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.