# Dielectric loss

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Dielectric loss quantifies a dielectric material's inherent dissipation of electromagnetic energy (e.g. heat). [1] It can be parameterized in terms of either the loss angleδ or the corresponding loss tangent tan δ. Both refer to the phasor in the complex plane whose real and imaginary parts are the resistive (lossy) component of an electromagnetic field and its reactive (lossless) counterpart.

In physics and engineering, a phasor, 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 more general concept called analytic representation, which decomposes a sinusoid into the product of a complex constant and a factor that encapsulates the frequency and time dependence. The complex constant, which encapsulates amplitude and phase dependence, is known as phasor, complex amplitude, and sinor or even complexor.

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis.

## 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. [2] 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

In radio-frequency engineering, a transmission line is a specialized cable or other structure designed to conduct alternating current of radio frequency, that is, currents with a frequency high enough that their wave nature must be taken into account. Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas, distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses.

Microstrip is a type of electrical transmission line which can be fabricated using printed circuit board technology, and is used to convey microwave-frequency signals. It consists of a conducting strip separated from a ground plane by a dielectric layer known as the substrate. Microwave components such as antennas, couplers, filters, power dividers etc. can be formed from microstrip, with the entire device existing as the pattern of metallization on the substrate. Microstrip is thus much less expensive than traditional waveguide technology, as well as being far lighter and more compact. Microstrip was developed by ITT laboratories as a competitor to stripline.

A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting expansion to one dimension or two. There is a similar effect in water waves constrained within a canal, or guns that have barrels which restrict hot gas expansion to maximize energy transfer to their bullets. Without the physical constraint of a waveguide, wave amplitudes decrease according to the inverse square law as they expand into three dimensional space.

${\displaystyle \varepsilon =\varepsilon '-j\varepsilon ''}$ .

If we assume that we have a wave function such that

${\displaystyle \mathbf {E} =\mathbf {E} _{o}e^{j\omega t}}$,

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

${\displaystyle \nabla \times \mathbf {H} =j\omega \varepsilon '\mathbf {E} +(\omega \varepsilon ''+\sigma )\mathbf {E} }$

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′ = ε0ε′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:

${\displaystyle \tan \delta ={\frac {\omega \varepsilon ''+\sigma }{\omega \varepsilon '}}}$ .

For dielectrics with small loss, this angle is ≪ 1 and tan δ  δ. 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

${\displaystyle P=P_{o}e^{-\delta kz}}$, where:
• Po is the initial power,
• ${\displaystyle k=\omega {\sqrt {\mu \varepsilon '}}={\tfrac {2\pi }{\lambda }}}$,
• 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

${\displaystyle \mu =\mu '-j\mu ''}$ ,

with the subsequent definition of a magnetic loss tangent

${\displaystyle \tan \delta _{m}={\frac {\mu ''}{\mu '}}}$ .

The electric loss tangent can be similarly defined: [3]

${\displaystyle \tan \delta _{e}={\frac {\varepsilon ''}{\varepsilon '}}}$,

upon introduction of an effective dielectric conductivity (see relative permittivity#Lossy medium).

## Discrete circuit perspective

For every discrete electrical circuit components, a capacitor is 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. [4] The ESR represents losses in the capacitor. In a low-loss capacitor the ESR is very small (the conduction is low leading to a high 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

A capacitor is a passive two-terminal electronic component that stores electrical energy in an electric field. The effect of a capacitor is known as capacitance. While some capacitance exists between any two electrical conductors in proximity in a circuit, a capacitor is a component designed to add capacitance to a circuit. The capacitor was originally known as a condenser or condensator. The original name is still widely used in many languages, but not commonly in English.

The lumped element model simplifies the description of the behaviour of spatially distributed physical systems into a topology consisting of discrete entities that approximate the behaviour of the distributed system under certain assumptions. It is useful in electrical systems, mechanical multibody systems, heat transfer, acoustics, etc.

Practical capacitors and inductors as used in electric circuits are not ideal components with only capacitance or inductance. However, they can be treated, to a very good degree of approximation, as being ideal capacitors and inductors in series with a resistance; this resistance is defined as the equivalent series resistance (ESR). If not otherwise specified, the ESR is always an AC resistance measured at specified frequencies, 100 kHz for switched-mode power supply components, 120 Hz for linear power-supply components, and at the self-resonant frequency for general-application components. Audio components may report "Q factor", incorporating ESR among other things, at 1000 Hz.

${\displaystyle \mathrm {ESR} ={\frac {\sigma }{\varepsilon '\omega ^{2}C}}}$

where C is the lossless capacitance.

When representing the electrical circuit parameters as vectors in a complex plane, known as phasors, 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

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, and b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.

${\displaystyle \tan \delta ={\frac {\mathrm {ESR} }{|X_{c}|}}=\omega C\cdot \mathrm {ESR} ={\frac {\sigma }{\varepsilon '\omega }}}$ .

Since the same AC current flows through both ESR and Xc, 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

${\displaystyle \tan \delta =\mathrm {DF} ={\frac {1}{Q}}}$ .

## Related Research Articles

A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material as they do in an electrical conductor but only slightly shift from their average equilibrium positions causing dielectric polarization. Because of dielectric polarization, positive charges are displaced in the direction of the field and negative charges shift in the opposite direction. This creates an internal electric field that reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axes align to the field.

The wave impedance of an electromagnetic wave is the ratio of the transverse components of the electric and magnetic fields. For a transverse-electric-magnetic (TEM) plane wave traveling through a homogeneous medium, the wave impedance is everywhere equal to the intrinsic impedance of the medium. In particular, for a plane wave travelling through empty space, the wave impedance is equal to the impedance of free space. The symbol Z is used to represent it and it is expressed in units of ohms. The symbol η (eta) may be used instead of Z for wave impedance to avoid confusion with electrical impedance.

