Cheap Price Headphones

1. TUTORIAL AES 120, Paris, May 2006 —– HEADPHONE FUNDAMENTALS Carl Poldy Philips Sound Solutions Vienna, Austria

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3. Contents 1 Introduction 1 1.1 Overview Of Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 SPL in Freefield of a Loudspeaker (LS) . . . . . . . . . . . . . . . . . . . . 3 1.3 SPL in a Pressure Chamber (Simplified HP) . . . . . . . . . . . . . . . . . 4 1.4 Types of Headphones and Leakages . . . . . . . . . . . . . . . . . . . . . . 5 2 Headphones Compared With Loudspeakers 7 2.1 Pressure Chamber & Freefield . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Harmonic Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.1 Less Harmonic Distortion for Headphone than Loudspeaker . . . . 8 2.2.2 Modulation Of Membrane Area . . . . . . . . . . . . . . . . . . . . 9 2.3 Effect of Headphone Size on Leakage Audibility . . . . . . . . . . . . . . . 12 2.3.1 Dipole Cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 What the Ear-Canal Does . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4.1 “Functional” Transfer Functions! . . . . . . . . . . . . . . . . . . . 14 2.5 The Ear Spectrum For Frontal Sound Source . . . . . . . . . . . . . . . . 16 2.5.1 A Tentative Assumption . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5.2 The Above Assumption Was Illogical . . . . . . . . . . . . . . . . . 17 2.6 More Reliable Stereo Image For Headphones . . . . . . . . . . . . . . . . . 18 2.7 Is the Ear Just a Pressure Detector? . . . . . . . . . . . . . . . . . . . . . 19 2.8 Audiometry With Headphones and Loudspeakers . . . . . . . . . . . . . . 19 3 An In-Ear Headphone 21 3.1 Bass Tube Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 The Main Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3 Earcanal Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.4 How the Bass Tube Works . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.5 Sharp Sound-Colour Of Far-field Leakage . . . . . . . . . . . . . . . . . . 28 iii

4. iv CONTENTS 4 Reference Points For Response Measurements 29 4.1 ERP and DRP: Ear and Drum Reference Points . . . . . . . . . . . . . . 29 4.2 Artificial Ear IEC711 Combined With Earcanal . . . . . . . . . . . . . . . 31 A Behind The Ear Canal Entrance 33 A.1 Tube Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 A.2 Eardrum Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 A.3 Lumped Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 B Harmonic Distortion 37 B.1 Two Superstitions Exposed . . . . . . . . . . . . . . . . . . . . . . . . . . 37 B.1.1 K2 K4 K6 etc Not Always Unsymmetrical . . . . . . . . . . . . . . 37 B.1.2 Apparent DC Value Can Be a Phantom . . . . . . . . . . . . . . . 39 B.2 Simulating Non-linear Distortions . . . . . . . . . . . . . . . . . . . . . . . 41 C Some Curious Observations 43 C.1 Bl Distortion In Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . 43 C.2 The Egg Does Come Before the Chicken. . . . . . . . . . . . . . . . . . . . 47 C.2.1 Potential and Motional Parameters . . . . . . . . . . . . . . . . . . 47 C.2.2 Does Intention Produce Action? . . . . . . . . . . . . . . . . . . . 47 C.2.3 Passive Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . 48 C.2.4 Pure Transduction Two-ports . . . . . . . . . . . . . . . . . . . . . 48 C.2.5 Piezoelectric Transducer Two-port . . . . . . . . . . . . . . . . . . 49 C.2.6 There Is No “Magnetic World” . . . . . . . . . . . . . . . . . . . . 50 C.3 Phase Delay in a Pressure Chamber . . . . . . . . . . . . . . . . . . . . . 51

5. Chapter 1 Introduction 1.1 Overview Of Concepts In Figure 1 on p. 2 the concepts on the left trigger in the mind the associations on the right. One point, not obvious to all, is that the magnitude of acceleration a for sinusoidal signals can easily be derived from velocity v by multiplying by ω (= 2πf), and from displacement x by multiplying by ω2. (For AC signals we normally use RMS values.) We can only hope here to cover a few aspects of headphones. Whether the ones chosen are of interest, depends on your point of view—whether you are a headphone user, or somebody nearby. Here we look at the physics relevant to both groups. There is also a third point of view—the protection of the headphone user from sound leaking in from the surroundings. This function of hearing protectors can also influence headphone design—such as fluid filled cushions etc. For such aspects, and others, I refer the reader to the recent AES tutorial by MR Avis and LJ Kelly [1]. 1

6. 2 CHAPTER 1. INTRODUCTION CONCEPT CHARACTERISTICS Constant SPL in pressure chamber Constant SPL in real headphone HP Constant SPL in LS Working range of Headphone Constant displacement x Constant velocity Constant acceleration Driver resonance within WR: Resistance- or stiffness controlled Above resonance: mass controlled Flat Imitates Free- or Diffuse-field Odd harmonics 3 5 7 etc f Symmetrical time signal. ?x (sometimes) ? 2 x (Newton’s law) Working range of LS Ideal Response of LS Ideal Response of HP Symmetrical Distortion e.g. Stiffness Bl profile F Bl x coil x Unsymmetrical Distortion EARCANAL 1 2 Even harmonics 2 4 6 etc f Unsymmetrical time signal possible. Infinite peaks at 3kHz 9kHz 15kHz for transfer function p2/p1 if terminated hard Damping of these peaks down to about 12dB Good for two-port analysis EARDRUM DRP Drum Reference Point in front of eardrum ERP Ear Reference Point in concha Problematic for analysis Figure 1: Some LS- and Headphone-relevant concepts.

7. 1.2. SPL IN FREEFIELD OF A LOUDSPEAKER (LS) 3 1.2 SPL in Freefield of a Loudspeaker (LS) For a simple monopole radiator the Sound Pressure Level (SPL1) at a distance is pro- portional to the volume acceleration of the radiating surface. As shown in Figure 2, if we consider the low-frequency behaviour of a LS2with closed rear cavity at low frequen- cies, its SPL appears at large distances to come from such a monopole radiator, which is omnidirectional. Mounting the LS in an infinite baffle reduces the full space to an infinite half space. The pressure is then double that shown in Figure 2. For progressively higher frequencies there is increased directionality. Off-axis SPL is reduced, but for a baffled circular radiator the on-axis still obeys the expression shown here (p should be doubled of course). p [Pa] d [m] Air density [kg/m 3 ] Volume acceleration [m3 /s2 ] 1,2 S × VolAcc p = 4 ? d 4 Normal acceleration [m/s 2 ] VolAcc = S × v. . Displacement x [m] × x . 1,2 × S p = ? d Figure 2: SPL at distance d from a monopole radiator. For a baffle the pressure p is dou- bled. All signals are AC and therefore RMS values. The magnitude of the acceleration ¨ x = ω2x, where ω = 2πf. 1SPL is usually expressed on a logarithmic scale dBSPL referred to the threshold of hearing. It has been agreed to define the threshold SPL (0 dB) at 20µPa. 2The abbreviations LS and HP are for loudspeaker(s) and headphone(s) respectively.

8. 4 CHAPTER 1. INTRODUCTION 1.3 SPL in a Pressure Chamber (Simplified HP) The equivalent of Figure 2 for a pressure chamber is Figure 3. AC pressure p [Pa] SPL distributed homogeneously d [m] Vol [m3] S Displacement x [m] 1 × (S×x) p = Vol[m3] ) 1,4×100000 ( Ratio of specific heats of air Cp/Cv = 1,4 acoust. Capacitor C Volume displacement [m3] ( current / j DC air pressure 100000 Pa ? ) Figure 3: SPL in pressure chamber, as for a simplified leak-free headphone at low frequencies for small piston displacement x. All signals are AC and therefore RMS values. An overview of R L C values for lumped elements is given in Figure 34 on p. 35. One may wonder how a homogeneous distribution of pressure is possible. Does sound propagate here at the speed of light or even faster3? To see what all this looks like also at higher frequencies, where standing waves are expected, see Figure 54 on p. 51. 3We are usually concerned with sine wave responses. Then we are interested in amplitude response and sometimes phase response and phase delay). Since, by definition, sine signals exist for ever causality is not violated. If we were concerned with energy propagation of impulses then we would of course expect causality to be respected. But then instead of phase delay we have signal delay, which equals group delay (-dφ/dω) only if the system is an all-pass.

