Editing Transformer (section)

==Principles==

'''Ideal transformer equations'''

By [[Faraday's law of induction]]:

{{NumBlk|:|<math>V_\text{P} = -N_\text{P} \frac{\mathrm{d}\Phi}{\mathrm{d}t}</math>|{{EquationRef|Eq. 1}}{{efn|With turns of the winding oriented perpendicularly to the magnetic field lines, the flux is the product of the [[magnetic flux density]] and the core area, the magnetic field varying with time according to the excitation of the primary. The expression <math>\mathrm{d}\Phi/\mathrm{d}t</math>, defined as the derivative of magnetic flux <math>\Phi</math> with time <math>t</math>, provides a measure of rate of magnetic flux in the core and hence of EMF induced in the respective winding. The negative sign in eq. 1 & eq. 2 is consistent with Lenz's law and Faraday's law in that by convention EMF "induced by an ''increase'' of magnetic flux linkages is ''opposite'' to the direction that would be given by the [[right-hand rule]]."}}<ref>{{cite book|last=Skilling|first=Hugh Hildreth|title=Electromechanics|year=1962|publisher=John Wiley & Sons, Inc.}} p. 39</ref>}}

{{NumBlk|:|<math>V_\text{S} = -N_\text{S} \frac{\mathrm{d}\Phi}{\mathrm{d}t}</math>|{{EquationRef|Eq. 2}}}}

where <math>V</math> is the [[Derivative|instantaneous]] [[voltage]], <math>N</math> is the [[number of turns]] in a winding, dΦ/dt is the [[derivative]] of the magnetic flux Φ through one turn of the winding over time ('''''t'''''), and subscripts <sub>'''P'''</sub> and <sub>'''S'''</sub> denotes primary and secondary.

Combining the ratio of eq. 1 & eq. 2:

{{NumBlk|:|Turns ratio <math>=\frac{V_\text{P}}{V_\text{S}} = \frac{N_\text{P}}{N_\text{S}}=a</math>|{{EquationRef|Eq. 3}}}}

where for a step-up transformer ''a'' < 1 and for a step-down transformer ''a'' > 1.<ref name="Brenner18-6"/>

By the law of [[conservation of energy]], [[Apparent power|apparent]], [[real power|real]] and [[reactive power|reactive]] power are each conserved in the input and output:

{{NumBlk|:|<math>S=I_\text{P} V_\text{P} = I_\text{S} V_\text{S}</math>|{{EquationRef|Eq. 4}}}}

where <math>S</math> is apparent power and <math>I</math> is [[Electric current|current]].

Combining Eq. 3 & Eq. 4 with this endnote{{efn|Although ideal transformer's winding inductances are each infinitely high, the square root of winding inductances' ratio is equal to the turns ratio.}}<ref name="Brenner18-1">{{harvnb|Brenner|Javid|1959|loc=§18-1 Symbols and Polarity of Mutual Inductance, pp.=589–590}}</ref> gives the ideal transformer [[identity function|identity]]:

{{NumBlk|:|<math>\frac{V_\text{P}}{V_\text{S}} = \frac{I_\text{S}}{I_\text{P}}=\frac{N_\text{P}}{N_\text{S}}=\sqrt{\frac{L_\text{P}}{L_\text{S}}}=a</math>|{{EquationRef|Eq. 5}}}}

where <math>L_\text{P}</math> is the primary winding [[self-inductance]] and  <math>L_\text{S}</math> is the secondary winding self-inductance.

By [[Ohm's law]] and ideal transformer identity:

{{NumBlk|:|<math>Z_\text{L}=\frac{V_\text{S}}{I_\text{S}}</math>|{{EquationRef|Eq. 6}}}}

{{NumBlk|:|<math>Z'_\text{L} = \frac{V_\text{P}}{I_\text{P}}=\frac{aV_\text{S}}{I_\text{S}/a}=a^2\frac{V_\text{S}}{I_\text{S}}=a^2{Z_\text{L}}</math>|{{EquationRef|Eq. 7}}}}

where <math>Z_\text{L}</math> is the load impedance of the secondary circuit & <math>Z'_\text{L}</math> is the apparent load or driving point impedance of the primary circuit, the superscript <math>'</math> denoting referred to the primary.

