Skin depth

When an electromagnetic wave interacts with a conductive material, mobile charges within the material are made to oscillate back and forth with the same frequency as the impinging fields. The movement of these charges, usually electrons, constitutes an alternating electric current, the magnitude of which is greatest at the conductor's surface. The decline in current density versus depth is known as the skin effect and the skin depth is a measure of the distance over which the current falls to 1/e of its original value. A gradual change in phase accompanies the change in magnitude, so that, at a given time and at appropriate depths, the current can be flowing in the opposite direction to that at the surface.

The skin depth is a property of the material that varies with the frequency of the applied wave. It can be calculated from the relative permittivity and conductivity of the material and frequency of the wave. First, find the material's complex permittivity, Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon_c

Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon_c={{\varepsilon}\left(1 - j{{\sigma}\over{\omega \varepsilon}}\right)} \qquad \qquad(1)

where:

Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon

= permittivity of the material of propagation

Failed to parse (Missing texvc executable; please see math/README to configure.): \omega

= angular frequency of the wave

Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma

= electrical conductivity of the material of propagation

Failed to parse (Missing texvc executable; please see math/README to configure.): j

= the imaginary unit

Thus, the propagation constant, Failed to parse (Missing texvc executable; please see math/README to configure.): k , will also be a complex number, and can be separated into real and imaginary parts.

Failed to parse (Missing texvc executable; please see math/README to configure.): k_c = {\omega}\sqrt{\mu\varepsilon_c} = \alpha + j\beta

= Failed to parse (Missing texvc executable; please see math/README to configure.): j\omega \sqrt {\mu \varepsilon \left( {1 - \frac{{j\sigma }}{{\omega \varepsilon }}} \right)} \qquad\qquad(2) 

The constants can also be expressed as^[1]

Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha = {\omega}\sqrt{{{\mu\varepsilon}\over2}\left(\sqrt{1 + \left({{\sigma}\over{\omega \epsilon}}\right)^2} + 1\right)}\qquad\qquad (3)

Failed to parse (Missing texvc executable; please see math/README to configure.): \beta = {\omega}\sqrt{{{\mu\varepsilon}\over2}\left(\sqrt{1 + \left({{\sigma}\over{\omega \epsilon}}\right)^2} - 1\right)}\qquad\qquad (4)

where:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mu

= permeability of the material

Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha

= attenuation constant of the propagating wave

The solution of the equation above is if it represent a uniform wave propagating in the +z-direction

Failed to parse (Missing texvc executable; please see math/README to configure.): E_x = E_0 e^{-\alpha z} e^{-j\beta z}\qquad\qquad (5)

The first term in the solution decreases as z increases and is for this reason an attenuation term where Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha

is an attenuation constant with the unit Np/m (Neper). If Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha = 1
then a unit wave amplitude decreases to a magnitude of Failed to parse (Missing texvc executable; please see math/README to configure.): e^{-1}
Np/m. 

It can be seen that the imaginary part of the complex permittivity increases with frequency, implying that the attenuation constant also increases with frequency. Therefore, a high frequency wave will only flow through a very small region of the conductor (much smaller than in the case of a lower frequency current), and will therefore encounter more electrical resistance (due to the decreased surface area).

A good conductor is per definition if Failed to parse (Missing texvc executable; please see math/README to configure.): 1\ll\sigma / \varepsilon \omega

why we can neglect 1 in equation (2) and it turns to

Failed to parse (Missing texvc executable; please see math/README to configure.): k_c = \sqrt j \, \sqrt {\mu \omega \sigma } = \frac{{1 + j}}{{\sqrt 2 }}\sqrt {\mu 2\pi f\sigma } = (1 + j)\sqrt {\pi f\mu \sigma }\qquad\qquad(6)

The skin depth is defined as the distance Failed to parse (Missing texvc executable; please see math/README to configure.): \delta

through which the amplitude of a traveling plane wave  decreases by a factor Failed to parse (Missing texvc executable; please see math/README to configure.): e^{-1}
and is therefore

Failed to parse (Missing texvc executable; please see math/README to configure.): \delta = \frac{1}{\alpha} \qquad\qquad(7)

and for a good conductor is it defined as

Failed to parse (Missing texvc executable; please see math/README to configure.): \delta = \frac{1}{\sqrt {\pi f\mu \sigma }} \qquad\qquad(8)

The term "skin depth" traditionally assumes ω real. This is not necessarily the case; the imaginary part of ω characterizes' the waves attenuation in time. This would make the above definitions for α and β complex, and so they would need to be redefined so that Failed to parse (Missing texvc executable; please see math/README to configure.): Im\{k_c\} = \beta .

The same equations also apply to a lossy dielectric. Defining

Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon_c={\left({\varepsilon'} - j{\varepsilon''}\right)}

replace Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon

with Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon' 

, and Failed to parse (Missing texvc executable; please see math/README to configure.): {\sigma\over{\omega\varepsilon}} with Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon''\over{\varepsilon'}

Examples

The electrical resistivity of a material is equal to 1/σ and its relative permeability is defined as Failed to parse (Missing texvc executable; please see math/README to configure.): \mu/\mu_0 , where Failed to parse (Missing texvc executable; please see math/README to configure.): \mu_0

is the magnetic permeability of free space. It follows that Equation (8) can be rewritten as

Failed to parse (Missing texvc executable; please see math/README to configure.): \delta = \frac{1}{\sqrt{\pi \mu_o}} \,\sqrt{\frac{\rho}{\mu_r f}} \approx 503\,\sqrt{\frac{\rho}{\mu_r f}}\qquad\qquad(9)

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \mu_0 =

4π×10^-7 H/m

Failed to parse (Missing texvc executable; please see math/README to configure.): \mu_r =

the relative permeability of the medium

Failed to parse (Missing texvc executable; please see math/README to configure.): \rho =

the resistivity of the medium in Ωm

Failed to parse (Missing texvc executable; please see math/README to configure.): f =

the frequency of the wave in Hz

If the resistivity of aluminium is taken as 2.8×10^-8 Ωm and its relative permeability is 1, then the skin depth at a frequency of 50 Hz is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \delta = 503 \,\sqrt{\frac{2.82 \cdot 10^{-8}}{1 \cdot 50}}= 11.9

mm

Iron has a higher resistivity, 1.0×10^-7 Ωm, and this will increase the skin depth. However, its relative permeability is typically 90, which will have the opposite effect. At 50 Hz the skin depth in iron is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \delta = 503 \,\sqrt{\frac{1.0 \cdot 10^{-7}}{90 \cdot 50}}= 2.4

mm

Hence, the higher magnetic permeability of iron more than compensates for the lower resistivity of aluminium and the skin depth in iron is therefore 5 times smaller. This will be true whatever the frequency, assuming the material properties are not themselves frequency-dependent.

Skin depth values for some common good conductors at a frequency of 10GHz (microwave region) are indicated below. Values are in micrometers (μm)

As one can see, in microwave frequencies most of the current in a good conductor flows in an extremely thin region near the surface of the latter. These microwave frequencies are the operating region for many modern devices such as Bluetooth, wireless, microwave ovens, and satellite television.

See also

• skin effect

References

• Ramo, Whinnery, Van Duzer (1994). Fields and Waves in Communications Electronics. John Wiley and Sons.