I. Introduction to Wave
Propagation
•
•
•
•
Waves on transmission lines
Plane waves in one dimension
Reflection and transmission at junctions
Spatial variations for harmonic time
dependence
• Impedance transformations in space
• Effect of material conductivity
July, 2003
© 2003 by H.L. Bertoni
1
Waves on Transmission Lines
• Equivalent circuits using distributed C and L
• Characteristic wave solutions
• Power flow
July, 2003
© 2003 by H.L. Bertoni
2
Examples of Transmission Lines
I(z,t)
Two-Wire Line
(Twisted Pair)
+
V(z,t) -
z
Coaxial Line
I(z,t)
+
V(z,t) Conductors
Dielectric
Strip Line
July, 2003
© 2003 by H.L. Bertoni
3
Properties of Transmission Lines (TL’s)
• Two wires having a uniform cross-section in one
(z) dimension
• Electrical quantities consist of voltage V(z,t) and
current I(z,t) that are functions of distance z along
the line and time t
• Lines are characterized by distributed capacitance
C and inductance L between the wires
– C and L depend on the shape and size of the conductors
and the material between them
July, 2003
© 2003 by H.L. Bertoni
4
Capacitance of a Small Length of Line
Open circuit
I(t)
+
E
V(t) l
The two wires act as a capacitor. Voltage applied to the wires
induces a charge on the wires, whose time derivative is the current.
Since the total charge, and hence the current, is proportional to
the length l of the wires. Let the constant of proportionality be
C Farads/meter. Then
dV (t)
I(t)  Cl
dt
July, 2003
© 2003 by H.L. Bertoni
5
Inductance of a Small Length of Line
B
I(t)
Short circuit
+
V(t) l
The wire acts as a one - turn coil. Current applied to the wires induces
a magnetic field throught the loop,
whose time derivative generates the
voltage. The amount of magnetic flux (magnetic field
 area), and hence
the voltage, is proportional to the length l of the wires. Let the constant
of proportality be L Henrys/meter. Then
dI(t)
V(t)  Ll
dt
July, 2003
© 2003 by H.L. Bertoni
6
C and L for an Air Filled Coaxial Line
C
a
b
2o
ln b a
L
o
ln b a
2
109
Permittivity of vacuum :  o 
Farads/m
36
Permeability of vacuum : o  4  107 Henrys/m
Suppose that a  0.5 mm and b  2 mm. Then
2 o
 40.1 pF/m
ln 4
Note that
C
and
1
1
8

 3 10 m/s and
LC
o o
July, 2003
L
o ln 4
 0.277 H/m
2
L ln b a o ln 4


377  83.2 
C
2
 o 2
© 2003 by H.L. Bertoni
7
C and L for Parallel Plate Line
w
h
z
w
h
Note that for air between the plates
C 
1
1

 3 10 8 m/s
LC
o o
July, 2003
h
w
  o and   o so that
L 
L h

C w
© 2003 by H.L. Bertoni
o h
 377 
o w
8
Two-Port Equivalent Circuit of Length Dz
I(z,t)
+
V(z,t)
z
I(z,t)
+
V(z,t)
z
+
LDz
Kirchhoff circuit equations
I(z,t)
V(z,t)  LDz
 V (z  Dz,t)
t
or
V(z  Dz,t)  V (z,t)
I(z,t)
 L
Dz
t
July, 2003
z+Dz
I(z +Dz,t)
C Dz V(z+Dz,t)
I(z,t)  CDz
V (z  Dz,t)
 I(z  Dz,t)
t
I(z  Dz,t)  I(z,t)
V (z  Dz,t)
 C
Dz
t
© 2003 by H.L. Bertoni
9
Transmission Line Equations
Taking the limit as
Dz  0 gives the Transmission Line Equations
V (z,t)
I(z,t)
 L
z
t
I(z,t)
V (z,t)
 C
z
t
These are coupled, first order, partial differential equations whose solutions
are in terms of functions F(t - z/v) and G(t  z /v) that are determined by
the sources. The solutions for voltage and current are of the form
1
V(z,t)  F(t - z/v) + G(t  z /v)
I(z,t)  F (t - z/v) - G(t  z /v)
Z
Direct substitution into the TL Equations,
and using the chain rule gives
1
1
 F'(t - z/v) - G'(t  z /v)  L F '(t - z/v) - G'(t  z /v)
v
Z
1
 F'(t - z/v) + G'(t  z /v)  CF'(t - z/v) + G'(t  z /v)
vZ
where the prime (' ) indicates differentiation with respect to the total variable
inside the parentheses of F or G.
July, 2003
© 2003 by H.L. Bertoni
10
Conditions for Existence of TL Solution
For the two equations to be satisfied
1 L
1