9. 1.4. TYPES OF HEADPHONES AND LEAKAGES 5 1.4 Types of Headphones and Leakages There are many different types of headphones4, ranging from small in-ear headphones (p. 21) to supra-aural and circum-aural headphones (Figure 4). Some are designed to be sealed. Doing this with fluid-filled cushions can give a reliable response even at low frequencies. But usually uncontrolled leaks make themselves felt (a). They can go directly to the outside, through hair and/or through porous cushions or through uncontrolled porosity in squashed foam (b and p. 21). Others are led in a controlled way to the outside: (c) or via the vented rear cavity (d). Normally leaks are desirable Figure 4: The various types of supra-aural and circum-aural headphone[2]. and a deliberate part of the acoustical design. As a result of controlled leaks the SPL in the cavity becomes more stable with respect to any additional chance leaks—the two are like resistances in parallel—whichever is lower will dominate. The working principle of headphones tends to be of the pressure chamber type. Mem- brane movement is largely stiffness controlled C or resistance controlled (damping R), but usually a combination of C and R. This is unlike loudspeakers, which are predom- inantly mass controlled L: Newton’s Law giving frequency-independent acceleration (Figure 1 on p. 2). See Figure 34 on p. 35 for an overview of the analogies used here. There are of course exceptions. For example some high fidelity headphones have more of the characteristics of a loudspeaker. This type has been designed as an acoustic dipole radiating the ear from close quarters, but not touching it. 4We shall confine this treatment to systems with the most common electro-dynamic transducers.

10. 6 CHAPTER 1. INTRODUCTION

11. Chapter 2 Headphones Compared With Loudspeakers 2.1 Pressure Chamber & Freefield Normally we want SPL to be frequency-independent1. It is then of interest to ask what happens to the membrane displacement x as frequency increases. It is helpful to write for SPL or p the expression: p = constant × x × fn. (2.1) For a simple cavity constant SPL requires that n = 0. The membrane is like a piston pumping on a pressure chamber (Figure 3 on p. 4). The relation between pressure p and total cavity volume (V ol) comes from the adiabatic law p × V olγ= const where γ = Cp/Cv, the ratio of specific heats at constant pressure and constant volume. For small compressional displacements xS there is a small increase of pressure p which does not depend on how fast the piston moves, as long as it is fast enough for adiabatic compression. This is normally the case for sound vibrations2. But if we want the same from a LS, we must set n = 2, because p ∝ acceleration (Figure 2 on p. 3). Fortunately Newton’s Law (Force = BlI = mass × acceleration) is quite obliging, and the LS engineer need make no special effort to fulfil this condition. This requires that displacement x must decrease with frequency. To get a real feel for this let us imagine for a LS that n=1 instead of n=2, which is untrue, but one step in the right direction. This would mean pressure is proportional to velocity. For n=1 the gradient in the zero crossings (Figure 5 on p. 8, red curves) reveals the constant velocity, and we see at a glance that the displacement would drop by 6 dB/octave. We 1Even though the ideal response of a headphone is not flat (e.g. freefield or diffuse-field calibrated headphone), for simplicity we shall assume flatness here. This is because such deliberate departures from flat response are more in the nature of local details within a confined frequency range, rather than a global phenomenon like a slope of xdB/octave over the whole audio range. 2The term “adiabatic” means no heat is exchanged with the surroundings. For a LS cabinet filled with cotton wool heat exchange does occur between the air and the fibres at low frequencies. Then we have isothermal compression and Boyle’s Law applies: (p×V ol = const). The pvsV ol curve is then less steep. The cavity is then acoustically softer. This makes it seem larger than it really is. One should not confuse the cavity volume V ol with the momentary volume displacement of the piston xS. 7

12. 8 CHAPTER 2. HEADPHONES COMPARED WITH LOUDSPEAKERS need no mathematics to grasp this. In reality n = 2 and displacement drops faster: 12 dB/octave, which agrees with observation: For low frequencies of double bass the large membrane displacements are visible. DISPLACEMENT = const * PRESSURE / FREQ^n PRESSURE CHAMBER: Pressure = const * DISPLACEMENT Just imagine it: Pressure = const * VELOCITY LS: Pressure = const * ACCELERATION ASSUME CONSTANT PRESSURE FOR ALL FREQUENCIES n = 0 n = 1 n = 2 100 Hz DISPLACEMENT 200 Hz 400 Hz TIME 800 Hz n=0 PRESSURE CHAMBER 100 Hz DISPLACEMENT 200 Hz 400 Hz TIME 800 Hz n=1 velocity constant (applies to nothing) 100 Hz DISPLACEMENT 200 Hz 400 Hz TIME 800 Hz n=2 LOUDSPEAKER (acceleration constant) Figure 5: Constant SPL for a pressure chamber with n = 0; for a LS with n = 2. 2.2 Harmonic Distortion 2.2.1 Less Harmonic Distortion for Headphone than Loudspeaker To supply the same output SPL a LS coil generally has to perform more excursion than that of a headphone. This applies especially at low frequencies since the coil

13. 2.2. HARMONIC DISTORTION 9 displacement rises with 12dB/octave while lowering frequency in the working range well above resonance, as we already saw (Section 2.1). This brings the LS into the non-linear region of any distortion mechanisms much faster. 2.2.2 Modulation Of Membrane Area The volume of air moved is proportional to effective membrane area S and displace- ment x. In LSs the modulation of S by x is usually unimportant, because the spring Cms is confined to the small pleat region at the edge of an otherwise stiff conical membrane. In headphones, which usually have torus-shaped membranes, the compliant spring is dis- tributed over a larger region between the coil and the edge of the membrane (Figure 6). S is reduced for coil displacement x to the right, and increased to the left. Effective area Figure 6: How coil displacement modulates effective membrane area. The effective area for moving air is a differential quantity: for any given DC displacement we need to ask how much air is moved by a superposed small AC displacement. For DC displacement to the right, the stretched outer part of the membrane contributes less. To demonstrate how this influences the harmonic distortion we simulate a pressure cham- ber headphone with a mechanical resonance of 300Hz, pumping on a 50cm3cavity. The circuit is shown in Figure 9 on p. 11. The small-signal AC response is shown in Figure 7 on p. 10. The cavity shifts the resonance up above 300Hz only slightly, because the large cavity does not pose much of a hindrance to coil displacement. The membrane spring is the one that counts.

14. 10 CHAPTER 2. HEADPHONES COMPARED WITH LOUDSPEAKERS AA54_28b_ONLY_AREA_AC.CIR 150.0u 100.0u RMS DISPLACEMENT 50.0u 0.0u 10 v(Displacement) (V) 100 1K 10K F (Hz) 140.0 120.0 100.0 dB SPL 80.0 60.0 40.0 10 dB(v(Pa_Cavity))+94 100 1K 10K F (Hz) Figure 7: Small signal AC simulation of circuit of Figure 9. The transient response is shown in Figure 10 on p. 11. The area was modulated using the macro of Figure 8. .PARAMETERS(area0,coeff1,coeff2,coeff3) Output is the momentary value of either 1) Bl [Tm], 2) Spring force [N] or 3) eff. Area of membrane X Voltage controlled voltage source Y+ Voltage Input is coil displacement [m] area .define mu (v(x)*1e6) Y- .define area ( area0 + coeff1*mu + coeff2*mu^2 + coeff3*mu^3 ) Figure 8: The macro NL Area of Figure 9 for modulating the effective membrane area. The area is expressed as a polynomial of the coil displacement in microns, not SI units3: area = area0 + coeff1 × mu + coeff2 × mu2+ coeff3 × mu3 The linear component is area0[m2]. coeff1 = −1 × 10−6, all others being set to 0. This value is chosen unrealistically large here because we want to see things happening. In Figure 10 on p. 11 (lowest curve) we see the area of 1cm2modulated by ±50%. We observe the following points: We choose here only one nonlinear coefficient • The coil displaces more to the right than to the left (lowest curve). This is con- firmed by the displacement time signal (top curve, black) being shifted up; and by the FFT (middle curve, black) having a small positive DC value. This is because the cavity is less of a hindrance to the coil movement when the area is smaller. • The SPL is much more distorted: It shows smaller positive values in the com- pression phase where the area is small. The DC value of its time signal does the opposite of that seen in the displacement: negative DC value in FFT. 3It is often advisable to replace the SI unit m by µm to avoid numerical inaccuracies in the simulation.

