Post on 03-Nov-2021
Electromagnetics II
Unit 2:Line Terminated by Resistive Load, Transmission Line Discontinuity
Lecturer: Professor Jean-Fu KiangText book: N. N. Rao, “Elements of Engineering
Electromagnetics,” sixth ed., Pearson, 2004.1
Unless noted, the course materials are licensed under Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan (CC BY-NC-SA 3.0)
Voltage and Current in Transmission Line
2
V1(z) V1e jk1z V1e
jk1z
I1(z) V1
Z01
e jk1z V1
Z01
e jk1z
V2 (z) V2e jk2z
I2 (z) V2
Z02
e jk2z
Reflection and Transmission Coefficients between Two Transmission Lines (1/2)
3
1
1
1 2/
1 1 2
1 2/
1 1 2
01 02 01 02
At 0, KVL: (0) (0)
11
KCL: (0 ) (0 )1
2
V
v v
Vv v
zV V
V V V
I IV V V
Z Z Z Z
Reflection and Transmission Coefficients between Two Transmission Lines (2/2)
4
v Z02 Z01
Z02 Z01
v 2Z02
Z02 Z01
solve 1 , 2
Example 1 - Three Transmission Lines Matched at source and load
5
Time-space Diagram of Example 1
6
Output Voltage with Delta Input in Example 1
7
0 0 2 0 22 3
0 20
4 4 4( ) ( ) ( 2 ) ( 4 ) ...9 9 9
4 1 ( 2 ) 9 9
( ): impulse response function
o
n
n
v t t T t T T t T T
t T nT
R t
vi (t) (t)
0 In general : ( ) ( ) ( )o iv t R t v d
Output Voltage with Harmonic Inputin Example 1
8
V o( ) (4 / 9)e jT0
1 19
e j 2T2
Frequency response function0 2
0 20
( 2 )
0
If ( ) cosConsider a dispersion-free system
4 1( ) cos ( 2 )9 9
4 1Re9 9
i
n
on
nj T nT j t
n
v t t
v t t T nT
e e
Transmission-line Discontinuity with Inserted Resistor
9
V1(z) V1e jk1z V1e
jk1z
I1(z) V1
Z01
e jk1z V1
Z01
e jk1z
V2 (z) V2e jk2z
I2 (z) V2
Z02
e jk2z
Evaluation of Reflection and Transmission Coefficients
10
B.C. at z 0, KVL: V1(0) V2 (0)
V1 V1 V2 1 v v (1)
KCL: I1(0) I2 (0) IL
IL
V2ZL
V1 V1
Z01
1
Z02
1ZL
1/ Z02||ZL
V2 /V1 1 v
Z01
v
Z02 || ZL
(2)
Equivalent Lumped-circuit of an Infinitesimal Parallel-plate Waveguide
11
v (ZL || Z02 ) Z01
(ZL || Z02 ) Z01
v 1 v
Time-domain Reflectometry
12
ZL 120, Z02 60ZL || Z02 40
v 40 6040 60
15
v 1 v 45
Example 3: Inductive Load
13
Example 3: Solution
14
0 0
0 0
0
0
0
, at
2 2
2( ) 0
( )2
L
t T z l
V V VdV Ldt Z Z
VdVL VZ dtI t T
VV t T
V V0
2 Ae(tT )/(L/Z0 ),
let L LZ0
I.C. : V0
2
V0
2 A
A V0
V V0
2V0e
(tT )/ L , t T ; I V0
2Z0
V0
Z0
e(tT )/ L , t T
} initial condition
Exercise: Capacitive Load
15
0
0 0
0 0
00
0
0
( )/( )0
0
( )/ ( )/0 0 00
0 0
( )
2 2
2
( ) 0 02
( ) (I.C.)2
2(I.C.)
, 2 2
c
c c
CT
t Z C
t t
t T z lV V VdC V
dt Z ZVdVCZ V
dtVV t T V
VV t T
VV Ae
A VV V VV V e I e
Z Z
Voltage and Current at z=l
16
tT
V0 / Z0
V0
I z l
V z l
Propagation of V and I Waves
17
V (z,0) 12
V (z,0) Z0I (z,0)
V (z,0) 12
V (z,0) Z0I (z,0)
{
V (z,0) V (z,0)V (z,0)Z0I (z,0) V (z,0)V (z,0)
t 0
V z,t
V z,t t
t
Propagation of V and I Waves
18
I z,t
I z,t
t
t
Example 4
19
Z0 50, RL 150
T 1s, V0 100V
At t 0 :
KVL:RLI V0 V
RLV
Z0
V0 V
00
0
00
25V
1 1 A2
L
L
ZV VR Z
I VR Z
Conservation of Energy
20
We 12
CV 20l
12
CV 20vpT
12
CV 20
1LC
T 12
V 20
Z0
T 104 J
Wm 12
LI 20l 0, at t 0
Wd dt0
V 2 (0,t)
RL
104 J
Copyright Declaration
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Page Work Licensing Author/Source
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Jean-Fu Kiang / National Taiwan UniversityThis work is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan
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Jean-Fu Kiang / National Taiwan UniversityThis work is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan
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Jean-Fu Kiang / National Taiwan UniversityThis work is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan
Copyright Declaration
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Page Work Licensing Author/Source
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Jean-Fu Kiang / National Taiwan UniversityThis work is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan
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Jean-Fu Kiang / National Taiwan UniversityThis work is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan
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Jean-Fu Kiang / National Taiwan UniversityThis work is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan
Copyright Declaration
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Page Work Licensing Author/Source
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Jean-Fu Kiang / National Taiwan UniversityThis work is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan
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Jean-Fu Kiang / National Taiwan UniversityThis work is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan
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Jean-Fu Kiang / National Taiwan UniversityThis work is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan
Copyright Declaration
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Page Work Licensing Author/Source
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Jean-Fu Kiang / National Taiwan UniversityThis work is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan
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Jean-Fu Kiang / National Taiwan UniversityThis work is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan
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Jean-Fu Kiang / National Taiwan UniversityThis work is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan
Copyright Declaration
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Page Work Licensing Author/Source
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Jean-Fu Kiang / National Taiwan UniversityThis work is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan