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# The Equivalent Circuit of a Transformer

Author: E.E. Kimberly

The leakage impedances of a transformer cause a voltage loss between line and load exactly the same as if they were not in the transformer but in the lines to or from the transformer. The exciting current Ie remains substantially the same for all loadings. If a circuit be made up of two branches in parallel, as in Fig. 17-2 (a), the voltage at the load and the current in the supply line will be the same as actually would occur if a transformer with a 1:1 ratio and the same values of X and R were used. Such a circuit is called the equivalent circuit of the transformer because its effect between the supply line and the load would be the same.

If the turns ratio is not 1:1, a modification in the circuit elements must be made to suit the turns ratio. For any definite secondary load, the primary current will be the same regardless of the turns ratio because N2I2 merely produces a magnetomotive force mmF2 which must be balanced by the force mmF1 at the available supply voltage. If the transformer had a turns ratio of 4:1 and both windings had the same size of wire, their resistances would be in the same ratio and R1 would equal 4R2. However, if the volt-ampere input and the volt-ampere output are to be equal, then I2 = 4I1 and the current-carrying capacity of the secondary coil conductor must be four times that of the primary coil conductor. As a result, the secondary conductor must have four times the cross-sectional area and, therefore, one-fourth of the resistance per turn. Hence,

Since

it follows that

 Fig. 17-2. Transformer Equivalent Circuits

Regardless of the value of the ratio

the secondary resistance is equal to the primary resistance divided by the square of the turns ratio, provided the mean length of turn is the same in the two windings and the number of circular mils per ampere is also the same. If R2 is to have the same effect on the line current with only I'1 flowing in it, then its magnitude must be modified by

in the equivalent circuit. Similar reasoning may be applied to X2 and its magnitude in the equivalent circuit must be

The secondary constants of the transformer as used in the equivalent circuit are called "reflected" constants. For use in the equivalent circuit the load constants RL and XL must be modified in the same way by multiplying them by

An equivalent circuit is shown in Fig. 17-2 (b) with reflected constants. The equivalent circuit is useful for calculating the voltage change on the secondary of a transformer when its load is varied or for calculating the voltage necessary at the primary terminals to provide rated voltage at the secondary terminals when the secondary is loaded.

Example 17-1. - A 5-kv-a transformer with a 4:1 turns ratio is to be used to supply 110 volts to a resistive load which draws full rated current from the transformer. The transformer constants are: R1 = 2.75 ohms, R2 = 0.172 ohm, X1 =4.0 ohms, and X2 = 0.25 ohm; and the load resistance is RL = 2.5 ohms. Calculate the necessary primary applied voltage.

Solution. - The rated currents are:

Primary........ .5000 /440 = 11.38 amp

Secondary....... 5000 / 110 = 45.45 amp

Also,

Then,

and

Last Update: 2010-10-06