15. 2.2. HARMONIC DISTORTION 11 coeff3=0 coeff2=0 coeff1=-1u area0=1e-4 .define Fres_mech 300 .define Cms 3m (1cm2) Displacement Area .define Mms (1/(twopi^2*Cms*Fres_mech^2)) Mms=93.816u X X Y+ Y+ NL Area Y- Y- charge sensor Mms Bl 0.5 Cms 50m in + (Tm) + + Rdc 8 volt=0.3 Hz=F1 Rmech N mech electr mech (V,Hz) Pa_Cavity Ac 1 .define F1 30 VOL IDEALTRA_nonlin m3=50u m3 Figure 9: The circuit for modulating the effective membrane area. Figure 10: The transient response of Figure 9 at 30Hz. One can expect this form of acoustic distortion never to appear on its own. It will always go hand in hand with a genuine mechanical distortion of the restoring force of the membrane spring. But physically, area modulation is an entirely different mechanism, only indirectly connected with non-linearity in the F vsx curve of the mechanical spring. Area modulation has its greatest influence when the coil diameter is small compared with the outer diameter of the torus membrane. It can be measured by combining Thiele-Small measurements in the mechanical domain (mass method using blue-tack→Bl) with those in the acoustical domain (inertance or acoustical mass method using a hollow tube or a cavity→Bl/S). The ratio of the results

16. 12 CHAPTER 2. HEADPHONES COMPARED WITH LOUDSPEAKERS gives the value of S. Each pair of measurements needs to be done with a different stationary displacement of the coil. It can be shifted by an electrical DC current (danger of heating) or static air pressure. Section B.2 on p. 41 describes how the other distortion mechanisms can be simulated. 2.3 Effect of Headphone Size on Leakage Audibility Experience of headphones at a distance demonstrates that large ones are noisier. To see in principle why this is so, or indeed whether it need necessarily be so, we shall drastically simplify the system, and change only one thing—the cavity volume. Comparing two realistic fully developed headphones of different size would not help understanding—we already know which will have more audible leakage. The design principles for different sizes are so different that we would be none the wiser after such a comparison. Rdc 15 Bl 0.3 Mms=8.443u .define Cms 3m (Tm) + + .define Fres_mech 1000 .define Mms (1/(twopi^2*Cms*Fres_mech^2)) (V,Hz) ac. 1 N mech electr mech Nvalue=1m (AREAm2,DISTm,deg) BACK FRONT Pa_FAR Pa_VOL deg volume=100.531u VOL RadiatorBaffledPascal AREAm2=999 DISTm=1 angledegrees=0 .define volume (area*length) m3 .define diam 80m .define area (piover4*diam^2) area=5.027m .define length (diam/4) length=20m Figure 11: The schematic for investigating the influence of size on audibility of leakage. Figure 11 shows the schematic for such a comparison. In Figure 12 on p. 13 we shall step the diameter from 20mm to 80mm, which increases the cavity volume, at the same time increasing the depth proportionally4. For simplicity we consider a closed headphone with only one leak, this being at the rear. The radiation impedance is neglected, by making the radiator area infinite: 999m2 (Figure 34 on p. 35 shows how the radiation impedance depends on area). Taking into account the radiation impedance would only dampen the resonance peak, which is not the issue here. The results would remain essentially the same. The area of the membrane is also kept constant (Nvalue of the ideal transformer). If we had increased this proportionally with the volume, the SPL in the cavity would have decreased. But the ratio (lower part of Figure 12 on p. 13) would still increase with size. 4But this is just a formality. We could just as well have increased the volume directly.

17. 2.3. EFFECT OF HEADPHONE SIZE ON LEAKAGE AUDIBILITY 13 AA16_4ZB_LEAKAGE_VS_VOL_ANALYSIS.CIR DIAM=20m...80m 140.0 20mm 120.0 100.0 80.0 dB SPL 60.0 80mm 80mm 40.0 20.0 20mm 0.0 200 dB(v(Pa_VOL))+94 1K 10K 20K dB(v(Pa_FAR))+94 F (Hz) 20.0 0.0 -20.0 80mm RATIO -40.0 diameter increase -60.0 -80.0 20mm -100.0 200 dB(v(Pa_FAR)/v(Pa_VOL)) 1K 10K 20K F (Hz) Figure 12: The larger the headphone the greater the audibility of leakage at a distance. In the results (Figure 12) the curve in the lower picture dB(RATIO) is of interest: dB(v(Pa FAR)/v(Pa V OL)). Summing up, we can say the following: • Headphone Cavity: With increased cavity size, the increase in far-field SPL comes just from the volume increase. The reason is that the combined system spring becomes more compliant, thus allowing more displacement x[mm] for larger vol- umes. This is because in the working range of frequencies a headphone tends to be stiffness controlled (Figure 1 on p. 2). The membrane area is unimportant. • Rear Cavity of LS Box: A LS also becomes louder with increase in box volume. But this is not because of the volume itself. The membrane area is the significant parameter. Increasing the box volume only shifts the resonance down, extending the frequency range. Since the working range of a LS is above resonance, where it is mass controlled (inertia) the cavity size does not significantly influence the strength of the output signal. 2.3.1 Dipole Cancellation In the above we had only one leak, for simplicity. As we shall see for the small in-ear headphone (Figure 21 on p. 24) the resultant leak is a vector sum of front and back leaks. The small headphone is essentially a dipole radiator: air current in at the back almost equals air current out of the front. This becomes increasingly true the lower the frequency is. If we add a front cavity or increase the size of any front cavity already present, part of the air current from the front of the membrane goes into compressing this air cushion, which reduces the air current out of the front leak. This unbalances the two dipole components, and there is less cancellation of the far-field sound. This contributes an additional mechanism to make large headphones noisier.

18. 14 CHAPTER 2. HEADPHONES COMPARED WITH LOUDSPEAKERS 2.4 What the Ear-Canal Does The earcanal is essentially a two-port tube (Figure 31 on p. 33). The transfer function p2/p1of any two-port can in principle depend only on the two-port itself and the output impedance (in this case that of the eardrum). The circuit for the earcanal transfer function is shown in Figure 13 on p. 15. Figure 14 shows the results. For hard termination we get infinite peaks due to standing waves for which the earcanal length equals λ/4, 3λ/4 etc. The eardrum impedance brings these down to about 12dB. Generally it would be a good question to ask “Of what is this really a transfer function?” The output is clearly just in front of the eardrum. But where is input? Is it a physical point at the entrance? Or are we dealing with a “functional” transfer function? 2.4.1 “Functional” Transfer Functions! An example is the “freefield transfer function of the head and ear”. When doing freefield measurements on real or artificial heads one often wants to know the transfer function of a system whose input is no longer physically present, because it has already been disturbed (diffraction) by the thing being measured. The output is clearly normally at the eardrum. But the input could be defined as the SPL at that point in space at the centre of the head, before the head had been put there. Resuming the inquiry about the ear-canal, we have a similar problem defining our earcanal input. Do we mean a point just behind the entrance, within the earcanal? If so, there is no problem and this is simply a physical transfer function. Or do we mean a point on the hard surface (where head diffraction effects are already taken into account) before the ear canal was opened5? There are two stages to this problem which should be dealt with separately and in sequence: • Step1 Diffraction Effects without openings. These yield a pressure source poon the surface of interest. If it is a head, then with plugged ear canal6. • Step2 Radiation impedance of the opening Now the surface is no longer hard, and the pressure source podevelops an internal impedance Zint, whose value is the radiation impedance Zradof the opening. Zraddepends mainly on the radiating area. Figure 34 on p. 35 (lowest compartment) shows a simplified formula for this Zrad. The real part of a radiation impedance represents a loss of energy no less real than that from the viscosity losses in an acoustic resistance. The energy lost by radiation need not heat anything. It simply does not return to the opening. When simulating a headphone response we need not explicitly deal with Step1. The whole circuit to the left of the point labelled “out1” (Figure 20 on p. 24) is implicitly a sound source with internal impedance depending on the details of the headphone. For a small headphone even step2 can be ignored, because the earcanal opening is not really radiating at all. So there is no energy loss through any supposed “real part” of an impedance. Lumped elements can be used (as in Figure 20 on p. 24). One of 5I do not mean ERP. That has its own problems, which are of a different nature (Figure 28 on p. 30). 6Or even better: The surface could be prepared with a rudimentary earcanal of depth 3mm. Then we have the luxury of ERPx being prepared in advance for plane waves.

