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25 Force and Torque from energy and co-energy

# 25 Force and Torque from energy and co-energy - Lecture 22...

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Lecture 22 Force and Torque from energy and co-energy We know from Lorentz’s law that the electromagnetic force is proportional to the cross product of the current and the magnetic field density f idl B × . Form magnetically linear materials this force may be expressed as 1 2 T d L f i i dx = , where ( ) L x i λ = if x is translational displacement of the moving part or ( ) L i λ θ = if θ is rotational displacement. In the latter case, the electromagnetic torque is 1 2 T d L T i i d θ = Recall that the co-energy W φ ’ stored in the magnetic field is ( ) ( ) ' 1 , 2 T W i i L i φ θ θ = . Therefore the torque, for constant current, can be defined as ' . i const W T φ θ = = Also, since ( ) L i λ θ = , then ( ) 1 i L θ λ = , and recalling that for linear devices the energy is equal to the co-energy, ' W W φ φ = , ( ) ( ) 1 , 2 T W i i L i φ θ θ = Substituting for λ , ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 , 2 1 , 2 T T T T W L L L W L φ φ θ λ λ θ θ θ λ θ λ λ θ λ = = and since L is symmetric, ( ) ( ) 1 1 , 2 T W L φ θ λ λ

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