dt
diL??
§ 11-1 Mutual inductance
1i
11? 21?
2i
12? 22? 2212 ?? ?
?? )( t?
)(ti
L
1121 ?? ?
1L 2L
1L 2L
A current i1 at L1 produces an
open circuit voltage at L2.
2?
12121 iM??
A current i2 at L2 produces an
open circuit voltage at L1.
1?
21212 iM??
The mutual inductance,M12=M21=M(H)
The coefficient of coupling:
1
21
?? LLMk
)0,.( 212,1 ?? ??DCiiIf
?
?
2?
1i
2i
?
?
1?
1L 2L
M
1L 2L
M
dt
diM
dt
d 1
21212 ??
??
dt
diM
dt
d 2
12121 ??
??
coupling
0 0.2 0.4 0.6 0.8 1.0
V(R1:2,R1:1)
0V
50mV
100mV
150mV
V+
P A R A M E T E R S,
L 2 = 1 m
V-
T X 1
{ L 2 }
R1
100
V1
3 1 1 V a c
0 V d c
P A R A M E T E R S,
c o u p l i n g = 1
R2
1k
,dot convention”
2L
?
2i
1i
1L
?
If i1,i2 (entering two coils) produce flux linkages which
are increase each other,then the two terminals(1,2 or 1’,
2’) are defined as,dot terminals”(“same name
terminals”),
1i
11? 21?
1L 2L
? ?2i
12? 22?
1 '1 '22
V
??
1i
1L 2L
L1--source
L2--voltage-meter
K--close,--upscale.V
The terminal connected (+) polarity of the
source and the terminal connected (+) polarity
of are defined as, dot terminals”V
Determine,dot terminals”
?
?
Determine the sign of the mutual voltage
dt
diM 1
2 ??
dt
diM 1
2 ???
dt
diM 1
2 ???
dt
diM 1
2 ??
? ?
?
?
2?
1i M
?
?
?
?
2?
1i M
??
?
?
2?
1i M
?
?
?
?
2?
1i M
A current entering the dotted terminal of one coil produces
an open-circuit voltage between the terminals of the second coil
which is sensed in the direction indicated by a positive voltage
reference at the dotted terminal of this second coil.
If a nonzero current will be flowing in each of the coils.
dt
diM
dt
diL 12
22 ???
dt
diM
dt
diL 21
11 ??? dt
diM
dt
diL 21
11 ???
dt
diM
dt
diL 12
22 ????
At a real frequency (sinusoidal)?
2111
??? ?? IMjILjV ??
1222
??? ?? IMjILjV ??
1222
??? ??? IMjILjV ??
2111
??? ?? IMjILjV ??
2i
2L1L
?
?
1?
? ?
?
?
2?
1i M 2i
2L1L
?
?
1?
? ?
?
?
2?
1i M
Example 2,Find each mesh current.
?
?
?
?
?
?
?
????
????
????
???
???
????
06382
08169
2975
321
321
1321
I)j(IjIj
IjIjIj
VIjIjI)j(
123211 )(2)(75:1
?????? ????? VIIjIIjIm e s h
0)(2)(2)(61)(7:2 123232212 ????????? ????????? IIjIIjIIjIjIIjm e s h
0)(23)(6:3 21323 ????? ????? IIjIIIjm e s h
Example 4,the parallel combination of the mutual inductance coils.
)1(211 ??? ?? IMjILjV ??
)2(221 ??? ?? ILjIMjV ??
)3(21 ??? ?? III
2
1
2
1
LjMj
MjLj
LjV
MjV
I
??
??
?
?
?
?
?
?
??
??
?? V
MLL
MjLjI
22
21
2
12
??
??
????
??
????? V
MLL
MLLjIII
22
21
2
2121 )2(
??
?
)2( 21
22
21
2
MLLj
MLL
I
VZ
??
????
?
?
?
??
MLL
MLLL
eq 2'
21
2
21
??
??
MLL
MLLL
eq 2''
21
2
21
??
??
(a) in side (同侧 ):
(b) out of side (异侧 ),
1
?I
2
?I
?I
?
?
?
V 2L1L
? ?
M
?
??
?? V
MLL
MjLj
22212
2
??
??
eqLjMLL
MLLj '
221
221
?? ??? ??
diL??
