Sinusoidal steady state analysis using phasors

Sinusoidal steady state analysis using phasors

In some of our studies of time varying electromagnetic fields, we will be considering sinusoidal steady state signals. In this case, the use of phasors greatly simplifies the analysis since in Maxwell’s equations

assuming an e jwt time dependence. In particular, we usually assume a cos(w t )time dependence by default.
What is a Phasor?
To answer this question, imagine we have a sinusoidally varying function of time
f( t)= f0 cos(wt+Æ)
By Euler’s identity
e jx= cos x++ j sin x

The quantity 0 F in this last expression is what is called a phasor. That is, the phasor F0 is a very simple and compact method of representing the:
. amplitude, and
. phase angle (i.e., time delay with respect to the source) of
the function f ((t )). These two properties are all that is needed to represent in a shorthand notation, of sorts, the time variation of a function in a linear system that has sinusoidal steady state excitation. In electromagnetics, our functions are generally vectors as well as phasors. Additionally, these “vector-phasors” are functions of space. Because of this, analyzing such problems can be complicated.
Example N7.1: Determine the phasor representation for

which is the phasor representation of `B (( y,t)
Example N7.2: Determine the phasor representation for Faraday’s law

Assuming a cos(wt)) time response, then `E((t)= Re [`Ee jwt] and `B((t)= Re [Be jwt . In these two expressions, `E and `B in the Re operators are (vector) phasors. Substituting into Faraday’s law

But, the Re operator commutes with the differentiation operator. Therefore,


Consequently, the phasor representation of Faraday’s law is
Ñ´` E =- jw`B
Maxwell’s Equations in Phasor Form
Applying the result of this last example, we can easily write Maxwell’s equations in phasor form as

and the continuity equation
Ñ.` J =-jwrv
Complex Numbers Aren’t Necessarily Phasors
Finally, note that a phasor is generally a complex number, but not every complex number is a phasor! For example, in circuit analysis:

Similarly, in electromagnetics:

Remember that a phasor is a shorthand representation for a function that has the following two time domain properties: 1. sinusoidal time dependence,
2. zero time average value