Transcript Introduction to Wave Propagation

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
T he two wires act as a capacitor. Volt age applied to t he wires
induces a charge on t he wires,whose tim e derivat ive is the current.
Since the t ot al charge,and hence the currentis, proport ional to
t he lengthl of t he wires. Let t he const ant of proport ionalit y be
C Farads/meter. T hen
I(t)  Cl
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dV(t)
dt
© 2003 by H.L. Bertoni
5
Inductance of a Small Length of Line
B
I(t)
Short circuit
+
V(t) l
T he wire act s as a one
- t urn coil. Current applied t o the wires induces
a magnet ic field t hrought the loop,
whose tim e derivat ive generat es t he
volt age. T he amount of m agnet ic flux (magnetic field
 area), and hence
t he volt age,is proportional to the length
l of t he wires. Let t he const ant
of proportality beL Henrys/met er. T hen
dI(t)
V(t)  Ll
dt
July, 2003
© 2003 by H.L. Bertoni
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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
P ermitt ivit y of vacuum
: o 
Farads/m
36
P ermeability of vacuum
: o  4  107 Henrys/m
Suppose t hat a  0.5 m m andb  2 m m. T hen
2 o
 ln 4
C
 40.1 pF/m
and L  o
 0.277 H/m
ln 4
2
Note t hat
1
1

 3 108 m /s and
LC
o o
July, 2003
L lnb a o ln 4


377 83.2 
C
2
 o 2
© 2003 by H.L. Bertoni
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C and L for Parallel Plate Line
w
h
z
w
h
L 
h
w
Note that for air bet ween t he plates
  o and   o so t hat
C 
1
1

 3 108 m/s
LC
o o
July, 2003
L h o h

 377 
C w o w
© 2003 by H.L. Bertoni
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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
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Transmission Line Equations
T aking the limit asDz  0 gives t he T ransmission Line Equat ions
V (z,t)
I(z,t)
 L
z
t
I(z,t)
V (z,t)
 C
z
t
T hese are coupled,first order,part ial differential equations whose solutions
are in terms of funct ions
F(t - z/v) and G(t  z /v) that are det ermined by
t he sources. T he solut ions for volt age and current are of t he form
1
F (t - z/v) - G(t  z /v)
Z
Direct substitution into the T L Equat ions,
and using t he chain rule gives
V(z,t)  F(t - z/v) + G(t  z /v)
I(z,t) 
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 t he prime (') indicates differentiat ion with respect t o the t otal variable
inside the parentheses ofF or G.

July, 2003
© 2003 by H.L. Bertoni
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Conditions for Existence of TL Solution
For the two equat ions to be sat isfied
1 L

v Z
and
1
C
vZ
1
LC
Mult iplying bot h sides of the two equat ions gives 2 
or
vZ
Z
1
v
m /s
LC
vZ
L
Dividing bot h sides of the t wo equat ions gives 
or
v ZC
L
Z

