Phasors Review

Introduction

This is a very sparse review of the motivation for and the use of phasors. It is not necessarily ready for prime time, and it may potentially have errors, typographical and otherwise. Also, there will be logical fallacies that, over time, will be corrected. But it is a start, and hopefully a worthwhile one!

Background

For circuits consisting of independent sources, dependent sources, reactive elements, and resistive elements, as long as certain conditions are met, if the independent sources consist of signals made up of one or more single-frequency sinusoids, the steady-state currents through and voltages across each element will also consist of signals made up of the same frequencies. "Steady-state" here refers to the time when whatever the "initial conditions" of the system are no longer relevant and the system is responding purely to the independent sources. "Certain conditions" include having resistive elements such that any perturbations from steady-state dissipate over time, avoiding doing anything "illegal" like trying to have an instantaneous change in voltage across a capacitor or instantaneous change in current through an inductor, and avoiding doing anything "weird" like having a voltage source in parallel with an inductor or a current source in series with a capacitor. Assuming those certain conditions are met and the circuit has been in place long enough to reach the AC steady state, you can examine the circuit one frequency at a time (because, linearity). And for that one frequency at a time, you can assume that every voltage and every current can be written as

\(x(t)=X\,\cos(\omega t+\phi_x)\)

where $$X$$ is the amplitude of the signal and $$\phi_x$$ is the phase of the signal. For a linear, time-invariant circuit with all the input signals at the same frequency $$\omega$$, all AC steady-state signals will also be at that frequency $$\omega$$. If one source has more than one frequency, if if there are several sources with a variety of different frequencies, as long as the system is LTI, you can solve at each frequency individually and then add the results together to get the outputs collectively.