Skin Depth in Good Conductors

This tutorial article focuses on the skin depth phenomenon in good conductors. In order to explain this concept, we begin with the uniform plane propagation, leading to the wave equations and their solutions in different media. Subsequently, the skin depth definition is presented and applied to good conductors. Finally, two important EMC applications of the skin depth concept are explained: (1) shielding using metallic conductors, and (2) current density
in conductors.

Uniform Plane Wave Propagation

Let’s begin with the concept of a uniform plane wave. Both descriptors in the name: uniform and plane are very important. The term plane means that the E and H vectors associated with the wave lie in a plane and as the wave propagates the planes defined by these vectors are parallel. The term uniform means that E and H vectors do not depend on the location within each plane i.e., the have the same amplitudes and directions over the entire plane. Such a wave, propagating in the +z direction, is shown in Figure 1.

It is customary to have the E field point in the positive x direction, as shown in Figure 1. That is

  (1)

Since the uniform plane wave is an electromagnetic wave, it must satisfy the Maxwell’s curl equations, which for the source-free media, in time domain, are given by

(2a)

(2b)

Uniformity assumption, combined with Equations (1) and (2a) reveals the fact that the H field is perpendicular to the E field in the plane and is pointing in the +y direction [1], as shown in Figure 1. Thus,

(3)

For arbitrary time variations, in any medium, equations (2a) and (2b) lead to the (uniform plane) wave equations

(4a)

(4b)

In sinusoidal steady-state Equations (4a) and (4b) become

(5a)

(5b)

where

(6)

is the propagation constant. The “hat” notation denotes a complex number () or the phasors ( and ) corresponding to the sinusoidal fields in time domain.

The solution of the wave equations (5a) and (5b) is well known and is given by [2],

 (7a)

 (7b)

where

 (8)

is the complex intrinsic impedance of the medium.

The solutions in Equations (7a) and (7b) consist of the superposition of two waves

 (9a)

 (9b)

where the forward propagating waves (in +z direction) are

 (10a)

 (10b)

and the backward propagating waves (in –z direction) are

 (11a)

 (11b)

Expressing the propagation constant  in terms of its real and imaginary parts

 (12a)

and the complex intrinsic impedance   in terms of its magnitude and angle

 (12b)

(α is the attenuation constant in Np/m and β is the phase constant in rad/m) we can write the solution in Equations (9a) and (b) as

 (13a)

 (13b)

In a lossless medium (ideal dielectric), σ = 0 , resulting in

 (14a)

 (14b)

The phasor solutions in Equations (13a) and (13b) become

 (15a)

 (15b)

Thus, in a perfect dielectric, the amplitudes of the forward
and backward propagating waves are not attenuated.

If the medium is considered a good conductor, then equations (13a) and (13b) apply, and the attenuation constant is given by

 (16)

Concept of a Skin Depth

Let’s consider a forward propagating wave, in a positive z direction, in a lossy medium. This wave is described by

 (17a)

 (17b)

The magnitudes (amplitudes) of the E and H fields associated with this wave are given by

 (18a)

 (18b)

Therefore, as the wave travels in a lossy medium, the amplitude of the E and H fields associated with it are attenuated by a factor e^-az. The distance δ through which the wave amplitude decreases by a factor of  is called skin depth of the medium.

Thus, we have

 (19)

or

 (20)

The expression in Equation (20) is valid in any lossy medium. Usually, the concept of skin depth is applied to good conductors. Using Equation (16) we obtain the well-known formula for skin depth in good conductors

 (21)

Note: The skin depth is inversely proportional to the square root of frequency. The higher the frequency, the lower the skin depth. Equation (21) also reveals that the skin depth of the ideal conductor is zero, since its conductivity is infinite. Practical conductors carry currents up to several skin depths as discussed later in this article.

EMC Applications – Shielding

Consider a metallic shield of thickness t surrounded on both sides by air (free space), as shown in Figure 2. It is assumed that the shield is in the xy plane and extends to infinity in the x and y directions (there are no waves travelling outside the shield in the z direction). It is also assumed that the free space on both side of the shield extends to infinity and is void of other objects.

Free space on both sides of the shield is described by the parameters μ₀, ε₀, σ = 0. The shield is described by its constitutive parameters μ, ε, σ.

Incident on the left surface of this shield is the uniform plane wave, normal to the shield boundary. The incident wave, (), will be partially reflected, (), and partially transmitted, () through the shield. The transmitted wave, (), upon arrival at the rightmost boundary will be partially reflected, (), and partially transmitted, () through the shield [1].

The incident wave is described by

 (22a)

 (22b)

The reflected wave is described by

 (23a)

 (23b)

The wave transmitted through the left interface is described by

 (24a)

 (24b)

The wave reflected at the right interface is described by

 (25a)

 (25b)

Finally, the wave transmitted through the right interface is described by

 (26a)

 (26b)

The purpose of the shield is to minimize (ideally eliminate) the transmitted wave described by Equations (26a) and (26b).

Let’s focus on the E₁ and H₁ fields as the wave gets transmitted through the first interface.

 (27a)

 (27b)

The amplitudes of these fields are

 (28a)

 (28b)

Since the shield is made of a good conductor, these magnitudes can be expressed in terms of the skin depth as

 (29a)

 (29b)

Thus, as the wave travels through a shield of thickness t, the associated field magnitudes become

 (30a)

 (30b)

This is illustrated in Figure 3. Equations (30a) and (30b) reveal that the thicker the shield, the higher the attenuation.

Note: In shielding theory Equations (30a) or (30b) lead to the expression for the absorption loss given by

 (31)

EMC Applications – Current Density in Conductors

Consider a typical circuit model shown in Figure 4.

Both the forward and the return differential-mode currents have the differential-mode EM waves, traveling in free space, associated with them, [3], as shown in Figure 5.

Let’s focus on the forward conductor. As the wave propagates in free space outside the conductor it suffers no attenuation, since free space is an ideal dielectric. The E and H fields associated with this wave penetrate the conductor and are attenuated as shown in Figure 6.

The current density inside the conductor is related to the electric field by

 (32)

Therefore, as the fields penetrate the
surface of the conductor, the current density decays exponentially, as shown in Figure 6.

References

1. Clayton R. Paul, Introduction to Electromagnetic Compatibility, Wiley, 2006.
2. Bogdan Adamczyk, Foundations of Electromagnetic Compatibility with Practical Applications, Wiley, 2017.
3. Bogdan Adamczyk, “Common-Mode Current Creation and Suppression,” In Compliance Magazine, August 2019.