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The application of a model to a particular real object usually requires an estimation of the free model parameters in such a way that the model describes the real object with maximal accuracy. With the linear models, straightforward techniques are available which may be applied for loudspeakers at small amplitudes.

Non-linear models require special techniques for the parameter identification. Static, quasi-static and full dynamic techniques have been developed to measure the force factor Bl(x), compliance parameter Cms(x) and inductance parameter Le(x). The dynamic techniques have the advantage that an audio-like ac signal is used for excitation and the loudspeaker is operated under working conditions.

The current version of the large signal identification module (LSI) in the Klippel Analyser performs a dynamic measurement using a noise stimulus [8]. The free model parameters are optimised to give the best fit between measured and modelled current and displacement. The LR-2 model is currently constrained so that the three lumped parameters vary with the same shape in x.

Although this assumption is a good approximation for most loudspeakers without shorting rings, one of the aims of this paper is to investigate the validity of this approximation for more elaborate loudspeakers using copper caps and aluminium rings for reducing the inductance nonlinearity.

Clearly the measurement of suspension stiffness using a dc offset gives significantly different results from a dynamic measurement due to creep, relaxation and other time dependent properties. Additionally, due to flux modulation, the force factor Bl(x) and the inductance Le(x) are dependent on the current i. Thus using an audio-like signal will produce far more meaningful data than measurement with extremely small input current (coil offset generated by external force or pressure) or extremely large currents (coil offset generated by a dc current).

Despite these limitations a quasi-static technique is a useful method for investigating the variation of the impedance with both frequency and displacement.

Measurements have been performed on two test loudspeakers. Both loudspeakers share the following specifications:

The magnet assembly of loudspeaker 1 has an aluminium shorting ring placed above the magnetic gap. The magnetic assembly of loudspeaker 2 has an aluminium shorting ring placed below the magnetic gap. The geometry of the two loudspeakers is shown in Fig. 3. The loudspeakers serve to illustrate the influence of the effect of aluminium rings on the variation of the voice coil inductance with displacement.

Mechanical Setup

A method for measurement of the displacement dependent impedance was developed at GP Acoustics (UK). The method is a simple modification to the standard Linear Parameter Measurement (LPM) of the Klippel Analyzer system. The measurements presented in this paper were performed by Klippel GmbH.

An additional spider is attached to the diaphragm.

The spider holds an inner clamping part made of aluminium which is secured to the lower rod (usually used for holding the microphone).

By shifting the lower rod a displacement may be imposed on the coil position. Displacing the coil will also change the other parameters such as Bl(x), Cms(x) and also loss factors.

The distortion analyser also provides a displacement meter [7]. This is used for measuring the original rest position of the cone and to measure the imposed static displacement. Throughout this paper the convention is that a positive displacement of the coil refers to movement out of the magnet assembly.

The loudspeaker under test is connected to distortion analyser allowing a simultaneous measurement of voltage, current and displacement signal.

Small signal measurements

The additional spider increases the stiffness of the total suspension and the resonance frequency. However, the modified loudspeaker still constitutes a mass, spring, damper system and can be represented by the equivalent circuit in Fig. 1.

The module Linear Parameter Measurement (LPM) of the Klippel Analyser system is used to measure the linear parameters at each prescribed displacement. The loudspeaker is excited by a sparse multitone signal of 0.5 V rms at the terminals. Since the voltage, current and displacement are measured simultaneously all of the linear parameters can be identified instantaneously. An additional measurement with a mechanical perturbation (additional mass or measurement in a test enclosure) is not required.

The multitone excitation signal allows assessment of the distortion generated by the loudspeaker. For the two loudspeakers the small signal measurements the maximum distortion occurred 20 dB below the fundamental lines in the current spectrum. This shows sufficiently linear operation of the loudspeaker [9].

Fitting of the inductance model

At first the linear parameters are measured at the rest position (x=0) and the different inductance models (Leach, Wright, LR-2) are used to describe the impedance response, measured up to 18 kHz.

Fig. 6 shows a measured curve and the fitted curves using the three models. The LR-2 and Wright models are able to describe this particular impedance curve very well. The Leach model causes minor deviations about 500 Hz and 5 kHz. Although Wright usually gives the best fit there are cases where the other models have provided a superior fit.

