Download 16.202: Magnetically Coupled Circuits: Mutual Inductance

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16.202: Magnetically Coupled Circuits: Mutual Inductance
• Background:
• Inductor: Passive element that resists change in current flow.
di
.
VL = L dt
• Consists of N turns of wire wound around a core material with
permeability µ;
2
• L = N lµA ; N : No. of Turns; A : Cross-Sectional Area; l :
Length.
• Amperes Law/ Biot-Savart : Current flowing in the coil creates
a magnetic field.
• A closed conducting loop carrying a current i when placed
in an external magnetic field, experiences a force. (causes a
torque/rotation) (Electric Motors)
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Prof.
K. Chandra, April 26, 2006
16.202: Circuit Theory II; ECE, UMASS Lowell
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• Symmetry Principle: (Faradays Law of Induction):
– Exerting a torque on a closed conducting loop in an external
magnetic field, causes electric current to flow in the loop
[Electric Generators]
– Induced EMF (Electromotive Force) appears when the number of magnetic field lines passing through the loop are
changing.
• Magnetic Flux: Number of Magnetic Lines that pass through
a surface bounded by the closed loop.
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Prof.
K. Chandra, April 26, 2006
16.202: Circuit Theory II; ECE, UMASS Lowell
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Mutual Inductance
φ11
+
i1 (t)
v1
+
φ12
L1
-
L2
v2
-
Figure 1:
• Self Inductances: L1 and L2 (N1 and N2 )turns.
• Current i1(t) flowing through L1 produces magnetic flux φ1
that consists of two components
φ11 : Links only L1
φ12 : Links both coils.
φ1 = φ11 + φ12
c
Prof.
K. Chandra, April 26, 2006
16.202: Circuit Theory II; ECE, UMASS Lowell
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• L1 and L2 are physically separated, but magnetically coupled.
dφ12
1 and v
=
N
• Induced voltages: v1 = N1 dφ
2
2
dt
dt
1 di1 = L di1
• In terms of source current i1(t): v1 = N1 dφ
1 dt
di1 dt
12 di1 = M di1
• v2 = N2 dφ
21 dt
di1 dt
1
Where L1 = N1 dφ
di1
12
Mutual Inductance M21 = N2 dφ
di1
• M21: Mut. Inductance of coil 2 with respect to coil 1. (H)
• Suffix 21 refers to induction in coil 2 due to current in coil 1
• Open-circuit mutual voltage (induced voltage): v2 = M21 didt1
c
Prof.
K. Chandra, April 26, 2006
16.202: Circuit Theory II; ECE, UMASS Lowell
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General Model
M
i1
i2
φ11
+
v1
φ21
φ12
L1
+
L2
-
v2
-
Figure 2:
di
• Finding polarity of mutual voltage M dt
.
• Depends on orientation of coil windings: Use Dot Convention
• Position of dot at one end of inductor gives direction of magnetic flux if current enters at the dotted terminal
c
Prof.
K. Chandra, April 26, 2006
16.202: Circuit Theory II; ECE, UMASS Lowell
6
• If current enters dotted terminal of one coil, reference polarity
of mutual voltage in the second coil is positive at the dotted
terminal of second coil
• If current leaves dotted terminal of one coil, reference polarity
of mutual voltage in the second coil is negative at the dotted
terminal of the second coil
M
i2
+
+
v1 = M di2 /dt
i2
v1 = −M di2 /dt
-
-
Figure 3:
c
Prof.
K. Chandra, April 26, 2006
16.202: Circuit Theory II; ECE, UMASS Lowell
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L2
L1
L1
i
i
L2
i
i
L = L1 + L2 − 2M
L = L1 + L2 + 2M
Figure 4:
c
Prof.
K. Chandra, April 26, 2006
16.202: Circuit Theory II; ECE, UMASS Lowell