# Condition for Zero and Maximum Voltage Regulation

Voltage regulation is given by this approximation

$\displaystyle V.R. = \biggl( \frac {I_R R cos(\phi_R) \pm I_R X sin (\phi_R)}{V_R} \biggr )$

• For Zero voltage regulation:  Zero voltage regulation means, Sending end voltage and Receiving end voltage become equal. This case is also known as ideal voltage regulation.

$\displaystyle \Rightarrow V.R. = 0$

$\displaystyle I_R R cos(\phi_R) + I_R X sin(\phi_R) = 0$

$\displaystyle R cos(\phi_R) = -X sin(\phi_R)$

$\displaystyle tan(\phi_R) = - \frac{R}{X}$

$\displaystyle \phi_R = - tan^{-1} \biggl (\frac{R}{X} \biggr )$

$\displaystyle \phi_R = cot^{-1} \biggl ( \frac{X}{R} \biggr )$

$\displaystyle cot^{-1} \biggl (\frac{X}{R} \biggr ) + tan^{-1} \biggl (\frac{X}{R} \biggr ) = \frac{\pi}{2}$

$\displaystyle cot^{-1} \biggl (\frac{X}{R} \biggr ) = \frac{\pi}{2} - tan^{-1} \biggl (\frac{X}{R} \biggr )$

$\displaystyle \phi_R = \frac{\pi}{2} - tan^{-1} \biggl (\frac{X}{R} \biggr )$

$\displaystyle tan^{-1} \biggl (\frac{X}{R} \biggr ) \in \biggl (-\frac{\pi}{2}, \frac{\pi}{2} \biggr )$

NOTE : Here Reference phasor is $I_R$

$\displaystyle \phi_R \in (-\pi , 0) \Rightarrow$  angle between $I_R$ and $V_R$ is negative, means leading power factor ( $I_R$ is leading the voltage $V_R$

• For maximum voltage regulation :

Condition for maximum V.R.

$\displaystyle \Rightarrow \frac{d V.R.}{d\phi_R} = 0$

$\displaystyle \frac{dV.R.}{d \phi_R} = -R sin(\phi_R) + X cos(\phi_R) = 0$

$\displaystyle tan(\phi_R) = \frac{X}{R}$

$\displaystyle \Rightarrow \phi_R = tan^{-1} \biggl ( \frac{X}{R} \biggr )$

$\displaystyle \phi \in \biggl (0, \frac{\pi}{2} \biggr )$  [Because Both X and R are Positive]

$\displaystyle \phi_R > 0$ [ Lagging power factor ≫ Current is lagging to Voltage ]