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Inelectrical engineering,dielectric loss quantifies adielectric material's inherentdissipation ofelectromagnetic energy (e.g. heat).^[1] It can be parameterized in terms of either theloss angleδ or the correspondingloss tangenttan(δ). Both refer to thephasor in thecomplex plane whose real and imaginary parts are theresistive (lossy) component of an electromagnetic field and itsreactive (lossless) counterpart.

For time-varyingelectromagnetic fields, the electromagnetic energy is typically viewed as wavespropagating either throughfree space, in atransmission line, in amicrostrip line, or through awaveguide. 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 theelectric andmagnetic field components of the propagating waves that satisfy the boundary conditions of the specific environment's geometry.^[2] In such electromagnetic analyses, the parameterspermittivityε,permeabilityμ, andconductivityσ represent the properties of themedia through which the waves propagate. The permittivity can havereal andimaginary components (the latter excludingσ effects, see below) such that

${\displaystyle \epsilon ={\epsilon }^{\prime }-j{\epsilon }^{″}.}$

If we assume that we have a wave function such that

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

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

${\displaystyle \mathrm{\nabla }\times \mathbf{H}=j\omega {\epsilon }^{\prime }\mathbf{E}+(\omega {\epsilon }^{″}+\sigma )\mathbf{E}}$

whereε′′ is the imaginary component of permittivity attributed tobound charge and dipole relaxation phenomena, which gives rise to energy loss that is indistinguishable from the loss due to thefree charge conduction that is quantified byσ. The componentε′represents the familiar lossless permittivity given by the product of thefree space permittivity and therelative real/absolute permittivity, or${\displaystyle {\epsilon }^{\prime }={\epsilon }_0{\epsilon }_r^{\prime }.}$

Theloss tangent is then defined as the ratio (or angle in a complex plane) of the lossy reaction to the electric fieldE in the curl equation to the lossless reaction:

${\displaystyle \tan \delta =\frac{\omega {\epsilon }^{″}+\sigma }{\omega {\epsilon }^{\prime }}.}$

Solution for the electric field of the electromagnetic wave is

${\displaystyle E=E_o{e}^{-jk\sqrt{1-j\tan \delta }z},}$

where:

• ${\displaystyle k=\omega \sqrt{\mu {\epsilon }^{\prime }}=\frac{2\pi }{\lambda },}$
• ω is the angular frequency of the wave, and
• λ is the wavelength in the dielectric material.

For dielectrics with small loss, square root can be approximated using only zeroth and first order terms of binomial expansion. Also,tanδ ≈δ for smallδ.

${\displaystyle E=E_o{e}^{-jk\left(1-j\frac{\tan \delta }{2}\right)z}=E_o{e}^{-kz\frac{\tan \delta }{2}}{e}^{-jkz},}$

Since power is electric field intensity squared, it turns out that the power decays with propagation distancez as

${\displaystyle P=P_o{e}^{-kz\tan \delta },}$

where:

• P_o is the initial power

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

${\displaystyle \mu ={\mu }^{\prime }-j{\mu }^{″},}$

with the subsequent definition of amagnetic loss tangent

${\displaystyle \tan {\delta }_m=\frac{{\mu }^{″}}{{\mu }^{\prime }}.}$

Theelectric loss tangent can be similarly defined:^[3]

${\displaystyle \tan {\delta }_e=\frac{{\epsilon }^{″}}{{\epsilon }^{\prime }},}$

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

Acapacitor is a discrete electrical circuit component typically made of a dielectric placed between conductors. Onelumped element model of a capacitor includes a lossless ideal capacitor in series with a resistor termed theequivalent 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 high leading to a low resistivity), and in a lossy capacitor the ESR can be large. Note that the ESR isnot simply the resistance that would be measured across a capacitor by anohmmeter. 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 thedipole relaxation typically dominates loss in a particular dielectric and manufacturing method. For the case of the conduction electrons being the dominant loss, then

${\displaystyle \mathrm{E}\mathrm{S}\mathrm{R}=\frac{\sigma }{{\epsilon }^{\prime }{\omega }^{2}C}}$

whereC is the lossless capacitance.

When representing the electrical circuit parameters as vectors in acomplex plane, known asphasors, a capacitor'sloss tangent is equal to thetangent 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

${\displaystyle \tan \delta =\frac{\mathrm{E}\mathrm{S}\mathrm{R}}{|X_c|}=\omega C\cdot \mathrm{E}\mathrm{S}\mathrm{R}=\frac{\sigma }{{\epsilon }^{\prime }\omega }}$ .

Since the sameAC current flows through bothESR andX_c, the loss tangent is also the ratio of theresistive power loss in the ESR to thereactive power oscillating in the capacitor. For this reason, a capacitor's loss tangent is sometimes stated as itsdissipation factor, or the reciprocal of itsquality factor Q, as follows

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

• Loss in dielectrics, frequency dependence

• Electromagnetism
• Electrical engineering

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