The inductance can again be calculated by integration and averaging of the vector potential A over the conductor. A more convenient alternative however is the inte- gration of the magnetic energy density w m = B · H 2 over the complete space. Here, this integral is simplified to 16 times the integral over the modelled volume V mod : LI 2 2 = 16 N 2 V mod w m d V (2.70)
36 2 Magnetic Induction with N the number of turns of one coil. The factor N 2 arises, as always, from the fact that I is divided by N to have an equal global current distribution and that the induced emf is perceived over N turns in series. For the Helmholtz coil in Fig. 2.14 , a total inductance L = 2 . 3 N 2 μH is calculated when both coils are connected in series and L = 575 N 2 nH when they are connected in parallel. 2.5.4 Mutual Inductance Thus far the developed FE models have been used to calculate self-inductance and where appropriate the equivalent series resistance. The same FE models can be used to calculate the mutual inductance of the modelled inductor with an arbitrary sec- ondary coil. Like for the self-inductance, usually a DC model suffices to calculate mutual inductances. Once the magnetic field induced by a certain source current is known, the mutual inductance can be calculated by simply integrating the magnetic flux density over the surface of interest (Eq. ( 2.28 )). This may involve exporting the FE solution to more advanced numerical computation packages, like M AT L AB . Integration over surfaces that do not conform to the symmetry of the original model, can also be conducted this way. Alternatively the vector potential can be integrated over a certain contour (Eq. ( 2.22 )). For a secondary coil with a certain winding thickness, the calculated flux should be averaged over its winding cross-section, as explained in Sect. 22.214.171.124 on page 27 . 2.6 Conclusions The electric field E inside of a conductor can be split up into two parts: the conserv- ative field −∇ V corresponding to a charge distribution ρ and the non-conservative field − jω A that is magnetically induced at non-zero frequencies. This latter field can be self-induced, i.e. the current through the conductor itself is responsible for it, or it can be induced by an external current flow. In the former case, the self- inductance L relates the magnetically induced emf across a piece of conductor with the current through it. In the latter case, it is the mutual inductance M that relates the generated emf to the external current distribution. The current density σ E in general is heterogeneously distributed over a conduc- tor’s cross-section as − jω A is heterogeneous. The heterogeneity of σ E rises with the frequency and as a result, since the current is pushed through smaller effective areas, the losses increase as well. This reflects in a higher equivalent series resis- tance (ESR) of the inductive element. In principle, also the inductance values are affected by this current redistribution. For typical wire-wound coils used in induc- tive powering applications however, the effect on the inductance is negligible.
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