EE361 Lab 2
Basic Transmission Lines in the Frequency Domain
In this laboratory experiment, you will use SPICE to study sinusoidal waves on lossless
transmission lines. Our goal is for you to become familiar with the basic behavior of
waves reflecting from loads in transmission lines, and compare the simulations with
numeric calculations and the Smith Chart.
2.1
Basic Transmission Line Model
There is a standard lossless transmission line model T, which is specified by several
parameters. We will need to specify two of the parameters:
 Z0, the characteristic impedance
 TD, the time delay, which is the length of the line in time units.
The length of the line L is related to the time delay through
L  u p TD
(2.1)
where up is the phase velocity of waves on the transmission line.
As we saw in lecture and in our text, the phase velocity and characteristic impedance may
be derived from the “lumped element” model of the transmission line. With L’ the
inductance per unit length, and C’ the capacitance per unit length, we have
1
(2.2)
up 
L' C '
L'
(2.3)
Z0 
C'
2.1.1 A standard coaxial cable
For common RG-58 coaxial cable, the characteristic impedance is Z0 = 50 Ω and the
phase velocity up = 2/3 c. (Note: c = speed of light = 3e8 m/s)
Question 1: For such a transmission line, what are the inductance and
capacitance per meter?
For lossless coaxial cables, the following formulas relate the differential inductance L’
and capacitance C’ to the radius of the inner conductor a and the outer conductor b:
L' 
 b
ln  
2  a 
(2.4)
C' 
2
b
ln  
a
(2.5)
Question 2: For a different coaxial cable, μ = μ0 and ε = 3ε0. What is b/a if Z0
= 50 Ω?
Question 3:
2.2
If b = 3 mm in question 2.2, what is a?
A SPICE model of a transmission line problem.
Using SPICE, create a (matched) Thevenin source VAC with 1 Volt amplitude and 50 Ω
source impedance, leading to a transmission line model T, terminated in a 100 Ω load.
Edit the transmission line so that it has a characteristic impedance of 50 Ω. Also, create
labels Input and Load at the ends of the transmission lines, so that you can measure the
voltages conveniently.
ZG
T1
Input
50
1Vac
0Vdc
ZL
Load
100
Z0 = 50
TD = {delay }
VG
PARAMETERS:
0
0
delay = 5ns
0
0
Figure 1. Circuit Schematic for Part 2.2
What we would like to do is to adjust the length of the transmission line and examine the
standing wave pattern at Input over one full wavelength at a frequency of 200MHz.
Question 4: At 200 MHz, and with up = 2/3 c, what is the wavelength in the
transmission line?
What is the time delay associated with λ/16? (Hint: Remember
L
L
that TD 
)

up   f
Question 5:
Use SPICE to simulate the steady state AC response of this transmission line for length 0,
λ/16, 2λ/16, …, 15λ/16, λ. Center your sweep on the frequency of interest and sweep
linearly.
Figure 2. Illustration of Transmission Line Length Change for Part 2.2
One way to make this easier is to use a parameter for TD. Place the special part
PARAM. Double click on it and then on New Column… Call it delay and set it to 5ns.
Assign {delay} (with the curly braces) to TD on the transmission line. When you create
your simulation profile, select the parametric sweep as an option. Choose Global
Parameter with a parameter of delay. Set the sweep range and increment based on your
TD calculations from above. Under “General Settings” set the sweep Range from Start
Frequency: 200Meg to End Frequency: 200Meg and increment Total Points: 1.
Using Excel, make a table of the voltage magnitudes and current magnitudes at nodes
Input and Load for each length.
Question 6: Use PSPICE, Excel, or Matlab to plot the magnitude of the voltage
at Input as a function of length. From the Voltage Values on the plot and
V
the relationship: VSWR  max , determine the VSWR, and from the
Vmin
VSWR calculate ||.
Question 7: Use PSPICE, Excel, or Matlab to plot the magnitude of the current at
Input as a function of length. From the Current Values on the plot,
determine the VSWR, and from the VSWR calculate ||. Do the voltage
and current yield the same VSWR and ||?
Question 8: Plot the magnitude of the impedance at Input as a function of
length using the data you collected with PSPICE. Plot the Real and
Imaginary Parts of the Impedance using PSPICE and also plot impedance
using a Smith Chart.
Question 9: Using the scales at the bottom of the Smith Chart, find the VSWR
and ||. Do they agree with your previous answers?
Question 10: Compute  and VSWR directly using equations (2.6) and (2.7)
below. Do these agree with your measurements from question 6, 7 & 8?
From class recall that:
VSWR 