C
v and Z are interpreted as t he 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 t hatG(t  z /v)  0
T hen the volt age 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 dist urbance
t raveling in t he posit ive
z direct ion with
velocityv.
a
-a
t>0
Note t hat t he current in t he conductor at
positive potential flows in the direction of
wave propagat ion.
July, 2003
z
© 2003 by H.L. Bertoni
V(z,t)=F[(-1/v)(z-vt)]
a+vt
z
-a+vt
vt
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G(t+z/v) Is a Wave Traveling in -z Direction
Assume t hatF(t  z /v)  0
T hen the volt age 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 dist urbance
t>0
t raveling in the negative
z direct ion with
velocityv.
Because of t he m inus sign in
I(z,t), t he
V(z,t)=G[(1/v)(z+vt)]
2a-vt
z
-vt
a-vt
physical current in t he conductor at posit ive
potential flows in t he direction of wave propagation.
July, 2003
© 2003 by H.L. Bertoni
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Example of Source Excitation
Excitation at one end of a semi
- infinit e lengt h of transmission line.
Source has open circuit volt age
VS (t) and int ernal resistance
RS .
Radiat ion condition requires t hat excited waves t ravel away from source.
T erminal condit ions 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
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
RS I(0,t)
VS(t) +
V(0,t)
∞
z
0
I(0,t)
∞
+
V(0,t)
0
© 2003 by H.L. Bertoni
RS
VS(t)
z
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Receive Voltage Further Along Line
Volt age observed on a high impedance scope at a distance
l from source.
Z
VS (t  l v)
Z  RS
Delayed version of the source voltageVS(t) +
V(l,t)  F (t  l v) 
Scope
RS
V(l,t)
∞
wit h t he semi- infinit e line acting as a
load resisor for the source.
V(l,t)  Gt  ( l v) 
z
0
Z
VS (t  l v)
Z  RS
Delayed version of the source voltage
∞
wit h t he semi- infinit e 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
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Power Carried by Waves
P(z,t)
Instantaneous powerP(z,t) carried past plane
perpendicular toz.
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 2
2
 F (t  z v)  G (t  z v)
Z
T he two waves carry power independent ly in t he direction of wave
propagat ion
For each wave,a t ransmission line extending to
z   acts as a resistor
of value Z,even though t he wires were assumed to have no resistance.
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© 2003 by H.L. Bertoni
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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
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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
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Electric Field and Voltage for Parallel Plates
w
x
Ex(z,t)
z
h
+
V(z,t)
-
y
T he elect ric field goes from t he posit ive plat e to t he negative plat e. If
w >> h, t he elect ric field out side of t he plates is very small. Bet ween
the plates it is nearly constant over the cross
- section wit h value
1
E x (z,t)   V (z,t) Volts/m or V (z,t)  hEx (z,t).
h
w
Recall t hat C   .
h
July, 2003
© 2003 by H.L. Bertoni
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Magnetic Field and Current for Parallel Plates
w
x
I(z,t)
h
z
y
Hy(z,t) or By(z,t)
T he magnet ic field links the currents in t he plat es.w If>> h, t he magnetic
field outside of the plat es is very small. Between the plat es it is nearly
constant over t he cross
- section, as if in a solenoid,with value
Hy (z,t) 
1

By (z,t) 
Recall t hat L  
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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
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Maxwell’s Equations in 1-D
Insert ing the foregoing expressions for
V(z,t), C, I(z,t) and L into t he
T ransmission Line equations

h 
hEx (z,t)    wHy (z,t)
z
 w t

w 
wHy (z,t)    hEx (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
T hese are t he two M axwell equations for linearly polarized wave propagating i
1- D. T hey are independent of
h,w)
( and refer to the fields.
We may think of t he plat es as being takenx,y)
t o (  so t hey need not be
considered.
T he 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
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Plane Waves: Solutions to Maxwell Equations
Maxwell's equations are form ally equivalent t o the T ransmission Line Equat ions
T he solut ion is therefore in term s of t wo wave traveling in opposit e directions
alongz .
1
E x (z,t)  F (t  z /v)  G(t  z /v)
H y (z,t)  F(t  z /v)  G(t  z /v)

In air v 
1
o o
 c  3 108 m /s is t he speed of light and 
o
 377 
o
is t he wave im pedance.
For waves in simple dielect ric medium,o is mult iplied by the relat ive dielectric
const ant r .
For normal m ediar  1, but it can be a funct ion of frequency. As and example,
in water at radio frequencies (below 20 GHz)
r  81, but at opt ical
frequencies r  1.78.
July, 2003
© 2003 by H.L. Bertoni
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Power Density Carried by Plane Waves
T otal instantaneous power carried in parallel plate line
P(z,t)  V (z,t)I(z,t)  hEx (z,t)wHy (z,t)
 hwEx (z,t)H y (z,t) wat ts
E
P ower densit y crossing any plane perpendicular
toz is
p(z,t)  P(z,t) hw  E x (z,t)H y (z,t) wat t/m
Direction of
propagation
2

