Electromechanical Dynamics (Part 1).0098

Electromechanical Dynamics (Part 1).0098 - 3.1.2 A-PDF...

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Unformatted text preview: 3.1.2 A-PDF Split DEMO : Purchase Electromechanical Coupling from www.A-PDF.com to remove the watermark the capacitance C(x) of plane parallel plates with area A and spacing x (Fig. 3.1.2). Then C(x) - A and (3.1.51) gives f" 2x 2 (3.1.54) .A (3.1.55) The minus sign tells us that fO acts on the upper plate (node) in the (-x) direction. This we expect, for positive charges on the top plate are attracted by negative charges on the bottom plate. We can obtain the same result by using the energy and the translational form of (3.1.27). f = From (3.1.53) and (3.1.54) f" = 2Ae (3.1.57) W(q, x) ax (3.1.56) In view of (3.1.49) and (3.1.54) this result and (3.1.55) are identical. Suppose, however, that we blindly apply (3.1.56) to the energy of (3.1.53) with q replaced by Cv. The magnitude of the resulting force will be correct, but the sign will be wrong. For electrically nonlinear systems the magnitude of the force will also be wrong if the partial differentiation is not carried out correctly. The generalized force and coenergy equations are summarized in Table 3.1. This table is intended to illustrate the generality of the equations and their interrelations. The general equations are not recommended for use in solving problems. It is better to rederive the equations in each case to make certain that fundamental physical laws are satisfied. Equations (k) to (n) in Table 3.1 for evaluating energy and coenergy are written by using a path of integration that brings each electrical variable from zero to its final value in sequence j = 1 toj = N. 3.1.2c Reciprocity The mathematical description of a conservative electromechanical coupling system must satisfy a reciprocity condition that is a generalization of the reciprocity conventionally discussed in electric circuit theory.* To illustrate reciprocity for a simple example, consider the magnetic field system of Fig. 3.1.1 for which the terminal relations are expressed as derivatives of stored * E. A. Guillemin, Introductory Circuit Theory, Wiley, New York, 1953, pp. 148-150 and 429. _____ ...
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This note was uploaded on 02/10/2012 for the course MECE 4371 taught by Professor Liu during the Fall '11 term at University of Houston.

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