===Ideal transformer===
An ideal transformer is [[Linearity|linear]], lossless and perfectly [[inductive coupling|coupled]]. Perfect coupling implies infinitely high core [[Permeability (electromagnetism)|magnetic permeability]] and winding [[inductance]] and zero net [[magnetomotive force]] (i.e. ''i''<sub>''p''</sub>''n''<sub>''p''</sub>&nbsp;−&nbsp;''i''<sub>''s''</sub>''n''<sub>''s''</sub>&nbsp;=&nbsp;0).<ref name="Brenner18-6">{{harvnb|Brenner|Javid|1959|loc=§18-6 The Ideal Transformer, pp. 598–600}}</ref>{{efn|This also implies the following: The net core flux is zero, the input impedance is infinite when secondary is open and zero when secondary is shorted; there is zero phase-shift through an ideal transformer; input and output power and reactive volt-ampere are each conserved; these three statements apply for any frequency above zero and periodic waveforms are conserved.<ref name="Crosby1958-145">{{harvnb|Crosby|1958|p=145}}</ref>}}

[[File:Transformer under load (alternative version).svg|left|thumb|upright=2|Ideal transformer connected with source ''V''<sub>''P''</sub> on primary and load impedance ''Z''<sub>''L''</sub> on secondary, where 0&nbsp;<&nbsp;''Z''<sub>''L''</sub>&nbsp;<&nbsp;∞.]]
[[File:Transformer3d col3.svg|left|thumb|upright=2|Ideal transformer and induction law{{efn|Direction of transformer currents is according to [[Right-hand rule#Rotations|the Right-Hand Rule.]]}}]]

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A varying current in the transformer's primary winding creates a varying magnetic flux in the transformer core, which is also encircled by the secondary winding. This varying flux at the secondary winding induces a varying [[electromotive force| electromotive force or voltage]] in the secondary winding. This electromagnetic induction phenomenon is the basis of transformer action and, in accordance with [[Lenz's law]], the secondary current so produced creates a flux equal and opposite to that produced by the primary winding.

The windings are wound around a core of infinitely high magnetic permeability so that all of the magnetic flux passes through both the primary and secondary windings. With a [[voltage source]] connected to the primary winding and a load connected to the secondary winding, the transformer currents flow in the indicated directions and the core magnetomotive force cancels to zero.

According to [[Faraday's law of induction|Faraday's law]], since the same magnetic flux passes through both the primary and secondary windings in an ideal transformer, a voltage is induced in each winding proportional to its number of turns. The transformer winding voltage ratio is equal to the winding turns ratio.<ref>Paul A. Tipler, ''Physics'', Worth Publishers, Inc., 1976 {{ISBN|0-87901-041-X}}, pp. 937–940</ref>

An ideal transformer is a reasonable approximation for a typical commercial transformer, with voltage ratio and winding turns ratio both being inversely proportional to the corresponding current ratio.

The load impedance ''referred'' to the primary circuit is equal to the turns ratio squared times the secondary circuit load impedance.<ref name="Flanagan1993-1">{{cite book| last = Flanagan| first = William M.| title = Handbook of Transformer Design & Applications| publisher = McGraw-Hill| year = 1993| edition = 2nd| isbn = 978-0-07-021291-6|url=https://archive.org/details/FlagananHandbookOfTransformerDesignApplications}} pp. 2-1, 2-2</ref>
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===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>
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====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.

===Transformer EMF equation===
{{anchor|Transformer universal EMF equation}}

If the flux in the core is purely [[sinusoidal]], the relationship for either winding between its '''[[root mean square|rms]] voltage''' ''E''<sub>rms</sub> of the winding, and the supply frequency ''f'', number of turns ''N'', core cross-sectional area ''A'' in m<sup>2</sup> and peak magnetic flux density ''B''<sub>peak</sub> in Wb/m<sup>2</sup> or T (tesla) is given by the universal EMF equation:<ref name="Say1983" />

: <math> E_\text{rms} = {\frac {2 \pi f N A B_\text{peak}} {\sqrt{2}}} \approx 4.44 f N A  B_\text{peak}</math>

===Polarity===
A [[dot convention]] is often used in transformer circuit diagrams, nameplates or terminal markings to define the relative polarity of transformer windings. Positively increasing instantaneous current entering the primary winding's 'dot' end induces positive polarity voltage exiting the secondary winding's 'dot' end. Three-phase transformers used in electric power systems will have a nameplate that indicate the phase relationships between their terminals. This may be in the form of a [[phasor]] diagram, or using an alpha-numeric code to show the type of internal connection (wye or delta) for each winding.

