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Transfomer Basics
A transformer is an electrical device that transfers energy from one circuit to another purely by magnetic coupling. Relative motion of the parts of the transformer is not required for transfer of energy. Transformers are often used to convert between high and low voltages, to change impedance, and to provide electrical isolation between circuits.

Introduction

The transformer is one of the simplest of electrical devices. Its basic design, materials, and principles have changed little over the last one hundred years, yet transformer designs and materials continue to be improved. Transformers are essential in high voltage power transmission providing an economical means of transmitting power over large distances. The simplicity, reliability, and economy of conversion of voltages by transformers was the principal factor in the selection of alternating current power transmission in the "War of Currents" in the late 1880's. In electronic circuitry, new methods of circuit design have replaced some of the applications of transformers, but electronic technology has also developed new transformer designs and applications.

Transformers come in a range of sizes from a thumbnail-sized coupling transformer hidden inside a stage microphone to gigawatt units used to interconnect large portions of national power grids, all operating with the same basic principles and with many similarities in their parts.

Transformers alone cannot do the following:

  • Convert DC to AC or vice versa
  • Change the voltage or current of DC
  • Change the AC supply frequency.

However, transformers are components of the systems that perform all these functions.

Basic principles

An analogy

The transformer may be considered as a simple two-wheel 'gearbox' for electrical voltage and current. The primary winding is analogous to the input shaft and the secondary winding to the output shaft. In this comparison, current is equivalent to shaft speed, voltage to shaft torque. In a gearbox, mechanical power (speed multiplied by torque) is constant (neglecting losses) and is equivalent to electrical power (voltage multiplied by current) which is also constant.

The gear ratio is equivalent to the transformer step-up or step-down ratio. A step-up transformer acts analogously to a reduction gear (in which mechanical power is transferred from a small, rapidly rotating gear to a large, slowly rotating gear): it trades current (speed) for voltage (torque), by transferring power from a primary coil to a secondary coil having more turns. A step-down transformer acts analogously to a multiplier gear (in which mechanical power is transferred from a large gear to a small gear): it trades voltage (torque) for current (speed), by transferring power from a primary coil to a secondary coil having fewer turns.

Flux coupling laws

An idealised step-down transformer showing resultant flux in the core
An idealised step-down transformer showing resultant flux in the core

A simple transformer consists of two electrical conductors called the primary winding and the secondary winding. If a time-varying voltage {v_P}\, is applied to the primary winding of N_P\, turns, a current will flow in it producing a magnetomotive force (MMF). Just as an electromotive force (EMF) drives current around an electric circuit, so MMF drives magnetic flux through a magnetic circuit. The primary MMF produces a varying magnetic flux \Phi_P\, in the core (shaded grey), and induces a back electromotive force (EMF) in opposition to {v_P}\,. In accordance with Faraday's Law, the voltage induced across the primary winding is proportional to the rate of change of flux  :

{v_P} = {N_P} \frac {d \Phi_P}{dt}

Similarly, the voltage induced across the secondary winding is:

{v_S} = {N_S} \frac {d \Phi_S}{dt}

With perfect flux coupling, the flux in the secondary winding will be equal to that in the primary winding, and so we can equate \Phi_P\, and \Phi_S\,. It thus follows that:

\frac{v_P}{v_S}=\frac{N_P}{N_S}.

Hence in an ideal transformer, the ratio of the primary and secondary voltages is equal to the ratio of the number of turns in their windings, or alternatively, the voltage per turn is the same for both windings. This leads to the most common use of the transformer: to convert electrical energy at one voltage to energy at a different voltage by means of windings with different numbers of turns.

The EMF in the secondary winding, if connected to an electrical circuit, will cause current to flow in the secondary circuit. The MMF produced by current in the secondary opposes the MMF of the primary and so tends to cancel the flux in the core. Since the reduced flux reduces the EMF induced in the primary winding, increased current flows in the primary circuit. The resulting increase in MMF due to the primary current offsets the effect of the opposing secondary MMF. In this way, the electrical energy fed into the primary winding is delivered to the secondary winding.

Neglecting losses, for a given level of power transferred through a transformer, current in the secondary circuit is inversely proportional to the ratio of secondary voltage to primary voltage. For example, suppose a power of 50 watts is supplied to a resistive load from a transformer with a turns ratio of 25:2.

