Phasors

Phasors are a powerful mathematical tool used to analyze alternating current (AC) circuits. They provide a complex number representation of sinusoids, with the magnitude of the phasor equating to the amplitude of the sinusoid and the angle of the phasor representing the phase measured from the positive x-axis.

One of the significant benefits of using phasors is that they simplify the analysis of AC circuits by eliminating the time dependence of the current and voltage. This transformation allows an AC circuit to be analyzed as if it were its equivalent direct current (DC) form, making calculations more straightforward.

Phasors can be represented in different forms - rectangular, polar, or exponential - by using Euler's identity, a fundamental formula in complex analysis that establishes a deep relationship between trigonometric and exponential functions.

To obtain the phasor of a sinusoid in sine form, one must first convert it into cosine form and then express it as the real part of a complex number. The phasor of this sinusoid equals the time-independent part of this complex number. Conversely, the sinusoid of a given phasor can be obtained by multiplying the phasor with a time factor and taking its real part.

In a graphical context, phasors can be visualized as rotating vectors, or 'sinors,' spinning in a counterclockwise direction on a complex plane with a constant angular frequency. The diagrams that depict these rotating sinors are known as phasor diagrams.

A key concept in understanding the phasor diagram is that the projection of the rotating sinors onto the real axis represents the sinusoids. This means that the horizontal position of the sinor at any point in time corresponds to the instantaneous value of the sinusoidal function it represents.