Exercise 31

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, 31), and First and Last would be your first and last name.  So my submission to Lan for this exercise would be titled “Ex31_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 periodic oscillations in TL circuits.  In this exercise, we will take code we have written in previous exercises and adjust to mimic the behaviour of the TLs we discussed in class.  This is going to require a little bit of a fudge, though.  Why?  Well, up to this point, we have been assuming we are injecting our signal from a source over a source resistance.  But as we discussed in class, the resonant TLs we are looking at today have either shorted or open terminals (on both input and at the load).  So this means that there is no easy way to inject a signal.  So this is what we are going to have to fudge...

Periodic Oscillations in TLs

What we are going to do to fudge our resonant TL circuits is launch a harmonic wave on the TL, but we are going to do this magically!  Namely, we are going to launch our wave at z=0, and not worry about how it got there.  Then we will be able to define either end of the TL as a short or open circuit. The systems we are going to study are shown below:

Let’s steal from our previous class exercise to get the basic framework and adjust from there... 
We can launch our wave as a propagating Sin wave at z=0 (this will be the product of our Sin wave with (1-HeavisideTheta) function).  We then can set our Gamma’s to be either -1 (short) or 1 (open), and we can see how our wave progresses. Let’s assume that our v=c=300m/μs, and our (Subscript[Z, o])=50Ω.  Our inputs are then going to be (Subscript[Γ, L]) (Load Reflection coefficient, either -1 or 1), (Subscript[Γ, o]) (Reflection coefficient at z=0, either -1 or 1), tlLo (the TL length), ω (the radial frequency of our Sine wave) and n (the number of round trips).

vresn[gamL_,gamo_,tlLo_,omega_,n_]:=(v=300;
vres=Sum[(gamL*gamo)^i*((1-HeavisideTheta[z-v*t+2*tlLo*i])*Sin[omega*(t-z/v+2*i*tlLo/v)])+gamL*(gamL*gamo)^i*HeavisideTheta[z+v*t-2*(i+1)*tlLo]*Sin[omega*(t+z/v-2*(i+1)*tlLo/v)],{i,0,n}];
Animate[Plot[vres/.{t->to},{z,0,tlLo}, PlotRange->{-5,5},PlotLabel->{gamL,gamo, to}],{to,0,2*n*tlLo/v},AnimationRepetitions->1, AnimationRate->1])

vresn[1,1,1200,Pi/8,3]

OK, so we have a lot of things to play around with here. Let’s start by considering the open/open configuration, with n=6.  Remember, an open TL will have a reflection coefficient of 1.  We’ll set our TL length to 1200m.  If our propagation velocity is v=c=300m/μs, then we would expect our resonant frequencies to be (Subscript[ω, n])=nπv/l=nπ/4.  So let’s start by inputting ω=(Subscript[ω, 1])/2=π/8.  What you’ll notice is that there doesn’t seem to really be any pattern to the behavior of this wave...It’s amplitude grows steadily (but this is because you are constantly inputting energy into the system).  But what happens if you choose ω=(Subscript[ω, 1])=π/4?  You see something very different.  Pretty quickly, your wave approaches a standing wave...it looks like a Sin or Cos wave fixed in space, but oscillating in time.  
Now check what happens when ω=(Subscript[ω, 2])=π/2.
You can see that these waveforms very closely mimic the resonant modes we analytically solved for in class.

vresn[1,1,1200,2*Pi/4,6]

Now look at the case where both ends are shorted.  From class, we would expect that our Voltage be pinned to zero at either side of the TL.

vresn[-1,-1,1200,2*Pi/4,6]

The voltage is definitely pinned at z=1200m, but not as much at z=0.  Again, this is because of our fudge.  We are constantly injecting the wave at z=0, so our set-up mandates that we have some signal there.  As the total amount of power carried on the line increases, the offset from V=0 at z=0 becomes smaller and smaller...
Finally, try one side shorted, one side open.  You should expect to see different resonant frequencies.  Use your animation to determine these resonant frequencies, and compare to the results from lecture.

vresn[-1,1,1200,Pi/8,6]

Finally, adjust your code to animate the current on the line.  Look at the current modes for each

iresn[gamL_,gamo_,tlLo_,omega_,n_]:=(
ires=;
Animate[Plot[ires/.{t->to},{z,0,tlLo}, PlotRange->{-.5,.5},PlotLabel->{gamL,gamo, to}],{to,0,2*n*tlLo/v},AnimationRepetitions->1, AnimationRate->1])

iresn[1,1,1200,2*Pi/4,6]

Now try plotting your I(z,t) and V(z,t) simultaneously (scale I by zo so that they have the same amplitude).  You should very quickly see the difference in the z-dependence of the waveforms, but look closely at the temporal (time) dependence of the waves.  What do you notice?  Are the in phase?  Out of phase?  If the latter, by how much?

viresn[gamL_,gamo_,tlLo_,omega_,n_]:=(
ires=;
vres=;
Animate[Plot[{ires/.{t->to},vres/.{t->to}},{z,0,tlLo}, PlotRange->{-5,5},PlotLegends->{"I(z,t)","V(z,t)"},PlotStyle->{Red,Blue},PlotLabel->{gamL,gamo, to}],{to,0,2*n*tlLo/v},AnimationRepetitions->1, AnimationRate->1])

viresn[1,1,1200,2*Pi/4,6]