  • P = E·I (power = electromotive force · current)

50 W = 2 V · 25 A in the primary circuit

  • Now with transformer change:

50 W = 25 V · 2 A in the secondary circuit.

In a practical transformer, the higher-voltage winding will have more turns,of smaller conductor cross-section, than the lower-voltage windings.

Since a DC voltage source would not give a time-varying flux in the core, no back EMF would be generated and so current flow into the transformer would be unlimited. In practice, the series resistance of the winding limits the amount of current that can flow, until the transformer either reaches thermal equilibrium or is destroyed.

The Universal EMF equation

If the flux in the core is sinusoidal, the relationship for either winding between its number of turns, voltage, magnetic flux density and core cross-sectional area is given by the universal emf equation:

E=4.44 \cdot f \cdot N \cdot a \cdot B

Where E\, is the sinusoidal root mean square voltage of the winding, f\, is the frequency in hertz, N\, is the number of turns of wire, a\, is the cross-sectional area of the core and B\, is the peak magnetic flux density in tesla. The value 4.44 collects a number of constants required by the system of units.

Invention

Those credited with the invention of the transformer include:

  • Michael Faraday, who invented an 'induction ring' on August 29, 1831. This was the first transformer, although Faraday used it only to demonstrate the principle of electromagnetic induction and did not foresee the use to which it would eventually be put.
  • Lucien Gaulard and John Dixon Gibbs, who first exhibited a device called a 'secondary generator' in London in 1881 and then sold the idea to American company Westinghouse. This may have been the first practical power transformer, but was not the first transformer of any kind. They also exhibited the invention in Turin in 1884, where it was adopted for an electric lighting system. Their early devices used an open iron core, which was later abandoned in favour of a more efficient circular core with a closed magnetic path.
  • William Stanley, an engineer for Westinghouse, who built the first practical device in 1885 after George Westinghouse bought Gaulard and Gibbs' patents. The core was made from interlocking E-shaped iron plates. This design was first used commercially in 1886.
  • Hungarian engineers Ottó Bláthy, Miksa Déri and Károly Zipernowsky at the Ganz company in Budapest in 1885, who created the efficient "ZBD" model based on the design by Gaulard and Gibbs.
  • Nikola Tesla in 1891 invented the Tesla coil, which is a high-voltage, air-core, dual-tuned resonant transformer for generating very high voltages at high frequency.

Practical considerations

Classifications

Single phase pole-mounted step-down transformer
Single phase pole-mounted step-down transformer

Transformers are adapted to numerous engineering applications and may be classified in many ways:

  • By power level (from fraction of a watt to many megawatts),
  • By application (power supply, impedance matching, circuit isolation),
  • By frequency range (power, audio, RF)
  • By voltage class (a few volts to about 750 kilovolts)
  • By cooling type (air cooled, oil filled, fan cooled, water cooled, etc.)
  • By purpose (rectifier, arc furnace, amplifier output, etc.).
  • By ratio of the number of turns in the coils
  • Step-up
The secondary has more turns than the primary.
  • Step-down
The secondary has fewer turns than the primary.
  • Isolating
Intended to transform from one voltage to the same voltage. The two coils have approximately equal numbers of turns, although often there is a slight difference in the number of turns, in order to compensate for losses (otherwise the output voltage would be a little less than, rather than the same as, the input voltage).
  • Variable
The primary and secondary have an adjustable number of turns which can be selected without reconnecting the transformer.
Three phase dry-type transformer with cover removed; rated about 200 KVA, 480 V
Three phase dry-type transformer with cover removed; rated about 200 KVA, 480 V

Losses

An ideal transformer would have no losses, and would therefore be 100% efficient. In practice energy is dissipated due both to the resistance of the windings (known as copper loss), and to magnetic effects primarily attributable to the core (known as iron loss). Transformers are in general highly efficient, and large power transformers (around 100 MVA and larger) may attain an efficiency as high as 99.75%. Small transformers such as a plug-in "power brick" used to power small consumer electronics may be less than 85% efficient.

