18 Energy and co-energy continuation

18 Energy and co-energy continuation - Lecture 20 Energy...

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Lecture 20 Energy and co-energy (Continuation) From previous lectures we know that n mutually coupled coils can be characterized by Li λ = . The energy stored in the magnetic field (W φ ) of the n coils can be expressed as 1 2 T Wi L i φ = If we consider the following magnetic circuit, Then d dW eidt i dt id dt == = Where N and therefore dW Nid And recalling from Ampere’s Law Ni = Hl then dW Hld Letting BA Φ= and substituting yields () dW Al HdB = Note that (Al) is the volume of the device, HdB is defined as the energy density, and BdH is defined as the co-energy density. Note that the energy is valid for the instantaneous value of i and L. Therefore L could be a function of a displacement parameter x which may depend on time.
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() 1 2 T Wi t L t i t φ = So, for example, if we consider again the relay magnetic circuit Then we know by Ampere’s Law cg Ni x =ℜ+ℜ Φ , where 0 2 g g x x A µ ℜ= and therefore the inductance is 2 0 2 c g N Lx x A = ℜ+
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18 Energy and co-energy continuation - Lecture 20 Energy...

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