solenoid_co_ener

# solenoid_co_ener - CO-ENERGY (AGAIN) In the linear case,...

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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? E XAMPLE : S OLENOID WITH MAGNETIC SATURATION . Previous solenoid constitutive equations assumed electromagnetic 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-energy page 1 © Neville Hogan

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M ODEL THAT PHENOMENON . Assume an electrical constitutive equation as follows: λ (i, x) = L(x) i λ s L(x) 2 i 2 + λ s 2 where L(x) is position-dependent inductance as before. For sufficiently small currents the relation is approximately linear. i << λ s L(x) λ L(x) i For sufficiently large currents the flux linkage reaches a limiting value, λ s . i >> λ s L(x) λ λ s Solenoid & co-energy page 2 © Neville Hogan
M ECHANICAL CONSTITUTIVE EQUATION may 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( λ , x) But flux cannot be specified arbitrarily. Realistic boundary conditions require current input. To find the relation between force and current, substitute. λ = λ (i, x) Solenoid & co-energy page 3 © Neville Hogan

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S TORED ELECTRICAL ENERGY (at a fixed displacement) E = i d λ Need to invert the relation between flux linkage and current. —In general, anything but straightforward. In this case, a little algebra yields the following. i = λ s L(x) λ λ s 2 λ 2 | λ | | λ s | U SE WITH CAUTION ! If λ > λ s this expression yields an imaginary number for the current. Solenoid & co-energy page 4 © Neville Hogan
A LITTLE CALCULUS (and some more algebra) yields an expression for energy. E(

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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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solenoid_co_ener - CO-ENERGY (AGAIN) In the linear case,...

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