ELECTROMAGNETIC THEORY AND TRANSMISSION LINES … · 2 0 1 G[E = r' . 4 0 U & 2 0 S 1 dS E = r' . 4...
Transcript of ELECTROMAGNETIC THEORY AND TRANSMISSION LINES … · 2 0 1 G[E = r' . 4 0 U & 2 0 S 1 dS E = r' . 4...
ELECTROMAGNETIC THEORY AND TRANSMISSION LINES
B.TECH IV SEM-ECE
Prepared ByDr.P.Ashok Babu,Professor
Mrs.A.Usha Rani,Assistant ProfessorMr.Murali Krishna, Assistant Professor
Dr.S.Pedda Krishna, Professor.
Introduction to Co-Ordinate System and
Electrostatics
UNIT -I
Spherical Coordinates
Cylindrical Coordinates
Cartesian Coordinates
P (x, y, z)
P (r, θ, Φ)
P (r, Φ, z)
x
y
zP(x,y,z)
Φ
z
rx
y
z
P(r, Φ, z)
θ
Φ
r
z
yx
P(r, θ, Φ)
Rectangular Coordinates
Or
X=r cos Φ,Y=r sin Φ,Z=z
X=r sin θ cos Φ,Y=r sin θ sin Φ,Z=z cos θ
• dx, dy, dz are infinitesimal displacements along X,Y,Z.
• Volume element is given by
dv = dx dy dz
• Area element is
da = dx dy or dy dz or dxdz
• Line element is
dx or dy or dz
Ex: Show that volume of a cube of edge a is a3.
P(x,y,z)
X
Y
Z
3
000
adzdydxdvV
aa
v
a
dx
dy
dz
Cartesian Coordinates
X
Y
Z
r
φ
Z
Cylindrical coordinate system
• dr is infinitesimal displacement
along r, r dφ is along φ and
dz is along z direction.
• Volume element is given by
dv = dr r dφ dz
• Limits of integration of r, θ, φ
are
0<r<∞ , 0<z <∞ , o<φ <2π
Ex: Show that Volume of a
Cylinder of radius ‘R’ and
height ‘H’ is π R2H .φ is azimuth angle
X
Y
Z
rφ
r dφ
dz
dr
r dφ
dr
dφ
Cylindrical coordinate system
Differential quantities:
Length element:
Area element:
Volume element:
dzardadraldzr
ˆˆˆ
rdrdasd
drdzasd
dzrdasd
zz
rr
ˆ
ˆ
ˆ
dzddrrdv
Limits of integration of r, θ, φ are 0<r<∞ , 0<z <∞ , o<φ <2π
• dr is infinitesimal displacementalong r, r dθ is along θ and rsinθ dφ is along φ direction.
• Volume element is given by
dv = dr r dθ r sinθ dφ
• Limits of integration of r, θ, φare
0<r<∞ , 0<θ <π , o<φ <2π
Ex: Show that Volume of a sphereof radius R is 4/3 π R3 .
P(r, θ, φ)
X
Y
Z
r
φ
θ
drP
r dθ
r sinθ dφ
r cos θ
r sinθ
Spherical Coordinate System
3
3
0 0
2
0
2
2
3
42.2.
3
sin
sin
RR
dddrr
dddrrdvV
R
v
Try Yourself:
1)Surface area of the sphere= 4πR2 .
Volume of a sphere of radius ‘R’
Spherical Coordinates: Volume element in space
System Coordinates dl1 dl2 dl3
Cartesian x,y,z dx dy dz
Cylindrical r, φ,z dr rdφ dz
Spherical r,θ, φ dr rdθ r sinθdφ
• Volume element : dv = dl1 dl2 dl3
• If Volume charge density ‘ρ’ depends only on ‘r’:
Ex: For Circular plate: NOTE
Area element da=r dr dφ in both the
coordinate systems (because θ=900)
drrdvQ
v l
2
4
Points to remember
6 - 12
The gradient is the closest thing to an ordinary derivative, taking a
scalar-valued function into a vector field.
The simplest geometric definition is “the derivative of a function with
respect to distance along the direction in which the function changes
most rapidly,” and the direction of the gradient vector is along that most-
rapidly changing direction.
The Gradient operator
ˆˆ ˆr e c ta n g u la r : = xx
1ˆˆ ˆc y lin d r ic a l: = r
1 1ˆ ˆˆs p h e r ic a l: = r
s in
y zy z
zr r z
r r r
g ra d ( ) ( )f r f r r r
6 - 13
Divergence in spherical coordinates
2
2
(s in )( )1 1 1
s in s in
rvvr v
d iv v vr r r r
r r
The coordinate system is orthogonal if the surfaces made by setting the
value of the respective coordinates to a constant intersect at right angles.
In the spherical example this means that a surface of constant r is a sphere.
A surface of constant θ is a half-plane starting from the z-axis. These
intersect perpendicular to each other. If you set the third coordinate, φ , to
a constant you have a cone that intersects the other two at right angles.
volumearea
6 - 14
Gauss’s Theorem
0 0
1 1lim limV V
d Vd iv v v d A
V d t V
rr r
Ñ
Fix a surface and evaluate the surface integral of over the surface:v
r
S
v d Arr
Ñ
Recall the original definition of the divergence of a vector field:
kVNow divide this volume into a lot of little volumes, with
individual bounding surfaces . If you do the surface integrals of over each of these pieces and add all of them, the result is the
original surface integral.
kS
v d A
rr
6 - 15
Stokes’ Theorem
The expression for the curl in terms of integrals is
Use exactly the same reasoning as that was used in the case of the Gauss’s theorem, this leads to
d A v
r rv r
1h A
1A
1nh
Let us first apply it to a particular volume, one that is very thin and small.
Take a tiny disk of height , with top and bottom area . Let be
the unit normal vector out of the top area. For small enough values of
these dimensions, is simply the value of the vector
inside the volume times the volume
16
Coulomb’s Law
• Coulomb‟s law is the “law of action” between charged bodies.
• Coulomb‟s law gives the electric force between two point charges in an otherwise empty universe.
• A point charge is a charge that occupies a region of space which is negligibly small compared to the distance between the point charge and any other object.
17
Coulomb’s Law
2
120
21
12
4ˆ
12
r
QQaF
R
Q1
Q212r
12F
Force due to Q1
acting on Q2
Unit vector in
direction of R12
18
Coulomb’s Law
• The force on Q1 due to Q2 is equal in
magnitude but opposite in direction to the
force on Q2 due to Q1.
1221 FF
19
Electric Field
• Consider a point chargeQ placed at the origin ofa coordinate system inan otherwise emptyuniverse.
• A test charge Qt broughtnear Q experiences aforce:
2
04ˆ
r
QQaF t
rQt
Q
Qt
r
20
Electric Field
• The existence of the force on Qt can be
attributed to an electric field produced by Q.
• The electric field produced by Q at a point in
space can be defined as the force per unit
charge acting on a test charge Qt placed at that
point.
t
Q
Q Q
FE
t
t0
lim
21
Electric Field
• The electric field describes the effect of a
stationary charge on other charges and is an
abstract “action-at-a-distance” concept, very
similar to the concept of a gravity field.
• The basic units of electric field are newtons
per coulomb.
• In practice, we usually use volts per meter.
22
Electric Field
• For a point charge at the origin, the electric
field at any point is given by
3
0
2
044
ˆr
rQ
r
QarE
r
23
Electric Field
• For a point charge located at a point P’
described by a position vector
the electric field at P is given by
rrR
rrR
R
RQrE
where
43
0
r
Q
P
r R
rO
24
Electric Field
• In electromagnetics, it is very popular to
describe the source in terms of primed
coordinates, and the observation point in terms
of unprimed coordinates.
• As we shall see, for continuous source
distributions we shall need to integrate over
the source coordinates.
Field due to Different Types of Charges
E = ?++
+
++ + ++ + +
+
Source
+
Continuous Charge Distributions
•Cut source into small (“infinitesimal”) charges dq
•Each produces
2
( )ˆ
e
d qd E k
r r
dE
r
dq
2e
d qd E k
r
or
Steps:•Draw a coordinate system on the diagram
•Choose an integration variable (e.g., x)
•Draw an infinitesimal element dx
•Write r and any other variables in terms of x
•Write dq in terms of dx
•Put limits on the integral
•Do the integral or look it up in tables.
Continuous Charge Distributions
2
0
1 λ d xE = r ' .
4 π ε r '
2
0 S
1 d SE = r ' .
4 π ε r '
2
0 V
1 d VE = r ' .
4 π ε r '
Charge distributed along a line:
Charge distributed over a surface:
Charge distributed inside a volume:
If the charge distribution is uniform, then , , and can be taken outside the
integrals.
Electric Flux Density
30
Electric Flux Density
Consider a point charge at the origin:
Q
31
Electric Flux Density of a Point Charge
(1) Assume from symmetry the form of the field
(2) Construct a family of Gaussian surfaces
rDaDrr
ˆ
spheres of radius r where
r0
spherical
symmetry
32
Electric Flux Density of a Point Charge
(3) Evaluate the total charge within the volume
enclosed by each Gaussian surface
V
evencldvqQ
33
Electric Flux Density of a Point Charge
Q
R
Gaussian surface
QQencl
34
Electric Flux Density of a Point Charge
(4) For each Gaussian surface, evaluate the
integral
DSsdD
S
24 rrDsdD
r
S
magnitude of D
on Gaussian
surface.
surface area
of Gaussian
surface.
35
Electric Flux Density of a Point Charge
(5) Solve for D on each Gaussian surface
S
QD
encl
24
ˆr
QaD
r
2
004
ˆr
Qa
DE
r
36
Electric Flux Density of a Spherical Shell of Charge
Consider a spherical shell of uniform charge density:
otherwise,0
,0
braqq
eva
b
Gauss Law, It’s Applications to
Symmetrical Charge Distributions
38
Gauss’s Law
• Gauss‟s law states that “the net electric flux
emanating from a close surface S is equal to
the total charge contained within the volume
V bounded by that surface.”
encl
S
QsdD
39
Gauss’s Law (Cont’d)
V
Sds
By convention, ds
is taken to be outward
from the volume V.
V
evencldvqQ
Since volume charge
density is the most
general, we can always write
Qencl in this way.
40
Applications of Gauss’s Law
• Gauss‟s law is an integral equation for the
unknown electric flux density resulting from a
given charge distribution.
encl
S
QsdD known
unknown
41
Applications of Gauss’s Law (Cont’d)
• In general, solutions to integral equations must
be obtained using numerical techniques.
