Voltage regulation of transformer for lagging load

Voltage regulation

Voltage regulation of a transformer is the change in secondary terminal voltage from no load to full load, expressed as a fraction (or percentage) of the no-load voltage, with the primary voltage held constant:

$$\%\text{Reg} = \frac{E_2 - V_2}{E_2} \times 100 = \frac{{}_0V_2 - V_2}{{}_0V_2}\times 100$$

where $E_2 = {}_0V_2$ is the no-load secondary voltage and $V_2$ is the full-load secondary voltage. A small regulation is desirable (power transformers: about 2-6%).

Expression for lagging load

Use the equivalent circuit referred to the secondary: $E_2 = V_2 + I_2(R_{02} + jX_{02})$. Take $V_2$ as reference; $I_2$ lags $V_2$ by $\phi$.

Phasor diagram (approximate):

                                   C
                                  /
                                 /  I2.X02
                                /   (perp. to I2)
  O -------------------- A     /
          V2              \   /
                           \ /
                    I2.R02  B
              (parallel to I2)

  E2 = OC (join O to C);  I2 lags V2 by phi

Construction: from O draw $V_2$ = OA. Draw $I_2R_{02}$ = AB parallel to $I_2$, and $I_2X_{02}$ = BC perpendicular to $I_2$ (leading it by 90°). Then OC = $E_2$. Drop a perpendicular from C to the extended OA line, meeting it at D.

$$\begin{aligned} OD &= V_2 + I_2R_{02}\cos\phi + I_2X_{02}\sin\phi \\ CD &= I_2X_{02}\cos\phi - I_2R_{02}\sin\phi \\ E_2 &= OC = \sqrt{OD^2 + CD^2} \end{aligned}$$

Because the angle between $E_2$ and $V_2$ is very small, $OC \approx OD$:

$$E_2 - V_2 \approx I_2R_{02}\cos\phi + I_2X_{02}\sin\phi$$

Therefore, for a lagging power factor:

$$\%\text{Reg} = \frac{I_2R_{02}\cos\phi + I_2X_{02}\sin\phi}{E_2}\times 100$$

or, in per-unit form,

$$\%\text{Reg} = \%R\cos\phi + \%X\sin\phi$$

where $\%R = \frac{I_2R_{02}}{E_2}\times100$ and $\%X = \frac{I_2X_{02}}{E_2}\times100$.

A more exact expression keeps the CD term:

$$\%\text{Reg} = \%R\cos\phi + \%X\sin\phi + \frac{(\%X\cos\phi - \%R\sin\phi)^2}{200}$$

Notes

• For lagging loads regulation is always positive ($V_2 < E_2$).
• For leading loads the sign of the $\sin\phi$ term becomes negative, so regulation can be zero or negative.
• Regulation is zero when $\tan\phi = -R_{02}/X_{02}$ (a leading pf), and maximum when $\tan\phi = X_{02}/R_{02}$ (a lagging pf).