Aes Tutorial 48 Headphone Fundamentals

7/31/2019 Aes Tutorial 48 Headphone Fundamentals

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TUTORIAL AES 120, Paris, May 2006

HEADPHONE FUNDAMENTALSCarl Poldy

Philips Sound SolutionsVienna, Austria

June 21, 2006


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ii


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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 . . . . . . . . . . . . . . . . . 172.6 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4 How the Bass Tube Works . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5 Sharp Sound-Colour Of Far-field Leakage . . . . . . . . . . . . . . . . . . 28

iii


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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


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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 onthe right. One point, not obvious to all, is that the magnitude of acceleration a forsinusoidal signals can easily be derived from velocity v by multiplying by (= 2f),and from displacement x by multiplying by 2. (For AC signals we normally use RMSvalues.)

We can only hope here to cover a few aspects of headphones. Whether the ones chosenare of interest, depends on your point of viewwhether 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 viewthe protection of the headphone user from soundleaking in from the surroundings. This function of hearing protectors can also influenceheadphone designsuch as fluid filled cushions etc. For such aspects, and others, I referthe reader to the recent AES tutorial by MR Avis and LJ Kelly [1].

1


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2 CHAPTER 1. INTRODUCTION

CONCEPT CHARACTERISTICS

Constant SPL in pressure chamber Constant displacement x

Constant SPL in real headphone HP Constant velocityx (sometimes)

Constant SPL in LS Constant acceleration2

x (Newtons law)

Working range of Headphone Driver resonance within WR:

Resistance- orstiffness controlled

Working range of LS Above resonance:

mass controlled

Ideal Response of LS Flat

Ideal Response of HP Imitates Free- or Diffuse-field

Odd harmonics 3 5 7 etc

f

Symmetrical Distortion

e.g. Stiffness Bl profile

F Bl

x coil x

Symmetrical time signal.

Even harmonics 2 4 6 etc

f

Unsymmetrical Distortion

Unsymmetrical time signal

possible.

EARCANAL 1 2 Infinite peaks at 3kHz 9kHz 15kHz

for transfer function p2/p1

if terminated hard

EARDRUM Damping of these peaks

down to about 12dB

DRP Drum Reference Pointin front of eardrum

Good for two-port analysis

ERP Ear Reference Point

in concha

Problematic for analysis

Figure 1: Some LS- and Headphone-relevant concepts.


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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 (SP L

1

) at a distance is pro-portional to the volume acceleration of the radiating surface. As shown in Figure 2, ifwe consider the low-frequency behaviour of a LS2 with closed rear cavity at low frequen-cies, its SP L appears at large distances to come from such a monopole radiator, whichis omnidirectional. Mounting the LS in an infinite baffle reduces the full space to aninfinite half space. The pressure is then double that shown in Figure 2. For progressivelyhigher frequencies there is increased directionality. Off-axis SP L is reduced, but for abaffled circular radiator the on-axis still obeys the expression shown here (p should bedoubled of course).

d [m]

p [Pa]

Air density [kg

/m

3]

VolAcc = S v.

Volume acceleration [m

3/

s

2]

Normal acceleration [m

/

s

2]

4 d

1,2p = VolAccS

Displacement x [m]

4 d

1,2 Sp =

. x

.

Figure 2: SP L 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 accelerationx = 2x, where = 2f.

1SP L is usually expressed on a logarithmic scale dBSPL referred to the threshold of hearing. It hasbeen agreed to define the threshold SP L (0 dB) at 20P a.

2The abbreviations LS and HP are for loudspeaker(s) and headphone(s) respectively.


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1.3 SPL in a Pressure Chamber (Simplified HP)

The equivalent of Figure 2 for a pressure chamber is Figure 3.

SPL distributed homogeneously

d [m]

AC pressure

p [Pa]

DC air pressure

100000 Pa

S

p = (Sx)1

Displacement x [m]

Ratio of specific heats

of air Cp/Cv = 1,4Volume displacement [m

3]

( current / j )

Vol [m3]

1 4100000

Vol[m3]

( )

acoust. Capacitor C

Figure 3: SP L in pressure chamber, as for a simplified leak-free headphone at lowfrequencies for small piston displacement x. All signals are AC and therefore RMSvalues. 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 soundpropagate here at the speed of light or even faster3? To see what all this looks like alsoat 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 responseand sometimes phase response and phase delay). Since, by definition, sine signals exist for ever causalityis not violated. If we were concerned with energy propagation of impulses then we would of courseexpect causality to be respected. But then instead of phase delay we have signal delay, which equalsgroup delay (-d/d) only if the system is an all-pass.


