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 ζ = 1 or 2 (Q 0.5 or = 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 ζ= 2 (Q 0.25) the slope angle in the frequency range of interest between 20 and 80 = 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.

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 + 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

60 +2

Magnitude [dB] 50

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

60 Magnitude [dB] 50 +2 40

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 ζ 2 the dynamic properties are = 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 resistive 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) = Vi = 1 GaGfb + 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) = Ra jωLa +

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) = Vo = − Vo = −Ra

Using this result in the Equation (1) the following closed-loop gain is obtained:

Vo Ga Gcl (4) = Vi = Gc 1 Ga − Ra

The term in the numerator would become 1 when Gc 0 and the closed-loop gain is = 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 = 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) = −

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) =

Which automatically leads to the conclusion that:

Ro GcGa (7) = − 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. One of the loudspeakers is reversed to cancel the reluctance force eﬀects.

2. Size of the theatre 20 metre wide and 5 metre high and 15 metre deep

3. Maximum (peak) sound intensity in the middle of the theatre:

• 100 dB @ 30 Hz 2.2 The Applied Loudspeaker Element 8

Figure 5: The used loudspeaker, Peerless XXLS 12.

• 106 dB @ 40 – 80 Hz

4. Aperiodic response (Q 0,7), no resonances nor overshoot. = 5. Size of the box less than 0.1 m3

6. No external vibrations exerted to the ﬂoor to avoid unwanted sounds from contact resonances.

7. Cost should be less than 1000 euro on material including power ampliﬁer.

The speciﬁed reduced levels at lower frequencies are based on the statistics that show that extremely low frequencies are never as strong as frequencies around 1 khz. As will be shown from the following sections this still will require large loudspeakers with very strong ampliﬁers.

2.2 The Applied Loudspeaker Element

The used loudspeaker by Peerless is shown in Figure5 and the main data are given in Figure6. This loudspeaker was selected because of its aluminium conical-shaped diaphragm which remains acting quite deterministic as a solid piston until around 1 kHz which allows its use in a model-based active control system. Also the size has to be as large as possible in order to realise the targeted sound intensity of 105 dB in the centre of the theatre. Important data for the design are the DC resistance, the resonance frequency, the force factor, the moving mass, the suspension compliance or its stiffness, the motor damping and the piston area. The loudspeaker has a long excursion range while having a longer voice coil (32 mm) than the air gap (8mm). This means that with a maximum motion amplitude of the membrane of +/- 12 mm the permanent magnet flux on the coil will remain 2.2 The Applied Loudspeaker Element 9 approximately unchanged which otherwise would cause harmonic distortions due to non-linearity As is explained before. The stiffness from the surround and spider is calculated from the resonance frequency and compared with the specified compliance, which should equal 1/k. The resonance frequency is:

ω0 2πf 2π 19.1 120 [rad/s] (8) = = · =

3 2 With m 130.6 10− and c mω the stiﬀness of the suspension is equal to: = · = 0

k 1.87 103 [N/m] (9) = · This is indeed almost equal to the inverse of the compliance in the data sheet of 3 0.53 10− [m/N]. The damping coeﬃcient is derived with Equation (26) from paper · “Low Frequency Sound Generation by Loudspeaker Drivers”

(BL)2 c 41 [Ns/m] (10) = R ≈

The resulting damping ratio is:

c 41 ζ 1.34 (11) = 2pkm = 2p1.87 131 = · With a corresponding Q of:

1 Q 0.37 (12) = 2ζ =

Specs:

