Definition of co-energy

If it's a linear system, energy is equal to coenergy.

Coenergy is defined as a convenience for computing force with current held constant.

F = -dW/dx with flux linkages held constant.
F = +dW'/dx with currents held constant

where W is energy and W' is coenergy.

W' = Sum(Lamdbak * Ik) - W

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F = -dW/dx (with lambda held constant) seems like an obvious result from conservation of energy.

If you have a linear system, we have the surprising result:

F = + dW/dx (with currents held constant). Seems to violate conservation of energy at first glance.

But there is electrical input energy to consider

P = dWe/dt = i * v = i * dLambda/dt
dWe = i dLambda

If you hold current constant while changing x which changes L, then lambda must change (since lambda = L*i). That means energy is input from the elctrical system. The amount of energy input from the electrical system is exactly twice the energy output to the mechanical system in a linear electromagnetic system.

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