Since the test loudspeaker is based on a woofer intended for frequencies below 200 Hz the models have been fitted using data only up to 2 kHz, Fig. 7. In this instance all models are able to give a good fit to the measured curve.

Fig. 7 Magnitude of electrical impedance of loudspeaker 1 measured and fitted by LR-2, Wright and Leach model up to 2 kHz.

Residual impedance ZL

The LPM module also calculates the amplitude and phase response of the residual impedance ZL(jw). The measured and the fitted curves are shown in Fig. 8 using the Leach model. The measured curves are calculated by subtracting the estimated dc resistance Re and the motional impedance (calculated from the estimated parameters Bl, MMS, RMS, and CMS) from the total electrical input impedance. The magnitude increases with frequency, usually with a slope less than 6 dB per octave.

The Leach model uses a constant slope corresponding with the exponent n in Equation (1). Close to loudspeaker resonance the magnitude of the calculated residual impedance varies significantly. In this region the motional impedance is very high (~100 Ohm), measurement and modelling error (in the order of 1 %) will be assigned to the residual impedance and makes accurate estimation of the residual impedance below 100 Hz impossible. The calculated phase of the residual impedance is about 68 degrees with a small decay to higher frequencies. The Leach model assumes a constant phase, which proves to be a good approximation for this particular loudspeaker.

Fig. 9 shows the residual impedance match using the LR-2 model. While the fitting above 1 kHz is good, at lower frequencies there are significant differences in both phase and amplitude. The LR-2 model, and also other shunted models using more Li and Ri elements (i > 2), behaves as an ideal inductance at very low frequencies giving a 6dB per octave slope and a phase shift of 90 degrees. This property corresponds to the observation that the eddy currents are frequency dependent and will vanish at very low frequencies. Though a small increase in the measured phase at low frequencies supports this observation, the phase shift of the LR-2 model begins at a higher frequency than the measured shift. Thus the LR-2 is usually limited to use over a frequency band of two decades. Using an additional shunted section (R3 and L3) improves the fit significantly and results in a good description over the whole audio band (three decades). The cascade of shunted inductances is a minimum-phase system and can be realised in the analogue or digital domain.

Fig. 10 Magnitude and phase of the residual impedance ZL(jw) of loudspeaker 1 measured (solid lines) and fitted by using the Wright model (dotted lines).

The residual impedance fitted by the Wright model is shown in Fig. 10. The magnitude response can be approximated by a smooth line having a different slope at low and high frequencies (corresponding with exponents Erm and Exm). Contrary to the Leach model, the phase is not constant but depends on all four parameters. The Wright model also considers the response below 100 Hz, where measurement errors and noise have corrupted the measurement, and generates a decrease of phase shift at very low frequencies. Since the Wright model is not bounded to be minimum phase and not composed from a system of lumped electrical elements it may be deceived by measurement artefacts when used to represent a measured curve. Thus a good match with the measured impedance curve does not guarantee that the parameters are meaningful.

Impedance versus displacement

Once measurements had been performed at rest position (x=0) an offset was imposed upon the coil using the lower rod in Fig. 5. The displacement meter at the hardware unit, distortion analyser, was used to measure the offset. The LPM module was then again used to measure the linear parameters. The residual impedance ZL is displayed for loudspeaker 1 in Fig. 11 for a negative offset of -8mm (coil in) and a positive offset of 7.5mm (coil out), the impedance at the rest position is also shown. The variation of the impedance with displacement may be clearly seen. It may be observed that the inductance of the coil is effectively reduced in the region near to the aluminium shorting ring, above the gap in this case. Conversely as the coil is moved into the magnetic assembly, the effective inductance is increased as the coil moves out from the magnetic gap.

Fig. 11 Magnitude of residual impedance ZL(jw) of Loudspeaker 1 measured at rest position (solid line), at –8 mm (dotted line) and 7.5 mm (dashed line)

The phase of the residual impedance ZL was also calculated and is shown in Fig. 12 at the same coil positions. Whereas the impedance magnitude varies by up to 40 % at higher frequencies the phase stays almost constant at 70 degrees.

