Phasors Review/Calculations Example

Introduction

Assume you have an independent voltage source connected to a series combination of an inductor and a resistor and you want to find the current through the mesh (labeled as the current through the inductor). Using phasor analysis, you can get the transfer function:

\(\begin{align*}\mathbb{H}(j\omega)&=\frac{\mathbb{I}_{\omega}}{\mathbb{V}_{\omega}}=\frac{1}{j\omega L + R}\end{align*}\)

where $$\mathbb{V}_{\omega}$$ represents the magnitude and phase of a single-frequency sinusoid of the voltage source at frequency $$\omega$$ and $$\mathbb{I}_{\omega}$$ represents the magnitude and phase of a single-frequency sinusoid of the steady state current at frequency $$\omega$$.

AC Steady State Calculations

If you further assume that the circuit has been in place for a long time and under the influence of one (or more) single-frequency sinusoidal voltages, you can find the AC steady state current using phasor analysis by noting that:

\(\begin{align*}\mathbb{I}_{\omega}=\mathbb{V}_{\omega}\,\mathbb{H}(j\omega)\end{align*}\)