19. 2.4. WHAT THE EAR-CANAL DOES 15 these is the end correction (Figure 34 on p. 35), which is actually the reactive part of the radiation impedance. But for circum-aural devices such as hearing protectors the radiation impedance above 500Hz approaches that of the unoccluded ear [9]. Presumably this applies to headphones too. For the unoccluded ear the first approximation at low and mid frequencies is the simple end correction (reactance jωL, no R). Now we are in a position to digest the red curve of Figure 14. As already noted, the main effect of the eardrum is to dampen the resonances. When we introduce the radiation impedance the resonance shifts to lower frequencies because of the mass loading of the input. The radiation impedance has been integrated into the two-port. This might be done if one is interested in freefield effects. Therefore, if the resonance is not exactly as expected for a certain tube, before looking for errors in tube length one should ask whether the radiation impedance has been included or not. TUBE HARD nonlossy volt=1 (L, D, N) length=24m diam=diam (V,Hz) .define diam 7.4m .define area (piover4*diam^2) area=43.008u earcanal TUBE DRUM nonlossy EARDRUMzwislocki volt=1 (L, D, N) length=24m diam=diam Groundnormally Groundnormally RadiatorBaffled AREAm2=area (V,Hz) DRUM_and_RADIMP (AREAm2) TUBE nonlossy EARDRUMzwislocki a a volt=1 (L, D, N) length=24m diam=diam (V,Hz) Figure 13: The circuit for earcanal TF. Radiation Z is normally earthed, but not here. AA22EARCANALTRANSFERFUNCTION_ANALYSIS.CIR 50. 40. 30. 20. 10. 0. -10. 200 db(v(HARD)) 1K 10K 20K db(v(DRUM)) db(v(DRUM_and_RADIMP)) F (Hz) Figure 14: drum (blue); with drum showing effect of radiation impedance (red). Earcanal transfer functions (for Figure 13): hard drum (black); with

20. 16 CHAPTER 2. HEADPHONES COMPARED WITH LOUDSPEAKERS 2.5 The Ear Spectrum For Frontal Sound Source Here we consider only the so-called monaural head-related transfer function for frontal incidence7. By this I mean we acknowledge the differences of individual ear geometry from person to person, but assume for the moment that the ears of any individual are identical. The characteristic features for a typical human ear are as drawn in Figure 15. The dips at 1kHz and 10kHz arise from the comb filtering effects of the shoulder and ~20dB EARDRUM ~10dB EARCANAL ENTRANCE 0dB 1kHz 3kHz 10kHz SHOULDER DIP 1kHz EAR CONCHA DIP 10kHz + + Figure 15: Typical diffraction features for frontal sound source. concha respectively—first minimum in each case. The 10dB mountain around 3kHz is due to the concave nature of the pinna region around the earcanal entrance. Diffrac- tion with multiple reflection in craters always give rise to a global amplification of the incoming sound. The pinna including concha gives approximately 10dB amplification. Superposed on this is the 10dB amplification due to the λ/4 resonance in the earcanal (as we saw in Figure 14 on p. 15). The result is a heavy colouration with a 20dB peak at about 3kHz. The individual has learnt to hear with his own ears and has got used to associating certain spectral characteristics with each visible direction8. 7We also neglect here second-order localisation effects which rely on such things as familiarity with the sound source, head movement and other temporal phenomena such as sound-source movement. We concentrate here on the first order effect which works for short high-frequency impulsive sounds, which need not be known to the listener. He can localise these even without head movement—indeed for short sounds there is no time for confirming first impressions by head movements. And as we know from experience these spontaneous first impressions are normally correct. 8For invisible directions the individual has no means of learning the spectral characteristics and

21. 2.5. THE EAR SPECTRUM FOR FRONTAL SOUND SOURCE 17 2.5.1 A Tentative Assumption We shall assume for the moment the absolute spectrum (such as Figure 15 on p. 16) of an individual to be the cue for frontal localisation. Irrespective of the correctness of this assumption, sharp colouration for a frontal sound source must be expected. If it is indeed localised frontally (“Gestalt” recognition) this colouration will not disturb him—association model of Theile[5]9. Otherwise the 20dB peak (free-field standard for headphones) is too sharp. Then a milder spectrum is needed, as for the diffuse field headphone standard (dashed line in Figure 1 on p. 2). Interestingly, the 3λ/4 earcanal resonance coincides approximately with the 10dB dip at 10kHz. This is the region for hissing sounds “S”. Thus there is a good chance of such sounds being heard despite the inherent attenuation—fragment of red curve around 10kHz in Figure 15 on p. 16. (The Creator forgot to patent this clever idea!) 2.5.2 The Above Assumption Was Illogical If the absolute monaural spectrum were the cue for localisation this would work reliably only if all sources radiate with flat frequency spectra. In reality sound sources have inherent colouration, which is present in the SPL spectra at any point in the transmission chain on the way to the eardrum. The brain has no means of knowing whether such linear distortions happened at the pinna or before. Indeed it could well be a property of the source. Since we can assume the clever Creator did not make such errors of logic, the essential cue for frontal localisation cannot be the absolute spectrum. No research is needed to establish this. Indeed, own attempts at equalising a rear LS to approximate the frontal spectrum of one of the ears were unsuccessful in achieving frontal localisation. So, what is there left? The interaural differences are unique for each individual and could not be a characteristic of the sound source. Faithfully reproducing these individual dif- ferences resulted in clear frontal localisation via headphones using DSP. This succeeded for two tested individuals (no statistics were needed). Exaggerating the differences as in Figure 16 on p. 18 enhanced frontal localisation. The fact that localisation can be im- proved in this way, departing from a faithful reproduction of the differences, agrees with the following surprising fact, which has often been reported: For some people, frontal lo- calisation is better achieved, not with their own spectra, but with those of some other individual. Presumably that other individual had similar interaural differences, but exaggerated. What’s more, interchanging the L and R spectra destroyed the frontal localisation even without exaggerating the differences. This is spectacular, because the original in- teraural differences were not large—less than 4dB. Therefore such interchange of L and R cannot result in any appreciable sound colour change in either ear. Auto-suggestion can be ruled out because the two subjective impressions: frontal- and in-head-localisation, were clear and unique even without knowing whether L and R were exchanged. associating them with known directions. These sounds are all dumped into the bin which we could label with various names such as “In Head Localisation”, “Rear Localisation”, “Elevation Effect”. In the case of the elevation effect the incorrectly localised phantom source comes from a visible direction, but the spectral errors are small enough not to result in totally wrong localisation such as inversion to the rear. 9The brain treats itself to an inverse filter, the inverse of that depicted in Figure 15 on p. 16, thereby “ironing out” the 20dB peak. so it is not subjectively perceived too strongly.

22. 18 CHAPTER 2. HEADPHONES COMPARED WITH LOUDSPEAKERS (a) Left and right ear spectra: as measured for frontal sound source (b) Left and right ear spectra: differences exaggerated for improved frontal localisation Figure 16: (a) Possible individual ear spectra for frontal source. (b) How to exaggerate the differences to enhance frontal localisation without changing the sound colour. Via headphones such exaggeration above 2kHz can be very effective. This agrees with the fact that some people experience better frontal localisation using the ear spectra of another individual than with their own. By analogy, the benefit of exaggeration applies also to stereo photography: To obtain a spectacular 3D impression of a landscape the effective “eye distance” needs to be exaggerated to several metres. The author knows of a person with practically identical ears and consequently identical ear spectra for frontal incidence. This person had the distinguishing feature that he was unable to localise frontally in the median plane under any conditions, even for real free-field sources. 2.6 More Reliable Stereo Image For Headphones Consider the soloist at centre stage. stereo signal. For a pair of loudspeakers at ±60 degrees the image, though in the median plane, is usually raised above where the phantom image should be (elevation effect). This is inherently a mono component of the For a real frontal sound source correct localisation is determined by the spectra of the transfer functions from free-field to the ears. Irrespective of whether the source is a single LS or a real musical instrument, these ear signals are automatically correctly localised, because the listener is using his own ears. In the electro-acoustic situation (LS or HP) we have the task of simulating these signals as well as possible. If we have done the necessary measurements for an individual, we can assume we know how his ear signals should look. Armed with this information there is a chance for correct frontal localisation in the electro-acoustical situation. The chances are as follows: HPs: very good—we are in control: (L signal→L ear); (R signal→R ear) LSs: modest (L signal→L ear); (R signal→R ear); (L signal→R ear); (R signal→L ear)

23. 2.7. IS THE EAR JUST A PRESSURE DETECTOR? 19 For LSs in the usual stereo set-up the ear signals come from ±60 degrees left and right, not 0 degrees. There might be a good chance if there existed a mechanism for frontal localisation for perfectly symmetrical ears and head (the tentative assumption of Sec- tion 2.5.1 on p. 17). This is because the shadowing and diffraction effects for all persons do have something in common. But frontal localisation of impulses relies on interaural differences in the ear spectra (Section 2.5.2 on p. 17 and Figure 16 on p. 18). These are chance phenomena of ear geometry. Moreover if these differences are known for 0 degrees, there is no reason what- soever for expecting the same for a pair of LSs at ±60 degrees, even if we were justified in neglecting the cross-feed components (L signal→R ear) and (R signal→L ear). Of course the cross-feed components make the situation even hazier. Thus we have the following ironical situation: • A LS (because of its genuine free-field acoustics) could provide perfect frontal localisation, but in the usual stereo set-up does not do so because the angle is wrong. • If we know the ideal ear spectra for an individual, we have in headphones (even with their non-free-field acoustics) a tool for making use of this knowledge directly, to achieve frontal localisation. The result can be exaggerated to be even better than with LS in freefield. 2.7 Is the Ear Just a Pressure Detector? One similar aspect of LSs and HPs is that the subjective sound impression for both comes from the SPL in the ear signals alone. Otherwise headphones would not have the localisation advantage over LS mentioned in Section 2.6. How the ear signals are produced—pressure chamber or freefield of a propagating wave—is unimportant. But we cannot say this without reservations (see also Section 2.8): Listeners tend to listen at higher SPL for HPs than for LSs to get the same subjective loudness. Many such phenomena have been reported in the literature (see references at the end of chapter 14 in [2]). We could put these under the collective term “Missing 6dB”, a ghost which has been repeatedly put to rest, apparently for ever (possible measurement errors revealed), and repeatedly dug up again. 2.8 Audiometry With Headphones and Loudspeakers Not only the subjective impression (the closed-in feeling), but also the hearing threshold, are not quite the same. They are influenced by secondary characteristics superimposed on the above pressure-detector mechanism. For example the hearing threshold in audiometry using a HP is higher than when using a LS. This comes from masking of the signals due to physiological noise, such as that from blood flow turbulences. The ear SPL of such body noises (and bone conduction), is raised in the case of headphones by the occlusion effect. This amplifies the SPL below 2kHz by up to 20dB ([2] p.651). This can easily be experienced by humming a low tone while closing one ear with a finger tip.

