What is Complex amplitude? — Physics

Definition 8.2 University Physics — Year 1 · Chapter 8 — Sinusoidal Steady State and Impedance

To the sinusoid $x(t) = X\cos(\omega t + \varphi)$ one associates the complex signal $\underline{x}(t) = X\eu^{j(\omega t + \varphi)}$ , so that $x = \Rea(\underline{x})$ , and the complex amplitude

\underline{X} = X\eu^{j\varphi}, \qquad \underline{x}(t) = \underline{X}\,\eu^{j\omega t},

which carries the amplitude $X = \abs{\underline{X}}$ and the phase $\varphi = \arg\underline{X}$ . Here $j$ denotes the imaginary unit, $j^2 = -1$ , the letter $i$ being reserved for currents. Drawn as a vector of length $X$ at angle $\varphi$ , $\underline{X}$ is a phasor (Fresnel vector).

Examples

Example 8.4 (Adding two sinusoids)

$3\cos\omega t + 4\sin\omega t$ : the complex amplitudes are $3$ and $4\eu^{-j\pi/2} = -4j$ (since $\sin\omega t = \cos(\omega t - \pi/2)$ ); their sum $3 - 4j$ has modulus $5$ and argument $-0.927\ \mathrm{rad}$ : $5\cos(\omega t - 0.927)$ . No trigonometric identity needed.

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Examples

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