Coplanar Waveguide Impedance

Z0 = (30π/√εeff) · K(k′)/K(k)

Quick answer

This calculator solves Z0 = (30π / √εeff) × K(k′)/K(k), k = S/(S+2W), εeff = (εr+1)/2 for Impedance. With Signal Width 10 mm, Gap Width 2.2 cm, Dielectric Constant 0.5, the result is 0.2123 kΩ.

Calculator

Impedance (Ω)
—

Formula

Z0 = (30π / √εeff) × K(k′)/K(k), k = S/(S+2W), εeff = (εr+1)/2

Description

A coplanar waveguide (CPW) carries an RF signal on a centre trace flanked by two coplanar ground planes, with the impedance set by the ratio of the signal width to the gap rather than the dielectric thickness. This makes CPW easy to fabricate and tune, and it is the workhorse transmission line for microwave PCBs and MMICs. The impedance follows from a conformal mapping that uses the ratio of complete elliptic integrals.

Variables

  • Z0 — Characteristic impedance (Ω)
  • S — Centre signal-trace width (m)
  • W — Gap to each ground (m)
  • εr — Substrate dielectric constant
  • εeff — Effective permittivity ≈ (εr+1)/2

Practical Notes

This is the impedance for CPW on a thick substrate without a bottom ground; a grounded CPW (with a ground plane below) reads lower. Widen the gap or narrow the trace to raise Z0.

Symbols and units

Symbol Meaning Default unit Accepted units
Z0 Impedance Ω mΩ, Ω, kΩ, MΩ
S Signal Width mm mm, cm, m, km
W Gap Width mm mm, cm, m, km
er Dielectric Constant — —

Step by step

  1. 1. What you know: Signal Width 10 mm, Gap Width 2.2 cm, Dielectric Constant 0.5
  2. 2. The relation: Z0 = (30π / √εeff) × K(k′)/K(k), k = S/(S+2W), εeff = (εr+1)/2
  3. 3. Substitute: S = 10 mm, W = 2.2 cm, er = 0.5
  4. 4. Result: Impedance = 0.2123 kΩ

Nearest standard component values

Series Nearest value Error
E12 0.22 kΩ 3.6%
E24 0.22 kΩ 3.6%

Worked examples

Minimum range

Given: Signal Width 0.1 mm, Gap Width 0.22 mm, Dielectric Constant 0.025

Result: Impedance = 0.2568 kΩ

Low range

Given: Signal Width 1 mm, Gap Width 2.2 mm, Dielectric Constant 0.25

Result: Impedance = 0.2325 kΩ

Typical

Given: Signal Width 10 mm, Gap Width 2.2 cm, Dielectric Constant 0.5

Result: Impedance = 0.2123 kΩ

High range

Given: Signal Width 10 cm, Gap Width 22 cm, Dielectric Constant 0.8

Result: Impedance = 0.1938 kΩ

Maximum range

Given: Signal Width 1 m, Gap Width 2.2 m, Dielectric Constant 8

Result: Impedance = 86.66 Ω

How Impedance changes with Signal Width

Signal Width (input) Impedance (result)
1 mm 0.3598 kΩ
2.5 mm 0.2985 kΩ
5 mm 0.254 kΩ
7.5 mm 0.2292 kΩ
10 mm 0.2123 kΩ
15 mm 0.1898 kΩ
2 cm 0.1749 kΩ
4 cm 143.1 Ω
7 cm 122 Ω
10 cm 110.6 Ω

The same result in other units

Unit Value
mΩ 212300
Ω 212.3
kΩ 0.2123
MΩ 0.0002123

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Frequently asked questions

How do you calculate Coplanar Waveguide Impedance?

Use Z0 = (30π / √εeff) × K(k′)/K(k), k = S/(S+2W), εeff = (εr+1)/2. With Signal Width 10 mm, Gap Width 2.2 cm, Dielectric Constant 0.5, Impedance works out to 0.2123 kΩ.

What is the formula for Coplanar Waveguide Impedance?

The formula is Z0 = (30π / √εeff) × K(k′)/K(k), k = S/(S+2W), εeff = (εr+1)/2. A coplanar waveguide (CPW) carries an RF signal on a centre trace flanked by two coplanar ground planes, with the impedance set by the ratio of the signal width to the gap rather than the dielectric thickness.

What units does Coplanar Waveguide Impedance use?

Impedance is shown in mΩ, Ω, kΩ, MΩ. Values are held in SI base units internally, so you can mix prefixes without converting by hand.

Related Concepts

All PCB & Traces formulas →
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