Wave impedance

Definition

The wave impedance is given by : Z = {E_0^-(x) \over H_0^-(x)} where E_0^-(x) is the electric field and H_0^-(x) is the magnetic field, in phasor representation. The impedance is, in general, a complex number.

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 (real) electric permittivity and σ is the electrical conductivity of the material the wave is travelling through (corresponding to the imaginary component of the permittivity multiplied by omega). In the equation, j is the imaginary unit, and ω is the angular frequency of the wave. Just as for electrical impedance, the impedance is a function of frequency. In the case of an ideal dielectric (where the conductivity is zero), the equation reduces to the real number : Z = \sqrt {\mu \over \varepsilon }.

In free space

Main article: Impedance of free space

In free space the wave impedance of plane waves is: : Z_0 = \sqrt{\frac{\mu_0} {\varepsilon_0}} (where ε0 is the permittivity constant in free space and μ0 is the permeability constant in free space). Now, since : c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} = 299,792,458\text{ m/s} (by definition of the metre), : Z_0 = \mu_0 c = \frac{1}{\varepsilon_0 c}.

The currently accepted value of Z_0 is

In an unbounded dielectric

In an isotropic, homogeneous dielectric with negligible magnetic properties, i.e. \mu = \mu_0 and \varepsilon = \varepsilon_r \varepsilon_0. So, the value of wave impedance in a perfect dielectric is : Z = \sqrt {\mu \over \varepsilon} = \sqrt {\mu_0 \over \varepsilon_0 \varepsilon_r} = {Z_0 \over \sqrt{\varepsilon_r}} \approx {377 \over \sqrt {\varepsilon_r} },\Omega, where \varepsilon_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 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 f**c is the cut-off frequency of the mode, and for transverse magnetic (TM) modes of propagation the wave impedance is: : Z = Z_{0} \sqrt{1 - \left( \frac{f_{c}}{f}\right)^{2}} \qquad \mbox{(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.

References

References

1. Pozar, David M.. (2012). "Microwave engineering". Wiley.