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EE 205 Dr. A. Zidouri Electric Circuits II Mutual Inductance Lecture #23 -1- EE 205 Dr. A. Zidouri The material to be covered in this lecture is as follows: o o o o Review of Self Inductance Concept of Mutual Inductance Magnetic coupling Dot Convention -2- EE 205 Dr. A. Zidouri After finishing this lecture you should be able to: Understand the concept of Mutual Inductance Determine the polarity of Induced Voltages Write Mesh Equations for a Circuit with mutual Inductance -3- EE 205 Dr. A. Zidouri Self Inductance • A magnetic field consists of force surrounding the current carrying conductor. • The voltage induced in the conductor is proportional to the number of lines that collapse into or cut the conductor. d v dt Faraday’s Law is i + is the flux linkage and is measured in Wb –turns Magnetic Lines of force Φ v - N turns Fig.31-1 Representation of a magnetic field linking an N turns coil -4- EE 205 Dr. A. Zidouri Self Inductance (Cont) • The flux linkage is the product of magnetic field ( ) measured in (Wb) and the number of turns linked by the field (N). N The magnetic flux is related to the magnitude of the coil current by the relationship: pNi where p is the permeance of the space occupied by the flux N is the number of turns i is the current -5- EE 205 Dr. A. Zidouri Self Inductance (Cont) Assuming the core material is non-magnetic then: v N d d (N ) dt dt d d ( pNi ) N dt dt pN N L L is the self inductance, given by i 2 di di L dt dt -6- EE 205 Dr. A. Zidouri Example 31-1: If an inductor with 100 turns and a current of 5 A produces a flux of 10 Wb the flux linkage equals N = 100x10 = 1000 Wb The inductance is therefore: L N 1000 2 00 W b A i 5 -7- EE 205 Dr. A. Zidouri Concept of Mutual Inductance (Cont): Mutual Inductance (M): Circuit Parameter that relates the voltage induced in one circuit to a time-varying current in a neighboring circuit, measured in henrys (H). Consider Two Magnetically Coupled Coils M L1 L2 Fig31-2 Circuit of Two Magnetically Coupled coils The magnetic field of the first coil also passes through the second coil and induces a voltage across it as well. We say that the two coils are magnetically coupled -8- EE 205 Dr. A. Zidouri Concept of Mutual Inductance (Cont): Case I: Coil 1 is energized by a time –varying current source that establishes a current i1 in N1 turns. Coil 2 is not energized and open. The Coils are wound on a magnetic core. Ф21 Coil Current The flux produced by i1 can be divided into two components: o Ф11 is the flux produced by i1 that links only the N1 turns. o Ф21 is the flux produced by i1 that links N2 and N1 turns. Coil Number The Total flux linking coil 1 is Ф1= Ф11 + Ф21 is 1 p1 N1i1 11 p11 N1i1 Ф21 i1 v1 Ф11 Ф11 v2 N2 N1 Ф21 Fig31-3 Magnetically Coupled coils with coil 1 excited and coil 2 open. 21 p21 N1i1 p1 p11 p21 The Voltages V1 and V2 are v1 v2 d 1 di di p1 N12 1 L1 1 dt dt dt d 2 di1 di1 dt p21 N 2 N1 dt M 21 dt -9- EE 205 Dr. A. Zidouri Concept of Mutual Inductance (Cont): Case II: Coil 2 is energized by a time –varying current source that establishes the current i2 in N2 turns. Coil 1 is not energized and open. The Coils are wound on a magnetic core. The flux produced by i2 can be divided into two components o Ф22 is the flux produced by i2 that links only the N2 turns. o Ф12 is the flux produced by i2 that links N1 and N2 turns. The Total flux linking coil 2 is Ф2= Ф22 + Ф12 Ф12 Ф22 v2 N2 N1 Coil Current Coil Number 2 p2 N2i2 22 p22 N2i2 12 p12 N2i2 i2 v1 Ф12 is Ф22 Ф12 Fig31-4 Magnetically Coupled coils with coil 2 excited and coil 1 open. The Voltages V2 and V1 are v2 v1 d 2 di di p2 N 22 2 L2 2 dt dt dt d 1 di2 di2 dt p12 N1 N 2 dt M 12 dt - 10 - EE 205 Dr. A. Zidouri Concept of Mutual Inductance (Cont.) Therefore v 2 M 21 di 1 dt if i 2 0, i 1 0 v 1 M 12 di 2 dt if i 1 0, i 2 0 For Nonmagnetic materials, the permeance p12 and p21 are equal and Therefore M12 M 21 M - 11 - EE 205 Dr. A. Zidouri The Polarity of Mutually Induced Voltages (The Dot Convention) 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. When the reference direction for a current leaves the dotted terminal of a coil, the reference polarity of the voltage that it induces in the other coil is negative at its dotted terminal. - 12 - EE 205 Dr. A. Zidouri Example 31-2 Consider the Circuit of Fig.32-5: R1 M Given the circuit with the dotted markings as shown in Fig.31-5 (a) Vg Arbitrarily choose the reference direction for each current Fig.31-5 (b) Fig.31-5 (a) Two Magnetically Coupled Coils R1 Determine the polarity of the mutually induced voltage, using the dot convention rule stated earlier Fig.31-5 (c) i1 i2 R2 L2 L1 Fig.31-5 (b) Coil Currents i1 and i2 and Dot Marking to Indicate Polarity of Induced Voltages for Circuit of Fig.31-4 (a) R1 (Mesh 1) Vg (Mesh 2) M Vg Write the mesh equations for i1 and i2 di di vg R1i1 L1 1 M 2 0 dt dt di di R2i2 L2 2 M 1 0 dt dt R2 L2 L1 i1 i2 M di M dt di L dt di L2 2 dt M di1 dt R2 Fig.31-5 (c) The Self and Mutually Induced Voltages appearing across the coils shown in Fig.31-4 (a) - 13 - v(t) Time x(t) y(t) i(t) C ++– dom ain EE 205 Dr. A. Zidouri Self Test: Which is closest in meaning to large inductance? a) large flux b) low current c) large flux linkage d) large flux linkage per ampere answer: d If you double the number of turns but keep the flux and the current the same; the inductance will a) stay the same b) halve c) double d) triple answer: c - 14 -