1.11 Transient signals on transmission lines
Department of Electrical and Computer Engineering
ECSE 352 Electromagnetic Waves
1.11 Transient signals on transmission lines References: Hayt and Buck 10.14
+
V
2R0
Vg
0.8 0.6
R0 -
1
3R0
0.4 0.2 2.5
5
7.5
10
12.5
time Module 1c: Digital signal transmission
15
17.5
20
Take note Assignment #2: • Due Tuesday in class before 10:30. • A penalty will apply on late assignments • Problem 2: 300 , not 300 W Class test Tuesday Oct 8 on module 1-chapter 10
1-2
Transmission line classes A. 1. 2. 3. 4. 5.
Transmission line basics Introduction Lossless transmission lines Harmonic waves on transmission lines Time independent (phasor) notation Power transmission and loss
B. 6. 7. 8. 9. 10.
Analog signal transmission Resistive loads and reflections Voltage standing wave ratio Finite lines and input impedance Smith chart calculation techniques Stub matching
C. Digital signal transmission 11. Transient analysis 12. Pulse propagation and initially charged lines 1-3
Transients on transmission lines With digital systems, often pulses are propagated down a transmission line (e.g. bit or clock signal) • What happens at a termination? • What is instantaneous voltage at the load? • What is the instantaneous voltage at other points on the line? V1+ + Vg
V1+
Zg
VL
ZL -
-
1-4
Contents • Applications of transient pulses • Reflection coefficients • Reflection diagrams
1-5
Transient pulses Signals examined thus far • Steady-state • Single frequency This is important for RF and analog systems However, there is a large range of systems in which transient pulses or edges are travelling. These are not steady state or single frequency
1-6
Time harmonic vs. transient signals Harmonic signal: • Single frequency component • Single propagation constant • Steady state
Transient pulse: • Multiple frequency components • Range of propagation constants
Signal
Spectrum
t
t
Vs z V0 e z e j z V0 e z e j z 1-7
Review: Types of termination There are 4 different possibilities for line termination +
Zg
Vg
+ Z0
Z0
Vg
-
Z0 -
Matched impedance + Vg
Zg
Open circuit
Zg
+ Z0
-
Vg
Zg Z0
ZL
-
Short circuit
Arbitrary termination
1-8
Voltage reflection coefficients Power efficiency
PL 4s 2 1 2 P 1 s
Load reflection coefficient L
Z L Z0 j L L L e Z L Z0 Generator reflection coefficient g
g
Z g Z0 Z g Z0
g e
j g
1-9
Propagating edge on lossless line Close switch to generate voltage edge V1+
R0 V1 V0 R0 Rg
Travels down line at speed v
v 1/ LC
Current pulse I1+ also generated
V V0 1 I1 R0 R0 Rg
V1+ + V0 -
Rg
I1+
Ro
RL
z
1-10
Voltage along line Voltage edge reaches position z1 at time t=z1/v V1+ +
Rg
V0
RL
Ro
-
z1
z
Voltage at z=z1 as a function of time: V(z1,t) V1+
0
z1/v
t 1-11
Example: Matched termination Parameters: V0=1, v=1, length=5 V + V0 -
2R0 R0
R0
z What is the value of V1+? What is I1+? Is the generator impedance matched? Are there any reflections? 1-12
Reflections at termination The edge reaches the line end at time T=l/v, where T is the propagation time for the line V1+ + V0 -
V1-
Rg
l
Ro
RL
z
A reflected edge is generated
V1 L V1
where
RL R0 L RL R0
The voltage at the load is thus
VL V1 V1 1-13
Reflections at generator At time t=2T=2l/v the reflected edge reaches the generator V1+ + V0 -
Rg
l
A second reflection is generated where
V1-
Ro
RL
z
V2 g V1 g
Rg R0 Rg R0
At time t=3T=3l/v this edge reaches the load 1-14
Example: Voltage reflections Parameters: V0=1, v=1, length=5
+ V0 -
V
2R0 R0
3R0
z
1-15
Voltage as function of time
H L
1
1 V z1 = €€€€ 3
0.8
VL=
+
1 €€€€ 2
2R0
V0
0.6
R0
-
0.4
3R0
0.2 21
0
4 2
6
38
4 10
125
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2 2.5
5
7.5
10
12.5
15
17.5
20
time
2.5
5
7.5
10
12.5
15
17.5
20
time 1-16
Current reflections We can also calculate the reflected current edge at the load V1 I1 R0 and at the generator
I2
V2 R0
1-17
Example: Current reflections Parameters: V0=1, v=1, length=5 I + V0 -
2R0 R0
3R0
z
1-18
Current as function of time (R0=1)
HL
1
I z1 =
0.8
0
IL=
+
0
2R0
V0
0.6
R0
-
0.4
3R0
0.2 0
21
4
26
83
10 4
12
5
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2 2.5
5
7.5
10
12.5
15
17.5
20
time
2.5
5
7.5
10
12.5
15
17.5
20
time 1-19
Final voltage at the load What is the load voltage as t ? By summation of all edges VL V1 V1 V2 V2 V3
V1 1 L g L g L 2 g 2 L 2 2 3 V1 1 L 1 g L g L g L
1 L VL V 1 g L 1
an n 0
1 1 a
a 1
1-20
Mathematical note N 1 a n a 1 a n 0 N 1
Proof N 1
N 1
N 1
n 0
n 0
n 0
a 1
1 a a n a n a n1
1 a a ... a 2
N 1
a a
2
... a
N
1 aN In the limit of N
1 a 1 a n 0 n
a 1 1-21
Solution 1 L Voltage at load VL V1 1 g L R0 Since V1 V0 R0 Rg
then
+
R0
V0
RL
-
RL VL V0 Rg RL
For the example given earlier (Rg=2R0, RL=3R0) VL=0.6V0
Rg
+
2R0 R0
V0
3R0
-
1-22
Current solution Similarly, the current at the load is
1 L V1 I L 1 g L R0 this reduces to V0 IL Rg RL
+
R0
V0 R0 V V 1 0 R0 Rg
Rg RL
-
Lossless transmission lines have no static resistance, but do possess a dynamic impedance.
1-23
Reflection diagrams Reflection diagrams are a simple method for determining current and voltage at any position and time on the line • Graphical technique • Aid to visualization • Space-time plot
1-24
Reflection diagram v 1/ LC V3+= L2 g2 V1+
t t5 t4
4T
t2
Find voltage distribution at z=z1
V2-= L2 g V1+ V2+=
t3
T l/v
3T
L g V1+
2T
1. Draw line to z1 2. Note the voltage event as they come with increasing time
V1-= L V1+
T
V1+ t1
0
z1
l
z 1-25
Voltage discontinuity Voltage discontinuity at z1 Time range
Voltage
Discontinuity
0≤t≤t1 (t1=z1/v)
0
0
t1≤t≤t2 (t2=2T-t1)
V1+
V1+ at t1
t2≤t≤t3 (t3=2T+t1)
V1+(1+L)
L V1+ at t2
t3≤t≤t4 (t4=4T-t1)
V1+(1+L+ g L)
g L V1+ at t3
t4≤t≤t5 (t5=4T+t1)
V1+(1+L+ g L+ g L2) g L2 V1+ at t4
1-26
Example Use reflection diagram to plot voltage and current at an arbitrary point z1 for a transmission line with R0=50, RL=3R0 and Rg=2R0 and generator voltage is V0=30V V1+=30/(2+1)=10 V L=(3-1)/(3+1)=0.5 g=(2-1)/(2+1)=0.333
+
Rg R0
V0
RL
-
What are the DC voltage and current values? Vz1,DC=18V Iz1,DC=0.12A 1-27
Voltage reflection diagram at z1 t V1
6l/v 5l/v
16.67
17.5 17.8
15
4l/v 10
3l/v 2l/v T=l/v
0
z1
l
0
T
t1=z1/v
2T
3T time
4T
5T
2T-t1 4T-t1 2T+t1 4T+t1 1-28
Current reflection diagram at z1 t I1
6l/v
0.2
5l/v 4l/v
0.133
3l/v
0.122 0.116
0.1
2l/v T=l/v
0
z1
l
0
T
t1=z1/v
2T
3T time
4T
5T
2T-t1 4T-t1 2T+t1 4T+t1 1-29
Review questions How long does it take for the front edge to reach the generator when the line length is l and the phase velocity is v? How do the voltage and current reflection diagrams differ? Does the impedance of the transmission line have any effect on the final (steady-state) result?
