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Phasor Notation

The complex amplitude $ {\cal A}\isdef A e^{j\phi}$ is also defined as the phasor associated with any sinusoid having amplitude $ A$ and phase $ \phi$ . The term ``phasor'' is more general than ``complex amplitude'', however, because it also applies to the corresponding real sinusoid given by the real part of Equations (1.9-1.10). In other words, the real sinusoids $ A\cos(\omega t+\phi)$ and $ A\cos(\omega nT+\phi)$ may be expressed as

\begin{eqnarray*}
A\cos(\omega t+\phi) &\isdef & \mbox{re}\left\{A e^{j(\omega t+\phi)}\right\} = \mbox{re}\left\{{\cal A}e^{j\omega t}\right\}\\
A\cos(\omega nT+\phi) &\isdef & \mbox{re}\left\{A e^{j(\omega nT+\phi)}\right\} = \mbox{re}\left\{{\cal A}e^{j\omega nT}\right\}\\
\end{eqnarray*}

and $ {\cal A}$ is the associated phasor in each case. Thus, we say that the phasor representation of $ A\cos(\omega t+\phi)$ is $ {\cal A}\isdef A e^{j\phi}$ . Phasor analysis is often used to analyze linear time-invariant systems such as analog electrical circuits.


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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA