-
8/9/2019 Electromagnetism and magnetic circuit 3
1/12
CHETAN UPADHYAY 1
Chapter 3
Magnetically coupled circuits
16-Apr-10
-
8/9/2019 Electromagnetism and magnetic circuit 3
2/12
CHETAN UPADHYAY 2
J
Mutual inductance
A single inductor:
dt
di
di
dN
dt
dNv
N
dt
dv
JJJJP
PP
!!@
!
!
flux:turns;ofnumber:N
linkageflux:
dt
dNLwhile
dt
diLv
J!
!@
16-Apr-10
-
8/9/2019 Electromagnetism and magnetic circuit 3
3/12
CHETAN UPADHYAY 3
J J
12111 JJJ !
dt
di
Ldt
d
Nv
1
1
1
11 !!
J
dt
di
di
dN
dt
dNv
1
1
12
2
12
22
122
JJ
JJ
!!
!
dt
dNMwhile
dt
diMv
12221212
J!!@
Mutual inductance ofM21 of coil 2 with respect to coil 1
16-Apr-10
-
8/9/2019 Electromagnetism and magnetic circuit 3
4/12
CHETAN UPADHYAY 4
J22
J21
N2N1
v2v1
i2(t)
22212 JJJ !
dt
diL
dt
dNv
22
222 !!
J
dt
di
di
dN
dt
dNv
2
2
211
2111
211
JJ
JJ
!!
!
dt
dNMwhile
dt
diMv
21112
2121
J!!@
MMM !! 2112 (for nonmagnetic cores)16-Apr-10
-
8/9/2019 Electromagnetism and magnetic circuit 3
5/12
CHETAN UPADHYAY 5
dt
diL
dt
diMv
dt
diM
dt
diLv
22
12
2111
!
!
dt
dNvand
dt
dNvand
coil
coil
coil
coil
2221222
1112111
JJJJ
JJJJ
!!
!!
16-Apr-10
-
8/9/2019 Electromagnetism and magnetic circuit 3
6/12
CHETAN UPADHYAY 6
dt
dNvand
dt
dNvand
coil
coil
coil
coil
2221222
1112111
JJJJ
JJJJ
!!
!!
dt
diL
dt
diMv
dt
diM
dt
diLv
22
12
2111
!
!
16-Apr-10
-
8/9/2019 Electromagnetism and magnetic circuit 3
7/12
CHETAN UPADHYAY 7
1L 2L
M
y y
1v
2
v
1
i2i
12
M
y
y
1v
2
v
1i 2i
dt
diL
dt
diMv
dt
diM
dt
diLv
2
2
1
2
2111
!
!
dt
diL
dt
diMv
dtdiM
dtdiLv
22
12
2111
!
!
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.
Dot convention
16-Apr-10
-
8/9/2019 Electromagnetism and magnetic circuit 3
8/12
CHETAN UPADHYAY 8
1L 2L
M
y y
1v
2v
1i
2i
Examples
1L 2L
M
y
y
1v
2
v
1i 2i
dt
di
Ldt
di
Mv
dt
diM
dt
diLv
2
2
1
2
2111
!
!
dt
di
Ldt
di
Mv
dt
diM
dt
diLv
2212
2111
!
!
How could we determine dot markings if we dont know?
16-Apr-10
-
8/9/2019 Electromagnetism and magnetic circuit 3
9/12
CHETAN UPADHYAY 9
Series connection
1 2
M
(a)mutually coupled coils in
series-aiding connection
LT=L
1+L
2+2M
1 2
M
(b)mutually coupled coils in
seriesopposing connection
LT=L
1+L
2-2M
Total inductance
16-Apr-10
-
8/9/2019 Electromagnetism and magnetic circuit 3
10/12
CHETAN UPADHYAY 10
Parallel connection
L1 L2
I+
V
M
L1 L2
I+
V
M
(a)mutually coupled coils in
parallel-aiding connection(b)mutually coupled coils in
parallelopposing connection
MLL
MLLL
e221
2
21
!
Equivalent inductance
MLL
MLLL
e
221
2
21
!
16-Apr-10
-
8/9/2019 Electromagnetism and magnetic circuit 3
11/12
CHETAN UPADHYAY 11
Coefficient of coupling
21LL
Mk !
10 ee k
The coupling coefficient k is a measure of the magnetic
coupling between two coils
k < 0.5 loosely coupled;
k > 0.5 tightly coupled.
16-Apr-10
-
8/9/2019 Electromagnetism and magnetic circuit 3
12/12
CHETAN UPADHYAY 12
SUMMARY Mutual inductance, M, is the circuit parameter relating
the
voltage induced in one circuit to a time-varying current in
another circuit.
The coefficient of coupling, k, is the measure of the degree
of magnetic coupling. By definition, 0k1
The relationship between the self-inductance of each
winding and the mutual inductance between the windings
is
The dot convention establishes the polarity of mutually
induced voltage
Reflected impedance is the impedance of the secondary
circuit as seen from the terminals of the primary circuit,
or
vise versa.
21LLkM !
16-Apr-10