• However, for certain symmetric charge
distributions closed form solutions to Gauss‟s
law can be obtained.
42
Applications of Gauss’s Law (Cont’d)
• Closed form solution to Gauss‟s law relies on
our ability to construct a suitable family of
Gaussian surfaces.
• A Gaussian surface is a surface to which the
electric flux density is normal and over which
equal to a constant value.
Electric Potential: Potential Field
Due To Different Types of Charges
44
Electrostatic Potential• An electric field is a force field.
• If a body being acted on by a force is moved from one point toanother, then work is done.
• The concept of scalar electric potential provides a measure of thework done in moving charged bodies in an electrostatic field.
• The work done in moving a test charge from one point to another ina region of electric field:
b
a
b
a
ba ldEqldFW
ab
q
F
ld
45
Electrostatic Potential
• In evaluating line integrals, it is customary to take the dl in the
direction of increasing coordinate value so that the manner in
which the path of integration is traversed is unambiguously
determined by the limits of integration.
3
5
ˆ dxaEqWxba
46
Electrostatic Potential
• The electrostatic field is conservative:
– The value of the line integral depends only on the end
points and is independent of the path taken.
– The value of the line integral around any closed path is
zero.
0C
ldE
C
47
Electrostatic Potential
• The work done per unit charge in moving a test charge
from point a to point b is the electrostatic potential
difference between the two points:
b
a
ba
abldE
q
WV
electrostatic potential difference
Units are volts.
48
Electrostatic Potential
• Since the electrostatic field is conservative we can write
aVbV
ldEldE
ldEldEldEV
a
P
b
P
b
P
P
a
b
a
ab
00
0
0
49
Electrostatic Potential
• Thus the electrostatic potential V is a scalar field that is
defined at every point in space.
• In particular the value of the electrostatic potential at any point
P is given by
P
P
ldErV
0 reference point
50
Electrostatic Potential
• The reference point (P0) is where the potential is zero
(analogous to ground in a circuit).
• Often the reference is taken to be at infinity so that the
potential of a point in space is defined as
P
ldErV
51
Electrostatic Potential and Electric Field
• The work done in moving a point charge from point a
to point b can be written as
b
a
abba
ldEQ
aVbVQVQW
52
Electrostatic Potential and Electric Field
• Along a short path of length Dl we have
lEV
lEQVQW
or
53
Electrostatic Potential and Electric Field
• Along an incremental path of length dl we have
• Recall from the definition of directional derivative:
ldEdV
ldVdV
>Thus
VE
Potential Gradient and the Dipole
field due to Dipole, Maxwell’s Two
Equations for Electrostatic Field
55
Charge Dipole
• An electric charge dipole consists of a pair of equaland opposite point charges separated by a smalldistance (i.e., much smaller than the distance atwhich we observe the resulting field).
d
+Q -Q
56
Dipole Moment
• Dipole moment p is a measure of the strength
of the dipole and indicates its direction
dQp +Q
-Q
d
p is in the direction from
the negative point charge to
the positive point charge
57
Electrostatic Potential Due to Charge Dipole
observation
point
d/2
+Q
-Q
z
d/2
P
Qdapz
ˆ
R
Rr
58
Electrostatic Potential Due to Charge Dipole (Cont’d)
R
Q
R
QrVrV
0044
,
cylindrical symmetry
59
Electrostatic Potential Due to Charge Dipole (Cont’d)
d/2
d/2
cos)2/(
cos)2/(
22
22
rddrR
rddrR
R
R
r
P
60
Electrostatic Potential Due to Charge Dipole in the Far-Field
• assume R>>d
• zeroth order approximation:
RR
RR
0V
not good
enough!
61
Electrostatic Potential Due to Charge Dipole in the Far-Field (Cont’d)
• first order approximation from geometry:
cos2
cos2
drR
drR
d/2
d/2
lines approximately
parallel
R
R
r
62
Electrostatic Potential Due to Charge Dipole in the Far-Field (Cont’d)
• Taylor series approximation:
cos2
111
cos2
11
cos2
11
cos2
111
r
d
rR
r
d
r
r
d
r
dr
R
1,11
:Recall
xnxxn
63
Electrostatic Potential Due to Charge Dipole in the Far-Field (Cont’d)
2
0
0
4
cos
2
cos1
2
cos1
4,
r
Qd
r
d
r
d
r
QrV
64
Electrostatic Potential Due to Charge Dipole in the Far-Field (Cont’d)
• In terms of the dipole moment:
2
0
ˆ
4
1
r
apV
r
Energy Density in Electrostatic
Field
Energy Density in Electrostatic Field
• To determine the energy that is present in an
assembly of charges
• let us first determine the amount of work
required to assemble them.
• Let us consider a number of discrete charges
Q1, Q2,......., QN are brought from infinity to
their present position one by one.
Energy Density in Electrostatic Field
• Since initially there is no field present, the
amount of work done in bring Q1 is zero.
• Q2 is brought in the presence of the field of Q1,
the work done W1= Q2V21 where V21 is the
potential at the location of Q2 due to Q1.
It takes no work to bring in first charges
1 0W
Work needed to bring in q2 is :
1 12 2 2
0 1 2 0 1 2
1 1[ ] ( )
4 4
q qW q q
R R
1 2 1 23 3 3
0 1 3 0 2 3 0 1 3 2 3
1 1 1[ ] ( )
4 4 4
q q q qW q q
R R R R
Work needed to bring in q3 is :
Work needed to bring in q4 is :
31 24 4
0 1 4 2 4 3 4
1[ ]
4
qq qW q
R R R
for q1
Total work
W=W1+ W2+ W3 +W4
1 3 2 3 3 41 2 1 4 2 4
0 1 2 1 3 2 3 1 4 2 4 3 4
1
4
q q q q q qq q q q q q
R R R R R R
0
1
1
4
nj
ii j
j
qV P
R
j i
01 1
1
4
n ni j
i ji j
q q
R
ij jiR R
01 1
1 1
2 4
n nj
ii j
i j
R
0
1 1
1
8
n ni j
i ji j
q q
R
1
1
2
n
i i
i
W q V P
j i j i
Where is the energy stored? In charge or in field ?
Both are fine in ES. But,it is useful to regard the energy
as being stored in the field at a density
2
02
E Energy per unit volume
The superposition principle,not for ES energy
201 1
2W E d
202 2 τ
2W E d
201 2( )
2to tW E E d
1 2 0 1 2( )W W E E d
2 201 2 1 2( 2 )
2E E E E d
Convection and Conduction Currents, Continuity Equation
and Relaxation Time
Current (in amperes) through a given area is the electric charge
passing through the area per unit time
Current density is the amount of current flowing through a surface,
A/m2, or the current through a unit normal area at that point
Current density
Convection and Conduction Currents
Currentd Q
Id t
Where
IJ
S
.S
I J d S
Depending on how the current is produced, there are different
types of current density
• Convection current density
• Conduction current density
• Displacement current density
- Current generated by a magnetic field
Does not involve conductors and does not obey Ohm‟s law
Occurs when current flows through an insulating medium such as
liquid, gas, or vacuum
Convection and Conduction Currents
Convection current density
Where v is the velocity vector of the fluid
Convection and Conduction Currents
v v y
Q yI S S v
t t
y v y
IJ v
S
Conduction current density
Current in a conductor
Obeys Ohm‟s law
Consider a large number of free electrons travelling in a
metal with mass (m), velocity (v), and scattering time (time
between electron collisions), τ
Convection and Conduction Currents
The carrier density is determined by the number of electrons, n,
with charge, e
m vF q E
vn e
Conduction current density can then be calculated as2
v
n eJ v E E
m
This relationship between current concentration and electric field is
known as Ohm‟s Law.
Continuity Equation
Continuity Equation and Relaxation Time
Where Qin is the total charge enclosed by the closed surface.
Invoking divergence theorem
.in
o u tS
d QI J d S
d t
. .v
S
J d S Jd v But,
in v
vv v
d Q ddd v d v
d t d t d t
Due to the principle of charge conservation, the time rate of
decrease of charge within a given volume must be equal to the net
outward current flow through the closed surface of the volume.
Thus, the current coming out of the closed surface is
From the above three equations, we can write as
which is called the continuity of current equation.
The continuity equation is derived from the principle of
conservation of charge
It states that there can be no accumulation of charge at any point
For steady currents,
Hence,
The total charge leaving a volume is the same as the total charge
entering it.
v
v v
v
J d v d vt
Jt
0v
t
0J
Continuity Equation and Relaxation Time
Utilizing the continuity equation and material properties such
as permittivity and conductivity, one can derive a time
constant
We start with Ohm‟s and Gauss‟ Laws
J
v
v v
E
Et
0v v
t
v
v
t
Relaxation Time
J E
0ln ln
v v
t
Continuity Equation and Relaxation Time
0
r
t
T
v ve
rT
Continuity Equation and Relaxation Time
is the initial charge density. The relaxation time(Tr) is the time
it takes a charge placed in the interior of a material to drop by e-1
(=36.8%) of its initial value.
For good conductors Tr is approx. 2*10-19 s.
For good insulators Tr can be days
Capacitance- Parallel plate, Co-axial and Spherical Capacitor
Capacitance is an intuitive characterization of a capacitor. It tells you,
how much charge a capacitor can hold for a given voltage
The property of a capacitor to „store electricity‟ may be called its
capacitance
Generally speaking, to have a capacitor we must have two (or more)
conductors carrying equal but opposite charges
Capacitance
Figure 1. Charge carriers of conductor with opposite polarity
Suppose we give Q coulomb of charge to one of the two plates of
capacitor, the potential difference V is established between the two
plates, then its capacitance is
The capacitance C is a physical property of the capacitor and in
measured in farads (F)
The charge Q on the surface of the plate and the potential difference
Vab between the plates can be represented in terms of electric field
Capacitance
a b
QC
V
.
.
b
a b
a
E d SQC
VE d l
.E d S Q
.
b
a b a b
a
V V V E d l
Therefore, the capacitance C can be written as
• Choose a suitable coordinate system.
