Transformer

    A transformer is a static electrical device that transfers energy by inductive coupling between its winding circuits. A varying current in the primary winding creates a varying magnetic flux in the transformer's core and thus a varying magnetic flux through the secondary winding. This varying magnetic flux induces a varying electromotive force (emf) or voltage in the secondary winding.

Transformers range in size from thumbnail-sized used in microphones to units weighing hundreds of tons interconnecting the power grid.  A wide range of transformer designs  used in electronic and electric power applications. Transformers are essential for the transmission, distribution, and utilization of electrical energy.

Basic principles

The ideal transformer

Ideal transformer circuit diagram

Consider the ideal, lossless, perfectly-coupled transformer shown in the circuit diagram at right having primary and secondary windings with N_P and N_S turns, respectively.

The ideal transformer induces secondary voltage E_S =V_S as a proportion of the primary voltage V_P = E_P and respective winding turns as given by the equation

,

where,

- V_P/V_S = E_P/E_S = a is the voltage ratio and N_P/N_S = a is the winding turns ratio, the value of these ratios being respectively higher and lower than unity for step-down and step-up transformers,^[3]^[4][lower-alpha 1][lower-alpha 2]

- V_P designates source impressed voltage,

- V_S designates output voltage, and,

- E_P & E_S designate respective emf induced voltages.[lower-alpha 3]


↵Any load impedance  connected to the ideal transformer's secondary winding causes current to flow without losses from primary to secondary circuits, the resulting input and output apparent power therefore being equal as given by the equation

.


↵Combining the two equations yields the following ideal transformer identity

.


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

The load impedance  is defined in terms of secondary circuit voltage and current as follows

.

The apparent impedance  of this secondary circuit load referred to the primary winding circuit is governed by a squared turns ratio multiplication factor relationship derived as follows^[5]^[6]

.




Induction law

The transformer is based on two principles: first, that an electric current can produce a magnetic field and second that a changing magnetic field within a coil of wire induces a voltage across the ends of the coil (electromagnetic induction). Changing the current in the primary coil changes the magnetic flux that is developed. The changing magnetic flux induces a voltage in the secondary coil.

Referring to the two figures here, current passing through the primary coil creates a magnetic field. The primary and secondary coils are wrapped around a core of very high magnetic permeability, usually iron,[lower-alpha 4] so that most of the magnetic flux passes through both the primary and secondary coils.  Any secondary winding connected load causes current and voltage induction from primary to secondary circuits in indicated directions.



Ideal transformer and induction law

The voltage induced across the secondary coil may be calculated from Faraday's law of induction, which states that:



where V_s = E_s is the instantaneous voltage, N_s is the number of turns in the secondary coil, and dΦ/dt is the derivative[lower-alpha 5] of the magnetic flux Φ through one turn of the coil. If the turns of the coil are oriented perpendicularly to the magnetic field lines, the flux is the product of the magnetic flux density B and the area A through which it cuts. The area is constant, being equal to the cross-sectional area of the transformer core, whereas the magnetic field varies with time according to the excitation of the primary. Since the same magnetic flux passes through both the primary and secondary coils in an ideal transformer,^[5]  the instantaneous voltage across the primary winding equals



Taking the ratio of the above two equations gives the same voltage ratio and turns ratio relationship shown above, that is,

.

The changing magnetic field induces an emf across each winding. ^[7] The primary emf, acting as it does in opposition to the primary voltage, is sometimes termed the counter emf.^[8] This is in accordance with Lenz's law, which states that induction of emf always opposes development of any such change in magnetic field.

As still lossless and perfectly-coupled, the transformer still behaves as described above in the ideal transformer.

Polarity

Instrument transformer, with polarity dot and X1 markings on LV side terminal

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 at the secondary winding's dot end.^[9]^[10]^[11][lower-alpha 6][lower-alpha 7]

The real transformer

Real transformer deviations from ideal

The ideal model neglects the following basic linear aspects in real transformers:

Core losses collectively called magnetizing current losses consisting of:^[12]

• Hysteresis losses due to nonlinear application of the voltage applied in the transformer core
• Eddy current losses due to joule heating in core proportional to the square of the transformer's applied voltage.

Whereas the ideal windings have no impedance, the windings in a real transformer have finite non-zero impedances in the form of:

• Joule losses due to resistance in the primary and secondary windings^[12]
• Leakage flux that escapes from the core and passes through one winding only resulting in primary and secondary reactive impedance.

