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Torque and Force in Inductively Coupled Circuits
It is evident from Eqs. 458 and 459 that the energy stored in the magnetic field associated with two or more magnetically coupled circuits can be varied by changing the current in one or more of the circuits, or by changing one or more of the selfinductances or mutual inductances. If all currents are held constant, mechanical work is involved in changing the stored energy; this necessitates a change in one or more of the inductances. The rotary electromagnet in Fig. 45 illustrates two inductively or magnetically coupled circuits and in which the reluctance of the iron is considered negligible in comparison with that of the air gaps between the stator and the rotor. When the currents i_{1} and i_{2} are flowing in circuits 1 and 2 in the directions shown, and with the angle θ smaller than 90°, the magnetic forces develop torque in the counterclockwise direction. Mechanical energy input is required to increase θ. This means that torque must be applied counterclockwise. If the angle θ is increased by the differential amount dθ, and the applied torque is dθ, then the mechanical differential energy input is
If, at the same time, there is an electromagnetic differential energy input dW_{em}, the energy stored in the field will be increased by the differential amount
When the differentiation in Eq. 456 is performed, the expression for the electromechanical differential energy becomes
It should be noted that the inductances L_{11}, L_{22}, and M are not constant but are functions of the angle θ. The energy stored in the field is given by Eq. 458 and it will be remembered that the energy stored in the field can be changed by a change in one or both of the currents as well as by a change in one or all of the inductances, so that the differential energy input to the magnetic field can be written as
When Eq. 463 is subtracted from 462, then, on the basis of Eq. 461, the mechanical differential energy input to the energy stored in the field is found to be
and, according to Eq. 460, the applied torque must be
The magnet develops a torque equal and opposite to the applied torque, that is
Similarly the developed force for a linear displacement is given by
When there are ncoupled circuits instead of 2coupled circuits, and there is only one movable member in the electromagnet, as in the case of conventional motors and generators, the torque is found on the basis of Eq. 459 to be
If the change in inductance takes place while the currents are held constant, onehalf of the electromagnetic energy input is stored in the magnetic field, whereas the other half is converted into mechanical energy. This is evident from Eqs. 463 and 464 because for constant currents, di_{1} and di_{2} must be zero. This equal division of stored energy and mechanical energy with constant currents is true only for linear circuits and does not hold for nonlinear magnetic circuits, i.e., where there is appreciable saturation. On the other hand, if the flux linkages are held constant while there is a change of inductance, the electromagnetic energy input must equal zero as is shown by Eq. 454 for dλ_{1} and dλ_{2} = 0. In that case energy stored in the field is given up to be converted into mechanical energy, i.e., the mechanical energy is abstracted from the energy absorbed in the field.


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