Quarter-Wave Filters | Qucs-S S-parameter Viewer Documentation

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# Quarter-Wave Filters

## Overview

Bandpass and bandstop filters constructed from quarter-wavelength transmission line sections with alternating short-circuit or open-circuit stubs. Simple topology with predictable performance.

## Principle

A cascade of λ/4 transmission lines with shunt stubs realizes a bandpass or bandstop response. The stub impedances are derived from the normalized lowpass prototype coefficients through a lowpass-to-bandpass transformation.

## Design Equations

### Quarter-Wave Transmission Line

\[λ/4 = c / (4 × fc)\]

All connecting lines are exactly λ/4 at the center frequency.

### Stub Impedances

Bandpass filter (short-circuit stubs):

\[Z_stub = (π × Z₀ × bw) / (4 × gₖ)\]

Bandstop filter (open-circuit stubs):

\[Z_stub = (4 × Z₀) / (π × bw × gₖ)\]

Where:

Parameter | Range
--- | ---
fc | Center frequency
bw = BW / fc | Fractional bandwidth
gₖ | Normalized lowpass prototype coefficient
Z₀ | System impedance

## Input Parameters

Parameter | Range | Default | Description
--- | --- | --- | ---
Center freq (fc) | — | 2 GHz | Filter center frequency
Bandwidth (BW) | — | 400 MHz | 3 dB bandwidth
Order (N) | 2 – 10 | 3 | Number of resonators
Filter type | BP / BS | BP | Bandpass or Bandstop

## Topology

### Bandpass

    Input ──[λ/4]──┬──[λ/4]──┬──[λ/4]──┬──[λ/4]── Output
                   │         │         │
                  SC        SC        SC
                 stub₁     stub₂     stub₃
                   │         │         │
                  GND       GND       GND

Each stub is short-circuited (grounded) at its far end.

### Bandstop

    Input ──[λ/4]──┬──[λ/4]──┬──[λ/4]──┬──[λ/4]── Output
                   │         │         │
                  OC        OC        OC
                 stub₁     stub₂     stub₃
                   │         │         │
                 OPEN      OPEN      OPEN

Each stub is open-circuited at its far end.

## Advantages

- Simple, symmetric layout

- Well-established design procedure

## Limitations

- Spurious passbands at odd harmonics (3fc, 5fc, …)

- Requires low stub impedances for wideband BP (difficult in microstrip)

- Large physical size at lower frequencies

## References

[1] David M. Pozar (2012). Microwave Engineering, pp. 437–440. Wiley.

## See Also

- RF Circuit Synthesis Tools
- RF Filter Design Tool