Sensorless Velocity Feedback Subwoofer Robert-H Munnig Schmidt © 2017 The author, RMS Acoustics & Mechatronics and Grimm Audio. All rights reserved. Copying of the complete document is allowed for personal use only. The author/publisher is not responsible for any problems that might arise by using the contents of this paper. Published by RMS Acoustics & Mechatronics and Linear Audio The Netherlands email:[email protected] www.rmsacoustics.nl Contents 2 Contents 1 Introduction3 2 Practical Design and First Modelling6 2.1 Target Specifications of the Prototype....................6 2.2 The Applied Loudspeaker Element......................8 2.3 Necessary Power................................. 11 2.4 Stiffness of the Enclosure............................ 12 2.5 Correction by Positive Current Feedback.................. 13 3 Further Improvements on Modelling 16 4 Experimental Validation 18 4.1 Frequency Response............................... 18 4.2 Measured Distortion............................... 19 5 Conclusions 21 3 1 Introduction 1When examining the dynamic properties of Loudspeakers it is known that the resonance of a loudspeaker is effectively damped by short circuiting the motion induced voltage from the actuator by using a voltage source amplifier with very low output impedance. This phenomenon is in fact a kind of proportional negative velocity feedback as the velocity is slowed down by a force generated by the motion voltage, which is proportional to that same velocity. This inherent negative velocity feedback phenomenon is determined by the total resistance of the electric loop shown in Figure1, where the motion voltage of the Lorentz actuator induces the current over this loop resistance causing the damping force. If it would be possible to reduce the total resistance below the resistance Rc of the loudspeaker itself, very high levels of damping could be obtained. This also would possibly further reduce the distortion by the increased negative velocity feedback and completely cancel the overshoot. In Figure2 the beneficial effect of such an increase of damping to ³ = 1 or 2 (Q 0.5 or Æ 0.25) is shown. The resonance peak is completely vanished while as a drawback the down slope starts at even a higher frequency albeit more gradual. In fact the system starts to act like a series of two first-order high-pass filters and it can be concluded that by further increasing the damping at a certain setting a simple first-order lag-lead network can be applied to compensate the amplitude and the phase. At a ³= 2 (Q 0.25) the slope angle in the frequency range of interest between 20 and 80 Æ Hz is about 20 dB per decade (+1) like a first order high pass filter so it should be possible to compensate that quite easily with a first-order filter. But how can the total resistance be reduced to below the resistance Rc of the loud- 1This Paper was earlier published in Linear Audio Volume 3 and 6. It Rs R + c Vs _ L Loud- + Amplifier speaker Vm (source) (load) _ Figure 1: By reducing the total resistance Rs Rc the effect of the motion voltage Vm on Å the current is increased. This creates a stronger damping force opposite to the velocity, which is negative velocity feedback. 4 120 ζ = 1 Q = 0.5 110 ζ = 1 Q = 0.5 ζ = 0.6 Q = 0.83 100 ζ= 0.6 Q = 0.83 90 80 70 60 +2 Magnitude [dB] 50 40 30 120 ζ = 2 Q = 0.25 110 ζ = 2 Q = 0.25 ζ = 0.6 Q = 0.83 100 ζ = 0.6 Q = 0.83 90 +1 80 70 60 Magnitude [dB] 50 +2 40 30 0 1 2 10 10 10 10 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Frequency [Hz] Time [sec] Figure 2: Frequency and step response of the closed-box system at different damping levels compared to the original damping factor. With ³ 2 the dynamic properties are Æ reduced to a series of two first-order high-pass filters with one corner frequency around 160 Hz and one around 16 Hz. speaker? With a normal audio amplifier with a very low source impedance (voltage source) the total resistance of the electric loop from Figure1 equals the coil resistance and the connecting wires only. As long as the resistance of these wires is less than 5% of the coil resistance their effect on the damping can be neglected. But it is possible to apply a special trick in the amplifier that causes it to supply more voltage when the load demands more current. This means that the amplifier has a negative resis- tive output impedance which effectively reduces the total resistance. The working principle is explained as follows: From electronics theory it is known that negative current feedback over an amplifier creates