Electrical Data Power handling Nominal impedance Zn 4 ohm 100h RMS noise test (IEC) -- W Minimum impedance Zmin 3 ohm Long-term Max System Power -- W Maximum impedance Zo 65.7 ohm (IEC) DC resistance Re 2.6 ohm Max linear SPL (rms) @ power -- dB/W Voice coil inductance Le 1.6 mH Short Term Max power -- W Capacitor in series with x ohm Cc -- uF Voice Coil and Magnet Parameters T-S Parameters Voice coil diameter 51 mm Resonance Frequency fs 19.1 Hz Voice coil height 32.6 mm Mechanical Q factor Qms 9.29 Voice coil layers 4 Electrical Q factor Qes 0.38 Height of the gap 8 mm Total Q factor Qts 0.37 Linear excursion +/- 13 mm Ratio fs/Qts F -- Max mech. excursion +/- -- mm Force factor Bl 10.3 Tm Flux density of gap -- mWb Mechanical resistance Rms 1.69 Kg/s Total useful ux 2.3 mWb Moving mass Mms 130.6 g Diameter of magnet 147 mm Suspension compliance Cms 0.53 mm/N Height of magnet 35 mm E ective cone diameter D 24.4 cm Weight of magnet 2.2 Kg 2 E ective piston area Sd 466 cm Equivalent volume Vas 159 ltrs Sensitivity 91.2 dB Ratio BL/ (Re) 6.4

Figure 6: Characteristics of the used loudspeaker. 2.2 The Applied Loudspeaker Element 10

110

100 Impedance On axis 32 90 ] Ω o 16 30 80 8

o 4

SPL [dB] 60 70

60 Real part of impedance [ Real

50 10 100 1000 10,000 Frequency [Hz]

Figure 7: Given response of the used loudspeaker measured in an anechoic room.

In the given characteristics the value for the electrical Q factor is 0.38, which is within reasonable tolerances for these kind of systems ( 1 dB 10 %). The Q factor < ≈ for the acoustical damping as found in the previous chapter would result in a value of around 70 indicating again the bad efficiency. The given mechanical Q factor (Qms) of 9.29 is determined by the spider, the surround and the air. This means that the mechanical guiding takes more energy than the acoustic damping although it determines still a very limited portion of the total damping. With a Q of 0.37 this loudspeaker represents an overly damped system conform the specified frequency response7. This response is normally measured with the loudspeaker mounted on an “infinite” baffle, which isolates the sound of the backside from the sound of the front side of the membrane without increasing the stiffness. The measuring is further done in an anechoic room so without reverberations. This situation is not representative for a real situation with the loudspeaker build in an enclosure and it will be shown why the manufacturer chose the parameters such that the damping is so high in the unmounted situation. Also the impedance of the loudspeaker is shown in equivalent ohms. In principle this is not really right as it is a complex impedance consisting of the resistance of the coil and the effect of the motion voltage on the coil by the velocity. After a minimum at 100 Hz at higher frequencies the self inductance starts to increase the impedance while at frequencies below 100 Hz the motion voltage of the motor starts to reduce the current in the system to reach a peak at the resonance frequency where the velocity and motion induced voltage are in phase with the amplifier voltage. Finally it is also shown that directional effects do not occur at the frequencies of interest ( 100Hz) < 2.3 Necessary Power 11

2.3 Necessary Power

The specified data state that one loudspeaker delivers 91 dB at 2.84 Vrms over 4 Ω = 2 W input power, when measured @ 1 metre distance under the condition that the power is radiated over a hemisphere. At 5 metre distance the surface of the hemisphere is 25 times larger so the intensity is 25 times smaller, which is -14dB, giving a total sound pressure of 77dB. To get 105 dB (+28dB) this would require an amplifier with 1250 W electrical power. The power related specifications of the loudspeaker are not presented in Figure6 but a similar loudspeaker from the same manufacturer with a paper diaphragm gives a maximum sustainable peak power per loudspeaker of 350 W and continuous power of 150 W. This means it is necessary to use at least two loudspeakers, which also reduces the necessary electrical power with a factor two based on the corresponding efficiency improvement by the larger total radiating surface. The now required amplifier power of 625 W for two loudspeakers is within the allowable specified peak power per loudspeaker. For reasons of availability and easy implementation the DS8.0 plate amplifier is chosen from Hypex, which can deliver 530W in 8 Ω, which equals the impedance of two loudspeakers in series. Although this is not equal to 625 W, the difference is only 0.7 dB, which is negligeable in comparison with other toterances. In practice the peak power level occurs very seldom and for a very limited time so even the 530 W will not often be necessary nor harm the loudspeaker unless the maximum excursion is continuously reached. This is checked with the following calculation: First the current is determined at the reference voltage level of 2.84 Vrms: V 2.83 Irms 1.09 [A] (13) = R = 2.6 =

