Download Document

Lecture 3
Homework
1. The fields of a traveling TEM wave inside the coaxial line shown left y
can be expressed as
a ρ
V0 r -g z
I 0 f -g z

E=
e ;
H=
e
r ln b / a
2pr
b
μ,
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).
y
x

w
2. For the parallel plate line shown left, derive the R, L, G, and C
parameters. Assume w >> d.
d
r
x
z
1.4 Field analysis of transmission lines
The Telegrapher equations Derived from Field Analysis
y
 The fields inside the coaxial line will satisfy
Maxwell's curl equations
  E   j H
a
b
  H  j E
ρ

x
μ,
Expanding the above equations in cylindrical coordinates and then gives the
following vector equations （TEM waves，Ez = Hz = 0, no ψdependence）
0
0
¶Er
¶E0
¶
f
-r
+f
+z
(r Ef ) = - jwm (rH r + f Hf )
¶z
¶z0
r¶r 0
0
¶H r
¶Hf
¶
-r
+f
+z
( r Hf ) = - jwe ( rEr + f Ef )
¶z
¶z
r¶r
Since the z components of these two equations must vanish and E = 0
at ρ = a and b, the components must have the forms
Ej = H r = 0.
Er =
h(z)
r
,
Hj =
g(z)
r
(p: 55-56)
Using the above equations, we obtain
h( z )
  jg ( z )
z
g ( z )
  jh( z )
z
Eliminate h(z) and g(z)
¶V (z)
wm ln b / a
=-j
I(z)
¶z
2p
¶I(z)
2p V (z)
= - jw (e '- je '')
¶z
ln b / a
Voltage between the two conductors:
V(z) =
b
b
a
a
ò Er (r, z)d r = h(z) ò
dr
r
= h(z)ln
b
a
Current on the inner conductor:
I(z) =
2p
Substitute the
coefficients by
L, G, C
ò Hj (a, z)adj = 2p g(z)
j =0
The same telegraphic equations as
derived from distributed theory.
¶V (z)
= - jw LI(z)
¶z
¶I(z)
= -(G + jeC)V (z)
¶z
Waves in lossless coaxial waveguides
h( z )
  jg ( z )
z
g ( z )
  jh( z )
z
Helmholtz equation
¶2 h(z)
2
+
w
me h(z) = 0
2
¶z
¶2 g(z)
2
+
w
me g(z) = 0
2
¶z
k = w me
A+
k
m
=
=
B+ wm
e
For one-way propagation, eg,
along +z axis
h(z) 1 + -ik×z
Er =
= Ae
r r
g(z) 1 + -ik×z
Hf =
= Be
r r
Electric and magnetic field (TEM):
Er =
Hf =
h(z) 1 + -ik×z - ik×z
= (A e + A e )
r r
g(z) 1 + -ik×z
= (B e + B-eik×z )
r r
Waves in lossless coaxial waveguides
Voltage between the two conductors at z = 0
V(0) =
b
b
ò Er (r, 0)d r = A ò
+
a
a
dr
Er =
V0
A =
ln b / a
b
= A ln
r
a
+
+
h(z)
V0
=
e-ik×z
r r ln b / a
Current on the inner conductor (Ampere’s circuit law) at z = 0:
I(0) =
2p
ò Hj (a, z)adj = 2p B
j =0
+
B+ =
I0
2p
Hf =
g(z) I 0 -ik×z
=
e
r 2pr
Propagation Constant, impedance and Power Flow
for the Lossless coaxial Line
 Propagation constant: b = k = w me = w LC (for a lossless medium)
TEM transmission lines have the same form of propagation
constant as that for plane waves in a lossless medium.
 Wave impedance:
Zw 
E
H
 Characteristic impedance:







