Wave impedance

The wave impedance is given by

${\displaystyle Z={E_{0}^{-}(x) \over H_{0}^{-}(x)}}$

where ${\displaystyle E_{0}^{-}(x)}$ is the electric field and ${\displaystyle H_{0}^{-}(x)}$ 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

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

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

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

In free space the wave impedance of plane waves is:

${\displaystyle Z_{0}={\sqrt {\frac {\mu _{0}}{\varepsilon _{0}}}}}$

(where ε₀ is the permittivity constant in free space and μ₀ is the permeability constant in free space) and:

${\displaystyle c_{0}={\frac {1}{\sqrt {\mu _{0}\varepsilon _{0}}}}=299,792,458{\text{ m/s}}}$ (by the SI definition of the metre)

hence, because the values of ${\displaystyle c_{0}}$ and ${\displaystyle \mu _{0}}$ are exact, the value of ${\displaystyle Z_{0}}$ in ohms is exactly:

${\displaystyle Z_{0}=\mu _{0}c_{0}=4\pi \times 10^{-7}{\text{ H/m}}\times 299,792,458{\text{ m/s}}=376.730313\ldots ~\Omega \approx 120\pi ~\Omega }$

In an isotropic, homogeneous dielectric with negligible magnetic properties, i.e. ${\displaystyle \mu =\mu _{0}=4\pi \times 10^{-7}}$ H/m and ${\displaystyle \varepsilon =\varepsilon _{r}\times 8.854\times 10^{-12}}$ F/m. So, the value of wave impedance in a perfect dielectric is

${\displaystyle Z={\sqrt {\mu \over \varepsilon }}={\sqrt {\mu _{0} \over \varepsilon _{0}\varepsilon _{r}}}={Z_{0} \over {\sqrt {\varepsilon }}_{r}}\approx {377 \over {\sqrt {\varepsilon _{r}}}}\,\Omega }$,

where ${\displaystyle \varepsilon _{r}}$ is the relative dielectric constant.

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:[1]

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

where f_c is the cut-off frequency of the mode, and for transverse magnetic (TM) modes of propagation the wave impedance is:[1]

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

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.

• Characteristic impedance
• Impedance (disambiguation)
• Impedance of free space

 This article incorporates public domain material from the General Services Administration document: "Federal Standard 1037C". (in support of MIL-STD-188)