Section 6 1 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


6.1
Transmission Line


6.1-3

Parallel-Plate Line

d
Vg t 

y

+
-

x

z

Neglect fringing (d << w).

w


6.1-4

E Ex  z , t ax ; H H y  z , t ay
z

+
x

H

-

+

+

, , 
-

+

-

+
E, Jc
-

d


6.1-5
d

V  z , t   Ex  z, t  dx
x 0

d Ex  z , t 

V  z, t 
 Ex  z , t  
d
I  z , t  w J s  z , t 

w H y  z , t 

I  z, t 
 H y  z, t  
w


6.1-6

H y
Ex
 
z
t
H y

Ex
  Ex  
z
t

 V

z  d

I

    
t  w 


I
V
    
z  w 
d

 V 


   
t  d 



6.1-7

V
  d  I
 

z
 w  t

Consider the circuit
I  z, t 

I
w
 
V
z
 d 

  w  V


d

 t

I  z   z, t 

L z

+
V  z, t 
z

-

+
G z

z

C z

V  z   z, t 
z  z


6.1-8

Then

V
I
 L
z
t

I
V
 GV  C
z
t

Transmission
Line
Equations

d
L=
, inductance per unit length
w

w
G=
, conductance per unit length
d
C =  w , capacitance per unit length
d


6.1-9

Transverse Electromagnetic (TEM) Waves

a

Ez Hz 0

b

V  z , t   E  z , t   d l
a

I  z , t   H  z , t   d l
C

C
x

b
z

y


6.1-10

Distributed Equivalent Circuit
L z

G z

L z
C z
G z

C z
z

Assumes perfect conductors but lossy dielectric.


6.1-11

Transmission-Line Equations
V
I
 L
z
t
I
V
 GV  C
z
t
L = Inductance per unit length (H/m)
C = Capacitance per unit length (F/m)
G = Conductance per unit length (S/m)


6.1-12

In general, conductors are also lossy. Then, the waves
are not exactly TEM waves.

V
I
 RI  L

z
t
I
V
 GV  C
z
t
R = Resistance per unit length (/m)


6.1-13

Lossless Line
(Perfect Conductors, Perfect Dielectric)

V
I
 L
z
t
I
V
 C
z
t
Combining, we obtain

2V
2V
LC 2

2
z
t

Wave
Equation


6.1-14

Solution:

V  z , t   Af t  z vp   Bg t  z vp 
    
    
  wave
  wave

1
1
vp 

LC


 velocity of propagation
Note, LC =


6.1-15

V
I
From
 L ,
z
t

I
1 V

t
L z
1
 Af t  z vp   Bg t  z vp 


Lvp 

1
I  z , t    Af t  z vp   Bg t  z vp 
Z0
L
where Z0 
 characteristic impedance of the line
C


6.1-16

Thus

V V



t  z v   V t  z v 
p



p

V  t  z vp  V  t  z vp 
I

Z0
Z0
    
    
  wave   wave
1
vp 
LC

L
Z0 
C

6.1-17

Parallel-Plate Line
d
L 
w
C  w
d

d
Z0 
w

w
d

d << w


6.1-18

Coaxial Cable

L  1n b
2
a
2 
C
b

1n
a


Z0  1n b
2
a

b

a


6.1-19

Parallel-Wire Line

L  cosh 1 d

a
C



cosh 1 d
a


Z0  cosh 1 d


a

a

a

2d


6.1-20

Parallel-Strip Line (d/w  1)
imbedded in a homogeneous medium
w
2d


Z0 






 1n 8d  w 

w 4d 