• Let the two conducting plates carry charges + Q and - Q
• Determine E using Coulomb's or Gauss's law and find V
• Finally, obtain C from
Capacitance
Procedure for Obtaining Capacitance
Capacitance can be determined using first method are as follows
Capacitance can be obtained for any given two-conductor
capacitance by following either of these methods:
• Assuming Q and determining V in terms of Q (involving Gauss's
law)
• Assuming V and determining Q in terms of V (involving solving
Laplace's equation)
QC
V
Parallel-Plate Capacitor
S
Q
A
the charge density is given by
The flux passing through the medium and flux density is given by
. .n S
S S
n S
Q
D d A Q d A
D
Figure 2. Parallel plate conductors
Parallel-Plate Capacitor
The charge density in terms of electric field as
D E
S
Q
A
Where,
But, we know,
SE
The above equation modifies to
Also, the relation between electric field and electric potential canbe written as
QE
A
0
.
d
QV E d l E d d
A
Q AC
Q dd
A
Thus, the parallel plate capacitor can be written as
Coaxial(Cylindrical) Capacitor
Consider length l of two coaxial conductors of inner radius a and
outer radius b (b > a) as shown in Figure 3
2
0 0
. .
.
( 2 )
S S
l
S
Q D d A E d A
Q E d A E d d l
Q E l
By applying Gauss's law to an arbitrary Gaussian cylindrical surface
of radius ρ (a < ρ < b), we obtain
Figure 3 Cylindrical conductors
Coaxial(Cylindrical) Capacitor
.2
ln2
ln ln ln ln2 2
ln2
a
b
a
b
Q dV E d r
l
Q
l
Q QV a b b a
l l
Q bV
l a
The potential difference between the inner and outer conductors
can be written as
Thus the capacitance of a coaxial cylinder is given by
2
ln ln2
Q Q lC
Q b bV
l a a
Spherical Capacitor
Consider the inner sphere of radius a and outer sphere of radius
b(b> a) separated by a dielectric medium with permittivity ε as
shown in Figure 4
22
0 0
2
. .
. s in
( 4 )
S S
S
Q D d A E d A
Q E d A E r d d
Q E r
Figure 4. Spherical conductor
By applying Gauss's law to an arbitrary Gaussian spherical surface
of radius r(a<r<b), we obtain
Spherical Capacitor
S
Q
A
Therefore, the potential difference between the inner and outersphere can be written as
2
Q.
4
Q 1
4
Q 1 1
4
a
b
a
b
d rV E d l
r
r
Va b
Thus, the capacitance of a spherical capacitor is given by4
1 1Q 1 1
4
Q QC
V
a ba b
By letting b ∞, C4πεa, which is the capacitance of aspherical capacitor whose outer plate is infinitely large
Magnetostatics and Time varying fields
UNIT II
Biot-Savart Law
The Law of Biot-Savart is
1 1 1 2
2
1 2
2
4
I dd
R
L aH
(A/m)
To get the total field resulting from a
current, you can sum the contributions
from each segment by integrating
2
4.
RId
R
L aH (A/m)
Note: The Biot-Savart law is analogous to the Coulomb‟s law
equation for the electric field resulting from a differential charge
92
Vector Magnetic Potential
• Vector identity: “the divergence of the curl of
any vector field is identically zero.”
• “If the divergence of a vector field is
identically zero, then that vector field can be
written as the curl of some vector potential
field.”
0 A
93
Vector Magnetic Potential (Cont’d)
• Since the magnetic flux density is solenoidal, it
can be written as the curl of a vector field
called the vector magnetic potential.
ABB 0
94
Vector Magnetic Potential (Cont’d)
• The general form of the B-S law is
• Note that
V
vdR
RrJrB
3
0
4)(
3
1
R
R
R
95
Vector Magnetic Potential (Cont’d)
• Furthermore, note that the del operator operates
only on the unprimed coordinates so that
R
rJ
rJR
RrJ
R
RrJ
1
1
3
96
Vector Magnetic Potential (Cont’d)
• Hence, we have
vdR
rJrB
V
4
0
rA
97
Vector Magnetic Potential (Cont’d)
• For a surface distribution of current, the vector
magnetic potential is given by
• For a line current, the vector magnetic potential is
given by
sd
R
rJrA
S
s
4)(
0
LR
ldIrA
4)(
0
98
Vector Magnetic Potential (Cont’d)
• In some cases, it is easier to evaluate the
vector magnetic potential and then use B
= A, rather than to use the B-S law to
directly find B.
• In some ways, the vector magnetic potential A
is analogous to the scalar electric potential V.
99
Vector Magnetic Potential (Cont’d)
• In classical physics, the vector magnetic
potential is viewed as an auxiliary function
with no physical meaning.
• However, there are phenomena in quantum
mechanics that suggest that the vector
magnetic potential is a real (i.e., measurable)
field.
Forces due to Magnetic Fields,
Ampere’s Force Law
101
Ampere’s Law of Force
• Ampere’s law of force is the “law of action”
between current carrying circuits.
• Ampere’s law of force gives the magnetic force
between two current carrying circuits in an
otherwise empty universe.
• Ampere‟s law of force involves complete
circuits since current must flow in closed
loops.
Inductances and Magnetic
Energy
103
Flux Linkage
• To discuss about inductance, first we have to
know the flux linkage
• Consider two magnetically coupled circuits
C1
I1
S1 S2 C2
I2
104
Flux Linkage (Cont’d)
• The magnetic flux produced I1 linking thesurface S2 is given by
• If the circuit C2 comprises N2 turns and thecircuit C1 comprises N1 turns, then the totalflux linkage is given by
2
2112
S
sdB
2
2121122112
S
sdBNNNN
105
Mutual Inductance
• The mutual inductance between two circuits is
the magnetic flux linkage to one circuit per
unit current in the other circuit:
1
1221
1
12
12
I
NN
IL
106
Mutual Inductance
2
2
21
1
21
21
1
21
1
1221
1
12
12
C
S
ldAI
NN
sdBI
NN
I
NN
IL
107
Mutual Inductance (Cont’d)
112
110
1
4C
R
ldIA
1 2
2
12
21210
21
1
21
12
4C C
C
R
ldldNN
ldAI
NNL
108
Mutual Inductance (Cont’d)
• The Neumann formula for mutual inductance
tells us that
– L12 = L21
– the mutual inductance depends only on the
geometry of the conductors and not on the current
109
Self Inductance
• Self inductance is a special case of mutual
inductance.
• The self inductance of a circuit is the ratio of
the self magnetic flux linkage to the current
producing it:
1
11
2
1
1
11
11
I
N
IL
110
Self Inductance (Cont’d)
• For an isolated circuit, we call the self
inductance, inductance, and evaluate it using
I
N
IL
2
111
Energy Stored in Magnetic Field
• The magnetic energy stored in a region
permeated by a magnetic field is given by
dvHdvHBW
VV
m 2
2
1
2
1
112
Energy Stored in an Inductor
• The magnetic energy stored in an inductor is
given by
2
2
1LIW
m
113
Ampere’s Law of Force (Cont’d)
Two parallel wires
carrying current in
the same direction
attract.
Two parallel wires
carrying current in
the opposite
directions repel.
I1 I2
F12F21
I1 I2
F12F21
114
Ampere’s Law of Force (Cont’d)
A short current-
carrying wire
oriented
perpendicular to a
long current-carrying
wire experiences no
force.
I1
F12 = 0
I2
115
Ampere’s Law of Force (Cont’d)
The magnitude of the force is inversely proportional
to the distance squared.
The magnitude of the force is proportional to the
product of the currents carried by the two wires.
116
Ampere’s Law of Force (Cont’d)
• The direction of the force established by theexperimental facts can be mathematicallyrepresented by
1212
ˆˆˆˆ12 RF
aaaa
unit vector in
direction of force on
I2 due to I1
unit vector in direction
of I2 from I1
unit vector in direction
of current I1
unit vector in direction
of current I2
117
Ampere’s Law of Force (Cont’d)
• The force acting on a current element I2 dl2 by
a current element I1 dl1 is given by
2
12
11220
12
12
ˆ
4 R
aldIldIF
R
Permeability of free space
0 = 4 10-7 F/m
118
Ampere’s Law of Force (Cont’d)
• The total force acting on a circuit C2 having a
current I2 by a circuit C1 having current I1 is
given by
2 1
12
2
12
12210
12
ˆ
4C C
R
R
aldldIIF
119
Ampere’s Law of Force (Cont’d)
• The force on C1 due to C2 is equal in
magnitude but opposite in direction to the
force on C2 due to C1.
1221FF
120
Force on a Moving Charge
• The total force exerted on a circuit C carrying
current I that is immersed in a magnetic flux
density B is given by
C
BldIF
121
Force on a Moving Charge
• A moving point charge placed in a magnetic
field experiences a force given by
BvQ
The force experienced
by the point charge is
in the direction into the
paper.
BvQFm
vQlId
122
Lorentz Force
• If a point charge is moving in a region where bothelectric and magnetic fields exist, then itexperiences a total force given by
• The Lorentz force equation is useful fordetermining the equation of motion for electronsin electromagnetic deflection systems such asCRTs.
BvEqFFFme
Faraday’s Law and Transformer EMF, Motional EMF
Steady current produces a magnetic field
In 1831, Michael Faraday discovery that a time-varying magnetic field
would produce an electric current
According to Faraday's experiments, a static magnetic field produces no
current flow, but a time-varying field produces an induced voltage
(electromotive force)
induced emf (in volts), in any closed circuit is equal to the time rale of
change of the magnetic flux linkage by the circuit(Faraday‟s law)
According to Faraday‟s, It can be expressed as
Where,N is the number of turns in the circuit is the flux through each
turn.
