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Home Inductance  Electromagnetic Energy Conversion Force and Torque in Variable Inductance Circuits  
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Force and Torque in a Circuit of Variable SelfInductance
The energy stored in an inductance, the magnetic circuit of which has constant permeability, is expressed by
which is true regardless of whether the inductance is constant or varies with time. The power required to change the energy stored in the field is related to the inductance and current as follows
It follows from Eq. 419 that the differential gain in reversible energy is
The expression for the electrical power input to the circuit during this change in the stored energy is
and the differential electrical energy input therefore must be
Equation 420 expresses the part of the electrical energy input that is stored in the magnetic field. In addition there is the energy that is converted directly into heat, namely Ri^{2} dt and that which is converted into mechanical energy. The mechanical energy output then is, as in the electromagnet of Fig. 325, f dx. The electrical differential energy input must therefore satisfy the following relationship
Equations 422 and 423 express the same amount of differential energy, and when equated to each other yield the following results
from which the expression for the force becomes
In Eq. 425, f is the force produced by the system. This means that when dL/dx is positive, i.e., if the inductance increases with displacement, motor action results. On the other hand, a decrease in the inductance with displacement makes dL/dx negative and an external force is required resulting in generator action. If the length of the air gap in the electromagnet shown in Fig. 325 is increased, it is necessary to apply a positive external force. An increase in the length of air gap corresponds to a decrease in the inductance, and dL/dx is then negative. Torque
In the case of rotary motion the mechanical energy is given by the product of torque and angular displacement as compared with that offeree and linear displacement for linear motion. Figure 42 shows a schematic diagram of a magnetic circuit with a winding on the stator, and with the magnetic axis of the rotor displaced from that of the stator by an angle Θ. The developed torque can therefore be related to the current and selfinductance by replacing the quantity f dx in Eq. 424 with TdΘ, where T = torque in newton meters This results in


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