1
F


2
H
(t  z /v)  G 2 (t  z /v)
Direct ion ofH y is such that turning a right hand screw in the
direction fromE x t o Hy advances t he screw in t he direct ion of
propagation
July, 2003
© 2003 by H.L. Bertoni
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Polarization
T he physical properties of a plane wave are independent of t he
coordinat e system.
For a plane wave traveing in one direction
:
Electric field vect orE must be perpedicular to t he direct ion of
propagation.
Magnet ic field vect orH must be perpedicular toE and t o t he
direction of propagation.
T he vect or cross productp  E  H wat t/m2 is in t he direction
of propagat ion.
T he rat io E H is t he wave im pedance .
July, 2003
© 2003 by H.L. Bertoni
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Examples of Polarization
Linear polaization ofE alongx
Linear polaizat ion of
E alongy
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 vect or alongx
ay = unit vect or alongy
H
z
E
y
H
y
Circular polarizat ion
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
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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
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Junctions Between Two Regions
T erm inal condt ions for t he
I(0-,t)
Junction of t wo T L'
s
V(0 ,t)  V (0 ,t)

TL 1

V(0-,t)
I(0+,t)
+
V(0+,t)
TL 2
I(0 ,t)  I(0 ,t)
0
Boundary conditions at t he
z
x
int erface of two m edia
E x (0,t)  E x (0 ,t)
Ex(0-,t)
Ex(0+,t)
Hy(0-,t)
Hy(0+,t)
Hy (0,t)  Hy (0 ,t)
P lane wave propagation and
boundary conditions are analogus
t o junct ioning of t wo TsL'
July, 2003
Medium 1
© 2003 by H.L. Bertoni
z
Medium 2
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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 creat es an incident wave whose elect ric field is given by the known
functionF1 (t - z/v1 ). Using the boundary conditions we solve for t he unknown
functions G1 (t +z/v1 ) and F2 (t - z/v2 ) for the electric fields of the reflected
and transmit ted 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)
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Reflection and Transmission Coefficients
Solution of t he boundary condit ion equations for
G1 (t) and F2 (t) in t erms ofF1 (t)
G1 (t)  F1 (t)
F2 (t)  F1 (t)
T he reflect ion coefficient
 and t ransm ission coefficient
 are given by:

 2  1
 2  1
  1  
2 2
 2  1
Exam ples:
I. Suppose medium 1 is air so t hat
1    o  o  377 and medium 2 has
relat ive dielectric const antr  4 so that 2  o ro  0.5. T hen going
from air- t o- dielectric ad 
0.5  
1
1 2
  and ad  1 
0.5  
3
3 3
July, 2003
© 2003 by H.L. Bertoni
30
Reflection and Transmission, cont.
II. Now suppose the wave is incident from the dielectric onto air so t hat medium 1
is t he dielect ric1  0.5  and medium 2 is air2  . T hen going from
dielect ic- t o- air, da 
  0.5
1
1 4

and ad  1 
  0.5
3
3 3
Note t hat:
1. da  ad
2. Since T is t he ratio of fields,
not power,it can be greater than 1.
July, 2003
© 2003 by H.L. Bertoni
31
Reflected and Transmitted Power
Instantaneous power carried by the incident wave
p In (z,t), t he reflect ed wave
pRe (z,t), and t he transmit t ed 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) 
Tr
Tr
Tr
p (z,t)  E x (z,t)H y (z,t) 
1
1
1
2
G12 (t  z v1 )
2
F2 (t  z v1 )
1 2
F1 (t) as well as
Z1
1
1
pTr (0 ,t)  F22 (t)   2 F1 2 (t)
Just on either side of t he interface
p In (0,t) 
pRe (0,t) 
July, 2003
1
1
G12 (t)   2
1
1
F12 (t)
and
© 2003 by H.L. Bertoni
2
2
32
Conservation of Power and Reciprocity