The negative sign shows to oppose the flux producing it( Lenz's law)
Faraday's Law of electromagnetic Induction
em f
d dV N
d t d t
Induced emf (For a single turn N=1) in terms of E and B can be written
as
The variation of flux with time may be caused in three ways
• By having a stationary loop in a time-varying B field
• By having a time-varying loop area in a static B field
• By having a time-varying loop area in a time-varying B field
Induced emf (For a single turn N=1) in
terms of E and B can be written as
e m fS
L
d dV E d l B d s
d t d t
Faraday's Law of electromagnetic Induction
This emf induced by the time-varying current (producing the
time-varying B field) in a stationary loop is often referred to as
transformer emf
A stationary conducting loop is in a time varying magnetic B
field
By applying Stokes's theorem to the middle term
S S
E d S B d st
BE
t
e m fS
L
d dV E d l B d s
d t d t
Faraday's Law of electromagnetic Induction
Moving Loop in Static B Field (Motional emf)
When a conducting loop is moving in a static B field, an emf is
induced in the loop
F q vB
We define the motional electric field Em asm
m
FE v B
Q
Moving with uniform velocity u as consisting
of a large number of free electrons, the emf
induced in the loop is
Faraday's Law of electromagnetic Induction
e m fL
L
dV E d l v B d l
d t
By applying Stokes's theorem
( )m
E v B
Faraday’s law of Induction:
This describes the creation of an electric field by a changing
magnetic flux
The law states that the emf, which is the line integral of the electric
field around any closed path, equals the rate of change of the
magnetic flux through any surface bounded by that path
One consequence is the current induced in a conducting loop placed
in a time-varying B
Maxwell’s equations in integral form
Modified Ampere’s law:
It describes the creation of a magnetic field by an electric field and
electric currents
The line integral of the magnetic field around any closed path is the
given sum
Bd
dd t
E s
IE
o o o
dd μ ε μ
d t
B s
Maxwell’s Equations in Point form and Integral form for Time-
Varying Fields
Maxwell equations in differential form
Maxwell’s equations in point form
,v
D
,t
BE
,0 B
,t
DJH
The two Gauss‟s laws are symmetrical, apart from the absence of the
term for magnetic monopoles in Gauss‟s law for magnetism
Faraday‟s law and the Ampere-Maxwell law are symmetrical in that
the line integrals of E and B around a closed path are related to the
rate of change of the respective fluxes
Maxwell’s equations in integral form
I
E A G a u s s 's la w e le c tr ic
B A 0 G a u s s 's la w in m a g n e tis m
E s F a ra d a y 's la w
B s A m p e re -M a xw e ll la w
oS
S
B
E
o o o
qd
ε
d
dd
d t
dd μ ε μ
d t
Gauss’s law (electrical):
The total electric flux through any closed surface equals the net
charge inside that surface divided by eo
This relates an electric field to the charge distribution that creates it
Maxwell’s equations in integral form
oS
qd
ε E A
Gauss’s law (magnetism):
The total magnetic flux through any closed surface is zero
This says the number of field lines that enter a closed volume must
equal the number that leave that volume
This implies the magnetic field lines cannot begin or end at any point
Isolated magnetic monopoles have not been observed in nature0
S
d B A
Faraday’s law of Induction:
This describes the creation of an electric field by a changing
magnetic flux
The law states that the emf, which is the line integral of the electric
field around any closed path, equals the rate of change of the
magnetic flux through any surface bounded by that path
One consequence is the current induced in a conducting loop placed
in a time-varying B
Maxwell’s equations in integral form
Modified Ampere’s law:
It describes the creation of a magnetic field by an electric field and
electric currents
The line integral of the magnetic field around any closed path is the
given sum
Bd
dd t
E s
IE
o o o
dd μ ε μ
d t
B s
UNIT-III
Uniform Plane Waves
Reflection and Refraction of
Uniform Plane Wave
Till now, we have studied plane waves in various medium
Let us try to explore how plane waves will behave at a media
interface
Electromagnetic waves are often at the interface of boundary may
be reflected or refracted or changes direction at the interface
When a radio wave reflects from a surface, the ratio of the two
(reflected wave/incident wave) is known as the „reflection
coefficient‟ of the surface
How much of the incident wave has been transmitted through the
material is given by another ratio(transmitted wave /incident wave)
known as „transmission coefficient‟
Reflection and Refraction of Uniform Plane Wave
Transmission and reflection ratio depends on the
• conductivity (σ),
• permittivity (ε) and
• permittivity (ε) of the material as well as material properties of
the air which the radio wave is incident
Also some part of the wave will be transmitted through the material
• how much of the incident wave has been transmitted through
the material is also dependent on the material parameters
mentioned above
• It is given by another ratio known as „transmission coefficient‟
• It is the ratio of the transmitted wave divided by the incident
wave
Reflection and Refraction of Plane Wave contd..,
In plane wave incident from media interface, consider two cases
• Normal incidence
• Obliquely incidence
The electric and magnetic field expressions
• in all the regions of interest and apply the boundary
conditions to get the values of the transmission and
reflection coefficients
• Type of boundary interfaces for the solutions of
transmission and reflection coefficients
dielectric –conductor interface (both normal and oblique
incidence)
• dielectric –dielectric interface (both normal and oblique
incidence)
Reflection and Refraction of Plane Wave contd..,
At an interface between two media, the angles of incidence,
reflection, and refraction are all measured with respect to the
normal.
• Incidence angle (θi): angle at which the
incident wave makes with the normal to
the interface
• Reflection angle (θr): angle at which the
reflected wave makes with the normal to
the interface
• Transmission or refraction angle (θt):
angle at which the transmitted
(refracted) wave makes with the normal
to the interface
Reflection and Refraction of Plane Wave contd..,
If a plane wave is incident obliquely upon a surface that is not a
conductor boundary, part of the wave is transmitted and part of it
reflected.
In this case the transmitted wave will be refracted; that is the
direction of propagation will be altered.
Consider a planar interface between two dielectric media. A
plane wave is incident at an angle from medium 1.
Incident
wave
x
z
z=0Medium 1
(ε1,μ1)
Medium 2
(ε2,μ2)
ani
anr ant
AA‟ B
O‟
O
Reflected
waveRefracted
wave
θtθi
θr
In the diagram incident ray 2 travels a
distance AO‟ where as the transmitted
ray 1 travels a distance OB, and reflected
ray 1 travels from O to A‟
Reflection and Refraction of Plane Wave contd..,
Snell‟s Law of refraction
• The angle of reflection is equal to the angle of incident
11 1 1 22 1 1 2
1 2 2 12 2 2 2
1
s in
s in
o
o r rt
i o r r o
u n
u n
n :
refractive
index
Reflection and Refraction of Plane Wave contd..,
Reflection of a Plane Wave at Normal
Incidence
-Dielectric Boundary
Normal Incidence -Dielectric Boundary
Reflected
wave
Medium 2
2 2( , )
Medium 1
1 1( , )
Hi
Ei
ani
Incident
wave
y
x
z
z=0
Hr
Era
nr
Ht
Et
ant
Transmitted
wave
Consider the boundary is an interface between two dielectrics
We will consider the case of normal incidence, when the incident
wave propagation vector is along the normal to the interface between
two media
Generally, consider a time harmonic x-polarized electric field
incident from medium 1 (µ1, ε1, σ1) to medium 2 (µ2, ε2, σ2)
For the case of perfect dielectric
• σ1= σ2=0, ε1≠ ε2
• No loss or absorption of power
We will assume plane waves with electric field vector oriented
along the x-axis and propagating along the positive z-axis
without loss of generality
For z<0 (we will refer this region as region I and it is assumed
to be a medium1)
When a plane electromagnetic wave incident on the surface of
perfect dielectric
• part of energy is transmitted
• part of it is reflected
Let the wave incident on medium 1
the incident, reflected and transmitted fields
shown in Figure
Medium 2
2 2( , )
Medium 1
1 1( , )
Hi
Ei
ani
Incident
wave
y
x
z
z=0
Hr
Era
nr H
t
Et
ant
Transmitted
wave
Normal Incidence -Dielectric Boundary contd..,
1
0ˆ( )
j z
i x iE z a E e
10
1
ˆ( )j zi
i y
EH z a e
Incident wave ( inside
medium 1 )
Reflected wave ( inside
medium 1 )1
0ˆ( )
j z
r x rE z a E e
10
1 1
1ˆ ˆ( ) ( ) ( )
j zr
r z r y
EH z a E z a e
Transmitted wave ( inside
medium 2 ) 2
0ˆ( )
j z
t x tE z a E e
20
2 2
1ˆ ˆ( ) ( )
j zt
t z t y
EH z a E z a e
Normal Incidence -Dielectric Boundary contd..,
The tangential components (the x-components) of the electric and
magnetic field intensities must be continuous. ( at interface z=0 )
1 ta n 2 ta nE E
1 ta n 2 ta nH H
(0 ) (0 ) (0 )i r t
E E E
io ro toE E E
(0 ) (0 ) (0 )i r t
H H H
1 2
1( )
to
io r o
EE E
The continuity of tangential components of the electric and
magnetic fields require that
Rearranging these two equations
2 1
2 1
r o ioE E
2
2 1
2
to ioE E
Normal Incidence -Dielectric Boundary contd..,
Normal Incidence -Dielectric Boundary contd..,
1 2: 0 , 1
2 1
2 1
r o
io
E
E
Reflection coefficient ( + or - ) ≤ 1
The reflection coefficient
Special cases
The transmission coefficient
2
2 1
2to
io
E
E
20 ( ) 1sh o r t E/H, E=0 perfect
conductor !!
2( ) 1o p en H(I)=0 No current !!
1
which is related to reflection coefficient
Normal Incidence - Dielectric Boundary contd..,
Further more, the reflection and transmission coefficient
1 2
2 1
r o r o
io io
H E
H E
1 1
2 2 1
2to to
io io
H E
H E
The permeabilities of all known dielectrics do not differ
appreciably from free space, so that μ1= μ2= μ0
0 0
2 1 1 2 12 1
2 1 0 0 1 2 1 2
2 1
2,
ro to
io io
E E
E E
2 1 2
1 2 1 2
2,
ro to
io io
H H
H H
Similarly,
Reflection of a Plane Wave Normal
Incidence
-Conducting Boundary
In z-direction ( x=constant )
9 0 o u t o f p h ase
1 1sin ( cos )
j t
y iE jA z e
n o is p ro p a g a te .a v
P
In x-direction ( z=constant )
1 1c o s ( c o s )
j t
x iH B z e
trave lin g w ave
1( sin )
1
ij t x
yE C e
1
1
1s in s in
x
i i
uu
1
1
s inx
i
C = f(z) , D = g(z)1
( s in )
1
ij t x
zH D e
Oblique Incidence - Conducting Boundary
Oblique Incidence -
Conducting Boundary
Oblique Incidence - Conducting Boundary
When EM Wave incident obliquely on the interface between two
media boundary, it can be decomposed into:
• Horizontal Polarization
• Vertical Polarization
Horizontal Polarization, or transverse electric (TE) polarization
The E Field vector is parallel to the boundary surface or
perpendicular to the plane of incidence
Vertical Polarization, or transverse magnetic (TM) polarization
The H Field vector is parallel to the boundary surface and the electric
vector is parallel to the plane of incidence
Let us consider the wave incident on a perfect conductor
• The wave is totally reflected with the angle of incidence equal to
the angle of reflection
Oblique Incidence - Conducting Boundary
Perpendicular Polarization
2( )
Medium 2
Perfect conductor
Medium 1
1( 0 )
Hi
Ei
ani
Hr
Er
an
r
Incident
wave
Reflected
wave
y.
