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multiport_capaci

# multiport_capaci - ENERGY-STORING COUPLING BETWEEN DOMAINS...

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ENERGY-STORING COUPLING BETWEEN DOMAINS M ULTI -P ORT E NERGY S TORAGE E LEMENTS Context: examine limitations of some basic model elements. E XAMPLE : open fluid container with deformable walls P = ρ g h h = A V V = C f P where C f = A ρ g —fluid capacitor But when squeezed, h (and hence P) may vary with time even though V does not. Seems to imply C f = C f (t) i.e., C f = A(t) ρ g —apparently a “modulated capacitor”

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P ROBLEM ! E p = V 2 2 C f V is constant, therefore no (pressure) work done dV = 0 PdV = 0 —yet (stored) energy may change This would violate the first law (energy conservation) P V where did this energy come from? initial stored energy —a BIG problem! M ODULATED ENERGY STORAGE IS PROSCRIBED !
S OLUTION Identify another power port to keep track of the work done to change the stored energy C P Q = dV/dt v = dx/dt F introduces a new network element: a multiport capacitor Mathematical foundations: Power variables: Each power port must have properly defined conjugate power and energy variables. Net input power flow is the sum of the products of effort and flow over all ports. P = i e i f i In vector notation: e _ _ e 1 e 2 · · · e n f _ _ f 1 f 2 · · · f n P = e t f = f t e

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