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1.4. TYPES OF HEADPHONES AND LEAKAGES 5

1.4 Types of Headphones and Leakages

There are many different types of headphones

4

, ranging from small in-ear headphones(p. 21) to supra-aural and circum-aural headphones (Figure 4). Some are designed tobe sealed. Doing this with fluid-filled cushions can give a reliable response even atlow frequencies. But usually uncontrolled leaks make themselves felt (a). They cango directly to the outside, through hair and/or through porous cushions or throughuncontrolled porosity in squashed foam (b and p. 21). Others are led in a controlledway 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 SP Lin the cavity becomes more stable with respect to any additional chance leaksthe twoare like resistances in parallelwhichever 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: Newtons 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 dipoleradiating the ear from close quarters, but not touching it.

4We shall confine this treatment to systems with the most common electro-dynamic transducers.


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Chapter 2

Headphones Compared WithLoudspeakers

2.1 Pressure Chamber & Freefield

Normally we want SP L to be frequency-independent1. It is then of interest to askwhat happens to the membrane displacement x as frequency increases. It is helpful towrite for SP L or p the expression:

p = constant x fn

. (2.1)

For a simple cavity constant SP L requires that n = 0. The membrane is like a pistonpumping on a pressure chamber (Figure 3 on p. 4). The relation between pressure pand total cavity volume (V ol) comes from the adiabatic law p V ol = const where = Cp/Cv, the ratio of specific heats at constant pressure and constant volume. Forsmall compressional displacements xS there is a small increase of pressure p which doesnot depend on how fast the piston moves, as long as it is fast enough for adiabaticcompression. 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 Newtons 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 forthis let us imagine for a LS that n=1 instead of n=2, which is untrue, but one stepin the right direction. This would mean pressure is proportional to velocity. For n=1the gradient in the zero crossings (Figure 5 on p. 8, red curves) reveals the constantvelocity, 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 calibratedheadphone), for simplicity we shall assume flatness here. This is because such deliberate departures fromflat response are more in the nature of local details within a confined frequency range, rather than aglobal phenomenon like a slope of x dB/octave over the whole audio range.

2The term adiabatic means no heat is exchanged with the surroundings. For a LS cabinet filledwith cotton wool heat exchange does o ccur between the air and the fibres at low frequencies. Then wehave isothermal compression and Boyles Law applies: (pV ol = const). The p vsV ol curve is then lesssteep. The cavity is then acoustically softer. This makes it seem larger than it really is. One should notconfuse the cavity volume V ol with the momentary volume displacement of the piston xS.

7


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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 thelarge membrane displacements are visible.

DISPLACEMENT = const * PRESSURE / FREQ^n

PRESSURE CHAMBER: Pressure = const * DISPLACEMENT n = 0

Just imagine it: Pressure = const * VELOCITY n = 1

LS: Pressure = const * ACCELERATION n = 2

ASSUME CONSTANT PRESSURE FOR ALL FREQUENCIES

100 Hz

200 Hz

400 Hz

800 Hz

DISPLACEMENT

TIME

PRESSURE CHAMBER n=0

100 Hz

200 Hz

400 Hz

800 Hz

DISPLACEMENT

n=1

TIME

velocity constant (applies to nothing)

100 Hz

200 Hz

400 Hz

800 Hz

DISPLACEMENT

n=2

TIME

LOUDSPEAKER (acceleration constant)

Figure 5: Constant SP L 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 SP L a LS coil generally has to perform more excursionthan that of a headphone. This applies especially at low frequencies since the coil


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2.2. HARMONIC DISTORTION 9

displacement rises with 12dB/octave while lowering frequency in the working range wellabove resonance, as we already saw (Section 2.1). This brings the LS into the non-linearregion 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 ofS by x is usually unimportant, because the spring Cmsis 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 displacementwe need to ask how much air is moved by a superposed small AC displacement. ForDC 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 50cm3 cavity. Thecircuit is shown in Figure 9 on p. 11. The small-signal AC response is shown in Figure 7on 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 isthe one that counts.


