Transcript Singlemode Fiber A Deeper look

CH 7. Time-Varying Fields
and Maxwell’s Equations
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7.1 Introduction
Fundamental Relations for Electrostatic and Magnetostatic Models
Fundamentl Relations
Electrostatic
Model
Governing equations
 E  0
D  
Constitutive relations
D E
(linear and isotropic media)
Magnetostatic
Model
B  0
 H  J
H 
1

B
In the static case, electric field vectors E and D
and magnetic field vectors B and H form separate and independent pairs.
In a conducting medium, static electric and magnetic fields may both exist and
form an electromagnetostatic field.
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7.1 Introduction
A static electric field in a conducting medium causes a steady current to flow that,
In turn, gives rise to a static magnetic field.
The electric field can be completely determined from the static electric charges or
potential distributions.
The magnetic field is a consequence; it does not enter into the calculation of the
electric field.
In this chapter we will see that a changing magnetic field gives rise to an electric field,
and vice versa.
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7.2 Faraday’s Law of Electromagnetic Induction
Michael Faraday, in 1831, discovered experimentally that a current was induced in a
conducting loop when the magnetic flux linking the loop changed.
The quantitative relationship between the induced emf and the rate of change of flux
linkage, based on experimental observation, is known as Faraday’s law.
Fundamental Postulate for Electromagnetic Induction
 E  
B

t
(7 - 1)
Equation 7-1 expresses a point-function relationship; that is, it applies to every point in
space, whether it be in free space or in a material medium.
The electric field intensity in a region of time-varying magnetic flux density is therefore
nonconservative and cannot be expressed as the gradient of a scalar potential.
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7.2 Faraday’s Law of Electromagnetic Induction
Taking the surface integral of both sides of Eq.(7-1) over an open surface and applying
Stokes’s theorem, we obtain
   E  
B
B
d s   E  d l  
 d s.
c
s
t
t
(7 - 2)
Equation (7-2) is valid for any surface S with a bounding contour C, whether or not a
physical circuit exists around C.
7–2.1 A STATIONARY CIRCUIT IN A TIME-VARYING MAGNETIC FIELD
For a stationary circuit with a contour C and surface S, Eq(7 - 2) can be written as
d
c E  d l   dt
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 B  d s.
(7 - 3)
s
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7.2 Faraday’s Law of Electromagnetic Induction
If we define  
 E  dl
= emf induced in circuit with contour C (V)
(7 – 4)
c
   B  d s = magnetic flux crossing surface S (Wb),
(7 – 5)
s
then Eq.(7 - 3) becomes    d 
dt
(V).
(7 – 6)
Equation (7 – 6) states that the electromotive force induced in a stationary closed circuit
is equal to the negative rate of increase of the magnetic flux linking the circuit.
This is a statement of Faraday’s law of electromagnetic induction.
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7.2 Faraday’s Law of Electromagnetic Induction
7-2.2 TRANSFORMERS
A transformer is an alternating-current (a-c) device that transforms voltage, currents,
and impendances.
For the closed path in the magnetic circuit
In Fig.7-1(a) traced by magnetic flux  ,
we have, from Eq.(6-101),
( Eq.(6-101),  N j I j   k  k . )

i (t )
i (t )
2
1
+
v1 (t )
_
N
1
N
2
+
v2 (t )
R
_

(a) Schematic diagram of a transformer.
FIGURE 7-1
back
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j
L
N i N i
1 1

2 2
l
.
S
 ,
k
(7 – 7)
(7 – 8)
( core of length l, cross-sectional area S,
permeability  )
Substituting Eq.(7-8) in Eq.(7-7)
l


N 1i1 N 2 i2 S . (7 – 9)
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7.2 Faraday’s Law of Electromagnetic Induction
i X
1
1
N N 2 R2
R
1
X i2
2
1
+
+
v
1
X
c
R
R
c
L
_
v
2
_
Ideal transformer
(b) An equivalent circuit.
FIGURE 7-1
a) Ideal transformer.
For an ideal transformer we assume that
Eq.(7 - 9)
N 1i1  N 2 i2 
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 
l
. becomes
S
i
i
1
2

