The condition for maximum regulation of a single-phase transformer with equivalent resistance, R_eq, equivalent reactance, X_eq, and load power factor angle, ϕ, is

Voltage regulation is the change in secondary terminal voltage from no load to full load at a specific power factor of load and the change is expressed in percentage.

E2 = no-load secondary voltage

V2 = full load secondary voltage

Voltage regulation for the transformer is given by the ratio of change in secondary terminal voltage from no load to full load to no load secondary voltage.

Voltage regulation \(= \frac{{{E_2} - {V_2}}}{{{E_2}}}\)

It can also be expressed as,

Regulation \(= \frac{{{I_2}{R_{02}}\cos {\phi _2} \pm {I_2}{X_{02}}\sin {\phi _2}}}{{{E_2}}}\)

Regulation = %R cos ϕ ± %X sin ϕ

+ sign is used for lagging loads

- ve sign is used for leading loads

Where %R = ohmic drop

%X = reactance drop

Condition for maximum regulation:

Regulation of transformer is maximum only at lagging pf loads.

%Reg = (%R) cos ϕ2 + (%X) sin ϕ2

\(\frac{d}{{d{\phi _2}}}\left( {V.R.} \right) = 0\)

\( \Rightarrow \tan {\phi _2} = \frac{{\% X}}{{\% R}}= \frac{{X_{eq}}}{{ R_{eq}}}\)

\( \Rightarrow {\phi _2} = {\tan ^{ - 1}}\left( {\frac{{\% X}}{{\% R}}} \right)\)

\( = {\tan ^{ - 1}}\left( {\frac{{{I_2}{X_{02}}/V_1^1}}{{{I_2}{R_{02}}/V_1^1}}} \right) = {\tan ^{ - 1}}\left( {\frac{{{X_{02}}}}{{{R_{02}}}}} \right)\)

For load impedance, \({\phi _2} = {\tan ^{ - 1}}\left( {\frac{{{X_L}}}{{{R_L}}}} \right)\)

At maximum regulation, \({\phi _2} = {\tan ^{ - 1}}\left( {\frac{{{X_{02}}}}{{{R_{02}}}}} \right) = {\tan ^{ - 1}}\left( {\frac{{{X_L}}}{{{R_L}}}} \right)\)

\( \Rightarrow \frac{{{X_L}}}{{{R_L}}} = \frac{{{X_{02}}}}{{{R_{02}}}}\)

ϕ2 is the load phase angle of transformer

θ is the internal impedance angle of the transformer

Power factor at maximum regulation \(\cos {\phi _2} = \frac{{\% R}}{{\% Z}}\)