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10 100 1K 10K0.0u

50.0u

100.0u

150.0u

v(Displacement) (V)F (Hz)

10 100 1K 10K40.0

60.0

80.0

100.0

120.0

140.0

dB(v(Pa_Cavity))+94F (Hz)

AA54_28b_ONLY_AREA_AC.CIR

RMS DISPLACEMENT

dB SPL

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 usingthe macro of Figure 8.

area.define mu (v(x)*1e6)

X Y+

Y-

.PARAMETERS(area0,coeff1,coeff2,coeff3)

.define area ( area0 + coeff1*mu + coeff2*mu^2 + coeff3*mu^3 )

Output is the momentary

value of either1) Bl [Tm],

2) Spring force [N] or

3) eff. Area of membrane

Voltage controlled voltage source

Voltage Input is coil displacement [m]

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 + coeff1mu + coeff2mu2 + coeff3mu3

The linear component is area0 [m2]. We choose here only one nonlinear coefficient

coeff1 = 1 106, all others being set to 0. This value is chosen unrealisticallylarge here because we want to see things happening. In Figure 10 on p. 11 (lowestcurve) we see the area of 1cm2 modulated by 50%. We observe the following points:

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 bythe FFT (middle curve, black) having a small positive DC value. This is becausethe cavity is less of a hindrance to the coil movement when the area is smaller.

The SP L 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 m to avoid numerical inaccuracies in the simulation.


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2.2. HARMONIC DISTORTION 11

X Y+

Y-

Rdc8

Mms+

charge sensor

(V,Hz)

volt=0.3

Hz=F1

m3VOL

m3=50u

Rmech

50m

N

mech

Ac1

IDEALTRA_nonlin

Cms

X Y+

Y-

NL Area

coeff1=-1ucoeff2=0

area0=1e-4

coeff3=0

(Tm)

mechelectr

++

Bl 0.5in

Pa_Cavity

Displacement

.define Fres_mech 300

.define Mms (1/(twopi^2*Cms*Fres_mech^2))

.define Cms 3m

Mms=93.816u

.define F1 30

Area

(1cm2)

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 willalways go hand in hand with a genuine mechanical distortion of the restoring force of themembrane spring. But physically, area modulation is an entirely different mechanism,only indirectly connected with non-linearity in the F vs x curve of the mechanical spring.

Area modulation has its greatest influence when the coil diameter is small comparedwith the outer diameter of the torus membrane.

It can be measured by combining Thiele-Small measurements in the mechanical domain(mass method using blue-tackBl) with those in the acoustical domain (inertance oracoustical mass method using a hollow tube or a cavityBl/S). The ratio of the results


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gives the value of S. Each pair of measurements needs to be done with a differentstationary displacement of the coil. It can be shifted by an electrical DC current (dangerof 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 seein principle why this is so, or indeed whether it need necessarily be so, we shall drasticallysimplify the system, and change only one thingthe cavity volume. Comparing tworealistic fully developed headphones of different size would not help understandingwealready 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(Tm)

mechelectr

++

Bl0.3

m3

VOL

ac.

1

N

mech

Nvalue=1m

(V,Hz)

(AREAm2,DISTm,deg)

deg

RadiatorBaffledPascalAREAm2=999DISTm=1angledegrees=0



.define Cms 3mMms=8.443u

Pa_VOL

.define area (piover4*diam^2)

Pa_FAR

.define diam 80m

area=5.027m

.define length (diam/4)

length=20m

.define volume (area*length)

volume=100.531u

FRONTBACK

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 stepthe diameter from 20mm to 80mm, which increases the cavity volume, at the same time

increasing the depth proportionally4

. For simplicity we consider a closed headphonewith 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 intoaccount the radiation impedance would only dampen the resonance peak, which is notthe issue here. The results would remain essentially the same.

The area of the membrane is also kept constant (Nvalue of the ideal transformer). If wehad increased this proportionally with the volume, the SP L in the cavity would havedecreased. 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.


