimum realizable impedance (e.g., 10 Ω)
Z_max | Maximum realizable impedance (e.g., 120 Ω)

## References

[1] “Handbook of Filter Synthesis”, Anatol I. Zverev. John Wiley and Sons Inc. 1967

## See Also

- RF Circuit Synthesis Tools
- RF Filter Design Tool