1-30
Before next class... Based on this class: – Example 10.11, 10.12 – Problem 10.37 Read section 10.14 Assignment #2 due Tuesday
1-31
Department of Electrical and Computer Engineering
ECSE 352 Electromagnetic Waves
1.12 Pulses and initially charged lines References: Section 10.14
V=-Vo
V=Vo
V=0
Zo l/2
Module 1c: Digital signal transmission
l
RL l
Overview In the previous class we introduced new concepts to deal with transient signals on transmission lines. We used reflection diagrams to describe the evolution of voltage and current for transient signals. Here we extend this analysis to consider pulses on transmission lines, where there is a trailing edge as well as a leading edge. We also consider the case of initially charged lines, which can be used to deliver pulses of a precise duration.
1-33
Transmission line classes A. 1. 2. 3. 4. 5.
Transmission line basics Introduction Lossless transmission lines Harmonic waves on transmission lines Time independent (phasor) notation Power transmission and loss
B. 6. 7. 8. 9. 10.
Analog signal transmission Resistive loads and reflections Voltage standing wave ratio Finite lines and input impedance Smith chart calculation techniques Stub matching
C. Digital signal transmission 11. Transient analysis 12. Pulse propagation and initially charged lines 1-34
Contents • Reflection diagrams • Pulses on transmission lines • Initially charged lines
1-35
Pulse excitation Often we are interested in pulse propagation E.g. clock pulses, digital signals V1+ + V0 -
Rg Ro
RL
1-36
Pulse representation A pulse is a combination of two step functions, 1 up + 1 down Step function u(t) Vg(t) u(t) V0
1 0
0 t 0 u t 1 t 0
t T0
Vg t V0 u t u t T0
1-37
Pulse propagation E.g. RL=3R0, Rg=2R0, T0=2 V +
2R0 R0
V0 -
3R0
z
1-38
Pulse propagation reflection diagram Add delayed negative edge
t 4T
Draw line at point of interest 2T+T0 Determine time of 2T interest Take sum positive and T0 negative pulses 0 present
3T
T
-1 +1 z1
l
z 1-39
Example An air-dielectric transmission line is 9 cm long and has a characteristic impedance of 50. The generator and load resistances are 25 and 100 respectively. The generator voltage is 12V. Plot the voltage reflection diagram for a pulse width of a) 0.1 ns b) 0.7 ns V1+ = 12(50)/(25+50) = 8V L = (100-50)/(100+50) = 0.333 g = (25-50)/(25+50) = -0.333 9cm 9cm/3e8m/s = 0.3 ns 1-40
a) Pulse width = 0.1ns VL
t (ns)
10.67
5T
0.013
3T
-1.2
0
T
T = l/v = 0.3 ns 0.1 ns 0
0.9
2T
3T
4T
5T time
z (m) 1-41
b) Pulse width = 0.7ns VL
t (ns)
10.67
5T
9.48
3T
-1.2
0
T
T = l/v = 0.3 ns 0.1 ns 0
0.9
2T
3T
4T
-1.17
5T time
z (m) 1-42
Lines with electrostatic charge The discharge of an initially charged line can also be considered V+ 1 I1+
Vg Rg
V0 + Ro
l
