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.