Transformer magnetic circuits with phasor diagram

Transformer with Winding Resistance but No Magnetic Leakage
An ideal transformer was supposed to possess no resistance, but in an actual transformer, there
is always present some resistance of the primary and secondary windings. Due to this resistance,
there is some voltage drop in the two windings. The result is that :
(i) The secondary terminal voltage V2 is vectorially less than the secondary induced e.m.f. E2
by an amount I2 R2 where R2 is the resistance of the secondary winding. Hence, V2 is equal to the
vector difference of E2 and resistive voltage drop I2 R2.
? V2 = E2 ? I2 R2 ...vector difference
(ii) Similarly, primary induced e.m.f. E1 is equal to the vector difference of V1 and I1 R1 where R1
is the resistance of the primary winding.
E1 = V1 ? I1 R1

transformer is shown whose primary and secondary windings have resistances of
R1 and R2 respectively. The resistances have been shown external to the windings.
Transformer 1133
It would now be shown that the resistances of the
two windings can be transferred to any one of the two
windings. The advantage of concentrating both the
resistances in one winding is that it makes calculations
very simple and easy because one has then to work in
one winding only. It will be proved that a resistance of
R2 in secondary is equivalent to R2/K2 in primary. The
value R2/K2 will be denoted by R2?? the equivalent
secondary resistance as referred to primary.
The copper loss in secondary is I2
2 R2. This loss is supplied by primary which takes a current of I1.
Hence if R2? is the equivalent resistance in primary which would have caused the same loss as R2 in
secondary, then
I1
2 R2? = I2
2R2 or R2? = (I2/I1)2R2
Now, if we neglect no-load current I0, then I2/I1 = I/K*. Hence, R2? = R2/K2
Similarly, equivalent primary resistance as referred to secondary is R1? = K2R1
In Fig. 32.24, secondary resistance has been transferred to primary side leaving secondary circuit
resistanceless. The resistance R1 + R2? = R1 + R2/K2 is known as the equivalent or effective resistance
of the transformer as referred to primary and may be designated as R01.
? R01 = R1 + R2? = R1 + R2 / K2
Similarly, the equivalent resistance of the transformer as referred to secondary is
R02 = R2 + R1? = R2 + K2 R1.
This fact is shown in Fig. 32.25 where all the resistances of the transformer has been concentrated
in the secondary winding.
R

1. a resistance of R1 in primary is equivalent to K2R1 in secondary. Hence, it is called equivalent
resistance as referred to secondary i.e. R1.
2. a resistance of R2 in secondary is equivalent to R2/K2 in primary. Hence, it is called the
equivalent secondary resistance as referred to primary i.e. R2?.
3. Total or effective resistance of the transformer as referred to primary is
R01 = primary resistance + equivalent secondary resistance as referred to primary
= R1 + R2? = R1 + R2/K2
4. Similarly, total transformer resistance as referred to secondary is,
R02 = secondary resistance + equivalent primary resistance as referred to secondary
= R2 + R1? = R2 + K2R1