, and
N
N
2
.
(7 – 10)
1
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7.2 Faraday’s Law of Electromagnetic Induction
i
i
1
2

N
N
2
. Eq.(7 - 10) states that the ratio of the currents in the primary and secondary
1
windings of an ideal transformer is equal to the inverse ratio of the numbers
of turns.
Faraday’s law tells us that
d
v1  N 1 dt
(7 - 11) and
v2  N 2
From Eqs. (7 – 11) and (7 – 12) we have
d
, (7 – 12)
dt
v
v
1
2

N
N
1
. (7 – 13)
2
Thus, the ratio of the voltages across the primary and secondary windings of an ideal
transformer is equal to the turns ratio.
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7.2 Faraday’s Law of Electromagnetic Induction
R
When the secondary winding is terminated in a load resistance
L, as shown in Fig.
7-1(a) click, the effective load seen by the source connected to primary winding is
2
(R1)
eff
(
N1 )
R
N2
,
L
or
(R1)
eff
 v1 
i
1
( N 1 / N 2 ) v2
( N 2 / N 1) i2
,
(7 – 14a)
v
For a sinusoidal source 1 (t ) and a load impedance Z L , it is obvious that the effective
2
load seen by the source is ( N 1 / N 2) Z L , an impedance transformation.
2
We have
(Z 1)
eff
(
N1 )
Z
N2
L
.
(7 – 14b)
b) Real transformer.
l


. Eq. (7 – 9), we can write the magnetic flux linkages
Referring to N 1 i1 N 2 i2
S
of the primary and secondary windings as
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7.2 Faraday’s Law of Electromagnetic Induction
1  N 1  
S
 N
S
2

2
l
l
( N 1 i1  N 1 N 2 i 2),
(7 – 15)
( N 1 N 2 i1  N 2 i 2).
(7 – 16)
2
2
Using Eqs. (7 - 15) and (7 - 16) in Eqs. (7 - 11) and (7 - 12), we obtain
di
di
v1  L1 1  L12 2 ,
dt
where
L1 
dt
S
N
2
1
,
(7 – 17)
di1  di2 ,

v2 L12
L2
dt
dt
(7 – 18)
(7 – 19) the self-inductance of the primary winding.
l
S 2
(7 – 20) the self-inductance of the secondary winding.

L2 l N 2 ,
S

L12 l N 1 N 2 . (7 – 21) the mutual inductance between the primary and
secondary windings.
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7.2 Faraday’s Law of Electromagnetic Induction
For an ideal transformer there is no leakage flux, and
For real transformers,
L
12
k
LL,
1
2
L
12
k<1,

