Phasors

J. C. Daly
9/15/04

Phasors are a complex numbers used to represent sinusoids.

Complex Numbers


Figure 1   Two dimensional representation of a complex number.
Recall Euler's formula relating exponential and sinusoidal functions,

w = x + jy = Re jq = R cos(q ) + j R sin(q )
where R is the magnitude and q is the phase.
R = sqrt(x2 + y2)

Phasors

Consider the sinusoidal function

v(t) = A cos( w t + q )
A phasor representation of this function is the complex number V with a magnitude A and a phase q.
V = A e j q
If we multiply V by e j wt and apply Euler's formula,
V e jw t = A e j q e j w t = V e i(w t + q ) = A cos(w t + q ) + jA sin(w t + q )
The real part of V e jw t is the desired sinusoidal function,
v(t) = A cos( w t + q )

Multiply the phasor by e jw t and take the real part to get the sinusoidal function.

Linear Operations

Addition

Consider two phasors representing two sinusoids at the same frequency

V1 = A1 e j q1
V2 = A2 e j q2
Multiply the sum of these two phasors by e j w t,
(V1 + V2) e j w t = A1 cos (w t + q1 ) + jA1 sin(w t + q1 ) + A2 cos (w t + q2 ) + jA2 sin(w t + q 2)
The real part of (V1 + V2 ) e j w t is the sum of the two sinusoidal functions.
A1 cos (w t + q1 ) +A2 cos (w t + q2 )

The phasor representation of the sum of sinusoidal functions is the sum of phasors representing individual sinusoidals.

Differentation

Consider the derivative of a phasor representing a sinusoid a multiplied by e j w t,

d/dt[V e jw t ] = d/dt[A e j qe jwt] = j w A e j qe jwt = j wV e jw t
j wV
is the phasor representing the derivative of the sinusoidal function.
The real part of the derivative of V e jw t is the derivative of the sinusoidal function
d/dt[V e jw t ] = d/dt[A cos(wt + q ) + j sin(wt + q )]

The phasor representation of the derivative of a sinusoidal is j w multiplied by the phasor representing the sinusoid.

Circuit Applications

Consider a capacitor

i = C dv/dt
If v is a sinusoidal function represented by the phasor V, dv/dt is represented by j w V. For a capacitor
I = jw C V
where I and V are the phasor representations of capacitor current and voltage. The impedance of the capacitor is the ratio of the voltage phasor to the current phasor.
V/I = 1/ j wC