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2.3. EFFECT OF HEADPHONE SIZE ON LEAKAGE AUDIBILITY 13

200 1K 10K 20K0.0

20.0

40.0

60.0

80.0

100.0

120.0

140.0

dB(v(Pa_VOL))+94F (Hz)

dB(v(Pa_FAR))+94

200 1K 10K 20K

-100.0

-80.0

-60.0

-40.0

-20.0

0.0

20.0

dB(v(Pa_FAR)/v(Pa_VOL))F (Hz)

AA16_4ZB_LEAKAGE_VS_VOL_ANALYSIS.CIR DIAM=20m...80m

dB SPL

RATIO

20mm

80mm80mm

20mm

80mm

20mm

diameter increase

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 F AR)/v(P a V OL)). Summing up, we can say the following:

Headphone Cavity: With increased cavity size, the increase in far-field SP L comesjust from the volume increase. The reason is that the combined system springbecomes more compliant, thus allowing more displacement x[mm] for larger vol-umes. This is because in the working range of frequencies a headphone tends tobe 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 significantparameter. Increasing the box volume only shifts the resonance down, extendingthe frequency range. Since the working range of a LS is above resonance, where itis mass controlled (inertia) the cavity size does not significantly influence thestrength 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-earheadphone (Figure 21 on p. 24) the resultant leak is a vector sum of front and backleaks. The small headphone is essentially a dipole radiator: air current in at the backalmost equals air current out of the front. This becomes increasingly true the lower thefrequency is. If we add a front cavity or increase the size of any front cavity alreadypresent, part of the air current from the front of the membrane goes into compressingthis air cushion, which reduces the air current out of the front leak. This unbalances thetwo dipole components, and there is less cancellation of the far-field sound. This

contributes an additional mechanism to make large headphones noisier.


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2.4 What the Ear-Canal Does

The earcanal is essentially a two-port tube (Figure 31 on p. 33). The transfer functionp2/p1 of any two-port can in principle depend only on the two-port itself and the outputimpedance (in this case that of the eardrum). The circuit for the earcanal transferfunction is shown in Figure 13 on p. 15. Figure 14 shows the results. For hard terminationwe get infinite peaks due to standing waves for which the earcanal length equals /4,3/4 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 physicalpoint 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 freefieldmeasurements on real or artificial heads one often wants to know the transfer functionof a system whose input is no longer physically present, because it has already beendisturbed (diffraction) by the thing being measured. The output is clearly normally atthe eardrum. But the input could be defined as the SP L at that point in space at thecentre of the head, before the head had been put there.

Resuming the inquiry about the ear-canal, we have a similar problem defining ourearcanal 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 wemean a point on the hard surface (where head diffraction effects are already taken intoaccount) before the ear canal was opened5? There are two stages to this problem whichshould be dealt with separately and in sequence:

Step1 Diffraction Effects without openings. These yield a pressure source po onthe 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 po develops an internal impedance Zint, whose value is theradiation impedance Zrad of the opening. Zrad depends mainly on the radiating

area. Figure 34 on p. 35 (lowest compartment) shows a simplified formula for thisZrad. The real part of a radiation impedance represents a loss of energy no lessreal than that from the viscosity losses in an acoustic resistance. The energy lostby 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. Thewhole circuit to the left of the point labelled out1 (Figure 20 on p. 24) is implicitly asound source with internal impedance depending on the details of the headphone.

For a small headphone even step2 can be ignored, because the earcanal opening isnot 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 of5I 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.


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2.4. WHAT THE EAR-CANAL DOES 15

these is the end correction (Figure 34 on p. 35), which is actually the reactive part ofthe radiation impedance.

But for circum-aural devices such as hearing protectors the radiation impedanceabove 500Hz approaches that of the unoccluded ear [9]. Presumably this applies toheadphones too.

For the unoccluded ear the first approximation at low and mid frequencies is thesimple end correction (reactance jL, no R).

