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Mutual Inductance
Section 41 described mutual inductance as a parameter that is associated with the flux linkage produced in one circuit by the current in another circuit.
Quantitatively, the mutual inductance between two circuits may be defined as the flux linkage produced in one circuit by a current of one ampere in the other circuit. In this discussion a circuit is considered as having one or more turns. Figure 44 shows a schematic diagram of two circuits that are coupled inductively, i.e., magnetic flux from one circuit links the other circuit. A current i_{1} in circuit 1 produces magnetic flux, some of which links some of the turns of circuit 2 and thus induces a voltage in circuit 2 when i_{1} changes. Let λ_{11} = flux linkage of circuit 1 produced by its own current i_{1} The unit of flux linkage is the weber turn and that of inductance is the henry, which can be expressed as a weber turn per ampere. Hence
The two circuits and the effect of magnetic coupling between them can be represented by the equivalent magnetic circuit and the windings N_{1} and N_{2} in Fig. 44(b), where N_{1} and N_{2} are the numbers of turns in circuits 1 and 2. Three air gaps having reluctances R_{1}, R_{2} and R_{1} are shown in the magnetic circuit, the shaded portion of which is assumed to have infinite permeability and, consequently, zero reluctance. The resistivity of the material in the shaded portions is considered infinite so that no eddy currents can flow. Assume a current of i_{1} to flow in circuit 1 while the current in circuit 2 is zero. Let Φ_{11} be an equivalent flux, which, in linking all N_{1} turns of circuit 1, produces the flux linkage λ_{11}. Then
In Fig. 44(b) the equivalent flux Φ_{11} is shown as having two components, Φ_{21} and Φ_{11}, in which Φ_{21} is the equivalent flux that, in linking all N_{2} turns of circuit 2, produces the flux linkage λ_{21}. Therefore
The remaining component Φ_{11} is the equivalent leakage flux of circuit 1 linking all N_{1} turns without linking any of the N_{2} turns of circuit 2. This means that
The magnetic circuit of Fig. 44(b) is comparable to the electric circuit of Fig. 44(c), which is the equivalent Tcircuit for any 3terminal arrangement of pure resistances. The resistances R_{1}, R_{2}, and R_{1} correspond to the reluctances R_{1}, R_{2} and R_{1} of the magnetic circuit, whereas the applied voltage V_{1}, is analogous to the mmf N_{1}i_{1} produced by the N_{1}turn winding. Therefore, it is evident, on the basis of elementary electric circuit theory, that
and that resulting flux linkage produced in circuit 2 by i_{1} is
The mutual inductance based on the flux linkage with circuit 2 due the current in circuit 1 is therefore
The mutual inductance L_{12}, based on the flux linkage with circuit 1 due the current in circuit 2, can be found by considering a current i_{2} in circuit 2 while the current i_{1} in circuit 1 is zero, and then going through the process used previously, or simply by interchanging the subscripts 1 and 2 in Eq, 443. This results in
Equations 443 and 444 show the mutual inductance between two electric circuits to be reciprocal when the circuits are coupled by a homogeneous magnetic medium of constant permeability, i.e.
In cases where there are only two magnetically coupled circuits, the letter M is used to represent mutual inductance, i.e.
Coefficient of coupling The mutual inductance between two circuits can be expressed in terms of their selfinductances L_{11}, L_{22} and their coefficient of coupling k, which is a function of the reluctance of the leakage flux path. It can be seen from Eq. 444 that for given values of R_{1} and R_{2} the mutual inductance increases with the reluctance R_{1} of the leakage path, becoming a maximum when R_{1} approaches infinity, the condition for perfect magnetic coupling between the two circuits. Let
and
from which the coefficient of coupling is, by definition
The selfinductance of circuit 1 is
and that of circuit 2 is
The relationship between the mutual inductance and the selfinductances is given by the ratio, based on Eqs. 444, 449, and 450.
A comparison of Eqs. 448 and 451 shows the mutual inductance to be
The coefficient of coupling k cannot exceed unity, although values as high as 0.998 are not unusual in ironcore transformers, whereas k in aircore transformers is generally smaller than 0.5. Energy in the field of coupled circuits The electromagnetic differential energy supplied to an electric circuit carrying a current i is given by
It should be remembered that electric energy over and above that required to supply the irreversible energy dissipated in the form of heat is required to maintain current in a circuit that is subjected to a variation of flux linkage. This is true whether the flux linkage λ is produced only by the current i in the circuit itself, or whether it is produced only by magnetic fields from other sources, or by fields from its own current in combination with those from other sources. The electromagnetic differential energy input to two magnetically coupled circuits is
where
from which
If there is no motion of one circuit relative to the other, and if there is no change in the configuration of the magnetic circuit, the selfinductances L_{11} and L_{22} as well as the mutual inductance M are constant, and no mechanical energy is supplied to or abstracted from the field. It is assumed, of course, that the inductances are independent of the currents i_{1} and i_{2} i.e., that the magnetic circuit is linear. Equation 456 can, therefore, be reduced to the form
Since this is also the differential energy stored in the field
and the energy present in the field due to the currents i_{1} and i_{2} is
Although the inductances were assumed constant while the currents were increased from zero to i_{1} and i_{2} in circuits 1 and 2, Eq. 458 expresses the energy stored in the field for the particular values of L_{11}, L_{22}, M, i_{1} and i_{2} whether the inductances are constant or variable. This means that the energy stored in the field, for given values of L_{11}, L_{22}, M, i_{1} and i_{2}, is unaffected by previous values these inductances might have had. The development that led to Eq. 458 can be generalized to include n instead of two magnetically coupled circuits expressing the energy stored in the field by


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