The losses arise from:

  • Winding resistance
Current flowing through the windings causes resistive heating of the conductors.
  • Eddy currents
Induced currents circulate in the core and cause its resistive heating.
  • Stray losses
Not all the magnetic field produced by the primary is intercepted by the secondary. A portion of the leakage flux may induce eddy currents within nearby conductive objects such as the transformer's support structure, and be converted to heat. The familiar hum or buzzing noise heard near transformers is a result of stray fields causing components of the tank to vibrate, and is also from magnetostriction vibration of the core.
  • Hysteresis losses
Each time the magnetic field is reversed, a small amount of energy is lost to hysteresis in the magnetic core. The level of hysteresis is affected by the core material.
  • Mechanical losses
The alternating magnetic field causes fluctuating electromagnetic forces between the coils of wire, the core and any nearby metalwork, causing vibrations and noise which consume power.
  • Magnetostriction
The flux in the core causes it to physically expand and contract slightly with the alternating magnetic field, an effect known as magnetostriction. This in turn causes losses due to frictional heating in susceptible ferromagnetic cores.
  • Cooling system
Large power transformers may be equipped with cooling fans, oil pumps or water-cooled heat exchangers designed to remove the heat caused by copper and iron losses. The power used to operate the cooling system is typically considered part of the losses of the transformer.

High frequency operation

The universal transformer emf equation indicates that at higher frequency, the core flux density will be lower for a given voltage. This implies that a core can have a smaller cross-sectional area and thus be physically more compact without reaching saturation. It is for this reason that the aircraft manufacturers and the military use 400 hertz supplies. They are less concerned with efficiency, which is lower at higher frequencies (mostly due to increased hysteresis losses), but are more concerned with saving weight. Similarly, flyback transformers which supply high voltage to cathode ray tubes operate at the frequency of the horizontal oscillator, many times higher than 50 or 60 hertz, which allows for a more compact component.

Construction

Cores

Steel cores

Laminated core transformer showing edge of laminations at top of unit.
Laminated core transformer showing edge of laminations at top of unit.

Transformers for use at power or audio frequencies have cores made of many thin laminations of silicon steel. By concentrating the magnetic flux, more of it is usefully linked by both primary and secondary windings. Since the steel core is conductive, it, too, has currents induced in it by the changing magnetic flux. Each layer is insulated from the adjacent layer to reduce the energy lost to eddy current heating of the core. A typical laminated core is made from E-shaped and I-shaped pieces, leading to the name "EI transformer".

Certain types of transformer may have gaps inserted in the magnetic path to prevent magnetic saturation. These gaps may be used to limit the current on a short-circuit, such as for neon sign transformers.

A steel core's magnetic hysteresis means that it retains a static magnetic field when power is removed. When power is then reapplied, the residual field will cause a high inrush current until the effect of the remanent magnetism is reduced, usually after a few cycles of the applied alternating current. Overcurrent protection devices such as fuses must be selected to allow this harmless inrush to pass. On transformers connected to long overhead power transmission lines, induced currents due to geomagnetic disturbances during solar storms can cause saturation of the core, and false operation of transformer protection devices.

Distribution transformers can achieve low off-load losses by using cores made with amorphous (non-crystalline) steel, so-called "metal glasses" - the high cost of the core material is offset by the lower losses incurred at light load, over the life of the transformer.

Solid cores

In circuits that operate above mains frequencies, up to a few tens of kilohertz, such as switch-mode power supplies, powdered iron cores are used. These materials combine a high magnetic permeability with a high bulk material resistivity. At even higher frequencies, typically radio frequencies, other types of core made of non-conductive magnetic materials, such as various ceramic materials called ferrites, are common. Some transformers in radio-frequency circuits have adjustable cores which allow tuning of the coupling circuit.

Air cores

High-frequency transformers may also use air cores. These eliminate the loss due to hysteresis in the core material. Such transformers maintain high coupling efficiency (low stray field loss) by overlapping the primary and secondary windings.

Toroidal cores

Toroidal transformers are built around a ring-shaped core, which is made from a long strip of silicon steel or permalloy wound into a coil, or from ferrite, depending on frequency. The strip construction ensures that all the grain boundaries are pointing in the optimum direction, making the transformer more efficient by reducing the core's reluctance. The ring shape eliminates the air gaps inherent in the construction of an EI core. The cross-section of the ring is usually square or rectangular, but more expensive cores with circular cross-sections are also available. The primary and secondary coils are wound concentrically to cover the entire surface of the core. This minimises the length of wire needed, and also provides screening to prevent the core's magnetic field from generating electromagnetic interference.

Ferrite cores are used at frequencies up to a few tens of kilohertz to reduce losses, particularly in switch-mode power supplies.