Conservation of power requires thatp (0 ,t)  p (0 ,t)  p (0 ,t) so that
In
1
1
F1 2 (t)   2
1
1
F12 (t)   2
1
2
F12 (t)
Re
or
1 2   2
Tr
1
2
T his relation is easily shown t o be satisfied from t he expressions,for
.
For waves going from medium 2 t o medium t1,
he reflect ion coefficient12 is
the negat ive of21 going from medium 1 to m edium 2. T hus for eit her
Re

pTr (0 ,t)
2
case the rat ios In 

and

1

are the same.
In

p (0 ,t)
p (0 ,t)
T herefore the sam e fract ion of the incident power is reflected from and
p (0 ,t)
2
t ransmit ted t hrough t he interface for waves incident from either medium.
T his result is an exam ple of a very general wave property called reciprocity.
July, 2003
© 2003 by H.L. Bertoni
33
Termination of a Transmission Line
T erminal condt ions
V(0,t)  RL I(0,t)
RL
F(t)  G(t) 
F(t)  G(t)
Z
Solving forG(t) in terms ofF (t),
I(0-,t)
TL
G(t)  F(t) where the reflection
coefficient is 
V(0-,t)
+
RL
0
z
RL  Z
RL  Z
Special cases:
1. M at ched termination,
RL  Z and   0. Simulates a semi- infinit e T L
2. Open circuit ,RL   and   1. T otal reflect ion wit hV (0,t)  2F (t).
3. Short circuit ,RL  0 and   1. T otal reflect ion wit hV (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- t ime diagram indicates t he relative amplit udes of t he electric field of
t he individual component s of t he mult iply reflected waves. In adding fields,
account must be t aken of the relat ive delay bet ween t he different component
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 t hat t he voltage and current (or
E x and H y fields) have harmonic t ime
dependenceexp( jt) at z  0. T hen
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 t he complex volt age and current at
z  0.
T he functionsF(t) and G(t) can sat isfy t hese equations only if they too have
harm onic time dependence. Hence
F(t)  V e jt
G(t=)V e jt
and
where V   12 V (0)  ZI(0) and V  
1
2
V (0)  Z I(0)
are t he com plex
volt age amplitudes of t he waves t raveling in 
the
z direct ions.
July, 2003
© 2003 by H.L. Bertoni
39
Traveling Wave Representation
At other locat ionsz  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 t he phasor volt age andI(z) is t he phasor currentwhich
,
give t he
spat ial variation for the implied t ime dependence
exp( jt).
I(z,t) 
1
Define the wave number (propagat ion constant)
k   v m . T hen
1   jk z
  jk z
  jk z
  jk z
V (z)  V e  V e
and I(z)  V e  V e 
Z
is t he traveling wave representation of phasor volt age and current.
July, 2003
© 2003 by H.L. Bertoni
40
Standing Wave Representation
Substit uting t he expressions forV  and V  in terms ofV (0) and I(0),
and rearranging terms gives t he st anding wave representation of the phasor
volt age and current:
V(z)  12 V (0)e jk z  e  jk z 12 Z I(0)e  jk z  e  jk z V (0)coskz  jZ I(0)sinkz
I(z) 
1
V (0)e  jk z  e  jk z 12 I(0)e  jk z  e  jk z I(0)coskz  j V (0)sinkz
Z
Z
1
2
T he wavenum ber is k   v 2f v  2  where  is t he
wavelength   v f  2 k
For plane waves in a dielect ric mediumk   
July, 2003
© 2003 by H.L. Bertoni
41
Variation of the Voltage Magnitude

For V  0 we have a pure t raveling For I(0) 0 we have a pure standing
  jk z
wave V (z)  V e
V (z)  V e jk z = V 
. T he magnit ude wave V (z)  V(0)cos kz. Its magnit ude
is independent V (z)  V (0) coskz is periodic wit h
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)
Perfect
P lane wave incident on a perfectly
conduticng plat e and t he equivalent
conductor
circuit of a shorted T L
0
ExRe(z)
z
T he st anding wave field is
E x (z)  12 ISC e  jk z  e  jk z
Reflected wave
ISC
, v
short
0
July, 2003
E x (0)  0 and H y (0)  I SC
  jI SC sinkz
T wo waves of equal amplitude and
t raveling in opposite direct ions creat e
a standing wave.
z
© 2003 by H.L. Bertoni
43
Standing Wave Before a Conductor, cont.
Plot of the magnit ude of the standing wave field
E x (z)  ISC sinkz
ISC
-z
Since k   v  2f v  2  the nodes (zeros) of t he field are
separat ed by a dist ancek Dz   or Dz   k   2
July, 2003
© 2003 by H.L. Bertoni
44
Period Averaged Power
For harmonic time dependence on a Tt L,
he time average over one period


of the instantaneous power is
P(z)  12 Re V (z)I (z)
 wat ts
Using the t raveling wave representat ion


