Electrical Machines · Transformers

Transformer Voltage Regulation and Equivalent Circuit

Dr. Mithun Mondal BITS Pilani, Hyderabad Campus Electrical Machines

Demonstrative Video

\(\bullet\) Transformer with resistance and leakage reactance

resistance reactance
Resistance Reactance
\[\begin{aligned} Z_{1} & =\sqrt{\left(R_{1}^{2}+X_{1}^{2}\right)}\\ Z_{2} & =\sqrt{\left(R_{2}^{2}+X_{2}^{2}\right)}\\ V_{1} & =E_{1}+I_{1}\left(R_{1}+jX_{1}\right)=E_{1}+I_{1}Z_{1}\\ E_{2} & =V_{2}+I_{2}\left(R_{2}+jX_{2}\right)=V_{2}+I_{2}Z_{2} \end{aligned}\]
\[\begin{array}{ccc} X_{2}^{'}=X_{2}/K^{2} & \mbox{and} & X_{1}^{'}=K^{2}X_{1}\\ X_{01}=X_{1}+X_{2}^{'} & \mbox{and} & X_{02}=X_{2}+X_{1}^{'} \end{array}\]
Leakage reactance can also be transferred from one winding to the other in the same way as resistance
total impedance
Total Impedance
\[\begin{aligned} Z_{01} & =\sqrt{\left(R_{01}^{2}+X_{01}^{2}\right)}\\ Z_{02} & =\sqrt{\left(R_{02}^{2}+X_{02}^{2}\right)} \end{aligned}\]
phasor impedance
Phasor Impedance
Voltage Drop in a Transformer
\[\begin{aligned} V_{1} & \thickapprox E_{1}\\ E_{2} & =KE_{1}=KV_{1}\\ E_{2} & =_{0}V_{2} \end{aligned}\]
At no load:
approx drop
Approx Drop
\[I_{2}R_{02}\cos\Phi\pm I_{2}X_{02}\sin\Phi\]
\(-\)\(+\)The approximate voltage drop is:
\[I_{1}R_{01}\cos\Phi\pm I_{1}X_{01}\sin\Phi\]
\[\begin{aligned} v_{r} & =\dfrac{I_{2}R_{02}}{_{0}V_{2}}\times100=\dfrac{I_{1}R_{01}}{V_{1}}\times100=\mbox{percentage resistive drop}\\ v_{x} & =\dfrac{I_{2}X_{02}}{_{0}V_{2}}\times100=\dfrac{I_{1}X_{01}}{V_{1}}\times100=\mbox{percentage reactive drop} \end{aligned}\]
\[\begin{aligned} = & \dfrac{I_{2}R_{02}cos\Phi\pm I_{2}X_{02}sin\Phi}{_{0}V_{2}}\times100\\ = & v_{r}cos\Phi\pm v_{x}sin\Phi \end{aligned}\]
Similarly, approximate voltage drop referred to primary is
Equivalent Circuit of a Transformer

Equivalent circuit is basically a diagram in which the resistance and leakage reactance of the transformer are imagined to be external to the winding

The equivalent circuit diagram of transformer is given below:-

equivalent circuit
Equivalent Circuit

The secondary circuit and its equivalent primary value

secondary equivalent
Secondary Equivalent

The total equivalent circuit is obtained by adding in the primary impedance

total equivalent
Total Equivalent

It can be simplified

simplified
Simplified

At last, the circuit is simplified by omitting \(I_0\) altogether

final equivalent
Final Equivalent
\[\begin{aligned} Z & =Z_{1}+Z_{m}||\left(Z_{2}^{'}+Z_{L}^{'}\right)\\ & =Z_{1}+\dfrac{Z_{m}\left(Z_{2}^{'}+Z_{L}^{'}\right)}{Z_{m}+\left(Z_{2}^{'}+Z_{L}^{'}\right)} \end{aligned}\]
\[V_{1}=I_{1}\left[Z_{1}+\dfrac{Z_{m}\left(Z_{2}^{'}+Z_{L}^{'}\right)}{Z_{m}+\left(Z_{2}^{'}+Z_{L}^{'}\right)}\right]\]
The total impedance between the input terminal: