ECE 391: Suggested Homework Problems
Reflections
1. Multiple digital devices wish to communicate over a shared 50Ω transmission line. Only one driver
will be enabled at any time and any receiver may be listening. Each bidirectional driver/receiver
pair on the line has an output impedance of (Zout < 2) and an input impedance of (Zin >
200KΩ). The input receivers logic one input threshold is V 2dd and should not see voltages above
Vdd. Determine what terminations if any are necessary at locations A,B,C and D, to create a
reflection-free network. Show a circuit diagram for your design.
Zo=50
Zo=50
B
A
Rsrc
Vsrc
+
Zo=100
A
Zo=50
C
D
Td=2.5ns
B
v,i
20
T1
−
2. For the transmission
0.5A
PULSE source parameters:
3.3v amplitude,Figure 1: Problem 1
1ns delay to first edge with 100pS tr and tf,
20ns pulse width,
line circuit
shown
in figure 2:
50ns cycle
time
Rsrc
Vs
30V
RL
40
Zo=50
A
75
15V
0
Td=250ps
10
20
Zo=75, V
B
Rg
A
T
open circuit
T1
Vg
Vs(t)=2V U(t)
50
0
Rg
A
Figure 2: Problem 2
B
Zo=75, Vp=0.66c
vA
150 t = 0ns through t = 1.75ns.
(a) draw a lattice diagram from
T1
Vg
RT
(b) simulate the circuit in
ngspice
rising edge at the
Vg(t)=30V
U(t) for 1.7ns showing the steps at the initial
driver end, and at the receiver end of the transmission line.
z
0
z r =20m
0
8V
5V
6
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Rsrc
A
B
75
Rg
A
open circuit
T1
Vs
3. Consider the transmission line circuit shown in figure 3 with Vg (t) = 30u(t)(VVg
), Rg = 50
150,
Vs(t)=2V U(t)
Z0 = 75, length zr = 20m, and velocity factor vp = 0.66c.
0
Rg
A
Vg
B
Zo=75, Vp=0.66c
150
vA
8V
T1
RT
Vg(t)=30V U(t)
5V
z
z r =20m
0
0
Figure 3: Problem 3
(a) Draw a lattice diagram and plot the voltage and current waveforms at z = 0, z = 0.75zr , and
z = zr for 0 ≤ t ≤ 430nsec for the cases (i) RT = 0, (ii)RT = 15 , and (iii) RT = 125 .
(b) Simulate the circuits in ngspice and compare with your results.
(c) What are the final values of voltage and current (i.e., for t → ∞) for the three terminations?
4. An uncharged, lossless coaxial transmission line with r = 4 is short-circuited at its far end
(z = zr ). At time t = 0, the near end of the cable (z = 0) is connected through a series resistance
Rg to an ideal source of Vg volts. A finite portion of the voltage observed at the input (near end)
of the line is shown below. A shorter portion of the corresponding current is also shown.
v,i
Rg
30V
RL
40
0.5A
15V
0
T1
Vg
10
20
30
t (usec)
z=0
z=z r
Zo=75, Vp=200m/usFigure 4: Problem 4
damage
Rg
en circuit
RT
A
B
Find the numerical values T1
of:
50
Rt
Vg (a) velocity of propagation, vp
50
(b) the length of the line, zr
(c) the T-line characteristic impedance, Z0
(d) the reflection
0 coefficient atd the generator and termination sides (Zg and Zt )
(e) the series
resistance
at the source Rg
vA
(f) the source voltage, V0 .
8V
(e) Draw a lattice diagram and complete the voltage and current waveforms up to time t = 25µsec.
5V
Specify the explicit
numerical values of voltage and current.
0
6
t (usec)
6
(f) Simulate the circuit in ngspice and compare your voltage and current waveforms obtained
earlier with the Spice simulation results.
(g) What are the final values of voltage and current (i.e., for t → ∞) at the near end and far end?
5. For the following circuit:
Rsrc
Vsrc
+
Zo=100
A
Td=2.5ns
B
v,i
20
PULSE source parameters:
3.3v amplitude,
1ns delay to first edge with 100pS tr and tf,
20ns pulse width,
50ns cycle time
Rsrc
A
30V
RL
40
T1
−
0.5A
15V
0
FigureZo=50
5: Schematic
Td=250ps
10
Zo=75, V
B
Rg
(a) Draw the lattice diagram75that shows the voltage at points ”A”
openand
circuit ”B” for 10ns. On your
T1
Vs
lattice diagram, show the magnitudes of the reflections for both current and voltage and the 50
total
Vg
Vs(t)=2V
voltages at both ”A” and
”B”.U(t)
(b) Using your lattice diagram, draw a voltage versus time graph of the voltage waveforms at
”A” and ”B”.
(c) Confirm the correctness of
your waveforms in part b by running an ngspice simulation.
Rg
vA
A 200Ω.
B
(d) Repeat all the above with RL =
Zo=75, Vp=0.66c
Vg
150
20
T1
RT
Vg(t)=30V U(t)
A
T
0
8V
5V
z
0
z r =20m
0
6
o=100
Td=2.5ns
B
v,i
Rg
30V
T1
h 100pS tr and tf,
6. A losslessR Ltransmission line cable (Z0 = 50, vp = 200m/sec) is suspected to be damaged at an
unknown40distance d from the
0.5Ainput. The cable is terminated in a matched resistance RT = 50Ω.
15V
Vg
In order to find the location of the damaged cable, a step voltage is applied
at the input at time
t = 0, and the voltage waveform is observed at the input of the cable. The step-voltage generator
is matched to the transmission
line (RG = 50).
t (usec)
0
10
o=50
20
30
z=0
Zo=75, Vp=200m/us
Td=250ps
B
damage
Rg
A
T1
Vg
50
Rt
50
d
0
vA
B
o=75, Vp=0.66c
B
T1
open circuit
8V
T1
RT
5V
z
z r =20m
0
t (usec)
6
Figure 6: Schematic and Waveform
(a) What are the voltage and current amplitudes of the first outgoing wave?
(b) Determine the generator voltage.
(c) Determine distance d at which the cable is damaged.
(d) Determine the voltage of the returning wave (reflected at the damaged location).
(e) What is the reflection coefficient at the location of the damaged cable?
(f) How would you model (equivalent circuit model) the damaged section of the cable?
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REVISION:
OF
DRAWN BY:
T1
z=
7. An engineer started the lattice diagram shown below but was distracted and never finished his
work. Given the information below, find:
(a) ρL
(b) RL
(c) Zo
(d) td
(e) Rs
(f) Vs
RS
Zo=?, td =?
Vs
RL
t=0ns
2.25V
0.03A
t=1ns
0.7425V
t=2ns
−0.3713V
t=3ns
t=4ns
Figure 7: Unfinished Lattice Diagram
8. For each waveform below at point ”A”, circle the correct circuit parameters. Dotted lines extending from a waveform indicate the waveform will continue its present behavior.
Figure 8: Reflections at ”A”
9. Consider the circuit and spice waveform for
point
”A”
below. The generator launches the rising
Lossy
Coaxial
Cable
Rs
edge at 5ns. Find:
(a) The flight time delay tf of the transmission Zo=75
line.
75
High−Powered
RL
(b) RS
T1
Transmitter
200
(c) RL
z’
’
z=0
Rs
V1
1V
Antenna
A
Zo=75
T1
RL
tf= ?
Vsrc
2Vrms
Vsrc
1V
@10Mhz
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10. Consider the circuit below. T1 is a lossless transmission line with Zo = 50Ω td = 5ns.
Rs
25
+
1u(t) volt
T1
Rt
80
Zo=50
td=5ns
−
(a)
z
1
Zr
0
(b)
(a) Determine ρs and ρl .
(b) Fill in the numerical voltage and current values for the first three wave components and add
the time and length scales in the Rs
lattice diagram shown below. Include units as appropriate.
a1
a2
z= Zo=75
z/meters
00
1u(t) volts
75
+
1
mystery
termination
T1
−
2nS
matching network
50
vin
25
a2
b1
Zo=50
td1 = 4ns
+
2u(t) volts
T1
a1
−
a1’
T2
b2
Zo=75
td2 = 2ns
b1’
a2’
225
b2’
time/nsec
(c) Sketch the voltage at the beginning and end of the line (z = 0) and (z = Zr ) for 0 ≤ t ≤ 15ns.
Include voltages and time on the axes. Indicate voltage levels that do not fall on the axis marks.
Rs
v(z=0,Zr)
2u(t) volts
+
T1
75
Rt
25
Zo=75
td=5ns
−
Lt
125nH
z
t/nsec
0
Zr