and
C
v Z
vZ
Multiplying both sides of the two equations gives
v
1
LC

or
2
vZ
Z
1
m/s
LC
Dividing both sides of the two equations gives
vZ
L

or
v ZC
L
Z

C
v and Z are interpreted as the wave velocity and wave impedance.
July, 2003
© 2003 by H.L. Bertoni
11
F(t-z/v) Is a Wave Traveling in +z Direction
Assume that G(t  z /v)  0
Then the voltage and current are
t=0
V(z,0)=F[(-1/v)(z)]
V(z,t)  F(t  z /v)  F (1 v)(z  vt)
I(z,t) 
1
1
F(t  z /v)  F (1 v)(z  vt)
Z
Z
F(t  z /v) represents a wave disturbance
traveling in the positive z direction with
velocity v.
a
-a
t>0
Note that the current in the conductor at
positive potential flows in the direction of
wave propagation.
July, 2003
z
© 2003 by H.L. Bertoni
V(z,t)=F[(-1/v)(z-vt)]
a+vt
z
-a+vt
vt
12
G(t+z/v) Is a Wave Traveling in -z Direction
Assume that F(t  z /v)  0
Then the voltage and current are
t=0
V(z,0)=G[(1/v)(z)]
V(z,t)  G(t  z /v)  G(1 v)(z  vt)
I(z,t) 
a
2a
z
1
1
G(t  z /v) 
G(1 v)(z  vt)
Z
Z
G(t  z /v) represents a wave disturbance
traveling in the negative z direction with
velocity v.
Because of the minus sign in
I(z,t), the
t>0
2a-vt
z
-vt
physical current in the conductor at positive
potential flows in the direction of wave propagation.
July, 2003
V(z,t)=G[(1/v)(z+vt)]
© 2003 by H.L. Bertoni
a-vt
13
Example of Source Excitation
Excitation at one end of a semi - infinite length of transmission line.
Source has open circuit voltage VS (t) and internal resistance RS .
Radiation condition requires that excited waves travel away from source.
Terminal conditions at z  0 :
VS (t)  RS I(0,t)  V (0,t)
1
 RS F(t)  F (t)
Z
Z
or F(t) 
VS (t)
Z  RS
RS I(0,t)
VS(t) +
V(0,t)
z
0
VS (t)  RS I(0,t)  V(0,t)
1
 RS G(t)  G(t)
Z
Z
or G(t) 
VS (t)
Z  RS
July, 2003
∞
I(0,t)
∞
+
V(0,t)
0
© 2003 by H.L. Bertoni
RS
VS(t)
z
14
Receive Voltage Further Along Line
Voltage observed on a high impedance scope at a distance
V(l,t)  F (t  l v) 
Z
VS (t  l v)
Z  RS
Delayed version of the source voltage
l from source.
Scope
RS
VS(t) +
V(l,t)
∞
with the semi - infinite line acting as a
z
load resisor for the source.
V(l,t)  Gt  ( l v) 
0
Z
V (t  l v)
Z  RS S
Delayed version of the source voltage
with the semi - infinite line acting as a
∞
load resisor for the source.
July, 2003
l
Scope
+
V(-l,t)
-l
© 2003 by H.L. Bertoni
RS
0
VS(t)
z
15
Power Carried by Waves
P(z,t)
Instantaneous power P(z,t) carried past plane
perpendicular to z.
I(z,t)
V(z,t)
P(z,t)  V (z,t)I(z,t)
z
1
 F(t  z v)  G(t  z v) F(t  z v)  G(t  z v)
Z
1
 F 2 (t  z v)  G 2 (t  z v)
Z
The two waves carry power independently in the direction of wave
propagation
For each wave, a transmission line extending to
z   acts as a resistor
of value Z, even though the wires were assumed to have no resistance.
July, 2003
© 2003 by H.L. Bertoni
16
Summary of Solutions for TL’s
• Solutions for V and I consists of the sum of the
voltages and current of two waves propagating in
±z directions
• For either wave, the physical current flows in the
direction of propagation in the positive wire
• Semi-infinite segment of TL appears at its
terminals as a resistance of value Z (even though
the wires are assumed to have no resistance)
• The waves carry power independently in the
direction of wave propagation
July, 2003
© 2003 by H.L. Bertoni
17
Plane Waves in One Dimension
• Electric and magnetic fields in terms of voltage
and current
• Maxwell’s equations for 1-D propagation
• Plane wave solutions
• Power and polarization
July, 2003
© 2003 by H.L. Bertoni
18
Electric Field and Voltage for Parallel Plates
w
x
Ex(z,t)
z
h
+
V(z,t)
-
y
The electric field goes from the positive plate to the negative plate. If
w >> h, the electric field outside of the plates is very small. Between
the plates it is nearly constant over the cross - section with value
1
E x (z,t)   V (z,t) Volts/m or V (z,t)  hE x (z,t).
h
w
Recall that C   .
h
July, 2003
© 2003 by H.L. Bertoni
19
Magnetic Field and Current for Parallel Plates
w
x
I(z,t)
h
z
y
Hy(z,t) or By(z,t)
The magnetic field links the currents in the plates. If
w >> h, the magnetic
field outside of the plates is very small. Between the plates it is nearly
constant over the cross - section, as if in a solenoid, with value
Hy (z,t) 
1

