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# Wave impedance

The wave impedance of an electromagnetic wave, is the ratio of the transverse components of the electric and magnetic fields (the transverse components being those at right-angles to the direction of propagation). For a transverse-electric-magnetic (TEM) plane wave travelling through a homogeneous medium, the wave impedance is everywhere equal to the intrinsic impedance of the medium. In particular, for a plane wave travelling through empty space, the wave impedance is equal to the impedance of free space. The symbol Z is used to represent it and it is expressed in units of ohms. The symbol η (eta) may be used instead of Z for wave impedance to avoid confusion with electrical impedance, although η is also the symbol for electromagnetic impedance, the light wave equivalent of wave impedance.

The wave impedance is given by

$Z = {Eo^-(x) \over Ho^- (x)}.$

In terms of the parameters of an electromagnetic wave and the medium it travels through, the wave impedance is given by

$Z = \sqrt {j \omega \mu \over \sigma + j \omega \varepsilon},$

where μ is the magnetic permeability, ε is the electric permittivity and σ is the conductivity of the material the wave is travelling through. In the equation, j is the imaginary unit, and ω is the angular frequency of the wave. In the case of a dielectric (where conductivity is zero), the equation reduces to

$Z = \sqrt {\mu \over \varepsilon }.$

## Wave impedance of free space

In free space, $\scriptstyle \mu=4\pi \times 10^{-7}$ H/m and $\scriptstyle \varepsilon = {1 \over 36\pi \times 10^{9}}$ F/m. So, the value of wave impedance in free space is

Z0 = 377Ω or 120π.

In a perfect dielectric, the wave impedance can be found by dividing Z0 into the refractive index. In anything else, the formula becomes larger and a complex number is the result.

## Wave impedance in an unbounded dielectric

In a perfect dielectric, $\scriptstyle \mu=4\pi \times 10^{-7}$ H/m and $\scriptstyle \varepsilon = {\varepsilon_r \over 36\pi \times 10^{9}}$ F/m. So, the value of wave impedance in a perfect dielectric is

$Z_0 = {377 \over \sqrt \varepsilon }\Omega.$

In a perfect dielectric, the wave impedance can be found by dividing Z0 into the refractive index. In anything else, the formula becomes larger and a complex number is the result.

## Wave impedance 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 f, but is the same throughout the guide. For transverse electric (TE) modes of propagation the wave impedance is

$Z = \frac{Z_{0}}{\sqrt{1 - \left( \frac{f_{c}}{f}\right)^{2}}} \qquad \mbox{(TE modes)},$

where fc is the cut-off frequency of the mode and for (TM) modes

$Z = Z_{0} \sqrt{1 - \left( \frac{f_{c}}{f}\right)^{2}} \qquad \mbox{(TM modes)}$

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.

## References

This article contains material from the Federal Standard 1037C (in support of MIL-STD-188), which, as a work of the United States Government, is in the public domain.