Toroidal transformers are more efficient (around 95%) than the cheaper laminated EI types. Other advantages, compared to EI types, include smaller size (about half), lower weight (about half), less mechanical hum (making them superior in audio amplifiers), lower exterior magnetic field (about one tenth), low off-load losses (making them more efficient in standby circuits), single-bolt mounting, and more choice of shapes. This last point means that, for a given power output, either a wide, flat toroid or a tall, narrow one with the same electrical properties can be chosen, depending on the space available. The main disadvantage is higher cost.

A drawback of toroidal transformer construction is the higher cost of windings. As a consequence, toroidal transformers are uncommon above ratings of a few kVA. Small distribution transformers may achieve some of the benefits of a toroidal core by splitting it and forcing it open, then inserting a bobbin containg primary and secondary windings.

When fitting a toroidal transformer, it is important to avoid making an unintentional short-circuit through the core. This can happen if the steel mounting bolt in the middle of the core is allowed to touch metalwork at both ends, which could result in a dangerously large current flowing in the bolt.

Windings

In most practical transformers, the primary and secondary conductors are coils of conducting wire because each turn of the coil contributes to the magnetic field, creating a higher magnetic flux density than would a single conductor.

The winding material depends on the application. Small power and signal transformers are wound with insulated solid copper wire, often enameled. Larger power transformers may be wound with wire, copper or aluminum rectangular conductors, or strip conductors for very heavy currents. High frequency transformers operating in the tens to hundreds of kilohertz will have windings made of Litz wire, to minimize the skin effect losses in the conductors. Large power transformers use multiply-stranded conductors as well, since even at low power frequencies non-uniform distribution of current would otherwise exist in large windings. Each strand is insulated from the others, and the strands are arranged so that either at certain points in the winding or throughout the winding, each portion occupies different relative positions in the complete conductor. This "transposition" equalises the current flowing in each strand of the conductor, and reduces eddy current losses in the winding itself. The stranded conductor is also more flexible than a solid conductor of similar size. ( see reference (1) below)

Windings on both primary and secondary of a power transformer may have external connections (called taps) to intermediate points on the winding to allow adjustment of the voltage ratio; taps may be connected to automatic on-load tap changer switchgear for voltage regulation of distribution circuits. Audio-frequency transformers used for distribution of audio to public address loudspeakers have taps to allow adjustment of power supplied to each speaker. A center-tapped transformer is often used in the output stage of an audio power amplifier in a push-pull circuit. Tapped transformers are also used as components of amplifiers, oscillators, and for feedback linearization of amplifier circuits.

Insulation

The conductor material must have insulation to ensure the current travels around the core, and not through a turn-to-turn short-circuit.

In power transformers, the voltage difference between parts of the primary and secondary windings can be quite large. Insulation is inserted between layers of windings to prevent arcing, and the transformer may also be immersed in transformer oil that provides further insulation. To ensure that the insulating capability of the transformer oil does not deteriorate, the transformer casing is completely sealed against moisture ingress. The oil serves as both cooling medium to remove heat from the core and coil and as part of the insulation system.

Shielding

Although an ideal transformer is purely magnetic in operation, the proximity of the primary and secondary windings can create a mutual capacitance between the windings. Where transformers are intended for high electrical isolation between primary and secondary circuits, an electrostatic shield can be placed between windings to minimize this effect.

Transformers may also be enclosed by magnetic shields, electrostatic shields, or both to prevent outside interference from affecting the operation of the transformer, or to prevent the transformer from affecting the operation of other devices (such as CRTs in proximity to the transformer).

Coolant

All transformers must have some circulation of coolant to remove the waste heat produced by losses. Small transformers up to a few kilowatts in size usually are adequately cooled by air circulation. Larger "dry" type transformers may have cooling fans. Some dry transformers are enclosed in pressurized tanks and are cooled by nitrogen or sulfur hexafluoride gas.

The windings of high-power or high-voltage transformers are immersed in transformer oil - a highly-refined mineral oil that is stable at high temperatures. Large transformers to be used indoors must use a non-flammable liquid. Formerly, polychlorinated biphenyl (PCB) was used as it was not a fire hazard in indoor power transformers and it is highly stable. Due to the stability of PCB and its environmental accumulation, it is no longer permitted in new equipment. Today, nontoxic, stable silicone-based oils or fluorinated hydrocarbons may be used, where the expense of a fire-resistant liquid offsets additional building cost for a transformer vault. Other less-flammable fluids such as canola oil may be used but all fire resistant fluids have some drawbacks in performance, cost, or toxicity compared with mineral oil.

