Derivation of Voltage Drop Formula

A short cable voltage drop in AC systems is provided by the formula

V_d = IR cos phi + IX sin phi

where:
V_d = Voltage drop per phase, volts
phi = Load power factor angle
IR cos phi = Voltage drop component on the cable resistance
IX sin phi = Voltage drop component on the cable reactance

You might be wondering where this formula came from. Actually, this formula is just an approximation. We shall going through the process of deriving this formula to better understand it.

Considering the following vector diagram:

Voltage - Current Phasor Diagram
Figure 2 Voltage – Current Phasor Diagram

where:
V_s = Sending end voltage per phase
V_r = Receiving end voltage per phase
I = Load current
R = Cable resistance
X = Cable reactance
phi = Load power factor angle
AB = IR cos phi
BE = CF = IR cos phi
BC = EF = IX sin phi
DF = IX cos phi
AC = AB + BC = AB + EF
AC = IR cos phi + IX sin phi
DC = DF - CF = DF - BE
DC = IX cos phi - IR sin phi

V_s = OD = sqrt{(OA + AB + BC)^2 + (DF - BE)^2}

Unless the cable is very long, the imaginary axis component of the voltage is very small compared to the real axis component.

(OA + AB + BC) gt gt (DF - BE)

thus the sending end voltage will be

V_s = (OA + AB + BC)
V_s = V_r + IR cos phi + IX sin phi,~{volts per phase}

From the above formula, the voltage drop on a per phase basis will be

V_d = V_sgt - V_r
V_d = IR cos phi + IX sin phi, ~volts per phase per unit length

That is how the voltage drop formula was derived.

Normally, voltage drop is expressed as a percentage of the sending end line-to-line voltage. The formula will be

%V_d={{sqrt{3}*I*(R cos phi + X sin phi)}/V}%

where:
V = sending end line-to-line voltage

R = r*l
X = x*l
r = the unit resistance in Omega / km
x = the unit reactance in Omega / km
l = cable length in km

Cable data will be taken from published Vendor data.