End-Coupled Bandpass Filter | Qucs-S S-parameter Viewer Documentation

Skip to content

# End-Coupled Bandpass Filter

## Overview

Bandpass filter using λ/4 or λ/2 resonators coupled through small gaps at their ends. The coupling is capacitive, realized by the fringing fields between adjacent resonator ends.

## Principle

Each resonator is approximately λ/2 long at the center frequency. Adjacent resonators are physically separated by a small gap, creating capacitive coupling. The gap capacitance acts as a series coupling element between resonators.

## Design Equations

### J-Inverter Parameters

The required coupling between resonators is expressed as J-inverters:

First section (external coupling):

\[J₀ = √(0.5 × π × bw / (g₀ × g₁))\]

Last section (external coupling):

\[Jₙ = √(0.5 × π × bw / (gₙ × gₙ₊₁))\]

Internal sections (inter-resonator coupling):

\[Jₖ = (0.5 × π × bw) / √(gₖ × gₖ₊₁)\]

### Susceptance and Capacitance

Each J-inverter is approximated by a series capacitor:

\[B = J / (1 - J²) C = B / ω₀\]

### Resonator Lengths

The resonator electrical length accounts for the phase shift introduced by the coupling capacitors:

\[θₖ = π - 0.5 × (arctan(2Bₖ₋₁) + arctan(2Bₖ)) lₖ = θₖ × λ_g0 / (2π)\]

Where:

Parameter | Description
--- | ---
bw = BW / fc | Fractional bandwidth
ω₀ = 2πfc | Center angular frequency
λ_g0 = c / fc | Wavelength at center frequency
gₖ | Normalized lowpass prototype coefficients
Bₖ | Susceptance of coupling capacitor k

## Input Parameters

Parameter | Range | Default | Description
--- | --- | --- | ---
Center freq (fc) | — | 2 GHz | Bandpass center frequency
Bandwidth (BW) | — | 400 MHz | 3 dB bandwidth
Order (N) | 2 – 10 | 3 | Number of resonators
Z₀ | — | 50 Ω | System impedance

## Topology

    Input ──[C₀]──[TL₁]──[C₁]──[TL₂]──[C₂]──[TLₙ]──[Cₙ]── Output
            gap     λ₁     gap    λ₂     gap    λₙ     gap

## Limitations

- Narrowband (5–10%)

- Gap capacitances may be difficult to control precisely

## References

[1] David M. Pozar (2012). Microwave Engineering, pp. 441–442. Wiley.

## See Also

- RF Circuit Synthesis Tools
- RF Filter Design Tool