Download EE205 - KFUPM Resources v3

EE 205 Dr. A. Zidouri
Electric Circuits II
Mutual Inductance
Lecture #23
-1-
EE 205 Dr. A. Zidouri
The material to be covered in this lecture is as follows:
o
o
o
o
Review of Self Inductance
Concept of Mutual Inductance
Magnetic coupling
Dot Convention
-2-
EE 205 Dr. A. Zidouri
After finishing this lecture you should be able to:
 Understand the concept of Mutual Inductance
 Determine the polarity of Induced Voltages
 Write Mesh Equations for a Circuit with mutual Inductance
-3-
EE 205 Dr. A. Zidouri
Self Inductance
• A magnetic field consists of force surrounding the current carrying conductor.
• The voltage induced in the conductor is proportional to the number of lines that collapse into or cut
the conductor.
d
v 
dt
Faraday’s Law is
i
+

is the flux linkage and is
measured in Wb –turns
Magnetic Lines of force
Φ
v
-
N turns
Fig.31-1 Representation of a magnetic field linking an N turns coil
-4-
EE 205 Dr. A. Zidouri
Self Inductance (Cont)

• The flux linkage
is the product of magnetic field (  ) measured in (Wb) and the number of turns
linked by the field (N).
 N
The magnetic flux

is related to the magnitude of the coil current by the relationship:
  pNi
where
p is the permeance of the space occupied by the flux
N is the number of turns
i is the current
-5-
EE 205 Dr. A. Zidouri
Self Inductance (Cont)
Assuming the core material is non-magnetic then:
v 
N
d  d (N  )

dt
dt
d
d ( pNi )
N
dt
dt
 pN
N
L

L is the self inductance, given by
i
2
di
di
L
dt
dt
-6-
EE 205 Dr. A. Zidouri
Example 31-1:
 If an inductor with 100 turns and a current of 5 A produces a flux of 10 Wb the flux linkage equals
N = 100x10 = 1000 Wb
 The inductance is therefore:
L
N  1000

 2 00 W b
A
i
5
-7-
EE 205 Dr. A. Zidouri
Concept of Mutual Inductance (Cont):
Mutual Inductance (M): Circuit Parameter that relates the voltage induced in one circuit to a time-varying
current in a neighboring circuit, measured in henrys (H).
Consider Two Magnetically Coupled Coils
M
L1
L2
Fig31-2 Circuit of Two Magnetically Coupled coils
The magnetic field of the first coil also passes through the second coil and induces a voltage across it
as well. We say that the two coils are magnetically coupled
-8-
EE 205 Dr. A. Zidouri
Concept of Mutual Inductance (Cont):
Case I:
 Coil 1 is energized by a time –varying current source that establishes a current i1 in N1 turns.
 Coil 2 is not energized and open.
 The Coils are wound on a magnetic core.
Ф21
Coil Current
 The flux produced by i1 can be divided into two components:
o Ф11 is the flux produced by i1 that links only the N1 turns.
o Ф21 is the flux produced by i1 that links N2 and N1 turns.
Coil Number
 The Total flux linking coil 1 is Ф1= Ф11 + Ф21
is
1  p1 N1i1
11  p11 N1i1
Ф21
i1
v1
Ф11
Ф11
v2
N2
N1
Ф21
Fig31-3 Magnetically Coupled coils with coil 1 excited and coil 2 open.
21  p21 N1i1
p1  p11  p21
The Voltages V1 and V2 are
v1 
v2 
d 1
di
di
 p1 N12 1  L1 1
dt
dt
dt
d 2
di1
di1
dt
 p21 N 2 N1
dt
 M 21
dt
-9-
EE 205 Dr. A. Zidouri
Concept of Mutual Inductance (Cont):
Case II:
 Coil 2 is energized by a time –varying current source that
establishes the current i2 in N2 turns.
 Coil 1 is not energized and open.
 The Coils are wound on a magnetic core.
 The flux produced by i2 can be divided into two components
o Ф22 is the flux produced by i2 that links only the N2 turns.
o Ф12 is the flux produced by i2 that links N1 and N2 turns.
 The Total flux linking coil 2 is Ф2= Ф22 + Ф12
Ф12
Ф22
v2
N2
N1
Coil Current
Coil Number
2  p2 N2i2
22  p22 N2i2
12  p12 N2i2
i2
v1
Ф12
is
Ф22
Ф12
Fig31-4 Magnetically Coupled coils with coil 2 excited
and coil 1 open.
The Voltages V2 and V1 are
v2 
v1 
d 2
di
di
 p2 N 22 2  L2 2
dt
dt
dt
d 1
di2
di2
dt
 p12 N1 N 2
dt
 M 12
dt
- 10 -
EE 205 Dr. A. Zidouri
Concept of Mutual Inductance (Cont.)
Therefore
v 2  M 21
di 1
dt
if
i 2  0, i 1  0
v 1  M 12
di 2
dt
if
i 1  0, i 2  0
For Nonmagnetic materials, the permeance p12 and p21 are equal and Therefore
M12  M 21  M
- 11 -
EE 205 Dr. A. Zidouri
The Polarity of Mutually Induced Voltages
(The Dot Convention)
 When the reference direction for a current enters the dotted terminal of a coil, the reference
polarity of the voltage that it induces in the other coil is positive at its dotted terminal.
 When the reference direction for a current leaves the dotted terminal of a coil, the reference
polarity of the voltage that it induces in the other coil is negative at its dotted terminal.
- 12 -
EE 205 Dr. A. Zidouri
 Example 31-2
Consider the Circuit of Fig.32-5:
R1
M
 Given the circuit with the dotted markings
as shown in Fig.31-5 (a)
Vg
 Arbitrarily choose the reference direction for
each current Fig.31-5 (b)
Fig.31-5 (a) Two Magnetically Coupled Coils
R1
 Determine the polarity of the mutually
induced voltage, using the dot convention
rule stated earlier Fig.31-5 (c)
i1
i2
R2
L2
L1
Fig.31-5 (b) Coil Currents i1 and i2 and Dot Marking to Indicate
Polarity of Induced Voltages for Circuit of Fig.31-4 (a)
R1
(Mesh 1)
Vg
(Mesh 2)
M
Vg
 Write the mesh equations for i1 and i2
di
di
vg  R1i1  L1 1  M 2  0
dt
dt
di
di
R2i2  L2 2  M 1  0
dt
dt
R2
L2
L1
i1
i2
M



di
M
dt

di
L
dt

di
L2 2
dt


M
di1
dt

R2
Fig.31-5 (c) The Self and Mutually Induced Voltages appearing
across the coils shown in Fig.31-4 (a)
- 13 -
v(t)
Time
x(t)
y(t)
i(t)
C
++–
dom
ain
EE 205 Dr. A. Zidouri
Self Test:
 Which is closest in meaning to large inductance?
a) large flux
b) low current
c) large flux linkage
d) large flux linkage per ampere
answer: d
 If you double the number of turns but keep the flux and the current the same; the inductance will
a) stay the same
b) halve
c) double
d) triple
answer: c
- 14 -