Detailed Notes & Questions Practice: Magnetically Coupled Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE) PDF Download

 Page 1


527
MAGNETICALLY COUPLED CIRCUITS
Enhancing Your Career
Career in Electromagnetics Electromagnetics is the
branch of electrical engineering (or physics) that deals with
the analysis and application of electric and magnetic ?elds.
In electromagnetics, electric circuit analysis is applied at low
frequencies.
The principles of electromagnetics (EM) are applied
in various allied disciplines, such as electric machines,
electromechanical energy conversion, radar meteorology,
remote sensing, satellite communications, bioelectromag-
netics, electromagnetic interference and compatibility, plas-
mas, and ?ber optics. EM devices include electric motors
and generators, transformers, electromagnets, magnetic lev-
itation, antennas, radars, microwave ovens, microwave
dishes, superconductors, and electrocardiograms. The de-
sign of these devices requires a thorough knowledge of the
laws and principles of EM.
EM is regarded as one of the more dif?cult disci-
plines in electrical engineering. One reason is that EM
phenomena are rather abstract. But if one enjoys working
with mathematics and can visualize the invisible, one should
consider being a specialist in EM, since few electrical
engineers specialize in this area. Electrical engineers who
specialize in EM are needed in microwave industries,
radio/TV broadcasting stations, electromagnetic research
laboratories, and several communications industries.
Telemetry receiving station for space satellites. Source: T. J. Mal-
oney, Modern Industrial Electronics, 3rd ed. Englewood Cliffs, NJ:
Prentice Hall, 1996, p. 718.
Page 2


527
MAGNETICALLY COUPLED CIRCUITS
Enhancing Your Career
Career in Electromagnetics Electromagnetics is the
branch of electrical engineering (or physics) that deals with
the analysis and application of electric and magnetic ?elds.
In electromagnetics, electric circuit analysis is applied at low
frequencies.
The principles of electromagnetics (EM) are applied
in various allied disciplines, such as electric machines,
electromechanical energy conversion, radar meteorology,
remote sensing, satellite communications, bioelectromag-
netics, electromagnetic interference and compatibility, plas-
mas, and ?ber optics. EM devices include electric motors
and generators, transformers, electromagnets, magnetic lev-
itation, antennas, radars, microwave ovens, microwave
dishes, superconductors, and electrocardiograms. The de-
sign of these devices requires a thorough knowledge of the
laws and principles of EM.
EM is regarded as one of the more dif?cult disci-
plines in electrical engineering. One reason is that EM
phenomena are rather abstract. But if one enjoys working
with mathematics and can visualize the invisible, one should
consider being a specialist in EM, since few electrical
engineers specialize in this area. Electrical engineers who
specialize in EM are needed in microwave industries,
radio/TV broadcasting stations, electromagnetic research
laboratories, and several communications industries.
Telemetry receiving station for space satellites. Source: T. J. Mal-
oney, Modern Industrial Electronics, 3rd ed. Englewood Cliffs, NJ:
Prentice Hall, 1996, p. 718.
13.1 INTRODUCTION
The circuits we have considered so far may be regarded as conductively
coupled, because one loop affects the neighboring loop through current
conduction. When two loops with or without contacts between them
affect each other through the magnetic ?eld generated by one of them,
they are said to be magnetically coupled.
The transformer is an electrical device designed on the basis of
the concept of magnetic coupling. It uses magnetically coupled coils to
transfer energy from one circuit to another. Transformers are key circuit
elements. They are used in power systems for stepping up or stepping
down ac voltages or currents. They are used in electronic circuits such as
radio and television receivers for such purposes as impedance matching,
isolating one part of a circuit from another, and again for stepping up or
down ac voltages and currents.
We will begin with the concept of mutual inductance and introduce
the dot convention used for determining the voltage polarities of induc-
tively coupled components. Based on the notion of mutual inductance,
we then introduce the circuit element known as the transformer. We will
consider the linear transformer, the ideal transformer, the ideal autotrans-
former, and the three-phase transformer. Finally, among their important
applications, we look at transformers as isolating and matching devices
and their use in power distribution.
