Voltage regulation expression for inductive load

Voltage regulation is the change in secondary terminal voltage from no load to full load (at the same primary voltage and a given power factor), expressed as a fraction of the no-load voltage:

$$\text{VR} = \frac{E_2 - V_2}{E_2}\times100\%$$

where $E_2$ is the no-load secondary voltage and $V_2$ the full-load terminal voltage.

Derivation (lagging power factor)

Refer all quantities to the secondary side: $R_{02} = R_2 + R_1/a^2$ and $X_{02} = X_2 + X_1/a^2$. With the exciting branch ignored, the circuit is a source $E_2$ in series with $R_{02} + jX_{02}$, feeding the load current $I_2$ at angle $\phi$ lagging $V_2$.

 Phasor diagram (inductive load, V2 as reference)

                  E2
                 /|
         I2X02  / |
               /  |
          V2  /___|
   O--------->  I2R02
    \ phi
     \ I2 (lags V2)

Using $V_2$ as reference: $V_2 = V_2\angle0°$, $I_2 = I_2\angle-\phi$.

$$\begin{aligned} \bar E_2 &= \bar V_2 + \bar I_2(R_{02} + jX_{02}) \\ &= (V_2 + I_2R_{02}\cos\phi + I_2X_{02}\sin\phi) + j(I_2X_{02}\cos\phi - I_2R_{02}\sin\phi) \end{aligned}$$

The quadrature term is small, so

$$E_2 \approx V_2 + I_2R_{02}\cos\phi + I_2X_{02}\sin\phi$$ $$E_2 - V_2 = I_2(R_{02}\cos\phi + X_{02}\sin\phi)$$

Hence

$$\boxed{\text{VR} = \frac{I_2R_{02}\cos\phi + I_2X_{02}\sin\phi}{E_2}\times100\%}$$

For the exact value: $E_2 = \sqrt{(V_2 + I_2R_{02}\cos\phi + I_2X_{02}\sin\phi)^2 + (I_2X_{02}\cos\phi - I_2R_{02}\sin\phi)^2}$.

For a leading load, the sign of the $X_{02}\sin\phi$ term changes (VR $= I_2(R_{02}\cos\phi - X_{02}\sin\phi)/E_2$), and the regulation can become negative. The maximum regulation occurs when $\cos\phi = R_{02}/Z_{02}$.