## Thiele Small Parameters

The Thiele-Small Parameters window is used to calculate the parameters for a drive unit from measurements of its impedance. To calculate all the parameters two measurements are required, one in "free air" and a second with either mass added to the cone or with the unit in a sealed (air tight!) enclosure (ideally with a volume a little less than the expected Vas). Note that the drive unit must be rigidly supported during the measurements, and ideally mounted vertically (i.e. so that the cone is firing horizontally as it would be in a typical loudspeaker installation). Some pre-conditioning of the unit with signals at medium levels helps to stabilise the behaviour and suspension compliance and reduce memory effects in the suspension from periods of storage or lack of use. Quiet conditions are important for good measurements, drive units act as microphones and pick up noise and vibration, affecting the results. The measurements should be made up to 20kHz so that the lossy inductance of the voice coil can be accurately modelled, and the impedance calibration step should be carried out before making measurements.

### An Example Run

To show the results of a TS parameter calculation a small bass-midrange drive unit was measured. It has an effective cone area of 137cm2. The plots below show the impedance measurements made in free air and then with a mass of 5g added to the cone. REW determines whether the secondary measurement is from a sealed box or added mass by looking at the resonant frequency, which is lower than free air for added mass and higher for a sealed box. A least squares fit of an electrical model of the drive unit impedance is carried out on the free air measurement to determine the model parameters. Another least squares fit is carried out on the secondary measurement to determine the modified motional parameters and the TS parameters are then calculated.

To calculate the TS parameters the two measurements are selected and the required values are entered:

- the DC resistance of the voice coil (RDC) in ohms. Accurate measurement of low resistances is unfortunately not easy (see footnote), but the impedance model REW uses can easily compensate for a DC resistance which is slightly lower than the actual, so it is recommended to err on the low side
- the effective area in square centimetres, most driver data sheets include an effective area figure but if this is not available REW can calculate the figure for you given the effective diameter, which is the diameter of the cone plus a proportion of the surround, typically 1/3 to 1/2, just click the calculator icon on the left hand side of the effective area box
- the air temperature in degrees Celsius
- the air pressure in millibar
- the volume of the sealed box in litres, or, if the second measurement was made with an added mass, the additional mass in grammes

The Calculate Parameters button is then clicked, with the following results.

The first column of results in the bottom of the window show the loudspeaker resistance RE, which is generally a little higher than the DC resistance; the minimum impedance Zmin after the peak and the frequency fmin at which it occurs; f3, which is the frequency at which the impedance has increased to sqrt(2)*Zfmin; the inductance at f3; the effective diameter and the effective area. The second column shows the resonant frequency fS; the mechanical (QMS), electrical (QES) and total (QTS) Q-factors and the FTS figure (fS/QTS). These parameters can also be calculated for any single measurement, without requiring a secondary measurement to be selected. The LP figure and the MMS, CMS, RMS, VAS, Bl and Eta (efficiency) figures in the third column can only be calculated using both measurements.

The "Compensate for leakage losses" and "Compensate for Air Load" check boxes are only applicable for sealed box measurements, they take into account the leakage loss of the sealed box (which would be shown as Ql at the bottom of the first column of results) and the air mass load due to the sealed box. These compensations use the Carrion-Isbert method described by Claus Futtrup in the documentation for his Driver Parameter Calculator application at http://www.cfuttrup.com/

The results can be copied to the clipboard by right-clicking on the results area, or written to a text file using the Write Parameters to File button. When writing to file the separator between values, labels etc is as defined in the File -> Export menu.

### FDD Electrical Model

REW uses a driver impedance model that incorporates elements that cater for Frequency-Dependent Damping. The model is described in detail in the paper by Thorborg, Tinggaard, Agerkvist & Futtrup, "Frequency Dependence of Damping and Compliance in Loudspeaker Suspensions" J. Audio Eng. Soc., vol. 58, pp. 472-486 (June 2010). The diagram below shows the components of the model.

The model is split into two parts. The part at the right hand side models the motional impedance due to the movement of the driver, with parameters RES, CMES, LCES and ΛES. This part reproduces the peak seen in the impedance plot. It differs from the classical model by the addition of a frequency-dependent resistance omega*ΛES in parallel with LCES. Note that the FDD model RES value is higher than that of the classical model due to the effect of omega*ΛES, which acts in parallel with RES.

The other part of the model deals with the blocked electrical impedance of the driver. It is based on a model developed by Thorborg and Unruh, described in “Electrical Equivalent Circuit Model for Dynamic Moving-Coil Transducers Incorporating a Semi-Inductor,” J. Audio Eng. Soc., vol. 56, pp. 696–709 (Sept 2008). That model begins with a drive unit resistance RE which is the DC resistance RDC followed by a small additional resistance dR which models the resistance contribution due to eddy currents. It is followed by a series inductance LEB and then a parallel combination of an inductance LE, a semi-inductance KE and a resistance RSS. LE represents the inductance of the part of the voice coil located inside the motor gap. LEB represents the part of the coil outside the motor gap. The semi-inductance KE has an impedance that varies with the square root of omega*j. It models the effects of eddy currents and skin depth in the pole piece. The parallel combination of LE and KE models the transition of the coil's behaviour from largely that of a conventional inductor at low frequencies to a semi-inductor at high frequencies. The RSS component models the effect of electrically conductive material in the magnet system, to be described in the paper by Thorborg and Futtrup "Electrodynamic Transducer Model Incorporating Semi-Inductance and Means for Shorting AC Magnetization", JAES Volume 59 Issue 9 pp. 612-627 (Sept 2011). The parameter values REW determines may be modified if desired and the effect on the modelled impedance and phase traces viewed on the graph, but the TS parameters which have been calculated will not be altered.

The plot below shows the modelled impedance traces (darker red and dashed) overlaying the measured values.

When TS parameters have been calculated the derived and simulated motional and blocked impedance magnitudes and phases can be plotted in addition to the total impedance traces. The simulated traces are produced using the model parameter values, the derived traces are produced by subtracting the model values from the measured values (for example, derived motional impedance is produced by subtracting the modelled blocked impedance from the total measured impedance).

### Simplified Model

As frequency-dependent component values are not supported by many circuit simulators REW also calculates values for an alternative blocked impedance model using two parallel resistor-inductor pairs in series, labelled R2-L2 and R3-L3, and the conventional RES, CMES, LCES motional impedance model without the frequency-dependent damping. The values of these components are shown in the "Simplified Model" box. This diagram shows the simplified model components.

#### Measuring the driver DC resistance

Accurate measurement of low resistances is challenging, LCR meters that are in calibration may have a suitable range and give good results. If you do not have access to a calibrated LCR meter an alternative is to get an accurate measurement of a higher value resistor, perhaps 50 ohm or so, or purchase a very high precision resistor (such as a Vishay bulk foil part) and form a voltage divider with a DC source, the reference resistor and the driver. A decent multimeter can provide accurate voltage measurements, measuring the voltage across the driver and the voltage across the reference resistor allows the driver resistance to be determined from (ref resistor) * (driver voltage) / (ref resistor voltage).

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