Side-Coupled (Parallel-Coupled) Bandpass Filter | Qucs-S S-parameter Viewer Documentation

Skip to content

# Side-Coupled (Parallel-Coupled) Bandpass Filter

## Overview

Bandpass filter using λ/4 coupled transmission line sections placed side-by-side. The electromagnetic coupling between parallel resonators realizes the filter response. Also known as parallel-coupled line filter or edge-coupled filter.

## Principle

The filter consists of N λ/4 coupled-line sections arranged in cascade. Each section acts as a resonator at the center frequency, and the coupling between adjacent sections is controlled by their physical spacing (gap). The coupling is primarily magnetic at the design frequency.

## Design Equations

### Admittance Inverters (J)

The required coupling between resonators is first calculated as J-inverters:

First section (input coupling, k=0):

\[J₀ = √(π × δ / (2 × g₁)) / Z₀\]

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

\[Jₖ = (π × δ / 2) / (√(gₖ × gₖ₊₁) × Z₀)\]

Last section (output coupling, k=N):

\[Jₙ = √(π × δ / (2 × gₙ × gₙ₊₁)) / Z₀\]

### Even and Odd Mode Impedances

Each coupled-line section is characterized by its even-mode (Z₀e) and odd-mode (Z₀o) impedances, which are derived from the J-inverter parameter:

\[Z₀e = Z₀ × (1 + J × Z₀ + (J × Z₀)²) Z₀o = Z₀ × (1 - J × Z₀ + (J × Z₀)²)\]

These impedances determine the required coupling strength. Tighter coupling (larger J) produces larger Z₀e and smaller Z₀o.

### Physical Length

All coupled-line sections are exactly λ/4 at the center frequency:

\[l = λ₀ / 4 = c / (4 × fc)\]

Where:

Parameter | Description
--- | ---
bw= BW / fc | Fractional bandwidth
fc | Center frequency
Z₀ | System impedance
gₖ | Normalized lowpass prototype coefficients
c | Speed of light

## 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 ──╪═══╪──╪═══╪──╪═══╪──╪═══╪── Output
            ║   ║  ║   ║  ║   ║  ║   ║
            ╪═══╪  ╪═══╪  ╪═══╪  ╪═══╪
           Section Section Section Section
              1      2      3      N

Each `╪═══╪` represents a λ/4 coupled-line section with two parallel traces.

## Advantages

- Fully planar, single-layer layout

- No vias or chip components required

## Limitations

- Fractional bandwidth limited by fabrication (typically 5–30%)

- Spurious passband at 3fc due to λ/4 resonance

- Large at lower frequencies

## References

[1] David M. Pozar (2012). Microwave Engineering, pp. 430–437. Wiley.

## See Also

- RF Circuit Synthesis Tools
- RF Filter Design Tool