Capacitively-Coupled Shunt Resonators Filter

Overview

Bandpass filter topology using λ/4 short-circuit stub resonators coupled through series capacitors. Each stub acts as a shunt resonator, and the coupling capacitors provide the inter-resonator and external coupling.

Principle

The filter consists of N short-circuited λ/4 stubs connected in shunt to the main transmission line. Series capacitors between each resonator and at the input/output ports provide coupling. The capacitor values and stub lengths are derived from the normalized lowpass prototype using admittance inverter transformations.

Design Equations

Admittance Inverters (J)

The coupling between resonators is realized through J-inverters, approximated by series capacitors.

First section (input coupling, k=0):

\[J₀ = √(π × δ / (4 × g₁)) / Z₀ C₀ = J₀ / (ω₀ × √(1 - Z₀² × J₀²))\]

Internal sections (inter-resonator coupling, k=1 to N-1):

\[Jₖ = (π × δ / 4) / (√(gₖ × gₖ₊₁) × Z₀) Cₖ = Jₖ / ω₀\]

Last section (output coupling, k=N):

\[Jₙ = √(π × δ / (4 × gₙ × gₙ₊₁)) / Z₀ Cₙ = Jₙ / (ω₀ × √(1 - Z₀² × Jₙ²))\]

Stub Lengths with Capacitive Loading Compensation

The shunt stubs are nominally λ₀/4 long, but must be adjusted to compensate for the loading effect of the coupling capacitors.

Capacitive loading:

\[ΔCₖ = -Cₖ₋₁ - Cₖ\]

Adjusted stub length:

\[lₖ = λ₀/4 + (Z₀ × ω₀ × ΔCₖ / (2π)) × λ₀\]

If lₖ < 0, add λ₀/4 to bring it into the physical range.

Where:

Parameter	Description
bw = BW / fc	fractional bandwidth
ω₀ = 2πfc	Center angular frequency
λ₀ = c / fc	Free-space wavelength at center frequency
gₖ	Normalized lowpass prototype coefficients
Z₀	System impedance

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

         C₀        C₁        C₂              Cₙ
Input ──[═]──┬──[═]──┬──[═]──┬── ... ──┬──[═]── Output
             │       │       │         │
           stub₁   stub₂   stub₃     stubₙ
            l₁      l₂      l₃        lₙ
             │       │       │         │
            GND     GND     GND       GND

All stubs have characteristic impedance Z₀ = ZS (typically 50 Ω).

Advantages

Compact, planar layout
All stubs have the same impedance.
Series capacitors easy to realize (chip caps or gap caps)

Limitations

Narrowband (typically 5–20% fractional BW)
Requires precise coupling values
Spurious passbands at odd harmonics (3fc, 5fc, …)
Via inductance can affect performance at high frequencies

References

[1] Pozar, D. M. (2012). Microwave Engineering (4th ed.), Section 8.7, pp. 443–448. Wiley.