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{{Short description|Device to couple energy between circuits}}{{about|the electrical device|other uses}}(File:Philips_N4422_-_power_supply_transformer-2098.jpg|thumb|An O-core transformer consisting of two coils of copper wire wrapped around a magnetic core)In electrical engineering, a transformer is a passive component that transfers electrical energy from one electrical circuit to another circuit, or multiple circuits. A varying current in any coil of the transformer produces a varying magnetic flux in the transformer's core, which induces a varying electromotive force (EMF) across any other coils wound around the same core. Electrical energy can be transferred between separate coils without a metallic (conductive) connection between the two circuits. Faraday's law of induction, discovered in 1831, describes the induced voltage effect in any coil due to a changing magnetic flux encircled by the coil.Transformers are used to change AC voltage levels, such transformers being termed step-up or step-down type to increase or decrease voltage level, respectively. Transformers can also be used to provide galvanic isolation between circuits as well as to couple stages of signal-processing circuits. Since the invention of the first constant-potential transformer in 1885, transformers have become essential for the transmission, distribution, and utilization of alternating current electric power.JOURNAL, Bedell, Frederick, History of A-C Wave Form, Its Determination and Standardization, Transactions of the American Institute of Electrical Engineers, 61, 12, 864, 10.1109/T-AIEE.1942.5058456, 1942, 51658522, A wide range of transformer designs is encountered in electronic and electric power applications. Transformers range in size from RF transformers less than a cubic centimeter in volume, to units weighing hundreds of tons used to interconnect the power grid.{{TOC limit|limit=3}}{{clear}}

Principles

{{anchor|Ideal transformer equations}}Ideal transformer equationsBy Faraday's law of induction:{{NumBlk|:|V_text{P} = -N_text{P} frac{mathrm{d}Phi}{mathrm{d}t}|{{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 mathrm{d}Phi/mathrm{d}t, defined as the derivative of magnetic flux Phi with time t, 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."}}BOOK, Skilling, Hugh Hildreth, Electromechanics, 1962, John Wiley & Sons, Inc., p. 39}}{{NumBlk|:|V_text{S} = -N_text{S} frac{mathrm{d}Phi}{mathrm{d}t}|{{EquationRef|Eq. 2}}}}where V is the instantaneous voltage, N 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 P and S denotes primary and secondary.Combining the ratio of eq. 1 & eq. 2:{{NumBlk|:|Turns ratio =frac{V_text{P}}{V_text{S}} = frac{N_text{P}}{N_text{S}}=a|{{EquationRef|Eq. 3}}}}where for a step-up transformer a < 1 and for a step-down transformer a > 1.By the law of conservation of energy, apparent, real and reactive power are each conserved in the input and output:{{NumBlk|:|S=I_text{P} V_text{P} = I_text{S} V_text{S}|{{EquationRef|Eq. 4}}}}where S is apparent power and I is 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.}}{{harvnb|Brenner|Javid|1959|loc=§18-1 Symbols and Polarity of Mutual Inductance, pp.=589–590}} gives the ideal transformer identity:{{NumBlk|:|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|{{EquationRef|Eq. 5}}}}where L is winding self-inductance.By Ohm's law and ideal transformer identity:{{NumBlk|:|Z_text{L}=frac{V_text{S}}{I_text{S}}|{{EquationRef|Eq. 6}}}}{{NumBlk|:|Z'_text{L} = frac{V_text{P}}{I_text{P}}=frac{aV_text{S}}{I_text{S}/a}=a^2frac{V_text{S}}{I_text{S}}=a^2{Z_text{L}}|{{EquationRef|Eq. 7}}}}where Z_text{L} is the load impedance of the secondary circuit & Z'_text{L} is the apparent load or driving point impedance of the primary circuit, the superscript ' denoting referred to the primary.

Ideal transformer

An ideal transformer is linear, lossless and perfectly coupled. Perfect coupling implies infinitely high core magnetic permeability and winding inductance and zero net magnetomotive force (i.e. i'p'n'p âˆ’ i's'n's = 0).{{harvnb|Brenner|Javid|1959|loc=§18-6 The Ideal Transformer, pp. 598–600}}{{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.{{harvnb|Crosby|1958|p=145}}}}File:Ideal transformer.svg|left|thumb|upright=2|Ideal transformer connected with source V'P on primary and load impedance Z'L on secondary, where 0 |url=https://archive.org/details/U.S._Army_Corps_of_Engineers_Engineering_and_Design_Hydroelectric_Power_Plants_E}}