Now we are in a position to digest the red curve of Figure 14. As already noted, the maineffect of the eardrum is to dampen the resonances. When we introduce the radiationimpedance the resonance shifts to lower frequencies because of the mass loading of theinput. The radiation impedance has been integrated into the two-port. This might bedone 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 askwhether the radiation impedance has been included or not.

aGrou

ndnormally (L, D, N)

nonlossy

earcanal

TUBE

length=24mdiam=diam

(V,Hz)

volt=1

EARDRUMzwislocki

(L, D, N)

nonlossyTUBE

length=24mdiam=diam

(V,Hz)

volt=1 (AREAm

2)

aGrou

ndnormally

RadiatorBaffled

AREAm2=area

(L, D, N)

nonlossyTUBE

length=24mdiam=diam(V,Hz)

volt=1

EARDRUMzwislocki

DRUM

DRUM_and_RADIMP

.define area (piover4*diam^2)

.define diam 7.4m

area=43.008u

HARD

Figure 13: The circuit for earcanal TF. Radiation Z is normally earthed, but not here.

200 1K 10K 20K-10.

0.

10.

20.

30.

40.

50.

db(v(HARD))F (Hz)

db(v(DRUM)) db(v(DRUM_and_RADIMP))

AA22EARCANALTRANSFERFUNCTION_ANALYSIS.CIR

Figure 14: Earcanal transfer functions (for Figure 13): hard drum (black); withdrum (blue); with drum showing effect of radiation impedance (red).


20/57


2.5 The Ear Spectrum For Frontal Sound Source

Here we consider only the so-called monaural head-related transfer function for frontalincidence7. By this I mean we acknowledge the differences of individual ear geometryfrom person to person, but assume for the moment that the ears of any individual areidentical. 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

10kHz

~10dB

3kHz1kHz

EARCANAL

ENTRANCE

0dB

+

EAR CONCHADIP 10kHz

+

SHOULDER

DIP 1kHz

Figure 15: Typical diffraction features for frontal sound source.

concha respectivelyfirst 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 theincoming sound. The pinna including concha gives approximately 10dB amplification.Superposed on this is the 10dB amplification due to the /4 resonance in the earcanal(as we saw in Figure 14 on p. 15). The result is a heavy colouration with a 20dB peakat about 3kHz.

The individual has learnt to hear with his own ears and has got used to associatingcertain spectral characteristics with each visible direction8.

7We also neglect here second-order localisation effects which rely on such things as familiarity withthe sound source, head movement and other temporal phenomena such as sound-source movement. Weconcentrate here on the first order effect which works for short high-frequency impulsive sounds, whichneed not be known to the listener. He can localise these even without head movementindeed for shortsounds there is no time for confirming first impressions by head movements. And as we know fromexperience these spontaneous first impressions are normally correct.

8For invisible directions the individual has no means of learning the spectral characteristics and


21/57

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) ofan individual to be the cue for frontal localisation. Irrespective of the correctness of thisassumption, sharp colouration for a frontal sound source must be expected.

If it is indeed localised frontally (Gestalt recognition) this colouration will not disturbhimassociation model of Theile[5]9. Otherwise the 20dB peak (free-field standard forheadphones) is too sharp. Then a milder spectrum is needed, as for the diffuse fieldheadphone standard (dashed line in Figure 1 on p. 2).

Interestingly, the 3/4 earcanal resonance coincides approximately with the 10dB dip at10kHz. This is the region for hissing sounds S. Thus there is a good chance of suchsounds being heard despite the inherent attenuationfragment 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 reliablyonly if all sources radiate with flat frequency spectra. In reality sound sources haveinherent colouration, which is present in the SP L spectra at any point in the transmissionchain on the way to the eardrum. The brain has no means of knowing whether suchlinear distortions happened at the pinna or before. Indeed it could well be a propertyof 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 isneeded to establish this. Indeed, own attempts at equalising a rear LS to approximatethe 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 couldnot be a characteristic of the sound source. Faithfully reproducing these individual dif-ferences resulted in clear frontal localisation via headphones using DSP. This succeededfor two tested individuals (no statistics were needed). Exaggerating the differences as inFigure 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 withthe 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 otherindividual. Presumably that other individual had similar interaural differences, butexaggerated.

Whats more, interchanging the L and R spectra destroyed the frontal localisation evenwithout exaggerating the differences. This is spectacular, because the original in-teraural differences were not largeless than 4dB. Therefore such interchange of L and Rcannot result in any appreciable sound colour change in either ear. Auto-suggestion canbe 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 labelwith various names such as In Head Localisation, Rear Localisation, Elevation Effect. In the caseof the elevation effect the incorrectly localised phantom source comes from a visible direction, but thespectral 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, therebyironing out the 20dB peak. so it is not subjectively perceived too strongly.


