Voltage regulation is given by this **approximation**

V.R. = [ I_{R }R Cosφ_{R }± I_{R} X Sinφ_{R }_{ }] / V_{R}

- 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.

When V.R. = 0

I_{R }R Cosφ_{R }+ I_{R} X Sinφ_{R }= 0

R Cosφ_{R }= – X Sinφ_{R}

tan φ_{R }= – ( R/X )

φ_{R }= – tan^{-1}( R/X )

φ_{R }= cot^{-1}( X/R )

**cot ^{-1}( X/R ) + tan^{-1}( X/R ) = Π / 2**

cot^{-1}( X/R ) = Π / 2 – tan^{-1}( X/R )

**φ _{R }= Π / 2 – tan^{-1}( X/R )**

**tan ^{-1}( X/R ) ∈ (-Π/2 , +Π/2 )**

*NOTE : Here Reference phasor is I _{R}*

φ_{R}∈ ( -Π , 0 ) , 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. ≫ ∂(V.R.) ∕ ∂φ_{R }= 0

∂(V.R.) ∕ ∂φ_{R }= [-R Sinφ_{R }+ X Cosφ_{R}] = 0

tan φ_{R }= X / R

φ_{R }=tan^{-1}( X / R )

**φ _{R }∈ ( 0, Π/2 ) [Because Both X and R are Positive]**

φ_{R }> 0 [ Lagging power factor ≫ Current is lagging to Voltage ]