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Phasor Analysis: Factoring a Complex Sinusoid into Phasor Times Carrier

The heart of the preceding proof was the algebraic manipulation

$\displaystyle \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

$\displaystyle A\cos(\omega t + \phi) =$   re$\displaystyle \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].
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