1 
1 
Z L  Z0
Z L  Z0
(2.6)
(2.7)
Question 11: Plot the voltage magnitude at Load as a function of length. How
does the voltage change with length? From this, how do you think the
power delivered to the load will change with length?
2.3
A shortcut, and more load impedances
SPICE has a nice mechanism for scanning in frequency, but does not directly scan the
length of the transmission line. The “electrical length” of a transmission line is βl,
l 
2

l
2f
l
up
(2.8)
Thus, changing the length of a transmission line from l to 10l achieves the same effect as
scanning the frequency from 10f to f. Or to put it differently, if a transmission line is 1 λ
at f0, then it is 0.5 λ long at 0.5f0 and 2 λ long at 2f0.
Question 12: If you have 1 meter of the coaxial cable described in question 4, at
what frequency does it have length λ/2? At what frequency does it
have length 2.5λ? (Note that we are NOT changing the physical
length of the line, only it’s “electrical length” as defined above.)
Using a 1-meter length of transmission line, adjust your SPICE simulation,
sweeping linearly in frequency from 0.5 to 2.5 wavelengths. In this simulation we
are not adjusting the Length of the Line. We are adjusting the frequency of the
system so as to produce similar effects to adjusting the length of the line.
ZG
T1
Input
ZL
Load
50
1Vac
0Vdc
100
Z0 = 50
TD = 5ns
VG
0
0
0
0
Figure 3. Circuit Schematic for Question 13 (Fixed Length)
Question 13: Plot the magnitude of the voltage at Input for the different
“lengths” (remember that you are really just adjusting the
frequency) properly relabeling the horizontal axis. (You can do
this by hand or by using text boxes in Pspice.) Does this agree
with your plot in question 6? What is the VSWR?
Replace the 100 Ω load with a 25 Ω load.
Question 14: Plot the magnitude of the voltage at Input, and compare to the
previous case of 100 Ω. From the plot, what is the VSWR? On a
Smith Chart, what similarity is there between the 100 Ω and 25 Ω
cases?
Replace the load with a “short circuit,” namely 0.001 Ω.
Question 15: Plot the magnitude of the voltage at Input. From the plot, find the
VSWR. From equations (2.6) and (2.7) calculate the VSWR. Do
these two results agree?
Replace the load with an “open circuit,” namely 1 MΩ. (remember that in
PSPICE, MEG = “mega”, M = “milli”)
Question 16: Plot the magnitude of the voltage at Input. Find the VSWR. Also,
calculate the VSWR. Do these two results agree?
Question 17: How are the plots from Question 15 and Question 16 similar? How
are these two impedances related on the Smith Chart?
2.4
Power Calculations
Calculating the actual power delivered to the load and the power in the incident and
reflected waves is a bit tedious. If you know the load impedance ZL and the generator
impedance ZG then you could first calculate the reflection coefficient at the load
L 
Z L  Z0
Z L  Z0
(2.9)
Then you can project the load upstream through the transmission line to find the input
impedance Zin
Z in  Z 0
1  L exp( 2 jl )
1  L exp( 2 jl )
(2.10)
Now we can use the voltage divider at the source to find the voltage being applied to the
transmission line
Vin  Vg
Z in
Z in  Z g
(2.11)
Now, Vin is also the sum of the two waves so that we can write
Vin = V+ exp(-jβl) + V- exp(jβl) = V+ (exp(-jβl) + Γexp(jβl))
(2.12)
So, putting this all together we have
V   Vg
Zin
1
Z in  Z g exp(  jl )   exp( jl )
(2.13)
Return to the case of 200MHz, set l = 1.65 m by adjusting TD, and set the load resistance
to 100Ω.
Question 18: Using the equations above and below (and possibly equations from
the book), calculate V+, V-, Vin and VL. How do the calculated Vin
and VL compare to your Pspice results?
Question 19: Calculate Pav (average power delivered to the load), Piav and Prav
(average power in the incident and reflected waves, respectively).
(Equations for Pav, Piav, and Prav are listed in Chapter 2 of the text.)
Show that conservation of power been satisfied.
Recall the total voltage and total current on the transmission line,
V ( z )  V  exp  jz   L exp  jz 
I ( z) 
(2.14)

V
exp  jz   L exp  jz 
Z0
(2.15)
This can be rearranged as
1
V z   Z 0 I z   V  exp  jz 
2
(2.16)
1
V z   Z 0 I z   V  exp  jz 
2
(2.17)
V ( z)  Z 0 I ( z)
 L exp( 2 jz )
V ( z)  Z0 I ( z)
(2.18)
Note that you can type equations into Probe in PSPICE to plot things like power, Γ,
VSWR. PSPICE can handle complex values. Also note that the exp(jβz) parts in the
above equations affect the phase only, not the magnitude. Thus the above equations are
ready to be input into PSPICE, by setting the Voltage and Current values to the
appropriate side of the transmission line (Input or Load). By default, Probe plots the
magnitude, but you can plot the real or imaginary parts, or the magnitude or phase by
using the functions provided in the dialog box that comes up when you add a trace.
Question 20: Use SPICE to find the magnitude of V+ and V- at Input. Use these
values to compute Piav and Prav. Do these match your answers in
Questions 18 and 19?