Exercise 21

As with all of the exercises you will complete in this course, you will be expected to turn your *.nb file in to the course TA before the next lecture.  Please do so by e-mailing your completed exercise file to Lan Yu (ece329.uiuc@gmail.com), make sure you name the file using the following notation: “Exn_Last_First.nb”, where n is the Exercise number (in this case, 21), and First and Last would be your first and last name.  So my submission to Lan for this exercise would be titled “Ex21_Wasserman_Dan.nb”.  You will receive full credit for your exercise if it turned in before the next lecture (Lecture n+1).  No late submissions will be accepted.

Objective/Overview:

We had our first true introduction to Phasor notation today, and we will use this exercise to familiarize ourselves with this VERY helpful way of writing monochromatic waves.

Basics of Phasor Notation

Just to remind you, phasor notation is based on Euler’s formula, which says (e^(iθ))=cosθ+isinθ.  This means that Re{(e^(iθ))}=cosθ, and Re{(ie^(iθ))}=-sinθ.

In class we said that we can write a wave described by (OverVector[E])(z,t)=(OverHat[x])(Subscript[E, o])cos(ωt∓βz) using Phasor notation as (OverVector[E])(z,t)=Re{(OverTilde[E])(e^(jωt))}, with (OverTilde[E])=(Subscript[E, o])(e^(∓jβz))(OverHat[x]).  Let’s show that this is true, starting with a forward propagating wave.

evector[eo_,w_,b_]:=eo*Cos[w*t-b*z]

remember that β=ω/v, so if we are in free space, v=c, and β and ω are linked.  Let’s animate this wave if v=c, and ω=2π×(10^6)rad/s

Animate[Plot[evector[1,2*Pi*10^6,(2/3)*10^-2]/.{t->to},{z,-2400,2400}],{to,0,10^-5}]

OK, that looks right.
Now let’s try writing our Electric Field in Phasor notation.

ephasor[eo_,b_]:=Exp[-I*b*z]

Plot[ephasor[1,(2/3)*10^-2],{z,-2400,2400}]

Why doesn’t this work?  Well, because our expression for ephasor is complex.  We know that we only want the real part, but Mathematica doesn’t!  So it get’s confused.

Plot[Re[ephasor[1,(2/3)*10^-2]],{z,-2400,2400}]

That’s better!
To plot the electric field (in time and space), we simply need to multiply our phasor by (e^(jωt)) and take the real part of the product:

Animate[Plot[Re[Exp[I*w*t]*ephasor[1,(2/3)*10^-2]/.{w-> 2*Pi*10^6,t->to}],{z,-2400,2400}],{to,0,10^-5}]

Excellent!  They are, and should be, exactly the same!
Use phasor notation to plot and animate a wave described by: (OverVector[E])(z,t)=(OverHat[y])(Subscript[E, o])cos(ωt+βz), where ω=2π×(10^9)rad/s, (Subscript[E, o])=4V/m and μ=(Subscript[μ, o]), ϵ=(Subscript[ϵ, o]).

ephasor1[eo_,w_,v_]

Animate[Plot[ephasor1[4,2*Pi*1000,300]/.{t->to},{z,-3,3}],{to,0,.01}]

Energy/Power with Phasor Notation

We showed, in class, that finding time averaged power using phasor notation is remarkably easy.  Let’s go back to our surface current sheet example, and solve using phasor notation.
If our sheet current  is surrounded by vacuum, located at z=0, and described using phasor notation, by (OverTilde[J])=(OverHat[x])A/m.  Let’s solve for (OverTilde[E])(z), (OverTilde[H])(z), (OverTilde[E])·(OverTilde[J]) and (OverTilde[S])(z)

jphas1={1,0,0}

hphas1plus={0,-jphas1[[1]]/2,0}

hphas1min={0,jphas1[[1]]/2,0}

ephas1=-120*Pi*jphas1/2

Now use our expressions for (OverTilde[S])(z) and  (OverTilde[E])·(OverTilde[J]) developed in class to determine the time averaged power flow and power generated by the current sheet:

sphas1plus=ephas1\[Cross]hphas1plus

sphas1min=ephas1\[Cross]Conjugate[hphas1min]

ejphas1=Dot[ephas1,Conjugate[jphas1]]

avesphas=0.5*Re[Abs[sphas1plus]+Abs[sphas1min]]

aveejphas=0.5*Re[ejphas1]

OK, now you try it, with (OverVector[J])=2sinωt (OverHat[y]) on z=0.   Solve for (OverTilde[J]), (OverTilde[E])(z), (OverTilde[H])(z), (OverTilde[E])·(OverTilde[J]) and (OverTilde[S])(z), as well as the time-averaged values for power.  Remember, writing Sin as a phasor requires an “I”, and so you will have to make sure you use the complex conjugates for the time-averaging.