Wave impedance

In free space

In free space the wave impedance of plane waves is:

${\displaystyle Z_0=\sqrt{\frac{{\mu }_0}{{\epsilon }_0}}}$

(where ε₀ is the permittivity constant in free space and μ₀ is the permeability constant in free space). Now, since

${\displaystyle c_0=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}=299792458\text{ m/s}}$ (by definition of the metre),

${\displaystyle Z_0={\mu }_0{c}_0}$.

Hence the value essentially depends on ${\displaystyle {\mu }_0}$.

The currently accepted value of ${\displaystyle Z_0}$ is 376.730313412(59) Ω.

In an unbounded dielectric

In an isotropic, homogeneous dielectric with negligible magnetic properties, i.e. ${\displaystyle \mu ={\mu }_0}$ and ${\displaystyle \epsilon ={\epsilon }_r{\epsilon }_0}$. So, the value of wave impedance in a perfect dielectric is

${\displaystyle Z=\sqrt{\frac{\mu }{\epsilon }}=\sqrt{\frac{{\mu }_0}{{\epsilon }_0{\epsilon }_r}}=\frac{Z_0}{\sqrt{{\epsilon }_r}}\approx \frac{377}{\sqrt{{\epsilon }_r}}\mathrm{\Omega }}$,

where ${\displaystyle {\epsilon }_r}$ is the relative dielectric constant.

In a waveguide

For any waveguide in the form of a hollow metal tube, (such as rectangular guide, circular guide, or double-ridge guide), the wave impedance of a travelling wave is dependent on the frequency ${\displaystyle f}$, but is the same throughout the guide. For transverse electric (TE) modes of propagation the wave impedance is:

${\displaystyle Z=\frac{Z_0}{\sqrt{1-{\left(\frac{f_c}{f}\right)}^2}}\text{(TE modes)},}$

where f_c is the cut-off frequency of the mode, and for transverse magnetic (TM) modes of propagation the wave impedance is:

${\displaystyle Z=Z_0\sqrt{1-{\left(\frac{f_c}{f}\right)}^2}\text{(TM modes)}}$

Above the cut-off (f > f_c), the impedance is real (resistive) and the wave carries energy. Below cut-off the impedance is imaginary (reactive) and the wave is evanescent. These expressions neglect the effect of resistive loss in the walls of the waveguide. For a waveguide entirely filled with a homogeneous dielectric medium, similar expressions apply, but with the wave impedance of the medium replacing Z₀. The presence of the dielectric also modifies the cut-off frequency f_c.

For a waveguide or transmission line containing more than one type of dielectric medium (such as microstrip), the wave impedance will in general vary over the cross-section of the line.