Z0 
(
¶Er
¶z
= - jwm H f )
V0 E ln b / a  ln b / a
 ln b / a



I0
2H
2
 2
 Power flow (computed from the Poynting vector):
2
b


V0 I 0
1
1
1

P   E H ds   

d

d


V
I
0
0
2S
2  0  a 22 ln b / a
2
(agreement with
circuit theory)
1.5 The terminated lossless transmission line
What is a voltage reflection coefficient?
V(z) = V + (z)+V - (z)
I(z) = I + (z)+ I - (z)
 Assume an incident wave ( V0 e  jz ) generated from a source at z < 0. We
have seen that the ratio of voltage to current for such a traveling wave is Z0,
the characteristic impedance. But when the line is terminated in an
arbitrary load ZL  Z0, the ratio of voltage to current at the load must be ZL.
Thus, a reflected wave must be excited with the appropriate amplitude to
satisfy this condition.
What is a voltage reflection coefficient?
Total voltage and current on the line
(superposition of incident and reflected
waves):
V ( z )  V0 e  jz  V0 e jz
V0  jz V0 jz
I ( z) 
e

e
Z0
Z0
(V0+: incident; V0-: reflected)
The total voltage and current at the load are related by the load
impedance, so at z = 0, we must have
V (0) V0  V0
ZL 

Z0
I (0) V0  V0
Voltage reflection coefficient Γ:
V0 Z L  Z 0
0   
V0
Z L  Z0
(Phase
difference: π)
What is a voltage reflection coefficient?
The total voltage and current waves on the line :
V (z) = V0+e- jb z +V0-e jb z = V0+ [e- jb z + Ge jb z ]
V0+ - jb z V0- jb z V0+ - jb z
I(z) =
e
e =
[e
- Ge jb z ]
Z0
Z0
Z0
V0G= +
V0
Consider the time-average power flow along the line at the point z:
2

V
1
1
2
0
Pav  Re[V ( z ) I ( z )* ] 
Re{1  *e  2 jz  e 2 jz   }
2
2 Z0
which can be simplified:
2

1 V0
2
Pav 
(1   )
2 Z0
• Constant average power flow at any point on the line;
• Total power delivered to the load = incident power – reflected power
What is a voltage reflection coefficient?
(a). Standing wave ( = -1)
(b). Voltage standing wave ratio (  < 1)
V ( z )  V0 1  e 2 jz  V0 1  e 2 jl
SWR 
Vmax 1  

Vmin 1  
RL  20 log 
(l = -z)
(1  SWR < ， where SWR=1 implied a match load.)
(dB) (return loss)
What is a voltage reflection coefficient?
(l)
The reflection coefficient at z = -l:
V0
(0)
Zin
ZL
Z0
V0 e  jl
(l )   jl  (0)e 2 jl
V0 e
V0
z
l
0
At a distance l = -z from the load, the input impedance seen looking
toward the load is

jl
 jl
1   e 2 jl
V (l ) V0 (e   e )
Z in 

Z0 
Z0
I (l ) V0 (e jl   e  jl )
1   e 2 jl
A more usable form of input impedance:
( Z L  Z 0 ) e j  l  ( Z L  Z 0 ) e  j l )
Z in  Z 0
( Z L  Z 0 ) e j  l  ( Z L  Z 0 ) e  j l )
 Z0
Z L cos l  jZ0 sin l
Z 0 cos l  jZ L sin l
 Z0
Z L  jZ0 tan l
Z 0  jZ L tan l
• Input impedance of oe portion of transmission
line with an arbitrary load impedance.
• Transmission line impedance Equation.
1.5The terminated lossless transmission line
Voltage and current in the line:
V ( z )  V0 [e  jz  e jz ]
V0  jz
I ( z) 
[e
 e jz ]
Z0
(1). Short circuit transmission line (ZL = 0)
voltage
Zin
d=l
0
current
impedance
(2). Open circuit transmission line ZL = 
voltage
Zin
current
impedance
(3). Quarter-wave transmission line
(l   / 4  n / 2,
n  1,2,3,...)
(4). Interface of two transmission lines
Reflection coefficient:
Transmission coefficient:
Insertion loss:
Grad, Div and Curl in Cylindrical Coordinates