13.2 MUTUAL INDUCTANCE
When two inductors (or coils) are in a close proximity to each other,
the magnetic ?ux caused by current in one coil links with the other coil,
thereby inducing voltage in the latter. This phenomenon is known as
mutual inductance.
i(t)
v
+
-
f
Figure13.1 Magnetic ?ux produced
by a single coil withN turns.
Let us ?rst consider a single inductor, a coil withN turns. When
currenti ?ows through the coil, a magnetic ?uxf is produced around it
(Fig. 13.1). According to Faraday’s law, the voltagev induced in the coil
is proportional to the number of turnsN and the time rate of change of
the magnetic ?uxf; that is,
v=N
df
dt
(13.1)
But the ?uxf is produced by currenti so that any change inf is caused
by a change in the current. Hence, Eq. (13.1) can be written as
v=N
df
di
di
dt
(13.2)
or
v=L
di
dt
(13.3)
which is the voltage-current relationship for the inductor. From Eqs.
(13.2) and (13.3), the inductanceL of the inductor is thus given by
L=N
df
di
(13.4)
Page 3


527
MAGNETICALLY COUPLED CIRCUITS
Enhancing Your Career
Career in Electromagnetics Electromagnetics is the
branch of electrical engineering (or physics) that deals with
the analysis and application of electric and magnetic ?elds.
In electromagnetics, electric circuit analysis is applied at low
frequencies.
The principles of electromagnetics (EM) are applied
in various allied disciplines, such as electric machines,
electromechanical energy conversion, radar meteorology,
remote sensing, satellite communications, bioelectromag-
netics, electromagnetic interference and compatibility, plas-
mas, and ?ber optics. EM devices include electric motors
and generators, transformers, electromagnets, magnetic lev-
itation, antennas, radars, microwave ovens, microwave
dishes, superconductors, and electrocardiograms. The de-
sign of these devices requires a thorough knowledge of the
laws and principles of EM.
EM is regarded as one of the more dif?cult disci-
plines in electrical engineering. One reason is that EM
phenomena are rather abstract. But if one enjoys working
with mathematics and can visualize the invisible, one should
consider being a specialist in EM, since few electrical
engineers specialize in this area. Electrical engineers who
specialize in EM are needed in microwave industries,
radio/TV broadcasting stations, electromagnetic research
laboratories, and several communications industries.
Telemetry receiving station for space satellites. Source: T. J. Mal-
oney, Modern Industrial Electronics, 3rd ed. Englewood Cliffs, NJ:
Prentice Hall, 1996, p. 718.
13.1 INTRODUCTION
The circuits we have considered so far may be regarded as conductively
coupled, because one loop affects the neighboring loop through current
conduction. When two loops with or without contacts between them
affect each other through the magnetic ?eld generated by one of them,
they are said to be magnetically coupled.
The transformer is an electrical device designed on the basis of
the concept of magnetic coupling. It uses magnetically coupled coils to
transfer energy from one circuit to another. Transformers are key circuit
elements. They are used in power systems for stepping up or stepping
down ac voltages or currents. They are used in electronic circuits such as
radio and television receivers for such purposes as impedance matching,
isolating one part of a circuit from another, and again for stepping up or
down ac voltages and currents.
We will begin with the concept of mutual inductance and introduce
the dot convention used for determining the voltage polarities of induc-
tively coupled components. Based on the notion of mutual inductance,
we then introduce the circuit element known as the transformer. We will
consider the linear transformer, the ideal transformer, the ideal autotrans-
former, and the three-phase transformer. Finally, among their important
applications, we look at transformers as isolating and matching devices
and their use in power distribution.
