Effective permittivity of microstrip line from frequency

The frequency-dependent effective permittivity of the microstrip line can be calculated from its frequency-indendent effective permittivity and physical dimensions.

Notation:

1. \(c\) (c) is speed_of_light.

effective_permittivity

Effective relative_permittivity of the microstrip line when frequency dependence is taken into account. See Effective permittivity of microstrip line.

Symbol:

epsilon_eff

Latex:

\(\varepsilon_\text{eff}\)

Dimension:

dimensionless

relative_permittivity

relative_permittivity of the dielectric substrate of the microstrip line.

Symbol:

epsilon_r

Latex:

\(\varepsilon_\text{r}\)

Dimension:

dimensionless

frequency

temporal_frequency of the signal.

Symbol:

f

Latex:

\(f\)

Dimension:

frequency

substrate_thickness

thickness of the substrate.

Symbol:

h

Latex:

\(h\)

Dimension:

length

width

Width (see length) of the microstrip line.

Symbol:

w

Latex:

\(w\)

Dimension:

length

independent_effective_permittivity

relative_permittivity of the microstrip line when frequency dependence is omitted. See Effective permittivity of microstrip line.

Symbol:

epsilon_eff0

Latex:

\(\varepsilon_{\text{eff}, 0}\)

Dimension:

dimensionless

law

epsilon_eff = ((sqrt(epsilon_r) - sqrt(epsilon_eff0)) / (1 + 4 / (4 * h * f * (1 + 2 * log(1 + w / h))^2 * sqrt(epsilon_r - 1) * 1 / (2 * c))^(3/2)) + sqrt(epsilon_eff0))^2

Latex:

\[\varepsilon_\text{eff} = \left(\frac{\sqrt{\varepsilon_\text{r}} - \sqrt{\varepsilon_{\text{eff}, 0}}}{1 + \frac{4}{\left(4 h f \left(1 + 2 \log \left( 1 + \frac{w}{h} \right)\right)^{2} \sqrt{\varepsilon_\text{r} - 1} \frac{1}{2 c}\right)^{\frac{3}{2}}}} + \sqrt{\varepsilon_{\text{eff}, 0}}\right)^{2}\]