Figure2.5shows the different frequency characteristics for the models proposedin Fig.2.4. Five segments are used for the segmented RLC model. The higher onegoes in frequency, the more segments are needed to end up with a valid model. In thelimit, for an infinite amount of segments, a continuous RLC distribution is obtained.An inductor split up into segments can also be considered as a multi-port net-work, with each node in the circuit constituting a port. Apart from the partial self-inductance of each segment, a mutual inductance is defined with all other segments[112]. This approach may be used to construct a model out of a given coil geometry.To account for skin and proximity effect, each segment is further divided into paral-lel filaments [74, 111]. From EM theory, a system of equations can be formulated tofind all elements of the impedance matrix of the multi-port network. This approachis followed in some inductance calculation routines, like FASTHENRY[74]. Thelatter one does not include capacitances. Hence, only equivalent RL models can beextracted from the calculated multi-port impedance matrix.As the coils in an inductive link are normally operated below, or in extremis atself-resonance, RLC models are sufficiently accurate. Well below self-resonance,even a plane RL model is acceptable. The inter-winding capacitance may be ab-sorbed into capacitors connected in parallel with the coil.
2.4 Inductor Models23Fig. 2.5The impedance of some distinct inductor modelling networks vs. frequencyfIn general, the inductances and especially the resistances in one-port lumpedmodels are frequency dependent, not because of the influence of parasitic capac-itances, but due to the fact that the electric current distribution depends on thefrequency. Section2.5.1elaborates on this phenomenon, referred to as skin andproximity effect. It can be stated already that the effect on the inductance is mi-nor as long as the wire cross-section is much smaller than the coil dimensions,which is the case for most inductive link applications. This is true because a cur-rent redistribution within the wire does not affect the magnetic flux density atlonger distances from the wire. The equivalent series resistance on the other handcan change several orders of magnitude with frequency. TheRvalues in Fig.2.4
242 Magnetic Inductiontherefore have to be specified as a function of frequency in order to be accu-rate.2.5 Finite Element ModellingMaxwell’s equations can be numerically solved for a given geometry and electro-magnetic source by means of a finite element (FE) package. In the examples furtheron in this section, COMSOLMULTIPHYSICShas been used [19].For translating a physical problem into a FE model, the inclusion of an externalsource current densityJecan be useful [21]. Maxwell’s fourth equation (2.7) thenbecomes:∇ ×H=σE+jωD+Je(2.39)The external current densityJestands separate from the conduction currentσE. Itmerely serves as an input for the model.