Exercise 36

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 (lanyu2@illinois.edu), make sure you name the file using the following notation: “Exn_Last_First.nb”, where n is the Exercise number (in this case, 36), and First and Last would be your first and last name.  So my submission to Lan for this exercise would be titled “Ex36_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 introduced the concept of VSWR (voltage standing wave ratio).  We showed that VSWR can be a powerful tool for characterizing TLs, but we mostly used the Smith Chart to discuss VSWR.  In today’s exercise, we will try to link the analytical description of VSWR to a physical picture of what is actually happening on a TL.

Clear[smith1]

VSWR

The VWSR of a TL is defined as VSWR=((V((Subscript[d, max])))/(V((Subscript[d, min]))))=((1+|(Subscript[Γ, L])|)/(1-|(Subscript[Γ, L])|)).  Let’s look at this on our Smith Chart.

p1=Graphics[Table[Circle[{r/(r+1),0},(1/(r+1))],{r,{0,.25,.5,1,2,5}}]];

p2=Graphics[Table[Circle[{1,1/x},Abs[1/x]],{x,{5,2,1,.5,.25,.01,-5,-2,-1,-.5,-.25,-.01}}]];

smith1=Show[p1,p2, Axes->True ,AxesLabel-> {(Subscript[Γ, r]),(Subscript[Γ, i])},PlotRange->{{-1,1},{-1,1}}]

We can plot our (Subscript[z, L]) and (Subscript[Γ, L]) on this chart, for a given load impedance (Subscript[Z, L]) and characteristic impedance (Subscript[Z, o]):

smith1[zo1_,zL1_]:=(zLnorm=zL1/zo1;gamL1=(zL1-zo1)/(zL1+zo1);
p3=Graphics[{PointSize[Large],Red,Point[{Re[gamL1],Im[gamL1]}]}];
p4=Graphics[{Red,Circle[{0,0},Abs[gamL1]]}];
p5=Graphics[Text[{"(Subscript[Γ, L])=",gamL1},{Re[gamL1],Im[gamL1]+.1}]];
p6=Graphics[Text[{"(Subscript[z, L])=",zLnorm},{Re[gamL1],Im[gamL1]+.25}]];
Show[p1,p2,p3,p4,p5,p6,Axes->True ,AxesLabel-> {(Subscript[Γ, r]),(Subscript[Γ, i])},PlotRange->{{-1,1},{-1,1}}])

smith1[50,50+I*50]

The position at which our V is maximum will occur when our d=(Subscript[d, max]), which will be when we are crossing the (Subscript[Γ, r]) axis on the (Subscript[Γ, r])>0 side.
The position at which our V is minimum will occur when our d=(Subscript[d, min]), which will be when we are crossing the (Subscript[Γ, r]) axis on the (Subscript[Γ, r])<0 side.
To determine this, we can make a Manipulate function:

smith2[zo1_,zL1_,d_]:=(zLnorm=zL1/zo1;gamL1=(zL1-zo1)/(zL1+zo1);
p3=Graphics[{PointSize[Large],Red,Point[{Re[gamL1],Im[gamL1]}]}];
p4=Graphics[{Red,Circle[{0,0},Abs[gamL1]]}];
p5=Graphics[Text[{"(Subscript[Γ, L])=",gamL1},{Re[gamL1],Im[gamL1]+.1}]];
p6=Graphics[Text[{"(Subscript[z, L])=",zLnorm},{Re[gamL1],Im[gamL1]+.25}]];
gamd1=gamL1*Exp[-2*I*2*Pi*d];
p7=Graphics[{PointSize[Large],Blue,Point[{Re[gamd1],Im[gamd1]}]}];
p8=Graphics[Text[{"d=",d},{0.2,1}]];
p9=Graphics[Text[{"(Subscript[Γ, d])=",gamd1},{0.2,.85}]];
Show[p1,p2,p3,p4,p5,p6,p7,p8,p9,Axes->True ,AxesLabel-> {(Subscript[Γ, r]),(Subscript[Γ, i])},PlotRange->{{-1,1},{-1,1}}])

Manipulate[smith2[50,50+I*50,d],{d,0,0.5,.001}]

For a (Subscript[Z, o])=50Ω and (Subscript[Z, L])=50+j50Ω, we see that (Subscript[d, max])=0.088λ and (Subscript[d, min])=0.338λ, again, showing that (Subscript[d, max])-(Subscript[d, min])=λ/4!