13.2 MUTUAL INDUCTANCE
When two inductors (or coils) are in a close proximity to each other,
the magnetic ?ux caused by current in one coil links with the other coil,
thereby inducing voltage in the latter. This phenomenon is known as
mutual inductance.
i(t)
v
+
-
f
Figure13.1 Magnetic ?ux produced
by a single coil withN turns.
Let us ?rst consider a single inductor, a coil withN turns. When
currenti ?ows through the coil, a magnetic ?uxf is produced around it
(Fig. 13.1). According to Faraday’s law, the voltagev induced in the coil
is proportional to the number of turnsN and the time rate of change of
the magnetic ?uxf; that is,
v=N
df
dt
(13.1)
But the ?uxf is produced by currenti so that any change inf is caused
by a change in the current. Hence, Eq. (13.1) can be written as
v=N
df
di
di
dt
(13.2)
or
v=L
di
dt
(13.3)
which is the voltage-current relationship for the inductor. From Eqs.
(13.2) and (13.3), the inductanceL of the inductor is thus given by
L=N
df
di
(13.4)
This inductance is commonly called self-inductance, because it relates
the voltage induced in a coil by a time-varying current in the same coil.
Now consider two coils with self-inductancesL
1
andL
2
that are in
close proximity with each other (Fig. 13.2). Coil 1 hasN
1
turns, while
coil 2 hasN
2
turns. For the sake of simplicity, assume that the second
inductor carries no current. The magnetic ?uxf
1
emanating from coil 1
has two components: one componentf
11
links only coil 1, and another
componentf
12
links both coils. Hence,
f
1
=f
11
+f
12
(13.5)
Although the two coils are physically separated, they are said to be mag-
netically coupled. Since the entire ?uxf
1
links coil 1, the voltage induced
in coil 1 is
v
1
=N
1
df
1
dt
(13.6)
Only ?uxf
12
links coil 2, so the voltage induced in coil 2 is
v
2
=N
2
df
12
dt
(13.7)
Again, as the ?uxes are caused by the currenti
1
?owing in coil 1, Eq.
(13.6) can be written as
v
1
=N
1
df
1
di
1
di
1
dt
=L
1
di
1
dt
(13.8)
whereL
1
= N
1
df
1
/di
1
is the self-inductance of coil 1. Similarly, Eq.
(13.7) can be written as
v
2
=N
2
df
12
di
1
di
1
dt
=M
21
di
1
dt
(13.9)
where
M
21
=N
2
df
12
di
1
(13.10)
M
21
is known as the mutual inductance of coil 2 with respect to coil 1.
Subscript 21 indicates that the inductanceM
21
relates the voltage induced
in coil 2 to the current in coil 1. Thus, the open-circuit mutual voltage
(or induced voltage) across coil 2 is
v
2
=M
21
di
1
dt
(13.11)
i
1
(t)
v
1
+
-
v
2
+
-
f
11
f
12
L
1
L
2
N
1
 turns N
2
 turns
Figure13.2 Mutual inductanceM
21
of
coil 2 with respect to coil 1.
v
1
+
-
v
2
+
-
i
2
(t)
f
22
f
21
L
1
L
2
N
1
 turns N
2
 turns
Figure13.3 Mutual inductanceM
12
of
coil 1 with respect to coil 2.
Suppose we now let currenti
2
?ow in coil 2, while coil 1 carries no
current (Fig. 13.3). The magnetic ?uxf
2
emanating from coil 2 comprises
?uxf
22
that links only coil 2 and ?uxf
21
that links both coils. Hence,
f
2
=f
21
+f
22
(13.12)
The entire ?uxf
2
links coil 2, so the voltage induced in coil 2 is
v
2
=N
2
df
2
dt
=N
2
df
2
di
2
di
2
dt
=L
2
di
2
dt
(13.13)
Page 4


527
MAGNETICALLY COUPLED CIRCUITS
Enhancing Your Career
Career in Electromagnetics Electromagnetics is the
branch of electrical engineering (or physics) that deals with
the analysis and application of electric and magnetic ?elds.
