電磁閥 資料.doc

CO-ENERGY (AGAIN)In the linear case, energy and co-energy are numerically equal.the value of distinguishing between them may not be obvious.Why bother with co-energy at all?EXAMPLE: SOLENOID WITH MAGNETIC SATURATION. Previous solenoid constitutive equations assumedelectromagnetic linearity.arbitrarily large magnetic fluxes could be generated.In reality flux cannot exceed saturation flux.For sufficiently high currents behavior is strongly nonlinear.Solenoid & co-energypage 1 Neville HoganMODEL THAT PHENOMENON.Assume an electrical constitutive equation as follows:where L(x) is position-dependent inductance as before.For sufficiently small currents the relation is approximately linear.For sufficiently large currents the flux linkage reachesa limiting value, s.Solenoid & co-energypage 2 Neville HoganMECHANICAL CONSTITUTIVE EQUATIONmay be found using the stored energy.Find the stored energy at a fixed displacement.Find force as the gradient with respect to displacement.That yields the relation between force and flux linkage.F = F(, x)But flux cannot be specified arbitrarily.Realistic boundary conditions require current input.To find the relation between force and current, substitute.= (i, x)Solenoid & co-energypage 3 Neville HoganSTORED ELECTRICAL ENERGY(at a fixed displacement)Need to invert the relation between flux linkage and current.In general, anything but straightforward.In this case, a little algebra yields the following.USE WITH CAUTION!If s this expression yields an imaginary number for the current.Solenoid & co-energypage 4 Neville HoganA LITTLE CALCULUS(and some more algebra) yields an expression for energy.Partial derivative with respect to displacement: ls2 l 2Substitute for flux linkage:Still more algebra simplifies this expression:Solenoid & co-energypage 5 Neville HoganCOMMENT:The assumed electrical constitutive equationwas chosen primarily for pedagogic simplicitya more realistic relation may be (far) less tractableusually no simple algebraic form existsConsequently a procedure requiring(a) inversion(b) integration with respect to flux linkage and(c) differentiation with respect to displacement may be impractical.Solenoid & co-energypage 6 Neville HoganAN ALTERNATIVE APPROACH:use co-energy instead of energy. Total stored energy:Electrical co-energy: (a Legendre transformation with respect to current) Force is the negative gradient of this co-energy.Solenoid & co-energypage 7 Neville HoganElectrical co-energy at a fixed displacement: KEY POINT:Co-energy may be found without inverting therelation between flux linkage and current.Partial differentiation with respect to displacement: No further algebra required.Solenoid & co-energypage 8 Neville HoganREMARKSIn general, co-energy functions are useful formultiport and/or nonlinear energy storage elements.Inverting constitutive equations may be avoided.In the linear case, energy and co-energy are numerically equal.the value of distinguishing between them may not be obvious.In the nonlinear case, the two are not equal. Distinguishing between them is important.In this example,energy is upper-bounded co-energy is not.In general,energy is conservedco-energy need not be.Solenoid & co-energypage 9 Neville HoganNOTE 1:* Integration to find energy:where y is a “dummy variable”.rearrange:substitution: defineSolenoid & co-energypage 10 Neville HoganDoes that make sense?In the limit as approaches sSolenoid & co-energypage 11 Neville HoganPlausible; implies that only a finite amount of energy maybe stored.In the limit as approaches 0, binomial series expansionof the root yields higher order termsas expected for a linear inductor.Solenoid & co-energypage 12 Neville HoganNOTE 2:* Integration to find co-energy:Does this make sense?In the limit asL(x) i sE* s iThis is the area of a rectangle of sides i and s. Note thatco-energy may increase without bound, whereas energy may not.In the limit asSolenoid & co-energypage 13 Neville HoganL(x) i sseries expansion of the square root + higherorder termsas expected for a linear inductor.Solenoid & co-energypage 14 Neville HoganNOTE 3:*Partial differentiation with respect to displacement:Solenoid & co-energypage 15 Neville