The oil cools the transformer, and provides part of the electrical insulation between internal live parts. It has to be stable at high temperatures so that a small short or arc will not cause a breakdown or fire. The oil-filled tank may have radiators through which the oil circulates by natural convection. Very large or high-power transformers (with capacities of millions of watts) may have cooling fans, oil pumps and even oil to water heat exchangers. Oil-filled transformers undergo prolonged drying processes, using vapor-phase heat transfer, electrical self-heating, the application of a vacuum, or combinations of these, to ensure that the transformer is completely free of water vapor before the cooling oil is introduced. This helps prevent electrical breakdown under load.

Oil-filled power transformers may be equipped with Buchholz relays - safety devices sensing gas buildup inside the transformer (a side effect of an electric arc inside the windings) and switching off the transformer.

Experimental power transformers in the 2 MVA range have been built with superconducting windings which eliminates the copper losses, but not the core steel loss. These are cooled by liquid nitrogen or helium.

Terminals

Very small transformers will have wire leads connected directly to the ends of the coils, and brought out to the base of the unit for circuit connections. Larger transformers may have heavy bolted terminals, bus bars or high-voltage insulated bushings made of polymers or porcelain. A large bushing can be a complex structure since it must both provide electrical insulation, and contain oil within the transformer tank.

Enclosure

Small transformers often have no enclosure. Transformers may have a shield enclosure, as described above. Larger units may be enclosed to prevent contact with live parts, and to contain the cooling medium (oil or pressurized gas).

Transformer designs

Autotransformers

An autotransformer has only a single winding, which is tapped at some point along the winding. AC or pulsed voltage is applied across a portion of the winding, and a higher (or lower) voltage is produced across another portion of the same winding. For voltage ratios not exceeding about 3:1, an autotransformer is less costly, lighter, smaller and more efficient than a two-winding transformer of a similar rating.

By exposing part of the winding coils and making the secondary connection through a sliding brush, an autotransformer with a near-continuously variable turns ratio can be obtained, allowing for very small increments of voltage.

Polyphase transformers

Wye and delta winding connections
Wye and delta winding connections

For three-phase power, three separate single-phase transformers can be used, or all three phases can be connected to a single polyphase transformer. The three primary windings are connected together and the three secondary windings are connected together. The most common connections are Y-Δ, Δ-Y, Δ-Δ and Y-Y. A vector group indicates the configuration of the windings and the phase angle difference between them. If a winding is connected to earth (grounded), the earth connection point is usually the center point of a Y winding. There are many possible configurations that may involve more or fewer than six windings and various tap connections.

Resonant transformers

A resonant transformer is one that operates at the resonant frequency of one or more of its coils. The resonant coil, usually the secondary, acts as an inductor, and is connected in series with a capacitor. If the primary coil is driven by a periodic source of alternating current, such as a square or sawtooth wave, each pulse of current helps to build up an oscillation in the secondary coil. Due to resonance, a very high voltage can develop across the secondary, until it is limited by some process such as electrical breakdown. These devices are therefore used to generate high alternating voltages. The current available from this type of coil can be much larger than that from electrostatic machines such as the Van de Graaff generator and Wimshurst machine. They also run at a higher operating temperature than standard units.

A voltage regulating transformer uses a resonant winding and allows part of the core to go into saturation on each cycle of the alternating current. This effect stabilizes the output of the regulating transformer, which can be used for equipment that is sensitive to variations of the supply voltage. Saturating transformers provide a simple rugged method to stabilize an ac power supply. However, due to the hysteresis losses accompanying this type of operation, efficiency is low.

Instrument transformers

Current transformers

Current transformers used as part of metering equipment for three-phase 400 ampere electricity supply
Current transformers used as part of metering equipment for three-phase 400 ampere electricity supply

A current transformer is designed to provide a current in its secondary which is accurately proportional to the current flowing in its primary.

Current transformers are commonly used in electricity meters to facilitate the measurement of large currents which would be difficult to measure more directly.

Care must be taken that the secondary of a current transformer is not disconnected from its load while current is flowing in the primary as in this circumstance a very high voltage would be produced across the secondary.

Current transformers are often constructed with a single primary turn either as an insulated cable passing through a toroidal core, or else as a bar to which circuit conductors are connected.

