Phasors Demystified

Phasors Demystified Swapnil Sunil Jain Aug 7, 2006

Phasors Demystified

Suppose the following integro-differential equation is given in the time-domain11This is not the most general integro-differential equation but it has all the basic elements required for this discussion and hence the reader can easily extend this discussion for the more generalized case.:

C1ddty(t)+C2-ty(t)𝑑t+C3y(t)=x(t) (1)

where y(t) and x(t) are sinusoidal waveforms of the same frequency. Now, since y(t) is a sinusoidal function it can be represented as Aycos(ωt+ϕy) and, similarly, x(t) can be represented as Axcos(ωt+ϕx). Furthermore, using the properties of complex numbers we can write


Now if we define the quantities Y~ as Ayejϕy and X~ as Axejϕx (where Y~ and X~ are called phasors), then we can write the above expression in a more compact form as


Now, using the above expression for y(t) and x(t) we can rewrite our original integro-differential equation as


Moving the derivative and the integral inside the operator we get


Equating the real partsMathworldPlanetmath above we get,

Y~jωC1+Y~C2jω+Y~C3-X~=0(for t-) (2)

Hence, we have now arrive at the phasor domain expression for (1). You can see from the analysis above that we aren’t adding or losing any information when we transform equation (1) into the ”phasor domain” and arrive at equation (2). One can easily get to (2) by using simple algebraic properties of real and complex numbersMathworldPlanetmathPlanetmath. Furthermore, since (2) can be derived readily from (1), in practice we don’t even bother to do all the intermediate steps and just skip straight to (2) calling this ”skipping of steps” as ”transforming the equation into the phasor domain.”

We can now continue the analysis even further and solve for y(t) which is the whole motivation behind the use of phasors. Solving for Y~ in (2) we get


Now, since


we have


The above equation makes sense because you can see that the output y(t) is given completely in terms of the variables Ax and ϕx (which depend only on the input sinusoid x(t)) and the constants C1, C2 and C3—as we expected! So by converting the integro-differential equation (1) into the phasor domain (2), all the complicated integration and differentiationMathworldPlanetmath operations become simple manipulation of complex variables—which is why phasors are so useful!

Title Phasors Demystified
Canonical name PhasorsDemystified1
Date of creation 2013-03-11 19:26:47
Last modified on 2013-03-11 19:26:47
Owner swapnizzle (13346)
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