Exercise 27

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, 27), and First and Last would be your first and last name.  So my submission to Lan for this exercise would be titled “Ex27_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 saw our first transmission lines (TLs).  We saw that for all intents and purposes, an excitation traveling on a transmission line is basically equivalent to a plane wave.  In today’s exercise we are basically going to hammer this point home by looking at a coax cable transmission line.  We already looked at parallel plate transmission lines in class, and though we will be focusing on these for most of the rest of the class, coax cables are probably to transmission line you will more frequently encounter in the real world.  As in our previous few exercises, we are going to use real numbers to describe our propagating waves.

The Coax Cable

In previous lectures, we determined the capcitance and inductance per unit length of a coax.  If you remember, for capacitance, we assumed a charge Q on the inner conductor, and determined the E field between the inner and outer conductor and took the line integral to find the voltage between the inner and outer condutors.  The ratio of charge to voltage (Q/V) is our capacitance, which we found to be  C=2πlϵ/ln(b/a), where l is the length of our coax and a,b are the inner and outer diameters of our cable.  The capacitance per unit length is then just 𝒞=2πϵ/ln(b/a).  For our inductance, we assumed a current I on the inner conductor, calculated the B field from this current, and then integrated the B field over the area spanning the distance between the conductors and the length of the wire to get a relationship between the B field flux and our current, with the ratio Ψ/I giving our inductance L=lμln(b/a)/2π, and our conductance per unit length ℒ=μln(b/a)/2π.
If our unknowns are μ,ϵ,a, and b, let’s write expression for the ℒ and 𝒞 of our coax:

muo=4*Pi*10^-7;epso=8.85*10^-12;

lcoax[mu1_,eps1_,a_,b_]:=mu1*muo*Log[b/a]/(2*Pi);

ccoax[mu1_,eps1_,a_,b_]:=2*Pi*eps1*epso/Log[b/a];

Our propagation velocity (which shouldn’t actual depend on geometry) is then

vpcoax[mu1_,eps1_,a_,b_]:=1/Sqrt[lcoax[mu1,eps1,a,b]*ccoax[mu1,eps1,a,b]]

and our characteristic impedance of the line is

zcoax[mu1_,eps1_,a_,b_]:=Sqrt[lcoax[mu1,eps1,a,b]/ccoax[mu1,eps1,a,b]]

If we assume our wave is traveling in the +(OverHat[z]) direction, with a frequency ω=2π*(10^10)rad/s (10GHz, RF frequency), then we should be able to plot V(z,t) and I(z,t).  Let’s first get an idea as to our wave’s period (the wavelength will depend on the material properties inbetween the conductors):

w=2*Pi*10^10; c=2.9979*10^8;
per=2*Pi/w

In free space, we can find the wavelength for this frequency wave:

vpcoax[1,1,.01,.05]

lamo=2*Pi*vpcoax[1,1,.01,.05]/w

This corresponds to a free-space wavelength of ~3cm.  Our V and I are then

vcoax[mu1_,eps1_,a_,b_,vo_]:=Re[vo*Exp[I*w*(t-z/vpcoax[mu1,eps1,a,b])]]

icoax[mu1_,eps1_,a_,b_,vo_]:=Re[(vo/zcoax[mu1,eps1,a,b])*Exp[I*w*(t-z/vpcoax[mu1,eps1,a,b])]]

Animate[Plot[{vcoax[1,1,.01,.04,1],icoax[1,1,.01,.04,1]}/.{t->to},{z,-10*10^-2,10*10^-2}],{to,0,10^-9}]

This, of course, just looks like a propagating plane wave.  Instead of E and H, we are plotting V and I.  In some ways this is easier to understand, since V and I are scalars, and E and H are othogonal vector fields.  However, it is important to recognize that V and I are linked to E and H.  For the coax, we know:
ΔV=∫(OverVector[E])·d(OverVector[r]) from r=a to r=b, where E=Q/2πϵlr or E=λ/2πϵr, where λ=Q/l is the charge density.  We can relate E and V through C, λ= 𝒞V.  Now we write: E=𝒞V/2πϵr, we can plot this in 2D by slicing through the center of the coax:

ecoax[mu1_,eps1_,a_,b_,vo_]:=If[a<Abs[x]<b,ccoax[mu1,eps1,a,b]*vcoax[mu1,eps1,a,b,vo]/(2*Pi*eps1*epso*Abs[x]),0]

Animate[DensityPlot[ecoax[1,1,.01,.03,1]/.{t->to},{z,-10*10^-2,10*10^-2},{x,-10*10^-2,10*10^-2},ColorFunction->"RedBlueTones", PlotPoints->100],{to,0,10^-9},AnimationRate->.1]

Using the same approach that we used for the electric field, plot the H field in the coax as a function of position, animated in time:

hcoax[mu1_,eps1_,a_,b_,vo_]:=

Animate[DensityPlot[hcoax[1,1,.01,.03,1]/.{t->to},{z,-10*10^-2,10*10^-2},{x,-10*10^-2,10*10^-2},ColorFunction->"RedBlueTones", PlotPoints->100],{to,0,10^-9},AnimationRate->.1]

Try to keep these plots in mind as we move forward in our discussion of transmission lines.  For the most part, we are going to ignore the E and H fields in our discussion of TLs, using only V and I.  However, it is important to remember that the Voltage and Current are directly related to the E and H in our TLs...as this is the physical picture of what is happening in the entire TL system.

Finally, many TLs are described using only the characteristic impedance of the line.  Using what we have done above, calculate geometries for coax cables that give Z=50 and 75Ω, assuming that the core dielectric is Polyethylene with (Subscript[ϵ, r])=2.3 and (Subscript[μ, r])=1. Try to use values that make sense (i.e. it wouldn’t be particularly convenient to have an inner conductor with radius of 4m!).  Note your values below.

zcoax[]