ENERGY-STORING COUPLING BETWEEN DOMAINSMulti-Port Energy Storage ElementsExample:—fluid capacitor—apparently a “modulated capacitor” Problem!This would violate the first law (energy conservation)—a BIG problem!Modulated energy storage is proscribed!SolutionMathematical foundations:Power variables:In vector notation:Energy variablesgeneralized displacementgeneralized momentumMulti-Port CapacitordefinitionBond graph notationStored energy:Coupling between ports.Mathematically:Coupling between ports must be symmetric. This is known as Maxwell's reciprocity condition.EXAMPLE: “CONDENSER” MICROPHONEHeads up!There’s an error in what follows — see if you can spot it.—essentially a moving-plate capacitor. Electrically a capacitor, but capacitance varies with plate separation. Mechanically, electric charge pulls the plates together. Device can store energy.Energy can change in two ways: mechanical displacementcharge displacement—a two-port capacitorlike a spring in the mechanical domainlike a capacitor in the electrical domainTwo constitutive equations neededboth relate effort to displacementAssuming electrical linearity:One constitutive equation is An easier way: energy is the same in all domains.compute energy in the electrical domaingradient with respect to plate separation = forceStored electrical energy:Gradient:ENERGY AND CO-ENERGYThere are two ways to integrate the capacitor constitutive equation.—only one of them is energy—the other is co-energyenergy:co-energy: The error was to confuse energy with co-energyStored electrical energy:gradient of energy with respect to plate separation:Sign error corrected, but ...this equation implies force is independent of plate separation. Is that physically reasonable?Shouldn’t electrostatic attraction weaken as plate separation increases? Would the plates pull together just as hard if they were infinitely far apart?A paradox?Cross-check:are the two constitutive equations reciprocal (symmetric)?partial derivativesWhat’s wrong? A “paradox” resolved:A clue: the electrical constitutive equationvoltage drop increases with plate separation. for a fixed charge, infinite separation requires infinite voltage.—not the usual arrangementreal devices cannot sustain arbitrarily large voltages.Change “boundary conditions” to input voltage:For fixed voltage, force between plates declines sharply with separation. —much more plausibleMechanically, a spring—albeit a highly nonlinear one. Key point:Boundary conditions profoundly influence behavior Causal assignmentdisplacement in, effort out on both ports—Integral causalityEnergy function:voltage in, charge out on the electrical port.—differential causality on the electrical port.Co-energy function:CO-ENERGY AND LEGENDRE TRANSFORMATIONSThis is the negative of a Legendre transformationCommonly used in thermodynamicsSingle-port capacitor:Two-port capacitor:Apply to the condenser microphoneEnergy:Legendre transform:SubstituteCo-energy:Note: mechanical force is the negative gradient of electrical co-energy with respect to displacement.—That fixes our sign error.Comment:REMARKS:Even with the idealizing assumptions above(no electrical saturation, no “fringe effects” in the electrostatic field)the multi-port constitutive equations are profoundly nonlinearfundamentally coupledThe condenser microphone is not well modeled by one-port energy storage elements in either the mechanical or the electrical domains.Because of inter-domain coupling, this device serves is both a sensor (a microphone) an actuator (a speaker)—It is commonly used for both purposes.It is an example of an energy-storing transducer.Energy may be stored or removed from either domainThus energy and power may be transferred between domains.INTRINSIC STABILITYMaxwell’s reciprocity constraintSTABILITYMathematically:Example: Condenser microphone (revisited)EnergyInverse capacitanceStability—Unstable for non-zero chargePHYSICALLY REASONABLE — stability requires something to oppose the attractive electrostatic force.Include elasticity of the supporting structureAssume elastic linearity (for simplicity)Energy (revised)Constitutive equationInverse capacitance (revised)Stability (revised)Sufficient condition for stability:PHYSICAL INTERPRETATIONWhy does the mechanical stiffness have to be any larger? Consider each sufficient condition in turnk > 0 means that increasing displacement requires increasing applied force—provided charge remains constant > 0 means that increasing charge requires increasing applied voltage—provided displacement remains constant Howeverincreasing charge requires increasing applied voltage—when both charge and displacement are free to changeENERGY-STORING COUPLING BETWEEN DOMAINS MULTI-PORT ENERGY STORAGE ELEMENTS Context: examine limitations of some basic model elements. EXAMPLE: open fluid container with deformable walls P = ρ g h h = A V V = Cf P where Cf = Aρ g —fluid capacitor But when squeezed, h (and hence P) may vary with time even though V does not. Seems to imply Cf = Cf(t) i.e., Cf = A(t)ρ g —apparently a “modulated capacitor”PROBLEM! Ep = V22 Cf 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) PVwhere didthis energycome from?initial stored energy —a BIG problem! MODULATED ENERGY STORAGE IS PROSCRIBED!SOLUTION Identify another power port to keep track of the work done to change the stored energy CPQ = dV/dtv = dx/dtF 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 = ∑iei fi In vector notation: e ∆__ ⎣⎢⎢⎡⎦⎥⎥⎤e1e2···en f ∆__ ⎣⎢⎢⎡⎦⎥⎥⎤f1f2···fn P = et f = ft eEnergy 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 + qo generalized momentum p = ⌡⌠e dt + po MULTI-PORT CAPACITOR 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