This current causes a force of:

Frms BI` 10.3 1.09 11.2 [N] (14) = = · =

At 100Hz the response is mainly determined by the mass and the acceleration equals:

F a xω2 [m/s2] (15) = m =

This means that the excursion of the membrane will be:

F 10 4 xrms 2.2 10− [m] (16) = mω2 = 0.130(628)2 ≈ ·

This eﬀective value of the motion of the membrane of 0.22 mm corresponds to a maximum motion amplitude of 0.3 mm. 300 W input power per loudspeaker would ≈ increase this amplitude to: Remark: the amplitude scales with the square root of the power! r 3 265 3 xd 0.3 10− 3.45 10− [m] (17) = · 2 = · 2.4 Stiﬀness of the Enclosure 12

The maximum linear excursion is specified to be 13 mm, which means that the lowest frequency, where 105 dB in the middle of the theatre is obtained is 100p3.45/12 ≈ 55 Hz, because the amplitude raises with the second power of the frequency (slope -2 of the mass spring transfer function). A higher excursion will lead to extreme distortion figures, because the coil is driven outside the magnetic air gap. Below 55 Hz the maximum output power will need to be limited by 12 dB per octave to restrict the maximum excursion magnitude. This is approximately a 10.5 dB reduction for 30 Hz and 17.5 dB reduction for 20 Hz! The real performance specifications become accordingly: Maximum (peak) sound intensity in the middle of the theatre:

• 82 dB @ 20 Hz

• 94 dB @ 30 Hz

• 105 dB @ 55 – 80 Hz

Adding two more loudspeakers could improve this performance but the cost (now already 400 euros excluding amplifier) would not allow that. Placing the subwoofer ≈ low in a corner is a better choice as it reduces the radiation field to a quarter of the hemisphere thereby increasing the acoustic power in the room up to a factor 4 (6 dB) for low frequencies where the wavelength is large in respect to the space, which is true for 20 Hz. Even with this step the previously mentioned “perpetuum mobile” approach is still applicable as the total efficiency remains low.

2.4 Stiﬀness of the Enclosure

2 5 With the surface area of the membrane Ad 0.047 m , an air pressure of 10 Pa and 2 = a volume of the enclosure Ven 0.06 m and a λ value of 1.2, the total stiﬀness of the = spring due to the enclosed air can be calculated with Equation ?? becomes:

5 2 P0 2 10 3 ke λSd 1.2 (0.047) 4.4 10 [N/m] (18) = Ve = · 0.06 ≈ ·

The compliance (1/k) of this spring is equally divided between the two masses. This means that each membrne will be coupled to a spring with twice the stiffness 3 (ka 2ke 8.8 10 [N/m]. Adding the stiffness of the diaphragm suspension results = = · in the total stiffness per membrane:

3 3 km (1.87 8.8) 10 11 10 [N/m] (19) = + · ≈ · Resulting in a resonance frequency of: s s ω 1 k 1 11 103 f0 · 46 [Hz] (20) = 2π = 2π m = 2π 0.131 = 2.5 Correction by Positive Current Feedback 13

120

110

100

Magnitude [dB] 50

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 8: Frequency and step response of the closed-box system when driven with a voltage source ampliﬁer. It shows a response with a ζ of 0.54 (Q 0.93). =

The damping coeﬃcient c, when applying a voltage source ampliﬁer, was calculated with Equation (10) and equals 41 Ns/m, resulting in a damping ratio of: ≈ c 41 ζ 0.54 (21) = 2pkm = 2p11 103 0.131 = · · With a corresponding Q of: 1 Q 0.93 (22) = 2ζ =

It can be concluded that the resonance frequency is signiﬁcantly higher than in the unmounted situation and the damping ratio is increased to a more optimal value. A value of Q ζ 0.7 would be the most optimal (aperiodic) value without overshoot = ≈ of the step response. In Figure8 it is shown that the step response with Q 0.93 = is indeed not without some overshoot but what is worse is that the lowest -3 dB frequency limit is a full octave above the targeted THX speciﬁcation of -3dB at 20 Hz.

