Conductor-backed coplanar waveguide Calculator

Online calculator Conductor-backed Coplanar Waveguide Analysis

Takayuki HOSODA
Dec. 26, 2022

Conductor-backed Coplanar Waveguide Analysis

Frequency MHz

Electrical length deg

Dielectric relative permittivity (ε_r )

Dielectric thickness (h ) μm

Conductor width (w)    μm

Conductor gap (g)    μm

Conductor thickness (t) μm

Impedance (Z₀) ≈ Ω

Effective permittivity (Ε_eff) ≈

Capacitance ≈    pF/m

Inductance ≈    nH/m

Velocity of propagation ≈

Phisical length ≈    mm

zcbcpw - Analyze Conductor-backed Coplanar Waveguide. Rev.0.82 (Feb. 15, 2023) (c) 2022 Takayuki HOSODA

Formulas used ^{[2], [3]}

$Z_0 &=& \frac{\eta_0}{2 \sqrt{\varepsilon_\mathrm{eff}}} \frac{1}{\displaystyle \frac{K(k)}{K(k')} + \frac{K(k_3)}{K(k'_3)}}\\ \varepsilon_\mathrm{eff} &=& \frac{1 + \varepsilon_{r} \displaystyle\frac{K(k')}{K(k)}\frac{K(k_3)}{K(k'_3)}} {1 + \displaystyle\frac{K(k')}{K(k)}\frac{K(k_3)}{K(k'_3)}}$

where

$k &=& \frac{w}{w + 2g}\\ k_3 &=& \frac{\tanh\left(\displaystyle\frac{ \pi w}{4 h}\right)}{\tanh\left(\displaystyle\frac{\pi (w + 2g)}{4 h}\right)}\\ k' &=& \sqrt{1 - k^2} \\ k'_3 &=& \sqrt{1 - k_3^2}$
η₀ : Impedance of free space 376.730313668(57) Ω
and K(k) is the complete elliptic integral of the first kind.

The above analytical solution is for a negligibly thin conductor thickness, but the conductor thickness used in an actual circuit board has a non-negligible effect. The following formulas are used to compensate the effect of conductor thickness to center strip width w and slot widths g.

$\delta &=& t\\ w & \leftarrow & w + \delta \\ s & \leftarrow & s - \delta \\ d & \leftarrow & d - \delta$

APPENDIX 1

Examples of electromagnetic field simulations results (ground plane width W = 12h + 2g + w, upper conductor height H = 8(h + t) Length unit in [ μm ].
w=220, g=100, h=200, t=18, ϵ_r=4.6 : Z₀=53.792 Ω w=300, g=200, h=200, t=18, ϵ_r=4.6 : Z₀=51.438 Ω
Pseudo color visualization of absolute value of the electric field of a cbcpw.

APPENDIX 2

The ratio of the complete elliptic integrals of the first kind K(k) and K(k') can be calculated by the following relation with the ratio of arithmetic-geometric means. The argument k is the elliptic modulus.
$\frac{K(k)}{K(k')} & = & \displaystyle\frac{\mathrm{agm}(1, k)}{\mathrm{agm}(1, k')}\\ \\ k' &=& \sqrt{1 - k^2}$

where agm(1, k) is the arithmetic-geometric mean of 1 and k.

REFERENCE

Victor Fouad Hanna, "Parameters of Couplanar Directional Couplers with Lower Ground Plane", 15h European Microwave Conference, 1985
Rainee N. Simons, "Coplanar Waveguide Circuits, Components, and Systems", A JOHN WILEY & SONS, INC., PUBLICATION, 2001
Brian C. Wadell, "Transmission Line Design Handbook", Artech House, Inc., 1991, ISBN 0-89006-436-9
Complete Elliptic Integral of the First Kind — Wolfram MathWorld
W. Hilberg, "From Approximations to Exact Relations for Characteristic Impedances", IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-17, No. 5 pp. 259-265. May 1969.

Frequency	MHz
Electrical length	deg
*Dielectric relative permittivity (ε_r* )**
Dielectric thickness (h )	μm
Conductor width (w)	μm
Conductor gap (g)	μm
Conductor thickness (t)	μm
Impedance (Z₀) ≈	Ω
Effective permittivity (Ε_eff) ≈
Capacitance ≈	pF/m
Inductance ≈	nH/m
Velocity of propagation ≈
Phisical length ≈	mm

Conductor-backed Coplanar Waveguide Analysis

Formulas used [2], [3]

APPENDIX 1

APPENDIX 2

REFERENCE

SEE ALSO

Formulas used ^{[2], [3]}