Exercise 22

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, 22), and First and Last would be your first and last name.  So my submission to Lan for this exercise would be titled “Ex22_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:

Today in class we discussed wave propagation in lossy media.  In particular, we discussed the effect of including a conductivity term in Maxwell’s equations.  We were able to develop expressions for the complex propagation constant γ=α+iβ and the complex impedence η.  In our next computer lab class, we are going to develop a CDF file that allows the user to model waves in lossy media with control over the ϵ, μ, and η.  In this exercise, I just want to make sure you understand the effect σ has on wave propagation.

Review of Lossless Medium, Phasor Notation

In a lossless medium, we know that our wave has a propagation constant γ=jβ=jω(Sqrt[μϵ]).  Let’s write the phasor for a wave propagating in the +(OverHat[z]) direction, with amplitude (Subscript[E, o]) and frequency ω.  Of course in phasor notation, we don’t worry about ω until we need to animate the wave...

e1phas[eo_,b_]:=eo*Exp[-I*b*z]

We can plot our E field vs for a given time t, but bear in mind that our β and ω are linked by the velocity of the wave, which is determined by μ and ϵ.  For today’s exercise, we don’t really care much about using real values for our frequency and speed...so let’s assume μ=ϵ=β=ω=v=1.  Keep in mind, these are not “realistic” values for an E&M wave, but they will make our lives easier for now as we try to understand the effects of losses.

e1[eo_,b_,w_,t_]:=Re[e1phas[eo,b]*Exp[I*w*t]]

Plot[e1[1,1,1,0],{z,-10,10}]

We can get an idea of how this moves by changing our value for “t”.  Use the above Plot command, changing t incrementally by 0.1s, to see the wave propagate. However, we can also animate the wave.  You should be familiar with this, so below, animate the wave to show its propagation in time...

OK, that looks right. What about the H-field?
Well, we know that our H field will be scaled by 1/η=(Sqrt[ϵ/μ])=1, again, not realistic...

h1[eo_,b_,w_,t_]:=Re[e1phas[eo,b]*Exp[I*w*t]]

Animate[Plot[h1[1,1,1,t]/.{t->to},{z,-10,10}],{to,0,10}]

Without any loss, and given our unrealistic parameters, we see that our H field is exactly the same as our E field (though we also kow that they will be orthogonal in direction (also something we aren’t worried about right now).

Introducing Loss

When we introduce loss to our medium, we change the wave propagation significantly.  From our lecture, we know that 
γ=(Sqrt[jωμ(σ+jωϵ)]) and η=(Sqrt[jωμ/(σ+jωϵ)])
So let’s re-write our expressions for E and H, assuming eo=μ=ϵ=ω=1, and allowing σ to be varied.

e2phas[sigma_]:=Exp[-Sqrt[I*1*1*(sigma+I*1*1)]*z]

e2[sigma_,t_]:=Re[e2phas[sigma]*Exp[I*1*t]]

Let’s plot this wave for varying sigmas.  It is important to note that because we expect an exponential term in our expression for the E field, (e^(-αz)), going in the -(OverHat[z]) direction will cause our solution to blow up, so let’s just assume our wave starts propagating at z=0.  We can plot our wave at t=0 for different sigma’s

Plot[e2[0.5,0],{z,0,30},PlotRange->Full]

We see that for increasing sigma’s, our wave amplitude very quickly dies out.  In fact, our amplitude for the wave will decay as (e^(-αz)), where α=Re{(Sqrt[jωμ(σ+jωϵ)])}

e2amp[sigma_]:=Exp[-Re[Sqrt[I*1*1*(sigma+I*1*1)]]*z]

Plot[{e2amp[.1],e2[.1,0]},{z,0,30},PlotRange->Full]

What about the H field?
Below, write an expression for the H field phasor and the H-field, and then plot the H-field with the E field at t=0 for varying sigmas.  Remember, your H field is scaled by η, which is now complex!!

h2phas[sigma_]:=

h2[sigma_,t_]:=

Plot[{e2[1,0],h2[1,0]},{z,0,10}, PlotRange->Full]

Play around with sigma for t=0.  What do you see?
It’s actually pretty interesting....The first thing you should notice, as you increase sigma from zero, is that the amplitude of your E field starts to decrease slightly, compared to your H-field.  This decrease comes from the change in the magnitude of η with increasing σ.  As you continue to increase σ, you will notice that the phase of your H field wave also changes with respect to that of the E field, this comes from the argument of η, which we write in class as ∠η.
Does the frequency change?  Well, we know that it doesn’t, since for any system we can use phasor notation with, we are assuming a single frequency, but let’s verify this by animating the waves:

Animate[Plot[{e2[.5,t]/.{t->to},h2[.5,t]/.{t->to}},{z,0,10},PlotRange->{-1.5,1.5}],{to,0,10}]

Cool!

Energy in a Lossy Medium

Really quickly,let’s plot the Poynting Vector.  We know that the Poynting Vector is the cross product of the E and H fields, so we should be able to quickly calculate this, knowing that the fields are orthogonal.

ClearAll[s2,s2phase]

s2[sigma_,t_]:=e2[sigma,t]*h2[sigma,t]

Animate[Plot[s2[.5,t]/.{t->to},{z,0,10},PlotRange->{-1.5,1.5}],{to,0,10}]

We see that the Poynting Vector decays as the wave moves into the lossy material.  Where is this energy going?