By (z,t) 
Recall that L  
July, 2003
1  
1

 I(z,t)   I(z,t) Amps/m or I(z,t)  wH y (z,t).
  w

w
h
.
w
© 2003 by H.L. Bertoni
20
Maxwell’s Equations in 1-D
Inserting the foregoing expressions for
Transmission Line equations

h 
hE x (z,t)    wHy (z,t)
z
 w t
V(z,t), C, I(z,t) and L into the

 w 
wH y (z,t)   
hE x (z,t)

z
 h t
or


E x (z,t)   H y (z,t)
z
t


H y (z,t)   E x (z,t)
z
t
These are the two Maxwell equations for linearly polarized wave propagating in
1- D. They are independent of ( h,w) and refer to the fields.
We may think of the plates as being taken to (
considered.
x,y)   so they need not be
The field are in the form of a plane wave,
which covers all space and is a simple
approximation for fields in a limited region of space,
such as a laser beam.
July, 2003
© 2003 by H.L. Bertoni
21
Plane Waves: Solutions to Maxwell Equations
Maxwell' s equations are formally equivalent to the Transmission Line Equations
The solution is therefore in terms of two wave traveling in opposite directions
along z .
E x (z,t)  F (t  z /v)  G(t  z /v)
In air
v
1
o o
H y (z,t) 
1

F(t  z /v)  G(t  z /v)
 c  3 108 m/s is the speed of light and

o
 377 
o
is the wave impedance.
For waves in simple dielectric medium,
o is multiplied by the relative dielectric
constant  r .
r  1, but it can be a function of frequency. As and example,
in water at radio frequencies (below 20 GHz)
r  81, but at optical
frequencies r  1.78.
For normal media
July, 2003
© 2003 by H.L. Bertoni
22
Power Density Carried by Plane Waves
Total instantaneous power carried in parallel plate line
P(z,t)  V (z,t)I(z,t)  hE x (z,t)wHy (z,t)
 hwE x (z,t)H y (z,t) watts
E
Power density crossing any plane perpendicular
to z is
p(z,t)  P(z,t) hw  E x (z,t)H y (z,t) watt/m

1
F


2
Direction of
propagation
2
H
(t  z /v)  G 2 (t  z /v)
Direction of H y is such that turning a right hand screw in the
direction from E x to Hy advances the screw in the direction of
propagation
July, 2003
© 2003 by H.L. Bertoni
23
Polarization
The physical properties of a plane wave are independent of the
coordinate system.
For a plane wave traveing in one direction :
Electric field vector E must be perpedicular to the direction of
propagation.
Magnetic field vector H must be perpedicular to E and to the
direction of propagation.
The vector cross product p  E  H watt/m 2 is in the direction
of propagation.
The ratio E H is the wave impedance  .
July, 2003
© 2003 by H.L. Bertoni
24
Examples of Polarization
Linear polaization of
E along x
Linear polaization of
E along y
E  a x cos (t  z /v)
E  a y sin  (t  z /v)
1
H  a y cos (t  z /v)
H  ax

E
1

sin  (t  z /v)
x
x
z
ax = unit vector along
ay = unit vector along
H
x
y
z
E
y
H
y
Circular polarization
E  a x cos (t  z /v)  a y sin  (t  z /v)
H
July, 2003
1
a y cos (t  z /v)  ax sin  (t  z /v)