§ 11-1 Mutual inductance
1i
11? 21?
2i
12? 22? 2212 ?? ?
?? )( t?
)(ti
L
1121 ?? ?
1L 2L
1L 2L
A current i1 at L1 produces an
open circuit voltage at L2.
2?
12121 iM??
A current i2 at L2 produces an
open circuit voltage at L1.
1?
21212 iM??
The mutual inductance,M12=M21=M(H)
The coefficient of coupling:
1
21
?? LLMk
)0,.( 212,1 ?? ??DCiiIf
?
?
2?
1i
2i
?
?
1?
1L 2L
M
1L 2L
M
dt
diM
dt
d 1
21212 ??
??
dt
diM
dt
d 2
12121 ??
??
coupling
0 0.2 0.4 0.6 0.8 1.0
V(R1:2,R1:1)
0V
50mV
100mV
150mV
V+
P A R A M E T E R S,
L 2 = 1 m
V-
T X 1
{ L 2 }
R1
100
V1
3 1 1 V a c
0 V d c
P A R A M E T E R S,
c o u p l i n g = 1
R2
1k
,dot convention”
2L
?
2i
1i
1L
?
If i1,i2 (entering two coils) produce flux linkages which
are increase each other,then the two terminals(1,2 or 1’,
2’) are defined as,dot terminals”(“same name
terminals”),
1i
11? 21?
1L 2L
? ?2i
12? 22?
1 '1 '22
V
??
1i
1L 2L
L1--source
L2--voltage-meter
K--close,--upscale.V
The terminal connected (+) polarity of the
source and the terminal connected (+) polarity
of are defined as, dot terminals”V
Determine,dot terminals”
?
?
Determine the sign of the mutual voltage
dt
diM 1
2 ??
dt
diM 1
2 ???
dt
diM 1
2 ???
dt
diM 1
2 ??
? ?
?
?
2?
1i M
?
?
?
?
2?
1i M
??
?
?
2?
1i M
?
?
?
?
2?
1i M
A current entering the dotted terminal of one coil produces
an open-circuit voltage between the terminals of the second coil
which is sensed in the direction indicated by a positive voltage
reference at the dotted terminal of this second coil.
If a nonzero current will be flowing in each of the coils.
dt
diM
dt
diL 12
22 ???
dt
diM
dt
diL 21
11 ??? dt
diM
dt
diL 21
11 ???
dt
diM
dt
diL 12
22 ????
At a real frequency (sinusoidal)?
2111
??? ?? IMjILjV ??
1222
??? ?? IMjILjV ??
1222
??? ??? IMjILjV ??
2111
??? ?? IMjILjV ??
2i
2L1L
?
?
1?
? ?
?
?
2?
1i M 2i
2L1L
?
?
1?
? ?
?
?
2?
1i M
Example 2,Find each mesh current.
?
?
?
?
?
?
?
????
????
????
???
???
????
06382
08169
2975
321
321
1321
I)j(IjIj
IjIjIj
VIjIjI)j(
123211 )(2)(75:1
?????? ????? VIIjIIjIm e s h
0)(2)(2)(61)(7:2 123232212 ????????? ????????? IIjIIjIIjIjIIjm e s h
0)(23)(6:3 21323 ????? ????? IIjIIIjm e s h
Example 4,the parallel combination of the mutual inductance coils.
)1(211 ??? ?? IMjILjV ??
)2(221 ??? ?? ILjIMjV ??
)3(21 ??? ?? III
2
1
2
1
LjMj
MjLj
LjV
MjV
I
??
??
?
?
?
?
?
?
??
??
?? V
MLL
MjLjI
22
21
2
12
??
??
????
??
????? V
MLL
MLLjIII
22
21
2
2121 )2(
??
?
)2( 21
22
21
2
MLLj
MLL
I
VZ
??
????
?
?
?
??
MLL
MLLL
eq 2'
21
2
21
??
??
MLL
MLLL
eq 2''
21
2
21
??
??
(a) in side (同侧 ):
(b) out of side (异侧 ),
1
?I
2
?I
?I
?
?
?
V 2L1L
? ?
M
?
??
?? V
MLL
MjLj
22212
2
??
??
eqLjMLL
MLLj '
221
221
?? ??? ??