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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
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