Learning goal: to understand the concepts explaining the operation of transformers.

one of the advantages of alternating current (ac) over direct current (dc) is the ease with which voltage levels can be increased or decreased. such a need is always present due to the practical requirements of energy distribution. on the one hand, the voltage supplied to the end users must be reasonably low for safety reasons (depending on the country, that voltage may be 110 volts, 220 volts, or some other value of that order). on the other hand, the voltage used in transmitting electric energy must be as high as possible to minimize losses in the transmission lines. a device that uses the principle of electromagnetic induction to increase or decrease the voltage by a certain factor is called a transformer.
the main components of a transformer are two coils (windings) that are electrically insulated from each other. the coils are wrapped around the same core, which is typically made of a material with a very large relative permeability to ensure maximum mutual inductance. one coil, called the primary coil, is connected to a voltage source; the other, the secondary coil, delivers the power. the alternating current in the primary coil induces the changing magnetic flux in the core that creates the emf in the secondary coil. the magnitude of the emf induced in the secondary coil can be controlled by the design of the transformer. the key factor is the number of turns in each coil.

consider an ideal transformer, that is, one in which the coils have no ohmic resistance and the magnetic flux is the same for each turn of both the primary and secondary coils. if the number of turns in the primary coil is and that in the secondary coil is , then the emfs induced in the coils can be written as
,

and therefore,
.

since both emfs oscillate with the same frequency as the ac source, the formula above can be applied to the instantaneous amplitude or the rms values of the emfs. moreover, if the coils have zero resistance (as we assumed), then for each coil the terminal voltage will be equal to the induced emf. therefore, we can write
.

note that if , then . this is a case of a step-up transformer. conversely, if , then . this is a case of a step-down transformer. without energy losses, the power in the primary and secondary coils is the same:
.

if the secondary circuit is completed by a resistance , then . combining this with the two equations above gives

dividing the first and last expressions by and then inverting gives

in other words, the current in the primary coil is the same as if it were connected directly to a resistance equal to . in a way, transformers "transform" resistances as well as voltages and currents. in reality, no transformer is ideal. there are always some energy losses. however, modern transformers have very high efficiencies, usually well exceeding 90%.

in answering the questions below, consider the transformer ideal unless otherwise noted.
a. the primary coil of a transformer contains 100 turns; the secondary has 200 turns. the primary coil is connected to a size aa battery that supplies a constant voltage of 1.5 volts. what voltage would be measured across the secondary coil?
zero

b. a transformer is intended to decrease the rms value of the alternating voltage from 500 volts to 25 volts. the primary coil contains 200 turns. find the necessary number of turns in the secondary coil.
=