In electromagnetics, electric circuit analysis is applied at low
frequencies.
The principles of electromagnetics (EM) are applied
in various allied disciplines, such as electric machines,
electromechanical energy conversion, radar meteorology,
remote sensing, satellite communications, bioelectromag-
netics, electromagnetic interference and compatibility, plas-
mas, and ?ber optics. EM devices include electric motors
and generators, transformers, electromagnets, magnetic lev-
itation, antennas, radars, microwave ovens, microwave
dishes, superconductors, and electrocardiograms. The de-
sign of these devices requires a thorough knowledge of the
laws and principles of EM.
EM is regarded as one of the more dif?cult disci-
plines in electrical engineering. One reason is that EM
phenomena are rather abstract. But if one enjoys working
with mathematics and can visualize the invisible, one should
consider being a specialist in EM, since few electrical
engineers specialize in this area. Electrical engineers who
specialize in EM are needed in microwave industries,
radio/TV broadcasting stations, electromagnetic research
laboratories, and several communications industries.
Telemetry receiving station for space satellites. Source: T. J. Mal-
oney, Modern Industrial Electronics, 3rd ed. Englewood Cliffs, NJ:
Prentice Hall, 1996, p. 718.
13.1 INTRODUCTION
The circuits we have considered so far may be regarded as conductively
coupled, because one loop affects the neighboring loop through current
conduction. When two loops with or without contacts between them
affect each other through the magnetic ?eld generated by one of them,
they are said to be magnetically coupled.
The transformer is an electrical device designed on the basis of
the concept of magnetic coupling. It uses magnetically coupled coils to
transfer energy from one circuit to another. Transformers are key circuit
elements. They are used in power systems for stepping up or stepping
down ac voltages or currents. They are used in electronic circuits such as
radio and television receivers for such purposes as impedance matching,
isolating one part of a circuit from another, and again for stepping up or
down ac voltages and currents.
We will begin with the concept of mutual inductance and introduce
the dot convention used for determining the voltage polarities of induc-
tively coupled components. Based on the notion of mutual inductance,
we then introduce the circuit element known as the transformer. We will
consider the linear transformer, the ideal transformer, the ideal autotrans-
former, and the three-phase transformer. Finally, among their important
applications, we look at transformers as isolating and matching devices
and their use in power distribution.
13.2 MUTUAL INDUCTANCE
When two inductors (or coils) are in a close proximity to each other,
the magnetic ?ux caused by current in one coil links with the other coil,
thereby inducing voltage in the latter. This phenomenon is known as
mutual inductance.
i(t)
v
+
-
f
Figure13.1 Magnetic ?ux produced
by a single coil withN turns.
Let us ?rst consider a single inductor, a coil withN turns. When
currenti ?ows through the coil, a magnetic ?uxf is produced around it
(Fig. 13.1). According to Faraday’s law, the voltagev induced in the coil
is proportional to the number of turnsN and the time rate of change of
the magnetic ?uxf; that is,
v=N
df
dt
(13.1)
But the ?uxf is produced by currenti so that any change inf is caused
by a change in the current. Hence, Eq. (13.1) can be written as
v=N
df
di
di
dt
(13.2)
or
v=L
di
dt
(13.3)
which is the voltage-current relationship for the inductor. From Eqs.
(13.2) and (13.3), the inductanceL of the inductor is thus given by
L=N
df
di
(13.4)
This inductance is commonly called self-inductance, because it relates
the voltage induced in a coil by a time-varying current in the same coil.
