Prof. David R. Jackson ECE Dept. Fall 2014 Notes 15 ECE 2317 Applied Electricity and Magnetism 1.
Prof. D. Wilton ECE Dept. Notes 27 ECE 2317 Applied Electricity and Magnetism Notes prepared by the...
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Prof. D. Wilton ECE Dept. Notes 27 ECE 2317 ECE 2317 Applied Electricity and Applied Electricity and Magnetism Magnetism Notes prepared by the EM group, University of Houston.
Transcript of Prof. D. Wilton ECE Dept. Notes 27 ECE 2317 Applied Electricity and Magnetism Notes prepared by the...
Prof. D. WiltonECE Dept.
Notes 27
ECE 2317 ECE 2317 Applied Electricity and MagnetismApplied Electricity and Magnetism
Notes prepared by the EM group,
University of Houston.
Mutual InductanceMutual Inductance
1n
I1
2n
I2
Two coils are in proximity of each other.
Current reference directions and unit normals are defined on both coils
(The unit normals are determined from the current reference directions, by the right-hand rule.)
Mutual Inductance (cont.)Mutual Inductance (cont.)
2
1 221
S
B n dS
2n
1n
21I1
In general, if coil 2 has multiple turns
2121
1
MI
21 2 2121
1 1
NM
I I
Coil 1 is energized. Coil 2 is left open-circuited
Define mutual inductance:
1
2 112
S
B n dS
2n
1n
12 I2
In general, if coil 1 has multiple turns,
1212
2
MI
12 1 1212
2 2
NM
I I
Define mutual inductance:
Coil 2 is energized. Coil 1 is left open-circuited
Mutual Inductance (cont.)Mutual Inductance (cont.)
Property: 12 21M M M
Mutual Inductance (cont.)Mutual Inductance (cont.)
Circuit Law for Coupled CoilsCircuit Law for Coupled Coils
1 21 1 12
2 12 2 21
di div L M
dt dtdi di
v L Mdt dt
+- v1
i1
+- v2
i2
M
L1L2
1 11 12
1 1 12 2L i M i
2 22 21
2 2 21 1L i M i
total flux through coil 1:
total flux through coil 2:
ExampleExample
1 1
12 1 1212
2 2
2 112 2
22 1
2 220 2 2 1 0 2 1
z
S S
z
s
NM
I I
B n dS B dS
B R
Nn I R I R
L
2212 1 0 1 H
s
NM N R
L
Bz2
R1
Find M12 , M21
Ls = length
R2
z
R1
I2
I1
Example (cont.)Example (cont.)
2 2
21 2 2121
1 1
21 221 1 1 1
2 2 11 0 1 1 1 0 1
z z
S S
s
NM
I I
B n dS B dS R B
NR n I R I
L
Bz1
R1
2 121 2 0 1
s
NM N R
L
R2
z
R1
I2
I1
Example (cont.)Example (cont.)
2212 1 0 1 H
s
NM N R
L
2 121 2 0 1
s
NM N R
L
2
112 21 0 1 2
s
RM M N N
L
R2
z
R1
I2
I1
Force on WireForce on Wire
charge:
wire:
F q v B
C
F I d B
qB
v
F
B
Idl
FThe contour C is in the direction of the reference direction for the current.
d dqdq v dq d Id
dt dt
ExampleExample
I2
I1
1
2
z
h
z = 0 z = L
x
2
2
2 12
12 0 2
C
C
F I z dz B
II z dz y
h
Note: there is no self-force on wire 2 due to the magnetic field produced by the current on wire 2.
Example (cont.)Example (cont.)
F
F
1 22 0
02
LI IF x dz
h
2
12 2 0 2C
IF I z dz y
h
1 22 0 N
2
I IF x L
h
