Transcript 1.4 Field analysis of transmission lines

Lecture 2
1.4 Field analysis of transmission lines
 Derive the transmission line parameters (R, L, G, C) in
terms of the electromagnetic fields
 Rederive the telegrapher equations using these parameters
Example:
Voltage : V0ejz
Current: I0ejz
Work (W) and power (P)
H* Multiplies the two sides of the first Maxell’s equation:
E Multiplies the two sides of the conjugated second Maxell’s equation:
Add the above two equations and utilize
We obtain (J=Js+σE):
Integrate the above formula in volume V and utilize divergence theory,
we have the following after reorganize the equation
Poynting law: PS  Po  Pl  2 j (Wm  We )
 Source power Ps: Ps  


1
(
E

J

H
 M s )dv

V
2
 Output power P0: Po  1 E  H   d s  1 S  d s
2 S
2 S
 Loss power Pl:
Pl 

2
(Time averaged)
V
2
E dv 

( ' ' E

2
V
2

(S  E  H )
2
  ' ' H )dv
2
1
 Stored magnetic energy Wm: Wm    ' H dv
4 V
2
1
 Stored electric energy We: We    ' E dv
4 V
Calculate magnetic energy
Calculate the time-average stored magnetic energy in an isotropic medium ( the
results valid for any media )
1
Wm 
2T
T
  Re[ H ]  Re[ B]dvdt
0V
1

2T
T
  H cos(t   )  B cos(t   ) cos dtdv
1
2
V 0
1
  H B cos  cos(1  2 )dv
4V

1
 Re  H  B dv
4 V
(cos   eH  eB )
( H  H e j1 eH ;
B  B e j2 eB )
Surface resistance and surface current of metal
Energy entering a conductor:

1
Pav  Re  E  H  nds
S0  S
2
The contribution to the integral from the surface S can be made zero by
proper selection of this surface. Therefore,

1
Pav  Re  E  H t  zds
S0
2
(H  n  E / )



From vector identity, we have z  ( E  H )  ( z  E )  H   H  H .
The energy absorbed by a conductor:
Rs
Pav 
2

S0
Rs
H t ds 
2
2

S0
2
J s ds
( Rs  Re( )  Re[(1  j )
Js  n H )

1
]
2
 s
1.4 Field analysis of transmission lines
Transmission line parameter: L
 The time-average stored magnetic energy for 1 m
long transmission line is
Wm 

4

 H  H ds
S
And circuit line gives W m L I 2 / 4 . Hence the self inductance could
be identified as
L

I0
2
H H
S

ds
( H / m)
Appendix 1:
Calculate the time-average stored magnetic energy in an isotropic medium ( the
results valid for any media )
1
Wm 
2T
T
  Re[ H ]  Re[ B]dvdt
0V
1

2T
T
  H cos(t   )  B cos(t   ) cos dtdv
1
2
V 0
1
  H B cos  cos(1  2 )dv
4V

1
 Re  H  B dv
4 V
(cos   eH  eB )
( H  H e j1 eH ;
B  B e j2 eB )
1.4 Field analysis of transmission lines
Transmission line parameter: C
 Similarly, the time-average stored electric energy per
unit length can be found as
We 


E  E ds

4
S
Circuit theory gives W e C V / 4, resulting in the following expression for the
capacitance per unit length:
2
C

V0

2
 E  E ds
S
( F / m)
1.4 Field analysis of transmission lines
Transmission line parameter: R
 The power loss per unit length due to the finite
conductivity of the metallic conductors is
Rs
Pc 
2
(Rs = 1/ is the surface resistance
and H is the tangential field)

 H  H dl
C1 C 2
The circuit theory gives P c  R I 0 / 2 , so the series resistance R
per unit length of line is
2
R
Rs
I0

2
 H  H dl
C1 C 2
( / m)
1.4 Field analysis of transmission lines
Transmission line parameter, G
 The time-average power dissipated per unit length in
a lossy dielectric is
Pd 
 ' '
2

 E  E ds.
S
Circuit theory gives P d  G V0 / 2 , so the shunt conductance
per unit length can be written as
2
G
 ' '
V0
2

 E  E ds
S
( S / m)
Homework
y
1. The fields of a traveling TEM wave inside the coaxial line shown left
can be expressed as
a
V0 
I 
E
e z ; H  0 e z
 ln b / a
2
where  is the propagation constant of the line. The conductors are
assumed to have a surface resistivity Rs, and the material filling the
space between the conductors is assumed to have a complex
permittivity  = ’ - j" and a permeability μ = μ0μr. Determine the
transmission line parameters (L,C,R,G).
b
ρ
φ
x
μ,
y

w
2. For the parallel plate line shown left, derive the R, L, G, and C
parameters. Assume w >> d.
d
r
x
z