2.5 Correction by Positive Current Feedback

In Figure9 the electronic circuit of the test set-up is shown. The first operational amplifier is an inverting amplifier where the input voltages and current feedback signal are added. To get the power in the loudspeaker an Hypex switched-mode Power Amplifier is used with normally a voltage source output with almost zero output impedance. A positive current feedback loop is added to create the negative output impedance . The coils of the two loudspeakers are connected in series which is allowed as they share the same dynamics, which means that all voltages and currents are in phase. The resulting current and the resulting damping force remain the same as in the situation with one loudspeaker because both the motion voltage and the total resistance is doubled. When neglecting the self inductance, the resulting 2.5 Correction by Positive Current Feedback 14

Bass boost 10k

Linear 50k 100k 15k Hypex DS 8.0 -Opa 500W @ 8 Ohm 2134 PA Voltage gain: 40 2 * subwoofer lsp. 10k + Peerless XXLS12 Positive 4 Ohm current 3k3 1k feedback 0.272 V/A + Lem module + 5k 1k8 - on Opa- Ii Io 2134 PA 68 - + + I = 4·10−3·I o i _

Figure 9: Electronic circuit of the test set-up showing the main components and some tun- able elements. With the upper switch the compensation circuit can be bypassed while the corresponding potentiometer can be adjusted to limit the gain difference at switching between linear and “bass-boost” to an acceptable level. The lower switch enables or cancels the positive current feedback. total resistance is 5.2 Ω and with a voltage-source amplifier the previously calculated value of the damping ratio (ζ = 0.6) would be obtained. This value has to be increased to at least a value of 2 and even more when possible to get the maximum controlled motion. To achieve this high damping ratio the maximum total resistance must be reduced to 5.2 0.6 1.56 Ω. This means that the amplifier must have a negative · 2 = output impedance of approximately -3.6 Ω. This is set with the 5 kΩ potentiometer. To determine the approximate setting, Equation (7) is used with Ga 40 and with = the potentiometer set at zero. In that situation Gc equals the total gain of the path from the current via the current-to-voltage converter and the difference amplifier to the input of the amplifier being Gc 0.272 1.8 0.15. This results in a negative = − · ≈ − output resistance of 6 Ω which is just too much but by increasing the value of the potentiometer the negative impedance can be reduced until the right value is obtained. In practice the selfinductance still shows to have a little bit of influence by increasing the impedance of the loudspeakers as the system is still just stable at a value of 6 for the negative output impedance. The second operational amplifier acts as the required first-order “bass-boost” filter. Above 16 Hz the gain will go down until 160 Hz where it becomes flat again. This compensates the first order +1 slope effect of the negative output impedance nicely as can be seen in Figure 10. A flat response with a low frequency -3 dB bandwidth limitation at 20 Hz has been realised. It is necessary to be aware of the additional amplification (gain) of this circuit caused by the positive current feedback. When using Equation (4) and Equation (7) an almost infinite gain is obtained as soon as the negative output impedance equals Ra. When the system is tuned close to this 15

120

110 Active 100 Passive 90

Magnitude [dB] 50

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 10: Response of the closed-box when driven with an amplifier with a negative output impedance compared with the passive uncontrolled response. The high damping with a first-order compensation for the low frequency roll off creates a system with a -3 dB bandwidth at 20 Hz. level the total gain becomes very sensitive for resistance and gain variations of the elements in the feedback loop which is also a reason not to tune the system too critical. For instance a damping ratio of 2 with the corresponding output impedance of 3.6 Ω requires Gc 0.15 3.6/6 0.09. The gain of the system without positive = − · = current feedback would increase with a factor 1/(1 GaGc/Ra). When filling in the − data this leads to an amount of 2.5 times the normal gain. This increased gain has to be taken into account when balancing the output of the subwoofer with the other loudspeaker systems in the surround set-up.

