Prof. David R. Jackson ECE Dept. Fall 2014 Notes 32 ECE 2317 Applied Electricity and Magnetism 1

Mutual Inductance I1I1 I2I2 Current reference directions and unit normals are defined on both coils. Two coils are in proximity of each other. (The unit normals are each determined from the corresponding current reference directions, by the right-hand rule.) 2

Mutual Inductance (cont.) In general, if coil 2 has multiple turns: Coil 1 is energized Coil 2 is left open-circuited Define mutual inductance: (calculated when I 2 = 0 )  21 I1I1 Note: For the figure shown,  Hence M 21 < 0. 3

 12 I2I2 In general, if coil 1 has multiple turns, Define mutual inductance: Coil 2 is energized Coil 1 is left open-circuited Mutual Inductance (cont.) 4

A general property (proof omitted) is that both mutual inductance components are always equal: Mutual Inductance (cont.) 5

Circuit Law for Coupled Coils + - v1v1 i1i1 + - v2v2 i2i2 M L1L1 L2L2 Total flux through coil 1: Total flux through coil 2: 6

Example Find M 12, M 21 L s = length For M 12 : 7 R2R2 z R1R1 I2I2 I1I1 Bz2Bz2 R1R1 R2R2 x y

Example (cont.) For M 21 : 8 L s = length R2R2 z R1R1 I2I2 I1I1 Note: B z 1 = 0 for  > R 1. Bz1Bz1 R1R1 R1R1 R2R2 x y

Example (cont.) Summary 9 L s = length R2R2 z R1R1 I2I2 I1I1

Dot Convention 10 This allows us to use mutual inductance without having to visually inspect how the coils are wound. The dots tell us how the voltage on each coil is related to the current on the other coil. Note: We use passive sign convention on both coils. The dots tell us: (1) where the + sign is for the voltage drops (2) which currents to use for the M terms

Dot Convention (cont.) 11 Coil 1 Coil 2 M + - v1(t)v1(t) i2(t)i2(t) i1(t)i1(t) v2(t)v2(t) + - Here is one possible dot arrangement:

Dot Convention (cont.) 12 Here is another possible dot arrangement: Coil 1 Coil 2 M + - v1(t)v1(t) i2(t)i2(t) i1(t)i1(t) v2(t)v2(t) - +