PREDICTION OF DISTORTION

If the non-linear parameters of the equivalent circuit in Fig. 1 are measured then the state variables (current, displacement) and the sound pressure output can be predicted for any excitation signal. This technique is useful for several applications: diagnostic testing of a loudspeaker; assessment or improvement of a new design; auralization of large signal performance with music or other test signals [1]. The simulation module (SIM) [13] has been used to predict the distortion based upon the measured non-linear parameters Le(x), R2(x) and L2(x) above.

The results of these simulations may be directly compared with the results of the 3D Distortion measurement module (DIS) [14], which provide a direct measure of the actual distortion of the loudspeaker. In this way we are able to validate the measured non-linear parameters of the LR-2 model by comparing predicted and measured distortion generated in the input current.

Measurement of distortion in the electrical input current reveals the distortion generated by the varying input impedance almost directly (provided the loudspeaker is connected to an amplifier with a suitably low output impedance). The distortion generated in the current will also appear in the displacement, velocity and sound pressure output. However, the distortion generated by displacement varying force factor Bl(x) and compliance CMS(x) appear in the current only close to the loudspeaker resonance where the velocity and induced voltage are high. Doppler distortion and any other radiation distortion will not be detected in the input current.

A very important aspect is the selection of the test stimulus. A single tone reveals only harmonic distortion which is, for displacement varying residual impedance ZL(jw,x), relatively low. This is because at low frequencies the variation of ZL(jw,x) with displacement x is relatively small, seen in Fig. 11 and Fig. 19 for both loudspeakers. In addition at high frequencies the voice coil displacement becomes very small and the variation of ZL(jw,x) with displacement is minimal.

However, a two-tone signal comprising a low frequency tone at frequency f2 and a probe tone at higher frequency f1 is a much more revealing signal [15]. The bass tone is set to a fixed frequency below resonance to produce a large displacement of ~5mm peak. The second ‘probe’ tone is varied from 200 Hz to 18 kHz to represent an audio signal. The varying impedance ZL generates not only harmonics of both tones but additionally difference and summed-tone intermodulation components, which may exceed the harmonics significantly.

There is a simple relationship between shape of a nonlinear parameter, such as L2(x), and the order of the resulting distortion component. A distinct asymmetry of the parameter, such as in Fig. 17 and Fig. 18, will generate dominant 2ndorder distortion, which will outweigh 3rdorder and higher distortion. Conversely a symmetrical curve, such as in L2(x) and R2(x) in Fig. 25 and Fig. 26, will generate strong 3rd-order and other odd-order distortion components.

Dynamic transient FEM

FEM provides an alternative means of predicting the distortion. Using the FEM software a full dynamic transient solution for an arbitrary input signal may be computed in the electrical and magnetic domains. In this analysis the coil is allowed to move as a single degree of freedom system with user defined stiffness and mass. This solution may be used to predict the distortion. For the purpose of this paper, the dynamic transient FEM solution appears for only one pilot frequency; the solution time is very long, as a full period of the complex signal is required, and use of this type of modelling is thus restrictive. The two-tone intermodulation stimulus must be solved with a small time step defined by the higher frequency over two periods of the lower frequency, provided that the frequencies are integral multiples. It is estimated that calculating intermodulation for several pilot frequencies would take 100 hours of processor time on a 2GHz PC.

As we have seen from section 4.2, the results of the stationary transient FEM correlate well with measured ZL(jw,x). The use of the measured non-linear parameters to calculate the distortion is dependent upon the LR-2 fitting and the assumption that the model is adequate to describe the behaviour of the system. Additionally, as previously discussed, the quasi-static measurement method may have a bearing upon the deduction of the LR-2 parameters and indeed the measured ZL(jw,x). The full dynamic transient FEM analysis does not have these limitations as it returns directly to the fundamental physical relationships in order to calculate the system output. It is also able to account for more complex phenomena such as the effect of voice coil current magnitude on the impedance response.

FEM allows application of specific laws of physics, in this case Maxwells’ equations, to model the behaviour; whereas lumped parameter models matches specific effects in such a way that the resulting non-linear system of equations behaves in a closely similar way to the loudspeakers measured behaviour. The dynamic transient FE method used here is also able to represent other non-linear relationships by means of user defined power series. This could be used to model the Cms(x) non-linearity for example.

It would be possible to use the ZL(jw,x) results from stationary transient FEM to determine the LR-2 parameters and predict the distortion by solving the resulting differential equations, for example with the SIM module [13]. This has not been done here since the results would be derived from almost identical data. In practice the time saving of this method is substantial and it is anticipated that further work will be done using this method. Where further understanding of the physics is an aim the detailed results of a direct FEM approach are likely to outweigh the time cost.

Results

Fig. 35 Second –order intermodulation distortion in the input current measured and predicted by using the lumped parameter method for loudspeaker 1 (ring above gap)

Fig. 36 Third –order intermodulation distortion in the input current measured and predicted by using the lumped parameter method for loudspeaker 1 (ring above gap

)

Fig. 35 and Fig. 36 show the measured and predicted 2nd-order and 3rd-order modulation distortion for loudspeaker 1, calculated as defined in [16]. The 2ndorder distortion is dominant and is caused by asymmetry of the parameters Le(x), L2(x) and R2(x).

Loudspeaker 2, with shorting ring below the gap, gives a slightly better performance than loudspeaker 1 for frequencies below 1 kHz. Here the 2nd-order distortion components are smaller but still dominant as the asymmetry remains in the non-linear parameters. At high frequencies the loudspeaker 2 generates similar intermodulation distortion levels to loudspeaker 1 which corresponds to the increasing asymmetry of the effective inductance Leff(x,jw) in Fig. 23.