===Effect of frequency===
The EMF of a transformer at a given flux increases with frequency.<ref name="Say1983"/> By operating at higher frequencies, transformers can be physically more compact because a given core is able to transfer more power without reaching saturation and fewer turns are needed to achieve the same impedance. However, properties such as core loss and conductor [[skin effect]] also increase with frequency. Aircraft and military equipment employ 400&nbsp;Hz power supplies which reduce core and winding weight.<ref>{{cite web | title = 400 Hz Electrical Systems | work = Aerospaceweb.org |url=http://www.aerospaceweb.org/question/electronics/q0219.shtml | access-date = May 21, 2007}}</ref> Conversely, frequencies used for some [[railway electrification system]]s were much lower (e.g. 16.7&nbsp;Hz and 25&nbsp;Hz) than normal utility frequencies (50–60&nbsp;Hz) for historical reasons concerned mainly with the limitations of early [[traction motor|electric traction motors]]. Consequently, the transformers used to step-down the high overhead line voltages were much larger and heavier for the same power rating than those required for the higher frequencies.

[[File:Power Transformer Over-Excitation.gif|thumb|Power transformer overexcitation condition caused by decreased frequency; flux (green), iron core's magnetic characteristics (red) and magnetizing current (blue).]]
Operation of a transformer at its designed voltage but at a higher frequency than intended will lead to reduced magnetizing current. At a lower frequency, the magnetizing current will increase. Operation of a large transformer at other than its design frequency may require assessment of voltages, losses, and cooling to establish if safe operation is practical. Transformers may require [[protective relay]]s to protect the transformer from overvoltage at higher than rated frequency.

One example is in traction transformers used for [[electric multiple unit]] and [[High-speed rail|high-speed]] train service operating across regions with different electrical standards. The converter equipment and traction transformers have to accommodate different input frequencies and voltage (ranging from as high as 50&nbsp;Hz down to 16.7&nbsp;Hz and rated up to 25&nbsp;kV).

At much higher frequencies the transformer core size required drops dramatically: a physically small transformer can handle power levels that would require a massive iron core at mains frequency. The development of switching power semiconductor devices made [[switch mode power supply|switch-mode power supplies]] viable, to generate a high frequency, then change the voltage level with a small transformer.

Transformers for higher frequency applications such as [[Switched-mode power supply|SMPS]] typically use core materials with much lower hysteresis and eddy-current losses than those for 50/60&nbsp;Hz. Primary examples are iron-powder and ferrite cores. The lower frequency-dependant losses of these cores often is at the expense of flux density at saturation. For instance, [[Ferrite core|ferrite]] saturation occurs at a substantially lower flux density than laminated iron.

Large power transformers are vulnerable to insulation failure due to transient voltages with high-frequency components, such as caused in switching or by lightning.

===Energy losses===
Transformer energy losses are dominated by winding and core losses. Transformers' efficiency tends to improve with increasing transformer capacity.<ref name="De Keulenaer2001">{{harvnb|De Keulenaer|Chapman|Fassbinder|McDermott|2001|}}</ref> The efficiency of typical distribution transformers is between about 98 and 99 percent.<ref name="De Keulenaer2001"/><ref>{{Cite book| last1 = Kubo| first1 = T.|last2 = Sachs| first2 = H.| last3 = Nadel| first3 = S.| title = Opportunities for New Appliance and Equipment Efficiency Standards| publisher = [[American Council for an Energy-Efficient Economy]] | at = p. 39, fig. 1| year = 2001|url=http://www.aceee.org/research-report/a016| access-date = June 21, 2009}}</ref>