2
2
1
   jk z
  jk z 1
  jk z
  jk z  


P(z)  Re V e  V e  V e  V e  
V V
Z

 2Z
Note t hat t he average power is t he algebraic sum of t he power carried by
1
2
t he incident and reflected waves,
and it is independent ofz.

1
wavesp(z)  E
2

y


For harmonic plane wavesp(z)  Re E x (z)H (z)
1
2
In term s of t raveling
July, 2003
In
x
2
© 2003 by H.L. Bertoni
E
Re
x
2
wat ts/m
2
45
Reflection From a Load Impedance
For a complex load impedanceZL
I(0)
ZL 
V  V 

Z
in t erms ofV  gives
V(0)  V   V   ZL I(0) 
Solving for V 

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 rat io of volt age t o current
can have som e valueV (0) I(0)  ZL
V(0) + ZL
ZIN
Using the formulas forV (z) and I(z)
we can compute t heir rat io at
z  l.
-l
0
z
Defining this ratio asZIN (l) we have
V (l) V (0)cos(-kl)  jZ I(0)sin(kl)

1
I(l)
I(0)cos(-kl)  j V (0)sin(-kl)
Z
Dividing numerat or and denominat or by
I(0) and rearranging gives
ZIN (l) 
ZIN (l)  Z
July, 2003
ZL cos(kl)  jZ sin(kl)
Z  jZ t an(kl)
Z L
Z cos(kl)  jZL sin(kl)
Z  jZL t an(kl)
© 2003 by H.L. Bertoni
49
Properties of the Impedance Transform
T he im pedance formula
ZIN (l)  Z
ZL cos(kl)  jZ sin(kl)
Z  jZ t an(kl)
Z L
Z cos(kl)  jZL sin(kl)
Z  jZL t an(kl)
shows t hat a length T L (or region of space) transforms an impedance
t o a different value.
Som e properties of t he transformation
:
1. For a matched loadZL  Z, the imput impedace is matched
ZIN  Z
2. T he impedance repeatsZIN (l)  ZIN (l  Dl) for k Dl   or
Dl   k   2
3. For quart er wave displacementl   4, kl   2 and impedance
invert sZIN ( 4)  Z 2 Z L
4. If ZL  0, then ZIN (l)  jZ t an(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 t he left . A -semi
infinit e T L (medium )
at it s t erminals (accessible surface) acts as a resistor so that
ZL   3 .
Impedance of the finit e segment of T L Zis  2 . W avenumber of t his
segment is k 2   v 2    r2 oo  ko  r2
where ko    oo is the wavenum ber of free space.
Input impedance at left surface of t he layer is t hen
 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 t he 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 wit h  r2  4
k2  2k o and  2 
o
 12  o
 r2 o
Reflection coefficient for t he 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
1 2
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 bew  30 cm so that 2k o w 
T hen p
Re
4 f
0.3  4fGHz
3 108
9sin 2 (4fGHz )
p  
64cos2 (4fGHz )  25sin 2 (4fGHz )
in
2
||

0
0.25 0.50
0.75
1.0
1.25
1.50 1.75 2.0
fGHz
Since there is no conductivity in t he brick wall,
t he fract ion of the incident
power t ransmitt ed t hrough the wall 1
is 
July, 2003
© 2003 by H.L. Bertoni
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
sin(k2 l)  sin( /2)  1
so that ZIN ( 2 /4)   22 / 3
July, 2003
© 2003 by H.L. Bertoni
55
Example 2: Quarter Wave Layers, cont.
For this value ofZIN
 22  13
we have   2
 2  1 3
22  1 3 , then   0 and no
If we choose t he layer material such t hat
reflect ion t akes place.
Suppose t hat medium 1 is air and m edium 3 is glass with relative
dielect ric constantg
: 22 
For no reflection
o o
o
 1 3 
o  g o
 r2 o
or r2   g
vo
1
v2


