Dielectric loss quantifies a
dielectric material's inherent dissipation of electromagnetic energy (e.g. heat). It can be parameterized in terms of either the loss angle ''δ'' or the corresponding loss tangent tan ''δ''. Both refer to the
phasor
In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and initial phase (''θ'') are time-invariant. It is related to ...
in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
whose real and imaginary parts are the
resistive (lossy) component of an electromagnetic field and its
reactive (lossless) counterpart.
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
Microstrip is a type of electrical transmission line which can be fabricated with any technology where a conductor is separated from a ground plane by a dielectric layer known as the substrate. Microstrip lines are used to convey microwave-frequ ...
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
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
Th ...
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 (the latter excluding ''σ'' effects, see below) 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 component 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'' real/absolute permittivity, or ''ε′'' = ''ε''
0''ε′''
r.
Loss tangent
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'' ≈ ''0''. After some further calculations to obtain the solution for the fields of the electromagnetic wave, it turns out that the power decays with propagation distance ''z'' as
:
where:
*''P
o'' is the initial power,
*
,
*''ω'' is the angular frequency of the wave, and
*''λ'' is the wavelength in the dielectric material.
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 magnetic permeability where
:
with the subsequent definition of a magnetic loss tangent
:
The electric loss tangent can be similarly defined:
:
upon introduction of an effective dielectric conductivity (see
relative permittivity#Lossy medium).
Discrete circuit perspective
A
capacitor is a discrete electrical circuit component typically made of a dielectric placed between conductors. The
lumped element model of a capacitor includes a lossless ideal capacitor in series with a resistor termed the
equivalent series resistance (ESR), as shown in the figure below.
The ESR represents losses in the capacitor. In a low-loss capacitor the ESR is very small (the conduction is high leading to a low resistivity), and in a lossy capacitor the ESR can be large. Note that the ESR is ''not'' simply the resistance that would be measured across a capacitor by an
ohmmeter. The ESR is a derived quantity representing the loss due to both the dielectric's conduction electrons and the bound dipole relaxation phenomena mentioned above. In a dielectric, one of the conduction electrons or the
dipole relaxation typically dominates loss in a particular dielectric and manufacturing method. For the case of the conduction electrons being the dominant loss, then
:
where ''C'' is the lossless capacitance.
When representing the electrical circuit parameters as vectors in a
complex plane, known as
phasors
In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and initial phase (''θ'') are time-invariant. It is related to a ...
, a capacitor's loss tangent is equal to the
tangent of the angle between the capacitor's impedance vector and the negative reactive axis, as shown in the adjacent diagram. The loss tangent is then
:
.
Since the same
AC current flows through both ''ESR'' and ''X
c'', the loss tangent is also the ratio of the
resistive power loss in the ESR to the
reactive power oscillating in the capacitor. For this reason, a capacitor's loss tangent is sometimes stated as its ''
dissipation factor'', or the reciprocal of its ''
quality factor Q'', as follows
:
References
{{reflist
Electromagnetism
Electrical engineering
External links
Loss in dielectrics frequency dependence