110 +1-slope

100 Impedance On axis 32 90 ] Ω o 16 30 80 8

o 4

SPL [dB] 60 70

60 Real part of impedance [ Real

50 10 100 1000 10,000 Frequency [Hz]

Figure 11: The impedance of the applied Peerless XXLS12 loudspeaker shows a less steep slope than the +1-slope that a self-inductance would give. 16

3 Further Improvements on Modelling

One of the assumptions in the modelling was that the self inductance could be neglected, due to the focus on very low frequencies. This assumption appeared to be not correct as practical measurements practical system showed a resonance around 90 Hz. Introducing a simple self inductance in the model, however, did not fully cancel the problem so there is more at stake. Ideally the dynamic impedance of the actuator of a loudspeaker can be modelled to consist of a resistor and an inductor connected in series with the motion induced voltage source. When looking at the impedance curves of the applied loudspeaker, shown in Figure7, at higher frequencies a pure inductor would normally give an increase in the absolute value of the impedance with a factor that is proportional to the frequency (Z jωL). This would graphically show-up as a +1 or 20dB/decade = slope above the frequency where the inductor impedance becomes larger than the resistance. In reality this slope shows to be less steep which is caused by the copper sleeve/ring around the core of the permanent magnet system, which manufacturers use to reduce the self-inductance and the reluctance force. This reduced effect is shown in Figure 11 and as a result the real impedance appears to be approximately half the impedance of a self-inductance, for which reason it is a named the "semi- inductance" by Knud Thorborg a.o. of Tymphany/Scanspeak as presented at the 122nd convention of the AES with an approximate +0.5 slope and -45 degrees phase between current and voltage instead of the -20 dB/decade with -90 degrees phase between current and voltage of a normal self inductance. In the LTspice model, shown in Figure 12, the semi-inductance is approximated as a series of four inductances with three parallel resistors. The application of this approximate semi-inductance model appeared to reduce but not fully cancel the resonance that was observed in LTSpice with a normal self-inductance. This meant that the measurement should also show this resonance or another source should be found. Fortunately a more refined measurement with noise averaging showed this resonance to occur, especially when the positive current feedback was maximised until instability occurred at 90 Hz. To solve this issue a compensating phase-lead-network was inserted in the current measurement branch. The two capacitors with series resistances also mimic a +0.5 slope transfer function after 100Hz. Even though this solved the issue in the model, the real setup still showed a strong oscillation occurring around 1 kHz, lowering to 300Hz at a higher positive current feedback loop-gain setting. After extensive measuring it could only be concluded that the positive feedback loop is so critically tuned with above settings that it is better to limit the higher frequency loop gain by means of an additional low-pass filter. A 0.1 uF capacitor over R3 appeared experimentally an optimal choice for this additional low-pass filter. The final modelled frequency response is shown in Figure 13. 17 SPL L11 G6 {1/Bl} {2*rho*Sd/(0.5*pi*d*2e-5)} Acoustical output Acoustical Xc 1m C8 L4 {Cmb*Bl**2} G5 {1.41/Bl} L2 {Cms*Bl**2} L12 C1 {Cmb*Bl**2} {Mms/Bl**2} L10 R2 {Cms*Bl**2} {Bl**2/Rms} C7 6 L5 R38 {Mms/Bl**2} 0.05mH R10 {Bl**2/Rms} 4 L3 R37 0.15mH 6 L9 R16 0.05mH Semi-inductance 4 L1 R36 0.6mH Electrical impedance Mechanical 4 L8 R15 0.15mH L6 Semi-inductance 0.15mH 4 L7 E3 80 R1 2.6 Hypex DS8.0 R14 0.6mH L21 0.15mH R9 2.6 U1 R3 470 C5 0.1µ R18 200 1 C4 0.22µ R13 R4 R17 1k 3.3k 1k R5 C3 0.15µ C2 100n Compensation network Compensation U2 R7 100k 10 10k R11 R12 LT-spice model of the full system where the semi-inductance is modelled assemi-inductance. a series of four inductances Based on initial model of J.M.Plantefève. with three parallel resistors. A compensation network is added to cancel the remaining effect of the R6 48 R8 15k V1 AC 0.5 F1 LEM E3 4m Figure 12: 18

108dB

99dB SPL 90dB

81dB

72dB

63dB

54dB

45dB

36dB Vamp

27dB

18dB

9dB 10Hz 100Hz 1KHz

Figure 13: The modelled frequency response of the sound power output and the output voltage of the power ampliﬁer shows the ﬂat response within +/-1 dB with a clear LF boost.

