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4-1). The difference in coupling is also true for the four-pole, two-phase machine (Fig. 4-2). Why? Although the above inductance matrices are rigorously derived in Chapter 2, we can get a “first look” from a simplistic consideration. We have previously defined the leakage, magnetizing, and mutual inductances; it is now the off-diagonal terms of Ls and Lr that are of concern. To explain these terms, let us first consider the coupling between the as and bs windings. They are displaced (2 / 3)π from each other.

The bulk of this energy is stored in the field (Wf), with a smaller amount transferred to the mechanical system, some of which is dissipated in the damper during the transient period while the remainder is stored in the spring. When the applied voltage is removed, the electrical and mechanical systems return to their original states. The change in Wm is small, increasing only slightly. Hence, during the transient period, there is an interchange of energy between the spring and mass that is finally dissipated in the damper.

3-65). 3-2. Electrostatic Force at Mechanical Input J fe (e f , x ) = ∑ e j =1 fj ∂q j ( e f , x )  ∂W f (e f , x )  − ∂x ∂x ∂Wc (e f , x ) fe (e f , x ) = ∂x ∂W f (q, x ) fe (q, x ) = − ∂x J  ∂e fj (q, x )  ∂Wc (q, x ) fe (q, x ) = − q j  + ∂x ∂x j =11  ∑ Note: For rotational systems, replace fe with Te and x with θ. 3-73) are readily verified by recalling the definitions of Wf and Wc that were obtained by holding x fixed (dx = 0). 3-1, the independent variables to be used are designated in each equation by the abbreviated functional notation.