Phasors of Complex Numbers
Phasors (phase vectors) is a very useful mathematical tool that converts a differential equation, with the time as the independent parameter, into algebraic form. Then the equation can be solved much easier and the solution is finally converted back to its initial form.
A phasor is actually a complex number that represents the amplitude and the phase of a sinusoid, with its time variable neglected. The amplitude and the angular velocity must be time invariant. Since the factor ωt is known, it can then be neglected. Two sinusoids and their phasors follow:
If we have an equation with both cosines and sines, we have to convert all of them into only one form.
Consider the equation:
is the phasor of f(t).
Taking the real part of f(t) we get:
So multiplying the phasor with the time variable ejωt and taking the real part of it, returns us the sinusoid that varies over time. This is how the phasor represents the actual sinusoid, which is constant in amplitude and in angular velocity.
Phasors in Differential Equations
A very important property of phasors is that they can convert a differential equation into algebraic form. To see how this works, we will differentiate the real part of f(t) we used before:
The time derivation of a sinusoid is equal to multiplying its representative phasor by jω. It can also be shown that the time integration of a sinusoid is equal to dividing its phasor by jω. So in a differential equation we factor ejωt in both sides of the equation, then we factored it out and we replace the deviations by jω and the integrations by 1/jω. The differential equation is now transformed into algebraic form. We solve it and finally, we multiply the algebraic solution by ejωt to return to it the time parameter.