Loss Tangent - Electromagnetic Field Perspective

Electromagnetic Field Perspective

For time varying electromagnetic fields, the electromagnetic energy is typically viewed as waves propagating either through free space, in a transmission line, in a microstrip line, or through a waveguide. Dielectrics are often used in all of these environments to mechanically support electrical conductors and keep them at a fixed separation, or to provide a barrier between different gas pressures yet still transmit electromagnetic power. Maxwell’s equations are solved for the electric and magnetic field components of the propagating waves that satisfy the boundary conditions of the specific environment's geometry. In such electromagnetic analyses, the parameters permittivity ε, permeability μ, and conductivity σ represent the properties of the media through which the waves propagate. The permittivity can have real and imaginary components such that

.

If we assume that we have a wave function such that

,

then Maxwell's curl equation for the magnetic field can be written as

where ε″ is the imaginary amplitude of permittivity attributed to bound charge and dipole relaxation phenomena, which gives rise to energy loss that is indistinguishable from the loss due to the free charge conduction that is quantified by σ. The component ε′ represents the familiar lossless permittivity given by the product of the free space permittivity and the relative permittivity, or ε′ = ε₀ ε_r. The loss tangent is then defined as the ratio (or angle in a complex plane) of the lossy reaction to the electric field E in the curl equation to the lossless reaction:

.

For dielectrics with small loss, this angle is ≪ 1 and tan δ ≈ δ. After some further maths to obtain the solution for the fields of the electromagnetic wave, it turns out that the power decays with propagation distance z as

, where

is the initial power,

,

ω is the angular frequency of the wave, and

λ is the wavelength in the dielectric.

There are often other contributions to power loss for electromagnetic waves that are not included in this expression, such as due to the wall currents of the conductors of a transmission line or waveguide. Also, a similar analysis could be applied to the permeability where

,

with the subsequent definition of a magnetic loss tangent

.