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.