Now consider two coils with self-inductancesL
1
andL
2
that are in
close proximity with each other (Fig. 13.2). Coil 1 hasN
1
turns, while
coil 2 hasN
2
turns. For the sake of simplicity, assume that the second
inductor carries no current. The magnetic ?uxf
1
emanating from coil 1
has two components: one componentf
11
links only coil 1, and another
componentf
12
links both coils. Hence,
f
1
=f
11
+f
12
(13.5)
Although the two coils are physically separated, they are said to be mag-
netically coupled. Since the entire ?uxf
1
links coil 1, the voltage induced
in coil 1 is
v
1
=N
1
df
1
dt
(13.6)
Only ?uxf
12
links coil 2, so the voltage induced in coil 2 is
v
2
=N
2
df
12
dt
(13.7)
Again, as the ?uxes are caused by the currenti
1
?owing in coil 1, Eq.
(13.6) can be written as
v
1
=N
1
df
1
di
1
di
1
dt
=L
1
di
1
dt
(13.8)
whereL
1
= N
1
df
1
/di
1
is the self-inductance of coil 1. Similarly, Eq.
(13.7) can be written as
v
2
=N
2
df
12
di
1
di
1
dt
=M
21
di
1
dt
(13.9)
where
M
21
=N
2
df
12
di
1
(13.10)
M
21
is known as the mutual inductance of coil 2 with respect to coil 1.
Subscript 21 indicates that the inductanceM
21
relates the voltage induced
in coil 2 to the current in coil 1. Thus, the open-circuit mutual voltage
(or induced voltage) across coil 2 is
v
2
=M
21
di
1
dt
(13.11)
i
1
(t)
v
1
+
-
v
2
+
-
f
11
f
12
L
1
L
2
N
1
 turns N
2
 turns
Figure13.2 Mutual inductanceM
21
of
coil 2 with respect to coil 1.
v
1
+
-
v
2
+
-
i
2
(t)
f
22
f
21
L
1
L
2
N
1
 turns N
2
 turns
Figure13.3 Mutual inductanceM
12
of
coil 1 with respect to coil 2.
Suppose we now let currenti
2
?ow in coil 2, while coil 1 carries no
current (Fig. 13.3). The magnetic ?uxf
2
emanating from coil 2 comprises
?uxf
22
that links only coil 2 and ?uxf
21
that links both coils. Hence,
f
2
=f
21
+f
22
(13.12)
The entire ?uxf
2
links coil 2, so the voltage induced in coil 2 is
v
2
=N
2
df
2
dt
=N
2
df
2
di
2
di
2
dt
=L
2
di
2
dt
(13.13)
whereL
2
=N
2
df
2
/di
2
is the self-inductance of coil 2. Since only ?ux
f
21
links coil 1, the voltage induced in coil 1 is
v
1
=N
1
df
21
dt
=N
1
df
21
di
2
di
2
dt
=M
12
di
2
dt
(13.14)
where
M
12
=N
1
df
21
di
2
(13.15)
which is the mutual inductance of coil 1 with respect to coil 2. Thus, the
open-circuit mutual voltage across coil 1 is
v
1
=M
12
di
2
dt
(13.16)
We will see in the next section thatM
12
andM
21
are equal, that is,
M
12
=M
21
=M (13.17)
and we refer toM as the mutual inductance between the two coils. Like
self-inductanceL, mutual inductanceM is measured in henrys (H). Keep
in mind that mutual coupling only exists when the inductors or coils are
in close proximity, and the circuits are driven by time-varying sources.
We recall that inductors act like short circuits to dc.
From the two cases in Figs. 13.2 and 13.3, we conclude that mutual
inductance results if a voltage is induced by a time-varying current in
another circuit. It is the property of an inductor to produce a voltage in
reaction to a time-varying current in another inductor near it. Thus,
Mutual inductanceistheabilityofoneinductortoinduceavoltage
across a neighboring inductor, measured in henrys (H).
