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.