Fig. 12 Phase of residual impedance ZL(jw) of Loudspeaker 1
measured at rest position (solid line), at –8 mm (dotted line) and 7.5
mm (dashed line)

Non-linear Parameters

Having identified the linear parameters for different values of coil offset, the displacement dependency of the parameters were calculated using the Math Processing Software (MAT), a free programmable (SCILAB or MATLAB) module [10] for the Klippel Analyser. This module imports all of the results measured by the LPM for all measured voice coil offsets and calculates the coefficients li, ri and li in equations (5) - (7).

Fig. 13 Inductance Le(x) and L2(x) of loudspeaker 1 versus displacement x.

Fig. 14 Resistance R2(x) of loudspeaker 1 versus displacement x.

Fig. 13 and Fig. 14 show the parameters Le(x), L2(x) and R2(x) of the LR-2 model versus displacement x for loudspeaker 1 (ring above the gap). The LR-2 model is used as the parameters have an analogue representation and are easy to interpret. Corresponding to our observations regarding the measured residual impedance, it can be seen that the fitted model also identifies the effective inductance as reducing when the coil is close to the aluminium shorting ring above the magnetic gap. It is interesting to observe that the shape of the parameter functions; Le(x), R2(x) and L2(x); is very similar in this instance. In this case the assumption of (8) appears to be valid.

Fig. 15 Effective inductance Leff(f,x) versus frequency f of loudspeaker 1 plotted for the rest position (solid line) and –7 and + 7mm displacement (dotted and dashed line, respectively).

Fig. 16 Effective resistance Reff(f,x) versus frequency f of loudspeaker 1 plotted for the rest position (solid line) and –7 and + 7mm displacement (dotted and dashed line, respectively).

Use of the effective resistance Reff(f,x) and the effective inductance Leff(f,x), as defined in equation (4) and illustrated in Fig. 2c, simplifies interpretation of the residual impedance. Fig. 15 shows the effective inductance Leff(f,x) for three different voice coil displacements based on the LR-2 model. It is clearly shown that the voice coil inductance decreases if the coil moves outwards and increases as the coil moves in.

At low frequencies the effective inductance Leff(f,x) is equal to the sum of Le(x) and L2(x) and the effective resistance Reff(f,x), Fig. 16, is close to zero. At high frequencies the Leff(f,x) is equal to Le(x) only and the effective resistance Reff(f,x) becomes equal to R2(x).

Fig. 17 and Fig. 18 show the variation of effective inductance and resistance versus displacement x for selected frequencies 133Hz, 1545Hz and 18kHz. Clearly all the curves have a distinct asymmetry and decrease with positive displacement.

A second loudspeaker has been made using the same suspension and motor structure but with the aluminium ring located below the magnetic gap.

The magnitude and phase of the residual impedance ZL(jw) of the second loudspeaker are shown in Fig. 19 and Fig. 20 for three coil displacements.

The effect of changing the location of the shorting ring can be seen most clearly in Fig. 21 & Fig. 22. Loudspeaker 1 exhibited an effective inductance that decreased as the coil moved out of the magnetic gap. This trend is close to the reverse for loudspeaker 2, the effective inductance of the coil remains almost constant as the coil moves out the gap and into free air. When the coil moves inward toward the now internally located ring, the effective inductance is seen to fall.

Fig. 23 Effective inductance Leff(f,x) versus displacement x of loudspeaker 2 (ring below) plotted for frequencies 133 Hz, 1545 Hz and 18 kHz.

Fig. 24 Effective resistance Reff(f,x) versus displacement x of loudspeaker 2 (ring below) plotted for frequencies 133 Hz, 1545 Hz and 18 kHz.

Fig. 25 Inductance Le(x) and L2(x) of the loudspeaker 2 (ring below) versus displacement x.

Fig. 26 Inductance R2(x) of the loudspeaker 2 (ring below) versus displacement x.

Fig. 25 and Fig. 26 show the estimated parameters Le(x), R2(x) and L2(x) of the LR-2 model. Contrary to the parameters of loudspeaker 1 both the inductance L2 and the shunt R2 decay symmetrically for positive and negative displacement. The inductance Le(x) is still asymmetric but increases in an unusual way if the coil moves outwards. This is a very interesting result as it demonstrates a case were the assumption of (8) would not be valid.

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