4 Experimental Validation

The complete system is realised in hardware and measured with the software pack- age ARTA, using a Beyerdynamic MM1 measurement microphone and a MOTU MicrobookII USB soundcard. It is important to note that the measurements were all done "near-ﬁeld" with the microphone located on a rubber foam support inside the cavity between both loudspeakers. This means that the overall radiated sound power is measured irrespective of the radiating direction and the eﬀect of the surroundig room. As a consequence this measurement is only valid in the frequency range where the subwoofer acts as a purely hemi-spherical radiator. In this case this approximation is allowed as the subwoofer is designed to operate in the lowest sound range of 16-150Hz.

4.1 Frequency Response

Another consequence of the near field measurement is the scaling of the magnitude axis. Because the measurement is only meaningful in a relative way, the absolute values on the vertical magnitude scale have been re-scaled to a corresponding averaged C-weighted SPL measurement of the system on a distance of 1 metre in a room of 30 m2 with an acoustic ceiling. The averaging was done to cancel the effects of standing waves in the reference. Figure 14 shows the measured frequency response of the system with pink-noise. The measurement was done at the maximum excitation level with the amplifier voltage just below clipping (500W for the total system), resulting in a loudspeaker-diaphragm excursion of approximately + 10 mm. With this small enclosure, the combination of the amplifier, actuator and the high 4.2 Measured Distortion 19

130

120

110

100

90 Magnitude [dB] 80

50 5 10 20 50 100 200 500 1k 5 10 20 50 100 200 500 1k Frequency [Hz] Frequency [Hz]

Figure 14: Near field measurement of Frequency response (left) and the corresponding output voltage of power amplifier (right) at near clipping level of the amplifier. stiffness by the large diaphragm surface (2*466 cm2) with the enclosed air of 30 litres per loudspeaker is not able to exceed the maximum allowed excursion of +/- 13 mm so active excursion limitation is not needed. The input low-pass filter of the Hypex DS8.0 plate amplifier module was set at 150 Hz. The influence of the coil temperature was investigated and appeared to have little influence on the measurements. The measurements of Figure 7 were done after 30 seconds at maximum power. In the cooled down situation, where the coil resistance is minimum, there could be a risk of instability but the positive current feedback gain was tuned such that the system remained stable. At higher temperature it would be expected that the damping of the fundamental frequency would be reduced but that effect remained unnoticeable. The only effect was observed on the sound output at 16 Hz which is slightly less at a high coil temperature, due to the higher resistance of the coil, resulting in less current at the maximum amplifier voltage. The output voltage of the amplifier further confirms the results of the Spice model. At 16 Hz the maximum gain is shown ( approx. +20 dB).

4.2 Measured Distortion

Positive current feedback increases the damping, because the motion induced voltage by the moving-coil actuator generates a current, which in its turn generates a force, directed opposite to the movement. For that reason positive current feedback acts as negative velocity feedback with the motion induced voltage as the measured value of the coil velocity . Unfortunately the velocity sensor is equal to the moving coil itself where the non-ideal F/I curve, as shown in Figure 9, will result in a non-linear velocity measurement at high excursion levels. In fact the measured velocity value is below the real value at large excursions due to the reduced captured magnetic field at the outer positions and as a result the negative velocity feedback will result in a slight increase of the excursion. This effect could possibly counteract the mechanical nonlinearity by the stiffness of the suspension which becomes significantly higher at large excursions. This would be especially beneficial at sub-resonant frequencies 4.2 Measured Distortion 20

a: b:

Figure 15: Distortion measurement at 20 Hz without feedback (a:), with feedback (b:) and with one inverted (rotated) loudspeaker (c:) shows a slight reduction of the higher harmonics.