Leakage flux

Main article: Leakage inductance



Leakage flux of a transformer

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.^[13] Such flux is termed leakage flux, and results in leakage inductance in series with the mutually coupled transformer windings.^[8] 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 (see Stray losses below), but results in inferior voltage regulation, causing the secondary voltage to not be directly proportional to the primary voltage, particularly under heavy load.^[13] Transformers are therefore normally designed to have very low leakage inductance. Nevertheless, it is impossible to eliminate all leakage flux because it plays an essential part in the operation of the transformer. The combined effect of the leakage flux and the electric field around the windings is what transfers energy from the primary to the secondary.^[14]

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 current it will supply.^[8] Leaky transformers may be used to supply loads that exhibit negative resistance, such as electric arcs, mercury vapor lamps, and neon signs or for safely handling loads that become periodically short-circuited such as electric arc welders.^[15]

Air gaps are also used to keep a transformer from saturating, especially audio-frequency transformers in circuits that have a DC component flowing through the windings.^[16]

Knowledge of leakage inductance is for example useful when transformers are operated in parallel. It can be shown that if the percent impedance (Z) and associated winding leakage reactance-to-resistance (X/R) ratio of two transformers were hypothetically exactly the same, the transformers would share power in proportion to their respective volt-ampere ratings  (e.g. 500 kVA unit in parallel with 1,000 kVA unit, the larger unit would carry twice the current). However, the impedance tolerances of commercial transformers are significant. Also, the Z impedance and X/R ratio of different capacity transformers tends to vary, corresponding 1,000 kVA and 500 kVA units' values being, to illustrate, respectively, Z ~ 5.75%, X/R ~ 3.75 and Z ~ 5%, X/R ~ 4.75.^[17]^[18]

Equivalent circuit

See also: 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.^[19]

Winding joule losses and leakage reactances are represented by the following series loop impedances of the model:

• Primary winding: R_P, X_P
• Secondary winding:  R_S, X_S.

In normal course of circuit equivalence transformation, R_S and X_S are in practice usually referred to the primary side by multiplying these impedances by the turns ratio squared, (N_P/N_S) ² = a².

Real transformer equivalent circuit

Core loss and reactance is represented by the following shunt leg impedances of the model:

• Core or iron losses: R_C
• Magnetizing reactance: X_M.

R_C and X_M 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.^[20] The finite permeability core requires a magnetizing current I_M 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.^[20] With sinusoidal supply, core flux lags the induced emf by 90°.  With open-circuited secondary winding, magnetizing branch current I₀ equals transformer no-load current.^[19]

The resulting model, though sometimes termed 'exact' equivalent circuit based on linearity assumptions, retains a number of approximations.^[19] 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 reactances 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,[lower-alpha 8] short-circuit test, winding resistance test, and transformer ratio test.

Basic transformer parameters and construction

Effect of frequency

↵Transformer universal emf equation

If the flux in the core is purely sinusoidal, the relationship for either winding between its rms voltage E_rms of the winding, and the supply frequency f, number of turns N, core cross-sectional area a in m² and peak magnetic flux density B_peak in Wb/m² or T (tesla)  is given by the universal emf equation:^[12]



If the flux does not contain even harmonics the following equation can be used for half-cycle average voltage E_avg of any waveshape:



The time-derivative term in Faraday's Law shows that the flux in the core is the integral with respect to time of the applied voltage.^[21] Hypothetically an ideal transformer would work with direct-current excitation, with the core flux increasing linearly with time.^[22] In practice, the flux rises to the point where magnetic saturation of the core occurs, causing a large increase in the magnetizing current and overheating the transformer. All practical transformers must therefore operate with alternating (or pulsed direct) current.^[22]

The emf of a transformer at a given flux density increases with frequency.^[12] 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 Hz power supplies which reduce core and winding weight.^[23] Conversely, frequencies used for some railway electrification systems were much lower (e.g. 16.7 Hz and 25 Hz) than normal utility frequencies (50 – 60 Hz) for historical reasons concerned mainly with the limitations of early electric traction motors. As such, the transformers used to step-down the high over-head line voltages (e.g. 15 kV) were much heavier for the same power rating than those designed only for the higher frequencies.

Power transformer over-excitation 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 transformer at other than its design frequency may require assessment of voltages, losses, and cooling to establish if safe operation is practical. For example, transformers may need to be equipped with 'volts per hertz' over-excitation relays to protect the transformer from overvoltage at higher than rated frequency.

One example of state-of-the-art design is traction transformers used for electric multiple unit and high speed train service operating across the,country border and using different electrical standards, such transformers' being restricted to be positioned below the passenger compartment. The power supply to, and converter equipment being supply by, such traction transformers have to accommodate different input frequencies and voltage (ranging from as high as 50 Hz down to 16.7 Hz and rated up to 25 kV) while being suitable for multiple AC asynchronous motor and DC converters & motors with varying harmonics mitigation filtering requirements.

Large power transformers are vulnerable to insulation failure due to transient voltages with high-frequency components, such as caused in switching or by lightning.^[24]