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Electromechanical Dynamics (Part 1).0097

Electromechanical Dynamics (Part 1).0097 - not equal An...

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Lumped-Parameter Electromechanics (a) (b) Fig. 3.1.8 Illustration of energy and coenergy: (a) electrically linear system; (b) electrically nonlinear system. We can compare this result with what we find if we integrate (3.1.13) along the path of Fig. 3.1.7b to find the energy W~ = v(q', x) dq', (3.1.52) which from (3.1.49) is q 2 W =(q, 2) 2 ) (3.1.53) Now, when we use (3.1.49) to eliminate q from this expression, we see that the coenergy and energy are numerically equal. This is a consequence of the electrical linearity, as may be seen by observing Fig. 3.1.8a, in which (3.1.48) and (3.1.52) are the areas in the q'-v' plane indicated. (Remember that, by definition, in our system with one electrical terminal pair W,' + W, = qv.) When the areas are separated by a straight line (3.1.49), the integrals are obviously equal. On the other hand, when the areas are not separated by a straight line, the system is electrically nonlinear and energy and coenergy
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Unformatted text preview: not equal. An example of electrical nonlinearity is shown in Fig. 3.1.8b. Energy and coenergy have the same numerical values in an electrically linear system. We have, however, consistently made use ofthe energy expressed as a function of (q, x) or (A, x) and the coenergy expressed as a function of (v, x) or (i, x). These functions are quite different in mathematical form, even when the system is electrically linear [compare (3.1.50) and (3.1.53)]. A word of caution is called for at this point. A partial derivative is taken with respect to one independent variable holding the other independent variables fixed. In order for this process to be correct, it is easiest to perform the differentiation when the function to be differentiated is written without explicit dependence on dependent variables. To be more specific, consider ,X) A-PDF Split DEMO : Purchase from www.A-PDF.com to remove the watermark...
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