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The current implementation of the method is based on the cylindricalcapacitor placed in the space inside the voice coil former and abovethe pole piece of magnet system. The inner surface (cylinder) of thecapacitor is attached to the top of the pole piece and outer surfaceinside the voice coil former. Capacitance of this capacitor changeslinearly according to the deflection of the cone. Principle of themeasurement method and it's utilization in motional feedback ispresented at the following figure (Parts added to "traditional"loudspeaker driver are coloured, insulators green, conductors red):
Figure 1: Principle of the measurement method
Capacitance of the measurement capacitor can be calculated usingthe following formula:
Formula 1
Where:
εo = permittivity of vacuum (8,8542 * 10-12 C2 * N-1 m-2)
εr = relative permittivity (for air approximately 1),
l = length of the capacitor (m),
r1 = inner radius of the capacitor (m),
r2 = outer radius of the capacitor (m).
In this formula, the length l is changed according to the movement of the loudspeaker cone.For calculating capacitance per unit of length, following formula can be used (l set to 1):
Formula 2
Measurement circuit is implemented with simple current tovoltage-converter. Providing that the capacitor in operationalamplifier circuit is small enough, voltagegenerated from the circuit isproportional to the change of the capacitance of themeasurement capacitor according to following formulas:
Formula 3
Formula 4
Formula 5
Where:
Uout = output voltage of the current to voltage converter (V),
R = value of the feedback resistor of the current to voltage converter (Ω),
Ucap = voltage applied to the measurement capacitor (V),
dQ/dt = derivative of the charge of the measurement capacitor (As/s (=A)),
dC/dt= derivative of the capacitance of the measurement capacitor (F/s (=A/V)),
dl/dt = derivative of the length of the measurement capacitor (m/s).
With sine signal the deflection of the cone and length ofthe measurement capacitor follow the following formula:
Formula 6
Where:
xmax = peak value of the deflection (one way),
l = length of the capacitor,
f = frequency of the sound.
The output voltage of the current to voltageconverter can be derived from the formulas 5 and 6 (for sine signal):
Formula 7
Shortly: You get an velocity of the loudspeaker cone fromthe operational amplifier output terminal. When voltage ofaround 1kV is connected to the outer surface of the measurementcapacitor, the feedback resistor of the op-amp is 1MΩ,excursion of the speaker is +-10mm and frequency 20Hz, youwill get output signal of around one volt.
Since the sound generated by loudspeaker is proportionalto the acceleration of the cone, either velocity measurementhave to be converted to acceleration (by differentation)or sound signal converted to velocity (by integration) to geta comparable signals for the correcting loop.
For more information, construction of the prototypes andmeasurements read the whole document (PDF, 403 kB).