Inductance - Inductance and Magnetic Field Energy

Inductance and Magnetic Field Energy

Multiplying the equation for vm above with imdt and summing over m gives the energy transferred to the system in the time interval dt,

\displaystyle \sum\limits_{m}^{K}i_{m}v_{m}dt=\sum\limits_{m,n=1}^{K}i_{m}L_{m,n}di_{n} \overset{!}{=}\sum\limits_{n=1}^{K}\frac{\partial W\left( i\right) }{\partial i_{n}}di_{n}.

This must agree with the change of the magnetic field energy W caused by the currents. The integrability condition

requires Lm,n=Ln,m. The inductance matrix Lm,n thus is symmetric. The integral of the energy transfer is the magnetic field energy as a function of the currents,

This equation also is a direct consequence of the linearity of Maxwell's equations. It is helpful to associate changing electric currents with a build-up or decrease of magnet field energy. The corresponding energy transfer requires or generates a voltage. A mechanical analogy in the K=1 case with magnetic field energy (1/2)Li2 is a body with mass M, velocity u and kinetic energy (1/2)Mu2. The rate of change of velocity (current) multiplied with mass (inductance) requires or generates a force (an electrical voltage).

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