Calculate Coplanar Waveguide Characteristic Impedance and Effective Permittivity
Results:
Impedance (Z0)
—
Ω (Ohms)
Effective Permittivity (εeff)
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Understanding Coplanar Waveguide (CPW) Design
A Coplanar Waveguide (CPW) is a type of electrical planar transmission line which can be fabricated using printed circuit board (PCB) technology. Unlike microstrip, CPW features the conductor strip and the ground planes on the same side of the dielectric substrate.
Key Parameters
Dielectric Constant (εr): The relative permittivity of the PCB material (e.g., 4.4 for FR4, 3.5 for Rogers).
Trace Width (w): The width of the signal-carrying center conductor.
Gap Width (s): The distance between the center trace and the side ground planes.
Substrate Height (h): The total thickness of the dielectric material.
Design Example
To achieve a characteristic impedance of approximately 50 Ohms on standard FR4 substrate (εr = 4.4, h = 1.6mm), a common configuration might use a trace width (w) of 1.0mm and a gap width (s) of 0.15mm. Reducing the gap width generally lowers the impedance, while increasing the trace width also decreases impedance.
Why use CPW?
CPW is highly favored in high-frequency RF and microwave applications because it provides excellent isolation, low radiation loss, and facilitates the easy mounting of shunt components without the need for via holes to a bottom ground plane.
function calculateCPW() {
var er = parseFloat(document.getElementById('cpw_er').value);
var h = parseFloat(document.getElementById('cpw_h').value);
var w = parseFloat(document.getElementById('cpw_w').value);
var s = parseFloat(document.getElementById('cpw_s').value);
if (isNaN(er) || isNaN(h) || isNaN(w) || isNaN(s) || er <= 1 || h <= 0 || w <= 0 || s <= 0) {
alert("Please enter valid positive numbers. Dielectric constant must be greater than 1.");
return;
}
// Calculation of k0 and k1
var k0 = w / (w + 2 * s);
// k1 factor for finite substrate thickness
var pi = Math.PI;
var arg1 = (pi * w) / (4 * h);
var arg2 = (pi * (w + 2 * s)) / (4 * h);
var k1 = Math.tanh(arg1) / Math.tanh(arg2);
// Elliptic integral ratio function approximation (Hilberg)
function getKRatio(k) {
var kp = Math.sqrt(1 – k * k);
if (k <= 0.7071) {
return pi / Math.log(2 * (1 + Math.sqrt(kp)) / (1 – Math.sqrt(kp)));
} else {
return Math.log(2 * (1 + Math.sqrt(k)) / (1 – Math.sqrt(k))) / pi;
}
}
var ratio0 = getKRatio(k0); // K(k0)/K(k0')
var ratio1 = getKRatio(k1); // K(k1)/K(k1')
// Effective Permittivity
var eeff = 1 + ((er – 1) / 2) * (ratio1 / ratio0);
// Impedance Z0
// Formula: Z0 = (30 * pi / sqrt(eeff)) * (1 / ratio0)
// Note: getKRatio returns K(k)/K(k'), we need K(k')/K(k) in the Z0 formula
var z0 = (30 * pi) / (Math.sqrt(eeff) * ratio0);
// Display results
document.getElementById('cpw_z0').innerText = z0.toFixed(2);
document.getElementById('cpw_eeff').innerText = eeff.toFixed(3);
document.getElementById('cpw_result_container').style.display = 'block';
}