© 2003 by H.L. Bertoni
25
Summary of Plane Waves
• Plane waves are polarized with fields E and H
perpendicular to each other and to the direction of
propagation
• Wave velocity is the speed of light in the medium
• ExH watts/m2 is the power density carried by a
plane wave
July, 2003
© 2003 by H.L. Bertoni
26
Reflection and Transmission at
Junctions
• Junctions between different propagation media
• Reflection and transmission coefficients for 1-D
propagation
• Conservation of power, reciprocity
• Multiple reflection/transmission
July, 2003
© 2003 by H.L. Bertoni
27
Junctions Between Two Regions
Terminal condtions for the
I(0-,t)
Junction of two TL' s
V(0 ,t)  V (0 ,t)
TL 1
I(0 ,t)  I(0 ,t)
V(0-,t)
I(0+,t)
+
V(0+,t)
TL 2
0
Boundary conditions at the
interface of two media
z
x
E x (0,t)  E x (0 ,t)
Ex(0-,t)
Ex(0+,t)
Hy(0-,t)
Hy(0+,t)
H y (0,t)  H y (0 ,t)
Plane wave propagation and
boundary conditions are analogus
Medium 1
z
Medium 2
to junctioning of two TL' s
July, 2003
© 2003 by H.L. Bertoni
28
Reflection and Transmission
Incident wave
x
ExIn(z,t)=F1(t-z/v1)
HyIn(z,t)
Transmitted wave
z
Reflected wave
v1 and 1
v2 and 2
A source creates an incident wave whose electric field is given by the known
function F1 (t - z/v 1 ). Using the boundary conditions we solve for the unknown
functions G1 (t + z/v 1 ) and F2 (t - z/v 2 ) for the electric fields of the reflected
and transmitted waves :
E x (0,t)  F1 (t) + G1 (t)  F2 (t)  E x (0 ,t)
Hy (0 ,t) 
July, 2003
1
1
F1 (t) - G1 (t)  
© 2003 by H.L. Bertoni
1
2
F2 (t)  H y (0 ,t)
29
Reflection and Transmission Coefficients
Solution of the boundary condition equations for
G1 (t) and F2 (t) in terms of F1 (t)
G1 (t)  F1 (t)
F2 (t)  F1 (t)
The reflection coefficient
 
 2 1
 2  1
Examples :
 and transmission coefficient  are given by :
2 2
  1  
 2  1
I. Suppose medium 1 is air so that
relative dielectric constant
from air - to - dielectric
July, 2003
1    o  o  377 and medium 2 has
r  4 so that 2  o ro  0.5. Then going
ad 
0.5  
1
1 2
  and ad  1 
0.5  
3
3 3
© 2003 by H.L. Bertoni
30
Reflection and Transmission, cont.
II. Now suppose the wave is incident from the dielectric onto air so that medium 1
is the dielectric 1  0.5  and medium 2 is air 2  . Then going from
  0.5
1
1 4
dielectic - to - air, da 
  and ad  1 
  0.5
3
3 3
Note that :
1. da  ad
2. Since T is the ratio of fields,
July, 2003
not power, it can be greater than 1.
© 2003 by H.L. Bertoni
31
Reflected and Transmitted Power
Instantaneous power carried by the incident wave
p In (z,t), the reflected wave
pRe (z,t), and the transmitted wave
pTr (z,t)
1
pIn (z,t)  E xIn (z,t)H yIn (z,t)  F12 (t  z v1 )
1
pRe (z,t)  E xRe (z,t)HyRe (z,t) 
pTr(z,t)  E xTr(z,t)H Tr
y (z,t) 
Just on either side of the interface
pRe (0,t) 
July, 2003
1
1
G12 (t)   2
1
1
F12 (t)
1
1
1
2
G12 (t  z v1 )
F22 (t  z v1 )
1 2
F1 (t) as well as
Z1
1
1
pTr (0 ,t)  F22 (t)   2 F1 2 (t)
p In (0,t) 
and
© 2003 by H.L. Bertoni
2
2
32
Conservation of Power and Reciprocity
p In (0,t)  p Re(0 ,t)  pTr(0 ,t) so that
Conservation of power requires that
1
1
F1 2 (t)   2
1
1
F12 (t)   2
1
2
F12 (t)
1  2   2
or
1
2
This relation is easily shown to be satisfied from the expressions for
For waves going from medium 2 to medium 1,
, .
the reflection coefficient
12 is
the negative of 21 going from medium 1 to medium 2. Thus for either
pRe (0,t)
pTr (0 ,t)
case the ratios

and
 1  2 are the same.
In

In

p (0 ,t)
p (0 ,t)
Therefore the same fraction of the incident power is reflected from and
2
transmitted through the interface for waves incident from either medium.
This result is an example of a very general wave property called reciprocity.
July, 2003
© 2003 by H.L. Bertoni
33
Termination of a Transmission Line
Terminal condtions
V(0,t)  RL I(0,t)
RL
F(t)  G(t) 
F(t)  G(t)
Z
Solving for G(t) in terms of F (t),
I(0-,t)
TL
V(0-,t)
+
RL
0 z
G(t)  F(t) where the reflection
R Z
coefficient is   L
RL  Z
Special cases :
1. Matched termination, RL  Z and   0. Simulates a semi - infinite TL
2. Open circuit, RL   and   1. Total reflection with V (0,t)  2F (t).
3. Short circuit, RL  0 and   1. Total reflection with V (0,t)  0.
July, 2003
© 2003 by H.L. Bertoni
34
Reflections at Multiple Interfaces
x
Incident wave
ExIn(z,t)=F1(t-z/v1)
Transmitted
waves
HyIn (z,t)
Reflected waves
0
l
z
Multiple
internal
reflections
v1 and 1
v2 and 2
v3 and 3
Multiple internal reflections occur within the finite thickness layer. These
internal waves generate multiple reflected waves in medium 1 and multiple
transmitted waves in medium 3.
July, 2003
© 2003 by H.L. Bertoni
35
Scattering Diagram for a Layer
1