a very high output impedance because the negative feedback will control the output current independent of the voltage. From the fact that negative current feedback increases the output impedance one can conclude that positive current feedback would decrease it. This is in a more schematic way shown in Figure3. The current through the load is measured by a current-to-voltage converter with a negative sign to compensate for the negative sign at the input of the amplifier, turning it into positive feedback. The effect on the feedback loop is as follows: The closed-loop gain of an amplifier with normal 5 Amplifier + + Ga + + + Ra I La a Vi Loudspeaker Vo + Vm Current-to-voltage - converter + Vc = Gc · Ia - - - Figure 3: Applying a positive current feedback will result in a negative output impedance. An increase of current induces an increase in output voltage. negative feedback is as follows: Vo Ga Gcl (1) Æ Vi Æ 1 GaGfb Å With Ga being the gain of the the amplifier in the forward path and Gfb the gain of the feedback path. This feedback gain can be derived from the gain of the current to voltage converter by calculating the current Ia as function of the output voltage Vo using Ohm’s law: Vo Vm Ia ¡ (2) Æ Ra j!La Å The self inductance La is as a first approximation neglected because of the low frequencies. Futher on it will be shown that it should partly be taken into account. The motion induced voltage source Va has no relation with the effect of the output voltage on the current and can be replaced by a short circuit to determine the feedback gain: Vc Gc Ia Gc Gfb (3) Æ Vo Æ¡ Vo Æ¡Ra Using this result in the Equation (1) the following closed-loop gain is obtained: Vo Ga Gcl (4) Æ Vi Æ Gc 1 Ga ¡ Ra The term in the numerator would become 1 when Gc 0 and the closed-loop gain is Æ equal the original gain of the amplifier. The closed-loop gain would become infinite 6 when Gc Ra/Ga. Between these values the system is stable and the closed-loop gain Æ is increased relative to the open-loop amplifier gain with a factor 1/(1 GaGc/Ra). This ¡ effect will be used to create a negative output impedance which can be understood from the following reasoning: The output impedance of a source is equal to the change of the output voltage due to the output current. In case the impedance is resistive and positive the output voltage decreases with an increase of load current:2 dVo Ro dIa (5) Æ¡ In the situation of Figure3 an increase of load current d Ia will create an increase of output voltage dVo as there is twice the minus term in the loop so: dVo dIaGcGa (6) Æ Which automatically leads to the conclusion that: Ro GcGa (7) Æ¡ By controlling both gains the negative output impedance can be adapted such that the total loop impedance is tuned to a suitable value to achieve the required damping. 2 Practical Design and First Modelling The theory is validated in a prototype system using the configuration of Figure2, while Figure4 shows some details of the construction. The main difference is the orientation of the two loudspeakers, while also the current is adapted to this orientation such that both membranes still move in opposite directions, necessary to achieve a high sound pressure in the opening. The reason for the different orientation is the attempt to cancel the reluctance force contribution to the distortion as these will alwas be directed towards the magnet system. 2.1 Target Specifications of the Prototype The presented system is designed according to the THX© specification of Lucasfilm for a university classroom which is quite a bit larger than the average living room but the design can fit in a larger living room giving ample headroom for impressive effects. 1. THX certified response from 20 – 80 Hz with a maximum deviation of +/-3 dB 2This should not be confused with Ohm’s law as that law describes the current and voltage over an impedance. In this case the output voltage is described as a function of the current to a load due to the output impedance of the amplifier. The positive voltage drop over the output impedance by Ohm’s law manifests itself as a negative voltage drop over the load. 2.1 Target Specifications of the Prototype 7 Symmetry axis 250 65 Loudspeaker 1 Open side Loudspeaker 2 3 30 18 0 0 2 1 0 3 3 4 500 3 18 0 56 0 60 0 9 2 Amplifier Figure 4: The designed enclosure of the subwoofer with two loudspeakers. A balanced configuration is chosen that balances the reaction forces of both loudspeakers and creates a perfect coupling of the sound pressure of the two loudspeakers.