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Download 16.202: Magnetically Coupled Circuits: Mutual Inductance
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1 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) c Prof. K. Chandra, April 26, 2006 16.202: Circuit Theory II; ECE, UMASS Lowell 2 • 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. c Prof. K. Chandra, April 26, 2006 16.202: Circuit Theory II; ECE, UMASS Lowell 3 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 4 • 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 5 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 7 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