Exercise 33

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, 33), and First and Last would be your first and last name.  So my submission to Lan for this exercise would be titled “Ex33_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 quarter and half wave transformer TLs.  Using simple descriptions of Voltage and Current signals on TLs as weighted combinations of forward and backward propagating waves, we were able to determine the unique properties of TLs with arbitrary loads and lengths equal to integrer multiples of quarter wavelengths.  We did all of this without worrying about what the waves actually looked like.  Today we are going to use Mathematica to better understand and visualize what the waveforms on the TLs actually look like.

Modelling Quarter Wave Transformers

Let’s start with a very simple system.  We are going to assume we have a TL with l=λ/4, v=c, and (Subscript[Z, o])=50Ω.  
Keep in mind that because we are considering a quarter wave transformer and v=c, we are automatically then setting β and ω.
β=2π/λ=2π/4ℓ=π/2ℓ
ω=2πc/λ=2πc/4ℓ=πc/2ℓ
Because we need to be able to consider complex loads, life will probably be easier if we use Phasor notation.  We know we can write the forward propagating phasor as (SuperPlus[(OverTilde[V])])(z)=(SuperPlus[V])(e^(jβz)) and our backward propagating phasor as (SuperMinus[(OverTilde[V])])(z)=(SuperMinus[V])(e^(-jβz)).  Remember, we are now labeling the load as z=0, and the input as z=λ/4.
How are (SuperPlus[V]) and (SuperMinus[V]) related?  Well, we actually know this...it must be a load reflection coefficient!!
(Subscript[Γ, L])=(((Subscript[Z, L])-(Subscript[Z, o]))/((Subscript[Z, L])+(Subscript[Z, o]))), but we have to recognize that our load can now be complex.
Let’s write an expression for our voltage wave assuming (SuperPlus[V])=1.

qw1[zo_,zL_,tlL_]:=(gamL=(zL-zo)/(zL+zo);beta=Pi/(2*tlL);omega=Pi*300/(2*tlL);
vphwq1=Exp[I*beta*z]+gamL*Exp[-I*beta*z];
 iphwq1=(1/zo)*(Exp[I*beta*z]-gamL*Exp[-I*beta*z]);
Animate[Plot[Re[(vphwq1*Exp[I*omega*t])/.{t->to}],{z,0,tlL}, PlotRange->{-3,3},PlotLabel->{gamL, to}],{to,0,2*2*Pi/omega},AnimationRepetitions->1, AnimationRate->1])

qw1[50,50,1200]

First try (Subscript[Z, L]) to be shorted or open...then try arbitrary real values for (Subscript[Z, L])...and finally, try complex (Subscript[Z, L])...Remember, in our coordinate system, z=0 is the load and z=tlL is the input...so as we have plotted above, the input is to the right of the load...
Let’s add a Label showing the ratio of V to I at the input of the TL...

qw1[zo_,zL_,tlL_]:=(gamL=(zL-zo)/(zL+zo);beta=Pi/(2*tlL);omega=Pi*300/(2*tlL);
vphwq1=Exp[I*beta*z]+gamL*Exp[-I*beta*z];
 iphwq1=(1/zo)*(Exp[I*beta*z]-gamL*Exp[-I*beta*z]);
Animate[Plot[Re[vphwq1*Exp[I*omega*t]]/.{t->to},{z,0,tlL}, PlotRange->{-3,3},PlotLabel->{gamL, "Zin=",(vphwq1/iphwq1)/.{z->tlL,t->to}}],{to,0,2*2*Pi/omega},AnimationRepetitions->1, AnimationRate->1])

qw1[50,25*I,1200]

You see that the input impedance of the TL is related to the load impedance by (Subscript[Z, IN])=(Subscript[Z, o]^2)/(Subscript[Z, L]).  Check to make sure this is the case.  
Determining the relationship between (Subscript[V, IN]) and (Subscript[V, L]) is harder, beause there is a factor of “i” in there, and the load current is related to the input voltage (and not the load resistance)

Modelling Half Wave Transformers

Now you try the same for a half-wave TL.
Keep in mind that because we are considering a half wave transformer and v=c, we are automatically then setting β and ω.
β=2π/λ=2π/2ℓ=π/ℓ
ω=2πc/λ=2πc/2ℓ=πc/ℓ

qw2[zo_,zL_,tlL_]:=(gamL=(zL-zo)/(zL+zo);beta=Pi/(tlL);omega=Pi*300/(tlL);
vphwq2=;
 iphwq2=;
Animate[Plot[Re[vphwq2*Exp[I*omega*t]]/.{t->to},{z,0,tlL}, PlotRange->{-3,3},PlotLabel->{gamL, "Zin=",(vphwq2/iphwq2)/.{z->1200,t->to}}],{to,0,2*2*Pi/omega},AnimationRepetitions->1, AnimationRate->1])

qw2[50,25*I,1200]

You should see the clear relationship between (Subscript[V, IN]) and (Subscript[V, L]), as well as the fact that  (Subscript[Z, IN])=(Subscript[Z, L]).  Make sure this is the case for a range of different (Subscript[Z, L])'s  both real and complex.

Abritrary Lengths

What is interesting (and powerful) about our approach is that we can very easily refine our code to model waves propagating on TLs of arbitrary length with arbitrary loads.  In class, we have so far only been able to tackle specific cases and solve these analytically.  In our next lecture, we will extend to understanding TLs of arbitrary length, but this is going to require the development of a new way of thinking about and solving for waves on these TLs.

Below, revise your code to model wave propagation on TLs of arbitrary length.  Make sure you show the (Subscript[Z, IN]) for your TL on the plot, as we did above.  Because we are now dealing with arbitrary lengths, we no longer have a relationship between tlL and beta, so we have to tell the function what our omega, lambda or beta is...let’s continue to assume v=c, and input our wavelength (lam).

qw3[zo_,zL_,tlL_,lam_]:=(gamL=;beta=;f=;omega=;
vphwq3=;
 iphwq3=;
Animate[Plot[Re[vphwq3*Exp[I*omega*t]]/.{t->to},{z,0,tlL}, PlotRange->{-3,3},PlotLabel->{gamL, "Zin=",(vphwq3/iphwq3)/.{z->tlL,t->to}}],{to,0,2*2*Pi/omega},AnimationRepetitions->1, AnimationRate->.5])

qw3[50,100,1200,300]