22/57


(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 exaggeratethe differences to enhance frontal localisation without changing the sound colour. Viaheadphones such exaggeration above 2kHz can be very effective. This agrees with thefact that some people experience better frontal localisation using the ear spectra ofanother individual than with their own. By analogy, the benefit of exaggeration appliesalso to stereo photography: To obtain a spectacular 3D impression of a landscape theeffective eye distance needs to be exaggerated to several metres.

The author knows of a person with practically identical ears and consequently identicalear spectra for frontal incidence. This person had the distinguishing feature that hewas unable to localise frontally in the median plane under any conditions, even for realfree-field sources.

2.6 More Reliable Stereo Image For Headphones

Consider the soloist at centre stage. This is inherently a mono component of thestereo 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).For a real frontal sound source correct localisation is determined by the spectra of thetransfer functions from free-field to the ears. Irrespective of whether the source is a singleLS 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 signalsas well as possible. If we have done the necessary measurements for an individual, wecan assume we know how his ear signals should look. Armed with this information thereis a chance for correct frontal localisation in the electro-acoustical situation.

The chances are as follows:

HPs: very goodwe are in control: (L signalL ear); (R signalR ear)

LSs: modest (L signalL ear); (R signalR ear); (L signalR ear); (R signalL ear)


23/57

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 frontallocalisation 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 personsdo 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 eargeometry. 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 justifiedin neglecting the cross-feed components (L signalR ear) and (R signalL ear). Ofcourse 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 frontallocalisation, but in the usual stereo set-up does not do so because the angle iswrong.

If we know the ideal ear spectra for an individual, we have in headphones (evenwith 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 betterthan 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 bothcomes from the SP L in the ear signals alone. Otherwise headphones would not havethe localisation advantage over LS mentioned in Section 2.6. How the ear signals areproducedpressure chamber or freefield of a propagating waveis unimportant. Butwe cannot say this without reservations (see also Section 2.8):

Listeners tend to listen at higher SP L for HPs than for LSs to get the same subjectiveloudness. Many such phenomena have been reported in the literature (see references atthe 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 superimposedon the above pressure-detector mechanism.

For example the hearing threshold in audiometry using a HP is higher than when usinga LS. This comes from masking of the signals due to physiological noise, such as that

from blood flow turbulences. The ear SP L of such body noises (and bone conduction),is raised in the case of headphones by the occlusion effect. This amplifies the SP L below2kHz by up to 20dB ([2] p.651). This can easily be experienced by humming a low tonewhile closing one ear with a finger tip.


24/57



25/57

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. Foamcover 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). Herewe 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 ordo we need them.

Why the leak signal for people standing nearby sounds sharp.

21


26/57

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 L cancompete with R even at low frequencies. This condition demands a large absolutevalue of d, the diameter1. But for a short tube of correct diameter the absolutevalue 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 17on p. 21, competing with the cable for space, and limiting design freedom. Sinceimpedance of sound constrictions depends mainly on length and diameternot somuch on the shapea curved tube is as good as a straight one. The circumferenceof 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 d2, whereas R d4.


27/57

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 areoutside the transducer. All was simplified to the bare essentials. Only one cavity isincluded, the rear one, all the others not being essential to the understanding. To avoidcluttering 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 resonance255Hz and mass Mms, the membrane compliance Cms is automatically adjusted for agiven mass Mms.

in

charge

out

Rdc (Tm)

mechelectr

++Blvalue

Mms CmsRms

acoust.1

N

mech

+in

charge

out

charge sensor

N=Amem

v [m/s]

AK2

EL2

EL1

AK1

.PARAMETERS(Fres_mech,Mms,Amem,Blvalue,Rms,Rdc)

.define Cms (1/(Mms*twopi 2*Fres_mech 2))displacement

BACKof membrane

FRONTof membrane

v [m3/s]

deltaP [N/m2]

Force [N]

Primary

Secondary

v [m/s]

mass complianceresistance

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 theeffect of the earcanal. Here we concentrate on the upper two curves. The response iscalculated 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 (/4, 3/4 etc.). The next thing we shall consider is the bass boost.


