Phasor Analysis: Factoring a Complex Sinusoid into Phasor Times Carrier

The heart of the preceding proof was the algebraic manipulation

$$\sum_{i=1}^N A_i e^{j(\omega t + \phi_i)} = e^{j\omega t} \sum_{i=1}^N A_i e^{j\phi_i}.$$

The carrier term $e^{j\omega t}$ ``factors out'' of the sum. Inside the sum, each sinusoid is represented by a complex constant $A_i e^{j\phi_i}$ , known as the phasor associated with that sinusoid.

For an arbitrary sinusoid having amplitude $A$ , phase $\phi$ , and radian frequency $\omega$ , we have

$$A\cos(\omega t + \phi) =$$   re$$\left\{(A e^{j\phi}) e^{j\omega t}\right\}.$$

Thus, a sinusoid is determined by its frequency $\omega$ (which specifies the carrier term) and its phasor ${\cal A}\isdef A e^{j\phi}$ , a complex constant. Phasor analysis is discussed further in [84].