As transformer losses vary with load, it is often useful to tabulate [[no-load loss]], full-load loss, half-load loss, and so on. Hysteresis and [[eddy current]] losses are constant at all load levels and dominate at no load, while winding loss increases as load increases. The no-load loss can be significant, so that even an idle transformer constitutes a drain on the electrical supply. Designing [[energy efficient transformer]]s for lower loss requires a larger core, good-quality [[Electrical steel|silicon steel]], or even [[Electrical steel#Amorphous steel|amorphous steel]] for the core and thicker wire, increasing initial cost. The choice of construction represents a [[trade-off]] between initial cost and operating cost.<ref name="Heathcote1998-41">{{harvnb|Heathcote|1998|pp=41–42}}</ref>

Transformer losses arise from:
; Winding joule losses
:Current flowing through a winding's conductor causes [[joule heating]] due to the [[electrical resistance|resistance]] of the wire. As frequency increases, skin effect and [[proximity effect (electromagnetism)|proximity effect]] causes the winding's resistance and, hence, losses to increase.
;[[magnetic core#Core loss|Core losses]]
:; Hysteresis losses
::Each time the magnetic field is reversed, a small amount of energy is lost due to [[Magnetic hysteresis|hysteresis]] within the core, caused by motion of the [[magnetic domain]]s within the steel. According to Steinmetz's formula, the heat energy due to hysteresis is given by
:::<math>W_\text{h}\approx\eta\beta^{1.6}_{\text{max}}</math> and,
::hysteresis loss is thus given by
:::<math>P_\text{h}\approx{W}_\text{h}f\approx\eta{f}\beta^{1.6}_{\text{max}}</math>
::where, ''f'' is the frequency, ''η'' is the hysteresis coefficient and ''β''<sub>max</sub> is the maximum flux density, the empirical exponent of which varies from about 1.4 to 1.8 but is often given as 1.6 for iron.<ref name="Heathcote1998-41"/> For more detailed analysis, see [[Magnetic core#Core losses|Magnetic core]] and [[Steinmetz's equation]].
:; Eddy current losses
:: [[Eddy current]]s are induced in the conductive metal transformer core by the changing magnetic field, and this current flowing through the resistance of the iron dissipates energy as heat in the core. The eddy current loss is a complex function of the square of supply frequency and inverse square of the material thickness.<ref name="Heathcote1998-41"/> Eddy current losses can be reduced by making the core of a stack of laminations (thin plates) electrically insulated from each other, rather than a solid block; all transformers operating at low frequencies use laminated or similar cores.
; Magnetostriction related transformer hum
:Magnetic flux in a ferromagnetic material, such as the core, causes it to physically expand and contract slightly with each cycle of the magnetic field, an effect known as [[magnetostriction]], the frictional energy of which produces an audible noise known as [[mains hum]] or "transformer hum".<ref name="PF (nd)">{{cite web|title=Understanding Transformer Noise|url=http://www.federalpacific.com/literature/drytrans/10transformernoise.pdf|publisher=FP|access-date=30 January 2013|url-status=dead|archive-url=https://web.archive.org/web/20060510231426/http://www.federalpacific.com/literature/drytrans/10transformernoise.pdf|archive-date=10 May 2006}}</ref> This transformer hum is especially objectionable in transformers supplied at [[utility frequency|power frequencies]] and in [[high-frequency]] [[flyback transformer]]s associated with television [[cathode-ray tube|CRTs]].
; {{anchor|stray}} Stray losses
:Leakage inductance is by itself largely lossless, since energy supplied to its magnetic fields is returned to the supply with the next half-cycle. However, any leakage flux that intercepts nearby conductive materials such as the transformer's support structure will give rise to eddy currents and be converted to heat.<ref name="nailen">{{Cite journal| last = Nailen| first = Richard| title = Why We Must Be Concerned With Transformers| journal = Electrical Apparatus| date = May 2005|url=http://www.blnz.com/news/2008/04/23/must_concerned_with_transformers_9639.html| url-status = dead| archive-url=https://web.archive.org/web/20090429031651/http://www.blnz.com/news/2008/04/23/must_concerned_with_transformers_9639.html| archive-date = 2009-04-29}}</ref>
; Radiative
:There are also radiative losses due to the oscillating magnetic field but these are usually small.
;Mechanical vibration and audible noise transmission
:In addition to magnetostriction, the alternating magnetic field causes fluctuating forces between the primary and secondary windings. This energy incites vibration transmission in interconnected metalwork, thus amplifying audible transformer hum.<ref name="Pansini1999-23">{{harvnb|Pansini|1999|p=23}}</ref>