Effective permittivity of coplanar transmission line when distance is greater than thickness¶
Under the conditions described below, the effective permittivity of a coplanar line can be calculated from the relative permittivity of the substrate and the physical dimensions of the system.
Conditions:
\(h < \frac{d}{4}\)
\(0 < \left( \frac{l}{d} \right)^2 \le \frac{1}{2}\)
\(0 < \left( \frac{\sinh{ \left((\pi l) / (4 h)\right) }}{\sinh{ \left((\pi d) / (4 h)\right) }} \right)^2 \le \frac{1}{2}\)
See below for symbol descriptions.
- effective_permittivity¶
Effective
relative_permittivityof the coplanar line. See Effective permittivity of coplanar line.
- Symbol:
epsilon_eff- Latex:
\(\varepsilon_\text{eff}\)
- Dimension:
dimensionless
- relative_permittivity¶
relative_permittivityof the dielectric substrate of the coplanar line.
- Symbol:
epsilon_r- Latex:
\(\varepsilon_\text{r}\)
- Dimension:
dimensionless
- electrode_distance¶
euclidean_distancebetween the first and last electrodes.
- Symbol:
d- Latex:
\(d\)
- Dimension:
length
- Symbol:
h- Latex:
\(h\)
- Dimension:
length
- Symbol:
l- Latex:
\(l\)
- Dimension:
length
- law¶
epsilon_eff = 1 + (epsilon_r - 1) / 2 * log(2 * (1 + (1 - (l / d)^2)^(1 / 4)) / (1 - (1 - (l / d)^2)^(1 / 4))) / log(2 * (1 + (1 - (sinh(pi * l / (4 * h)) / sinh(pi * d / (4 * h)))^2)^(1 / 4)) / (1 - (1 - (sinh(pi * l / (4 * h)) / sinh(pi * d / (4 * h)))^2)^(1 / 4)))- Latex:
- \[\varepsilon_\text{eff} = 1 + \frac{\varepsilon_\text{r} - 1}{2} \frac{\log \left( \frac{2 \left(1 + \left(1 - \left(\frac{l}{d}\right)^{2}\right)^{\frac{1}{4}}\right)}{1 - \left(1 - \left(\frac{l}{d}\right)^{2}\right)^{\frac{1}{4}}} \right)}{\log \left( \frac{2 \left(1 + \left(1 - \left(\frac{\sinh{\left(\frac{\pi l}{4 h} \right)}}{\sinh{\left(\frac{\pi d}{4 h} \right)}}\right)^{2}\right)^{\frac{1}{4}}\right)}{1 - \left(1 - \left(\frac{\sinh{\left(\frac{\pi l}{4 h} \right)}}{\sinh{\left(\frac{\pi d}{4 h} \right)}}\right)^{2}\right)^{\frac{1}{4}}} \right)}\]