The relative permittivity of a material is its (absolute) permittivity expressed as a ratio relative to the vacuum permittivity.

In electromagnetism, absolute permittivity, often simply called permittivity, usually denoted by the Greek letter ε (epsilon), is the measure of capacitance that is encountered when forming an electric field in a particular medium. More specifically, permittivity describes the amount of charge needed to generate one unit of electric flux in a particular medium. Accordingly, a charge will yield more electric flux in a medium with low permittivity than in a medium with high permittivity. Permittivity is the measure of a material's ability to store an electric field in the polarization of the medium.

In physics, screening is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases, electrolytes, and charge carriers in electronic conductors . In a fluid, with a given permittivity ε, composed of electrically charged constituent particles, each pair of particles interact through the Coulomb force as

Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of continuum mechanics. The fundamental "linearizing" assumptions of linear elasticity are: infinitesimal strains or "small" deformations and linear relationships between the components of stress and strain. In addition linear elasticity is valid only for stress states that do not produce yielding. These assumptions are reasonable for many engineering materials and engineering design scenarios. Linear elasticity is therefore used extensively in structural analysis and engineering design, often with the aid of finite element analysis.

In electromagnetism, displacement current density is the quantity D/∂t appearing in Maxwell's equations that is defined in terms of the rate of change of D, the electric displacement field. Displacement current density has the same units as electric current density, and it is a source of the magnetic field just as actual current is. However it is not an electric current of moving charges, but a time-varying electric field. In physical materials, there is also a contribution from the slight motion of charges bound in atoms, called dielectric polarization.

In physics, the dissipation factor (DF) is a measure of loss-rate of energy of a mode of oscillation in a dissipative system. It is the reciprocal of quality factor, which represents the "quality" or durability of oscillation.

In electricity (electromagnetism), the electric susceptibility is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applied electric field. The greater the electric susceptibility, the greater the ability of a material to polarize in response to the field, and thereby reduce the total electric field inside the material. It is in this way that the electric susceptibility influences the electric permittivity of the material and thus influences many other phenomena in that medium, from the capacitance of capacitors to the speed of light.

In physics, the electric displacement field, denoted by D, is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in the related concept of displacement current in dielectrics. In free space, the electric displacement field is equivalent to flux density, a concept that lends understanding to Gauss's law. In the International System of Units (SI), it is expressed in units of coulomb per meter squared (C⋅m−2).

The Havriliak–Negami relaxation is an empirical modification of the Debye relaxation model in electromagnetism. Unlike the Debye model, the Havriliak–Negami relaxation accounts for the asymmetry and broadness of the dielectric dispersion curve. The model was first used to describe the dielectric relaxation of some polymers, by adding two exponential parameters to the Debye equation:

In classical electromagnetism, reciprocity refers to a variety of related theorems involving the interchange of time-harmonic electric current densities (sources) and the resulting electromagnetic fields in Maxwell's equations for time-invariant linear media under certain constraints. Reciprocity is closely related to the concept of Hermitian operators from linear algebra, applied to electromagnetism.

The physical constant ε0, commonly called the vacuum permittivity, permittivity of free space or electric constant or the distributed capacitance of the vacuum, is an ideal, (baseline) physical constant, which is the value of the absolute dielectric permittivity of classical vacuum. It has an exactly defined value that can be approximated as

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:

Dynamic modulus is the ratio of stress to strain under vibratory conditions. It is a property of viscoelastic materials.

When an electromagnetic wave travels through a medium in which it gets attenuated, it undergoes exponential decay as described by the Beer–Lambert law. However, there are many possible ways to characterize the wave and how quickly it is attenuated. This article describes the mathematical relationships among:

In physics, the slowly varying envelope approximation is the assumption that the envelope of a forward-travelling wave pulse varies slowly in time and space compared to a period or wavelength. This requires the spectrum of the signal to be narrow-banded—hence it also referred to as the narrow-band approximation.

A microwave cavity or radio frequency (RF) cavity is a special type of resonator, consisting of a closed metal structure that confines electromagnetic fields in the microwave region of the spectrum. The structure is either hollow or filled with dielectric material. The microwaves bounce back and forth between the walls of the cavity. At the cavity's resonant frequencies they reinforce to form standing waves in the cavity. Therefore, the cavity functions similarly to an organ pipe or sound box in a musical instrument, oscillating preferentially at a series of frequencies, its resonant frequencies. Thus it can act as a bandpass filter, allowing microwaves of a particular frequency to pass while blocking microwaves at nearby frequencies.

Surface plasmon polaritons (SPPs) are infrared or visible-frequency electromagnetic waves that travel along a metal–dielectric or metal–air interface. The term "surface plasmon polariton" explains that the wave involves both charge motion in the metal and electromagnetic waves in the air or dielectric ("polariton").

Effective permittivity and permeability are averaged dielectric and magnetic characteristics of a microinhomogeneous medium. They are subject of Effective medium theory. There are two widely used formulae. They both were derived in quasi-static approximation when electric field inside a mixture particle may be considered as homogeneous. So, these formulae can not describe the particle size effect. Many attempts were undertaken to improve these formulae.

## References

1. Ramo, S.; Whinnery, J.R.; Van Duzer, T. (1994). Fields and Waves in Communication Electronics (3rd ed.). New York: John Wiley and Sons. ISBN   0-471-58551-3.
2. Chen, L. F.; Ong, C. K.; Neo, C. P.; Varadan, V. V.; Varadan, Vijay K. Microwave Electronics: Measurement and Materials Characterization. eq. (1.13).
3. "Considerations for a High Performance Capacitor".