(<42 Hz) where the stiffness determines the excursion level. A second beneficial effect at sub-resonant frequencies could be related to the increased damping of the fundamental resonance frequency. It is to be expected that the second harmonic of a frequency at half the fundamental resonance frequency (21 Hz) will be reduced by this damping. Several measurements were done to validate these qualitative statements, using the same equipment and setup as with the sound power measurements. Figure 15 shows one example for 20 Hz excitation at +/- 5 mm diaphragm excursion, corresponding to approximately 100 dB at 1 metre distance (see Figure 14). At maximum excursion (+/- 10 mm) the distortion equals approximately four times these values, corresponding with the squared character of the distortion sources. Without positive current feedback the measured THD equals 7.4% with second and third harmonic as almost equal main contributors. Adding feedback shows no reduction of the second harmonic distortion and only a 3 dB reduction of the third harmonic distortion which does not support the statement that the increased damping will result in less excitation by harmonics around the fundamental frequency (42 Hz). More improvement is achieved with one inverted loudspeaker, which reduces the second harmonic distortion with a factor two (6 dB), indicating that the reluctance force 21

Table 1: Distortion measurements in % THD at diﬀerent frequencies for the three situations.

Frequency (Hz) Without With feedback Inverted feedback loudspeaker 20 7.4 6.0 2.3 30 1.8 1.0 0.8 40 1.0 0.7 0.7 50 0.26 0.3 0.3 60 0.21 0.13 0.2 80 0.12 0.18 0.2 100 0.16 0.29 0.2 of the actuator accounts for approximately 50% of this distortion. Unfortunately the beneﬁcial eﬀects are only observed sub-resonant as can be seen from Table1 for the same output level as with the example at 20Hz from Figure 15. It seems even that above 60 Hz the distortion rises slightly, mainly in higher harmonics. Consid- ering the limited accuracy of the measurements and the fact that these numbers are already quite moderate and measured at a very high sound power level, the application of positive current feedback appears to give only a very limited reduction of the distortion.

5 Conclusions

Application of positive current feedback is useful to create a stable robust subwoofer with simple first order correction filters, when taking the complex impedance of the loudspeaker actuator into account. Especially the “partial” self-inductance, called "semi-inductance" must be accounted for in the design as the actuator current is strongly influenced by the impedance. The method enables to extend the low frequency range to 16 Hz, while slightly reducing the harmonic distortion especially at sub-resonant frequencies. This reduction is however very limited. The main reason for this lack of significant distortion reduction is the fact that the velocity sensor is the same coil that drives the system. This means that in measuring it suffers the same problems regarding position dependency as the actuator. Furthermore it also detects the eddy-currents that run in the eddy-current ring, which is coupled like transformer windings to the actuator coil. the eddy-current ring also has a self-inductance, which slows down changes of current, and this means that the effect of eddy-currents will be detected after the cause of the eddy-current is terminated. This implies a delayed response. Aside from these issues there is the third item on the problem list as described in the paper “Distortion Sources in Loudspeaker Drivers”, which is the current dependency on the temperature and the self-inductance, which is severely increased by the positive current feedback. After all, if the current decreases due to a higher 22 temperature, the positive feedback will also decrease, thereby further decreasing the current. This can only be solved by sensing or real-time modelling the temperature of the coil and automatically correct the settings of the controller. This requires digital control and the charm of the simplicity of the design is lost. In practice the measured effect of this temperature rise was a slight change of the overall gain (compression) after an extended testing period and a bit less damping, all in the range of 1-2 dB. Overall its performance was perceived acceptable in demonstrations in a classroom at the university. Nevertheless the distortion levels are still unacceptably high and demand an alternative method that fully addresses the main causes for distortion. This method requires active feedback of the diaphragm motion and is described in the paper “Motional Feedback in a Nutshell” As a last remark for those readers who would like to experiment with the system, it should be mentioned that all active feedback controlled systems require a kind of limiter to protect the loudspeakers from damage due to extreme motions at subsonic frequencies and to prevent the amplifier from severe clipping caused by the compensation network.