Capacitively-Coupled Shunt Resonators Filter | Qucs-S S-parameter Viewer Documentation

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# 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.

## See Also

- RF Circuit Synthesis Tools
- RF Filter Design Tool