Although mutual inductanceM is always a positive quantity, the
mutual voltageMdi/dt may be negative or positive, just like the self-
induced voltageLdi/dt. However, unlike the self-inducedLdi/dt,
whose polarity is determined by the reference direction of the current and
the reference polarity of the voltage (according to the passive sign con-
vention), the polarity of mutual voltageMdi/dt is not easy to determine,
because four terminals are involved. The choice of the correct polarity for
Mdi/dt is made by examining the orientation or particular way in which
both coils are physically wound and applying Lenz’s law in conjunction
with the right-hand rule. Since it is inconvenient to show the construction
details of coils on a circuit schematic, we apply the dot convention in cir-
cuit analysis. By this convention, a dot is placed in the circuit at one end
of each of the two magnetically coupled coils to indicate the direction of
the magnetic ?ux if current enters that dotted terminal of the coil. This is
illustrated in Fig. 13.4. Given a circuit, the dots are already placed beside
the coils so that we need not bother about how to place them. The dots
are used along with the dot convention to determine the polarity of the
mutual voltage. The dot convention is stated as follows:
Page 5


527
MAGNETICALLY COUPLED CIRCUITS
Enhancing Your Career
Career in Electromagnetics Electromagnetics is the
branch of electrical engineering (or physics) that deals with
the analysis and application of electric and magnetic ?elds.
In electromagnetics, electric circuit analysis is applied at low
frequencies.
The principles of electromagnetics (EM) are applied
in various allied disciplines, such as electric machines,
electromechanical energy conversion, radar meteorology,
remote sensing, satellite communications, bioelectromag-
netics, electromagnetic interference and compatibility, plas-
mas, and ?ber optics. EM devices include electric motors
and generators, transformers, electromagnets, magnetic lev-
itation, antennas, radars, microwave ovens, microwave
dishes, superconductors, and electrocardiograms. The de-
sign of these devices requires a thorough knowledge of the
laws and principles of EM.
EM is regarded as one of the more dif?cult disci-
plines in electrical engineering. One reason is that EM
phenomena are rather abstract. But if one enjoys working
with mathematics and can visualize the invisible, one should
consider being a specialist in EM, since few electrical
engineers specialize in this area. Electrical engineers who
specialize in EM are needed in microwave industries,
radio/TV broadcasting stations, electromagnetic research
laboratories, and several communications industries.
Telemetry receiving station for space satellites. Source: T. J. Mal-
oney, Modern Industrial Electronics, 3rd ed. Englewood Cliffs, NJ:
Prentice Hall, 1996, p. 718.
13.1 INTRODUCTION
The circuits we have considered so far may be regarded as conductively
coupled, because one loop affects the neighboring loop through current
conduction. When two loops with or without contacts between them
affect each other through the magnetic ?eld generated by one of them,
they are said to be magnetically coupled.
The transformer is an electrical device designed on the basis of
the concept of magnetic coupling. It uses magnetically coupled coils to
transfer energy from one circuit to another. Transformers are key circuit
elements. They are used in power systems for stepping up or stepping
down ac voltages or currents. They are used in electronic circuits such as
radio and television receivers for such purposes as impedance matching,
isolating one part of a circuit from another, and again for stepping up or
down ac voltages and currents.
We will begin with the concept of mutual inductance and introduce
the dot convention used for determining the voltage polarities of induc-
tively coupled components. Based on the notion of mutual inductance,
we then introduce the circuit element known as the transformer. We will
consider the linear transformer, the ideal transformer, the ideal autotrans-
former, and the three-phase transformer. Finally, among their important
applications, we look at transformers as isolating and matching devices
and their use in power distribution.
13.2 MUTUAL INDUCTANCE
When two inductors (or coils) are in a close proximity to each other,
the magnetic ?ux caused by current in one coil links with the other coil,
thereby inducing voltage in the latter. This phenomenon is known as
mutual inductance.
i(t)
v
+
-
f
Figure13.1 Magnetic ?ux produced
by a single coil withN turns.