x
z
z=0
i
r
Oblique Incidence - Conducting Boundary
Perpendicular Polarization Assume the wave is propagating obliquely along an arbitrary
direction ar and the electric field vector is along y-direction(normal
to the plane of incidence x-z plane) and direction of propagation
Using direction cosines the direction of propagation of incidence
wave can be written as
ˆ ˆ ˆs in c o sn i x i z i
a a a
Similarly the direction of propagation of reflected wave (-z
direction )can be written as
ˆ ˆ ˆs in c o sn i x i z i
a a a
Oblique Incidence - Conducting Boundary
Incident wave ( inside medium 1 )
1
1ˆ( , ) ( , )
i n i iH x z a E x z
1ˆ
0ˆ( , ) n i
j a R
i y iE x z a E e
1( s in c o s )0
1
ˆ ˆc o s s in i ij x zi
x i z i
Ea a e
1( sin cos )
0ˆ i i
j x z
y ia E e
Reflected wave ( inside medium 1 )
1( s in co s )
0ˆ( , ) r r
j x z
r y rE x z a E e
Boundary condition, z = 0
1( , 0 ) ( , 0 ) ( , 0 )
i rE x E x E x
Oblique Incidence - Conducting Boundary
0 0&
r i i rE E
1 1sin sin
0 0ˆ ( ) 0i r
j x j x
y i ra E e E e
, for all x
Snell’s law of reflection the angle of reflection equals the angle of incidence
1( s in co s )
0ˆ( , ) r r
j x z
r y rE x z a E e
1( s in co s )
0
( sn e ll's la w o f re fle c tio n )
ˆ
i ij x z
y ia E e
Oblique Incidence - Conducting Boundary
Similarly, Magnetic field of the reflected wave can be written as
1( s in co s )
0ˆ( , ) r r
j x z
r y rE x z a E e
1( s in c o s )
0
( sn e ll's la w o f re fle c tio n )
ˆ
i ij x z
y ia E e
1
1ˆ( , ) ( , )
r n r rH x z a E x z
1( s in c o s )0
1
ˆ ˆ c o s s in i ij x zi
x i z i
Ea a e
Oblique Incidence - Conducting Boundary
Total field in medium 1 can be written as(standing wave in terms of
electric and magnetic field)
1( , ) ( , ) ( , )
i rE x z E x z E x z
1sin
0 1ˆ 2 sin ( cos ) i
j x
y i ia j E z e
1
1
s in0
1 1
1
s in
1
ˆ( , ) 2 c o s c o s ( c o s )
ˆ s in s in ( c o s )
i
i
j xi
x i i
j x
z i i
EH x y a z e
a j z e
1 1 1cos cos sin
0ˆ ( )i i i
j z j z j x
y ia E e e e
Oblique Incidence - Conducting Boundary
In z-direction ( x=constant )
9 0 o u t o f p h ase
1 1sin ( cos )
j t
y iE jA z e
n o is p ro p a g a te .a v
P
In x-direction ( z=constant )
1 1c o s ( c o s )
j t
x iH B z e
trave lin g w ave
1( sin )
1
ij t x
yE C e
1
1
1s in s in
x
i i
uu
1
1
s inx
i
C = f(z) , D = g(z)1
( s in )
1
ij t x
zH D e
Oblique Incidence - Conducting Boundary
Oblique Incidence - Conducting Boundary
Oblique Incidence - Conducting Boundary
Parallel Polarization
2( )
Medium 2
Perfect conductor
Medium 11
( 0 )
Hi
Ei
ani
Hr
Eranr
Incident
wave
Reflected
wave
y.
x
z
z=0
i
r
Parallel Polarization Assume the wave is propagating obliquely along an arbitrary
direction ar and the electric field vector is parallel to the plane of
incidence (x-z plane) and the magnetic field vector is normal to
the plane of incidence
Using direction cosines the direction of propagation of incidence
wave can be written as
ˆ ˆ ˆs in c o sn i x i z i
a a a
Similarly the direction of propagation of reflected wave (-z
direction ) can be written as
ˆ ˆ ˆs in c o sn i x i z i
a a a
Oblique Incidence - Conducting Boundary
Incident wave ( inside medium 1 )
Reflected wave ( inside medium 1 )
Boundary condition, z = 0
1ˆ
0ˆ ˆ( , ) ( cos sin ) n i
j a R
i i x i z iE x z E a a e
1( s in c o s )0
1
ˆ( , ) i ij x zi
i y
EH x z a e
1( s in c o s )
0ˆ ˆ( c o s s in ) i i
j x z
i x i zE a a e
1( sin cos )
0ˆ ˆ( , ) ( cos sin ) r r
j x z
r r x i z rE x z E a + a e
1( s in c o s )0
1
ˆ( , ) r rj x zr
r y
EH x z a e
Oblique Incidence - Conducting Boundary
ro io r i
E E
0 1ˆ2 [ c o s s in ( c o s )
i x i iE a j z
1s in0
1
1
ˆ 2 c o s ( c o s ) ij xi
y i
Ea z e
1s in
1ˆ s in c o s ( c o s )] i
j x
z i i+ a z e
1( , ) ( , ) ( , )
i rE x z E x z E x z
1( , ) ( , ) ( , )
i rH x z H x z H x z
1 1s in s in
( c o s ) ( c o s ) 0i rj x j x
io i r o rE e E e
( for all x)
Total field in medium 1 can be written as(standing wave in terms of
electric and magnetic field)
Oblique Incidence - Conducting Boundary
In z-direction : standing-wave1,
xE
1 yH
1 1/ s in
x iu u
The wave is non uniform plane wave
1,
zE
1 yH
Oblique Incidence - Conducting Boundary
Reflection of a Plane Wave Oblique Incidence -Interface between
dielectric media
x
z=0
Oθi
θr
iH
iE
•
The electric field phasors for the perpendicular polarization, with
reference to the system of coordinates in the figure
Let us assume that the incident wave propagates in the
first quadrant of x-z plane and propagation makes an
angle θi with the normal
1( sin cos )
( , ) i ij x z
yi ioE x z a E e
1( s in c o s )
1
( , ) ( i ij x zio
i x zi i
EH x z a c o s + a s in )e
Let us assume that the reflected wave propagates in
the second quadrant of xz plane and reflection makes
an angle θr with the normal
1( sin cos )
( , ) r rj x z
yr roE x z a E e
1( s in c o s )
1
( , ) ( i ij x zr o
r x zr r
EH x z a c o s + a s in )e
Oblique Incidence -Interface between dielectric media
Perpendicular Polarization
θt
Medium 1
Medium2
Similarly the transmitted fields are
Equating the tangential components of electric field
• electric field has only Ey component and it is tangential at the
interface z=0)
2( sin cos )
( , ) t tj x z
yt toE x z a E e
2( s in c o s )
2
( , ) ( i ij x zto
t x zi i
EH x z a c o s + a s in )e
Similarly the magnetic field
• magnetic field has two components: Hx and Hz , but only Hx is
tangential at the interface z=0)
( , ) ( , ) ( , )iy ry ty
E x o E x o E x o
( , ) ( , ) ( , )ix r x txH x o H x o H x o
Oblique Incidence -Interface between dielectric media
Boundary at z=0 gives
1 21s in s ins in
i trj x j xj x
e e e
1 21s in s ins in
1 1 2
1c o s c o s c o si tr
j x j xj x
i r te e e
If Ex and Hy are to be continuous at the interface z = 0 for all x,
then, this x variation must be the same on both sides of the
equations (also known as phase matching condition)
1 1 2s in s in s in
i r t
i r Follows Snell's law
Oblique Incidence -Interface between dielectric media
Now we can simplify above two equations by applying Snell‟s
1 1 2
1c o s c o s c o s
i r t
The above two equations has two unknowns τ and Г and it can be
solved easily as follows
1
2
1 1
1c o s c o s
c o si r
t
1
Oblique Incidence -Interface between dielectric media
Solving the equations and rearranging, we get
For normal incidence, it is a particular case and put
2 1
2 1
c o s c o s
c o s c o s
r o i t
io i t
E
E
2
2 1
2 c o s
c o s c o s
to i
io i t
E
E
1
0i r t
Oblique Incidence -Interface between dielectric media
In this case, electric field vector lies in the x-z plane Since the magnetic field is transversal to the plane of incidence such
waves are also called as transverse magnetic (TM) waves So let us start with H vector which will have only y component
Similar to the previous case of perpendicularpolarization, we can write the reflected magneticfield and electric field vectors as
Parallel Polarization
θr
z=0
Oθi
iH
iE
•
rH
rE
θtx
1( s in c o s )
1
( , ) i ij x zio
i y
EH x z a e
1( sin cos )
( , ) ( i ij x z
i x zio i iE x z E a cos a sin )e
1( s in c o s )
1
( , ) r rj x zr o
r y
EH x z a e
1( sin cos )
( , ) ( r rj x z
r x zro r iE x z E a cos a sin )e
Oblique Incidence -Interface between dielectric media
Similarly the transmitted fields are
Equating the tangential components of magnetic field
• electric field has only Hy component and it is tangential at the
interface z=0
Similarly the magnetic field
• magnetic field has two components: Ex and Ez , but only Ex is
tangential at the interface z=0)
( , ) ( , ) ( , )iy ry ty
E x o E x o E x o
( , ) ( , ) ( , )ix r x txH x o H x o H x o
2( sin cos )
( , ) ( t tj x z
t x zto t tE x z E a cos a sin )e
2( s in c o s )
2
( , ) t tj x zto
t y
EH x z a e
Oblique Incidence -Interface between dielectric media
Boundary at z=0 gives
1 21s in s ins in
c o s c o s c o si trj x j xj x
i r te e e
1 21s in s ins in
1 1 2
1i tr
j x j xj xe e e
If Ex and Hy are to be continuous at the interface z = 0 for all x,
then, this x variation must be the same on both sides of the
equations (also known as phase matching condition)
1 1 2s in s in s in
i r t
i r Follows Snell's law
Oblique Incidence -Interface between dielectric media
Now we can simplify above two equations by applying Snell‟s
c o s c o s c o si r t
The above two equations has two unknowns τ and Г and it can be
solved easily as follows
1 2
11
co s co s
co s
i r
t
2
1
1
Therefore,
2 1
2 1
c o s c o s
c o s c o s
ro t i
io t i
E
E
2
2 1
2 c o s
c o s c o s
to i
io t i
E
E
Note that cosine terms multiplication in the above equations is
different from the previous expression for perpendicular polarization
Oblique Incidence -Interface between dielectric media
Brewster Angle, Critical Angle
Critical Angle
When light travels from one medium to another it changes speed and
is refracted.
