Editing Transformer (section)

===Real transformer===
[[File:Transformer Flux.svg|thumb|Leakage flux of a transformer|300x300px]]

====Deviations from ideal transformer====
The ideal transformer model neglects many basic linear aspects of real transformers, including unavoidable losses and inefficiencies.<ref>{{cite book|isbn=0-03-061758-8|title=Electrical Engineering: An Introduction|publisher=Saunders College Publishing|year=1984|page=610}}</ref>

(a) Core losses, collectively called magnetizing current losses, consisting of<ref name="Say1983"/>
* [[Magnetic hysteresis|Hysteresis]] losses due to nonlinear magnetic effects in the transformer core, and
* [[Eddy current]] losses due to joule heating in the core that are proportional to the square of the transformer's applied voltage.

(b) Unlike the ideal model, the windings in a real transformer have non-zero resistances and inductances associated with:
* [[Joule heating|Joule losses]] due to resistance in the primary and secondary windings<ref name="Say1983"/>
* Leakage flux that escapes from the core and passes through one winding only resulting in primary and secondary reactive impedance.

(c) similar to an [[inductor]], parasitic capacitance and self-resonance phenomenon due to the electric field distribution. Three kinds of parasitic capacitance are usually considered and the closed-loop equations are provided<ref>L. Dalessandro, F. d. S. Cavalcante, and J. W. Kolar, "Self-Capacitance of High-Voltage Transformers," IEEE Transactions on Power Electronics, vol. 22, no. 5, pp. 2081–2092, 2007.</ref>
* Capacitance between adjacent turns in any one layer;
* Capacitance between adjacent layers;
* Capacitance between the core and the layer(s) adjacent to the core;

Inclusion of capacitance into the transformer model is complicated, and is rarely attempted; the [[#Real transformer equivalent circuit figure|'real' transformer model's equivalent circuit shown below]] does not include parasitic capacitance. However, the capacitance effect can be measured by comparing open-circuit inductance, i.e. the inductance of a primary winding when the secondary circuit is open, to a short-circuit inductance when the secondary winding is shorted.

====Leakage flux====
{{Main|Leakage inductance}}

The ideal transformer model assumes that all flux generated by the primary winding links all the turns of every winding, including itself. In practice, some flux traverses paths that take it outside the windings.<ref name="McLaren1984-68">{{harvnb|McLaren|1984|pp=68–74}}</ref> Such flux is termed ''leakage flux'', and results in [[leakage inductance]] in [[series and parallel circuits|series]] with the mutually coupled transformer windings.<ref name="calvert2001"/> Leakage flux results in energy being alternately stored in and discharged from the magnetic fields with each cycle of the power supply. It is not directly a power loss, but results in inferior [[voltage regulation]], causing the secondary voltage not to be directly proportional to the primary voltage, particularly under heavy load.<ref name="McLaren1984-68"/> Transformers are therefore normally designed to have very low leakage inductance.

In some applications increased leakage is desired, and long magnetic paths, air gaps, or magnetic bypass shunts may deliberately be introduced in a transformer design to limit the [[Short circuit|short-circuit]] current it will supply.<ref name="calvert2001">{{cite web| last = Calvert| first = James| title = Inside Transformers| publisher = University of Denver| year = 2001|url=http://www.du.edu/~jcalvert/tech/transfor.htm| access-date = May 19, 2007| url-status = dead| archive-url=https://web.archive.org/web/20070509111407/http://www.du.edu/~jcalvert/tech/transfor.htm| archive-date = May 9, 2007}}</ref> Leaky transformers may be used to supply loads that exhibit [[negative resistance]], such as [[electric arc]]s, [[mercury-vapor lamp|mercury-]] and [[sodium-vapor lamp|sodium-]] vapor lamps and [[neon sign]]s or for safely handling loads that become periodically short-circuited such as [[arc welding|electric arc welders]].<ref name="Say1983"/>{{rp|485}}

[[Air gap (magnetic)|Air gaps]] are also used to keep a transformer from saturating, especially audio-frequency transformers in circuits that have a DC component flowing in the windings.<ref>{{cite book|last=Terman|first=Frederick E.|title=Electronic and Radio Engineering|url=https://archive.org/details/electronicradioe00term|url-access=registration| edition=4th |year=1955|publisher=McGraw-Hill|location=New York|pages=[https://archive.org/details/electronicradioe00term/page/15 15]}}</ref> A [[saturable reactor]] exploits saturation of the core to control alternating current.