Let us ?rst consider a single inductor, a coil withN turns. When
currenti ?ows through the coil, a magnetic ?uxf is produced around it
(Fig. 13.1). According to Faraday’s law, the voltagev induced in the coil
is proportional to the number of turnsN and the time rate of change of
the magnetic ?uxf; that is,
v=N
df
dt
(13.1)
But the ?uxf is produced by currenti so that any change inf is caused
by a change in the current. Hence, Eq. (13.1) can be written as
v=N
df
di
di
dt
(13.2)
or
v=L
di
dt
(13.3)
which is the voltage-current relationship for the inductor. From Eqs.
(13.2) and (13.3), the inductanceL of the inductor is thus given by
L=N
df
di
(13.4)
This inductance is commonly called self-inductance, because it relates
the voltage induced in a coil by a time-varying current in the same coil.
Now consider two coils with self-inductancesL
1
andL
2
that are in
close proximity with each other (Fig. 13.2). Coil 1 hasN
1
turns, while
coil 2 hasN
2
turns. For the sake of simplicity, assume that the second
inductor carries no current. The magnetic ?uxf
1
emanating from coil 1
has two components: one componentf
11
links only coil 1, and another
componentf
12
links both coils. Hence,
f
1
=f
11
+f
12
(13.5)
Although the two coils are physically separated, they are said to be mag-
netically coupled. Since the entire ?uxf
1
links coil 1, the voltage induced
in coil 1 is
v
1
=N
1
df
1
dt
(13.6)
Only ?uxf
12
links coil 2, so the voltage induced in coil 2 is
v
2
=N
2
df
12
dt
(13.7)
Again, as the ?uxes are caused by the currenti
1
?owing in coil 1, Eq.
(13.6) can be written as
v
1
=N
1
df
1
di
1
di
1
dt
=L
1
di
1
dt
(13.8)
whereL
1
= N
1
df
1
/di
1
is the self-inductance of coil 1. Similarly, Eq.
(13.7) can be written as
v
2
=N
2
df
12
di
1
di
1
dt
=M
21
di
1
dt
(13.9)
where
M
21
=N
2
df
12
di
1
(13.10)
M
21
is known as the mutual inductance of coil 2 with respect to coil 1.
Subscript 21 indicates that the inductanceM
21
relates the voltage induced
in coil 2 to the current in coil 1. Thus, the open-circuit mutual voltage
(or induced voltage) across coil 2 is
v
2
=M
21
di
1
dt
(13.11)
i
1
(t)
v
1
+
-
v
2
+
-
f
11
f
12
L
1
L
2
N
1
 turns N
2
 turns
Figure13.2 Mutual inductanceM
21
of
coil 2 with respect to coil 1.
v
1
+
-
v
2
+
-
i
2
(t)
f
22
f
21
L
1
L
2
N
1
 turns N
2
 turns
Figure13.3 Mutual inductanceM
12
of
coil 1 with respect to coil 2.
Suppose we now let currenti
2
?ow in coil 2, while coil 1 carries no
current (Fig. 13.3). The magnetic ?uxf
2
emanating from coil 2 comprises
?uxf
22
that links only coil 2 and ?uxf
21
that links both coils. Hence,
f
2
=f
21
+f
22
(13.12)
The entire ?uxf
2
links coil 2, so the voltage induced in coil 2 is
v
2
=N
2
df
2
dt
=N
2
df
2
di
2
di
2
dt
=L
2
di
2
dt
(13.13)
whereL
2
=N
2
df
2
/di
2
is the self-inductance of coil 2. Since only ?ux
f
21
links coil 1, the voltage induced in coil 1 is
v
1
=N
1
df
21
dt
=N
1
df
21
di
2
di
2
dt
=M
12
di
2
dt
(13.14)
where
M
12
=N
1
df
21
di
2
(13.15)
which is the mutual inductance of coil 1 with respect to coil 2. Thus, the
open-circuit mutual voltage across coil 1 is
v
1
=M
12
di
2
dt
(13.16)
We will see in the next section thatM
12
andM
21
are equal, that is,
M
12
=M
21
=M (13.17)
and we refer toM as the mutual inductance between the two coils. Like
self-inductanceL, mutual inductanceM is measured in henrys (H). Keep
in mind that mutual coupling only exists when the inductors or coils are
in close proximity, and the circuits are driven by time-varying sources.
