Engineering

The instantaneous values of two alternating voltages are represented

respectively by v1 = 60 sinθ volts and v2 = 40 sin(θ − π/3) volts. Derive

an expression for the instantaneous value of:

(a) the sum;

(b) the difference of these voltages.

(a) It is usual to draw the phasors in the position corresponding to θ = 0,*

i.e. OA in Fig. 9.20 is drawn to scale along the x-axis to represent 60 V, and

OB is drawn π/3 radians or 60° behind OA to represent 40 V. The diagonal

OC of the parallelogram drawn on OA and OB represents the phasor sum of

OA and OB. By measurement, OC = 87 V and angle φ between OC and the

x-axis is 23.5°, namely 0.41 rad; hence:

Instantaneous sum of the two voltages = 87 sin(θ − 23.5°) V

Alternatively, this expression can be found thus:

Horizontal component of OA = 60 V

Horizontal component of OB = OD = 40 cos 60° = 20 V