24. 20 CHAPTER 2. HEADPHONES COMPARED WITH LOUDSPEAKERS

25. Chapter 3 An In-Ear Headphone Resistive: R Small holes Inductive: L Mass of air plug Figure 17: The essential components of a small headphone shown schematically. Foam cover usually deactivates half the output holes. Red: bass tube, thick black line: cable. This is the most popular and most common type of headphone today (Figure 17). Here we consider the following aspects: • How the bass tube can be implemented in a “design-friendly” way. • What other leaks there are, and do we just tolerate them for practical reasons or do we need them. • Why the leak signal for people standing nearby sounds sharp. 21

26. 22 CHAPTER 3. AN IN-EAR HEADPHONE 3.1 Bass Tube Design The length of the bass tube shown schematically in Figure 17 on p. 21 is not exaggerated. To have an effective bass tube we need: • A large L/R ratio of acoustic mass L to resistive damping R, so that ωL can compete with R even at low frequencies. This condition demands a large absolute value of d, the diameter1. But for a short tube of correct diameter the absolute value of L would be too low. The same applies to the bass-reflex LS. • In many products the bass tube is incorporated in the vertical portion of Figure 17 on p. 21, competing with the cable for space, and limiting design freedom. Since impedance of sound constrictions depends mainly on length and diameter—not so much on the shape—a curved tube is as good as a straight one. The circumference of the transducer is large enough to accommodate the bass tube (Figure 18). Figure 18: The bass tube on the circumference of the capsule. 1To see why, refer to Figure 34 on p. 35: L ∝ d−2, whereas R ∝ d−4.

27. 3.2. THE MAIN FREQUENCY RESPONSE 23 3.2 The Main Frequency Response Figure 20 on p. 24 shows the circuit for those parts of a small headphone which are outside the transducer. All was simplified to the bare essentials. Only one cavity is included, the rear one, all the others not being essential to the understanding. To avoid cluttering the diagram the contents of the transducer are put into a macro fed by Thiele- Small parameters (Figure 19), which also has no cavity. For a desired vacuum resonance 255Hz and mass Mms, the membrane compliance Cmsis automatically adjusted for a given mass Mms. .PARAMETERS(Fres_mech,Mms,Amem,Blvalue,Rms,Rdc) .define Cms (1/(Mms*twopi^2*Fres_mech^2)) displacement charge charge mass resistance compliance charge sensor Mms Cms Rms v [m/s] Blvalue Rdc EL1 + (Tm) + + out out in in electr mech Force [N] EL2 v [m/s] N mech Secondary N=Amem Primary acoust. 1 v [m3/s] AK1 AK2 BACK of membrane FRONT of membrane deltaP [N/m2] Figure 19: The Thiele-Small Macro for the small transducer. 3.3 Earcanal Influence The artificial ear is described in Section 4.2 on p. 31. Figure 21 on p. 24 shows the effect of the earcanal. Here we concentrate on the upper two curves. The response is calculated at the earcanal entrance (out1) and at the far end DRP of the IEC711 coupler (out2drum). We see the characteristic resonances (Figure 14 on p. 15) of the earcanal at 3.5kHz etc (λ/4, 3λ/4 etc.). The next thing we shall consider is the bass boost.

28. 24 CHAPTER 3. AN IN-EAR HEADPHONE (AREAm2,DISTm) (AREAm2,DISTm) back holes 1k 400e5 far_front volt=0.126 AREA=888 DIST=distance AREA=0.5e-4 DIST=distance (V,Hz) out1 out2drum far_backholes foam 4e5 bass tube (AREAm2,DISTm) ear canal 0 IEC711 10 front holes 200e5 20k switch DRP ERP 100 IEC_711_with_ear_canal_noEC LS ThieleSmall_LS Fres_mech=255 Mms=17.5u Blvalue=0.46 Rdc=14.4 AREA=888 DIST=distance m3 Vol_back m3=0.15u far_SUM far_basstube Rms=0 .define distance 30m Amem=0.8e-4 Figure 20: The equivalent circuit of a small in-ear headphone. Figure 21: Frequency responses for small in-ear headphone. Upper two black curves: at ear canal entrance (dotted line); at output DRP of IEC711 (full line).

29. 3.4. HOW THE BASS TUBE WORKS 25 3.4 How the Bass Tube Works AA14MINIBASSTUBE_ANALYSIS.CIR SWITCH=0...90T 110. 100. 90. 80. 70. 60. 50. 40. 30. 20. 20 db(v(out2drum))+94 db(v(far_sum))+94 100 db(v(far_basstube))+94 1K 10K 20K db(v(far_backholes))+94 db(v(far_front))+94 F (Hz) Figure 22: Bass tube normal (upper curve) and closed (lower curve). AA15MINIBACKHOLES_ANALYSIS.CIR SWITCH=0...90T 130. 120. 110. 100. 90. 80. 70. 60. 50. 40. 30. 20 db(v(out2drum))+94 db(v(far_sum))+94 100 db(v(far_basstube))+94 1K 10K 20K db(v(far_backholes))+94 db(v(far_front))+94 F (Hz) Figure 23: Back holes open (normal) and closed (sharp dip at 1kHz). Figure 22 shows what happens when the bass tube is closed. There are also other rear leaks. Switching them off must make a “purer” bass tube. Figure 23 shows that this is not desirable. The sharp minimum indicates that the coil hardly moves (lower curve, LS.displacement) at 1kHz where the bass tube is allowed to act “undiluted”.

30. 26 CHAPTER 3. AN IN-EAR HEADPHONE It is as if the compliant rear cavity were replaced by an infinitely hard wall intimately in contact with the rear surface of the membrane. This comes from the parallel resonance of the cavity compliance (condenser Cvol) and bass tube (Lbasstube), as shown in Figure 24. bass tube 19k deg Vol_back m3=0.15u .define Cvol (.15u/(1.4*atmospressure)) (AREAm2,DISTm,deg) m3 Cvol=1.071p Mode2 atmospressure=100K Pa 4kHz Resonance 19k Mms Cvol dominates stiffness Cvol/2 Cvol/2 Cmem 19k 1kHz DIP Coil and membrane stand still Helmholtz resonance infinite impedance Cvol 19k Mms 200Hz bass boost soft membrane dominates stiffness Mode1 Cvol Cmem no compression of air cushion Figure 24: When the rear holes are closed the Helmholtz resonance (1kHz) of bass tube with rear cavity presents an infinite impedance for membrane motion (sharp dip in Figure 23 on p. 25). This resonance is in the middle between two other resonance modes. The R of the basstube is here omitted for simplicity. Of course the imagined partition for 4kHz divides the cavity into equal halves Cvol/2 and Cvol/2 only for equal masses. Normally they are unequal. Also we have a peak at 4kHz. This is the second “bassreflex” resonance, where the currents of q1 and q2 into the rear cavity (Figure 25 on p. 27) are in phase (Figure 26 on p. 27). The basstube and membrane both pump into the rear cavity with the same phase above 1kHz. At the resonance 4kHz we can imagine an immovable partition dividing

31. 3.4. HOW THE BASS TUBE WORKS 27 the cavity spring into two smaller and stiffer parts: one for the bass tube and one for the membrane. Since they are each stiffer than the undivided cavity stiffness the resonance must be higher than the main Helmholtz resonance at 1kHz. The bass boost around 200Hz has nothing to do with the cavity. The equivalent of Figure 23 on p. 25, but without any cavity is shown in Figure 27 on p. 28. (AREAm2,DISTm) back holes 0 1k 400e5 switch far_front volt=0.126 AREAm2=0.5e-4 DISTm=distance (AREAm2,DISTm) AREAm2=999 DISTm=distance (V,Hz) q1 q2 out1 out2drum far_backholes foam 4e5 current current current current bass tube current sensor current sensor ear canal IEC711 10 front holes 200e5 20k DRP ERP out out out out in in in LS ThieleSmall_LS Fres_mech=255 Mms=17.5u Blvalue=0.46 Rdc=14.4 in 100 IEC_711_with_ear_canal_noEC (AREAm2,DISTm) AREAm2=999 DISTm=distance far_SUM m3 far_basstube Vol_back m3=0.15u Rms=0 .define distance 30m Amem=0.8e-4 Figure 25: The circuit for revealing the phases of currents q1 and q2 into the rear cavity. Figure 26: The phase difference of the currents q1 and q2 in Figure 25.