When the switch is turned on, a current flows counter clockwise that equals to I1 V1 Z 0 It is associated with a voltage wavefront that propagates R0 V 1 V0 VR V0 V1 I1 Rg Rg V1 R0 Rg Z0 1-43
Example A charged transmission line has a characteristic impedance of 50and V0=12V. The generator resistance is 25 . Plot the reflection diagrams for voltage and current at generator Rg
+ V0 -
Ro l
V1+ = -12(50)/(25+50) = -8V L = (-50)/(+50) = 1 g = (25-50)/(25+50) = -0.333 1-44
V&I reflection diagrams t (ns) 5T Vg, Ig 4 0.053
3T
0.44 0.0176
-1.33
-0.16
T
0
2T
3T
4T
5T time
V0= 12V
0
T
l
z (m) 1-45
Special case R=R0 • Initial voltage wavefront V1+ propagates with negative jump and reduces the voltage along the line by a factor of two • When the pulse reaches the end line, =1 • The reflected pulse V1- cancels the remaining voltage on its way back until reaching the generator load -Vo/2 V1+ Rg
Ro=R
+ V0 -
1-46
Before next class... Based on this class • Example 10.12 • Problem 10.39 • Tutorial questions Thursday class: 11.1 Wave propagation in free-space Friday class: Quick review of transmission lines
1-47
End of module 1 Class test Tuesday Oct 8 Covering chapter 10 of the book, module 1 of the slides 50 minutes in duration Standard calculators, compass, ruler allowed Closed book Formula sheet provided Smith chart provided 1-48
End of module 1
Frozen-wave generator A frozen wave generator is shown below. Both switches are closed simultaneously at t = 0. Construct an appropriate voltage reflection diagram for the case in which RL = Z0. Determine and plot the load voltage as a function of time V=-Vo
V=Vo
V=0
Zo l/2
l
RL l
1-50
Frozen-wave generator Closing the switches sets up a total of four voltage wavefronts . Note that the first and second waves from the left are of magnitude V0, since in fact we are superimposing voltage waves from the −V0 and +V0 charged sections acting alone. The reflection diagram is drawn and is used to construct the load voltage with time by accumulating voltages up the right hand vertical axis. V=-Vo
V=Vo
V=0
Zo l/2
l
RL l 1-51
Frozen-wave generator
1-52
1-53
a) Pulse width = 0.1ns
1-54
ECSE 352 Electromagnetic Waves
1.11 Transient signals on transmission lines References: Hayt and Buck 10.14
+
V
2R0
Vg
0.8 0.6
R0 -
1
3R0
0.4 0.2 2.5
5
7.5
10
12.5
time Module 1c: Digital signal transmission
15
17.5
20
Take note Assignment #2: • Due Tuesday in class before 10:30. • A penalty will apply on late assignments • Problem 2: 300 , not 300 W Class test Tuesday Oct 8 on module 1-chapter 10
1-2
Transmission line classes A. 1. 2. 3. 4. 5.
Transmission line basics Introduction Lossless transmission lines Harmonic waves on transmission lines Time independent (phasor) notation Power transmission and loss