2l/v2
 
  
4l/v2
  
  
  
   
  
t
l
z

   
   
Space - time diagram indicates the relative amplitudes of the electric field of
the individual components of the multiply reflected waves. In adding fields,
account must be taken of the relative delay between the different components.
July, 2003
© 2003 by H.L. Bertoni
36
Summary of Reflection and Transmission
• The planar interface between two media is analogous to the
junction of two transmission lines
• At a single interface (junction) the equation T = 1 +  is a
statement of the continuity of electric field (voltage)
• The ratio of reflected to incident power = 
• Power is conserved so that the ratio of transmitted to incident
power = 1 - 
• The reciprocity condition implies that reflected and transmitted
power are the same for incidence from either medium
• At multiple interfaces, delayed multiple interactions complicate
the description of the reflected and transmitted fields for
arbitrary time dependence
July, 2003
© 2003 by H.L. Bertoni
37
Spatial Variations for Harmonic
Time Dependence
• Traveling and standing wave representations of
the z dependence
• Period average power
• Impedance transformations to account for layered
materials
• Frequency dependence of reflection from a layer
July, 2003
© 2003 by H.L. Bertoni
38
Harmonic Time Dependence at z = 0
Suppose that the voltage and current (or
E x and H y fields) have harmonic time
dependence exp( jt) at z  0. Then
V(0,t)  V (0)e jt  F(t)  G(t)
1
j t
I(0,t)  I(0)e  F(t)  G(t)
Z
where V (0) and I(0) are the complex voltage and current at
z  0.
The functions F(t) and G(t) can satisfy these equations only if they too have
harmonic time dependence. Hence
F(t)  V e jt
and
where V   12 V (0)  ZI(0) and V  
G(t) = V e jt
1
2
V (0)  Z I(0)
voltage amplitudes of the waves traveling in the
July, 2003
© 2003 by H.L. Bertoni
are the complex
 z directions.
39
Traveling Wave Representation
At other locations
z 0
V(z,t)  F(t  z v)  G(t  z v)  V exp  j  (t  z v)  V exp  j (t  z v)


 V e  jz v  V e  jz v e jt  V (z)e jt
1
1
F(t  z v)  G(t  z v)  V  exp  j (t  z v)  V  exp  j  (t  z v)
Z
Z
1   j z v
 V e
 V e  j z v e jt  I(z)e jt
Z
Here V (z) is the phasor voltage and I(z) is the phasor current, which give the
I(z,t) 
spatial variation for the implied time dependence
Define the wave number (propagation constant)
exp( jt).
1
k   v m . Then
1   jkz
  jkz
V e V e 

Z
is the traveling wave representation of phasor voltage and current.
  jkz
V (z)  V e
July, 2003
  jkz
V e
and
I(z) 
© 2003 by H.L. Bertoni
40
Standing Wave Representation
Substituting the expressions for V  and V  in terms of V (0) and I(0),
and rearranging terms gives the standing wave representation of the phasor
voltage and current :
V(z)  12 V (0)e jkz  e  jkz  12 Z I(0)e  jkz  e  jkz  V (0)cos kz  jZI(0)sin kz
I(z) 
1
V (0)e  jkz  e  jkz  12 I(0)e  jkz  e  jkz  I(0)cos kz  j V (0)sin kz
Z
Z
1
2
The wavenumber is k   v 2f v  2  where  is the
wavelength   v f  2 k
For plane waves in a dielectric medium
July, 2003
k   
© 2003 by H.L. Bertoni
41
Variation of the Voltage Magnitude
For V   0 we have a pure traveling
For I(0)  0 we have a pure standing
wave V (z)  V e  jkz . The magnitude
wave V (z)  V(0)cos kz. Its magnitude
V (z)  V e jkz = V 
V (z)  V (0) coskz is periodic with
is independent
period  k   2.
of z.
V (z)
V (z)
|V+|
V (0)
z
July, 2003
© 2003 by H.L. Bertoni
0
 z
42
Standing Wave Before a Conductor
Incident wave
x
ExIn(z)
HyIn(z)
Plane wave incident on a perfectly
Perfect
conduticng plate and the equivalent
conductor
circuit of a shorted TL
0
ExRe(z)
z
The standing wave field is
E x (z)  12 ISC e  jkz  e  jkz 
Reflected wave
  jI SC sin kz
Two waves of equal amplitude and
ISC
, v
short
0
July, 2003
E x (0)  0 and H y (0)  I SC
traveling in opposite directions create
a standing wave.
z
© 2003 by H.L. Bertoni
43
Standing Wave Before a Conductor, cont.
Plot of the magnitude of the standing wave field
E x (z)  ISC sin kz
ISC
-z
Since k   v  2f v  2  the nodes (zeros) of the field are
separated by a distance
July, 2003
k Dz   or Dz   k   2
© 2003 by H.L. Bertoni
44
Period Averaged Power
For harmonic time dependence on a TL,
of the instantaneous power is
the time average over one period