Knowledge of leakage inductance is also useful when transformers are operated in parallel. It can be shown that if the [[Per-unit system|percent impedance]]{{efn|Percent impedance is the ratio of the voltage drop in the secondary from no load to full load.<ref name="Heathcote1998-4">{{harvnb|Heathcote|1998|p=4}}</ref>}} and associated winding leakage reactance-to-resistance (''X''/''R'') ratio of two transformers were
the same, the transformers would share the load power in proportion to their respective ratings.  However, the impedance tolerances of commercial transformers are significant. Also, the impedance and X/R ratio of different capacity transformers tends to vary.<ref name="Knowlton6-97">{{cite book|editor-last=Knowlton|editor-first=A.E. |title=Standard Handbook for Electrical Engineers|edition=8th|year=1949|publisher=McGraw-Hill|page=see esp. Section 6 Transformers, etc, pp. 547–644}} Nomenclature for Parallel Operation, pp. 585–586</ref>
{{clear}}

====Equivalent circuit====
{{See also|Induction motor#Steinmetz equivalent circuit|l1=Steinmetz equivalent circuit}}

Referring to the diagram, a practical transformer's physical behavior may be represented by an [[equivalent circuit]] model, which can incorporate an ideal transformer.<ref name="daniels1985-47">{{harvnb|Daniels|1985|pp=47–49}}</ref>

Winding joule losses and leakage reactance are represented by the following series loop impedances of the model:
* Primary winding: ''R''<sub>P</sub>, ''X''<sub>P</sub>
* Secondary winding: ''R''<sub>S</sub>, ''X''<sub>S</sub>.
In normal course of circuit equivalence transformation, ''R''<sub>S</sub> and ''X''<sub>S</sub> are in practice usually referred to the primary side by multiplying these impedances by the turns ratio squared, (''N''<sub>P</sub>/''N''<sub>S</sub>)<sup>&nbsp;2</sup>&nbsp;=&nbsp;a<sup>2</sup>.

{{anchor|Real transformer equivalent circuit figure}}
[[Image:Transformer equivalent circuit.svg|thumb|upright=2|Real transformer equivalent circuit]]
Core loss and reactance is represented by the following shunt leg impedances of the model:
* Core or iron losses: ''R''<sub>C</sub>
* Magnetizing reactance: ''X''<sub>M</sub>.
''R''<sub>C</sub> and ''X''<sub>M</sub> are collectively termed the ''magnetizing branch'' of the model.

Core losses are caused mostly by hysteresis and eddy current effects in the core and are proportional to the square of the core flux for operation at a given frequency.<ref name="Say1983">{{cite book | last = Say | first = M. G. | title = Alternating Current Machines| edition = 5th| publisher = Pitman| year = 1983| location = London | isbn = 978-0-273-01969-5}}</ref>{{rp|142–143}} The finite permeability core requires a magnetizing current ''I''<sub>M</sub> to maintain mutual flux in the core. Magnetizing current is in phase with the flux, the relationship between the two being non-linear due to saturation effects. However, all impedances of the equivalent circuit shown are by definition linear and such non-linearity effects are not typically reflected in transformer equivalent circuits.<ref name="Say1983"/>{{rp|142}} With [[sinusoidal]] supply, core flux lags the induced EMF by&nbsp;90°. With open-circuited secondary winding, magnetizing branch current ''I''<sub>0</sub> equals transformer no-load current.<ref name="daniels1985-47"/>
[[File:Instrument Transformer_LV_terminals.jpg|thumb|Instrument transformer, with [[Polarity (mutual inductance)|polarity dot]] and X1 markings on low-voltage ("LV") side terminal]]
The resulting model, though sometimes termed 'exact' equivalent circuit based on [[linearity]] assumptions, retains a number of approximations.<ref name="daniels1985-47"/> Analysis may be simplified by assuming that magnetizing branch impedance is relatively high and relocating the branch to the left of the primary impedances. This introduces error but allows combination of primary and referred secondary resistances and reactance by simple summation as two series impedances.

Transformer equivalent circuit impedance and transformer ratio parameters can be derived from the following tests: [[open-circuit test]], [[short-circuit test]], winding resistance test, and transformer ratio test.