Day 3 - Phase Vectors aka Phasors

Systems that consist of only resistors, capacitors, inductors, and wires can be described using linear differential equations.

In these problems we will assume that there were no initial conditions or we have waited a long time such that the transients die out.

The response of a linear system to the sum of scaled sinusoidal inputs is a sum of the same frequencies as the inputs (magnitudes and phases may not be the same).

Therefore the output is represented as scaled, time-shifted versions of the same sinusoids as the ones that comprise the input.

The steady-state output is:

Where A is the maximum amplitude and B is the phase angle.

Now, if we have the source voltage Vm*cos(wt  + theta), we can compute the phasor:

If we are dealing with sin, then we need to subtract 90 (or pi/2) since sin(x) = cos(x - pi/2).

Inverse Phasor Transform
Going form a phasor to a time domain function

When solving an equation with the source voltage we just need to deal with the real parts.