B. 6. 7. 8. 9. 10.
Analog signal transmission Resistive loads and reflections Voltage standing wave ratio Finite lines and input impedance Smith chart calculation techniques Stub matching
C. Digital signal transmission 11. Transient analysis 12. Pulse propagation and initially charged lines 1-3
Transients on transmission lines With digital systems, often pulses are propagated down a transmission line (e.g. bit or clock signal) • What happens at a termination? • What is instantaneous voltage at the load? • What is the instantaneous voltage at other points on the line? V1+ + Vg
V1+
Zg
VL
ZL -
-
1-4
Contents • Applications of transient pulses • Reflection coefficients • Reflection diagrams
1-5
Transient pulses Signals examined thus far • Steady-state • Single frequency This is important for RF and analog systems However, there is a large range of systems in which transient pulses or edges are travelling. These are not steady state or single frequency
1-6
Time harmonic vs. transient signals Harmonic signal: • Single frequency component • Single propagation constant • Steady state
Transient pulse: • Multiple frequency components • Range of propagation constants
Signal
Spectrum
t
t
Vs z V0 e z e j z V0 e z e j z 1-7
Review: Types of termination There are 4 different possibilities for line termination +
Zg
Vg
+ Z0
Z0
Vg
-
Z0 -
Matched impedance + Vg
Zg
Open circuit
Zg
+ Z0
-
Vg
Zg Z0
ZL
-
Short circuit
Arbitrary termination
1-8
Voltage reflection coefficients Power efficiency
PL 4s 2 1 2 P 1 s
Load reflection coefficient L
Z L Z0 j L L L e Z L Z0 Generator reflection coefficient g
g
Z g Z0 Z g Z0
g e
j g
1-9
Propagating edge on lossless line Close switch to generate voltage edge V1+
R0 V1 V0 R0 Rg
Travels down line at speed v
v 1/ LC
Current pulse I1+ also generated
V V0 1 I1 R0 R0 Rg
V1+ + V0 -
Rg
I1+
Ro
RL
z
1-10
Voltage along line Voltage edge reaches position z1 at time t=z1/v V1+ +
Rg
V0
RL
Ro
-
z1
z
Voltage at z=z1 as a function of time: V(z1,t) V1+
0
z1/v
t 1-11
Example: Matched termination Parameters: V0=1, v=1, length=5 V + V0 -
2R0 R0
R0
z What is the value of V1+? What is I1+? Is the generator impedance matched? Are there any reflections? 1-12
Reflections at termination The edge reaches the line end at time T=l/v, where T is the propagation time for the line V1+ + V0 -
V1-
Rg
l
Ro
RL
z
A reflected edge is generated
V1 L V1
where
RL R0 L RL R0
The voltage at the load is thus
VL V1 V1 1-13
Reflections at generator At time t=2T=2l/v the reflected edge reaches the generator V1+ + V0 -
Rg
l
A second reflection is generated where
V1-
Ro
RL
z
V2 g V1 g
Rg R0 Rg R0
At time t=3T=3l/v this edge reaches the load 1-14
Example: Voltage reflections Parameters: V0=1, v=1, length=5
+ V0 -
V
2R0 R0
3R0
z
1-15
Voltage as function of time
H L
1
1 V z1 = €€€€ 3
0.8
VL=
+
1 €€€€ 2
2R0
V0
0.6
R0
-
0.4
3R0
0.2 21
0
4 2
6
38
4 10
125
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2 2.5
5
7.5
10
12.5
15
17.5
20
time
2.5
5
7.5
10
12.5
15
17.5
20
time 1-16
Current reflections We can also calculate the reflected current edge at the load V1 I1 R0 and at the generator
I2
V2 R0
1-17
Example: Current reflections Parameters: V0=1, v=1, length=5 I + V0 -
2R0 R0
3R0
z
1-18
Current as function of time (R0=1)
HL
1
I z1 =
0.8
0
IL=
+
0
2R0
V0
0.6
R0
-
0.4
3R0
0.2 0
21
4
26
83
10 4
12
5
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2 2.5
5
7.5
10
12.5
15
17.5
20
time
2.5
5
7.5
10
12.5
15
17.5
20
time 1-19
Final voltage at the load What is the load voltage as t ? By summation of all edges VL V1 V1 V2 V2 V3
V1 1 L g L g L 2 g 2 L 2 2 3 V1 1 L 1 g L g L g L
1 L VL V 1 g L 1
an n 0
1 1 a
a 1
1-20
Mathematical note N 1 a n a 1 a n 0 N 1
Proof N 1
N 1
N 1
n 0
n 0
n 0
a 1
1 a a n a n a n1
1 a a ... a 2
N 1
a a
2
... a
N
1 aN In the limit of N
1 a 1 a n 0 n
a 1 1-21
Solution 1 L Voltage at load VL V1 1 g L R0 Since V1 V0 R0 Rg
then
+
R0
V0
RL
-
RL VL V0 Rg RL
For the example given earlier (Rg=2R0, RL=3R0) VL=0.6V0
Rg
+
2R0 R0
V0
3R0
-
1-22
Current solution Similarly, the current at the load is
1 L V1 I L 1 g L R0 this reduces to V0 IL Rg RL
+
R0
V0 R0 V V 1 0 R0 Rg
Rg RL
-
Lossless transmission lines have no static resistance, but do possess a dynamic impedance.
