Skin depth

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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, $\varepsilon_c$

$\varepsilon_c={{\varepsilon}\left(1 - j{{\sigma}\over{\omega \varepsilon}}\right)} \qquad \qquad(1)$

where:

$\varepsilon$ = permittivity of the material of propagation

ω

= angular frequency of the wave

σ

= electrical conductivity of the material of propagation

Thus, the propagation constant, $k$ , will also be a complex number, and can be separated into real and imaginary parts.

$k_c = {\omega}\sqrt{\mu\varepsilon_c} = \alpha + j\beta$ = $j\omega \sqrt {\mu \varepsilon \left( {1 - \frac{{j\sigma }}{{\omega \varepsilon }}} \right)} \qquad\qquad(2)$

The constants can also be expressed as

$\alpha = {\omega}\sqrt{{{\mu\varepsilon}\over2}\left(\sqrt{1 + \left({{\sigma}\over{\omega \epsilon}}\right)^2} - 1\right)}\qquad\qquad (3)$

$\beta = {\omega}\sqrt{{{\mu\varepsilon}\over2}\left(\sqrt{1 + \left({{\sigma}\over{\omega \epsilon}}\right)^2} + 1\right)}\qquad\qquad (4)$

where:

μ

= permeability of the material

α

= attenuation constant of the propagating wave

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

$E_x = E_0 e^{-\alpha z} e^{-j\beta z}\qquad\qquad (5)$

The first term in the solution decreases as z increases and is for this reason an attenuation term where $α$ is an attenuation constant with the unit Np/m (Neper). If $α = 1$ then a unit wave amplitude decreases to a magnitude of $e - 1$ 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 $1\ll\sigma / \varepsilon \omega$ why we can neglect 1 in equation (2) and it turns to

$k_c = \sqrt j \, \sqrt {\mu \omega \sigma } = \frac{{1 + j}}{{\sqrt 2 }}\sqrt {\mu 2\pi f\sigma } = (1 + j)\sqrt {\pi f\mu \sigma }\qquad\qquad(6)$

The skin depth is defined as the distance $δ$ through which the amplitude of a traveling plane wave decreases by a factor $e - 1$ and is therefore

$\delta = \frac{1}{\alpha} \qquad\qquad(7)$

and for a good conductor is it defined as

$\delta = \frac{1}{\sqrt {\pi f\mu \sigma }} \qquad\qquad(8)$

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 $I m {k c} = β$ .

The same equations also apply to a lossy dielectric. Defining

$\varepsilon_c={\left({\varepsilon'} - j{\varepsilon''}\right)}$

replace $\varepsilon$ with $\varepsilon'$ , and ${\sigma\over{\omega\varepsilon}}$ with $\varepsilon''\over{\varepsilon'}$

[edit] Examples

Skin depths for some metals

The electrical resistivity of a material is equal to 1/σ and its relative permeability is defined as $μ / μ 0$ , where $μ 0$ is the magnetic permeability of free space. It follows that Equation (8) can be rewritten as

$\delta = \frac{1}{\sqrt{\pi \mu_o}} \,\sqrt{\frac{\rho}{\mu_r f}} \approx 503\,\sqrt{\frac{\rho}{\mu_r f}}\qquad\qquad(9)$

where

μ 0 =

4π×10^-7 H/m

μ r =

the relative permeability of the medium

ρ =

the resistivity of the medium in Ωm

f =

the frequency of the wave in Hz

If the resistivity of aluminium is taken as 2.82×10^-8 Ωm and its relative permeability is 1, then the skin depth at a frequency of 50 Hz is given by

$\delta = 503 \,\sqrt{\frac{2.82 \cdot 10^{-8}}{1 \cdot 50}}= 11.9$ 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

$\delta = 503 \,\sqrt{\frac{1.0 \cdot 10^{-7}}{90 \cdot 50}}= 2.4$ 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.