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:

${\epsilon }_{\text{eff}}$

Dimension:

dimensionless

relative_permittivity

relative_permittivity of the dielectric substrate of the microstrip line.

Symbol:

epsilon_r

Latex:

${\epsilon }_{\text{r}}$

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:

${\epsilon }_{\text{eff},0}$

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:

$${\epsilon }_{\text{eff}}={(\frac{\sqrt{{\epsilon }_{\text{r}}}-\sqrt{{\epsilon }_{\text{eff},0}}}{1+\frac{4}{{\left(4hf{(1+2\log (1+\frac{w}{h}))}^2\sqrt{{\epsilon }_{\text{r}}-1}\frac{1}{2c}\right)}^{\frac{3}{2}}}}+\sqrt{{\epsilon }_{\text{eff},0}})}^2$$