Editing Magnetic core (section)

== Core loss ==
When the core is subjected to a ''changing'' magnetic field, as it is in devices that use AC current such as [[transformer]]s, [[inductor]]s, and [[AC motor]]s and [[alternator]]s, some of the power that would ideally be transferred through the device is lost in the core, dissipated as [[heat]] and sometimes [[noise]].  Core loss is commonly termed ''iron loss'' in contradistinction to [[copper loss]], the loss in the windings.<ref>{{cite book|last1=Thyagarajan|first1=T.|last2=Sendur Chelvi|first2=K.P.|last3=Rangaswamy|first3=T.R.|title=Engineering Basics: Electrical, Electronics and Computer Engineering|date=2007|publisher=New Age International|isbn=9788122412741|edition=3rd|pages=184–185|url=https://books.google.com/books?id=oY-pcBq0z48C&pg=PA185}}</ref><ref>{{cite book|last1=Whitfield|first1=John Frederic|title=Electrical Craft Principles|volume=2|date=1995|publisher=IET|isbn=9780852968338|edition=4th|page=195|url=https://books.google.com/books?id=b31Z2x1ZlUEC&pg=PA195}}</ref>  Iron losses are often described as being in three categories:

===Hysteresis losses===
{{main|Magnetic hysteresis}}
When the magnetic field through the core changes, the [[magnetization]] of the core material changes by expansion and contraction of the tiny [[magnetic domain]]s it is composed of, due to movement of the [[Domain wall (magnetism)|domain wall]]s.  This process causes losses, because the domain walls get "snagged" on defects in the crystal structure and then "snap" past them, dissipating energy as heat.  This is called [[hysteresis loss]].  It can be seen in the graph of the ''B'' field versus the ''H'' field for the material, which has the form of a closed loop. 
The net energy that flows into the inductor expressed in relationship to the B-H characteristic of the core is shown by the equation<ref name="Kluwer Academic Publishers">{{cite book|last1=Erickson|first1=Robert|last2=Maksimović|first2=Dragan|title=Fundamentals of Power Electronics, Second Edition|date=2001|publisher=Kluwer Academic Publishers|isbn=9780792372707|page=506}}</ref>
:<math>W=\int{\left(nA_c\frac{dB(t)}{t}\right)\left(\frac{H(t)l_m}{n}\right)dt}=(A_cl_m)\int{HdB}</math>
This equation shows that the amount of energy lost in the material in one cycle of the applied field is proportional to the area inside the [[hysteresis loop]].  Since the energy lost in each cycle is constant, hysteresis power losses increase proportionally with [[frequency]].<ref>{{cite book|last1=Dhogal|first1=P.S.|title=Basic Electrical Engineering, Volume 1|date=1986|publisher=Tata McGraw-Hill Education|isbn=9780074515860|page=128}}</ref> The final equation for the hysteresis power loss is<ref name="Kluwer Academic Publishers"/>
:<math>P_H=(f)(A_cl_m)\int{HdB}</math>

===Eddy-current losses===
If the core is electrically [[electrical conductivity|conductive]], the changing magnetic field induces circulating loops of current in it, called [[eddy current]]s, due to [[electromagnetic induction]].<ref>{{cite book|last1=Kazimierczuk|first1=Marian K.|title=High-frequency magnetic components|date=2014|publisher=Wiley|location=Chichester|isbn=978-1-118-71779-0|page=113|edition=Second}}</ref>  The loops flow perpendicular to the magnetic field axis.  The energy of the currents is dissipated as heat in the resistance of the core material.  The power loss is proportional to the area of the loops and inversely proportional to the resistivity of the core material. Eddy current losses can be reduced by making the core out of thin [[lamination]]s which have an insulating coating, or alternatively, making the core of a magnetic material with high electrical resistance, like [[ferrite (magnetic core)|ferrite]].<ref>{{cite book|last1=Erickson|first1=Robert|last2=Maksimović|first2=Dragan|title=Fundamentals of Power Electronics, Second Edition|date=2001|publisher=Kluwer Academic Publishers|isbn=9780792372707|page=507}}</ref> Most magnetic cores intended for power converter application use ferrite cores for this reason.

===Anomalous losses===
By definition, this category includes any losses in addition to eddy-current and hysteresis losses.  This can also be described as broadening of the hysteresis loop with frequency.  Physical mechanisms for anomalous loss include localized eddy-current effects near moving domain walls.

=== Legg's equation ===
An equation known as Legg's equation models the [[magnetism|magnetic material]] core loss at low [[flux]] densities. The equation has three loss components: hysteresis, residual, and eddy current,<ref>{{harvnb|Arnold Engineering Company|n.d.|p=70}}</ref><ref>{{Citation|last=Legg|first=Victor E.|title=Magnetic Measurements at Low Flux Densities Using the Alternating Current Bridge|date=January 1936|url=http://www3.alcatel-lucent.com/bstj/vol15-1936/articles/bstj15-1-39.pdf|journal=[[Bell System Technical Journal]]|volume=15|issue=1|pages=39–63|publisher=Bell Telephone Laboratories|doi=10.1002/j.1538-7305.1936.tb00718.x}}</ref><ref>{{Cite book|title=Soft ferrites : properties and applications|last=Snelling|first=E.C.|date=1988|publisher=Butterworths|isbn=978-0408027601|edition=2nd|location=London|oclc=17875867}}</ref> and it is given by

:<math>\frac{R_{\text{ac}}}{\mu L} = a B_{\text{max}} f + c f + e f^2</math>

where

* <math>R_{ac}</math> is the effective core loss resistance (ohms),
* <math>\mu</math> is the [[Permeability (electromagnetism)|material permeability]],
* <math>L</math> is the [[inductance]] (henrys),
* <math>a</math> is the hysteresis loss coefficient,
* <math>B_{\text{max}}</math> is the maximum flux density (gauss),
* <math>c</math> is the residual loss coefficient,
* <math>f</math> is the frequency (hertz), and
* <chem>e</chem> is the eddy loss coefficient.

=== Steinmetz coefficients ===
{{main|Steinmetz's equation}}
Losses in magnetic materials can be characterized by the Steinmetz coefficients, which however do not take into account temperature variability. Material manufacturers provide data on core losses in tabular and graphical form for practical conditions of use.