LL.
1
2
(7 -22)
Where k is called the coefficient of coupling.
For real transformers we have the following real-life conditions.
the existence of leakage flux ( k < 1 ),
noninfinite inductances, nonzero winding resistances,
the presence of hysteresis and eddy-current losses.
The nonlinear nature of the ferromagnetic core further compounds the difficulty of
an exact analysis of real transformers.
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7.2 Faraday’s Law of Electromagnetic Induction
Eddy currents.
When time-varying magnetic flux flows in the ferromagnetic core, an induced emf will
result in accordance with Faraday’s law.
This induced emf will produce local currents in the conducting core normal to the
magnetic flux.
These currents are called eddy currents.
Eddy currents produce ohmic power loss and cause local heating.
This is the principle of induction heating.
In transformers, eddy-current power loss is undesirable and can be reduced by using
core materials that have high permeability but low conductivity (high  and low  ).
For low-frequency, high-power applications an economical way for eddy-current
power loss is to use laminated cores.
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7.2 Faraday’s Law of Electromagnetic Induction
7-2.3 A MOVING CONDUCTOR IN A STATIC MAGNETIC FIELD.
☉
☉
☉
☉
2
u
dl
☉
☉
☉
☉
u
☉ B ☉
1
☉
☉
Figure 7-2
A conducting bar moving in a magnetic field.
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A force F  qu  B will cause the
m
freely movable electrons in the conductor to
drift toward one end of the conductor and
leave the other end positively charged.
This separation of the positive and negative
charges creates a Coulombian force of attraction.
The charge-separation process continues
until the electric and magnetic forces
balance each other and a state of equilibrium is reached.
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7.2 Faraday’s Law of Electromagnetic Induction
To an observer moving with the conductor there is no apparent motion, and the
magnetic force per unit charge F m / q  u  B can be interpreted as an induced
electric field acting along the conductor and producing a voltage
V
2
21
  (u  B)  d l.
(7 – 23)
1
If the moving conductor is a part of a closed circuit C, then the emf generated
around the circuit is
    (u  B)  d l
(V).
(7 – 24)
c
This is referred to as a flux cutting emf or a motional emf. Obviously, only the part
of the circuit that moves in a direction not parallel to the magnetic flux will
contribute  in Eq. (7 – 24).
'
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7.2 Faraday’s Law of Electromagnetic Induction
7-2.4 A MOVING CIRCUIT IN A TIME-VARYING MAGNETIC FIELD.
Lorentz’s force equation
F  q( E  u  B).
(7 – 31)
The force on q can be interpreted as caused by an electric field E , where
E  E  u  B
(7 – 32)
or
E  E   u  B.
(7 – 33)
Hence, when a conducting circuit with contour C and surface S moves with a velocity
in a field ( E, B), we use Eq. (7 – 33) in Eq. (7 – 2) to obtain
u
B
(V).
(7 – 34)
 d s   (u  B)  d l
s t
c
B
B


E


d
s

E

d
l



 t
c
s t  d s. )
 E  d l  
c
( Eq. 7-2
Eq. (7 – 34) is the general form of Faraday’s law for a moving circuit in a time-varying
magnetic field.
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7.2 Faraday’s Law of Electromagnetic Induction
C that moves from C1 at time t to C
t  t in a changing magnetic field B .
Let us consider a circuit with contour
C
B
S
at time
The time-rate of change of magnetic flux
through the contour is
2
C
dS3
1
1
d d
  Bds
dt dt s
ut
dl
S
2
 lim
t 0
1 
.
B
(
t


t
)

d

B
(
t
)

d
s2 s1
s1
t  s2
2
(7 – 35)
FIGURE 7-5
A moving circuit in a time-varying magnetic field.
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7.2 Faraday’s Law of Electromagnetic Induction
B(t  t ) in Eq. (7 - 35) can be expanded as a Taylor’s series:
B(t  t )  B(t ) 
 B(t )
t  H .O.T .,
t
(7 – 36)
Substitution of Eq. (7 – 36) in Eq. (7 – 35) yields
d
B
1 

B

d
s


d
s

lim
B

d

B

d

H
.
O
.
T
.
,
s
s




2
1


s
s

t

0
s1
dt
t
t  s2

An element of the side surface is
d s3  d l  ut.
(7 – 37)
(7 – 38)
Apply the divergence theorem for B at time t to the region sketched in Fig. 7 – 5 :
   Bdv  sB  d s  sB  d s  sB  d s ,
V
2
2
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1
1
3
(7 – 39)
3
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7.2 Faraday’s Law of Electromagnetic Induction
Using Eq. (7 – 38) in Eq. (7 – 39) and noting that   B  0 , we have
sB2  d s2  sB1  d s1  t c(u  B)  d l.
(7 – 40)
Combining Eqs. (7 – 37) and (7 – 40), we obtain
d
dt
B
s B  d s  s t  d s  c(u  B)  d l ,
(7 – 41)
which can be identified as the negative of the right side of Eq. (7 – 34).
If we designate
    E  d l
c
 emf induced in circuit C measured in the moving frame,
Eq. (7 – 34) can be written simply as
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d
Bds

s
dt
d
(V),

dt
(7 – 42)
  
