Input Electric Energy We. The input electrical energy can be calculated by integrating the coil voltage and current over time, as follows.
Substituting Faraday's law, Eq. (1.20), into Eq. (1.33) shows that the input electrical energy is equal to the product of the coil magnetizing current I and the flux linkage X.
Stored Magnetic Field Energy. As can be seen in Fig. 1.5, the flux linkage X is a function of the magnetizing current I and depends on the material properties or the magnetization curve. The stored magnet field energy is calculated by integrating Eq. (1.34) over the magnetization curve. By inspection of Eq. (1.34), the area of integration lies above the magnetization curve, as shown in Fig. 1.6.
The stored magnetic field energy can be calculated for linear materials by substituting Eq. (1.31) into Eq. (1.34) as follows. Linear materials are characterized by a constant value of inductance L or permeability |i.
where
FIGURE 1.6 Stored magnetic field energy and magnetic coenergy.
FIGURE 1.7 Graphical visualization of electromechanical energy conversion.
The change in electric energy and the change in stored magnetic field energy are defined as follows.
The electric energy and the stored magnetic field energy for states 1 and 2 can be obtained by using the applicable regions above and below the magnetization curve designated as A, B, C, D, E, F, and G in Fig. 1.7.
Electric energy:
Stored magnetic field energy:
The resulting change in the mechanical energy is obtained by rewriting the energy balance, Eq. (1.32), and substituting the results from Eqs. (1.52) and (1.56), as follows.
The magnetic coenergy for states 1 and 2 can also be obtained by using the applicable regions below the magnetization curve from Fig. 1.7, as follows.
A comparison of the results from Eqs. (1.59) and (1.63) shows that the change in mechanical energy is equal to the change in magnetic coenergy.
The mechanical force can be calculated as follows, by substituting Eq. (1.38) into
Eq. (1.65).
The mechanical torque can be calculated using Eq. (1.66) and the radius r that relates the force F, the torque T, the linear displacement X, and the angular displacement 6.