We recall that inductors act like short circuits to dc.
From the two cases in Figs. 13.2 and 13.3, we conclude that mutual
inductance results if a voltage is induced by a time-varying current in
another circuit. It is the property of an inductor to produce a voltage in
reaction to a time-varying current in another inductor near it. Thus,
Mutual inductanceistheabilityofoneinductortoinduceavoltage
across a neighboring inductor, measured in henrys (H).
Although mutual inductanceM is always a positive quantity, the
mutual voltageMdi/dt may be negative or positive, just like the self-
induced voltageLdi/dt. However, unlike the self-inducedLdi/dt,
whose polarity is determined by the reference direction of the current and
the reference polarity of the voltage (according to the passive sign con-
vention), the polarity of mutual voltageMdi/dt is not easy to determine,
because four terminals are involved. The choice of the correct polarity for
Mdi/dt is made by examining the orientation or particular way in which
both coils are physically wound and applying Lenz’s law in conjunction
with the right-hand rule. Since it is inconvenient to show the construction
details of coils on a circuit schematic, we apply the dot convention in cir-
cuit analysis. By this convention, a dot is placed in the circuit at one end
of each of the two magnetically coupled coils to indicate the direction of
the magnetic ?ux if current enters that dotted terminal of the coil. This is
illustrated in Fig. 13.4. Given a circuit, the dots are already placed beside
the coils so that we need not bother about how to place them. The dots
are used along with the dot convention to determine the polarity of the
mutual voltage. The dot convention is stated as follows:
i
1
f
21
f
11
f
22
f
12
v
1
+
-
i
2
Coil 1 Coil 2
v
2
+
-
Figure13.4 Illustration of the dot convention.
If a current enters the dotted terminal of one coil, the reference
polarityofthemutualvoltageinthesecondcoilis positive
at the dotted terminal of the second coil.
Alternatively,
If a current leaves the dotted terminal of one coil, the reference
polarityofthemutualvoltageinthesecondcoilis negative
at the dotted terminal of the second coil.
+
-
M
i
1
v
2 
= M
di
1
dt
(a)
+
-
M
i
1
v
2 
= –M
di
1
dt
v
1 
= –M
di
2
dt
(b)
+
-
M
(c)
(d)
i
2
v
1 
= M
di
2
dt
+
-
M
i
2
Figure13.5 Examples
illustrating how to apply the
dot convention.
Thus, the reference polarity of the mutual voltage depends on the refer-
ence direction of the inducing current and the dots on the coupled coils.
Application of the dot convention is illustrated in the four pairs of mu-
tually coupled coils in Fig. 13.5. For the coupled coils in Fig. 13.5(a),
the sign of the mutual voltagev
2
is determined by the reference polarity
forv
2
and the direction ofi
1
. Sincei
1
enters the dotted terminal of coil
1 andv
2
is positive at the dotted terminal of coil 2, the mutual voltage is
+Mdi
1
/dt. For the coils in Fig. 13.5(b), the currenti
1
enters the dot-
ted terminal of coil 1 andv
2
is negative at the dotted terminal of coil 2.
Hence, the mutual voltage is-Mdi
1
/dt. The same reasoning applies to
the coils in Fig. 13.5(c) and 13.5(d). Figure 13.6 shows the dot conven-
tion for coupled coils in series. For the coils in Fig. 13.6(a), the total
inductance is
L=L
1
+L
2
+ 2M (Series-aiding connection) (13.18)
For the coil in Fig. 13.6(b),
L=L
1
+L
2
- 2M (Series-opposing connection) (13.19)
Now that we know how to determine the polarity of the mutual
voltage, we are prepared to analyze circuits involving mutual inductance.