Note t hat t he layer t hickness lis  2 /4 
4 f 4 f r 2o o 4 f r 2
or l 
July, 2003
o
4 4 g
where  o is t he wavelengt h 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 t he material between the plat e conduct s electricity,
t here will be a
conduct anceG mho/m in addit ion t o the capacit ance
C farads/m
and inductanceL henry/m.
T he conduct ivit y of a material is give by the parameter
 mho/m
Expressions for t he circuit quantities are
:
w
w
G
C 
h
h
July, 2003
© 2003 by H.L. Bertoni
L 
h
w
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 asDz  0 the Kirchhoff circuit equations for the phasor
volt age and current give t he T L equat ions for harmonic t ime 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
+
I(z)
h
V(z)
z
Ex(z)
y
Hy(z)
If w >> h, t he fields between the plat es are nearly constant over the -cross
section,
so that the phasor circuit quant ities are
V (z)  hE x (z) and I(z)  wH y (z).
Substit uting these exprsssions in t he T L equations for harm onic time dependence,
along wit h t he 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
M axwell's equations for 1- D propagat ion of
With minor manipulat ion,
harmonic waves in a medium wit h conduct ion loss can be writ ten
dH y (z)
dE x (z)
ˆ E x (z)
  j
and
  j H y (z)
dz
dz
ˆ is given by

T he com plex equivalent dielectric const ant
ˆ  ro  j    o  r  j   o 

ˆ   o r  j"
Let "   o . T hen 
In other m at ierials atomic processes lead t o a complex dielect ric of the
form o  r  j". T hese processes have a different frequency
dependence for", but have t he 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 dielect ric const ant
M axwell'
,
s equations
have t he same form as when no loss (conduct ivit y) is present.
T he solut ions therefore have t he same mathematical formwit h
ˆ.
replaced by 
For example,t he traveling wave solut ions in a mat erial are
1
E x (z)  V e  jk z  V e  jk z and
H y (z)  V e jk z  V e  jk z

ˆ    o r  j" and  
Here k   



ˆ
 o r  j"
are complex quantities.
July, 2003
© 2003 by H.L. Bertoni
64
Wave Number and Impedance
T he com plex wavenumberk will have real and im aginary parts
k    j    o r  j"
If " is less than about  r 10, we may use t he approxim at ions
    or
and
    or
"
2r
 
1 j " 

Similarly, for " sm all,  

 o r  j"
 or  2r 

July, 2003
© 2003 by H.L. Bertoni
65
Effect of Loss on Traveling Waves
For a wave t raveling in the positive
z direction
E x (z)  V e  jk z  V  exp j(  j)z  V  exp(-jz)exp(z)
T he presence of loss (conduct ivit y) result s in a finite value of the
att enuation const ant
. T he att enuation (decay) lengt h1is.
T he magnitude of t he field depends on z as given by

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,t he at t enuat ion in units of deci
- Bells is found from
E x (z) 
V  exp(z) 
Attn 20log10 
 20log10 


V
E x (0) 


 20z log10 e  8.67z
T hus t he att enuat ion rat e of the wave in a m edium8.67
is  dB/m
July, 2003
© 2003 by H.L. Bertoni
67
Effect of Loss on Traveling Waves, cont.
T he instantaneous field of the wave has bot h sinusoidal variat ion over a
wavelength   2  and the decay over t he at tenuat ion length
1 .
For real amplit udeV , the spat ial 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 t he brick wall,
t he int erference minima are
reduced and t he reflect ion coefficient approaches that of t he
  o
first air- brick interface or  B
 1 3
B  o
T he fract ion of t he incident power t ransmitt ed through the
wall is  1 
July, 2003
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