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Coupled lines calculator

Two lines that behave like four

Two strips side by side are not two independent lines. Driven in phase they have one characteristic impedance, driven out of phase another, and it is the difference between the two that decides how much power crosses from one to the other. Every coupler, hybrid and coupled-line filter is built on this.

slZ₀eZ₀o
The coupled length is a quarter wave; Z₀ = √(Z₀e·Z₀o), and the difference between them sets the coupling
What to compute
Required coupling (dB)
System impedance Z₀ (Ω)
Frequency (MHz)
Effective permittivity εeff
Result
Even-mode impedance Z₀e
Odd-mode impedance Z₀o
Coupling coefficient k
Coupling
Characteristic impedance Z₀ = √(Z₀e·Z₀o)
Coupled section length (λg/4)

📐 The formulas​

Forward problem — for a wanted coupling:

k = 10^(−C/20)
Z₀e = Z₀ × √( (1 + k) / (1 − k) )
Z₀o = Z₀ × √( (1 − k) / (1 + k) )

Inverse — from known mode impedances:

k = (Z₀e − Z₀o) / (Z₀e + Z₀o)
C[dB] = −20 · log10(k)
Z₀ = √(Z₀e × Z₀o)

Length of the coupled section: a quarter wave in the line, λg/4.

A check point:​

10 dB coupling in a 50 Ω system → k = 0.3162, Z₀e = 69.37 Ω, Z₀o = 36.04 Ω.

Where this calculator stops:
  • The calculation assumes a homogeneous medium, such as stripline. In microstrip the even and odd modes have different εeff, and the coupling comes out weaker than calculated
  • Coupling tighter than 6 dB on a single board layer is barely achievable: it needs a gap of microns
  • The geometry (widths and gap) for given Z₀e and Z₀o cannot be found analytically — that needs a field solver
  • The bandwidth of a coupled section is roughly ±20 % of the centre frequency

🎯 In practice​

  • A directional coupler for an SWR or power meter.
  • A 3 dB hybrid for an equal split with 90° of phase difference.
  • Bandpass filters on coupled resonators.
Practical advice:
  1. For an SWR meter use 20 dB coupling or weaker: then the insertion loss is negligible.
  2. In microstrip the coupling will come out weaker than calculated — allow margin and provide for adjustment.
  3. Several λ/4 sections in cascade widen a coupler's bandwidth several times over.
  4. You can check a finished pair with this same calculator in reverse — measure Z₀e and Z₀o with an analyser.

© 2026 UR3PKI · CyberDev.Space · Content licensed under CC BY-NC-SA 4.0.

How to cite this calculator
UR3PKI. «Coupled lines calculator». CyberDev.Space. https://cyberdev.space/en/radio/calculators/lines/coupled_lines (licence CC BY-NC-SA 4.0).

The licence lets you use this material freely, including in teaching materials, but only with attribution to the author and a link to the source.

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