Kolmogorov continuity theorem - Revision history

2001:638:501:526C:30BB:72F5:CDA7:530A at 13:15, 12 August 2014

2014-08-12T13:15:51Z

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	~~'''Dielectric loss''' quantifies a [[dielectric material]]'s inherent dissipation of electromagnetic energy into, e~~.g., heat.<ref>http://www.ece.rutgers.edu/~orfanidi/ewa/ch01.pdf</ref> 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 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.		Hi there. Allow me start by introducing the author, her title is Sophia. To climb is something I really appreciate doing. Distributing production is how he tends to make a residing. Ohio is where her home is.<br><br>My website ... psychic chat online ([http://m-card.co.kr/xe/mcard_2013_promote01/29877 http://m-card.co.kr/])

	~~==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.<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> In such electromagnetic analyses, the parameters [[permittivity]] ''ε'', [[Permeability (electromagnetism)\|permeability]] ''μ'', and [[Electrical conductivity\|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

	~~:<math> \epsilon = \epsilon' - j \epsilon'' </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 equation for the magnetic field can be written as~~

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

	~~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 permittivity, or ''ε′'' = ''ε<sub>0</sub> ε′<sub>r</sub>''. 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:~~

	~~:<math> \tan \delta = \frac{\omega \epsilon'' + \sigma} {\omega \epsilon'} </math>~~ .

	~~For dielectrics with small loss, this angle~~ is ≪ 1 and tan ''δ'' ≈ ''δ''. After some further maths to obtain the solution for the fields of the electromagnetic wave, it turns out that the power decays with propagation distance ''z'' as
	~~:<math>P = P_o e^{-\delta k z}</math>,~~ where
	~~:<math>P_o</math>~~ is ~~the initial power,~~
	:<~~math~~>~~k = \omega \sqrt{\mu \epsilon'} = \frac {2 \pi} {\lambda}~~<~~/math~~>,
	~~:''ω'' is the angular frequency of the wave, and~~
	~~:''λ'' is the wavelength in the dielectric~~.

	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 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>~~[http://~~books.google~~.~~com~~.br/~~books?id=1vmUdUXlBNIC&lpg=PA8&ots=CWTrAbLoae&dq=loss%20is%20the%20angle%20of%20permittivity%20in%20complex%20plane&pg=PA8#v=onepage&q&f=false], eq. (1.13)<~~/~~ref>~~

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

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

	~~==Discrete circuit perspective==~~

	For 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.<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 \|publisher=Cartage.org.lb \|date= \|accessdate=2011-11-08}}</ref> The ESR represents losses in the capacitor. In a low-loss capacitor the ESR is very small, 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, only one of either the conduction electrons or the [[Dielectric spectroscopy#Dipole relaxation\|dipole relaxation]] typically dominates loss. For the case of the conduction electrons being the dominant loss, then

	~~<math> \mathrm{ESR} = \frac {\sigma} {\epsilon' \omega^2 C} </math>,~~

	~~where <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.]]

	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 diagram to the right. ~~The loss tangent is then~~

	~~<math> \tan \delta = \frac {\mathrm{ESR}} {\|X_{c}\|} = \omega C \cdot \mathrm{ESR} = \frac {\sigma} {\epsilon' \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 = DF = \frac {1} {Q} </math> .~~

	~~==References==~~
	~~{{reflist}}~~

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

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2013-03-16T04:00:25Z

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← Older revision		Revision as of 06:00, 16 March 2013
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	~~Hi there~~, ~~I am Andrew Berryhill~~. ~~To perform lacross~~ is ~~something I truly appreciate doing~~. ~~Her family members life~~ in ~~Alaska but her spouse desires~~ them to ~~transfer~~. ~~Distributing manufacturing has been his occupation~~ for some ~~time~~.<br><br>~~Here~~ is ~~my page~~ :: [~~http~~://~~cpacs~~.~~org~~/~~index~~.~~php?document_srl~~=~~90091&mid~~=~~board_zTGg26 online psychic reading~~]		'''Dielectric loss''' quantifies a [[dielectric material]]'s inherent dissipation of electromagnetic energy into, e.g., heat.<ref>http://www.ece.rutgers.edu/~orfanidi/ewa/ch01.pdf</ref> 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 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 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.<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> In such electromagnetic analyses, the parameters [[permittivity]] ''ε'', [[Permeability (electromagnetism)\|permeability]] ''μ'', and [[Electrical conductivity\|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

			:<math> \epsilon = \epsilon' - j \epsilon'' </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 equation for the magnetic field can be written as

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

			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 permittivity, or ''ε′'' = ''ε<sub>0</sub> ε′<sub>r</sub>''. 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:

			:<math> \tan \delta = \frac{\omega \epsilon'' + \sigma} {\omega \epsilon'} </math> .

			For dielectrics with small loss, this angle is ≪ 1 and tan ''δ'' ≈ ''δ''. After some further maths to obtain the solution for the fields of the electromagnetic wave, it turns out that the power decays with propagation distance ''z'' as
			:<math>P = P_o e^{-\delta k z}</math>, where
			:<math>P_o</math> is the initial power,
			:<math>k = \omega \sqrt{\mu \epsilon'} = \frac {2 \pi} {\lambda}</math>,
			:''ω'' is the angular frequency of the wave, and
			:''λ'' is the wavelength in the dielectric.

			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 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>[http://books.google.com.br/books?id=1vmUdUXlBNIC&lpg=PA8&ots=CWTrAbLoae&dq=loss%20is%20the%20angle%20of%20permittivity%20in%20complex%20plane&pg=PA8#v=onepage&q&f=false], eq. (1.13)</ref>

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

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

			==Discrete circuit perspective==

			For 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.<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 \|publisher=Cartage.org.lb \|date= \|accessdate=2011-11-08}}</ref> The ESR represents losses in the capacitor. In a low-loss capacitor the ESR is very small, 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, only one of either the conduction electrons or the [[Dielectric spectroscopy#Dipole relaxation\|dipole relaxation]] typically dominates loss. For the case of the conduction electrons being the dominant loss, then

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

			where <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.]]

			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 diagram to the right. The loss tangent is then

			<math> \tan \delta = \frac {\mathrm{ESR}} {\|X_{c}\|} = \omega C \cdot \mathrm{ESR} = \frac {\sigma} {\epsilon' \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 = DF = \frac {1} {Q} </math> .

			==References==
			{{reflist}}

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

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2011-10-16T16:00:40Z

added Category:Probability theorems using HotCat

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