Phasors

Phasor Representation

\( \begin{align} v(t)&=V_{max}\cos(\omega t+\phi)=\Re\left\{V_{max}e^{j(\omega t+\phi)}\right\}=\Re\left\{V_{max}e^{j\omega t}e^{j\phi)}\right\}=\Re\left\{\mathbb{V}e^{j\omega t}\right\}\\ \mathbb{V}&=\mathbb{V}_{max}=V_{max}e^{j\phi}\\ \end{align} \)

RMS Values

\( \begin{align} V_{rms}&=\sqrt{\frac{1}{T}\int_{t_0}^{t_0+T}\left(x(\tau)\right)^2~d\tau} \end{align} \)

And for a pure sinusoid,

\( \begin{align} v(t)&=V_{max}\cos(\omega t+\phi)\\ V_{rms}&=\sqrt{\frac{1}{T}\int_{t_0}^{t_0+T}\left(\cos(\omega\tau+\phi)\right)^2~d\tau}=\frac{1}{\sqrt{2}}V_{max} \end{align} \)

meaning

\( \begin{align} \mathbb{V}&=\mathbb{V}_{max}=V_{max}e^{j\phi}=(\sqrt{2})V_{rms}e^{j\phi}\\ \mathbb{V}_{rms}&=V_{rms}e^{j\phi}=\frac{V_{rms}}{\sqrt{2}}e^{j\phi} \end{align} \)

Note - The Giorgio Rizzoni book uses

\( \begin{align} \tilde{\mathbf{V}} \end{align} \)

to express the RMS version of the phasor, so

\( \begin{align} \tilde{\mathbf{V}}\Longleftrightarrow\mathbb{V}_{rms} \end{align} \)

Using Phasor Notation for Voltage and Current to Calculate Average Power

\( \begin{align} \mbox{Complex Power: }&\mathbb{S}=\mathbb{V}_{rms}\mathbb{I}_{rms}^*=\frac{\mathbb{V}_{max}\mathbb{I}_{max}}{2}\\ \end{align} \)

Note - the potentially complex number \(\mathbb{S}\) is not a phasor. It is not a representation of a magnitude and phase for a sinusoid. It is merely a mathematical tool for calculating the average real and reactive power for an AC circuit.