1-23
Reflection diagrams Reflection diagrams are a simple method for determining current and voltage at any position and time on the line • Graphical technique • Aid to visualization • Space-time plot
1-24
Reflection diagram v 1/ LC V3+= L2 g2 V1+
t t5 t4
4T
t2
Find voltage distribution at z=z1
V2-= L2 g V1+ V2+=
t3
T l/v
3T
L g V1+
2T
1. Draw line to z1 2. Note the voltage event as they come with increasing time
V1-= L V1+
T
V1+ t1
0
z1
l
z 1-25
Voltage discontinuity Voltage discontinuity at z1 Time range
Voltage
Discontinuity
0≤t≤t1 (t1=z1/v)
0
0
t1≤t≤t2 (t2=2T-t1)
V1+
V1+ at t1
t2≤t≤t3 (t3=2T+t1)
V1+(1+L)
L V1+ at t2
t3≤t≤t4 (t4=4T-t1)
V1+(1+L+ g L)
g L V1+ at t3
t4≤t≤t5 (t5=4T+t1)
V1+(1+L+ g L+ g L2) g L2 V1+ at t4
1-26
Example Use reflection diagram to plot voltage and current at an arbitrary point z1 for a transmission line with R0=50, RL=3R0 and Rg=2R0 and generator voltage is V0=30V V1+=30/(2+1)=10 V L=(3-1)/(3+1)=0.5 g=(2-1)/(2+1)=0.333
+
Rg R0
V0
RL
-
What are the DC voltage and current values? Vz1,DC=18V Iz1,DC=0.12A 1-27
Voltage reflection diagram at z1 t V1
6l/v 5l/v
16.67
17.5 17.8
15
4l/v 10
3l/v 2l/v T=l/v
0
z1
l
0
T
t1=z1/v
2T
3T time
4T
5T
2T-t1 4T-t1 2T+t1 4T+t1 1-28
Current reflection diagram at z1 t I1
6l/v
0.2
5l/v 4l/v
0.133
3l/v
0.122 0.116
0.1
2l/v T=l/v
0
z1
l
0
T
t1=z1/v
2T
3T time
4T
5T
2T-t1 4T-t1 2T+t1 4T+t1 1-29
Review questions How long does it take for the front edge to reach the generator when the line length is l and the phase velocity is v? How do the voltage and current reflection diagrams differ? Does the impedance of the transmission line have any effect on the final (steady-state) result?
1-30
Before next class... Based on this class: – Example 10.11, 10.12 – Problem 10.37 Read section 10.14 Assignment #2 due Tuesday
1-31
Department of Electrical and Computer Engineering
ECSE 352 Electromagnetic Waves
1.12 Pulses and initially charged lines References: Section 10.14
V=-Vo
V=Vo
V=0
Zo l/2
Module 1c: Digital signal transmission
l
RL l
Overview In the previous class we introduced new concepts to deal with transient signals on transmission lines. We used reflection diagrams to describe the evolution of voltage and current for transient signals. Here we extend this analysis to consider pulses on transmission lines, where there is a trailing edge as well as a leading edge. We also consider the case of initially charged lines, which can be used to deliver pulses of a precise duration.
1-33
Transmission line classes A. 1. 2. 3. 4. 5.
Transmission line basics Introduction Lossless transmission lines Harmonic waves on transmission lines Time independent (phasor) notation Power transmission and loss
