Phasors are a complex numbers used to represent sinusoids.
Complex Numbers
Figure 1 Two dimensional representation of a complex number. |
w = x + jy = Re ^{jq} = R cos(q ) + j R sin(q )where R is the magnitude and q is the phase.
R = sqrt(x^{2} + y^{2})
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
V_{1} = A_{1} e ^{j q1}Multiply the sum of these two phasors by e ^{j w t},
V_{2} = A_{2} e ^{j q2}
(V_{1} + V_{2}) e ^{j w t} = A_{1} cos (w t + q_{1} ) + jA_{1} sin(w t + q_{1} ) + A_{2} cos (w t + q_{2} ) + jA_{2} sin(w t + q _{2})The real part of (V_{1} + V_{2} ) e ^{j w t} is the sum of the two sinusoidal functions.
A_{1} cos (w t + q_{1} ) +A_{2} cos (w t + q_{2} )
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 q}e ^{jwt}] = j w A e ^{j q}e ^{jwt} = j wV e^{ jw t }The real part of the derivative of V e^{ jw t } is the derivative of the sinusoidal function
j wV is the phasor representing 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/dtIf v is a sinusoidal function represented by the phasor V, dv/dt is represented by j w V. For a capacitor
I = jw C Vwhere 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