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Mutual Inductance and Dot Convention in Coils

Learn about mutual inductance between two coils and how the dot convention is used to determine the voltage and current direction. Understand the application of these concepts in circuit analysis.

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Mutual Inductance and Dot Convention in Coils

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  1. ECE 3318 Applied Electricity and Magnetism Spring 2019 Prof. David R. Jackson Dept. of ECE Notes 33 Mutual Inductance

  2. Mutual Inductance Two coils are in proximity of each other. Current reference directions and unit normal vectors are defined on both coils. (The unit normal vectors are each determined from the corresponding current reference directions, by the right-hand rule for inductor flux.) Right-hand rule for inductor flux: Fingers are in the direction of the current Iin the coil, and the thumb gives the direction of the unit normal (the reference direction for the flux).

  3. Mutual Inductance (cont.) Coil 1 is energized Coil 2 is left open-circuited Meaning: “the flux through coil 2 due to coil 1” These two numbers go together. Define mutual inductance: Note: For the figure shown, 21<0 if I1 > 0. Hence, M21< 0. In general, if coil 2 has multiple turns:

  4. Mutual Inductance (cont.) Coil 2 is energized Coil 1 is left open-circuited Meaning: “the flux through coil 1 due to coil 2” These two numbers go together. Define mutual inductance: Note: For the figure shown, 12<0 if I2> 0. Hence, M12< 0. In general, if coil 1 has multiple turns:

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

  6. Circuit Law for Coupled Coils Total flux through coil 1: Total flux through coil 2: Faraday’s law: Faraday’s law: Note: We use passive sign convention on both coils. + - + -

  7. Example FindM12 , M21 Right-hand rule for inductor flux: For M12:

  8. Example (cont.) For M21: y Bz1 R1 R1 x Note: Bz1= 0 for  > R1. R2

  9. Example (cont.) Summary:

  10. Dot Convention This allows us to use mutual inductance M without having to visually inspect how the coils are wound. The dots tell us where to put the + sign for the voltage on one coil, and where the current enters the other coil. Note: The dot convention is chosen to be consistent with the passive sign convention, to be consistent with how we use self inductance, as you can see in the following figures.

  11. Dot Convention (cont.) Here is one possible dot arrangement: Coil 2 + - - + Coil 1

  12. Dot Convention (cont.) Here is another possible dot arrangement: + + Note: The M here is the negative of the M on the last slide (if the coils are the same). - - Coil 1 Coil 2

  13. Example Write down KVL phasor-domain mesh-current equations to describe this circuit. Note: M could be either positive or negative. + - Phasor voltage

  14. Example (cont.) + - - + + - + - + - - + - + - + - +

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