# Mutual Inductance

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## mutual inductance

[′myü·chə·wəl in′dək·təns]
(electromagnetism)
Property of two neighboring circuits, equal to the ratio of the electromotive force induced in one circuit to the rate of change of current in the other circuit.

## Mutual Inductance

a quantity that characterizes the magnetic coupling between two or more electric circuits. If there are two conducting circuits, (1) and (2), some of the lines of magnetic induction generated by the current flowing through the first circuit will penetrate the area within the second circuit (see Figure 1)—that is, they will be coupled with the second circuit. The magnetic flux Φ12 through circuit (2) generated by the current I in circuit (1) is directly proportional to the current:

Φ12 = M12I1

The proportionality coefficient M2 depends on the dimensions and form of circuits (1) and (2), on the distance between the circuits, on their relative positions, and on the magnetic permeability of the surrounding medium. This coefficient is called mutual inductance or the coefficient of mutual induction. In the SI system of units, mutual inductance is measured in henrys.

Figure 1

If current I2 flows through circuit (2), the magnetic flux Φ12 through the area of circuit (1) is also proportional to the current:

Φ21 = M21I2

where M21 = M12.

The existence of a magnetic coupling between the circuits is manifested by the fact that, upon a change of current in one circuit, an induced electromotive force appears in the adjacent circuit. According to the law of electromagnetic induction,

where E2 and E1 are induced electromotive forces arising in circuits (2) and (1) and ΔΦ]2 and ΔΦ21 are the change in magnetic fluxes through the respective circuits over time Δt.

The mutual energy W12 of the magnetic field is expressed with the aid of mutual inductance for the currents I1 and I2 as follows:

Wl2 = ±M12I1I2

### REFERENCE

Kalashnikov, S.G. Elektrichestvo. Moscow, 1970. Chapter 10. (Obshchiikurs fiziki, vol. 2.)

G. IA. MIAKISHEV

References in periodicals archive ?
(iv) [M.sub.22]([nS.sub.v]) is the mutual inductance between two turns of a rectangular small loop separated between them at a distance of [nS.sub.v];
It can be seen from (3) that, regardless of the value of three mutual inductances [M.sub.1], [M.sub.2], and [M.sub.3], "dead spot" never exists compared to the traditional symmetrical magnetic circuit, ensuring that the load on any position can achieve efficient power transmission.
(a) Conversion of mutual inductance; and (b) internal node elimination process.
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In the present approach we calculate the electromagnetic interactions using conventional and well-tested approximation formulae for inductances and mutual inductances, which do not require iterating.
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MAIN PARAMETERS OF THE ANALYZED PMSM Outer Rotor Characteristics Rated power 1 kW Rated torque 23 Nm Rated speed 379 r/min Outer diameter 220 mm Axial length 36 mm Air gap length 1 mm Number of phases 5 Poles pairs 26 Rated phase voltage 18 V Max current 10 A Rated electrical frequency 164 Hz Resistance per phase 0.1 [ohm] Self-inductance per phase 107 [micro]H Mutual inductance between adjacent phases 15 [micro]H Mutual inductance between nonadjacent phases 10 [micro]H Number of stator slots 55 Number of coils per phase 11 Number of turns per coil 10 Number of wires in parallel per turn 10 Number of magnets in the rotor 52 Permanent magnets material SmCo Permanent magnets remanence 0.8 T TABLE II.
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For mutual inductances between rotor windings and stator phases, the stator reference frame is used.
The coupling coefficient changes continuously with the change of rotor position [[theta].sub.r] and machine model is described by differential equations with time varying mutual inductances. To simplify the problem solution, any three phase induction machine can be represented by an equivalent two phase machine with stator direct and quadrature axes [d.sup.s] - [q.sup.s] and rotor direct and quadrature axes [d.sup.r] - [q.sup.r] [1, 2].

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