 convert/phaseamp - Help

convert/phaseamp

convert expressions to phase-amplitude form

 Calling Sequence convert( expr, phaseamp, t )

Parameters

 expr - any Maple expression t - name; the name of the dependent variable

Description

 • The phaseamp conversion attempts to locate subexpressions of the form $aF\left(t\right)\mathrm{cos}\left(\mathrm{omega}t\right)+bF\left(t\right)\mathrm{sin}\left(\mathrm{omega}t\right)$ in expr, and converts them to the form $\sqrt{{a}^{2}+{b}^{2}}F\left(t\right)\mathrm{cos}\left(\mathrm{omega}t-\mathrm{arctan}\left(b,a\right)\right)$. Here, the coefficients $a$ and $b$ are independent of the variable $t$, which must be passed as a parameter to the conversion.
 • The strength of the conversion is sensitive to the values of the environment variables Testzero and Normalizer.

Examples

This example demonstrates the basic conversion that is applied.

 > $\mathrm{convert}\left(a\mathrm{exp}\left(ct\right)\mathrm{cos}\left(\mathrm{\omega }t\right)+b\mathrm{exp}\left(ct\right)\mathrm{sin}\left(\mathrm{\omega }t\right),\mathrm{phaseamp},t\right)$
 $\sqrt{{{a}}^{{2}}{+}{{b}}^{{2}}}{}{{ⅇ}}^{{c}{}{t}}{}{\mathrm{cos}}{}\left({\mathrm{\omega }}{}{t}{-}{\mathrm{arctan}}{}\left({b}{,}{a}\right)\right)$ (1)

The following two examples illustrate how the dependent variable affects the conversion.

 > $\mathrm{convert}\left(a\left(s\right)\mathrm{exp}\left(st\right)\mathrm{cos}\left(st\right)+b\left(s\right)\mathrm{exp}\left(st\right)\mathrm{sin}\left(st\right),\mathrm{phaseamp},t\right)$
 $\sqrt{{{a}{}\left({s}\right)}^{{2}}{+}{{b}{}\left({s}\right)}^{{2}}}{}{{ⅇ}}^{{s}{}{t}}{}{\mathrm{cos}}{}\left({s}{}{t}{-}{\mathrm{arctan}}{}\left({b}{}\left({s}\right){,}{a}{}\left({s}\right)\right)\right)$ (2)
 > $\mathrm{convert}\left(a\left(s\right)\mathrm{exp}\left(st\right)\mathrm{cos}\left(st\right)+b\left(s\right)\mathrm{exp}\left(st\right)\mathrm{sin}\left(st\right),\mathrm{phaseamp},s\right)$
 ${a}{}\left({s}\right){}{{ⅇ}}^{{s}{}{t}}{}{\mathrm{cos}}{}\left({s}{}{t}\right){+}{b}{}\left({s}\right){}{{ⅇ}}^{{s}{}{t}}{}{\mathrm{sin}}{}\left({s}{}{t}\right)$ (3)