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 !
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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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Energy variables Energy variables are defined as in the scalar case as time integrals of the flow and effort vectors respectively. generalized displacement q = f dt + q o generalized momentum p = e dt + p o M ULTI -P ORT C APACITOR A “vectorized” or multivariable generalization of the one-port capacitor. definition A multiport capacitor is defined as an entity for which effort is a single-valued (integrable) function of displacement . e = Φ ( q ) The vector function Φ (·) is the capacitor constitutive equation. —a vector field (in the mathematical sense of the word). A multi-port capacitor is sometimes called a C-field or capacitive field.
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B OND GRAPH NOTATION By convention, power is defined positive into all ports. C f 1 = dq 1 /dt e 2 f 2 = dq 2 /dt f 3 = dq 3 /dt e 1 e 3 An alternative notation: C n n denotes the number of ports. More on this multi-bond notation later. S TORED ENERGY : determined by integrating the constitutive equation. E p - E po = e t f dt = e t d q = Φ ( q ) t d q = E p ( q ) potential energy, as it is a function of displacement —a function of as many displacements as there are ports.
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C OUPLING BETWEEN PORTS . Each effort may depend on any or all displacements. e i = Φ i (q 1 ,q 2 ,... q n ) all i This coupling between ports is constrained. Mathematically: Energy stored is a scalar function of vector displacement. Stored energy is a scalar potential field. The effort vector is the gradient of this potential field. e = q E p ( q ) Therefore the constitutive equation, Φ (·), must have zero curl.
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.141 taught by Professor Nevillehogan during the Fall '06 term at MIT.

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multiport_capaci - ENERGY-STORING COUPLING BETWEEN DOMAINS...

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