B. 6. 7. 8. 9. 10.
Analog signal transmission Resistive loads and reflections Voltage standing wave ratio Finite lines and input impedance Smith chart calculation techniques Stub matching
C. Digital signal transmission 11. Transient analysis 12. Pulse propagation and initially charged lines 1-34
Contents • Reflection diagrams • Pulses on transmission lines • Initially charged lines
1-35
Pulse excitation Often we are interested in pulse propagation E.g. clock pulses, digital signals V1+ + V0 -
Rg Ro
RL
1-36
Pulse representation A pulse is a combination of two step functions, 1 up + 1 down Step function u(t) Vg(t) u(t) V0
1 0
0 t 0 u t 1 t 0
t T0
Vg t V0 u t u t T0
1-37
Pulse propagation E.g. RL=3R0, Rg=2R0, T0=2 V +
2R0 R0
V0 -
3R0
z
1-38
Pulse propagation reflection diagram Add delayed negative edge
t 4T
Draw line at point of interest 2T+T0 Determine time of 2T interest Take sum positive and T0 negative pulses 0 present
3T
T
-1 +1 z1
l
z 1-39
Example An air-dielectric transmission line is 9 cm long and has a characteristic impedance of 50. The generator and load resistances are 25 and 100 respectively. The generator voltage is 12V. Plot the voltage reflection diagram for a pulse width of a) 0.1 ns b) 0.7 ns V1+ = 12(50)/(25+50) = 8V L = (100-50)/(100+50) = 0.333 g = (25-50)/(25+50) = -0.333 9cm 9cm/3e8m/s = 0.3 ns 1-40
a) Pulse width = 0.1ns VL
t (ns)
10.67
5T
0.013
3T
-1.2
0
T
T = l/v = 0.3 ns 0.1 ns 0
0.9
2T
3T
4T
5T time
z (m) 1-41
b) Pulse width = 0.7ns VL
t (ns)
10.67
5T
9.48
3T
-1.2
0
T
T = l/v = 0.3 ns 0.1 ns 0
0.9
2T
3T
4T
-1.17
5T time
z (m) 1-42
Lines with electrostatic charge The discharge of an initially charged line can also be considered V+ 1 I1+
Vg Rg
V0 + Ro
l
When the switch is turned on, a current flows counter clockwise that equals to I1 V1 Z 0 It is associated with a voltage wavefront that propagates R0 V 1 V0 VR V0 V1 I1 Rg Rg V1 R0 Rg Z0 1-43
Example A charged transmission line has a characteristic impedance of 50and V0=12V. The generator resistance is 25 . Plot the reflection diagrams for voltage and current at generator Rg
+ V0 -
Ro l
V1+ = -12(50)/(25+50) = -8V L = (-50)/(+50) = 1 g = (25-50)/(25+50) = -0.333 1-44
V&I reflection diagrams t (ns) 5T Vg, Ig 4 0.053
3T
0.44 0.0176
-1.33
-0.16
T
0
2T
3T
4T
5T time
V0= 12V
0
T
l
z (m) 1-45
Special case R=R0 • Initial voltage wavefront V1+ propagates with negative jump and reduces the voltage along the line by a factor of two • When the pulse reaches the end line, =1 • The reflected pulse V1- cancels the remaining voltage on its way back until reaching the generator load -Vo/2 V1+ Rg
Ro=R
+ V0 -
1-46
Before next class... Based on this class • Example 10.12 • Problem 10.39 • Tutorial questions Thursday class: 11.1 Wave propagation in free-space Friday class: Quick review of transmission lines
1-47
End of module 1 Class test Tuesday Oct 8 Covering chapter 10 of the book, module 1 of the slides 50 minutes in duration Standard calculators, compass, ruler allowed Closed book Formula sheet provided Smith chart provided 1-48
End of module 1
Frozen-wave generator A frozen wave generator is shown below. Both switches are closed simultaneously at t = 0. Construct an appropriate voltage reflection diagram for the case in which RL = Z0. Determine and plot the load voltage as a function of time V=-Vo
V=Vo
V=0
Zo l/2
l
RL l
1-50
Frozen-wave generator Closing the switches sets up a total of four voltage wavefronts . Note that the first and second waves from the left are of magnitude V0, since in fact we are superimposing voltage waves from the −V0 and +V0 charged sections acting alone. The reflection diagram is drawn and is used to construct the load voltage with time by accumulating voltages up the right hand vertical axis. V=-Vo
V=Vo
V=0
Zo l/2
l
RL l 1-51
Frozen-wave generator
1-52
1-53
a) Pulse width = 0.1ns
1-54
