This article presents a concept of the loss tangent of the medium, which is often used to determine if a medium is a good conductor. A good conductor is defined as a medium in which the conduction current density is much greater than the displacement current density, or equivalently, the loss tangent of a medium (σ ⁄ ωε >>1). The loss tangent of the medium is used to obtain an approximate solution for the shielding effectiveness in the far field, which in turn leads to the formulas for the shielding effectiveness in the near field [1].
Uniform Plane Wave Propagation in a Lossy Medium
To define the loss tangent of a medium, we begin with the uniform plane wave propagation in a lossy medium. The solution of the plane wave equations leads to the definition of the intrinsic impedance of the medium, which in turn leads to the concept of the loss tangent of the medium.
Consider a uniform plane wave propagating in a lossy medium, as shown in Figure 1.
The wave equations for sinusoidal variations are given by [2].
(1.1a)
(1.1b)
where
(1.2)
is the propagation constant of the medium, where α is the attenuation constant and β is the phase constant.
The solutions of Eqs. (1.1) consist of forward and backward propagating waves, [2], and are given by
(1.3a)
(1.3b)
where
(1.4)
is the intrinsic impedance of the medium.
Intrinsic Impedance of the Medium
Utilizing Eqs. (1.2) and (1.4) we get [1].
(2.1)
thus,
(2.2)
or
(2.3)
and finally
(2.4)
Note that Eq. (2.4) expresses the intrinsic impedance in terms of σ ⁄ ωε, which is termed loss tangent of the medium (a very important concept in shielding), and is the subject of the next section.
The magnitude of the intrinsic impedance is
(2.5)
while the angle is
(2.6)
It follows
(2.7)
or
(2.8)
showing that the angle of the intrinsic impedance is
(2.9)
Loss Tangent of the Medium
Let’s start with Ampere’s Law (Maxwell’s equation) in a source-free medium, [1,2],
(3.1)
or
(3.2)
The first term on the right-hand side of Eq. (3.2) is the conduction current density
(3.3)
while the second term is in displacement current density
(3.4)
The conduction current represents an energy loss while the displacement current represents energy storage. The ratio of the magnitude of the conduction current density to that of the displacement current density is
(3.5)
or
(3.6)
where tan θ is known as the loss tangent and θ is the loss angle of the medium, as illustrated in Figure 2.
Loss tangent of the medium is a measure of the lossy nature of the material and provides a meaningful way of classifying different media, as shown in Figure 3.
What does much larger (>>) or much smaller (<<) mean? Like many concepts in engineering, these two are not precisely defined. In mathematics, much larger or much smaller usually corresponds to the ratio of at least two orders of magnitude, i.e., the ratio of 100 or more. In engineering, this ratio often equals at least one order of magnitude, or 10.
Note that the loss tangent is a function of frequency and thus at one frequency a medium can be classified as a good conductor and at another frequency as a lossy medium or a good dielectric.
Calculations of the propagation constant and the intrinsic impedance can be simplified if the medium can be classified as a good conductor, as the following derivations show.
Recall, the propagation constant was defined by Eq. (1.2), repeated here
(3.7)
which can be expressed in terms of the lost tangent of the medium as follows
(3.8)
For good conductors we have
(3.9)
And thus, the propagation constant in Eq. (3.8) can be approximated by
(3.10)
The intrinsic impedance is related to the propagation constant by Eq. (2.2) as
(3.11)
Utilizing the result in Eq. (3.10) in Eq. (3.11), we get
(3.12)
In shielding, it is desirable to use good conductors as shields. Loss tangent of the medium is often used to compare different metallic shields at the frequency of interest.
Another way of describing a metallic shield is by comparing its intrinsic impedance, 
, to that of free space, 
. Loss tangent and intrinsic impedance of the medium are related, as shown next.
From Eq. (2.9), we get
(3.13)
Comparing it with Eq. (3.6) reveals that
(3.14)
i.e., the loss angle is twice the angle of the intrinsic impedance.
Finally, let’s show the relationship between the loss tangent and the complex permittivity of the medium, often encountered when discussing dielectric media vs. conducting media.
From Eq. (3.1), we obtain
(3.15)
or
(3.16)
where the complex permittivity of the medium, 
, is
(3.17)
or
(3.18)
with the real part of it equal to
(3.19a)
and the imaginary part expressed by
(3.19b)
We observe that the ratio of ε″ to ε′ is the loss tangent of the medium, that is,
(3.20)
For (non-conducting) dielectric medium the loss tangent is defined as
(3.21)
while for a conductor the loss tangent is
(3.22)
Loss Tangent and Far-Field/Near‑Field Shielding Formulas for Good Conductors
The shielding effectiveness, SE, of a thick (thickness of a shield is much greater than skin depth), conducting shield in the far field was derived in [1, 2], as
(4.1)
where R is the reflection loss and A is the absorption loss, and δ is the skin depth. For good conductors, with the loss tangent
(4.2)
the reflection loss and absorption loss in Eq. (4.1) can be approximated by [3]
(4.3)
(4.4)
where μr is the relative permeability, σr is the relative conductivity (with respect to copper), and t is the shield thickness in meters.
The far-field formula for the reflection loss in Eq. (4.3) is valid for both the electric and magnetic fields.
In the near field, the reflection loss for the electric field is [2,3]
(4.5)
while the reflection loss for the magnetic field is
(4.6)
The absorption loss for both fields is the same as in the far field.
In conclusion, let us emphasize that the far-field and near-field shielding effectiveness formulas were derived under the assumption of a good conductor, i.e., a conductor with the loss tangent satisfying the condition
(4.7)
References
- Bogdan Adamczyk, Principles of Electromagnetic Compatibility – Laboratory Exercises and Lectures, Wiley, 2023.
- Clayton R. Paul, Introduction to Electromagnetic Compatibility, Wiley, 2006.
- Henry W. Ott, Electromagnetic Compatibility Engineering, Wiley, 2009