Note that the distortion increases with frequency similarly to the increase in the variation of the impedance ZL with x shown in Fig. 11.

Dynamic transient FEM results

Fig. 39 Acceleration response spectra of kinematic model with twotone excitation at 12.5Hz & 2kHz. Level of 2kHz 2.83v, 12.5Hz level adjusted to give same excursion as lumped parameter model. Loudspeaker 2.

The discretized geometry of loudspeaker 2 was solved using a dynamic transient solution with a two tone input signal in order to reveal intermodulation distortion components as previously discussed. The model was excited with an input signal comprising a 12.5 Hz tone and a 2Khz tone.

Fig. 39 shows, by means of an FFT, the acceleration response spectra of the model to this signal. The two excitation tones may be clearly seen on the spectra at 12.5Hz and 2kHz. In addition, due to the non-linearities of the model, distortion artefacts have been produced. These artefacts are both harmonic and intermodulation products resulting from the interaction of the two excitation signals.

Fig. 40 Acceleration response spectra of dynamic transient model with two-tone excitation at 12.5Hz & 2kHz – detail of 4kHz region. Loudspeaker 2.

An enlarged view of the spectra is shown in Fig. 40. This view clearly shows that the product visible at 4kHz is composed from three distinct peaks. These peaks would appear to be the results of additional intermodulation products as well as the second harmonic of the 2kHz tone.

Fig. 41 Total current flow in aluminium ring inside magnet, excited with two-tone signal at 12.5Hz & 2kHz. Loudspeaker 2.

The dynamic transient FEM solution allows generation of additional data such as the total current flow in the aluminium ring as shown in Fig. 41. It is difficult to think how else this type of detailed information may be generated. FE methods are excellent at providing data for situations that would be extremely difficult to measure or otherwise predict.

A number of observations may be made from Fig. 41. Firstly the induced current in the aluminium ring is predominantly at high frequencies, the eddy currents & skin depth increase in severity with frequency as their induction is proportional to current flux. Additionally the modulation of the current with voice coil position is clearly shown.

Fig. 42 Total current flow in pole piece and top plate, excited with two-tone signal at 12.5Hz & 2kHz. Loudspeaker 2.

Fig. 42 shows a similar result for the current in the pole. In this case a far greater proportion of low frequency current flows. The overall level of the current is significantly higher than that of the aluminium ring even though the resistivity of steel is higher. This is thought to be a consequence of the close proximity of coil and poles as well as the high magnetic permeability in the poles.

Discussion of modelling

Throughout this paper stationary transient FEM and lumped parameter models have been applied to model static voice coil impedance at a constant temperature. This leaves only the magnetic and electrical domains to determine the coil impedance. Helpfully FEM software considering both these two domains is available [12]. All that is required is the appropriate material properties and geometry.

Using FEM, such as dynamic transient FEM, it may soon become possible to deduce the final linear and non-linear behaviour of a loudspeaker based upon the geometry and the material properties. FEM allows additional data and visualisations to be computed that are not possible to deduce using measurements or other simulations. Most usefully FEM may also be applied to conceptual loudspeakers. However, describing the loudspeaker using one mammoth FEM model requires massive computation, furthermore the results will inevitably be complex and difficult to interpret.

Even with the relatively simple case considered in this paper the disadvantages of an exclusively FE method are clear. Calculation of the intermodulation distortion using the SIM module can be performed in ~30 seconds, faster than an equivalent measurement, the dynamic transient FE solution for the seven other pilot tones used for the ZL(jw,x) modelling is estimated to take in the region of 100 hours solving time on an equivalent PC. The lumped parameter method is faster by factor of around 10,000. Applying Moores’ law [17], the current dynamic transient FEM computation is likely to be able to match the current lumped parameter computation speed in approximately 20 years.

With lumped parameter models the question ‘which physical domains must we consider?’ is replaced by a requirement to understand the physics involved so that appropriate simplifications may be made. This allows development of a simplified system with appropriate parameters for representation of an effect. The system and parameters may be defined in numerous ways to suit the users requirements in order to provide the required accuracy, complexity, practicality etc.

The ability of FEM to allow display of the fields concerned may well prove informative to the engineer. However, while FEM provides extensive information, the overall principles by which a loudspeaker operates are best illustrated using lumped parameter models. Using FEM to produce data for a lumped parameter model combines these viewpoints to give an engineer the most enlightening information in a reasonable time.

A number of FEM models may be used to provide values for the various model parameters. Furthermore these FEM results show how the fields concerned change for the parameter modelled. Even without modelling the result of the non-linearities, such as the intermodulation distortion, the stationary transient FEM ZL(jw,x) is extremely useful to the engineer. Another very significant result if that if one aspect of the design is altered only the appropriate model need be re-solved.

Bl(x) may be deduced from static magnetic FEM. Cms(x) may be derived from non-linear static structural FEM. Mechanical impedance and acoustic transfer function from vibro-acoustic steady state FEM [6]. ZL(jw,x) can be deduced as demonstrated. These results may be combined using a lumped parameter model such as that in the SIM module [13] to give a useful design tool. At the time of writing the thermal time constants are the least tractable using the FEM approach [6]

As shown here for the voice coil impedance, with careful consideration and investigation, the most important information can be compressed into a few meaningful parameters and an appropriate lumped model. The benefits of an integrated modelling approach, such as that described in [6], cannot be emphasised too strongly. Linking various FEM results & measurements using assumptions and relationships from lumped parameter modelling is the most puissant method currently available.

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