28/57


400e5 1k

200e5

(V,Hz)

volt=0.126

foam

4e5

20k switch0

m3

Vol_backm3=0.15u

LSThieleSmall_LSFres_mech=255Mms=17.5u

Amem=0.8e-4

Blvalue=0.46

Rms=0Rdc=14.4

100

(AREAm2,DISTm)

AREA=0.5e-4DIST=distance

(AREAm2,DISTm)

AREA=888DIST=distance

(AREAm2,DISTm)

AREA=888DIST=distance

ear

canal

ERPDRP

IEC711

IEC_711_with_ear_canal_noEC

10 front holes

out2drum

far_basstube

far_backholes

far_front

.define distance 30m

far_SUM

back holes

bass tube

out1

Figure 20: The equivalent circuit of a small in-ear headphone.

Figure 21: Frequency responses for small in-ear headphone. Upper two black curves: atear canal entrance (dotted line); at output DRP of IEC711 (full line).


29/57

3.4. HOW THE BASS TUBE WORKS 25

3.4 How the Bass Tube Works

20 100 1K 10K 20K20.

30.

40.

50.

60.

70.

80.

90.

100.

110.

db(v(out2drum))+94

F (Hz)

db(v(far_basstube))+94 db(v(far_backholes))+94 db(v(far_front))+94db(v(far_sum))+94

AA14MINIBASSTUBE_ANALYSIS.CIR SWITCH=0...90T

Figure 22: Bass tube normal (upper curve) and closed (lower curve).

20 100 1K 10K 20K30.

40.

50.

60.

70.

80.

90.

100.

110.

120.

130.

db(v(out2drum))+94

F (Hz)

db(v(far_basstube))+94 db(v(far_backholes))+94 db(v(far_front))+94db(v(far_sum))+94

AA15MINIBACKHOLES_ANALYSIS.CIR SWITCH=0...90T

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 rearleaks. Switching them off must make a purer bass tube. Figure 23 shows that thisis 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.


30/57


It is as if the compliant rear cavity were replaced by an infinitely hard wall intimately incontact with the rear surface of the membrane. This comes from the parallel resonance ofthe cavity compliance (condenser Cvol) and bass tube (Lbasstube), as shown in Figure 24.

19k

m3Vol_back

m3=0.15u

Cvol

(AREAm2,DISTm,deg)

deg

Cvol/2 Cmem

Mms19k

Cvol Cmem

Mms

19k

19k

Cvol/2

Cvol=1.071p

bass tube

.define Cvol (.15u/(1.4*atmospressure))atmospressure=100K

soft membranedominatesstiffness

200Hz bass boost

4kHz Resonance

1kHz DIPCoil and membrane

stand still

infinite impedance

Mode1

Mode2

Cvol dominatesstiffness

Helmholtz resonance

Pa

no compression of air cushion

Figure 24: When the rear holes are closed the Helmholtz resonance (1kHz) of basstube with rear cavity presents an infinite impedance for membrane motion (sharp dipin Figure 23 on p. 25). This resonance is in the middle between two other resonancemodes. The R of the basstube is here omitted for simplicity. Of course the imaginedpartition for 4kHz divides the cavity into equal halves Cvol/2 and Cvol/2 only for equalmasses. Normally they are unequal.

Also we have a peak at 4kHz. This is the second bassreflex resonance, where thecurrents of q1 and q2 into the rear cavity (Figure 25 on p. 27) are in phase (Figure 26 onp. 27). The basstube and membrane both pump into the rear cavity with the same phaseabove 1kHz. At the resonance 4kHz we can imagine an immovable partition dividing


31/57

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 themembrane. Since they are each stiffer than the undivided cavity stiffness the resonancemust be higher than the main Helmholtz resonance at 1kHz.