Visualizing VSWR

This is all great, but what does it mean?  Well, in order to understand that, it might help to visualize the waves on this TL.  Let’s assume (SuperPlus[V])=1V.

v1[zo1_,zL1_,t_]:=(gamL1=(zL1-zo1)/(zL1+zo1);
vz1=Exp[I*2*Pi*d]*(1+gamL1*Exp[-I*4*Pi*d]);
vzt1=Re[vz1*Exp[I*t]];
Plot[vzt1,{d,0,1},PlotRange->{-2,2}])

Animate[v1[50,50,to],{to,0,20},AnimationRepetitions->1]

Remember, our load is at 0, so for this plot the load is on the LEFT side of the plot, at d=0.
It’s still not entirely obvious what is going on, so let’s plot V as a function of time for various values of “d”, assuming (Subscript[Z, o])=50Ω, but allowing us to manipulate (Subscript[Z, L]).

v2[zo1_,zLre_,zLim_,d_]:=(gamL1=((zLre+I*zLim)-zo1)/((zLre+I*zLim)+zo1);
vz2=Exp[I*2*Pi*d]*(1+gamL1*Exp[-I*4*Pi*d]);
vzt2=Re[vz2*Exp[I*t]];
Plot[vzt2,{t,0,20},PlotRange->{-2,2}])

Manipulate[v2[50,zLre,zLim,d],{d,0,1},{zLre,50},{zLim,50}]

OK, so the plot above is going to require some explaining.
1) First, set (Subscript[Z, L])=50.  This is the case where we are impedance matched.  As you change d, what do you see?  You should see a harmonic wave that does not change amplitude.  This is because you only have a forward propagating wave, no reflection, so the amplitude of the wave as a function of time, at all d’s, will simply be (SuperPlus[V]).
2)  Now set (Subscript[Z, L])=1000000.  This is basically infinite impedance.  What do you notice about V(t) as you change d?  At different points along the TL, you see that V(t), while still a harmonic wave, has very different amplitudes.  In fact, at some points, the amplitude nasically goes to zero, while at some points, your amplitude is basically 2*(SuperPlus[V])=2!  The ratio of the largest amplitude you see and the smallest you see is the VSWR of the TL.
3)  Now try (Subscript[Z, L])=100+j50Ω.  What is the largest and smallest amplitude you see?  What is their ratio? And at what positions do you see these amplitudes?  You may want to zoom in by changing the functions we use to plot above...

Answers
Largest V(t) amplitude:
positions of largest V(t) amplitude:
Smallest V(t) amplitude:
positions of smallest V(t) amplitude:

Now go back to the “smith2” function.  Find the positions (d) where your Γ(d)=|(Subscript[Γ, L])| and  Γ(d)=-|(Subscript[Γ, L])|.  Also, use this to determine the VSWR of the line.

Manipulate[smith2[,,d],{d,0,0.5,.001}]

Answers
(Subscript[d, max])=
(Subscript[d, min])=
VSWR=

What do you notice about these values, compared to the values you determined in the last highlighted section?
Finally, go back to your animation, and input the above values ( (Subscript[Z, L])=100+j50Ω,  (Subscript[Z, o])=50Ω).  Do you see the VSWR in the behavior of your animation?

Animate[v1[50,100+I*50,to],{to,0,20},AnimationRepetitions->1]