FEM MODELLING

Coil impedance modelling FEM allows the geometry and material properties to determine physical behaviour by applying the appropriate physical laws. This gives a wealth of information about how a device behaves under particular circumstances.

In this case FEM is being used to derive the frequency dependence of the stationary coil impedance. It is necessary to solve in the time domain to allow nonlinear behaviour to be modelled.

A discretized model of the magnet assembly was produced using Flux2D, a programme supplied by Cedrat [12]. The model domain was axisymmetric with second order elements, the physical domains considered are magnetic and electrical. The model boundary is defined with an outer annular ‘infinite region’. In this region the elements have a modified co-ordinate system in which the outer nodes are at an infinite distance. This technique avoids the errors due to modelling only a small region of space [11][12].

Evaluating the impedance requires knowledge of the current and voltage through the voice coil. This has been achieved by means of stationary transient FEM in which a voltage source has been coupled to the voice coil region [11]. The result we are seeking is the current at each time step.

In the coil region of this FEM model the current is restricted to flow uniformly since the coil is a stranded conductor. In the other conductive regions the magnetic forces produced by the current are allowed to force the current flow into a skin on the surface of the conductor. The discretization is critical for this analysis and a suitable ‘skin’ of quadrilateral elements must be formed on the outer surfaces of conductive regions [11]. After sufficient time-steps for the starting transient to settle, a steady-state waveform of current versus time may be extracted from the solution files, along with the driving voltage. This analysis includes the effect of eddy currents induced in the pole and the aluminium ring; these may be seen in Fig. 27 and Fig. 28. The effect of the eddy currents is to produce a field opposing the voice coil flux. This has the effect of reducing the voice coil impedance.

In both loudspeakers the lines of equal power density are dense in the steel pole surface and more widely spaced in the aluminium rings. This is largely due to permeability of the steel being very much higher. With increasing frequency both the extent and thickness of the skin reduce.

The current waveform flowing through the coil is extracted by evaluating the current through the coil at each timestep. A typical result is shown in Fig. 29.

Subsequently the voice coil’s ‘blocked’ or stationary electrical impedance may be calculated by applying Ohm’s law to the fundamental components of the waveforms.

Stationary transient FEM results

Fig. 30 shows the calculated impedance compared to the measured data for loudspeaker 1 with the coil in the rest position. At low frequencies the small errors in the modelled mechanical impedance cause some significant artefacts in the measured data. At higher frequencies the agreement is very good.

At around 1kHz the measured value is consistently below the stationary transient FEM value. To determine the cause of this discrepancy a conventional impedance measurement of loudspeaker 2 was made. The loudspeaker voice coil was then glued in position and a direct measurement made of the blocked impedance. The result in Fig. 31 clearly shows that at 1kHz the blocked impedance is higher than the free impedance. This curious difference is thought to be an artefact resulting from the motional impedance due to nonpistonic diaphragm motion.

The impedance results are illustrated below, together with the equivalent data from the measured values using the method described in 3.5. Each loudspeaker was solved for seven frequencies in five positions. This analysis took approximately 6 hours calculation time on a 2GHz PC for the 35 data points for each loudspeaker. The positions were chosen to be coincident with the measurement positions. To allow the large number of results to be compared contour plots have been used. The contour plots allow the magnitude of the full ZL(jw,x) matrix to be plotted for a single loudspeaker. The coil displacement is shown on the vertical axis and the frequency on the horizontal axis, in the usual manner. Impedance magnitude is described by the shaded regions as indexed by the right hand colour map. The data points are interpolated by the graphing software for improved visualisation. A loudspeaker that did not have impedance variation with displacement would have vertical contour lines.

It is evident that in both cases the stationary transient FEM yields the same trend of impedance variation as the measurements. The measured data exhibits some noise around the fundamental resonance as mentioned in residual impedance. There is also evidence of nonpistonic behaviour which has resulted in anomalous impedance variations.

Fig. 34 shows the real and imaginary parts of the stationary transient FEM derived impedance for loudspeaker 2. The difference in symmetry of the
impedance mentioned in 3.6 can be clearly seen.

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