Editing Dissipation factor (section)

==Explanation==
[[Electrical potential energy]] is dissipated in all [[dielectric]] materials, usually in the form of [[heat]].  In a [[capacitor]] made of a dielectric placed between conductors, the typical [[lumped element model]] includes a lossless ideal capacitor in series with a resistor termed the [[equivalent series resistance]] (ESR) as shown below.<ref>{{cite web |url=http://www.cartage.org.lb/en/themes/sciences/physics/electromagnetism/electrostatics/Capacitors/Applications/BasicConsiderations/BasicConsiderations.htm |title=Basic Considerations: DF, Q, and ESR |accessdate=2008-11-29 |url-status=dead |archiveurl=https://web.archive.org/web/20090822191220/http://www.cartage.org.lb/en/themes/sciences/physics/electromagnetism/electrostatics/Capacitors/Applications/BasicConsiderations/BasicConsiderations.htm |archivedate=2009-08-22 }}</ref>  The ESR represents losses in the capacitor.  In a good capacitor the ESR is very small, and in a poor capacitor the ESR is large. However, ESR is sometimes a minimum value to be required.  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 with physical origins in both the dielectric's conduction electrons and dipole relaxation phenomena.  In dielectric only one of either the conduction electrons or the dipole relaxation typically dominates loss.<ref>S. Ramo, J.R. Whinnery, and T. Van Duzer, ''Fields and Waves in Communication Electronics'', 3rd ed., (John Wiley and Sons, New York, 1994).  {{ISBN|0-471-58551-3}}</ref>  For the case of the conduction electrons being the dominant loss, then

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

where

* <math> \sigma </math> is the dielectric's bulk [[electrical conductivity|conductivity]],
* <math> \varepsilon </math> is the lossless [[permittivity]] of the dielectric, and
* <math> \omega = 2\pi f</math> is the [[angular frequency]] of the AC current ''i'',
* <math> C </math> 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.]]

If the capacitor is used in an [[alternating current|AC]] circuit, the dissipation factor due to the non-ideal capacitor is expressed as the ratio of the [[Electrical resistance|resistive]] power loss in the ESR to the [[reactance (electronics)|reactive]] power oscillating in the capacitor, or

: <math> \text{DF} = \frac{i^2 \text{ESR}}{i^2 \left|X_c\right|} = \omega C\, \text{ESR} = \frac{\sigma}{\varepsilon\omega} = \frac{1}{Q} </math>

When representing the electrical circuit parameters as vectors in a [[Complex number|complex]] plane, known as [[Phasor (sine waves)|phasors]], a capacitor's dissipation factor 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.  This gives rise to the parameter known as the [[loss tangent]] tan ''δ'' where

: <math> \frac{1}{Q} = \tan(\delta) = \frac{\text{ESR}}{\left|X_c\right|} = \text{DF} </math>

Alternatively, <math>\text{ESR}</math> can be derived from frequency at which loss tangent was determined and capacitance

: <math> \text{ESR} = \frac{1}{\omega C}\tan(\delta) </math>

Since the <math>\text{DF}</math> in a good capacitor is usually small, <math>\delta \sim \text{DF}</math>, and <math>\text{DF}</math> is often expressed as a percentage. {{Citation needed|date=November 2021}}

<math>\text{DF}</math> approximates to the [[power factor]] when <math>\text{ESR}</math> is far less than <math>X_c</math>, which is usually the case.

<math>\text{DF}</math> will vary depending on the dielectric material and the frequency of the electrical signals.  In low [[dielectric constant]] ([[low-κ dielectric|low-κ]]), temperature compensating ceramics, <math>\text{DF}</math> of 0.1–0.2% is typical.  In high dielectric constant ceramics, <math>\text{DF}</math> can be 1–2%.  However, lower <math>\text{DF}</math> is usually an indication of quality capacitors when comparing similar dielectric material.