Force and Torque in a Circuit of Variable Self-Inductance

The energy stored in an inductance, the magnetic circuit of which has constant permeability, is expressed by

[4-18]

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

[4-19]

It follows from Eq. 4-19 that the differential gain in reversible energy is

[4-20]

The expression for the electrical power input to the circuit during this change in the stored energy is

[4-21]

and the differential electrical energy input therefore must be

[4-22]

Equation 4-20 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 Ri² dt and that which is converted into mechanical energy. The mechanical energy output then is, as in the electromagnet of Fig. 3-25, f dx. The electrical differential energy input must therefore satisfy the following relationship

[4-23]

Equations 4-22 and 4-23 express the same amount of differential energy, and when equated to each other yield the following results

[4-24]

from which the expression for the force becomes

[4-25]

In Eq. 4-25, 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. 3-25 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

Figure 4-2. Rotary electromagnet

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 4-2 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 self-inductance by replacing the quantity f dx in Eq. 4-24 with TdΘ,

where

T = torque in newton meters
Θ = angular displacement in radians

This results in

[4-26]