Mechatronic design optimisation of subwoofer system. P6-project spring 2010 Group MCE6-621

Transcription

1 Mechatronic design optimisation of subwoofer system P6-project spring 2010 Group MCE6-621

2 Title: Mechatronic design optimisation of subwoofer system Semester: 6th semester 2010 Semester theme: Mechatronic System Design Project period: to ECTS: 18 Supervisor: Michael Møller Bech & Henrik C. Pedersen Project group: MCE6-621 SYNOPSIS: Mads H. Andersen Wisam El-khatib Mikkel P. Ehmsen Rune Wiben In this project the design process of a subwoofer for a home cinema systems is presented. A mathematical model of driver and enclosure is developed and experiments are carried out in a laboratory and in an anechoic chamber with different driver configurations to verify the model. The possibility to use closed loop control of the driver position is investigated. Finally an optimum driver and enclosure configuration is chosen for the subwoofer through an optimisation routine. The design is chosen in order to meet requirements on frequency response while favoring a small enclosure size. The final optimum design consist of a 10 [inch] JBL driver in a vented enclosure of approximately 20 [litre] with an optional positive current feedback circuit. Copies: 8 Pages, total: 92 Appendix: 21 Supplements: CD By signing this document, each member of the group confirms that all participated in the project work and thereby all members are collectively liable for the content of the report.

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4 Preface This report is written by group MCE6-621 at The University of Aalborg during the period 1th of February 2010 to 28th of May The report is part of the sixth semester assignment at the Energy department, and is directed to engineers and engineering students. The theme of the semester is Mechatronic System Design covering design of mechatronic systems and components, and evaluating the applicability of different solution principles. The background picture on the front page is from [1]. The report uses the Vancouver method for citations. On the back cover of the report is a CD, where all data sheets and MATLAB R m-scripts used in the report are found. iii

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6 TABLE OF CONTENTS Table of contents 1 Summary 1 2 Introduction 3 3 Initial problem statement 5 4 Problem analysis The propagation of sound The subwoofer system Modeling of subwoofer Specifications and limitations Closed loop control Problem statement 39 6 Problem solution Model verification Design optimisation Conclusion 63 8 Future Work Feedback control A Appendix 69 A.1 Sound pressure test A.2 Impedance test A.3 Interview with Hi-Fi Klubben A.4 Small signal parameters and Thiele-Small parameters A.5 Sensors TABLE OF CONTENTS v

7 A.6 Test of Sharp distance sensor A.7 Driver A.8 Contents of CD

8 Summary 1 The later years development in home entertainment have created an increasing interest in high quality audio equipment. This include reproducing a sound image in the full frequency range of the human hearing, [Hz]. Most loudspeakers have a cut-off frequency above [Hz], which calls for a subwoofer for the lower frequency sound reproduction. Often a powerful subwoofer with a low cut-off frequency involves a large enclosure which is not desirable in a living room. The initial problem in this project is to investigate the possibility for minimising the enclosure of the subwoofer without compromising the sound quality. For this purpose different criteria defining high quality sound reproduction are investigated. This has led to the following initial constraints for the final subwoofer system: The system should have a flat frequency response down to at least 40 [Hz]. Furthermore both harmonic and transient distortion should be minimised. In order to meet these requirements a model-based design optimisation is carried out. A description of sound propagation and the mode of operation of a subwoofer system is made in order to model the system. It is decided to limit the choice of enclosure types to the closed box and the vented box (bass reflex) types. The amplifier used in the project (a 250 [W] amplifier taken from an active subwoofer) is treated as a black box as the main focus is on the subwoofer driver and enclosure. A mathematical model of the complete subwoofer system, including the driver and the enclosure, is derived and suitable design variables are chosen. The requirements to the system are reformulated in terms of these variables. The possibility of including feedback control to the system. Both position and current feedback is considered and some advantages of each type of control is investigated. These advantages include control of the damping of the system, compensation for non-linearities in the system and compensation for parameter variations. The mathematical model is verified experimentally by testing two different pre-built subwoofers. The model of both closed and vented box subwoofer systems are verified. The verification is made using SPL measurements of the two subwoofers and by measuring impedance curves for the two subwoofers as well. The model is concluded suitable for further use in the optimisation process. The design optimisation is carried out for both closed box and vented box subwoofer systems based on a cost function which minimises the enclosure volume while lowering the cut-off frequency of the system. The optimisation is carried out using a built in optimisation tool called fmincon. This optimisation tool is well suited for solving non-linear constrained optimisation problems. A sequential quadratic programming algorithm is used chosen with requirements of generality, ease of use and efficiency. Initially the subwoofer system is tuned as a Butterworth filter but this shows high demands for the damping of the system. Therefore a second optimisation is made where a 3 [db] resonance peak is allowed. In this way the demand for the damping of the system is lowered. 10 different drivers are chosen for individual optimisation and their optimum designs are compared by comparing the value of the cost function for each optimisation. Both optimisation strategies favour the same driver for both closed and vented enclosures. This driver is the 10 [inch] JBL GTO1014. For both optimisation strategies a vented enclosure yields the lowest cost function. The resulting design 1 Summary 1

9 yields a subwoofer with an enclosure volume of approximately 20 [litre] and a cut-off frequency at 28 [Hz]. Due to limited amount of time the construction and testing of the subwoofer design has not been carried out. 2 1 Summary

10 Introduction 2 The development in electronics for home entertainment has always moved towards higher performance and lower prices. Most danish homes are today equipped with devices like CD players, DVD players and TV sets. As an example 84% of danish households owned a DVD player in 2008 [2]. Resently flatscreen TV s and Blu-ray players have been the big hit in home entertainment. The two combined offer the possibility for High Defenition picture quality, and the Blu-ray disk also contain multichannal uncompressed sound, that can be enjoyed with the right sound equipment. To fully enjoy the sound quality the different medias offer, an audio system containing an amplifier and speakers of a certain quality is needed. Many speakers, especially small models, are not capable of reproducing low frequency sound under [Hz]. To improve the overall performance of the sound, a subwoofer can be added to the system. In this report a subwoofer is defined as a loudspeaker used to produce low frequency sound. The therm subwoofer is used as a description of the whole subwoofer system, containing a driver and an enclosure. Subwoofers on the market for home use has a frequency range from as low as 20 [Hz], to about 140 [Hz] in the high end [3]. The use of a subwoofer not only widens the frequency range of the normal speakers, it also complete the sound picture in multi channel sound, offered especially by DVD and Blu-ray medias by handling the low frequency effects (LFE). One drawback however is that high quality subwoofers capable of reproducing sound at very low frequencies often demand large enclosures. As the size of a subwoofer may be a problem it would be desirable to design a subwoofer capable of reproducing low frequency sound in high quality while minimizing the volume. Defining good quality of sound systems is difficult because the perception of sound is different according to the listener and different listeners might have different preferences. Research has shown that most listeners prefer a flat response of the sound pressure for frequencies down to approximately 40 [Hz], low harmonic distortion and little or no transient distortion [4, p. 6], [5, p ]. Maintaining a small enclosure volume will generally conflict with these desirable criteria as a small enclosure volume will oppose large displacements of the loudspeaker diaphragm which is necessary for producing high sound pressure at low frequencies. When designing a subwoofer many parameters must be considered as the performance of the system is a function of both electrical parameters, mechanical parameters and the geometry of the system. Therefore computer based optimisation might be a useful tool when designing the system. This report will focus on subwoofers used to handle the LFE-channel in home cinema systems. 2 Introduction 3

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12 Initial problem statement 3 The introduction leads to the following initial problem statement: Investigate the possibility of minimizing the size of a subwoofer system without compromising the sound quality. 3 Initial problem statement 5

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14 Problem analysis 4 This section describes the concepts used to describe and model the subwoofer system. First the propagation of sound is described along with the concept of piston range. Afterwords the components in the subwoofer system is described individually. When the subwoofer is described the basic equations used for a mathematical model of the subwoofer is derived. The emphasis is on the transfer functions for the subwoofer system, both the impedance and voltage-to-sound pressure transfer functions. After the modeling a section is devoted to specifying which limitations are set both by the equations in the model and by the mechanical and electrical components of the driver. Finally a suggestion for implementing feedback control is explained. 4.1 The propagation of sound There are many definitions of sound, the acoustic definition is: Sound may be described as the passage of pressure fluctuations through an elastic medium as the result of a vibrational impetus imparted to that medium [6, p. 13]. Figure 4.1 illustrates how sound travels through a medium with particles pushing to the neighbouring particles, propagating a variation in particle density and thereby a variation in pressure. Density of fluid Pressure Figure 4.1: Soundwave. The velocity of this propagation is known as the speed of sound and is widely dependent on the material in which the propagation occurs. In this report the focus will be on the sound traveling through air as the media. The formula for the speed of sound, c [m/s], affected by the temperature of the air, T air [K], is seen in equation (4.1). Where R is the thermodynamic constant characteristic of the gas [N m/kg K] and γ is the thermodynamic ratio of specific heats C p/c v of the gas which is unitless [6, p. 17]. The speed of sound in 4 Problem analysis 7

15 4.1 The propagation of sound air at 20 [ C] is calculated in equation (4.2). This value will be used throughout the rapport. c = γ R T air [ m s ] (4.1) c = ( ) = [ m s ] (4.2) The audible acoustic range of the human ear is called the minimum audible field (MAF) and is generally defined as the frequency band from 20 [Hz] to 20 [khz] [6, p. 215]. The relation between the length of a sound wave, λ [m], the speed of sound c [m/s] and the frequency f [Hz] is seen in equation (4.3). λ = c f [m] (4.3) Calculating the wavelength for sound in the MAF of the human ear, reveals wavelengths between 17 [m] at low frequencies, and [m] at high frequencies. The propagation of sound is very dependent on the ratio of the wavelength to the circumference of the source of sound. For a moving piston the piston range is defined as the frequency range for which the circumference of the piston is smaller than the wavelength. In this range the radiated sound will be approximately non-directional - i.e the sound will propagate equally in all directions. Often the wave number, k, defined as: k ω c = 2π f c = 2π λ (4.4) is used to determine whether the frequency of the sound wave is inside the piston range. If the product of the radius of the source and the wave number, k, is less than one, then the radiated wave is inside the piston range. 8 4 Problem analysis

16 4.2 The subwoofer system 4.2 The subwoofer system In this section the construction and basic mode of operation of a subwoofer system is described. The subwoofer system consists of a driver, an enclosure and an amplifier. To model the subwoofer system it is of the essence to understand the basic structure of these The loudspeaker driver The loudspeaker driver works as a transducer transforming electrical energy (voltage and current) to mechanical energy (pressure and volume velocity). The driver mainly consist of an electric motor, a diaphragm and a suspension system. The motor is composed by a permanent magnet, a back and front plate, an air gap, a voice coil and a pole piece. Its purpose is to transform the electrical current supplied by the amplifier into force acting on the diaphragm. The diaphragm consist of a cone and a dust cap and its purpose is to transform the applied force from the motor (and the resulting movement) into sound pressure and volume velocity radiated to the surrounding air. The suspension consists of a spider and a surround which keeps the diaphragm and voice coil centered in the air gap and drives the diaphragm towards its equilibrium position. A cross sectional sketch of the loudspeaker driver is shown in figure 4.2. Figure 4.2: Cross sectional sketch of a loudspeaker driver. The driver is to be thought of as circular when seen from the front [7, p. 3] 4 Problem analysis 9

17 4.2 The subwoofer system The mathematical equations describing the drivers electrical and mechanical equations will be laid down in section 4.3. The resulting mathematical model will only be valid for frequencies inside the piston range but some further parameters limit the scope of the model. These are the so-called large signal parameters which determine the mechanical and electrical constraints of the driver. The large signal parameters include [8, p. 277]: P e,max : The maximum thermal power handling of the driver which determines the maximum electrical input power to the driver. x max : The maximum displacement of the voice coil possible in order to avoid damage of the mechanical system. x max : The maximum displacement of the voice coil allowed in order to maintain constant flux linkage in the system linking the voice coil and the flux from the permanent magnets. The flux density is constant if the number of voice coil turns inside the air gap is constant. If the number of coil turns in the air gap change then the force acting on the voice coil will be a non-linear function of the current which will introduce harmonic distortion. x max is illustrated in figure 4.3. It can be calculated as: x max = l coil h g 2 [m] (4.5) where, l coil is the voice coil length in the travel direction and h g is the height of the air gap. V d : The peak volume displacement. The peak volume displacement is the product of the effective surface area of the diaphragm and x lim. V d limits the sound pressure produced at low frequencies and will be investigated in section 4.4. lcoil h g Figure 4.3: Segment of figure 4.2 showing x max. [7, p. 5] The large signal parameters will often be stated in the data sheet of the driver along with the small signal parameters and Thiele-Small parameters. These parameters are sufficient in order to describe the driver under normal operation. A brief description of the small signal and Thiele-Small parameters and how they will be implemented in this project is given in appendix A Problem analysis

18 4.2 The subwoofer system The loudspeaker enclosure This section is based on [9]. In this section different types of enclosures will be discussed. The main purpose of the enclosure is to separate the front and the back side of the driver from each other. By doing so the driver is capable of producing pressure difference between the front and the back. As mentioned sound is defined as variations in pressure, so if there is no pressure variation around the driver no sound is produced. If the driver is located in free air and is feeded with a low frequency signal about 20 [Hz] the slow pressure build-up at the front will level out to the back of the driver and therefore the pressure around the driver will be equalized and the amplitude of the sound will be low. A particularly desirable enclosure would be an infinite baffle in which the driver is placed, see figure 4.4. This type of enclosure is of course not practically possible but it has the capability of completely shielding the front and rear of the driver without causing an increase in pressure on the rear of the diaphragm when this is retracted. Subwoofers are often enclosed by very large boxes in order to approximate an infinite baffle shielding as the volume of the box affects the pressure build-up inside the box. A smaller enclosure would often be desirable but the smaller the volume of the box the more sensitive it is to movement of the diaphragm as the air inside the box is compressed. The low frequency response is particularly affected by the pressure build-up inside the box as the diaphragm has to move further at low frequencies in order to maintain the same sound pressure. As the diaphragm is moved the air will be compressed and act as a spring on the diaphragm pushing it towards its equilibrium position. Closed box Infinite baffle Vented box Port Figure 4.4: Driver mounted as infinite baffle, driver in closed enclosure and driver in a vented enclosure. Many different enclosures exist but in this project the modeling and design of the subwoofer system will be limited to the two most common enclosure types: The closed box enclosure and the vented box (or bass reflex) enclosure. The closed box is, as the name suggests, a closed box in which the driver is placed. The vented box on the other hand has an opening in the enclosure which has two purposes. One 4 Problem analysis 11

19 4.2 The subwoofer system is that the pressure build-up inside the box will be reduced as the box is not pressure tight and the other purpose of the opening is to act like a second diaphragm producing sound pressure The amplifier Subwoofers can be divided into two categories when considering amplification: Passive and active subwoofers. Passive subwoofers are connected to an external amplifier which is not necessarily designed to collaborate with the subwoofer. In this case the interaction between the subwoofer and the other loudspeakers in the complete audio system will be determined by the external amplifier. An active subwoofer has a built-in amplifier often with extensive possibilities for adjusting the performance of the subwoofer. Often the relative power level delivered to the subwoofer and the rest of the loudspeakers can be altered along with cross-over frequency, phase modulation and cross-over topology. The amplifier used in this project is a 250 [W] amplifier originally used in a KTH produced active subwoofer called PSW2010. The specifications for the amplifier/subwoofer can be seen in appendix A.8. A factor that can affect the system performance is the output impedance of the amplifier. For the used amplifier the impedance has been measured to be 0.2 [Ω]. The driver to be used in the subwoofer is expected to have a nominal impedance of 4-8 [Ω], and hence the output impedance of the amplifier will be neglected in future calculations. The amplifier will be treated as a black box and only considered as a power source in this report Problem analysis

20 4.3 Modeling of subwoofer 4.3 Modeling of subwoofer In this section a mathematical model of a subwoofer system is made. This model will be the basis for the design optimisation of the system. Initially the loudspeaker driver will be modeled as it will be identical for the two types of subwoofers. The mechanical model will be divided into two separate cases, one modeling the closed box enclosure and one modeling the vented enclosure. The model will only be valid for frequencies within the piston range of the driver unless stated otherwise. Furthermore the it will only model the subwoofer system within the limits of P e,max and x max Modeling of loudspeaker driver The driver is essentially an electric motor driving the diaphragm of the loudspeaker and thereby transforming electrical energy into mechanical energy. The motor has a resistance, R e, and an inductance, L e, in the coil. The motor also has a back electromotive force introducing a voltage drop, ε(t), caused by the motion of the diaphragm. The coupling link to the mechanical model is the output force from the motor and the back electromotive force resulting from the moving coil. According to Lorentz law the back electromotive force, ε(t), is determined by the magnetic flux density, B, of the magnetic field caused by the permanent magnets, the length, l, and velocity, dx(t) dt, of the coil moving in the magnetic field: ε(t) = B l dx(t) dt (4.6) A diagram of the electrical system is shown in figure 4.5. R e L e v (t) s i(t) dx(t) Bl dt Figure 4.5: Diagram of the electrical system. Applying KVL on the system and Laplace transforming yields: V s (s) = (s L e + R e )I(s) + s B l X(s) (4.7) 4 Problem analysis 13

21 4.3 Modeling of subwoofer The mechanical model As mentioned the mechanical model will be divided into two separate models - one for the closed box system and one for the bass reflex system Closed box system In a closed box system the driver is placed in a pressure tight box in order to shield the front of the diaphragm from the rear. When using a pressure tight box as an enclosure the moving diaphragm will cause a pressure build up when the air inside the box is compressed and a pressure drop when the air is rarefied, acting like a spring on the diaphragm. This is undesirable as the diaphragm displacement will be large at low frequencies in order to maintain a high sound pressure. At these large displacements the air inside the box will counteract the diaphragm movement resulting in an attenuation of the low frequency sound pressure. In addition to the spring effect of the air, the mass of the air load on the rear of the diaphragm must also be considered in the modeling of the system as the air on each side of the diaphragm must be accelerated when the diaphragm accelerates. When sound pressure is radiated from the diaphragm the movement of the air will introduce friction which is modeled as a damper with a frequency dependent damping coefficient. The resulting mechanical system is sketched in figure 4.6. x(t) m,β,k,s ms ms ms d m box,k box F m r,β r Figure 4.6: The mechanical system modeling a driver enclosed by a box. In figure 4.6 x(t) is the displacement of the diaphragm, m ms is the mass of the diaphragm and coil, m box is the mass of the air load presented to the rear of the diaphragm, m r is the mass of the air load presented to the front of the diaphragm (the radiation mass), β ms is the damping coefficient of the surround and spider, β r is the damping coefficient of the air that sound is radiated into (the radiation resistance), k ms is the spring coefficient of the suspension, k box is the spring coefficient of the air inside the box, S d is the surface area of the diaphragm and F is the force from the electric motor driving the diaphragm. According to Lorentz law F is given by: 14 4 Problem analysis

22 4.3 Modeling of subwoofer F = Bl i(t) [N] (4.8) where, i(t) is the current through the voice coil and Bl is the force factor - a product of the flux density of the magnetic field in the air gap, B and the length of the voice coil conductor, l, enclosed by the magnetic field. Values of m ms, β ms, k ms and S d are given in the data sheet for the driver. The remaining parameters will be determined by the acoustic impedances of the system. Acoustic impedance, Z A, is defined as rms sound pressure, p, divided by rms volume velocity, U [5, p. 11]: Z A = p U [ ] N s m 5 (4.9) The diaphragm acts as a mechano-acoustic transducer converting the velocity, dx(t) dt, of the diaphragm and the force, F, acting on it to volume velocity, U, and sound pressure, p, according to the following relationships: F = S d p (4.10) U = S d dx(t) dt (4.11) Using equation (4.9), (4.10) and (4.11) all acoustic impedances can be converted into equivalent mechanical impedances, Z M : Z M = Ḟ x = Z A S 2 d [ ] N s m (4.12) Equation (4.12) can be used to convert acoustic impedances to equivalent mechanical impedances. Converting the impedances reveals the parameters m r, β r, m box and k box as it will be described below. When a driver is placed in an enclosure it will have different acoustic impedances on each side of the diaphragm. The impedance on the front will be called the radiation impedance, Z R, and the impedance on the rear will be called the box impedance, Z B. When deriving the acoustic impedances of a system the wave equation is used. When setting up the wave equations for the exact geometry of a loudspeaker system the resulting equations will be very complex. Therefore the acoustic impedance of some standard geometries is used to approximate the system. The diaphragm will be modeled as a rigid, vibrating piston although it is actually cone shaped and elastic - this assumption however is valid for wavelengths within the piston range and therefore the resulting expressions for acoustic impedances will only be valid within this restriction. When choosing a radiation impedance many texts suggest using the impedance of a piston vibrating in an infinite baffle as many 4 Problem analysis 15

23 4.3 Modeling of subwoofer enclosures will be large enough to make this approximation valid [5, p ], [9, p. 23], [10, p. 15]. Since one of the goals in this project is to minimize the volume of the enclosure it is assumed that the infinite baffle approximation is not valid. According to [5, p ] the radiation impedance of a closed box subwoofer is more accurately approximated by the radiation impedance of piston vibrating in the end of a long tube when the dimensions of the enclosure is less than 0.6 x 0.6 x 0.6 [m] - therefore this approximation is used. Even the approximated geometry yields a complex mathematical expression for the radiation impedance and therefore the following simplified expression for the radiation impedance is used: Z R = ρ ω ρ + jω 4 π c a (4.13) where ρ is the density of air, c is the speed of sound in air, ω is the angular frequency of the sound wave and a is the radius of the diaphragm. Converting the ( radiation ) impedance to its equivalent mechanical impedance using equation (4.12) shows that the term ρ ω 2 corresponds to a damper with damping coefficient: 4 π c β r = S2 d ρ ω2 4 π c ( ) Similarly the term jω ρ a corresponds to an extra mass of [ ] N s m (4.14) m r = S2 d ρ a [kg] (4.15) in the mechanical system. This is a result of the air load that must be accelerated by the diaphragm as it vibrates. The acoustic impedance of a closed tube is used to model the box impedance. In the conversion from acoustic to mechanical impedance it is assumed that the input to the radiation impedance is sinusoidal and therefore in steady state excitation jω = s. The box impedance for a closed box enclosure, Z B, depends on whether the inside of the box is reflective (without acoustic absorption) or lined with sound absorbing material. The acoustic absorption is normally used to prevent standing waves inside the box which occur when the depth of the box is half a wavelength [5, p. 219]. It is assumed that the box dimensions will be small enough to avoid this phenomenon. In this assumption harmonics are not considered. Acoustic lining in the box can also be used to dampen the system but other and better methods of achieving further damping of the system will be used if necessary. This will be discussed in section 6.2. For frequencies where the wavelength is larger than 8 times the smallest dimension of the box the acoustic impedance can be described as: Z B = jω Bc ρ π a + 1 jω γ P 0 V box (4.16) 16 4 Problem analysis

24 4.3 Modeling of subwoofer where a is the radius of the diaphragm, γ is the thermodynamic ratio of specific heats C p/c v of the gas, P 0 is the atmospheric pressure and V box is the volume of the enclosure. B c is a correction factor dependent on the ratio of the diaphragm area, S d, to the area, L 2 of the side in which the driver is mounted - the value of B c as a function of S d/l 2 is seen in figure 4.7. Figure 4.7: B c as a function of S d/l 2 [5, p. 218]. A curve fit to the curve seen in figure 4.7 has been made using MATLAB R resulting in an approximated relationship between S d/l 2 and B c given by: B c = ( ) 2 ( ) Sd Sd L L (4.17) The MATLAB R m-file to calculate the curve fit can be found in appendix A.8. Converting the( acoustic) box impedance in equation (4.16) to an equivalent mechanical impedance shows that the term 1 jω γ P 0 V box corresponds to an extra spring constant, k box, in the mechanical system with a value of: k box = S2 d γ P 0 V box (4.18) This extra spring effect is due to the compression of the air inside the box. Again the signal is assumed sinusoidal. ( ) The other term in the acoustic impedance jω Bc ρ π a corresponds to an extra mass of m box = S2 d B c ρ π a (4.19) 4 Problem analysis 17

25 4.3 Modeling of subwoofer in the mechanical system. This is due to the mass of the air that has to be moved by the diaphragm (not to be confused with the total mass of air inside the box). The total spring coefficient, k, total damping coefficient, β, and total mass, m, of the mechanical system will be: k = k ms + k box β = β ms + β r m = m ms + m r + m box (4.20) This gives the mass-spring-damper system seen in figure 4.8 k m B. l. i(t) β x(t) Figure 4.8: Free body diagram of the mechanical system. Applying Newtons 2nd law on the mass and Laplace transforming yields: B l I(s) = ( s 2 m + s β + k ) X(s) (4.21) It will be shown later that the frequency dependent term, β r, can be neglected. In order to obtain an impedance curve of the system (showing the total impedance seen from the amplifier) a current-to-voltage transfer function of the system is made by isolating X(s) in equation (4.21), inserting into equation (4.7) and solving for V s(s) I(s). This yields the following impedance function: [ V s (s) m L e s3 + [β L e + m R e ] s2 + k L e + β R e + (B l) 2] s + k R e I(s) = m s 2 + β s + k (4.22) where k, β and m are given by equation (4.20). The impedance curve will later be used to verify the model Problem analysis

26 4.3 Modeling of subwoofer Sound pressure for closed box system When analysing a loudspeaker system the final goal is usually to calculate the rms sound pressure produced in the room in which the loudspeaker is placed as this is what the listener will perceive. The directivity pattern for the radiation impedance of the system (approximated as a vibrating piston in the end of a tube) illustrates how the sound pressure will be radiated into the surrounding air. This directivity pattern is seen in figure 4.9 [5, p. 104] for different values of k a (k is the wave number and a is the radius of the piston): Figure 4.9: Directivity patterns for a piston vibrating in the end of a tube as a function of k a. It is seen that for low frequencies or small radii the loudspeaker is nearly non directional - the radiation of sound is equal in all directions. Therefore the subwoofer may be treated as though it was a simple spherical source - this approximation is valid as long as the radius of the diaphragm is smaller than the 4 Problem analysis 19

27 4.3 Modeling of subwoofer wavelength [5, p. 222]. In this case the sound pressure at the point A is independent of the angle, θ, between the source and A but dependent on the distance, r, from A to the source. The magnitude of the rms sound pressure produced at distance r is calculated as [5, p. 222 eq. 8.16]: p = jω U0 ρ 4 π r (4.23) where U 0 is the rms volume velocity at the surface of the source. This is equal to the velocity of the diaphragm multiplied by the area of the diaphragm. Thereby equation (4.23) is rewritten as: p = jω Sd ẋ ρ 4 π r (4.24) the variables p and ẋ are sinusoidal and for steady-state sinusoidal excitation s = jω. Laplace transforming equation (4.24) yields: P(s) = s 2 Sd ρ X(s) (4.25) 4 π r A voltage-to-position transfer function is derived by isolating I(s) in equation (4.21), inserting into equation (4.7) and solving for X(s) V s (s). This yields the following voltage-to-position transfer function: X(s) V s (s) = B l m L e s 3 + [β L e + m R e ] s 2 + [k L e + β R e + (B l) 2] (4.26) s + k R e In order to obtain a voltage-to-sound pressure transfer function equation (4.26) is inserted into equation (4.25). This yields the following voltage-to-sound pressure transfer function: P(s) V s (s) = S d ρ B l 4 π r s 2 m L e s 3 + [β L e + m R e ] s 2 + [k L e + β R e + (B l) 2] (4.27) s + k R e Often the transfer function stated in equation (4.27) is simplified by neglecting the inductance of the voice coil as the impedance caused by the inductance is only significant at high frequencies. It will later be shown that this simplification is valid for frequencies considered in this report. Neglecting L e simplifies equation (4.27) to a 2nd order high-pass filter: P(s) V s (s) = S d ρ Bl 4 π m R e r [ m k [ m k ] s 2 ] s 2 + [ β Re +(B l) 2 k R e ] s + 1 (4.28) It is convenient to express the ratio of radiated rms sound pressure, p, to a reference sound pressure, p re f, 20 4 Problem analysis

28 4.3 Modeling of subwoofer in decibel. This ratio is called the sound pressure level or SPL. In this report a reference sound pressure of 20 [µpa] will be used. This is the threshold of human hearing. The voltage-to-spl transfer function is: SPL(s) V s (s) = S d ρ Bl 4 π m R e r p re f [ m k [ m k ] s 2 ] s 2 + [ β Re +(B l) 2 k R e ] s + 1 (4.29) Equation (4.29) is a 2nd order high pass filter with passband gain, K pb, cut-off frequency, ω c, and damping factor, ζ, given by: S d ρ Bl K pb = 4 π m R e r p re f [ ] SPL V (4.30) ω c = k m [ ] rad s (4.31) and ζ = β R e + (Bl) 2 2 R e k m (4.32) Bass reflex system In the analysis of a bass reflex loudspeaker system the mechanical system considered so far will not be adequate as the vent (or port) in the enclosure will add a degree of freedom to the system. The resulting mechanical system is illustrated in figure 4.10: 4 Problem analysis 21

29 4.3 Modeling of subwoofer x(t) m,β,k,s ms ms ms d m box,k box F m r,β r m p,β p,s m r,β r p x (t) p Figure 4.10: The mechanical system of a vented box loudspeaker system. In figure 4.10 m ms, m box, m r, β ms, β r, k ms, k box, S d, F and x(t) are the same as for the closed box case. m p is the mass of the air in the port, β p is the damping factor of the port due to viscous friction, S p is the cross-sectional area of the port and x p (t) is the displacement of the air in the port. It is assumed that the masses m ms, m box and m r move in phase. The resulting coupled system is seen in figure k ms β p+ βr m + m p r k box m + m + m ms r box B. l. i(t) β + β ms r x (t) p x(t) Figure 4.11: Diagram of bass reflex loudspeaker system. For vented box systems the produced sound pressure will be radiated from both the diaphragm and the port, and according to [5, p ] the total radiated sound pressure is a function of the volume velocity from the diaphragm, U 0 (s) = S d s X(s), subtracted the volume velocity from the port, U p (s) = S p s X p (s), i.e. the total produced sound pressure will be a function of the air necessary to compress the air inside the box, U b (s) = U 0 (s) U p (s). When solving for U b (s) it is much more convenient to express 22 4 Problem analysis

30 4.3 Modeling of subwoofer the system seen in figure 4.11 as its equivalent circuit diagram seen in figure 4.12 [11, ch. 1] [5, p. 241] as compared to the method used for the closed box system. x β m ms m ms β r m 1 box r m r β r k ms m p B. l. i(t) x =x-x p b 1 kbox x p β p Figure 4.12: Equivalent mechanical circuit diagram for the system seen in figure The electrical system can be included in the circuit diagram by converting the voltage source and the electric impedances to the mechanical side using equation (4.8) and (4.6). The resulting mechanical circuit diagram is seen in figure x (Bl) 2 R e βms 1 m ms β r m r m box k ms m r β r m p Bl V R e s x =x-x p b 1 kbox x p β p Figure 4.13: Equivalent mechanical circuit diagram for the mechanical and electrical system. In figure 4.13 the inductance of the voice coil is neglected as in the closed box case. Now all the mechanical impedances can be converted into its equivalent acoustic impedances using equation (4.12) to convert the force, F, and velocity, ẋ, to sound pressure, p, and volume velocity, U. The resulting equivalent acoustic circuit diagram is seen in figure U m β ms m ms β ar m (Bl) 2 S2 abox d ar Sd 2 Re S2 S2 d d k ms m ar β ar m av BlV s S d R e U =U-U p b C abox U p β av Figure 4.14: Equivalent acoustic circuit diagram for the mechanical and electrical system. 4 Problem analysis 23

31 4.3 Modeling of subwoofer In figure 4.14 β ar is the acoustic radiation resistance, m ar is the acoustic radiation mass, m abox is the acoustic mass of the air load of the box, C abox is the acoustic compliance of the box, m av is the acoustic air mass of the vent and β av is the acoustic resistance of the port. β ar, m ar, m abox and C abox are determined from equation (4.13) and (4.16) and stated below: β ar = ρ ω2 4 π c (4.33) m ar = ρ a (4.34) m abox = B c ρ π a (4.35) C abox = V box γ P 0 (4.36) The values of m ap and β ap are determined from the impedance of the port. The acoustic impedance of an open tube is used to model the port. This acoustic impedance, Z p, is [5, p ]: Z p = ρ 2 ω µ π a 2 p ( 1 + l ) p + jω ρ(l p + l c ) a p π a 2 p (4.37) where ρ is the density of air, µ is the kinematic coefficient of viscosity which varies with temperature and pressure (at 20 C and atmospheric pressure, µ = [ m 2 /s] ), ap is the radius of the tube, l p is the length of the tube and l c is an end-correction factor for the inner end of the tube which is a p [5, p ]. m av and β av are determined from equation (4.37): m av = ρ (l p + l c ) π a 2 p (4.38) and β av = ρ 2 ω µ π a 2 p ( 1 + l ) p a p (4.39) Now all the values in figure 4.14 are known and the volume velocity U b can be found using traditional circuit analysis techniques. In order to simplify the result some parameters are collected and the resulting 24 4 Problem analysis

32 4.3 Modeling of subwoofer circuit diagram is seen in figure U β at C as m at m ap BlV s S d R e U =U-U p b C abox U p β ap Figure 4.15: Simplified acoustic circuit diagram. In figure 4.15 the new circuit parameters are: m at is the total acoustic mass that moves in phase with the diaphragm: m at = m ms S 2 d + m ar + m abox β at is the total acoustic resistance working against the movement of m at : β at = (Bl)2 S d R + β 2 ms + β e Sd 2 ar C as is the acoustic compliance due to the spring constant of the suspension: C as = S2 d k ms m ap is the total acoustic mass moving in phase with the port air mass: m ap = m ar + m av β ap is the total acoustic resistance working against the movement of m ap : β ap = β ar + β av Solving for U b in the system seen in figure 4.15 gives the following voltage-to-volume velocity transfer function: U b (s) V s (s) = Bl R e m at S d A 1 s 3 + A 2 s 2 B 1 s 4 + B 2 s 3 + B 3 s 2 + B 4 s + 1 (4.40) where R e is the resistance of the voice coil, S d is the surface area of the diaphragm and A 1, A 2, B 1, B 2, B 3 and B 4 are given by: A 1 = B 1 = m at m ap C abox C as A 2 = m ap β ap C abox C as B 2 = m at β ap C abox C as + m ap β at C ab C as (4.41) B 3 = m at C as + β at β ap C abox C as + m ap C ab + m ap C as B 4 = β ap C abox + β ap C as + β at C as The SPL produced by the diaphragm and port is calculated as [5, p ]: 4 Problem analysis 25

33 4.3 Modeling of subwoofer SPL(s) V s (s) = s ρ Ub(s) 4 π r p re f V s (s) = Bl ρ A 1 s 4 + A 2 s 3 4 π m AT R e S d r p re f B 1 s 4 + B 2 s 3 + B 3 s 2 + B 4 s + 1 (4.42) It is observed that the acoustic resistance (damping) caused by the viscous friction of the air in the port, β av, and the radiation resistance from the port and diaphragm, β ar, is negligible compared to the other acoustic impedances. A bode plot of the voltage-to-spl transfer function for a subwoofer used for testing is seen in figure the blue curves includes β av and β ar and the red curve does not. The data used in the transfer function is from a Peerless SLS 12 driver in an 51 [litre] enclosure. The data sheet for the driver can be found in appendix A.8 Figure 4.16: Bode plot of equation (4.42) with β av and β ar (blue) and without β av and β ar (red) for frequencies between 0.1 and 200 [Hz] using a Peerless SLS 12 driver in an 51 [litre] enclosure. As seen from the bode plot in figure 4.16 the port and radiation damping is only significant at very low frequencies, which is outside the audible range of the human ear. This frequency response only describes one specific driver in one specific enclosure, but measurements on 50 different loudspeakers have shown that this simplification does not cause significant errors [12, p ]. Neglecting β av reduces equation (4.42) to a standard 4th order high-pass filter: SPL(s) V s (s) = K α 1 s 4 pb α 1 s 4 + α 2 s 3 + α 3 s 2 + α 4 s + 1 (4.43) with the coefficients K pb, α 1, α 2, α 3 and α 4 given by: 26 4 Problem analysis

34 4.3 Modeling of subwoofer where β at now only includes Bl ρ K pb = 4 π m at R e S d r p re f ( (Bl) 2 S 2 d R e α 1 = m at m ap C abox C as α 2 = m ap β at C abox C as (4.44) α 3 = m at C as + m ap C abox + m ap C as α 4 = β at C as ) ( ) and βm. Sd 2 The voltage-to-spl transfer function stated in equation (4.43) has two cut-off frequencies as it is composed by two 2nd order systems. They are determined by the resonance frequency of each of the two moving masses. The two cut-off frequencies are determined as [12, p. 185]: ω c1 = ω c2 = 1 mat C as (4.45) 1 map C abox (4.46) In the passband (in this case this will be at higher frequencies than the highest cut-off frequency) the gain is K pb, given in equation (4.44). As for the closed box system an impedance function is made for the vented box system in order to verify the model. For this purpose two impedance function are made - one including the inductance of the voice coil, L e, the viscous friction of the air in the port, β av, and the radiation resistance from the port and diaphragm, β ar, and another function neglecting these terms. The impedance function is found by analysis of the system seen in figure Applying Newton s 2nd law on the mass m = m ms + m r + m box and Laplace transforming yields: B l I(s) + k box X p (s) = ( s 2 m + s β + k ) X(s) (4.47) where β = β ms + β r and k = k ms + k box. Applying Newton s 2nd law on the mass m v = m p + m r and Laplace transforming yields: k box X(s) = ( s 2 m v + s β p + k box ) Xp (s) (4.48) Isolating X p (s) in equation (4.48) and inserting into equation (4.47), isolating X(s), inserting into equa- 4 Problem analysis 27

35 4.3 Modeling of subwoofer tion (4.7) and solving for V s (s) yields the impedance function (the function is including L e, β av and β ar ): V (s) I(s) = C 1 s 5 +C 2 s 4 +C 3 s 3 +C 4 s 2 +C 5 s +C 6 D 1 s 4 + D 2 s 3 + D 3 s 2 + D 4 s + D 5 (4.49) with the coefficients C 1, C 2, C 3, C 4, C 5, C 6, D 1, D 2, D 3, D 4 and D 5 given by: C 1 = L e m m p C 2 = m m p R e + L e m β p + L e m p β C 3 = R e m β p + R e m p β + (Bl) 2 m p + L e m k box + L e m p k ms + L e β β p C 4 = R e m k box + R e m p k ms + R e β β p + (Bl) 2 β p + L e β k box + L e β p k ms C 5 = R e β k box + R e β p k ms + (Bl) 2 k box + L e k ms k box + L e kbox 2 (4.50) C 6 = R e k ms k box + R e kbox 2 D 1 = m m p D 2 = m β p + m p β D 3 = m k box + m p k ms + β β p D 4 = β k box + β p k ms D 5 = k ms k box + kbox 2 The impedance function given by equation 4.49 is seen in the bode plot in figure 4.17 together with a plot of the same function with the terms L e, β av and β ar neglected. Again the Peerlees driver in a 51 [litre] enclosure is used. Figure 4.17: Bode plot of equation (4.49) with L e, β av and β ar (blue) and without L e, β av and β ar (red) for frequencies between 10 and 200 [Hz] using a Peerless SLS 12 driver in a 51 [litre] enclosure. The derived equations will form the basis of the design optimisation of a subwoofer system Problem analysis

36 4.4 Specifications and limitations 4.4 Specifications and limitations In this section it will be investigated how the demands for the subwoofer system stated in chapter 2 can be reformulated as specific design criteria based on the equations derived in section 4.3. The demands stated in chapter 2 are represented here for convenience: The subwoofer system should be capable of reproducing sound with a flat frequency response down to 40 [Hz]. Furthermore is should have low harmonic and transient distortion, while keeping the volume of the subwoofer as small as possible. The design limitations resulting from the requirements will be determined in this section, along with some other limitations concerning the physical limits of the system and the scope of the model. The design criteria will be divided into different categories and will be stated for both closed and vented box enclosures Flat frequency response As described in section 4.3 the closed and vented box systems may be described by high-pass filters of 2nd and 4th order respectively. In order to achieve a flat frequency response down to the resonance frequency of the system both filters must be properly dampened. However there is a trade-off between bandwidth and flatness of the frequency response, as increased damping and thereby increased flatness will result in decreased bandwidth [13, p. 415] and vice versa. One special class of filters result in maximum flatness in the passband - these filters are called Butterworth filters [7, p. 12] and it will be especially desirable to design the subwoofer system as a Butterworth filter. Closed box: For the closed box system where the voltage-to-spl transfer function is a 2nd order highpass filter it will be necessary to match the coefficients of the filter to the coefficients of a 2nd order Butterworth filter which has the form: SPL(s) V s (s) = K pb where K pb is the passband gain and ω c is the cut-off frequency of the system. ( ωc s 1 ) ω c s + 1 (4.51) Comparing equation (4.51) to equation (4.29) gives the criterion for maximal flatness of the frequency response of the closed box subwoofer: This is the Butterworth criterion for the closed box system. ζ = β R e + (Bl) 2 2 R e k m = 1 2 (4.52) Vented box: As mentioned in section 4.3 the vented box system yields a voltage-to-spl transfer function with two cut-off frequencies, one determined by the driver and one determined by the box and port. 4 Problem analysis 29

37 4.4 Specifications and limitations In order to design the system as a Butterworth filter these two frequencies must be equal. If the system experiences two cut-off frequencies, then the response will not be maximally flat. From equation (4.43) to (4.46) it is observed that the criteria for making the two cut-off frequencies equal is: 1 1 = mat C as map C abox m at C as = m ap C abox (4.53) When the two cut-off frequencies are matched the coefficients of the voltage-to-spl transfer function describing the vented box system can be matched to the coefficients of a 4th order Butterworth filter which has the form [12, p. 186]: SPL(s) V s (s) = K pb ( ωc s ) ( ωc s 1 ) 3 ( ωc s where ω c is the cut-off frequency of the system which is equal to ω c1 and ω c2. ) 2 ( ωc ) (4.54) s + 1 The voltage-to-spl transfer function equation (4.43) is rewritten in order to be able to compare it to equation (4.54): SPL(s) V s (s) = K 1 pb 1 α 1 s + α 4 4 α 1 s + α 3 3 α 1 s + α 2 2 α + 1 (4.55) 1 s Matching of coefficients of equation (4.54) to equation (4.55) shows that the criteria for maximal flatness of the frequency response of the vented box subwoofer is: 1 α 1 = ω 4 c (4.56) α 4 α 1 = ω 3 c (4.57) α 3 α 1 = ω 2 c (4.58) α 2 α 1 = ω c (4.59) Insertion of equation (4.44) into equation (4.56) to (4.59) reveals that equation (4.56) to (4.59) is synonymous to: 30 4 Problem analysis

38 4.4 Specifications and limitations Q T = mat C as = 1 β at (4.60) C as C abox = (4.61) ( The term Q T is the called the total quality factor of the system and Cas is called the compliance ratio. Equation (4.53), (4.60) and (4.61) are the Butterworth criteria for the vented box system. C abox ) System passband The passband of the system will be defined as the frequency band for which the magnitude of the voltageto-spl transfer function describing the system is K pb or less than 3 [db] below K pb. The low frequency limit of the passband is determined by the cut-off frequency of the system. In order to meet the requirement of a flat frequency response down to 40 [Hz] the cut-off frequency must be lower or equal to this frequency. The cut-off frequency is not constrained but it is wanted as low as possible. As the designed subwoofer is intended to be used as part of a home cinema system only the frequencies present in the low frequency effects (LFE) channel will be considered. The LFE channel is an optional band-limited effects channel that delivers bass-only information in e.g. 5.1 or 7.1 channel audio recordings. According to Dolby Laboratories: The LFE channel for cinema applications has a range that extends up to 120 [Hz]. [14, p. 4-7]. Therefore the maximum system frequency considered in this project is limited to 120 [Hz]. Frequencies above this limit are expected to be filtered out by a cross-over network. The model developed in section 4.3 is only valid for frequencies within the piston range of the loudspeaker driver. The size of the most commonly used loudspeaker drivers in subwoofers are between 8 to 15 [inch] (approximately 0.20 to 0.38 [m]) [3]. This means that the wavelengths considered must be larger than 1.2 [m] for 8 [inch] drivers and 2.28 [m] for 15 [inch] drivers in order to be able to predict the behavior of the subwoofer using the model. This corresponds to frequencies below 286 [Hz] for 8 [inch] drivers and 151 [Hz] for 15 [inch] drivers. As frequencies above 120 [Hz] are not considered in the model this will not be a problem Enclosure The scope of the model introduces some constraints on the enclosure dimensions. The acoustic radiation impedance of the diaphragm is approximated as the radiation impedance of a piston vibrating in the end of a long tube (equation (4.13)). This approximation only describes the diaphragm well when the dimensions of the box are less than 0.6 x 0.6 x 0.6 [m]. In figure 4.18 a sketch of the enclosure is seen where the driver is to be placed in the hole in the xy-surface. 4 Problem analysis 31

39 4.4 Specifications and limitations y x z Figure 4.18: The driver position according to x, y and z coordinates. This sets up the following criteria for the enclosure: V box = [ m 3 ] (4.62) There is also a minimum length for x and y since they should be able to house the driver. It is decided that there should be at least 1 [cm] free on each side of the driver. That is: x y D s [m] (4.63) where D s is the outer diameter of the driver. The acoustic impedance of the box is approximated as the impedance of a closed tube. This approximation is only valid when the smallest dimension of the enclosure is smaller than one eighth wavelength. As the smallest wavelength considered in this project is 2.86 [m] (corresponding to a frequency of 120 [Hz]) the smallest dimension of the enclosure must be less than [m] i.e: x y z [m] (4.64) As for x and y the dimensions of the driver determines the minimum length of z. It is decided that there should be at least 4 [cm] free behind the driver. That is: 32 4 Problem analysis

40 4.4 Specifications and limitations z depth driver [m] (4.65) where depth driver is the depth of the driver. For the vented box there are some additional constraints. As both the driver and port will radiate sound equally in all directions the port can be placed in all sides of the enclosure but it is decided that it cannot be placed in the front (where the driver is placed) or in the rear side. It is assumed that the port length, l p, will be larger than the port diameter, d p, therefore the port opening will be in the smallest enclosure wall. 1 [cm] of free space is wanted in each side of the port - therefore: d p x y z [m] (4.66) When the radius of the port decreases the velocity of the air flow through it will increase. If the flow velocity is too high a windy noise will result from the flow. As a rule of thumb the minimum diameter of the port is given as [7, p. 51]: d p where V d is the peak volume displacement of the driver. ωc 2 π V d [m] (4.67) There is no constraint on the length of the port, l p, as the port can be bended inside the box [7, p. 54] if the port length is longer than the longest side length Driver The driver used in the subwoofer system is - due to the time frame of this project - limited to an already manufactured driver.this choice influences the optimisation of the subwoofer since the driver parameters are given and cannot be altered independently of each other. 10 drivers which are already in production are chosen for further investigation. The drivers are chosen to give a broad spectrum of drivers available on the market, they are described in appendix A.7. In the selection of the 10 drivers considered in the project a limitation of the driver diameter has been made. The maximum sound pressure produced by the driver is limited by the peak volume displacement, V d = S d x max [8, p ], [15]. A minimum requirement for the peak volume displacement of the driver can be calculated from [15, p. 596]: V d = ( ( ( ))) SPL r=1 m 40 log( f ) 20 5+log π ρ 2 [m 3 ] (4.68) 4 Problem analysis 33

41 4.4 Specifications and limitations where SPL r=1 m is the required SPL measurered at 1 [m], f is the frequency of the wave for which the SPL is measured and ρ is the density of air. Equation (4.68) is only valid for a driver placed in an infinite baffle but the equation is used as an approximation for the actual system and is only used to set up a minimum requirement for the driver. The following minimum requirement for the driver is set up: The mechanical limitations should not constrain the system from producing a SPL of at least 90 [db] at 1 [m] from the driver at a frequency of 40 [Hz]. Inserting this into equation (4.68) yields a minimum peak displacement of 75.4 [ cm 3]. When the necessary peak volume displacement, V d, is known, the required linear displacement, x max, can be calculated as: x max = V d S d [m] (4.69) Equation (4.69) is plotted in figure 4.19 using typical values of S d for drivers of different diameters. Figure 4.19: The graph shows driver diameter vs. required linear driver displacement at a peak volume displacement of 75.4 [ cm 3]. It follows from the graph that drivers with diameters lower than 6.5 [inches] requires maximum displacements beyond known fabricated driver specifications.[16][17] Transient distortion Transient distortion is a term used to describe the unwanted distortive effect that occurs when the transient response of a loudspeaker has a large time constant. When a steady sound field is terminated the sound pressure will not drop instantaneously because of the reverberation time of the room [5, p ], [18, p ]. The reverberation time is defined as the time it takes for the sound pressure in a room to drop 60 [db] from its original value [18, p. 198]. The time it takes for the diaphragm velocity to die out (or, for the vented box system, the velocity of the diaphragm minus the velocity of the air in the port) will be perceived as an extension of the reverberation time. According to [18, p. 198] an added 34 4 Problem analysis

42 4.4 Specifications and limitations reverberation time of [s] due to the transient hangover of the subwoofer will not be perceived by the listener. At low sound pressure level or at low frequencies near the limit of the MAF (around 20 [Hz]) the reverberation time perceived by the listener will be shorter than the time it takes for the sound pressure to fall 60 [db] from its initial value because the human ear simply cannot detect low sound pressure especially at low frequencies. The frequency of the ringing in the transient period after a sound source is terminated will be approximately the cut-off frequency and the reverberation time can be expressed as a number of cycles of this frequency. In [18, p. 199] a table of calculated reverberation times for different kinds of filters is given and the reverberation time for the closed and vented box systems are: Closed box system: This is a 2nd order Butterworth filter - the reverberation time is 1.63 times the period of the cut-off frequency. Vented box system: This is a 4th order Butterworth filter - the reverberation time is 2.87 times the period of the cut-off frequency. Butterworth filters have very good transient response and it is observed that the cut-off frequency can be as low as 8.15 [Hz] for the closed box system and [Hz] for the vented box system in order to keep the reverberation time under 0.2 [s]. It is assumed that the cut-off frequencies considered in this project will be higher than [Hz] and all systems will be tuned as a Butterworth alignment, therefore no further consideration of transient distortion will be taken Harmonic distortion When using transfer functions in the design optimisation the system is assumed linear. However the operation of subwoofers is non-linear, especially when the diaphragm displacement become large. Harmonic distortion can occur when non-linearities are present in the system. Some of the non-linear effects that can cause harmonic distortion are listed below based on [19]: The restoring force on the diaphragm caused by the suspension is a non-linear function of the diaphragm displacement. This causes harmonic distortion especially at large displacements. The force factor, Bl, is a non-linear function of the driver displacement when the displacements are larger than x max as the number of windings enclosed by the magnetic flux from the permanent magnets will change beyond this limit. Therefore the diaphragm displacements should be limited in order to maintain the linearity between the force factor and the diaphragm displacement. As seen from equation (4.14) the damping caused by the radiation resistance of the diaphragm is a non-linear function of the frequency of the radiated sound wave. The flow resistance in the port of vented enclosures is a non-linear function of the velocity of the flow through the port and thereby a non-linear function of the radiated sound wave - see equation (4.39). The non-linearities stated above cannot be altered. However, minimising the excursion of the diaphragm can reduce the harmonic distortion caused by the suspension and the force factor. Reduction of the diaphragm excursion can be obtained by choosing a driver with a large effective diaphragm area, S d, as 4 Problem analysis 35

43 4.4 Specifications and limitations the produced SPL is a function of the volume displacement. A certain sound pressure can be produced by smaller displacement of the diaphragm if the effective area is equivalently larger. Another way of reducing the diaphragm excursion is by choosing a vented enclosure as the produced SPL will be radiated from both the diaphragm and the port. According to equation (4.42) the produced SPL is a function of the velocity of the diaphragm minus the velocity of the port, i.e. the port relieves the diaphragm and the resulting harmonic distortion from the suspension non-linearity will be reduced. However, when a vented enclosure is used harmonic distortion will be present due to the non-linear flow resistance. The reduced harmonic distortion achieved by reducing the diaphragm movement will most often justify the added harmonics due to the non-linear flow resistance as the total harmonic distortion will be reduced, especially if the vent diameter is sufficiently large (as stated in equation (4.67)) [7, p ] [5, p ]. A possible way of compensating for the non-linearities of the system could be by implementing position feedback control. This will be investigated further in section Cost Since the subwoofer is only a prototype the price is of low significance. However the price for materials is set to a maximum of 4500 DKr. according to an interview with Hi-Fi-klubben seen in appendix A Problem analysis

44 4.5 Closed loop control 4.5 Closed loop control As mentioned in section 4.4 some of the non-linearities of the system can be compensated for using a position feedback control. When applying position feedback the displacement of the diaphragm is controlled. The harmonic distortion caused by the non-linearities in the suspension could be suppressed in this way along with other unwanted disturbances which normally would cause displacement errors. When using position feedback the voltage-to-position transfer function of the system (see equation (4.26) as an example) determine the reference diaphragm position from the applied input voltage. Different kinds of sensors can be used to measure the position of the diaphragm which can be compared to the reference. The position can also be calculated from the velocity or acceleration of the diaphragm - this expands the possible sensor-topologies that can be implemented. Different kinds of sensors for this purpose are described in appendix A.5. A block diagram of a position controlled subwoofer system is seen in figure Audio signal Preamplifier V in Power amplifier X K. X ref X error V s Controller K V s Subwoofer system X Sensor Figure 4.20: Block diagram showing position feedback control in a subwoofer system. In figure 4.20 the constant K denotes the gain of the power amplifier which must be accounted for when calculating the reference position X re f. The block Controller can be designed to meet the requirements of the system. Besides the non-linearities described in section 4.4 a position feedback control can compensate for other unwanted disturbances. Manufactures of subwoofers using position feedback technologies claim that it reduces many different problems related to accurate sound reproduction [20]. These problems being for example: Thermal induced distortion, spider and surround distortion and frequency response dependent of driver impedance. As an example, the thermal induced distortion, or thermal compression is considered. This kind of distortion occurs as the coil in the driver heats up when the current passes through it. The resistance of a conductor varies approximately linearly with temperature within a limited temperature range [21, p.843]. The formula for the variation of resistance caused by temperature is expressed as: R = R 0 [1 + α(t T 0 )] [Ω] (4.70) Where R 0 is the resistance at a reference temperature, T 0 (usually at 20 [ C]) and α is the temperature coefficient of resistivity. For copper α is [( C) 1 ]. When the temperature in the coil builds up, the resistance increases, this results in less electric power delivered to the driver, hence a reduction in the SPL. This is illustrated in figure 4.21 (blue). By implementing a closed loop with the position of the diaphragm as the feedback signal the voltage to the driver will be raised until expected movement is achieved, thereby eliminating the thermal compression. This is illustrated in figure 4.21 (red) as a signal 4 Problem analysis 37

45 4.5 Closed loop control with a constant amplitude. Figure 4.21: Example of a constant output over time. As the temperature in the coil builds up, the output decreases (blue). With position closed loop control (red). Although position feedback is useful for reducing harmonic distortion it will not be included in the optimisation process, but will be left as an option for future improvement of the system Problem analysis

46 Problem statement 5 The problem analysis leads to the following problem statement: Optimise the design of a subwoofer system using the mathematical models developed in section 4.3. The design optimum should be based on the wish for low enclosure volume and low resonance frequency while satisfying the constrains stated in section 4.4. Based on this statement the following is required: Verify the closed box- and vented box model using data achieved by laboratory tests. Optimise closed and vented enclosure design for 10 selected drivers. Compare the optimum designs for the 10 drivers in both closed and vented enclosures. 5 Problem statement 39

47 40 5 Problem statement

48 Problem solution Model verification A verification of the mathematical model is performed as the design optimisation will be based on this model Verification using SPL tests To verify the model a test of two different subwoofer systems are performed in an anechoic chamber at Aalborg University, see appendix A.1 for a detailed description of the test. The SPL is measured at a distance of 1.57 [m] from the subwoofer for frequencies going from 10 to 200 [Hz]. This SPL is denoted, SPL 1.57[m]. Using the measured results, frequency response graphs are plotted for the two systems and then compared with the output of the mathematical model. The first subwoofer tested is a Vifa 10 [inch] driver in a 24 [litre] closed box environment - the data sheet for the driver is seen in appendix A.8. The frequency response of this subwoofer is seen in figure 6.1. Figure 6.1: SPL 1.57[m] test of a 10 [inch] driver in closed box at different input voltages. Test 1 at 20 [V ] peak to peak and test 2 at 40 [V ] peak to peak. Test 1 (green) shows the SPL using an amplifier output of 20 [V ] peak to peak. There is no clear cut-off frequency which makes comparison to the simulation difficult. The frequency response from approximately 50 [Hz] to 100 [Hz] settles at an amplitude around 90 [db] for test 1 and 97 [db] for test 2 (with amplifier output of 40 [V ] peak to peak), until it drops again at the high frequencies. At f 33 [Hz] there 6 Problem solution 41

49 6.1 Model verification is a peak in the amplitude which can be explained by possible reflections in the room at this frequency - because the room is not anechoic below 200 [Hz] - causing standing waves at certain frequencies and therefore a peak in the SPL-curve. In test 2 (red) the amplifier output is increased to 40 [V ] peak to peak. As expected the SPL 1.57[m] is raised in the whole frequency range but the graph still roughly follows the same progress. The output of the 10 [inch] driver in closed box is then simulated in the model at 20 [V ] peak to peak. Figure 6.2 shows the plot of the output from the model (blue) compared with the measured output from test 1 (green). Figure 6.2: SPL test 1 of the 10 [inch] driver at 20 [V ] peak to peak compared with the output of the simulation. The simulated curve follows the measured curve quite well at low frequencies from 20 to 60 [Hz] when considering that the model does not simulate any effects from the room causing peaks and dips in the SPL-curve. At frequencies from 60 to 100 [Hz] the model is 2-4 [db] higher in SPL. The second subwoofer tested was a 12 [inch] Peerless driver in a bass reflex enclosure of 51 [litre] - the data sheet for the driver is seen in appendix A.8. This subwoofer was tested as a bass reflex with different port lengths. Furthermore it was tested in the same manner as the 10 [inch] subwoofer, that is as a closed box configuration, by blocking the port. The frequency response of three different tests can be seen in figure Problem solution

50 6.1 Model verification Figure 6.3: SPL 1.57[m] test of the 12 [inch] driver. Test 3 in closed box and test 4 and 5 in bass reflex at 2 different port lengths. In these tests a clear cutoff frequency is seen. For test 4 and 5 in bass reflex configuration the cut-off frequency is approximately 63 [Hz]. For test 4 in closed box, the resonant frequency is a bit lower around 51 [Hz]. Again in these tests there is a peak in the SPL around 33 [Hz] which has to be caused by the same phenomenon as in the previous tests. The SPL in these tests reaches a magnitude of approximately 95 [db] at high frequencies, which was 2-3 [db] lower than the 10 [inch] subwoofer at the same input voltage. This can partly be explained by looking at the data sheets for the two drivers: The 10 [inch] has a nominal impedance of 4 [Ω], whereas the 12 [inch] is a 8 [Ω] driver. Below 30 [Hz] the frequency responses simply do not behave according too expectations. After simulating test 3 of the 12 [inch] in a closed box and again as test 4 in a vented box with a port length of 0.29 [m] the results are plotted together with the test as seen in figure 6.4 and 6.5. Figure 6.4: SPL test 3 as closed box at 40 [V ] peak to peak of the 12 [inch] driver compared with the output of the simulation. 6 Problem solution 43

51 6.1 Model verification Figure 6.5: SPL test 4 as vented box at 40 [V ] peak to peak of the 12 [inch] driver compared with the output of the simulation. The verification of the model using the SPL test shows a difference in the magnitude. The output from the model is generally higher than the outputs measured at the tests - especially for the 12 [inch] driver. This can be explained by the fact that the enclosure was lined with acoustic damping material which is not modeled in the simulation. This damping material reduces the efficiency of the system and thereby the produced SPL [12, p. 182]. Due to the influence on the SPL from the room it is concluded that a further verification of the model is necessary. Therefore a test method less susceptible to be influenced of the surroundings will be used Verification using impedance tests To verify the mathematical model, the measured impedance of the driver is compared with the impedance found from the current to voltage transfer functions in the model, derived in section 4.3. The measurements of the impedances for the closed box and vented box systems will be comparable with the calculated impedances from the model. The electrical impedance of the enclosed driver is dependent on the electrical, the mechanical and the acoustic system and therefore an impedance test can be used to verify the complete model of the system Problem solution

52 6.1 Model verification Figure 6.6: Impedance test 2 of the 10 [inch] driver in a closed box, compared with the output of the simulation. The impedance test 2 of the 10 [inch] driver in a closed box in figure 6.6 shows a peak impedance of 33 [Ω] at 60 [Hz]. The peak in the impedance is located at the resonance frequency of the subwoofer system [7, p ]. Comparing the output of the model with the test shows a higher resonance frequency for the model. The peak from the model is 37.2 [Ω] at 64.2 [Hz]. A possible explanation for this is the fact the 10 [inch] Vifa driver has been in use for several years. Supposing that the suspension decays over time resulting in a lower spring constant, k ms, of the driver, this will lower the resonance frequency as it is depending on k ms, as seen in formula Overall the impedance output of the model is acceptable compared with the test. Figure 6.7: Impedance test 4 of 12 [inch] driver in a closed box, compared with the output of the simulation. The impedance test of the 12 [inch] driver in a closed box, illustrated on figure 6.7, has an impedance peak of 36.8 [Ω] at 56 [Hz]. The simulated value 78 [Ω] at 55.3 [Hz] is very accurate on the frequency 6 Problem solution 45

53 6.1 Model verification but deviates on the peak amplitude. A possible explanation on the high peak value from the model is that damping material is present in the enclosure. This is not modeled in the simulattion. The effect of the damping material can be seen by observing that the free air impedance peak is larger than the impedance peak which occurs when the driver is enclosed which is not the case for the 10 [inch] driver (see appendix A.2). This is seen in figure 6.8. Figure 6.8: Peerless SLS 12, free air and closed box. In test 3 f peak = 29.5 [Hz] and Z peak = 50.9 [Ω] In test 4 f peak = 56 [Hz] and Z peak = 36.8 [Ω] A small change in the damping can have a large impact on the peak value and the higher damping in the measurements is ascribed to the acoustic damping material. Therefore the deviation in the impedance peaks is concluded to be of low significance. Figure 6.9: Impedance test 6 of 12 [inch] driver in vented box with 0.29 [m] port, compared with the output of the simulation. Figure 6.9 shows the plots of tested and simulated impedance for the 12 [inch] driver in a vented box. Here there are two resonance frequencies, one for the driver and one for the port. Again it is seen that 46 6 Problem solution

54 6.1 Model verification the simulated peak impedance values are higher than the tested values. The tested peak values are 25.5 [Ω] at 15.8 [Hz] and 30.9 [Ω] at 66.5 [Hz]. The resonance peaks from the model are 68.4 [Ω] at 16.2 [Hz] and 75.1 [Ω] at 67.2 [Hz]. It is again seen that the simulated frequency values are close to the actual test values but with deviations in the magnitude of the resonance peaks as for the closed box test. Using the impedance tests for verification generally shows good similarity between the tests and the output from the model. The frequency values at the resonance points are very close to the tested values, and also the impedance values outside the resonance areas shows good results. Figure 6.1 shows the calculated deviations in percent for the resonance frequencies and for two extra frequencies that are chosen outside the resonance area, that is: 30 [Hz] and 100 [Hz]. 10" closed box Z peak [Ω] f peak [Hz] Z [Ω] (at 30Hz) Z [Ω] (at 100Hz) Test Model Deviation[%] " closed box Test Model Deviation[%] " vented box Test 6 low 25.5 high 30.9 low 15.8 high Model low 68.4 high 75.1 low 16.2 high Deviation[%] low high low 2.5 high Table 6.1: Deviations between test and simulations. Based on the SPL and impedance tests the model is concluded verified and suitable for the design optimisation. 6 Problem solution 47

55 6.2 Design optimisation 6.2 Design optimisation This section is based on [22]. Based on the limitations and specifications described in section 4.4 an optimised design of the subwoofer system is to be achieved. This is done by formulating the problem statement as a mathematical function called the cost function which is to be minimised. The optimisation of the subwoofer will be divided into an optimum design for the closed box system and one for the vented box system. The cost function will be the same for both enclosure types, so that they can be compared after optimisation Design variables Design variables are the parameters that can be altered to affect the system and thereby the cost function. A significant variable considered for the closed box system is the volume of the enclosure, V box. The volume is divided into the three side lengths x, y and z which are the three design variables for the system. The optimisation for the vented box include the same three design variables x, y and z and two additional variables; l p which is the length of the port and a p the radius of the port. In this project a readily available driver will be used in the construction of the subwoofer. Therefore all the driver parameters cannot be independently chosen. Instead a selection of 10 drivers are chosen for further investigation - relevant data for the 10 drivers is seen in appendix A.7. An optimisation will be made using the data from each driver and the different optima will be compared. The design that yields the lowest value of the cost function - no matter if it is a closed box or vented box system - will be the optimal design The cost function Based on the problem statement the cost function is formulated as a scalar function which is to be minimised: f = k 1 V box + k 2 f c (6.1) In the cost function the volume of the enclosure inclusding port volume and driver volume is wanted as small as possible, therefore V box is a positive term given in [litre]. The cut-off frequency, f c, is wanted as small as possible since this increases the passband of the system as previously described, the frequency is given in [Hz]. The reason for the units in the cost function not being SI-units is that to easily compare the results to other subwoofers where these units are used. The two constants, k 1 and k 2 are weighting constants which can be altered to weigh the different terms in the cost function. To determine values for the weighing constants it is necessary to define how important the terms in the cost function are compared to each other. The volume of the enclosure is used as a base value because the implications of changing this variable is easily comprehended, therefore k 1 = 1. The natural frequency is weighted so that changing the frequency 1 [Hz] corresponds to a volume change of 1 [litre], ergo k 2 = Problem solution

56 6.2 Design optimisation Minimizing f will determine the design resulting in a weighted optimum of small enclosure volume and low cut-off frequency Constraints Based on the limitations and specifications, some necessary constraints for the system are identified. These constraints determine the feasible regions of the designs - if the constraints are not fulfilled then the design is not feasible. The constraints can be divided into two groups, equality and inequality constraints where equality constraints determine feasible values and inequality constraints determine feasible regions of the design. The constraints are set up as mathematical equations and can be seen in table 6.2. The general constraints are the constraints that apply to both the closed box and vented box system. Specific constraints for each of the enclosure types are stated below the general contraints. General constraints: Inequality constraints Equation Description 0 D s x The side length x must be 2 cm larger than driver diameter, D s. 0 V box The enclosure volume V box must be smaller than 60 x 60 x 60 [ cm 3]. 0 D s y Side length y must be 2 cm larger than driver diameter, D s. 0 depth driver z Side length z must be 4 cm larger than driver depth, depth driver. 0 min(x,y,z) Smallest side length must maximum be [cm]. Only for closed box: Equation ζ = 1/ 2 Equality constraints Description The frequency response must obey the Butterworth criteria. Only for vented box: Equality constraints Equation Description m at C as = m ap C abox The cut-off frequencies must equal = Q T The quality factor must obey the Butterworth criterion = C as/c abox The compliance ratio must obey the Butterworth criterion. Inequality constraints Equation Description 0 2 a p min(x,y,z) Diameter of the port must be smaller than smallest box side length f c V d 2 a p Diameter of the port must be larger than f c V d [m]. Table 6.2: Equality and inequality constraints for the closed box and vented box subwoofer system. The constraints are set up according to section 4.4. The side length are to be thought of as the inner measurement of the enclosure. 6 Problem solution 49

57 6.2 Design optimisation Finding local minima of the cost function The optimisation problem is solved using MATLAB R. A built in optimisation tool, fmincon, is used. This is a constrained non-linear optimization tool. The solver algorithm can be chosen specifically in the program. The Active-set algorithm based on sequential quadratic programming (SQP) is used. The solver is chosen with requirements of generality, ease of use and efficiency. The solver must be able to contain both inequality and equality constraints and is effective on problems with non-smooth constraints - this is the generality requirement. The ease of use requirement implies that the parameters used to configure the solver are easily selected. The algorithm can take large steps which adds speed - this is useful as two optimisations will be required for each driver, one for the closed box enclosure and one for the vented box enclosure Closed box The cost function for the closed box and for the vented box is optimised using different start values for the design variables and the optimum values giving the lowest cost function value are noted, along with enclosure volume, cut-off frequency and the value of the cost function. The optimum for each of the 10 chosen drivers is noted in table 6.3 for the closed box system (relevant parameters for each driver is seen in appendix A.7). Driver Optimum design Enclosure Cut-off Cost function variables volume* frequency value [cm] [litre] [Hz] [.] x y z V box ω c f opt Table 6.3: Simulation results that shows the optimum values for each driver in a closed box design. *The reason for the volume enclosure not corresponding to the product of the side lengths is that the volume occupied by the driver subtracted from the volume. However the volume to be minimised (in the cost function) is the total volume including the volume occupied by the driver. The driver volume is estimated as V driver = 1/2 a 2 π depth driver. This equation is used because the driver is assumed to occupy approximately half the volume of a cylinder with a radius equal to the effective driver radius, the rest is assumed to be air. The driver which yields the lowest cost function value is the JBL GTO " (number 6) because it delivers the best trade-off between small enclosure volume and low cut-off frequency Problem solution

58 6.2 Design optimisation To compare the cost function values they are plotted along with the enclosure volume (in [litre]) and the cut-off frequency (in [Hz]) in the graph of figure Figure 6.10: Plot showing the value of the cost function (green), the cut-off frequency in [Hz] (red) and the enclosure volume in [litre] (blue) for the 10 drivers in a closed box environment. Each driver is used in its own optimised environment. It can be seen that driver number 6 not necessarily is the driver with the lowest volume or cut-off frequency, but the driver with the best trade-off between the two. Weighing the cost function parameters differently could favor another driver in the optimisation. Another observation to be made from table 6.3 is that driver 5 and 10 has a very large volume as their optimum compared with the drivers with some of the best cost function values. This is most likely because of difficulty in satisfying the Butterworth criteria. To compare the system frequency response, the closed box system voltage-to-spl frequency response of each driver in their specific optimum design is shown in figure Figure 6.11: Plots of the output of the 10 different drivers in their optimum closed box design. The figure shows that the passband gain varies with approximately 6 [db] for the 10 drivers, with the 6 Problem solution 51

59 6.2 Design optimisation lowest driver passband gain of 71 [db]. The magnitude of the SPL-curves for each driver is not comparable as the drivers have different impedances. As the simulations are made with a sinusoidal input voltage with an amplitude of 1 [V ] the drivers with low impedance will generally produce a higher SPL as they will consume more power than the drivers with high impedance. T Vented box The cost function is also minimised for the vented box with the extra constraints necessary. It is observed that the vented box system cannot be tuned to a Butterworth alignment as the quality factor of the system, Q T, cannot be altered using the selected design variables x, y, z, l p and a p. According to equation (4.60) the quality factor is determined by: mat Q T = C as 1 = β at m ms S 2 d ρ a + B c ρ π a Sd 2 k ms 1 (Bl) 2 S 2 d R e + β ms S 2 d The only term of equation (6.2) that can be altered is m at as B c is a function of the side lengths x and y but the necessary adjustment of Q T cannot be achieved in this way. Therefore different methods of changing Q T - and in particular the damping of the system - are investigated. (6.2) If the system is too dampened a small series resistance can be connected in series with the voice coil resistance of the driver resulting in a decrease in the damping caused by the electrical system to 2 S d (R 2 e+r s ) (Bl) where R s is the series resistance. If however the system is insufficiently dampened which is the case for the majority of the drivers considered more extensive methods must be applied in order to increase the damping. A commonly used method of increasing the damping of the system is to line the box with acoustic absorbing material. The damping caused by adding acoustic material to the inside of the box is due to the flow resistance of the material. The flow resistance causes a pressure drop across the material proportional the the flow velocity through it, resulting in a power loss from the system as the some of the energy of the sound waves are converted into heat. The two methods described both decrease the efficiency of the system which is undesirable as the efficiency of loudspeakers in general is in the vicinity of 0.4 to 4 % [12, p. 183]. Two methods of increasing (or decreasing) the damping of the system without significant power losses will be considered. One method is by inserting a filter in cascade with the subwoofer which is represented by a 4th order high pass filter. If the system is too dampened an active band pass filter can be combined with the subwoofer boosting the input voltage at low frequencies. If, on the other hand, the system is insufficiently dampened an active high pass filter can boost the input voltage at the frequencies above the resonance peak or a band rejection filter can attenuate the voltage in the vicinity of the resonance frequency. A disadvantage of this method for shaping the frequency response is that the filters do not take into account that parameter variations might occur. Parameter variations can alter the damping or the resonance frequency of the system. If, as an example, the filter is designed to attenuate the input voltage near the resonance 52 6 Problem solution

60 6.2 Design optimisation frequency and a variation in the parameters yield a different resonance frequency, then the resonance peak would not be dampened but an unwanted dip would occur in another frequency band. Another method of adjusting the damping of the system is by applying current feedback. By applying current feedback to the system the input voltage to the driver will be a function of the current through the driver. If the system is too dampened negative current feedback will be applied and the input voltage will fall when the current rises thereby acting like a resistance in series with the voice coil resistance but without power losses (apart from the small power dissipation in the feedback loop) as the output voltage from the amplifier is decreased rather than using a resistor to make a voltage drop. If the system is insufficiently dampened a limited amount of positive current feedback can be applied to compensate for the voltage drop across the voice coil resistance. When using positive current feedback the current through the voice coil is measured and the input voltage to the driver will rise as a function of the current. Obviously the feedback gain must be designed carefully in order to avoid instability. A block diagram of the feedback system is seen in figure 6.12 (the amplifier is approximated as a gain). Audio signal Preamplifier V in Power amplifier K V s 1 Z e i R s Figure 6.12: Block diagram of positive current feedback system. In figure 6.12 V in is the voltage that would be applied to the power amplifier input channel if no feedback loop was used. K denotes the amplifier gain, Z e is the total electrical impedance of the subwoofer system which varies with frequency, R s is the gain of the feedback loop (incuding sensor gain) and V s is the input voltage to the driver. Without the feedback loop V s would be K V in and with the feedback loop it is K (V in + R s i). In this case the impedance seen from the terminals of the driver when looking back into the amplifier will be negative (the output impedance of the amplifier is neglected as it will normally be very low) as the voltage will rise when the current rises. Therefore the system can be considered as the diagram seen in figure K Rs K. V in K. V + K. in R. s i Ze Figure 6.13: Circuit diagram modeling the effect of the positive current feedback as a negative resistance. 6 Problem solution 53

61 6.2 Design optimisation Observation of the circuit in figure 6.13 shows that the system is stable if K R s < Z e. As the impedance of the subwoofer system never falls below the DC resistance of the voice coil, the system will be stable if K R s < R e. The system seen in 6.13 corresponds to a damping of S d (R from the electrical system. The damping effect of the feedback loop can be understood by observing a typical impedance curve for a vented 2 e R s ) box subwoofer like the curve seen in figure 6.9. The impedance of such a system will be highest at the resonance frequencies. Therefore the current through the system is small at these frequencies compared to the other frequencies and the result is that extra voltage will be applied to the driver terminals at all frequencies but this extra voltage is reduced at the resonance frequencies. Variations in cut-off frequency will not compromise the usefulness of the feedback loop as would be the case for a predetermined filter. (Bl) 2 To be able to design the vented box system as a Butterworth alignment the current feedback will be applied and the value of K R s will be chosen as a further design variable among the other design variables x, y, z, l p and a p. If the optimisation yields a positive value for K R s negative current feedback can be applied and if a negative value of K R s is necessary positive current feedback will be applied. How to implement the current feedback to the system is described in section 8.1 along with some further considerations which must be taken into account when the feedback is used in practice. Implementing the positive current feedback in the design optimisation makes it possible to find local minima for the cost function. The optimum design for the vented box with an added design variable labeled, K R s is noted in table 6.4. For the vented box the port length and radius, l p and a p, are also noted. Driver Optimum design Enclosure Cut-off Cost function variables volume* frequency value [cm] [Ω] [litre] [Hz] [.] x y z a p l p K R s V box ω c f opt Table 6.4: Simulation results that shows the optimum values for each driver in a vented box design. As for the closed box system, the volume occupied by the driver is subtracted from the effective volume of the enclosure but also the volume occupied by the port is subtracted. The best design is achieved by using driver number 6 as for the closed box system but in the vented box system the cost function is 54 6 Problem solution

62 6.2 Design optimisation even lower yielding a volume of [litre] and a cut-off frequency of [Hz]. These values are significantly lower than the closed box values. To compare the cost function values they are plotted along with the enclosure volume (in [litre]) and the cut-off frequency (in [Hz]) in the graph of figure Figure 6.14: Plot showing the value of the cost function (green), the cut-off frequency in [Hz] (red) and the enclosure volume in [Litre] (blue) for the 10 drivers when enclosed by a vented box. In general the vented box environment produces lower optimised cut-off frequencies as it was expected, but some of the volumes are larger to make room for the port volume. So the cost functions for the vented box are not all lower than the closed box case. For example driver number 9 yields the lowest cost function value in the closed box. The vented box system voltage-to-spl frequency response of each driver in their specific optimum design is shown in figure The passband gain varies with approximately 6 [db] as for the closed box system. Driver number 6 is the one with the lowest passband gain at around 71 [db]. The Butterworth criteria is also satisfied for the vented box. Figure 6.15: Plots of the output of the 10 different drivers in their optimum vented box. 6 Problem solution 55

63 6.2 Design optimisation In the optimisation approach using K R s the value of K R s is assumed constant. However if the gain of the power amplifier, K, is changed the value of K R s changes and the optimisation approach is not valid 1. One method that could be used to avoid this problem is changing the gain of the feedback loop according to the change of power amplifier gain. This could be accomplished by using a dual potentiometer to change the gain of the power amplifier. This potentiometer could change the gain of the feedback loop simultaneously. Then the block diagram seen in figure 6.12 is modified to the one seen in figure Audio signal Preamplifier V in Power amplifier K V s 1 Z e i Rs K Figure 6.16: Block diagram of positive current feedback system using a dual potentiometer to control the feedback gain. And V s reduces to: V s = K V in + R s i (6.3) [db] peak criterion During the design approach it has been realised that tuning the system to a Butterworth alignment results in strict if not impossible demands to the damping of the system using conventional tuning techniques. For the vented box system feedback control has been shown necessary in order to satisfy the Butterworth constraints. Therefore an alternative treatment of the optimisation problem is presented which lowers the demands for damping and eliminates the need for feedback. Furthermore there is no constraint on the tuning of the cut-off frequencies of the vented box system as for the previous optimisation approach. In this alternative optimisation approach a 3 [db] resonance peak in the SPL curve is allowed effectively reducing the demands for damping. The resulting peaks could be removed by an equalizer. The results from this optimisation approach is not directly comparable to the previously determined cost function values since system does not obey the Butterworth criteria but serves as an alternative tuning method. The cost function will be the same as for the previous optimisations with f c determined as the frequency where the SPL is 3 [db] down from the passband SPL. The constraints for the closed and vented box system using this optimisation approach is seen in table Not all active subwoofer systems have variable gain 56 6 Problem solution

64 6.2 Design optimisation General constraints: Inequality constraints Equation Description 0 (SPL max 3[dB]) SPL pb The resonance peak must be smaller than 3 [db]. 0 D s x The side length x must be 2 cm larger than driver diameter, D s. 0 V box The enclosure volume V box must be smaller than 60 x 60 x 60 [ cm 3]. 0 D s y Side length y must be 2 cm larger than driver diameter, D s. 0 depth driver z Side length z must be 4 cm larger than driver depth, depth driver. 0 min(x,y,z) Smallest side length must maximum be [cm]. Only for vented box: Inequality constraints Equation Description 0 2 a p min(x,y,z) Diameter of the port must be smaller than smallest box side length f c V d 2 a p Diameter of the port must be larger than f c V d [m]. Table 6.5: Equality and inequality constraints for the closed box and vented box subwoofer system. In table 6.5 SPL pb denotes the passband SPL at 1 [V ] input voltage and SPL max is the maximum SPL at 1 [V ] input voltage - i.e. the SPL at the resonance peak if there is any. The optimum design is found for each driver in both closed and vented box enclosures Closed box To compare the best configuration in closed and vented box with each other, they should be analysed using the same criteria. For that reason the optimisation of the closed box is analysed again with the 3[dB] criterion. Driver Optimum design Enclosure Cut-off Cost function variables volume frequency value [cm] [liter] [Hz] [.] x y z V box ω c f opt Table 6.6: Simulation results that shows the optimum values for each driver in a closed box design using the 3 [db] criterion. 6 Problem solution 57

65 6.2 Design optimisation The 3 [db] constraint effectively reduces the volume of the enclosure which is smaller for all drivers than was the case for the closed box optimisation with the Butterworth criteria. The cut-off frequency however, is raised for all drivers between approximately 2 and 13 [Hz], but all in all the cost function values are lower when the 3 [db] constraint is used. Also there are no drivers which stand out with cost function values that deviate with over 100 % as it was the case for the closed box with Butterworth constraint. Figure 6.17: Plot showing the value of the cost function (green), the cut-off frequency in [Hz] (red) and the enclosure volume in [litre] (blue) for the 10 drivers in a closed box and with the 3 [db] criterion. The closed box system voltage-to-spl frequency response of each driver in their specific optimum design is shown in figure Figure 6.18: Plots of the output of the 10 different drivers in their optimum closed box design with the 3 [db] constraint. On the graph of figure 6.18 some of the drivers does not have a resonance peak of 3 [db] which implies that the system is dampened more than the limit requires Problem solution

66 6.2 Design optimisation Vented box As for the closed box the vented box is also optimised with the 3 [db] constraint. The optimum values for each driver can be seen in table 6.7. Driver Optimum design Enclosure Cut-off Cost function variables volume* frequency value [cm] [liter] [Hz] [.] x y z a p l p V box ω c f opt Table 6.7: Simulation results that shows the optimum values for each driver in a vented box design using the 3 [db] criterion. Driver number 6 is still the optimum choice and with and even lower cost function value than the closed box system. The smallest cost function value though still belongs to driver number 6 in the vented box design obeying the Butterworth constraint with the positive current feedback. The deviation in cost function values for this design is however smaller. Figure 6.19: Plot showing the value of the cost function (green), the cut-off frequency in [Hz] (red) and the enclosure volume in [litre] (blue) for the 10 drivers in a vented box and with the 3 [db] criterion. In the graph of figure 6.19 driver number 3 has a peak in volume of [litre] making it the driver with the largest volume. At the same time this driver has the lowest cut-off frequency of 24 [Hz] with as much 6 Problem solution 59

67 6.2 Design optimisation as 15 [Hz] up to the average value of the driver cut-off frequencies. To have such a low cut-off frequency the port volume for this driver is similarly large with a port length of [cm] and port radius of 4.72 [cm]. The plot of the SPL outputs in figure 6.20 have included the frequency response down to 5 [Hz] to illustrate how the 3 [db] criterion move the resonance frequency for the port down for some of the drivers, compared to when the Butterworth criteria is used. Figure 6.20: Plots of the output of the 10 different drivers in their optimum vented box and with the 3 [db] criterion. For clarity the SPL output of the 3 drivers with the lowest cost function value according to table 6.7 are shown in figure Figure 6.21: Plots of the output of the 3 best drivers in their optimum vented box and with the 3 [db] criterion Problem solution

68 6.2 Design optimisation Evaluation of optimisation The cost function values from the simulation results are now to be considered.the lowest overall value is 52.09, scored by the driver 6, JBL GTO ", in a vented box enclosure using positive current feedback. Since driver 6 also has best cost function values in the closed box configurations satisfying the Butterworth criteria, it is concluded that this is the best driver among the 10 selected drivers. It is important to emphasise that while it is found to be the best driver for a subwoofer with a small enclosure and a linear frequency response, it might not be the best choice if the demands were different. The selected driver is seen in figure Figure 6.22: The JBL GTO1014 driver. As for the best configuration it is interesting to compare it with the vented box with the 3 [db] resonance peak constraint remembering although that the cost function values are not directly comparable. The vented box with positive current feedback has a volume of [litre], and a cut-off frequency at [Hz]. At a slightly higher cost function values comes the vented box without any feedback control, but with the less restrictive 3 [db] criteria. It has a volume of 20.7 [litre], and a cut-off frequency at [Hz]. It is seen that the values are very similar, which makes it difficult to announce a clear winner. In figure 6.23 the plots of the SPL output of the two configurations are seen. It is clear to see the peak permitted by the 3 [db] constraint, but otherwise the outputs are very similar. 6 Problem solution 61

69 6.2 Design optimisation Figure 6.23: Plots of the SPL output of driver 6 in a vented box with the 3 [db] constraint (red), and the same driver in a vented box with the Butterworth constraint (blue). An interesting question is, if a subwoofer build on the Butterworth criteria and with a positive current feedback circuit design would perform noticeable better, than the 3 [db] constraint solution. Especially if the extra cost in development time, and the cost in extra materials to build the control cuircuit is considered. The question will stand unanswered in this this report, as the time frame do not permit further investigation Problem solution

70 Conclusion 7 The objective of this project has been to develop a subwoofer with a small volume and a flat frequency response. Initially in the Problem Analysis the definition of sound was presented. This led to a general presentation of a loudspeaker driver and the enclosure required to construct a subwoofer. It was decided to limit the choice of enclosures types to only two types in this project: The closed box and the vented box enclosure. The amplifier used for the active subwoofer is treated as a "black box" in the report, as an available 250 [W] amplifier is used. A mathematical model of the driver and the closed and vented enclosure was derived. The driver was modeled as an electrical system, and the movement of the diaphragm in the enclosure as a mechanical mass-spring-damper system. From the equations, transfer functions expressing SPL (Sound Pressure Level) and impedance for the systems were derived. Next the specifications for the desired subwoofer was outlayed. Main concern being a flat frequency response with a cutoff frequency as low as possible while minimising the volume of the enclosure. It was decided to design the subwoofer system as a Butterworth filter, as this class of filters has a maximum flatness in the passband. The purpose of the project has not been to develop a driver, whereas to select the best driver/enclosure combination for the presented demands. For that 10 different available drivers were selected, spanning from 6.5 [inch] to 12 [inch], to be used in the later design optimisation process. Finally closed loop control of the diaphragm is presented as a possible solution to further improve the performance of the subwoofer. In the Problem Solution a verification of the derived mathematical model was performed. Two available subwoofers were tested. A 10 [inch] Vifa driver in a closed box, and a 12 [inch] Peerless driver in a closed and a vented box. Initially a SPL test was made in an anechoic chamber. The test showed large variation in the amplitude in the low frequencies range, which was most likely caused by the test room, as it was not anechoic below 200 [Hz]. Comparison with the output of the model showed a good approximation in the frequency response, however the magnitude of the output from the model was generally higher than from the test. It was concluded that the difference in magnitude was partly due to damping material present in the enclosure which was not included in the model. Further it was decided to verify the model from derived impedance transfer functions as the SPL verification did not yield a clear consensus between model and tests. A lab-test of the two drivers impedance values throughout the frequency range was performed. The result showed satisfying results for the resonance frequency and for the impedance values outside the resonance area. There was however a deviation in the impedance value at the resonance frequencies, again this was explained by the damping material in the enclosure. As a result of the test the model was concluded valid for the optimisation process. The Problem Statement was formed as a mathematical function which was to be minimised. Based on the desired limitations and specifications, some constraints were set up for the optimisation. The design with the lowest value of the cost function was chosen as the optimum design. The built-in optimisation tool, fmincon, in MATLAB R was used to solve the optimisation problem. The driver with the lowest minimum cost function value for the closed box configuration was the driver number 6, the JBL 7 Conclusion 63

71 GTO ". The optimum design variables was: A side length x of 28.8 [cm], a side length y of 28.8 [cm] and a side length z of [cm] yielding a volume of the enclosure, V box = [litre]. The optimisation for the vented box environment without modifications proved to be impossible if the Butterworth criteria were to be obeyed. An optimum design for the vented box with the Butterworth criteria was found with the use of positive current feedback. The cost function value for driver number 6 proved again to be the lowest of the 10 drivers, this value was which was lower than for the closed box optimum design. The vented box optimum design variables was: A side length x of [cm], a side length y of [cm], a side length z of [cm], a port radius of 2.86 [cm], a port length of [cm] and a R s K of [Ω]. This design has an effective volume of, V box = [litre]. As the positive current feedback is difficult to implement, the system frequency response for a design with a constraint on the maximum resonance peak of 3 [db], which is less strict than the Butterworth criteria, was investigated. The cost function value for this design was 52.49, almost the same as the design with the positive current feedback circuit. This could also be seen on the almost identical enclosure volumes and cut-off frequencies for the to designs. From the optimisation it was determined that a build of a vented box design with a JBL GTO " driver and the optimum design variables from the optimisation with the Butterworth criteria would result in the theoretically best design in the request for a flat frequency response and lowest possible enclosure volume Conclusion

72 Future Work Feedback control As discussed in section 4.5 the position feedback could eliminate different issues in the subwoofer system such as thermal induced distortion, spider and surround distortion and frequency response dependent of driver impedance. For future investigation a description on how to implement the closed loop in the subwoofer system will be derived. The closed loop configuration can be achieved either by digital components or by analogue components. The maximum displacement of the driver has to be considered before building the controller Analogue feedback control A suggestion to implementation of closed loop control using analogue components is sketched in figure 8.1. The working principle of the circuit is as follows: Starting from the left side a reference is connected to the positive input of the summing amplifier. The negative input is connected with the feedback loop from the sensor. The summing amplifier is configured as a differential amplifier used to compare the two inputs and amplify the difference between them. The controller is based on a PI-regulator in this example but in theory any kind of closed loop controller could be implemented. The inverter is inserted to compensate for the 180 [deg] phase shift from the controller (this inverter is only used if there is a phase shift from the controller). Controller X ref GND 7 Summing amplifier GND I-regulator GND Inverter Power amplifier Driver Sensor P-regulator GND 6 5 Feedback from sensor 8 Figure 8.1: 8 Future Work 65

73 8.1 Feedback control Digital feedback control To implement the digital version of a closed loop control system a digital micro controller could be used. For the micro controller to control the movement of the diaphragm, a digitalisation of the analog audio input signal, and the analog feedback from the sensor, have to be implemented, using a ADC. Once in digital form, the signal can be processed in the micro controller, and features like gain, filters and maximum travel of the diaphragm can be implemented in the software code. The output from the micro controller is then converted back to an analog signal via a DAC. The schematics of the signal path is illustrated in figure 8.2. Analog signal IN ADC Micro Controller DAC Analog signal OUT Sensor feedback signal ADC Amplifier Figure 8.2: Micro controller used in the closed loop control system. For this purpose a test have been made with the Arduino Duemilanove micro processor board, which features an Atmel ATmega328 single chip processor, running at 16 [MHz] [23]. The processor has 6 build-in 10 bits A/D converters and 6 build-in 8 bits PWM outputs. In the test a 200 [Hz] sine signal was sampled at 2 [khz] via the A/D input. After down conversion of the signal to an 8 bit signal, it is sent through the PWM port and further passed through a low-pass filter to remove the PWM switching signal. The output resulted in a sine wave with some distortion from the PWM based D/A converter. It is concluded that the micro processor board can be used in future development of feedback control to the subwoofer system, if it is upgraded with an external D/A converter with better performance Analogue implementation of positive current feedback The discussion of positive current feedback in the optimisation section which could lower the Q T of the system can be constructed with the use of analogue components. As mentioned in section the positive current feedback loop will increase the voltage across the driver. In practice a current feedback alone does not work because it is configured with an open loop that has a close to infinity gain. The problem is that it is sensitive to radio-signals and other interference. In practice a combination of negative voltage feedback and positive current feedback is used. An example of how to implement the feedback loop is shown in the circuit of figure 8.3. The inverting input to the op-amp is connected to the voltage feedback where the ratio between R1 and R2 determines the negative gain. The non inverting input is connected to the audio signal. The positive current feedback is a function of the current through R3 and is a voltage connected to the ground of the audio signal. In this way the current through R3 determines the potential of the audio signal. Therefore the voltage across R3 is 66 8 Future Work

74 8.1 Feedback control summed with the audio signal. If the ratio between R1 and R2 is smaller than the ratio between the loudspeaker impedance, Z tot and R3 the system will oscillate. So the comparison between these ratios indicate the limit of the output impedance seen from the driver. + Audio signal - 1 Inverting amplifier 2 Power amplifier 3 Driver R1 5 4 R2 R3 GND GND Figure 8.3: The mixed feedback circuit To calculate the different components the desired output impedance value has to be specified. As the optimisation indicated the need for K R = 1 [Ω]. As derived in section the absolute loudspeaker impedance value must not be equal to or below the absolute output impedance value seen from the loudspeaker. The driver DC resistance is 4 [Ω]. To obtain an output impedance of -1 [Ω], the driver impedance, Z s and R3 is set to 1 [Ω] and 0.2 [Ω] respectively. R2 R1 + R2 > R3 Z s + R3 R2 R1 + R2 > R1 ( ) 1 R2 > 1 R R2 > 5 (8.1) R1 has to be 5 times larger than R2 to obtain a output impedance of -1 [Ω] and to avoid oscillation. 8 Future Work 67

75 8.1 Feedback control 68 8 Future Work

76 Appendix A A.1 Sound pressure test A.1.1 Objective The objective of this experiment is to verify the model of closed box and vented box loudspeakers established in section 4.3. The verification is necessary as the design optimization will be based on this model. In this experiment the sound pressure level (SPL) produced by the subwoofer at different frequencies will be measured at a known distance from the source. The measurements will be comparable with the calculated SPL for the closed box and vented box system respectively. A.1.2 Theory The model established in section 4.3 does not take into account the acoustics of the room in which the subwoofer is placed. Therefore an anechoic chamber is required when accurate SPL-measurements are to be taken. If no reflective waves are present in the room then the SPL can be calculated according to formula (4.28) for the closed box system. For the vented box system the SPL can be calculated according to formula (4.43) in section 4.3. The SPL will be measured using a SPL-meter. The reference pressure, p re f, used by the SPL-meter is 20 [µpa]. As it is assumed that the subwoofer system is linear in the operating points considered and only single-frequency sine waves are applied to the amplifier connected to the subwoofer. It is assumed that the output will be a single-frequency sine wave as well. Therefore there is no need for a bandpass filter in the SPL-meter. A.1.3 Apparatus To ensure reproducibillity of the test, follows a list of the equipment used and the procedure followed. Two different subwoofers are tested: Subwoofer 1: Closed box subwoofer with vifa 10 M26WR driver in a 24 liter enclosure - data sheet for the driver can be found in appendix A.8. Subwoofer 2: Varibel bass-reflex subwoofer with Peerless SLS 12 driver in a 51 liter enclosure - data sheet for the driver can be found in appendix A.8. The enclosure is lined with acoustic damping material. Equipment used to test the subwoofers are listed in the table below. A Appendix 69

77 A.1 Sound pressure test Equipment Description AAU Number Rotel RB-976MKII Philips PM 3050 GW Instek Function Generator GFG-8216 Six channel power amplifier, with three sockets. Only channel A is used Oscilloscope. Used to measure output voltage of the amplifier. The peak voltage was secured at fixed values of 10 [V ] for test 1 and 3, and 20 [V ] for the rest of the tests. Function generator. Used to make frequency sweep from 10 to 200 [Hz] 1000 [Hz] Generator Calibration equipment to calibrate the sound level meter at f = 1000[Hz] and distance r = 0 [m] to a sound pressure level value of SPL = 93.6 [db] Modular Precision Sound Level Meter Sound level meter. Used to measure the sound pressure level at frequencies between 10 and 200 [Hz]. Settings: Time-weighting: Slow (using maximum cycles for calculating the rms SPL), Frequency-weighting: Linear (uses the reference sound pressure p re f = 20[µPa] throughout the frequency range), Frontal/random: Frontal (uses frontal sound direction to the microphone) AUC LVNR Institute 8 AUC LBNR Institute 8 AUC LBNR Institute 14 B4-109-A-2 AAU AUC LBNR Institute 14 A.1.4 Procedure Two subwoofers are tested, one with a closed box enclosure and one with a variable bass reflex enclosure, variable meaning that the port length can be adjusted. The bass reflex subwoofer can also be modified to behave as an closed box type by sealing the port. The subwoofers are positioned in the middle of a 4.5x5 [m] room with sound absorbing material on all sides. The subwoofers are pointing towards one corner and the SPL is measured on-axis at a distance of 1.57 [m]. The position of the subwoofer and microphone (connected to the SPL-meter) is seen in figure A A Appendix

78 A.1 Sound pressure test Subwoofer Position 500 cm Subwoofer 157 cm Microphone 450 cm Figure A.1: Test set up. Before the tests are carried out the SPL-meter is calibrated using the calibration equipment, so that is displays 93.6 [db] at f = 1000 [Hz], the subwoofer and the microphone is connected to a separate control room using cables. In the control room the loudspeaker is connected to the amplifier, which is connected to the signal generator. The oscilloscope is connected to the output of the amplifier. The microphone is connected to the SPL-meter. During each test the following 3 steps are performed: 1. Set the signal of the signal generator to deliver a sine waveform with a specified frequency starting with a frequency of 10 [Hz]. 2. 2: Adjust the amplifier, so that it supplies the subwoofer with a peak-to-peak voltage as stated in table A : Note the SPL and repeat step 1 with a higher frequency. From 10 to 30 [Hz] the frequency is stepped by 1 [Hz], from 30 to 100 [Hz] the frequency is stepped by 2 [Hz] and from 100 to 200 [Hz] the frequency is stepped by 5 [Hz]. A.1.5 Results and data processing The raw test data can be found in appendix A.8 Test Enclosure Driver Peak-to-peak voltage Port length 1 Closed box 10 E G = 20 [V ] - 2 Closed box 10 E G = 40 [V ] - 3 Closed box 12 E G = 40 [V ] - 4 Bass reflex 12 E G = 40 [V ] l P = 0.29[m] 5 Bass reflex 12 E G = 40 [V ] l P = 0.20[m] Table A.1: Driver tests A Appendix 71

79 A.1 Sound pressure test In figure A.2 and A.3 the data has been processed and plotted. Figure A.2: Measured SPL-curve for Vifa M26WR 10" driver in closed box. Figure A.3: Measured SPL-curve for Peerless SLS 12" driver in closed and vented enclosure. A.1.6 Source of error 1. The sound damping material in anechoic room is made of damping foam with a length of l = 40 [cm], this makes the room anechoic to frequencies higher than approximately f = [Hz]. All tests were carried out in frequencies below this limit, in the range of [Hz]. The anechoic room was however not unnecessary as the sound pressure level was measured by a sound level meter. Any external sources of sound would have effected the measured sound pressure level, if the subwoofer would not have been placed in the anechoic room. 2. Standing waves at different frequencies could cause measurement errors, as the subwoofer is to be tested in anechoic environment. This source of error was sought reduced in placing the subwoofer in the middle of the room and the microphone on the diagonal. 3. Heat generation from the driver voice coil resistance. The voice coil resistance is linear dependent with temperature and will rise as the temperature rises. 72 A Appendix

80 A.1 Sound pressure test A.1.7 Uncertainty of measurements 1. The distance between the SPL-meter and the radiation source was measured to 1.57 [m] however small deviations in this distance will increase or decrease the measured SPL as these are directly proportional. As an example an increase in radiation distance of 10 % will correspond to a drop in SPL of 0.83 [db] according to 20 log( ) = 0.83 [db]. A Appendix 73

81 A.2 Impedance test A.2 Impedance test A.2.1 Objective The objective of this experiment is to verify the model of closed box and vented box loudspeakers established in section 4.3. The verification is necessary as the design optimisation will be based on this model. In this experiment the impedance of the driver in free air, in a closed and a vented box is measured at different frequencies. The measurements will be comparable with the calculated impedances for the closed box and vented box system respectively. A.2.2 Theory The free air test of both drivers was made to compare the result with the data sheet from the manufacturer for each driver and from this verify the impedance test. The electrical impedance of the driver is dependent on the mechanical and the acoustic (the acoustic system is looked at as a mechanical mass and added the mechanical system) system. So by measuring the impedance of the driver in a box, the behavior of the entire system can be investigated. The impedance can be calculated as: Z tot = V a V r I total (A.1) I total = V r R10 (A.2) Where V a is the rms output voltage of the amplifier, V r is the rms voltage drop across resistor R10 and I tot is the rms current through resistor R10 and the driver. For more information see figure A.4. A.2.3 Apparatus To ensure reproducibillity of the test, follows a list of the equipment used and the procedure followed. Two different drivers are tested: Driver 1: Vifa 10 M26WR data sheet for the driver can be found in appendix A.8. Driver 2: Peerless SLS data sheet for the driver can be found in appendix A.8. Equipment used to test the drivers are listed in the table below. 74 A Appendix

82 A.2 Impedance test Equipment Description AAU Number Amcron DC-300A series 2 Teletronix DPO 2014 Digital GW Instek Function Generator GFG-8216 Two channel power amplifier. Only channel A is used Oscilloscope. Used to measure output voltage of the amplifier. The rms voltage was secured at fixed values of 1 [V ] for test 3 and 4, and 2 [V ] for the remaining tests. Function generator. Used to make frequency sweep from 10 to 20,000 [Hz] Hewlett Packard 34401A Multimeter. Used to measure voltage drop over R10 Resistor ± 5 % 10Ω. Used to create a voltage division between Z Speaker and R10 N/A AUC LBNR AUC LBNR Institute 14 AUC LBNR N/A A.2.4 Procedure Both drivers are tested in free air and in a box. The Vifa driver is tested in a closed box and the Peerless driver is tested in a variable bass reflex enclosure, variable meaning that the port length can be adjusted. The bass reflex subwoofer can also be modified to behave as an closed box type by sealing the port. In free air the volume is considered as infinitely big. As seen in figure A.4 the voltage drop over R10 was measured and recorded while the frequency was changed from 10 to 20,000 [Hz]. As seen in table A.2 below, 6 test was carried out. Test Enclosure Driver Rms voltage Port length 1 Free air 10 V a = 2 [V ] - 2 Closed box 10 V a = 2 [V ] - 3 Free air 12 V a = 2 [V ] - 4 Closed box 12 V a = 2 [V ] - 5 Bass reflex 12 V a = 2 [V ] l P = 0.20[m] 6 Bass reflex 12 V a = 2 [V ] l P = 0.29[m] Table A.2: Driver tests A Appendix 75

83 A.2 Impedance test Amplifier Ocilloscope Function generator out in out V a in Driver Resistor R10 Multimeter V r in Figure A.4: Test set up for impedance measurement. 1. Connect the equipment as shown in figure A Set the signal of the signal generator to deliver a sine waveform with a specified frequency, starting with a frequency of 10 [Hz] 3. Adjust the amplifier, so that it supplies the subwoofer with a rms voltage as stated in table A Note the multimeter reading and repeat step 1 with a higher frequency. From 10 to 30 [Hz] the frequency is stepped by 1 [Hz], from 30 to 100 [Hz] the frequency is stepped by 2 [Hz], from 100 to 200 [Hz] the frequency is stepped by 5 [Hz] and from 200 to 20,000 [Hz] the frequency is stepped by 100 [Hz]. A.2.5 Results and data processing The raw test data can be found in appendix A.8. In figure A.5, A.6 and A.7 the data has been processed and plotted. 76 A Appendix

84 A.2 Impedance test Figure A.5: Vifa M26WR 10", free air and closed box. In test 1 f peak = 26 [Hz] and Z peak = 24.7 [Ω]. In test 2 f peak = 60 [Hz] and Z peak = 33.3 [Ω]. Figure A.6: Peerless SLS 12", free air and closed box. In test 3 f peak = 29.5 [Hz] and Z peak = 50.9 [Ω] In test 4 f peak = 56 [Hz] and Z peak = 36.8 [Ω]. A Appendix 77

85 A.2 Impedance test Figure A.7: Peerless SLS 12", vented box 0.2 [m] and 0.29 [m]. In test 5 f peaklow = 17.3 [Hz], f peakhigh = 69 [Hz] and Z peaklow = 28.3 [Ω], Z peakhigh = 29.3 [Ω]. In test 6 f peaklow = 15.8 [Hz], f peakhigh = 66.5 [Hz] and Z peaklow = 25.5 [Ω], Z peakhigh = 30.9 [Ω]. A.2.6 Source of error 1. Heat generation from the driver voice coil resistance. The voice coil resistance is linear dependent with temperature and will rise as the temperature rises. A.2.7 Uncertainty of measurements In figure A.6 it is seen that the impedance value of the speaker is lower than its DC resistor value. Since this is not possible, the error most originate from measurements. Changes in input voltage, V a and the R10 resistor value has large impact on the impedance calculation. At the low frequencies the system impedance decreases and causes the amplifier output voltage, V a to decrease as it will be heavier loaded. This decrease in voltage was not compensated for. 78 A Appendix

86 A.3 Interview with Hi-Fi Klubben A.3 Interview with Hi-Fi Klubben In the initial phase of developing a new product, it is important to investigate competing products in the marked. In the project of developing a subwoofer, an interview with the manager of Hi-Fi Klubben in Aalborg, Denmark, have been arranged. Hi-Fi Klubben is a chain of shops specialized in Hi-Fi music equipment and other kind of home entertainment, and consist of more than 70 shops in Denmark, Norway and Sweden [3]. In march 2010 their selection of subwoofers counted 31 different models, from 6 different brands. The price range being between 1,499 Dkr. and 34,900 Dkr. The interview with the manager, Lars Bo Pedersen, have been conducted with the purpose of finding out what types of subwoofers are in the marked today, and what are the costumers main concern when they are looking to buy one. The interview is presented with the questions, followed by the answers of Lars Bo Pedersen (LBP). The answers have been translated to English and edited by the authors of this report. The answers are therefore not direct citations of Lars Bo Pedersen. A Interview: What kind of subwoofer categories are there in the home consumer marked? LBP: Subwoofers for home entertainment can be divided into tree different categories: 1. Combined sets of small satellite speakers and a subwoofer. The small speakers are not able to play bas notes by themselves because of their limited size. Often these kinds of sets are not high quality, and it is not unusual to see a reduced performance in the frequency range between the subwoofer and the satellite speakers. 2. Subwoofers for surround systems. To complete a system with for example 5 or 7 speakers, you need one or two subwoofers. It is important for the customers that it is a powerful subwoofer. They want one that can really move the air so you can feel the bass. 3. Subwoofers for Hi-Fi stereo. Customers that focus on Hi-Fi sound often don t use a subwoofer. They sometimes try, but after a lot of adjusting they switch it of, because it can be very difficult to make the system perform well together. A subwoofer that interact well with an existing system and is easy to adjust, would have a lot of potential customers. Is the location of the subwoofer important? LBP: The location is very important. It can perform very different depending on location and the acoustic coupling with the room. What is most commonly used, closed or vented boxes? LBP: There is a tendency towards closed boxes. They require expensive powerful amplifiers to perform well. The development of cheap powerful ICE-power amplifiers have made it more attractive to build closed boxed subwoofers today. Generally a small closed box sounds better in the sense of Hi-Fi sound. Also they are easier to place in the room for optimum performance. A Appendix 79

87 A.3 Interview with Hi-Fi Klubben Does the costumers focus on design and color, or on the sound performance? LBP: The design is very important. Subwoofers in different shapes than the traditional box, and in white or black glossy colors gets a lot of interest from the costumers. Most people want a very small subwoofer, but as they have a limited output, the costumers often prefers a bigger subwoofer with improved performance. A 10 inch, 30x30x30cm box is an acceptable size. A big box, measuring for example 60x50x45, is almost impossible to sell, and therefore there is not a lot of big box sizes in the market. Figure A.8: Scandyna subwoofer, appealing to costumers with focus on design.[3] What is an acceptable price? LBP: A price range of Dkr. is the most popular. Therefore the most models in the market are found within this range. For most costumers 4500 Dkr. is the maximum price. What frequency range is required? LBP: In the high end it should not go higher than 80 Hz since higher notes from a subwoofer sounds bad. In the low end 20 Hz seams to be desirable, but at that frequency many things in the room will start vibrating and thereby making noise. Also many rooms are too small to fully take advantage of frequency s at 20 Hz because of their long wavelength. We have good experience with a range from Hz in the low end, and up to Hz where especially large speakers can take over without any problems. Any good advice how to improve the performance of a subwoofer? LBP: In many new surround amplifiers there is a build in room correction with a microphone feedback. That works very well for subwoofers. A good approach is to place the subwoofer where it performs best, and then calibrate it. That way you get a lot of work of the amplifier, and minimizes the distortion. A nice feature found on some models, are a remote control. That makes it easy for the costumer to quickly recalibrate the volume if that is necessary. A good subwoofer is expected to play a dry and crisp bass sound. Since the sound is influenced by many 80 A Appendix

88 A.3 Interview with Hi-Fi Klubben things, like the driver, the box and the amplifier, it is difficult to say exactly what makes a difference between a good and a bad subwoofer. A Appendix 81

89 A.4 Small signal parameters and Thiele-Small parameters A.4 Small signal parameters and Thiele-Small parameters This appendix is based on [8]. In most data sheets for loudspeaker drivers some basic parameters necessary to describe the behaviour of the driver is listed. Apart from the large signal parameters described in section 4.2 which desribes the physical limits of the driver, the small signal parameters and/or the Thiele-Small parameters are given. The small signal parameters describe the behavioiur of the driver under normal operation conditions and are listed below: The moving mass M ms which includes the voice coil and the diaphragm. The acoustic mass corresponding to the mass of air in connection with the diaphragm of the subwoofer can be included. The mechanical resistance, R ms, of the moving mass caused by the suspension. A mechanical resistance is the same as a damping factor. The mechanical compliance, C ms, of the moving mass resulting from the suspension. A mechanical compliance is the reciprocal of a spring constant. The DC-resistance of the voice-coil R e. The inductance of the voice-coil L e. The magnetic field in the air gap of the driver, B, and the length of the voice-coil conductor present in the air gap, l, are often collected in one term called the magnetic force factor, Bl. Often the above-mentioned small signal parameters are given indirectly as Thiele-Small parameters. These parameters where developed in the 1970s by the autralians Neville Thiele and Richard Small - two pioneers in the analysis and design of low frequency sound reproduction systems. The Thiele-Small parameters have become very popular as they give information about suitable box size and enclosure type using very limited calculations. Furthermore they give information about the shape of the frequency response of the driver. The Thiele-Small parameters are listed below. The resonance frequency of driver, f s, in a no baffle environment. The frequency is determined as: f s = 2 π Cms M ms [Hz] (A.3) The quality factor of the driver at the resonance frequency describing the damping of the driver. It consists of the eletrical quality factor, Q es, and the mechanical quality factor, Q ms 1 : Q ts = Q es Q ms Q es + Q ms [.] (A.4) where the electrical quality factor, Q es, is determined as: Q es = 2 π f s M ms R e (Bl) 2 [.] (A.5) and the mechanical quality factor, Q ms, is determined as: Q ms = 2 π f s M ms R ms [.] (A.6) 1 The subscript s denotes that the parameters apply to the driver and do not include information of the enclosures effect on the quality factors. 82 A Appendix

90 A.4 Small signal parameters and Thiele-Small parameters A volume of air, V as, which - when acted upon by a piston with cross-sectional area S d - has the same compliance as the driver s suspension. V as is determined as: V as = ρ c 2 S 2 d C ms [ m 3 ] (A.7) (A.8) where ρ is the density of air and c is the speed of sound in air. Based on the small signal parameters and the Thiele-Small parameters all the information of each driver necessary in the design optimisation is available. A Appendix 83

91 A.5 Sensors A.5 Sensors To archive the closed loop feedback from the driver, a sensor signal is needed. For that purpose different sensing technics should be investigated. A.5.1 Optically distance measuring The Sharp GP2D120XJ00F A.8 is an opto-electronic distance measuring sensor with integrated signal processing and analog voltage output. It has a distance range from 4 to 30 [cm]. The 37x19[mm] sensor housing is emitting light from a LED in one side, and receives the reflected light with an optic sensor in the other side of the housing. To measure the movement of the diaphragm the sensor should be mounted on the frame of the driver, and on the diaphragm there should be mounted a white light reflecting material. The output from the sensor is an analog voltage changing according to the distance measured. The output voltage difference, V o is typical 2.25[V ] within the distance range. Using the sensor together with a microprocessor with a 10 [bit] (1024) 0-5[V] A/D converter the range in use will be: This yields a theoretical resolution for an ideal sensor of: = 461 [bit] (A.9) 260 = 0.56 [mm 461 bit ] (A.10) The GP2D120XJ00F sensor have been tested with a Arduino microprocessor board, see appendix A.6. The test reveled an unexceptable high variation in the measured distance. Based on the test this sensor is evaluated to be ill-suited for the servo subwoofer. Another type of optical sensing may be suited for the purpose. A.5.2 Capacitive sensor The principles for capacitive sensing is basically a capacitor where the capacitance may change[24]. The capacitance for a plate capacitor is given by: C = ε o ε A d (A.11) Where A is the area of the plates, d is the distance, ε is the relative permittivity or dielectric constant of the isolating material and ε o is the permittivity of vacuum, 8.85[pF/m]. The basic capacitor is illustrated in figure A.9. For measuring purpose the capacitance can vary by varying the area, the distance or the permittivity. For distance measuring an obvious choice would be to vary the distance. Another possibility is to mount one sensor plate at the fixed reference point, and the other on the moving object, then by dislocating the plates horizontally the common area will vary and thereby the capacitance. 84 A Appendix

92 A.5 Sensors Metal plate ε A d Figure A.9: Basic view of a capacitor. Capacitive sensors with the ability to resolve measurements below 25 [nm] can be purchased [25]. The disadvantage is that the working distance is rarely greater than 15 [mm], thereby eliminating this sensor type for the purpose of diaphragm position as it is expected to move more than ± 10 [mm]. A.5.3 Acceleration measurement There are different types of sensors for acceleration measurement. They are based on different kinds of sensors sensing the effect of an accelerating moving mass, e.g. piezo-electric and strain gauges sensors[24]. The piezo-electric sensor are relative inexpensive, comes in IC based housings, have a measuring range up to ± 1000 times the gravitational acceleration (9.82 [ m/s 2] ) and a bandwidth up to 125 [khz]. The working principle of the sensor is based on a piezo-elecric material that generates an electric charge when it is compressed. Inside the sensor the material is mounted between a mass and the accelerating body. Due to the inertia of the mass the force on the piezo-electric material is altered. Figure A.10 illustrates the basic principle of a piezo-electric acceleration sensor. Housing Inner sleeve Mass Direction of acceleration Piezo-electric material Accelerating body Figure A.10: Basic principle of a piezo-elecric acceleration sensor. The disadvantage of a acceleration sensor mounted on the diaphragm is the lag of a reference point. A difference in the measured acceleration back and forth will over time be accumulated and force the diaphragm out of its center position. Countermeasures against this problem should be taken if a acceleration sensor is to be used. A Appendix 85

93 A.5 Sensors A.5.4 Back EMF Lorentz law states the relationship between the back electromotive force, ε(t), the magnetic flux density, B and the length, l and velocity, dx(t) dt, of the coil moving in the magnetic field, see section 4.6. The back EMF from a moving coil in a known magnetic field can therefore be used to measure the velocity of coil. To use this method to measure the velocity of the diaphragm, an additional coil have to be mounted beside the voice coil on the driver. This method of feedback is already on the marked in a servo subwoofer system [20]. The disadvantage of this type of sensor is the very limited amount of available drivers in the market with an extra coil for feedback. 86 A Appendix

94 A.6 Test of Sharp distance sensor A.6 Test of Sharp distance sensor To test the Sharp GP2D120XJ00F optical distance sensor an Arduino Duemilanove micro processor board, with a Atmel ATmega328 running at 16 [MHz] have been used [23]. The purpose of the sensor in a closed loop circuit is to feed back the position of the diaphragm. Since the diaphragm only move +/1-2 [cm], only a part of the sensors working range of 26 [cm] is used. A program in C have been written to read out the measurements to a LCD display: The sensor is mounted at a fixed distance, and should therefore read out a constant distance. The program A Appendix 87