KindiakMath

The Phasor Method

Define the phasor of the sine wave r \sin(t + \theta) by r\angle \theta \equiv re^{i\theta}.

Problem 1. Prove that r \angle \theta \cdot s \angle \omega = (rs) \angle (\theta + \omega).

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Solution. Using the laws of exponents,

\displaystyle r \angle \theta \cdot s \angle \omega = re^{i \theta} \cdot se^{i \omega} = (rs) \cdot e^{i(\theta + \omega)} = (rs) \angle (\theta + \omega).

Problem 2. Prove that r \sin \theta = \mathrm{Im}(r \angle \theta).

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Solution. By Euler’s formula,

r\angle \theta = re^{i \theta} = r (\cos \theta + i \cdot \sin \theta) = (r \cos\theta) + i \cdot (r \sin \theta).

Hence, \mathrm{Re}(r\angle \theta) = r \cos \theta and \mathrm{Im}(r\angle \theta) = r \sin \theta.

Problem 3. Suppose r_1 \angle \theta_1 + r_2 \angle \theta_2 = r_3 \angle \theta_3. Prove that for any t \in \mathbb R,

r_1 \sin(t + \theta_1) + r_2 \sin(t + \theta_2) = r_3 \sin(t + \theta_3).

This result is called the phasor method.

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Solution. By Problems 1 and 2,

\begin{aligned} r_1 \sin(t + \theta_1) + r_2 \sin(t + \theta_2) &= \mathrm{Im}(r_1 \angle (t + \theta_1)) + \mathrm{Im}(r_2 \angle (t + \theta_2)) \\ &= \mathrm{Im}(r_1 \angle (t + \theta_1) + r_2 \angle (t + \theta_2)) \\ &= \mathrm{Im}(r_1 \angle \theta_1 \cdot 1 \angle t + r_2 \angle \theta_2 \cdot 1 \angle t) \\ &= \mathrm{Im}( ( r_1 \angle \theta_1 + r_2 \angle \theta_2 ) \cdot 1 \angle t) \\ &= \mathrm{Im}( r_3 \angle \theta_3 \cdot 1 \angle t) \\ &= \mathrm{Im}( r_3 \angle (t + \theta_3) ) = r_3 \sin(t + \theta_3). \end{aligned}

In various engineering applications, the phasor method simplifies computations that require the addition of sine or cosine waves.

—Joel Kindiak, 23 Aug 25, 2209H

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