P(z)  Re V (z)I (z) watts
1
2
Using the traveling wave representation

2
2
1
   jkz
  jkz 1
  jkz
  jkz  


P(z)  Re V e  V e  V e  V e  
V V
Z

 2Z
Note that the average power is the algebraic sum of the power carried by
the incident and reflected waves, and it is independent of z.
1
2
For harmonic plane waves
In terms of traveling waves
July, 2003

1
p(z)  E
2

y


p(z)  Re E x (z)H (z) watts/m
1
2
In
x
2
© 2003 by H.L. Bertoni
E
Re
x

2
2
45
Reflection From a Load Impedance
For a complex load impedance ZL
Z
V(0)  V   V   ZL I(0)  L V   V 
Z
Solving for V  in terms of V  gives

I(0)
V(0) + ZL

0
V  V where the complex
reflection coefficient  is

ZL  Z
ZL  Z
V+
Reflected power
2
1 2 1
2
Re

P 
V 
V   P In
2Z
2Z
July, 2003
z
© 2003 by H.L. Bertoni
V-
ZL
0
z
46
Summary of Spatial Variation for Harmonic
Time Dependence
• Field variation can be represented by two traveling waves
or two standing waves
• The magnitude of the field for a pure traveling wave is
independent of z
• The magnitude of the field for a pure standing wave is
periodic in z with period 
• The period average power is the algebraic sum of the
powers carried by the traveling waves
• The period average power is independent of z no matter if
the wave is standing or traveling
• The fraction of the incident power carried by a reflected
wave is ||
July, 2003
© 2003 by H.L. Bertoni
47
Impedance Transformations
in Space
• Impedance variation in space
• Using impedance for material layers
• Frequency dependence of reflection from a brick
wall
• Quarter wave matching layer
July, 2003
© 2003 by H.L. Bertoni
48
Defining Impedance Along a TL
I(0)
At z  0 the ratio of voltage to current
can have some value
Using the formulas for
V (0) I(0)  ZL
V(0) + ZL
ZIN
V (z) and I(z)
we can compute their ratio at z  l.
Defining this ratio as
ZIN (l) we have
-l
0
z
V (l) V (0)cos(-kl)  jZI(0)sin( kl)

1
I(l)
I(0)cos(-kl)  j V (0)sin(-kl)
Z
Dividing numerator and denominator by
I(0) and rearranging gives
Z cos(kl)  jZsin( kl)
Z  jZ tan( kl)
ZIN (l)  Z L
Z L
Z cos(kl)  jZL sin(kl)
Z  jZL tan(kl)
ZIN (l) 
July, 2003
© 2003 by H.L. Bertoni
49
Properties of the Impedance Transform
The impedance formula
Z cos(kl)  jZsin( kl)
Z  jZ tan( kl)
Z IN (l)  Z L
Z L
Z cos(kl)  jZL sin(kl)
Z  jZL tan(kl)
shows that a length TL (or region of space) transforms an impedance
to a different value.
Some properties of the transformation :
1. For a matched load ZL  Z, the imput impedace is matched
2. The impedance repeats
Z IN  Z
Z IN (l)  Z IN (l  Dl) for k Dl   or
Dl   k   2
3. For quarter wave displacement
l   4, kl   2 and impedance
inverts ZIN ( 4)  Z 2 Z L
4. If ZL  0, then ZIN (l)  jZtan( kl)
July, 2003
© 2003 by H.L. Bertoni
50
Using Transform for Layered Media
x
Incident wave
ExIn(z)
ExTR(z)
Transmitted
wave
HyIn (z)
0
ExRe(z)
l
z
Reflected wave
v1 , 1
v2 , 2
v3 , 3
Z= 2
ZIN(l)
July, 2003
ZL =  3
© 2003 by H.L. Bertoni
51
Circuit Solution for Reflection Coefficient
Medium 3 acts as a load on the layer to the left. A semi - infinite TL (medium)
at its terminals (accessible surface) acts as a resistor so that
ZL   3 .
Impedance of the finite segment of TL is
segment is
Z   2 . Wavenumber of this
k 2   v 2    r2 oo  ko  r2
where ko    oo is the wavenumber of free space.
Input impedance at left surface of the layer is then
 cos(k2 l)  j 2 sin( k 2 l)
ZIN (l)   2 3
2 cos(k2 l)  j 3 sin(k 2 l)
Reflection coefficient for the wave incident from medium 1 is
ZIN (l)  1  2 3  1 cos(k 2 l)  j( 22  13 )sin( k 2 l)


