Section 6 2 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


6.2
Line Terminated
by Resistive Load


6.2-3

Notation

1
I  z , t    V  t  z v p   V  t  z v p 
Z0

V  z , t  V  t  z v p   V  t  z v p 


V V  V



I I  I







V
V
I   , I  
Z0
Z0


6.2-4

I+, I–
+
V+, V–

P+, P–

I+, I–

z

–



 V 
V

 

P V I V 

Z0
 Z0 

 2

V 

 V 
P V I V  
 
Z0
 Z0 












2


6.2-5

I+ = 0.1A
+
V+

= 5V

0.1 A
P+ = 0.5 W



–

0.5 W

5V
–

I+ = –0.2A
+
V+ =

–

+

–10 V

0.2 A
P+ = 2 W



–
10 V
+
0.2 A

2W


6.2-6

I– = –0.12 A
+

0.12 A


V– = 6 V

P– =0.72

0.72W
P
W



–

+
6V
–

I– = 0.08A
+
V–
–

0.72 W

= –4 V

0.08 A
P– = –0.32W



–
4V
+
0.08 A

0.32 W


6.2-7

Excitation by Constant Voltage Source
Semi-infinite Line, No Source Resistance

V0

t=0
z=0

V – 0

Z0, vp


z

V  z , t  V  t  z v p 
1 
I  z , t   V t  z v p 
Z0


6.2-8

V 0, t  V0u t 

V  t  V0u t 

V  z , t  V  t  z v p 

V0u t  z vp 

V0 for t  z vp   0


0 for t  z vp   0
V0 for t  z vp

0 for t  z vp
V0 for z  vp t

0 for z  vp t


6.2-9

V0

V(z)

0
V0

t

z/vp

V(t)

0

vpt

z


6.2-10

Example:

S
t=0

V0

Z0, vp

z=0

V0 10 V

10

z

Z0 50 
vp 3 10 8 m s
z = 150 m

V, V

0

to 

0.2
0.5

t, s

0

I, A

0.5

t, s


6.2-11

t = 1 s
10

V, V

0

I, A

0.2

300

z, m

t, ms

0

[V ]z150 m, V
10

300

z, m

[V ]t 1 s , V
10

0

0.5 t, ms

0

300

z, m


6.2-12

Effect of Source Resistance

Rg
Vg

t=0
–
+

+

I+

V+
–

z=0
Vg  I  Rg  V  0

B.C.



V
I 
Z0

(+) Wave


6.2-13

V
Vg –
Rg – V  0
Z0
Rg



Vg V   1
Z0

V  Vg

Z0
Rg  Z0


Vg
V
I 


Z0 Rg  Z0

I+
Rg
Vg

V+

+
Z0
–

z=0


6.2-14

Line Terminated by Resistance

S
Rg
V0

t=0

Z0, vp

z=0

z =l

t = 0+

I+
Rg
Vg

Z0
z=0

RL

+ +
V
–

I 

V0

Rg  Z0

V   I  Z0


6.2-15

t l v p

B.C:





V

V

R
I

I

 L

I+ + I–
+

V +V
z =l

–

RL



I  V
Z0



I   V
Z0


– 

V
V
V   V –  RL  –

Z 0
Z 0 

 RL 
RL

–

V 1 
V  – 1
 Z0 
Z0




6.2-16

RL – Z0
–

V V
RL  Z0
Define Voltage Reflection Coefficient,
V – RL – Z 0


V  RL  Z 0
Then, Current Reflection Coefficient
I – – V – Z0
V–
   
–  – 
I
V Z0
V


6.2-17

t 2l vp
Rg
V0

I+ + I– + I–+
+

V+ + V– + V–+
–
z=0

V   V   V   V0  Rg  I   I   I   






V
V
V


I  , I 
, I 
Z0
Z0
Z0






V  V  V



Rg 


V0 
V

V

V


Z0


6.2-18



V 1 


Rg 

–
 V 1 
Z 0 


Rg 


Z 0 

Rg


–
V0  V  – 1
Z0

V0
+
But V =
Z0
Rg + Z0

–
 V 1 


Rg 
Rg


–
V  – 1
Z0 
Z0


V – Rg – Z0

V–
Rg  Z0


6.2-19

(–)
(–+)

Rg

t   (steady state)

z=0

VSS V  1  R  R S        
2
R

2
S

2
R

2
S

V0 Z0
 1  R S  2R 2S  

Rg  Z0 
R 1  R S     
2
R

2
S


6.2-20

V0 Z0
1  R

Rg  Z0 1 – R S
RL – Z0 
1  

R

Z
V0 Z0
 L
0 

Rg  Z0
RL – Z0 Rg –

1  

RL  Z0 Rg 
V0

RL
RL  Rg

Z0 
Z0 
