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((short description|Amount of electromagnetic energy dissipated by a dielectric material))
((More citations needed|date=October 2014))

In [[electrical engineering]], '''dielectric loss''' quantifies a [[dielectric material]]'s inherent [[dissipation]] of [[electromagnetic energy]] (e.g. heat).<ref>((Cite web |title=Maxwell’s Equations |url=https://www.ece.rutgers.edu/~orfanidi/ewa/ch01.pdf |access-date=2023-11-06 |website=www.ece.rutgers.edu))</ref>  It can be parameterized in terms of either the '''loss angle''' ((mvar|δ)) or the corresponding '''loss tangent''' ((math|tan(''δ''))).  Both refer to the [[phasor]] in the [[complex plane]] whose real and imaginary parts are the [[Electrical resistance|resistive]] (lossy) component of an electromagnetic field and its [[Reactance (electronics)|reactive]] (lossless) counterpart.

==Electromagnetic field perspective==

For time-varying [[electromagnetic field]]s, the electromagnetic energy is typically viewed as waves [[Wave propagation|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 field|electric]] and [[magnetic field]] components of the propagating waves that satisfy the boundary conditions of the specific environment's geometry.<ref>((cite book|first1=S. |last1=Ramo |first2=J.R. |last2=Whinnery |first3=T. |last3=Van Duzer |title=Fields and Waves in Communication Electronics |edition=3rd |publisher=John Wiley and Sons |location=New York |date=1994 |isbn=0-471-58551-3))</ref>  In such electromagnetic analyses, the parameters [[permittivity]] ((mvar|ε)), [[Permeability (electromagnetism)|permeability]] ((mvar|μ)), and [[Electrical conductivity|conductivity]] ((mvar|σ)) represent the properties of the [[Optical medium|media]] through which the waves propagate.  The permittivity can have [[Real number|real]] and [[Imaginary number|imaginary]] components (the latter excluding ((mvar|σ)) effects, see below) such that

:<math> \varepsilon = \varepsilon' - j \varepsilon'' .</math>

If we assume that we have a wave function such that

:<math> \mathbf E = \mathbf E_{o}e^{j \omega t},</math>

then Maxwell's [[Curl (mathematics)|curl]] equation for the magnetic field can be written as:

:<math> \nabla \times \mathbf H = j \omega \varepsilon' \mathbf E + ( \omega \varepsilon'' + \sigma )\mathbf E </math>

where ((mvar|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 ((mvar|σ)).  The component ((mvar|ε′))represents the familiar lossless permittivity given by the product of the ''free space'' permittivity and the ''relative'' real/absolute permittivity, or <math>\varepsilon' = \varepsilon_0 \varepsilon'_r.</math>

=== <span id="LossTan"></span>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 ((math|'''E''')) in the curl equation to the lossless reaction:
:<math> \tan \delta = \frac{\omega \varepsilon'' + \sigma} {\omega \varepsilon'} .</math>

Solution for the electric field of the electromagnetic wave is 
:<math>E = E_o e^{-j k \sqrt{1 - j \tan \delta} z},</math>
where: 
*<math>k = \omega \sqrt{\mu \varepsilon'} = \tfrac {2 \pi} {\lambda} ,</math>
*((mvar|ω)) is the angular frequency of the wave, and
*((mvar|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, ((math|tan ''δ'' ≈ ''δ'')) for small ((mvar|δ)). 

:<math>E = E_o e^{- j k \left(1 - j \frac{\tan \delta}{2}\right) z} = E_o e^{-k\frac{\tan \delta}{2} z} e^{-j k z},</math>

Since power is electric field intensity squared, it turns out that the power decays with propagation distance ((mvar|z)) as
:<math>P = P_o e^{-k z \tan \delta},</math>
where:
*((mvar|P<sub>o</sub>)) 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

:<math> \mu = \mu' - j \mu'' ,</math>

with the subsequent definition of a '''magnetic loss tangent'''

:<math> \tan \delta_m = \frac{\mu''} {\mu'} .</math>

The '''electric loss tangent''' can be similarly defined:<ref>((cite book|last1=Chen|first1=L. F.|url=https://books.google.com/books?id=1vmUdUXlBNIC&pg=PA8 | title=Microwave Electronics: Measurement and Materials Characterization|last2=Ong|first2=C. K.|last3=Neo|first3=C. P.|last4=Varadan|first4=V. V.|last5=Varadan |first5=Vijay K.|date=19 November 2004|isbn=9780470020456|at=eq. (1.13)|author-link4=Vasundara Varadan))</ref>

:<math> \tan \delta_e = \frac{\varepsilon''} {\varepsilon'} ,</math>

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. One [[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.<ref>((cite web|url=http://www.reliablecapacitors.com/consider.htm |archive-url=https://web.archive.org/web/20081119040541/http://www.reliablecapacitors.com/consider.htm |url-status=dead |archive-date=2008-11-19 |title=Considerations for a High Performance Capacitor))</ref> 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 [[Dielectric spectroscopy#Dipole relaxation|dipole relaxation]] typically dominates loss in a particular dielectric and manufacturing method. For the case of the conduction electrons being the dominant loss, then

:<math> \mathrm{ESR} = \frac {\sigma} {\varepsilon' \omega^2 C} </math>

where ''C'' is the lossless capacitance.

[[Image:Loss tangent phasors 1.svg|frame|A real capacitor has a lumped element model of a lossless ideal capacitor in series with an equivalent series resistance (ESR). The loss tangent is defined by the angle between the capacitor's impedance vector and the negative reactive axis.]]

When representing the electrical circuit parameters as vectors in a [[Complex number|complex]] plane, known as [[Phasor (sine waves)|phasors]], a capacitor's '''loss tangent''' is equal to the [[tangent (trigonometric function)|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

:<math> \tan \delta = \frac {\mathrm{ESR)) {|X_{c}|} = \omega C \cdot \mathrm{ESR} = \frac {\sigma} {\varepsilon' \omega} </math> .

Since the same [[alternating current|AC]] current flows through both ''ESR'' and ''X<sub>c</sub>'', the loss tangent is also the ratio of the [[Electrical resistance|resistive]] power loss in the ESR to the [[Reactance (electronics)|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

:<math> \tan \delta = \mathrm{DF} = \frac {1} {Q} .</math>

==References==
((reflist))

[[Category:Electromagnetism]]
[[Category:Electrical engineering]]

==External links==
*[https://www.doitpoms.ac.uk/tlplib/dielectrics/loss.php Loss in dielectrics], frequency dependence

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