Voltage transformers

Voltage transformers (also known as potential transformers) are used in the electricity supply industry to measure accurately the voltage being supplied. They are designed to present negligible load to the voltage being measured.

Pulse transformers

A pulse transformer is a transformer that is optimised for transmitting rectangular electrical pulses (that is, pulses with fast rise and fall times and a constant amplitude). Small versions called signal types are used in digital logic and telecommunications circuits, often for matching logic drivers to transmission lines. Medium-sized power versions are used in power-control circuits such as camera flash controllers. Larger power versions are used in the electrical power distribution industry to interface low-voltage control circuitry to the high-voltage gates of power semiconductors such as TRIACs, IGBTs, thyristors and MOSFETs. Special high voltage pulse transformers are also used to generate high power pulses for radar, particle accelerators, or other pulsed power applications.

To minimise distortion of the pulse shape, a pulse transformer needs to have low values of leakage inductance and distributed capacitance, and a high open-circuit inductance. In power-type pulse transformers, a low coupling capacitance (between the primary and secondary) is important to protect the circuitry on the primary side from high-powered transients created by the load. For the same reason, high insulation resistance and high breakdown voltage are required. A good transient response is necessary to maintain the rectangular pulse shape at the secondary, because a pulse with slow edges would create switching losses in the power semiconductors.

The product of the peak pulse voltage and the duration of the pulse (or more accurately, the voltage-time integral) is often used to characterise pulse transformers. Generally speaking, the larger this product, the larger and more expensive the transformer.

RF transformers

For radio frequency use, transformers are sometimes made from configurations of transmission line, sometimes bifilar or coaxial cable, wound around ferrite cores. This style of transformer gives an extremely wide bandwidth, however only a limited number of ratios (such as 1:9, 1:4 or 1:2) can be achieved with this technique. The ferrite increases the inductance dramatically while also lowering its Q factor. The cores of such transformers help performance at the lower frequency end of the band. This style of transformer is frequently used as an impedance matching balun to convert from 300 ohm balanced to 75 ohm unbalanced in FM receivers.


Audio transformers

Traditionally, in the valve amplifier, the function of the output transformer was to convert the low alternating current music signal (that had been imposed on top of the high-voltage direct current from the plate electrode of the final output tube) into a useable high-current/low-voltage level for conversion by the loudspeakers.

In early transistor amplifiers, such transformers were also used.

Uses of transformers

  • Electric power transmission over long distances.
  • High-voltage direct-current HVDC power transmission systems
  • Large, specially constructed power transformers are used for electric arc furnaces used in steelmaking.
  • Rotating transformers are designed so that one winding turns while the other remains stationary. A common use was the video head system as used in VHS and Beta video tape players. These can pass power or radio signals from a stationary mounting to a rotating mechanism, or radar antenna.
  • Sliding transformers can pass power or signals from a stationary mounting to a moving part such as a machine tool head. An example is the linear variable differential transformer,
  • Some rotary transformers are precisely constructed in order to measure distances or angles. Usually they have a single primary and two or more secondaries, and electronic circuits measure the different amplitudes of the currents in the secondaries, such as in synchros and resolvers.
  • Small transformers are often used to isolate and link different parts of radio receivers and audio amplifiers, converting high current low voltage circuits to low current high voltage, or vice versa.
  • Balanced-to-unbalanced conversion. A special type of transformer called a balun is used in radio and audio circuits to convert between balanced circuits and unbalanced transmission lines such as antenna downleads. A balanced line is one in which the two conductors (signal and return) have the same impedance to ground: twisted pair and "balanced twin" are examples. Unbalanced lines include coaxial cables and strip-line traces on printed circuit boards. A similar use is for connecting the "single ended" input stages of an amplifier to the high-powered "push-pull" output stage.

External links

References

  • Daniels, A.R. (1985). Introduction to Electrical Machines, Macmillan. ISBN 0-333-19627-9.
  • Heathcote, MJ (1998). J&P Transformer Book, 12th ed., Newnes. ISBN 0-7506-1158-8.
  • Hindmarsh, J. (1984). Electrical Machines and their Applications, 4th ed., Pergamon. ISBN 0-08-030572-5.
  • Shepherd,J; Moreton,A.H; Spence,L.F. (1970). Higher Electrical Engineering, Pitman Publishing. ISBN 0-273-40025-8.
Retrieved and changed from: http://en.wikipedia.org/wiki/Transformer
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