ZIN (l)  1 2  3  1 cos(k 2 l)  j( 22  1 3 )sin( k2 l)
July, 2003
© 2003 by H.L. Bertoni
52
Example 1: Reflection at a Brick Wall
Medium 1 and medium 3 are air
1   3   o 
w
IN
Ex
H
IN
y
o
o
Medium 2 is brick with
 r2  4
k2  2k o and  2 
o
 12  o
 r2 o
Reflection coefficient for the wave incident from air is
2 3  1 cos(k2 w)  j(22  1 3 )sin( k2 w)

 2  3  1 cos(k2 w)  j(22  13 )sin( k 2 w)
j 14 o   o sin( 2k o w)
2
2
j 43 sin( 2ko w)
 2

2
2
1
2o cos(2ko w)  j 4 o  o sin( 2k o w) 2cos(2k o w)  j 54 sin( 2ko w)
July, 2003
© 2003 by H.L. Bertoni
53
Example 1: Reflection at a Brick Wall, cont.
Let the wall thickness be
Then
p
Re
w  30 cm so that 2k o w 
4 f
8 0.3  4 fGHz
3 10
9sin 2 (4 fGHz )
p  
64 cos2 (4 fGHz )  25sin 2 (4 fGHz )
in
2
||

0
0.25 0.50
0.75
1.0
1.25
Since there is no conductivity in the brick wall,
power transmitted through the wall is
July, 2003
1 
© 2003 by H.L. Bertoni
1.50 1.75 2.0
fGHz
the fraction of the incident
2
54
Example 2: Quarter Wave Layers
x
Incident wave
ExIn(z)
ExTR(z)
Transmitted
wave
HyIn (z)
ExRe(z)
0
z
Reflected wave
v1 , 1
v2 , 2
v3 , 3
l=k2)=
cos(k 2 l)  cos(k 2 2 4)  cos( /2)  0 and
so that
July, 2003
sin (k2 l)  sin(  /2)  1
ZIN (  2 /4)   22 / 3
© 2003 by H.L. Bertoni
55
Example 2: Quarter Wave Layers, cont.
For this value of ZIN we have
 22  13
 2
 2  1 3
If we choose the layer material such that
22  1 3 , then   0 and no
reflection takes place.
Suppose that medium 1 is air and medium 3 is glass with relative
dielectric constant g
For no reflection :
22 
o  o
o
 1 3 
o  g o
 r2 o
Note that the layer thickness is
or l 
July, 2003
o
4 4 g
or r2   g
vo
1
v2


l   2 /4 
4 f 4 f r 2o o 4 f r 2
where  o is the wavelength in air.
© 2003 by H.L. Bertoni
56
Summary of Impedance Transformation
• The impedance repeats every half wavelength in
space, and is inverted every quarter wavelength
• Impedances can be cascaded to find the impedance
seen by an incident wave
• Reflection from a layer has periodic frequency
dependence with minima (or maxima) separated
by Df = v2/(2w)
• Quarter wave layers can be used impedance
matching to eliminate reflections
July, 2003
© 2003 by H.L. Bertoni
57
Effect of Material Conductivity
• Equivalent circuit for accounting for conductivity
• Conductivity of some common dielectrics
• Effect of conductivity on wave propagation
July, 2003
© 2003 by H.L. Bertoni
58
G, C, L for Parallel Plate Line
w
h
z
If the material between the plate conducts electricity,
there will be a
conductance G mho/m in addition to the capacitance
C farads/m
and inductance L henry/m.
The conductivity of a material is give by the parameter
 mho/m
Expressions for the circuit quantities are :
w
w
h
G
C 
L 
h
h
w
July, 2003
© 2003 by H.L. Bertoni
59
Equivalent Circuit for Harmonic Waves
+
I(z)
V(z)
z
I(z)
z+Dz
z
+
V(z)
jLDz
j C Dz
G
+
I(z +Dz)
V(z+Dz)
In the limit as Dz  0 the Kirchhoff circuit equations for the phasor
voltage and current give the TL equations for harmonic time dependence
dV (z)
  jL I(z)
dz
July, 2003
dI(z)
  G  jC V (z)
dz
© 2003 by H.L. Bertoni
60
Harmonic Fields and Maxwell’s Equations
w
x
h
+
I(z)
V(z)
z
Ex(z)
y
Hy(z)
If w >> h, the fields between the plates are nearly constant over the cross
so that the phasor circuit quantities are
- section,
V (z)  hE x (z) and I(z)  wH y (z).
Substituting these exprsssions in the TL equations for harmonic time dependence,
along with the expressions for
G, C, L gives Maxwell' s equations
dE x (z)
  j H y (z)
dz
July, 2003
dH y (z)
   j   E x (z)
dz
© 2003 by H.L. Bertoni
61
Maxwell’s Equations With Medium Loss
With minor manipulation,
Maxwell' s equations for 1 - D propagation of
harmonic waves in a medium with conduction loss can be written
dE x (z)
and
  j H y (z)
dz
The complex equivalent dielectric constant
dH y (z)
ˆ E x (z)
  j
dz
ˆ is given by