As the angle of incidence increases so does the angle of refraction.
When the angle of incidence reaches a value known as the critical
angle the refracted rays travel along the surface of the medium or in
other words are refracted to an angle of 90°.
The critical angle for the angle of incidence in glass is 42°.
Surface ImpedanceIt is defined as the ratio of the tangential component of the electric
field to the surface current density at the conductor surface.
It is given by
where Etan is the tangential component parallel to the surface of the
conductor.
And Js is the surface current density.
ta n
s
s
EZ
J Ω
Critical Angle
> When the angle of incidence of the light ray reaches the critical
angle (420) the angle of refraction is 900. The refracted ray travels
along the surface of the denser medium in this case the glass.
> According to the law of refraction
1 2s in s in
i tn n
1 1
2 2
s in
s in
t
i
n
n
2i c t
i f th e n
1 12 2
1 1
s in s inc
no r
n
Total Internal Reflection
>When the angle of incidence of the light ray is greater than the
critical angle then no refraction takes place. Instead, all the light is
reflected back into the denser material in this case the glass. This is
called total internal reflection.
Brewster AngleFor unpolarized waves the electric fields are in many directions as shown in figure, e.g. unpolarized light
Where as in polarized waves the electric field vector is either vertical or horizontal as shown in figure below.
Brewster Angle
We can convert unpolarized light into polarized light by passing the light through a polarizing filter.
When unpolarized wave is incident obliquely at Brewster angle θB, only the component with perpendicular polarization (Horizontal polarization) will be reflected, while component with parallel polarization will not be reflected.
Brewster AngleIt is also called as Polarizing angle.
At Brewster angle, the angle between reflected ray and refracted ray is 900.
Normal Incidence - Conducting Boundary
Consider the boundary is an interface with a perfect conductor
The incident wave travels in a lossless medium (Medium 1 )
2( )
Medium 2
Perfect
conductor
Medium
11( 0 )
Hi
Ei
an
i
Hr
Er
an
r
Incident
wave
Reflected wave
y.
x
z
z=0
1
0ˆ( )
j z
i x iE z a E e
10
1
ˆ( )j zi
i y
EH z a e
Incident wave ( inside
medium 1 )
1( 0 )
1( )
Reflected wave ( inside
medium 1 )1
0ˆ( )
j z
r x rE z a E e
1
1ˆ( ) ( )
r n r rH z a E z
1
1ˆ( ) ( )
z ra E z
1
0
1
1ˆ
j z
y ra E e
2 20, 0
E H
Normal Incidence -Conducting Boundary contd..,
2( )
Medium
2
Perfect
conductor
Mediu
m 11( 0 )
Hi
Ei
ani
Hr
Er
anr
Incident
wave
Reflected wave
y
x
z
z=0
The boundary is an interface with a perfect
conductor(medium 2) both electric and magnetic fields
vanish.
No wave is transmitted across the boundary into the z >
0 i.e.
Total wave in medium
1
1 1
1 0 0ˆ( ) ( ) ( ) ( )
j z j z
i r x i rE z E z E z a E e E e
Continuity of tangential component of the E-field at the
boundary z = 0
1 0 0 2ˆ(0 ) ( ) (0 ) 0
x i rE a E E E
0 0r iE E
Normal Incidence - Conducting Boundary contd..,
Incident and reflected field are of equal amplitude, so all incident
energy is reflected by a perfect conductor
Negative sign indicates that the reflected field is shifted in phase
by 180° relative to incident field at the boundary
The magnetic field must be reflected without phase reversal, If
both were reversed there would be no reversal of direction .
In medium 1, the total field can be written as
(α=0)
Normal Incidence - Conducting Boundary contd..,
0
1 1
1
ˆ( ) ( ) ( ) 2 c o s
i
i r y
EH z H z H z a z
1 1
1 0ˆ( ) ( )
j z j z
x iE z a E e e
0 1ˆ 2 s in
x ia j E z
The space-time behavior of the total field in medium
1(multiply with )
After multiplying a time factor and take a real instantaneous
part of the fields in medium 1 as a function of z and t can be
written as
1 1 0 1ˆ( , ) R e[ ( ) ] 2 sin sin
j t
x iE z t E z e a E z t
0
1 1 1
1
ˆ( , ) R e [ ( ) ] 2 c o s c o sj t i
y
EH z t H z e a z t
From the total fields in medium 1, we conclude that there is a
standing wave in medium 1, which does not progress
The magnitude of field varies sinusoidal with distance from the
reflecting plane
Electric field has zero at the surface and multiple of half wave
length from the surface and maxima for magnetic field
1
1
1
Z e ro s o f ( , ) o c c u r a t , o r , 0 ,1, 2 , .. .
2M a x im a o f ( , )
E z tz n z n n
H z t
1
1
1
M a x im a o f ( , ) o c c u r a t ( 2 1) , o r ( 2 1) , 0 ,1, 2 , .. .
2 4 Z e ro s o f ( , )
E z tz n z n n
H z t
Normal Incidence - Conducting Boundary contd..,
It has a maximum value of twice the electric field strength of the
incident wave at the distance from the odd multiple of the quarter
wave length and zero for magnetic field
The totally reflected wave combines with
the incident wave to form a standing wave
The total wave in medium 1 is not a
traveling wave
It stands and does not travel, it consists of
two traveling waves Ei and Er of equal
amplitudes but in opposite directions.
z
x
2
0t
5 / 4t
3 / 2t
/ 4 , 3 / 4t
/ 2t
z
0t
/ 4t
/ 2t
3 / 4t
t
4
3
4
E1 versus z
H1 versus z
1 1 0 1ˆ( , ) R e[ ( ) ] 2 sin sin
j t
x iE z t E z e a E z t
0
1 1 1
1
ˆ( , ) R e [ ( ) ] 2 c o s c o sj t i
y
EH z t H z e a z t
Normal Incidence - Conducting Boundary contd..,
1 1 0 1ˆ( , ) R e[ ( ) ] 2 sin sin
j t
x iE z t E z e a E z t
The equations also shows that there is a 90° out of time
phase between the electric and magnetic fields
0 1ˆ 2 s in co s ( )
2x i
a E z t
0
1 1 1
1
ˆ( , ) R e [ ( ) ] 2 c o s c o sj t i
y
EH z t H z e a z t
Normal Incidence - Conducting Boundary contd..,
Brewster AngleIt is also called as Polarizing angle.
At Brewster angle, the angle between reflected ray and refracted ray is
900.
UNIT-IV
Transmission line Characteristics
If we are familiar with low frequency circuits and the circuit consists
of lumped impedance elements (R,L,C), we treat all lines(wires)
connecting the various circuit elements as perfect wires, with no
voltage drop and no impedance associated to them.
As long as the length of the wires is much smaller than the wavelength
of the signal and at any given time, the measured voltage and current
are the same for each location on the same wire.
Let us try to explore what happens, when the signal propagates as a
wave of voltage and current along the line at sufficiently high
frequencies
For sufficiently high frequencies, wavelength is comparable to the
length of a conductor, so the positional dependence impedance
properties (position dependent voltage and current) of wire can not be
neglected, because it cannot change instantaneously at all locations.
Transmission line equivalent circuit
So our first goal is to represent the uniform transmission line as a
distributed circuit and determine the differential voltage and current
behavior of an elementary cell of the distributed circuit.
Once that is known, we can find a global differential equation that
describes the entire transmission line by considering a cascaded
network (subsections) of these equivalent models.
So a uniform transmission line is represented as a distributed
circuit shown below
L series inductance
R series resistance
C shunt capacitance
G shunt conductance
Transmission line equivalent circuit
Let us assume a differential length dz of transmission line ,V(z),I(z)
are voltage and current at point P and V(z)+dV, I(z)+dI are voltage
and current at point Q
QP
The series impedance and shunt admittance determines the variation of
the voltage and current from input to output of the cell shown below
Transmission line equivalent circuit
For a differential length dz of transmission line ,the series
impedance and shunt admittance of the elemental length can be
written as
( )
( )
R j L d z
G j C d z
The corresponding circuit voltage and current equation is
( )
( )
V d V V I R j L d z
I d I I V G j C d z
From which we obtain a first order differential equation
( )
( )
d VI R j L
d z
d IV G j C
d z
Transmission line equations
We have a system of coupled first order differential equations that
describe the behavior of voltage and current on the transmission line
It can be easily obtain a set of uncoupled equations by differentiating
with respect to the coordinate z2
2
2
2
( )
( )
d V d IR j L
d zd z
d I d VG j C
d zd z
Substitute , in the above equations ,we obtain a second order
differential equations2
2
2
2
( )( )
( )( )
d VG j C R j L V
d z
d IG j C R j L I
d z
Transmission line equations
γ is the complex propagation constant, which is function of
frequency
α is the attenuation constant in nepers per unit length, β is the
phase constant in radians per unit length
Let the constant term can be represented as propagation constant,
which is written as
The transmission line equations can be written as
( )( )j G j C R j L
2
2
2
2
2
2
d VV
d z
d II
d z
Transmission line equations
Where, a,b,c, and d are the constants
Above equations represent the standard solutions of the wave
equations, which are similar to the solution of uniform plane wave
equations.
The terms and can be represented as backward and
forward wave along z-direction.
