Transformers

For many practical purposes, it is necessary to increase or decrease the magnitude of an alternating current (or voltage). A transformer is an electrical device for converting low voltage to high voltage or vice versa by using the principle of mutual induction. A wide range of transformer designs are encountered in electronic and electric power applications.

Above we see what happens when too much energy is transferred through induction, the systems heats up, raising the resistance of the wires, and eventually the material can no longer support the requisite current, causing the transformer to blow.

A transformer consists of two coils, which are insulated from each other, have an unequal number of turns (at times there may be equal number of turns), and are wound on a soft-iron core. The coil to which energy is supplied is called the primary coil and the coil from which energy is drawn is called secondary coil. The numbers of turns in each are denoted by \(N_p\) and \(N_s,\) respectively.

The design of a simple transformer is shown in the following figure:

Schematic Diagram of Transformer

\[\] Picture of a Transformer

A transformer works on the principle of mutual induction. When an alternating voltage is applied to the primary coil, varying current creates a varying magnetic flux and thus a varying magnetic field impinging on the secondary coil. According to Faraday's law, the varying magnetic field induces a varying EMF in secondary coil. For an ideal transformer having infinitely high magnetic permeability, it may be assumed that all of the magnetic flux passes through both primary and secondary coils.

If \(\epsilon_p\) and \(\epsilon_s\) are the instantaneous values of the EMF induced in the primary and secondary coils, respectively, and \(\phi\) is the flux linked through each turn of either coil, then the EMF induced in the secondary coil with \(N_s\) turns is given by

\[\epsilon_{s}=-N_{s}\dfrac{d\phi }{dt}\]

and the EMF induced in the primary coil with \(N_p\) turns is given by

\[\epsilon_{p}=-N_{p}\dfrac{d\phi }{dt}.\]

Dividing the two relations, we find the following equation for mutual induction:

\[\dfrac{\epsilon_s}{\epsilon_p}=\dfrac{N_s}{N_p}.\]

Let \(i_p\) and \(i_s\) be the current at any instant in the primary and secondary coils, respectively. Since ideal transformer behavior is assumed, the power input must be equal to the power output (perfect efficiency):

\[\begin{align} (\text{Power Input})&=(\text{Power Output})\\ \epsilon_p i_p&=\epsilon_s i_s\\ \dfrac{\epsilon_s}{\epsilon_p}&=\dfrac{i_p}{i_s}. \end{align}\]

From the relations for mutual induction and power, we find

\[\dfrac{\epsilon_{s}}{\epsilon_{p}}=\dfrac{N_{s}}{N_{p}}=\dfrac{i_{p}}{i_{s}}\implies i_{s}=i_p\left [\dfrac{N_{p}}{N_{s}}\right ]=\dfrac{i_{p}}{K}.\]

and

\[\epsilon_{s}=\epsilon_{p}\left [\dfrac{N_{s}}{N_{p}}\right ]=\epsilon_{p} K,\]

where \(K\) is the transformer ratio \(\dfrac{N_{s}}{N_{p}}.\)

An old radio set operates at 6 V DC. A transformer with 18 turns in the secondary coil is used to step down the input 220 V AC emf to 6 V AC emf. This AC emf is then rectified by another circuit to give 6 V DC, which is fed to the radio. Find the number of turns in the primary coil.

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We have

\[\dfrac{\epsilon_{s}}{\epsilon_{p}}=\dfrac{N_{s}}{N_{p}},\]

or \[N_{p}=\dfrac{\epsilon_{p}}{\epsilon_{s}}\times N_{s}=\dfrac{220}{6}\times 18=660.\]

Therefore, our final answer is \(N_p = 660\). \(_\square\)

A transformer has 50 turns in the primary coil and 100 turns in the secondary coil. If the primary coil is connected to 220 V DC supply, what will be the voltage across the secondary coil?

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Transformers are used in AC supplies only. Since the current is unchanging, there will not be any flux and thus no mutual induction. The voltage across secondary coil will be 0 V. \(_\square\)

Step Up Transformer:

• A transformer which increases the voltage is called step up transformer.
• \(K>1\) or \(N_{s} > N_{p}\).
• Since the secondary current is less than the primary current, the primary coil is made up of thick wire to sustain high current.

Step Down Transformer:

• A transformer which decreases the voltage is called step down transformer.
• \(K<1\) or \(N_{s} < N_{p}\).
• As the secondary current is higher, the secondary coil is made up of thick wire to support the high current.

The efficiency of a transformer is defined as the ratio of output power to input power. It is denoted by Greek letter \(\eta\):

\[\eta=\dfrac{\text{Output Power}}{\text{Input power}}=\dfrac{\epsilon_{s} i_{s}}{\epsilon_{p} i_{p}}.\]

For an ideal transformer, \(\eta=1\). However, due to losses, realistic efficiencies are less than 1, although efficiencies of order 0.99 can be achieved easily.

Flux Leakage:

There is always some flux leakage, i.e., not all of the flux from the primary coil passes through the secondary coil either due to poor coupling or air gaps in the core. It can be reduced by winding coils one over the other.

Resistance of the Windings:

The wire used for the windings has some internal resistance, so energy is lost due to heat produced in the wire \((I^2R)\). In high current/low voltage windings, these losses are minimized by using thick wire.

Eddy Currents:

The alternating magnetic flux induces turbulent eddy currents in the iron core and causes heating. The effect is reduced by having a laminated core.

Hysteresis:

The magnetization of the core is repeatedly reversed by the alternating magnetic field. The resulting expenditure of energy in the core appears as heat and is kept to a minimum by using a magnetic material which has a low hysteresis loss.

Since the invention of the first constant potential transformer in 1885, transformers have become essential for the AC transmission, distribution and utilization of electrical energy. Some of many applications of transformer are listed as follows:

• in radio, telephones, loudspeakers
• in voltage regulators for TV, refrigerators, air conditioners, computers, etc.
• in stabilized power supplies
• for obtaining large current for electrical welding
• for production of X-rays
• for transmission of electricity.