ˆ  ro  j    o  r  j   o 

ˆ   o r  j"
Let "   o . Then 
In other matierials atomic processes lead to a complex dielectric of the
form o  r  j". These processes have a different frequency
dependence for ", but have the same effect on a hamonic wave
July, 2003
© 2003 by H.L. Bertoni
62
Constants for Some Common Materials
When conductivity exists, use complex dielectric constant given by
 = o(r - j") where " = o and o  10-9/36
Material*
Lime stone wall
Dry marble
Brick wall
Cement
Concrete wall
Clear glass
Metalized glass
Lake water
Sea Water
Dry soil
Earth
r
7.5
8.8
4
4-6
6.5
4-6
5.0
81
81
2.5
7 - 30
mho/m)
0.03
" at 1 GHz
0.54
0.22
0.02
0.36
0.3
0.08
1.2
0.005 - 0.1
2.5
45
0.013
0.23
3.3
59
--0.001 - 0.03 0.02 - 0.54
* Common materials are not well defined mixtures and often contain water.
July, 2003
© 2003 by H.L. Bertoni
63
Incorporating Material Loss Into Waves
Using the equivalent complex dielectric constant,
Maxwell' s equations
have the same form as when no loss (conductivity) is present.
The solutions therefore have the same mathematical form with
ˆ.
replaced by 

For example, the traveling wave solutions in a material are
1
E x (z)  V e  jkz  V e  jkz and
H y (z)  V e jkz  V e  jkz 



ˆ
Here k       o r  j" and  

ˆ

 o r  j"
are complex quantities.
July, 2003
© 2003 by H.L. Bertoni
64
Wave Number and Impedance
The complex wavenumber
k will have real and imaginary parts
k    j    o r  j"
If " is less than about  r 10, we may use the approximations
"
    or
and     or
2r
Similarly,
July, 2003
for " small,
 
1 j " 



 o r  j"
 or  2r 

© 2003 by H.L. Bertoni
65
Effect of Loss on Traveling Waves
For a wave traveling in the positive
z direction
E x (z)  V e  jkz  V  exp j(  j )z  V  exp(- jz)exp(z)
The presence of loss (conductivity) results in a finite value of the
attenuation constant  . The attenuation (decay) length is
The magnitude of the field depends on z as given by
1 .

E x (z)  V exp( z)
|V+|
|V+| e
z
July, 2003
© 2003 by H.L. Bertoni
66
Attenuation in dB
For a traveling wave, the attenuation in units of deci
- Bells is found from
E x (z) 
V  exp(z) 
Attn 20log 10 
 20log 10 


V
E x (0) 


 20 z log10 e  8.67z
Thus the attenuation rate of the wave in a medium is
July, 2003
© 2003 by H.L. Bertoni
8.67 dB/m
67
Effect of Loss on Traveling Waves, cont.
The instantaneous field of the wave has both sinusoidal variation over a
wavelength   2  and the decay over the attenuation length 1  .
For real amplitude V , the spatial variation is given by
Re E x (z)e
j t
 V

Re exp  j(t  z)exp( z) 
or
V+
V cos( t - z)exp( z)
V+e
z


July, 2003
© 2003 by H.L. Bertoni
68
Loss Damps Out Reflection in Media
Traveling wave
amplitude
Incident wave
Reflecting
boundary
Reflected wave
z
IN
x

E (z)  V exp(z)
July, 2003
Re
x
E (z)  V  exp(z)
© 2003 by H.L. Bertoni
69
Effect of Damping on the || for a Wall
||


0
0.25 0.50
0.75
1.0
1.25
1.50 1.75 2.0
fGHz
With absorption in the brick wall, the interference minima are
reduced and the reflection coefficient approaches that of the
  o
first air - brick interface or   B
 1 3
B  o
The fraction of the incident power transmitted through the
wall is
July, 2003
 1 
2
© 2003 by H.L. Bertoni
70
Summary of Material Loss
• Conductivity is represented in Maxwell’s equations
by a complex equivalent dielectric constant
• The wavenumber k = j and wave impedance
 then have imaginary parts
• The attenuation length = 1/
• Loss in a medium damps out reflections within a
medium
July, 2003
© 2003 by H.L. Bertoni
71