The solution of the second order transmission line equation isz z
z z
V a e b e
I c e d e
Transmission line equations
ze ze
Determination of constant
terms A and B
Determination of the constant terms A and B
To determine the constants a,b,c, and d , the above equations can be
written in terms of hyperbolic functions, where substitute
Substitute above equations in the solutions of transmission line
equations V and I
z z
z z
V a e b e
I c e d e
(c o sh c o sh ) (c o sh c o sh )
(c o sh c o sh ) (c o sh c o sh )
V a z z b z z
I c z z d z z
c o sh c o sh
c o sh c o sh
z
z
e z z
e z z
Let the solutions of the transmission line wave equations can written as
The constants a+b, a-b, c+d, and c-d can be replaced by another
constant terms A,B,C, and D respectively
So, the above equations can be written as
In order to reduce the four constant terms to two constant terms, we write
the relation between V and I by considering the following basic
differential equations
( ) c o sh ( ) s in h
( ) c o sh ( ) s in h
V a b z a b z
I c d z c d z
c o s h s in h
c o s h s in h
V A z B z
I C z D z
( )
( )
d VI R j L
d z
d IV G j C
d z
Determination of the constant terms A and B
( co sh s in h )( )
d A z B zI R j L
d z
Substitute V in
( s in h co sh ( )A z B z I R j L
Differentiating in terms of z, we obtain
So, the current I can be written in terms of constants A and B as
( s in h c o s h( )
A z B z IR j L
Where γ is propagation constant, which is
( )( )G j C R j L
Determination of the constant terms A and B
Substitute γ in current equation, we obtain
( )( )( s in h co sh
( )
G j C R j LA z B z I
R j L
0
s
s
V A
I Z B
0
1( s in h c o s hA z B z I
Z
The above equation can be simplified and written as
Where Z0 is another constant and can be called as characteristic
impedance along the line
0
( )
( )
R j LZ
G j C
Determination of the constant terms A and B
So, the voltage and current wave equation in terms of constants A and
B can be re written as
0
c o s h s in h
1( s in h c o s h )
V A z B z
I A z B zZ
Now the constants A and B can be obtained by applying initial
conditions of the transmission line at Z=0
Let Vs and Is be the source voltage and current respectively. At the
source end, Z=0 the voltage V= Vs, current I= Is, then the above
equations can be simplified as
0
c o s h ( 0 ) s in h ( 0 )
1( s in h ( 0 ) c o s h ( 0 ))
s
s
V A B
I A BZ
0
s
s
V A
I Z B
Determination of the constant terms A and B
Substitute A and B values in basic voltage and current equationswe obtain
0
0
c o s h s in h
s in h c o s h
s s
s
s
V V z I Z z
VI z I z
Z
These equations are called transmission line equations. The canrepresent voltage and current at any point z from the sourcevoltage and current.
0
s
s
V A
I Z B
The constants A and B can be simplified as
Determination of the constant terms A and B
Substitute A and B values in basic voltage and current equations
we obtain0
0
c o s h s in h
s in h c o s h
s s
s
s
V V z I Z z
VI z I z
Z
These equations are called transmission line equations. The can
represent voltage and current at any point z from the source voltage
and current.
0
s
s
V A
I Z B
The constants A and B can be simplified as
Determination of the constant terms A and B
Input Impedance of Transmission Line from Load
Impedance
If the source voltage and source current is known to us, input
impedance of a transmission line is derived from source end as
Similar expression can be derived from terminating voltage or current
is known
This can be derived from a transmission line length l with known
terminating voltage and current
Transmission Line with Load Impedance
0
0
0
tan h
tan h
R
in
R
Z l ZZ Z
Z Z l
This can be start from basic transmission line equations with two
unknown constants A and B
0
c o s h s in h
1( s in h c o s h )
V A z B z
I A z B zZ
Now the terminating voltage and current at a distance z=l can be written
as
To derive the constants A the above voltage and current equations can
be multiplied with coshγl and sinhγl
Transmission Line with Load Impedance
0
c o s h s in h
1( s in h c o s h )
R
R
V A l B l
I A l B lZ
Then we get the constants A as
2
2
0
co sh co sh s in h co sh
1s in h ( s in h s in h co sh )
R
R
V l A l B l l
I l A l B l lZ
0c o sh s in h
R RV l Z I l A
Similarly, the constant B can be obtained by multiplying voltage
equation with sinhγl and current equation with coshγl , and adding
both the equations, we get B as
Transmission Line with Load Impedance
Then substitute A and B constants in basic equation. we obtain voltage
and current at any point from the terminating point as
0c o sh s in h
R RV l Z I l B
0
0
c o sh ( ) s in h ( )
s in h ( ) c o sh ( )
R
R
R
R
VV V l z l z
Z
VI l z I l z
Z
Where, l-z is the distance from the terminating end
Now from the above expressions, the input impedance z=0 can be
written as
Transmission Line with Load Impedance
0
0
c o s h ( 0 ) s in h ( 0 )
s in h ( 0 ) c o s h ( 0 )
S R R
R
S R
V V l I Z l
VI l I l
Z
0
0
c o s h s in h
s in h c o s h
S R R
RSR
V V l I Z l
VIl I l
Z
0
0
0
tan h
tan h
R
in
R
Z l ZZ Z
Z Z l
Infinite Transmission Line
Infinite Transmission Line
>If a line of infinite length is considered then all the power fed into it
will be absorbed. The reason is as we move away from the input
terminals towards the load, the current and voltage will decrease
along the line and become zero at an infinite distance, because the
voltage drops across the inductor and current leaks through the
capacitor.
>By considering this hypothetical line of infinite line an important
terminal condition is formed.
Infinite Transmission LineLet Vs be the sending end voltage and Is be the sending end current and
Zs be the input impedance which is given by
Current at any point distance x from sending end is given by
The value of c & d can now determined by considering an infinite line.
At the sending end x=0 and I =Is.
At the receiving end, I=0 and x=∞.
Zs =Vs/Is
-PxPxde+ce=I
d+cIs
c=0 0
Infinite Transmission Line
Therefore
If then
Therefore
The above equation gives current at any point of an infinite line
And
Similarly the above equation gives voltage at any point of an infinite line
Px
seVV
0c
0cdI
s
Px
seII
Px
seVV
Infinite line terminated in its Z0
When a finite length of line is joined with a similar kind of infinite line,
their total input impedance is same as that of infinite line itself,
because they together make one infinite line however the infinite line
alone presents an impedance Zo at its input PQ because the input
impedance of an infinite line is Zo.
It is therefore concluded that a finite line has an impedance Zo when it is
terminated in Zo.
Or A finite line terminated by its Zo behaves as an infinite line.
Let a finite length of „l‟ is terminated by its characteristic impedance Zo
and is having voltage and current VR and IR at terminating end.
Infinite line terminated in its Z0
Therefore
Dividing VR by IR we get Zo
R
R
I
V=Zo
have weequations, generalin I =I,V =Vl,= xPuttingRR
sinhpl Z
V - coshpl I=I
O
S
SR
Infinite line terminated in its Z0
Multiplying right hand side numerator and denominator by Z0, we get
Therefore
But Vs/Is is equal to the input impedance.
Thus the input impedance of a finite line terminated in its characteristicimpedance is the characteristic impedance of the line. Since theinput impedance of an infinite line is the characteristic impedance ofthe line.
Characteristic Impedance, Phase Velocity
The characteristic impedance, Z0 can be defined as:
0
R j L R j LZ
G j C
characteristic impedance (Zo) is the ratio of voltage to
current in a forward travelling wave, assuming there is no
backward wave
• Zo determines relationship between voltage and
current waves
• Zo is a function of physical dimensions and r
• Zo is usually a real impedance (e.g. 50 or 75
ohms)
Characteristic impedance, Z0 :
Characteristic impedance, Z0 :
• Voltage waveform can be expressed in time domain as:
• The factors V0+ and V0
- represent the complex quantities. The Φ± is
the phase angle of V0±. The quantity βz is called the electrical length
of line and is measured in radians.
0 0, c os c os
z zv z t V t z e V t z e
Phase Velocity
• Phase Velocity For a sinusoidal wave, or a waveform comprised of
many sinusoidal components that all propagate at the same velocity,
the waveform will move at the phase velocity of the sinusoidal
components We‟ve seen already that the phase velocity is
• vp=ω/k
Group Velocity
• When the various frequency components of a waveform have
different phase velocities
• The phase velocity of the waveform is an average of these
velocities (the phase velocity of the carrier wave), but the waveform
itself moves at a different speed than the underlying carrier wave
called the group velocity
Group vs Phase velocity
• An analogy that may be useful for understanding the difference
comes from velodrome cycling
• Riders race as a team and take turns as leader with the old leader
peeling away and going to the back of the pack
• As riders make their way from
the rear of the pack to the front
they are moving faster than
the group that they are in.
Group vs Phase velocity
• The phase velocity of a wave is
V=vp = λ T = fλ = ω/k
and comes from the change in the position of the wavefronts as a
function of time
• The waveform moves at a rate that depends on the relative position
of the component wavefronts as a function of time. This is the group
velocity and is
• vg = dω /dk
Condition for distortion less and
minimum attenuation
Condition for Distortion Less Line
• A transmission line is said to be distortion less if the attenuation
constant is frequency independent.
• The phase constant is linearly dependent of frequency.
• From the general equations of α&β, The distortion less line results,
if the line parameters are such that,
R/L=G/C
The attenuation constant and phase constant are
α= √RG and β= ω √LC
Condition for Minimum Attenuation
• From the equation the attenuation of a line is expressed by
attenuation constant DC as
α = ½
• It is observed that depends on the four primary constants in
addition to the frequency. The Value of L for minimum attenuation.
• Let us assume the three line constants C,G and R including are
constant and only L may be varied .
• Therefore, differentiating above value of , W.R.T L and equating
it to zero, we get
Condition for Minimum Attenuation:
then by solving above equation we get
R/L=G/C
This is the condition for minimum attenuation and
distortion less line.
= 0
= 0
Condition for Distortion Less Line
CONCLUSIONS:• The phase velocity is independent of frequency because, the phase
constant of linearly depends upon frequency
• Both VP and Z0 remains the same for loss less line.
• A loss less line is also a distortion less line, but a distortion less line
is not necessarily loss less.
• Loss less lines are desirable in power transmission telephone lines
are to b distortion less.
UNIT-V
UHF Transmission Lines and Applications
UHF Transmission Lines
• UHF spectrum is used worldwide for land mobile radio systems for
commercial, industrial, public safety, and military purposes.
• Many personal radio services use frequencies allocated in the UHF
band, although exact frequencies in use differ significantly between
countries.
• Major telecommunications providers have deployed voice and data
cellular networks in UHF/VHF range. This allows mobile phones
and mobile computing devices to be connected to the public
switched telephone network and public internet.
• UHF radars are said to be effective at tracking stealth fighters, if not
stealth bombers
UHF Transmission Lines• Ultra High Frequency lines commonly abbreviated as
U.H.F lines are one of the types of the transmission lines.
• Ultra high frequency lines have operational frequency
range from 300 to 3000 MHz or wavelength from 100 cm
to 10 cm.
• Under normal frequencies the transmission lines are used
as wave guides for transferring power and information
from one point to another.
UHF Transmission Lines
• At Ultra High Frequencies, the transmission lines can be used as
circuit elements like capacitor or an inductor .
• It means they can be used in circuits like a capacitor or an inductor.
Applications:
• UHF Television Broad casting fulfilled the demand for additional
over-the-air television channels in urban areas. Today, much of the
bandwidth has been reallocated to land mobile, trunked radioand
mobile telephone use. UHF channels are still used for digital
television
UHF Transmission Lines
• UHF spectrum is used worldwide for land mobile radio systems for
commercial, industrial, public safety, and military purposes.
• Many personal radio services use frequencies allocated in the UHF
band, although exact frequencies in use differ significantly between
countries.
• Major telecommunications providers have deployed voice and data
cellular networks in UHF/VHF range. This allows mobile phones
and mobile computing devices to be connected to the public
switched telephone network and public internet.
• UHF radars are said to be effective at tracking stealth fighters, if not
stealth bombers
Input impedance Relations; SC and OC lines
Let the voltage and current transmission line equations at any point on
the line from the source end can be written as
A transmission line, which is terminated with some load impedance
ZR at a distance „l „from the load
The voltage and current at the terminating end is VR and IR
Input Impedance Relations
0
0
c o s h s in h
s in h c o s h
s s
s
s
V V z I Z z
VI z I z
Z
At z= l, the voltage and current can be written as
0
0
c o s h s in h
s in h c o s h
R s s
s
R s
V V l I Z l
VI l I l
Z
Now the load impedance from the terminating point can be written as
By solving the above equation, we obtain
Input Impedance Relations
0
0
c o s h s in h
s in h c o s h
s sR
R
sRs
V l I Z lVZ
VIl I l
Z
Where ZS is the source impedance, also called input impedance
Above expression can be written in terms of hyperbolic tan
functions as
0
0
0
s in h co sh
co sh s in h
R
s
R
Z l Z lZ Z
Z l Z l
0
0
0
tan h
tan h
R
in s
R
Z l ZZ Z Z
Z Z l
For a loss less line the input impedance can be written as
Input Impedance Relations
0
0
0
tan h
tan h
R
in s
R
Z j l ZZ Z Z
Z Z j l
A special cases from the above general lossless line input impedance
relations
• When the line is terminated with characteristics impedance, ZR=Z0
• When the line is terminated with open circuited, ZR=infinity(∞)
• When the line is terminated with short circuited, ZR=0
0
0
0
tan
tan
R
in s
R
jZ l ZZ Z Z
Z jZ l
Lossless Transmission line(α=0)
For a loss less line the input impedance can be written as
Input Impedance Relations
0
0
0
tan h
tan h
R
in s
R
Z j l ZZ Z Z
Z Z j l
The line is terminated with characteristic impedance, the input
impedance is equal to the characteristic impedance
This condition is called matched load condition
There is no reflections on the line
0 0
0 0
0 0
tan
tanin
jZ l ZZ Z Z
Z jZ l
Matched Load ZR=Z0
For a loss less line the input impedance can be written as
Input Impedance Relations
0
0
0
tan
tan
R
in
R
jZ l ZZ Z
Z jZ l
0
0
0
ta n 1
ta n
R
in O C
R
Zj l
ZZ Z Z
Zj l
Z
Open Circuited, i.e. ZR=infinity(∞)
0c o t
in O CZ Z jZ l
Short Circuited, i.e. ZR=0
0ta n
in S CZ Z jZ l
If the line is terminated with open circuit or short circuit, the input
impedance can be purely imaginary
0
0
0
tan
tan
R
in
R
jZ l ZZ Z
Z jZ l
From the short circuit and open circuit impedance relations, the
characteristic impedance can be written as
Input Impedance Relations
0
0( ta n )
ta nO C S C
ZZ Z jZ l
j l
The characteristic impedance of the line can be measured from open and
short circuit Transmission line
2
0
0
O C S C
O C S C
Z Z Z
Z Z Z
Impedance Matching, Single Stub Matching
Impedance matching is one of the important aspects of high
frequency circuit analysis.
To avoid reflections and power loss from transmission line sections
impedance matching techniques can be used
• Quarter Wavelength Transformer,
• Stub Matching
Single Stub Matching
Double Stub Matching
Impedance Matching
The quarter wave transformer needs special line of characteristics
impedance for every pair of resistances to be matched
To add quarter wave transformer, cut the main line
To avoid above difficulties, either open or short circuited transmission
line attached at some position parallel to line
Impedance Matching
Adding of either open or short line in parallel to main line is call stub
The main advantage of this stub matching is
• The length and characteristic impedance remains same
• Since the stub is added in shunt, there is no need to cut the main line
• The susceptance of the stub can be adjusted for perfect matching
A short-circuited stub is less prone to leakage of electromagnetic
radiation and is somewhat easier to realize.
Open circuited stub may be more practical for certain types of
transmission lines, for example microstrips where one would have to
drill the insulating substrate to short circuit the two conductors of the
line
Single Stub Matching
Based on the number of parallel stubs connected to the line stub
matching techniques can be classified into two types
• Single-Stub Matching Technique
• Double- Stub Matching Technique
A short-circuited section (stub) of a
transmission line connected in parallel to
the main transmission line is shown in
Figure
The stub is connected in parallel, so it is
easy to deal with admittance analysis
instead of impedance
Single Stub Matching
Single Stub Matching
There are two design parameters for single stub matching
• The location of the stub with reference to the load can be
represented as dstub
• The length of the stub line Lstub
For proper impedance matching, the admittance at the location of the
stub can be written as0
0
1(d ) Y
A s tu b s tu bY Y Y
Z
Location of the stub(dstub)
Single Stub Matching
To determine the location of the short-circuited stubs, the input
impedance of a lossless transmission line and convert it to admittance
0
ta n
1 ta n
in R
in
R
Y y j ly
Y jy l
2 2
2 2
2 2
2 2 2 2
ta n ta n ta n
1 ta n
1 ta n 1 ta n
1 ta n 1 ta n
R R R
in
R
R R
R R
y y l jy l j ly
y l
y l j y l
y l y l
0
0
0
ta n
ta n
R
in
R
Y jY lY Y
Y jY l
The normalized admittance can be written as
Separating the real and imaginary parts by rationalizing
Single Stub Matching
0R e [ ] 1
in inY Y o r y
2
2 2
1 ta n1
1 ta n
R s tu b
R s tu b
y d
y d
2 2 2
2
2
1
1 0
1 ta n 1 ta n
1 1ta n
1 1ta n
2ta n
R s tu b R s tu b
R
s tu b
RR R
s tu b
R
s tu b
R
y d y d
yd
yy y
d
y
Yd
Y
For no reflection, at a distance l=dstub the real part of the admittance
is unity
At l=dstub
Simplifying the above expression, we obtain
Single Stub Matching
2 2
2 2 2 2
1 ta n 1 ta nIm [ ]
1 ta n 1 ta n
R R s tu b
in s
R R s tu b
y l y dy b
y l y d
2 2
2 2 2 2
2
2
2
0 0
0
0
1 ta n 1 ta n
1 ta n 1 ta n
11
1 1
1 (1 )1
1
. .
R R s tu b
s
R R s tu b
R
RR R
s
R R RR
R
R
R
s
R R
y l y db
y l y d
yyy y
b
y y yy
y
Y
Y Y Yi e b
Y Y Y
Y
At this location the imaginary part of susceptance of bs can be
written as
Therefore at length dstub the input admittance is
Length of the stub(Lstub)
Single Stub Matching
To determine the length of the short-circuited stub consider the short
circuited impedance and write in terms of admittance
That is
0
c o ts
s
Bb l
Y
0
0
0
0
c o t
ta n
R
s tu b
R
R
s tu b
R
Y YL
Y Y
Y YL
Y Y
0
0
tan
co t
sc s
sc s
Z jX jZ l
Y jB jY l
The required normalized suspetance of the short circuited stub
Equating with ysc
Single Stub Matching
0
0
1
ta n1 1
R
s tu b
R
Z ZL
Z Z
0
0
R
R
Z Z
Z Z
01
0
01
0
ta n2
ta n2
R
s tu b
R
R
s tu b
R
Z ZL
Z Z
Z ZL
Z Z
Converting into impedance
Therefore, the length of short circuited stub is
for
for
The drawback of this approach is that if the load is changed, the
location of insertion may have to be moved
Smith Chart
Smith Chart
It is a polar plot of the complex reflection coefficient.
It is the transformation of complex impedance into
reflection coefficient plane.
Figure 1: Transformation of Z into
Smith Chart
• Construction of smith chart:-
• It consist of r-circles and x-circles
Fig-2: r-circles Fig-3: x-circles
Smith Chart
• Points to remember regarding smith chart:
The value of r is always positive, x can be positive (for inducatance
impedance) and negative (for capacitance impedance)
Apart from the r and x circles, we can draw the VSWR-circles or (S-circles)(ALWAYS NOT SHOWN ON THE CHART)
Figure 4: Some points on smith chart
Smith Chart At point Psc on the chart r = 0 and x = 0, it represents short circuit point. Similarly
Poc on the chart r = infinity and x = infinity represents open circuit point.
A complete revolution around the smith chart represents a distance of on the
line(360 degress).
Clockwise movement on the chart is regarded as moving toward the generator.
Similarly, counter clockwise movement on the chart corresponds to moving towards
load.
There are 3 scales around periphery of the smith chart
> Outermost scale used to determine distance on the from generator end in terms
of wave length.
> The next scale determines distance from the load end interms of WL
> Outermost scale used to determine distance on the from generator end in terms
of wave length.
> The innermost scale is a protractor and is primarily used in determining angle of
reflection coefficient
Smith Chart
To the horizontal line upper part is inductive in nature and bottom part is capacitive in nature.
As shown in figure (4):
> The left most point A on the smith chart corresponds to and therefore represents ideal short-circuit load.
> The right most point B on the Smith chart corresponds to , and therefore represents ideal open circuit load.
>The upper most point C represents a pure inductive load of unity reactance and the lower most point D represents a pure capacitive load of unity reactance.
Voltage Vmax occurs where impedance is maximum i.e point A and Vminoccurs where impedance is minimum i.e point B.