MyOpenMath/Complex phasors

• For more help with phasors visit Phasor algebra

Time average of the product of two signals

Here we multiply two signals with the same angular frequency but with different phases:

$I(t)=I_0\cos (\omega t+{\varphi }_i)=\frac{1}{2}{I}_0\exp i(\omega t+{\varphi }_i)+cc$

$V(t)=V_0\cos (\omega t+{\varphi }_v)=\frac{1}{2}{V}_0\exp i(\omega t+{\varphi }_v)+cc$,

where $cc$ denotes complex conjugate. For example,

$e^{i\theta }+cc=e^{i\theta }+e^{-i\theta }=2\cos \theta .$

Define $P=IV$, and make the algebra easier to follow by defining two phases:

${\mathrm{\Phi }}_I=\omega t+{\varphi }_i,$   and,   ${\mathrm{\Phi }}_V=\omega t+{\varphi }_v$.

Note that $PV$ is the product of two binomials, which yield four terms:

$IV=\frac{1}{4}{I}_0{V}_0(e^{i{\mathrm{\Phi }}_I}+e^{-i{\mathrm{\Phi }}_I})(e^{i{\mathrm{\Phi }}_V}+e^{-i{\mathrm{\Phi }}_V})$

When the two binomials are multiplied we obtain four terms. We group them according to whether they involve the sum or difference between the two phases, ${\mathrm{\Phi }}_I$ and ${\mathrm{\Phi }}_V$, because whether it is a sum or difference affects the time-dependence as follows:

${\mathrm{\Phi }}_I+{\mathrm{\Phi }}_V=2\omega t+{\varphi }_i+{\varphi }_v$

${\mathrm{\Phi }}_I-{\mathrm{\Phi }}_V={\varphi }_i-{\varphi }_v$

These terms can be grouped into real and imaginary parts, expressed in terms of the sine and cosine functions:

$IV=\frac{1}{4}{I}_0{V}_0\left[\underset{2\cos ({\mathrm{\Phi }}_I-{\mathrm{\Phi }}_V)}{\underbrace{e^{i({\mathrm{\Phi }}_I-{\mathrm{\Phi }}_V)}+e^{i({\mathrm{\Phi }}_V-{\mathrm{\Phi }}_I)}}}\right]+[(\underset{2\cos ({\mathrm{\Phi }}_I+{\mathrm{\Phi }}_V)}{\underbrace{e^{i({\mathrm{\Phi }}_I+{\mathrm{\Phi }}_V)}+e^{-i({\mathrm{\Phi }}_V+{\mathrm{\Phi }}_V)}}}]$

File:Leistung-uip-real English.svg

Graphs of current Template:Color, voltage Template:Color, and power Template:Color for an ac circuit with a phase shift between current and voltage.

With ac circuits it is customary to average over one period,

$T$

, defined by the expression

$\omega T=2\pi$

. Using the overbar notation to denote this time average, we have:

$\overline{IV}=\frac{1}{2}{I}_0{V}_0\cos ({\varphi }_i-{\varphi }_v)=I_{\text{rms}}{V}_{\text{rms}}\cos ({\varphi }_i-{\varphi }_v)$