Section 6 3 TRƯỜNG ĐIỆN TỪ

Slide Presentations for ECE 329,
Introduction to Electromagnetic Fields,
to supplement “Elements of Engineering
Electromagnetics, Sixth Edition”
by

Nannapaneni Narayana Rao

Edward C. Jordan Professor of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign, Urbana, Illinois, USA
Distinguished Amrita Professor of Engineering
Amrita Vishwa Vidyapeetham, Coimbatore, Tamil Nadu, India


6.3
Transmission-Line
Discontinuity


6.3-3

Transmission-Line Discontinuity
(+)

Z01, v p1

(++)

(–)

I + + I–

I ++
+ +

V+ + V–

V ++

– –

Z 02, vp2


6.3-4

V   V – V  
B.C.

I   I –  I  

–

V
V
V
I 
, I– –
, I  
Z01
Z 01

Z02

V  V  V  V 


Z01 Z01
Z02
Z02 

V V 
V

V

Z01






 Z02 

  Z02
V 1
 1
 V 
 Z01 
 Z01 


6.3-5

V – Z02 – Z0l
  
V
Z02  Z0l

Z01

(+)

Z02


6.3-6

Define Voltage Transmission Coefficient,
V  V   V –
V–
V   
1  

V
V
V

 V 1  
Current Transmission Coefficient,

I  I   I –
C   
1 

I
I

 C 1 – 

I–
I


6.3-7

Note that

P



 

V I



V V   C I



1   V  1    I


1  2 V  I 
1    P
2

 P






6.3-8

Three Lines in Cascade
50 

Z 0l = 50 

Z 02 = 100 

T1 = 2 s

T2 = 2 s

+
Z03 = 50 

50 
–V o
T3 = 2 s
–

(t)
 = 1/3
 = 0  V = 4/3
1/2
0

 = –1/3  = –1/3
V = 2/3 V = 2/3  = 0
2/3

4
8
t, s

12

4/9

–2/9
–2/81

2/27
2/243

4/81

6

4/9

10
4/92
4/729
14
4/93


6.3-9

Vg t   t 
n

4  1
V0 t       t  2nT2  T0 
9 n 0  9 
T0 T1  T2  T3


4/9
0

6

4/92
10

4/93
14

and so on

t, s


6.3-10

(t)

V g(t)

System

System

h(t)

– Vg (t –  ) h( ) d


6.3-11

For Vg (t) = cos t ,


V0 t   cos  t   



n

4 1
       2nT2  T0  d
9 n 0  9 


4 1
  
9 n 0  9 




n



 cos  t   


  2nT2  T0  d
n

4 1
    cos  t  2nT2  T0 
9 n 0  9 


6.3-12

n

4
 1   j 2 nT2 T0 
V 0       e
9 n 0  9 


4  jT0  1  j 2T2 
 e

 e

9

n 0  9
4  jT0
e
9

1  j 2T2
1 e
9


n

6.3-13

Vo ( ) 

49

1 – j2T
2
1– e
9
49
Vo ( ) max 
0.5
1–19
49
Vo ( ) min 
0.4
119


6.3-14

Vo ( )

0.5
0.4

0

 2T2

 T2 3  2T2 2 T2




Junction of Three Lines

Line 1

Li Z
ne 0 
2 50


6.3-15

P



Z0 = 50 

00
1

Line 1

Z0



3
ne
Li

Z0 50 

50  100 


6.3-16

100 3  50
50
1



100 3  50
250
5
4
V 1   
5
6
 C 1   
5

100
 C eff 2  C 
50  100
2
2 6 12
 C   
3
3 5 15


6.3-17

 C eff 3

50
 C 
50  100
1
1 6 6
 C   
3
3 5 15

1
2
Pref1  P  P
25

Ptrans2 =V  Ceff2 P
4 12

48
  P P
5 15
75


6.3-18

Ptrans3 =V  Ceff3 P
4 12
24
  P P
5 15
75
1
48 24
Note that


1
25 75 75