(7 – 43)
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7.2 Faraday’s Law of Electromagnetic Induction
d
Eq. (7 – 6).
dt
If a circuit is not in motion,   reduces to  , and Eqs. (7 – 43) and (7 – 6) are exactly
the same.
Eq. (7 – 43) is of the same form as   
Faraday’s law that the emf induced in a closed circuit equals the negative time-rate of
increase of the magnetic flux linking a circuit applies to a stationary circuit as well
as a moving one.
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7.3 Maxwell’s Equations
The fundamental postulate for electromagnetic induction assures us that a timevarying magnetic field gives rise to an electric field.
 E  0
Time-varying case:
 E  
B
.
t
The revised set of two curl and two divergence equations from Table 7 – 1:
 E  
B
,
t
  D  ,
(7 – 47a)
 H  J,
(7 – 47b)
(7 – 47c)
  B  0.
(7 – 47d)
The mathematical expression of charge conservation is the equation of continuity :
 J  

.
t
(7 – 48)
Divergence of Eq. (7 – 47b) :   (  H )  0    J ,
(7 – 49)
(null identity)
since Eq. (7 – 48) asserts that   J does not vanish in a time-varying situation,
Eq. (7 – 49) is, in general, not true.
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7.3 Maxwell’s Equations
How should Eqs. (7 – 47a, b, c, d) be modified so that they are consistent with
Eq. (7 – 48)?
First of all, a term  / t must be added to the right side of Eq. (7 – 49) :
  (  H )  0    J 

.
t
(7 – 50)
Using Eq. (7 – 47c) in Eq. (7 – 50), we have
  (  H )    ( J 
which implies that
 H  J 
D
.
t
D
),
t
(7 – 51)
(7 – 52)
Eq. (7 – 52) indicates that a time-varying electric field will give rise to a magnetic field,
even in the absence of a current flow.
The additional term  D / t is necessary to make Eq. (7 – 52) consistent with the
principle of conservation of charge.
The term  D / t is called displacement current density.
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7.3 Maxwell’s Equations
In order to be consistent with the equation of continuity in a time varying situation,
both of the curl equations in Table 7 – 1 must be generalized.
The set of four consistent equations to replace the inconsistent equations,
Eqs. (7 – 47a, b, c, d), are
 E  
B
,
t
 H  J 
D
,
t
(7 – 53a)
(7 – 53b)
  D  ,
(7 – 53c)
  B  0.
(7 – 53d)
They are known as Maxwell’s equations.
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7.3 Maxwell’s Equations
7-3.1 INTEGRAL FORM OF MAXWELL’S EQUATIONS.
The four Maxwell’s equations in (7 – 53a, b, c, d) are differential equations that are
valid at every point in space.
In explaining electromagnetic phenomena in a physical environment we must deal
with finite objects of specified shapes and boundaries.
It is convenient to convert the differential forms into their integral-form equivalents.
We take the surface integral of both sides of the curl equations in Eqs. (7 – 53a, b)
over an open surface S with contour C and apply Stokes’s theorem to obtain
B
ds
s t
 E  d l  
(7 – 54a)
D
cH  d l  s( J  t )  d s.
(7 – 54b)
c
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7.3 Maxwell’s Equations
Taking the volume integral of both sides of the divergence equations in Eqs. (7 – 53c,
d) over a volume V with a closed surface S and using divergence theorem, we have
 D  d s   dv
s
v
 B  d s  0.
(7 – 54c)
(7 – 54d)
s
The set of four equations in (7 – 54a, b, c, d) are the integral form of Maxwell’s
equations.
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7.3 Maxwell’s Equations
Maxwell’s Equations
Differential Form
 E  
B
t
 H  J 
D
t
Integral Form
 E  dl  
c
Significance
d
dt
Faraday’s law
D
ds
s t
 H  dl  I  
c
Ampere’s circuital law
 D  
 Dds  Q
Gauss’s law
B  0
 Bds  0
No isolated magnetic charge
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