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 ]

voltage-regulation5

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