32. 28 CHAPTER 3. AN IN-EAR HEADPHONE AA16_4MINIBASSTUBE_ANALYSISNOCAVITY.CIR SWITCHBACKHOLES2=0...1.1T SWITCHBACKHOLES=0...1.1T 130. 120. 110. 100. 90. 80. 70. 60. REAR CAVITY REMOVED 50. 40. 30. 20 db(v(out2drum))+94 100 db(v(out2drum_nocavity))+94 1K 10K 20K F (Hz) Figure 27: Like Figure 23 on p. 25, but cavity deactivated. Unlike in a bass reflex LS, the bass tube still works. 3.5 Sharp Sound-Colour Of Far-field Leakage Each opening except the bass tube radiates into the surroundings with a bass deficit (Figure 21 on p. 24). This fact alone would explain why a headphone sounds “sharp” to bystanders. But the signal at the summation node SUM is even “sharper”. The individual components are vectorially added. We must remember that the air in front and behind the membrane cannot experience much “compression” at mid and low frequencies where the reactance ωL of all acoustic masses L resembles a short circuit. Thus the total current in at the back must almost equal that out of the front. Since their influences at a distance are in anti-phase, we get progressive cancellation as the frequency is lowered. The headphone is a dipole radiator.

33. Chapter 4 Reference Points For Response Measurements 4.1 ERP and DRP: Ear and Drum Reference Points Tracing the transmission chain from a sound source to the inner ear the last point for normal analysis1is DRP (Drum Reference Point) 3mm in front of the eardrum (Figure 28 on p. 30). What the brain receives is of course a spectral impression of the signal at DRP. So obviously this is the one we should be primarily interested in. But some measurements refer to ERP (Ear Reference Point) in the concha. Results involving ERP can be problematic, depending on what you want to do with them. If you are interested in SPL at ERP, you can measure it and you have what you want. But there are standardised tabular data for (dBSPLERP − dBSPLDRP), which are intended for transforming from one to the other. The implicit assumption is that this transfer function (TF) is unique—independent of diffraction effects to the left of the input. This only applies if the stretch of air between these two points is a genuine two-port, which is true for ERPx → DRP, but not for ERP → DRP (Figure 28). 1“Normal” here means the SPL [Pa] is proportional to other stages before this point. Behind the eardrum we have movements of bone levers and nerve impulses. These cannot be treated as signals in linear two-ports. For example, even for a sinusoidal sound signal the nerve impulses are spikes of impulse probability, firing rhythmically once per period with the zero-crossings of the acoustic signal. Between these spikes nothing happens. Thus there is nothing sinusoidal about them. 29

34. 30 CHAPTER 4. REFERENCE POINTS FOR RESPONSE MEASUREMENTS ? ) ( SBIG = S / cos ( vSMALL = v × cos qERP ambiguous ? ) ? vSMALL DRP ERPx ? q2 ERP q2 = vSMALL ×SBIG q1and q2 are equal. qERP velocity v [m/s] DRP ERPx ERP q1 [m3/s] q1 = v × S Unambiguous two-port Figure 28: The transfer function from ERP to DRP is not unique. For a genuine two-port the current at the input depends only on the two-port itself and its output load. In general, we might be able to define the pressure p at the input of a system. But this is not always sufficient for defining the current. If the input is 3mm just behind the earcanal entrance (ERPx) the current is also defined because the input impedance of everything to the right is unique. It is easier to deal with ERPx and DRP, because the region around them is inherently one-dimensional and of small cross-section. But that around ERP comprises the concha and pinna etc. with a 3-dimensional geometry2. Defining the current q means choosing a small area element Snormalor Sbig, through which the current is supposed to flow, and integrating the particle velocity v over this area: q =Rv dS. Figure 28 shows why the concept of volume current q is unique only inside the earcanal. The value of q must be independent of the exact choice of angle θ for the area S. Within the earcanal this is fulfilled for all reasonably small values of θ. 2For measurements involving circum-aural headphones, ERP is surrounded by the headphone cavity. For free-field measurements the head and shoulders are part of the surroundings of ERP and influence the transfer function.

35. 4.2. ARTIFICIAL EAR IEC711 COMBINED WITH EARCANAL 31 4.2 Artificial Ear IEC711 Combined With Earcanal In the simulation circuit of Figure 20 on p. 24 we saw a macro of the IEC711 coupler combined with an earcanal. The content of this is shown in Figure 29. Obviously the short stretch 10mm cannot represent a real earcanal. Part of the IEC711 already contains several tubes in series, representing the second half of the earcanal and eardrum impedance. The equivalent circuit using lumped elements can be much improved above about 2kHz by introducing transmission lines for the tube sections (Figure 30). EAR CANAL IEC711 TUBE ear drum nonlossy in DRP ERP in DRP ERP (L, D, N) length=10m diam=7.4m N=1 Figure 29: A short stretch of earcanal combined with the IEC711 coupler. IEC711 TUBE length=4.7m diam=7.4m TUBE length=0.6m diam=7.4m TUBE length=4m diam=7.4m TUBE length=2.7m diam=7.4m HOLE N=1 N=1 N=1 N=1 D1 D D2 (L, D, N) (L, D, N) (L, D, N) (L, D, N) IN DRP ERP Ra7 311e5*1.2 nonlossy nonlossy nonlossy nonlossy Ra5 506e5 table2 L (L,D,N,D1,D2) length=0.2m diam=1m N=19 D1=7.4m D2=7.4m Ma5 94e2 Ma7 9.838e2*0.8 Ca7 2.1p Ca5 1.9p Figure 30: The improved simulation of the IEC711 coupler. The Head And Torso Simulator HATS comprises a Silicone rubber pinna and earcanal terminated by the eardrum impedance. This is suitable for headphone measurements. But if one wants to compare measurements with computer simulations, one needs a realistic equivalent circuit for the artificial ear. The manufacturer of the IEC711 supplied an equivalent circuit consisting of lumped elements R, L (for constrictions) and C (for cavities). But the IEC711 consists also of several tube sections for which lumped elements are inaccurate. Figure 30 shows an improved circuit using transmission line two-ports as described in Figure 31 on p. 33.

36. 32 CHAPTER 4. REFERENCE POINTS FOR RESPONSE MEASUREMENTS

37. Appendix A Behind The Ear Canal Entrance A.1 Tube Transfer Function length [m] p2 p1 S [m2] q2 q1 q1 q2 A11 A21 A12 A22 p1 p2 Figure 31: The chain matrix nomenclature for a tube as a two-port. ?p1 q1 ? ? ? ?p2 q2 ? cos(kl) (j/Zw)sin(kl) j Zwsin(kl) cos(kl) = (A.1) × p1= cos(kl) × p2+ j Zwsin(kl) × q2 q1= (j/Zw)sin(kl) × p2+ cos(kl) × q2 (A.2) (A.3) The input and output currents (Figure 31) are both to the right, as is convenient for combining such chain matrices. These equations apply to a non-lossy tube. The para- meters are k = ω /c where ω = 2πf; l = tube length; Zw= ρc/S, where ρ is air density and c the speed of sound propagation. For a hard eardrum the transfer function of the earcanal would be p2/p1= 1/cos(kl) (setting q2= 0 in equ. A.2). This would show an infinite peak where cos(kl) = 0. The frequency of this peak is 3500 Hz1. In reality the peak is only about 12dB because the eardrum impedance is not infinite. 1This means k l = π/2. c=340000mm/s and l=24mm we get f = 340000/(4 × 24) = 3500Hz. Since k = 2πf/c, we get 2πfl/c = π/2 giving f = c/(4l). Setting 33

38. 34 APPENDIX A. BEHIND THE EAR CANAL ENTRANCE A.2 Eardrum Impedance Figure 32 shows a highly simplified equivalent circuit of the eardrum impedance using lumped elements. Its frequency response is shown in Figure 33, where it is compared with that from the model of Zwislocki [3], where ossicle effects, such as the decoupling of the malleus from the eardrum, are taken into account. Eardrum impedance simplified as RLC circuit a Resistance_acoustic 300e5 1000 Mass_acoustic 6.5p Compliance_acoustic Figure 32: Very simplified equivalent circuit for eardrum impedance. Figure 33: The real and imaginary parts of the eadrum impedance. The results of the circuit of Figure 32 compared with those from Zwislocki’s circuit [3].

39. A.3. LUMPED ELEMENTS 35 A.3 Lumped Elements Figure 34 shows the equivalent circuits of acoustical constrictions, cavities, the resonant core of a LS, and a simplified radiation impedance. Flow velocity profile f <fo Equivalent Circuit R 8 η L ρ C HOLE l l 4 × ? ? 4 2 3 d d ? ? 4 π π l ? ? 53 2 2 d f − 4 , 2 04 l × 0 , 7 58 10 l d = = 2 4 d d ρ f >fo l η 8 l f η 64 ρ = f ! × - * 2 4 0 f 2 d d π + ( 0 d π π , ) d 2 2 − 4 d × ' $ , 3 16 10 3 0 − = × % " , 7 58 10 l f , 1 l 1 . 2 = = % " d 4 1 . & # 2 / 2 d d η ρ SLIT d<<b f = π f <fo 12 l 6 l × 3 b 5 b d b d l ? < 9 6 d − 4 , 1 44 l 7 4 = : , 2 23 10 l = = > ; 7 4 d b 3 8 5 b d l η 36 ρ f >fo ρ l η 12 l f 0 2 d 3 f b d b d 0 − 4 × E B K H , 1 = 78 10 − d 4 2 , 1 l × C @ , 2 23 10 l f I F 2 = = d C @ J G 3 D A b d f b d 0 CAVITY compliance Volume V = MEMBRANE PLUS COIL area S res= Radiating piston (approx.) area S = V γ atmos P V Q N O L V = O L P M 5 × 2 4 , 1 10 m S ≈ 0 2 = = , m mass S 2 2 π 4 m f res f resonance ρ ρ c0 , 0 85 S 408 π , 0 S 58 W T ] Z U R = [ X V S = \ Y S S The acoustic elements, their electrical equivalent circuits, and R, L, C values expressed in terms of mechanical dimensions: l, b, d [m], S [m2], V [m3], frequency f [Hz]. Effective total length of a constriction in flow direction is l. For a circular hole the total effective length l equals the geometrical length l0plus the end correction 0.85d. The viscosity coefficient under normal conditions is η = 1.86 × 10-5 kg/(ms). It is proportional to pressure and roughly proportional to absolute temperature [K]. Air density is ρ = 1.2kg/m3. Speed of sound c0 = 340m/s. Specific heat ratio γ = 1.4 for air. Patmos = 105 Pascal. The impedance analogy is used, where sound pressure [Pa] is equivalent to voltage [V], and volume velocity [m3/s] is equivalent to current [A]. The equivalences for L, R, and C are 1H ≡ 1kg/m4, 1Ω≡ 1kg/(m4s), 1F ≡ 1m4s2/kg. Figure 34: The electro-acoustical equivalent circuits of lumped elements.

40. 36 APPENDIX A. BEHIND THE EAR CANAL ENTRANCE

41. Appendix B Harmonic Distortion B.1 Two Superstitions Exposed B.1.1 K2 K4 K6 etc Not Always Unsymmetrical Many of us have come to believe the rule of thumb: (1) Even harmonics K2 K4 K6 etc1cause unsymmetrical time signals; (2) Odd harmonics K3 K5 K7 etc cause symmetrical time signals. Indeed the circuit of Figure 35 with Nharm=3 gives a symmetrical time signal (Figure 36 on p. 38), as expected. IHD means Individual Harmonic Distortion referred to F1 in %. in in2 sum SineVoltHzPha Hz=F1 (V,Hz) .define F1 1k SineVoltHzPha Hz=(F1*Nharm) .define Nharm 3 volt=1 deg=0 (V,Hz) volt=.25 deg=180 SUBJECT HERE IS: 1 K2 NOT NECESSARILY UNSYMMETRICAL (Nharm=2) 2 DC NOT NECESSARILY VISIBLE IN TIME SIGNAL Battery_for_DC_value 0 Figure 35: Circuit for revealing two superstitions. Now let us set Nharm=2. We get an unsymmetrical time signal (Figure 37 on p. 38) as expected. But if we set the phase to 90 degrees (Figure 38 on p. 39) we are not so sure. The question of symmetry or non-symmetry in the displacement direction is only meaningful if the signal looks the same when the time axis is reversed. Otherwise we optically can get a sawtooth waveform and we cannot know from its shape whether even or odd harmonics are responsible. 1This is German nomenclature (K stands for Klirrfaktor). For example, K2 is 2nd-order harmonic distortion, expressed in % or dB relative to the fundamental. 37

42. 38 APPENDIX B. HARMONIC DISTORTION 5_SYMM_DC_K3SYMM.CIR 1.00 0.00 -1.00 0.0m v(in) (V) 2.0m 4.0m 6.0m 8.0m 10.0m v(in2) (V) t (Secs) 2.00 0.00 -2.00 0.0m v(sum) (V) 2.0m 4.0m 6.0m 8.0m 10.0m t (Secs) 1.2 0.6 0.0 0.0K HARM(v(sum)) 1.0K 2.0K 3.0K 4.0K 5.0K F (Hz) 45.0 22.5 0.0 0.0K IHD(harm(v(sum)),F1) (%) 1.0K 2.0K 3.0K 4.0K 5.0K F (Hz) Figure 36: Time signal with K3 is symmetrical, as expected. 6_SYMM_DC_K2UNSYMM.CIR 1.00 0.00 -1.00 0.0m v(in) (V) 2.0m 4.0m 6.0m 8.0m 10.0m v(in2) (V) t (Secs) 2.00 0.00 -2.00 0.0m v(sum) (V) 2.0m 4.0m 6.0m 8.0m 10.0m t (Secs) 1.2 0.6 0.0 0.0K HARM(v(sum)) 1.0K 2.0K 3.0K 4.0K 5.0K F (Hz) 45.00 22.50 0.00 0.0K IHD(harm(v(sum)),F1) (%) 1.0K 2.0K 3.0K 4.0K 5.0K F (Hz) Figure 37: Time signal with K2 (phase=180 deg) is unsymmetrical, as expected.

43. B.1. TWO SUPERSTITIONS EXPOSED 39 7_SYMM_DC_K2NOTREALLYUNSYMM.CIR 1.00 0.00 -1.00 0.00m v(in) (V) 2.00m 4.00m 6.00m 8.00m 10.00m v(in2) (V) t (Secs) 2.00 0.00 -2.00 0.00m v(sum) (V) 2.00m 4.00m 6.00m 8.00m 10.00m t (Secs) 1.2 0.6 0.0 0.0K HARM(v(sum)) 1.0K 2.0K 3.0K 4.0K 5.0K F (Hz) 45.0 22.5 0.0 0.0K IHD(harm(v(sum)),F1) (%) 1.0K 2.0K 3.0K 4.0K 5.0K F (Hz) Figure 38: Time signal with K2 (phase=90 deg) is not really unsymmetrical. As hinted in Figure 1 on p. 2, if the time signal shows symmetrical distortion we can say the distortion mechanism had no even harmonics. But if the distorted time signal is unsymmetrical we can say there must be at least one even harmonic component (unsymmetrical distortion mechanism such as spring harder in one direction than the other). But such deductive statements can only be made if the signal is reversible: in other words, reversing time does not change the signal shape. By contrast, in the case of a saw-tooth type of distortion signal (e.g. faster ramp up and slower drop) no deductions of the above type can be made from a qualitative inspection of signal form alone. B.1.2 Apparent DC Value Can Be a Phantom Now, still for Nharm=2 let us set the phase back to 180 deg and look at the FFT instead of IHD, so we get the sign of each component. This can have negative values. A pure real negative value means phase=180 deg, which is one of the two possible phase values for DC. We get Figure 39 on p. 40. Using the battery with a real DC value of -0.5V shifts the time signal down further (Figure 40 on p. 40). Now there is a visible negative DC value, which is not a phantom.

44. 40 APPENDIX B. HARMONIC DISTORTION 8_SYMM_DC_K2AGAINFFTDC0.CIR 1.0 0.5 0.0 -0.5 -1.0 0.0m v(in) (V) 2.0m 4.0m 6.0m 8.0m 10.0m v(in2) (V) t (Secs) 2.0 1.0 0.0 -1.0 -2.0 0.0m v(sum) (V) 2.0m 4.0m 6.0m 8.0m 10.0m t (Secs) 300.0 150.0 0.0 -150.0 -300.0 0.0K (FFT(v(sum))) 1.0K 2.0K 3.0K 4.0K 5.0K F Figure 39: K2 again. Phantom DC value due to unsymmetry not really there (look at FFT: lowest curve). 9_SYMM_DC_K2AGAINFFTDCMP5.CIR 1.0 0.5 0.0 -0.5 -1.0 0.0m v(in) (V) 2.0m 4.0m 6.0m 8.0m 10.0m v(in2) (V) t (Secs) 2.0 1.0 0.0 -1.0 -2.0 0.0m v(sum) (V) 2.0m 4.0m 6.0m 8.0m 10.0m t (Secs) 300.0 150.0 0.0 -150.0 -300.0 0.0K (FFT(v(sum))) 1.0K 2.0K 3.0K 4.0K 5.0K F Figure 40: With real DC value (-0.5V) in battery it is no longer a phantom.

45. B.2. SIMULATING NON-LINEAR DISTORTIONS 41 B.2 Simulating Non-linear Distortions The same approach as shown in Figure 9 on p. 11 for area modulation can be followed for the other causes of non-linearity: non-linear membrane stiffness (F vsx) non-flat Bl profile (Blvsx). For a Bl profile combining two Fermi functions (a concept used in semiconductor- and solid-state physics) is useful for describing a hump (lowest curve in Figure 45 on p. 45). This is better than a polynomial because it is guaranteed never to go negative: .define xstep1 (xmid-xwidth/2) .define xstep2 (xmid+xwidth/2) .define Fermi1 (1/(exp((x-xstep1)/smoothness1)+1)) .define Fermi2 (1/(exp((x-xstep2)/smoothness2)+1)) .define stepup (-Fermi1+1) .define stepdown (Fermi2) .define hump (stepup*stepdown) The coil position at rest can easily be adjusted relative to the hump. The spring can be described by a polynomial (which will go haywire at large amplitudes): Force = (1/Cms) × x + coeff2 × x2+ coeff3 × x3+ coeff4 × x4+ coeff5 × x5 The linear component is 1/Cms, where Cmsis the compliance. Note the opposite sign of the charges for the charge sensor and the place where the Cmscapacitor was, before non- linear elements were introduced. The charge sensor is essentially a piece of wire, which does not disturb the circuit. The shift in charge to the right represents displacement of the coil to the right. This charges the imaginary capacitor with an excess positive charge on the left plate (purple). The voltage across this bridge, supplied by the output of the two-port NLCompliance, works against the restoring force of the spring. Figure 41 shows how all three mechanisms can be combined in a single circuit. coeff3=0 coeff2=-1 Cms=Cms Force Y+ Y+ X X NL Compliance .define Fres_mech 300 .define Cms 3m coeff3=0 Y- Y- coeff2=0 coeff1=-1u area0=1e-4 .define Mms (1/(twopi^2*Cms*Fres_mech^2)) Mms=93.816u (1cm2) A capacitor Cms would be here for linear case Area X X Y+ Y+ .define F1 30 Displacement NL Area volt=0.3 Hz=F1 charge sensor Y- Y- Mms Bl_NONLIN + - + + B B C C + Rdc 8 N mech electr mech (V,Hz) A A D D Bl_value Pa_Cavity Ac 1 Pin_for_Bl Pin_for_Bl (Tm) (m3/140000) .define m3 50u IDEALTRA_nonlin Y+ Y+ X X Blvalue_at_rest=0.5 NL Bl Fermi Y- Y- ShiftCoilOut=0.1m Figure 41: Combining 3 distortion mechanisms in a circuit for a simplified headphone.

46. 42 APPENDIX B. HARMONIC DISTORTION

47. Appendix C Some Curious Observations C.1 Bl Distortion In Resonance Using the technique described in Section B.2 on p. 41 we simulate now a non-linear symmetrical Bl profile (circuit: Figure 42) for a LS with 300Hz resonance (Figure 43). We perform first a transient analysis at a very low frequency 30Hz, where inertial mass plays no role (Figure 44 on p. 45). As expected the coil displacement clips where the field becomes weak. As also expected the distortion for sound at a distance is much worse than that of displacement (displacement x being differentiated twice for acceleration, which determines SPL). .define Fres_mech 300 .define Cms 3m Mms=93.816u displacement .define Mms (1/(twopi^2*Cms*Fres_mech^2)) charge sensor Mms Cms 40m Bl_NONLIN volts + volt=4 Hz=F1 + + Rdc 8 Rmech electr mech N mech (V, Hz, Deg) (AREAm2,DISTm) 1e-4 acoust. 1 Bl_value SPL (Tm) .define F1 30 NL Bl Fermi Blvalue_at_rest=0.5 ShiftCoilOut=0 DISTm=2 Figure 42: LS circuit in which Bl value is modulated by coil displacement. We now choose a generator frequency of 300Hz. Since this is exactly in the resonance, the displacement must be much larger, as confirmed in Figure 45 on p. 45. But the distortion has just about disappeared. Normally we expect distortions to increase with displacement. But this applies only to distortions of mechanical or acoustical impedances within the system itself. The Bl profile is not a mechanical impedance. building blocks as spring compliances which definitely distort if stretched beyond the linear limit. It cannot properly be classed with such 43

48. 44 APPENDIX C. SOME CURIOUS OBSERVATIONS 2BLAC.CIR 70.0 60.0 50.0 40.0 30.0 20.0 10.0 10 dB(v(SPL))+94 100 1K 10K F (Hz) 5.0m 4.0m 3.0m 2.0m 1.0m 0.0m 10 v(displacement) (V) 100 1K 10K F (Hz) Figure 43: AC analysis of Figure 42. What we see here is an irregularity in the force generator, which need not necessarily be revealed in the reaction of the mechanical system, if the latter has a high quality factor. A pendulum clock shows a very similar behaviour. It receives an impulse every time it goes through the mid point. And yet its oscillation is perfectly sinusoidal. The LS simulated here does the same because the damping resistance Rms=40mΩ is not very high. But, as seen in Figure 43 the Q value is fairly typical of an average LS.

49. C.1. BL DISTORTION IN RESONANCE 45 3BL30HZ.CIR 5.0 0.0 -5.0 0.0m v(volts) (V) 20.0m 40.0m 60.0m 80.0m 100.0m T (Secs) 1.0m 1.5m 0.0m 0.0m -1.0m -1.5m 0.0m v(displacement) (V) 20.0m 40.0m 60.0m 80.0m 100.0m v(SPL) (V) T (Secs) 500.0m 250.0m 0.0m -2.0m v(Bl_value) (V) -1.0m 0.0m 1.0m 2.0m 3.0m v(displacement) (V) Figure 44: Transient analysis of Figure 42 at 30Hz. 4BL300HZ.CIR 5.0 0.0 -5.0 0.0m v(volts) (V) 20.0m 40.0m 60.0m 80.0m 100.0m T (Secs) 50.0m 1.5m 0.0m 0.0m -50.0m -1.5m 0.0m v(displacement) (V) 20.0m 40.0m 60.0m 80.0m 100.0m v(SPL) (V) T (Secs) 500.0m 250.0m 0.0m -2.0m v(Bl_value) (V) -1.0m 0.0m 1.0m 2.0m 3.0m v(displacement) (V) Figure 45: Transient analysis of Figure 42 at 300Hz.

50. 46 APPENDIX C. SOME CURIOUS OBSERVATIONS Figure 46 shows the frequency responses of THD etc. And finally Figure 47 shows the time signals of Figure 44 on p. 45 with coil shifted out of the magnet gap by 0.3mm. THD AND K2 TILL K5 12.0 8.0 4.0 0.0 30 Y_Level(THD(HARM(v(Pa)),F1),1,1,11*F1) Y_Level(IHD(HARM(v(Pa)),F1),1,1,3*F1) Y_Level(IHD(HARM(v(Pa)),F1),1,1,4*F1) Y_Level(IHD(HARM(v(Pa)),F1),1,1,5*F1) 100 400 Y_Level(IHD(HARM(v(Pa)),F1),1,1,2*F1) F1 -20.0 -40.0 -60.0 -80.0 -100.0 -120.0 30 Y_Level(dB(HARM(v(Pa))),1,1,F1) Y_Level(dB(HARM(v(Pa))),1,1,4*F1) Y_Level(dB(HARM(v(Pa))),1,1,5*F1) 100 400 Y_Level(dB(HARM(v(Pa))),1,1,2*F1) Y_Level(dB(HARM(v(Pa))),1,1,3*F1) F1 Figure 46: Quasi-AC analysis of same LS. Lower figure black: linear SPL component. 6BL30Hzcoiloutby300u.CIR 5.0 0.0 -5.0 0.0m v(volts) (V) 20.0m 40.0m 60.0m 80.0m 100.0m T (Secs) 1.0m 1.5m 0.0m 0.0m -1.0m -1.5m 0.0m v(displacement) (V) 20.0m 40.0m 60.0m 80.0m 100.0m v(SPL) (V) T (Secs) 500.0m 250.0m 0.0m -2.0m v(Bl_value) (V) -1.0m 0.0m 1.0m 2.0m 3.0m v(displacement) (V) Figure 47: As in Figure 44, but coil shifted out of the magnet gap by 0.3mm.