Skin depth

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Skin depth is a term used for the depth at which the amplitude of an electromagnetic wave attenuates to 1/e of its original value. It also has applications in numerous other areas, such as seismic exploration.

The skin depth can be calculated from the relative permittivity and conductivity of the material and frequency of the wave. First, find the material's complex permittivity, $\varepsilon_c$

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

where:

$\varepsilon$ = permittivity of the material of propagation

ω

= angular frequency of the wave

σ

= 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

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

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

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 conductivity, implying that the attenuation constant also increases with in conductive materials. 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 $1<<\sigma / \varepsilon \omega$ 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

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

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] Applications

Sometimes is equation (8) 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

$\mu_0 = 4\pi \cdot 10^{-7}$

μ r =

the relative permeability of the medium

ρ =

the resistivity of the medium

f =

the frequency

For aluminium, the resistivity is $2.82\cdot 10^{-8}$ and the relative permeability is 1. Assuming the frequency is of the order of 50 Hz, we insert these values into equation (9) and get:

$\delta = 503 \sqrt{\frac{2.82 \cdot 10^{-8}}{1 \cdot 50}}= 0.0119 m$

For iron, however, the resistivity, $ρ$ , is $1.0 \cdot 10^{-7}$ . While the relative permeability is, in general, a function of temperature and magnetic field, it can be estimated to be 90.