The bass boost around 200Hz has nothing to do with the cavity. The equivalent ofFigure 23 on p. 25, but without any cavity is shown in Figure 27 on p. 28.

in

current

out

in

current

out

400e5 1k

200e5

(V,Hz)

volt=0.126

foam

4e5

20k

switch0

m3

Vol_back

m3=0.15u

LSThieleSmall_LSFres_mech=255Mms=17.5u

Amem=0.8e-4

Blvalue=0.46

Rms=0Rdc=14.4

100

ear

canal

ERPDRP

IEC711

IEC_711_with_ear_canal_noEC

in

current

out

current sensor

in

current

out

current sensor

(AREAm2,DISTm)

AREAm2=999DISTm=distance

(AREAm2,DISTm)

AREAm2=0.5e-4DISTm=distance

(AREAm2,DISTm)

AREAm2=999DISTm=distance

10 front holes

out2drum

far_basstube

far_backholes

far_front

.define distance 30m

far_SUM

back holes

bass tube

out1q1 q2

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.


32/57


20 100 1K 10K 20K30.

40.

50.

60.

70.

80.

90.

100.

110.

120.

130.

db(v(out2drum))+94F (Hz)

db(v(out2drum_nocavity))+94

AA16_4MINIBASSTUBE_ANALYSISNOCAVITY.CIR SWITCHBACKHOLES2=0...1.1TSWITCHBACKHOLES=0...1.1T

REAR CAVITY REMOVED

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 sharpto bystanders. But the signal at the summation node SUM is even sharper. Theindividual components are vectorially added. We must remember that the air in front andbehind the membrane cannot experience much compression at mid and low frequencieswhere the reactance L of all acoustic masses L resembles a short circuit. Thus the totalcurrent in at the back must almost equal that out of the front. Since their influences ata distance are in anti-phase, we get progressive cancellation as the frequency is lowered.

The headphone is a dipole radiator.


33/57

Chapter 4

Reference Points For ResponseMeasurements

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 fornormal analysis1 is DRP (Drum Reference Point) 3mm in front of the eardrum (Figure 28on p. 30). What the brain receives is of course a spectral impression of the signal atDRP. So obviously this is the one we should be primarily interested in.

But some measurements refer to ERP (Ear Reference Point) in the concha. Resultsinvolving ERP can be problematic, depending on what you want to do with them.

If you are interested in SP L at ERP, you can measure it and you have what you want.But there are standardised tabular data for (dBSPLERP dBSPLDRP), which areintended for transforming from one to the other. The implicit assumption is that thistransfer function (TF) is uniqueindependent of diffraction effects to the left of theinput. This only applies if the stretch of air between these two points is a genuinetwo-port, which is true for ERPx DRP, but not for ERP DRP (Figure 28).

1Normal here means the SP L [P a] is proportional to other stages before this point. Behind theeardrum we have movements of bone levers and nerve impulses. These cannot be treated as signals inlinear two-ports. For example, even for a sinusoidal sound signal the nerve impulses are spikes of impulseprobability, firing rhythmically once per period with the zero-crossings of the acoustic signal. Betweenthese spikes nothing happens. Thus there is nothing sinusoidal about them.

29


34/57

30 CHAPTER 4. REFERENCE POINTS FOR RESPONSE MEASUREMENTS

DRP

ERP

q2 = vSMALL SBIG

q1 = v S

q2

q1 [m /s]

ERP

DRP

qERP

qERP ambiguous

?

q1

and q2

are equal.

Unambiguous

two-port

vSMALL

( SBIG = S / cos )

velocity v [m/s]

ERPx

ERPx

( vSMALL = v cos )

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 andits output load. In general, we might be able to define the pressure p at the input of asystem. 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 inputimpedance of everything to the right is unique.

It is easier to deal with ERPx and DRP, because the region around them is inherentlyone-dimensional and of small cross-section. But that around ERP comprises the conchaand pinna etc. with a 3-dimensional geometry2.

Defining the current q means choosing a small area element Snormal or Sbig, throughwhich the current is supposed to flow, and integrating the particle velocity v over thisarea: q =

v 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 for the area S. Within the earcanal this is fulfilled for all reasonably small values of .

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 influencethe transfer function.


35/57

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 couplercombined with an earcanal. The content of this is shown in Figure 29. Obviouslythe short stretch 10mm cannot represent a real earcanal. Part of the IEC711 alreadycontains several tubes in series, representing the second half of the earcanal and eardrumimpedance. The equivalent circuit using lumped elements can be much improved aboveabout 2kHz by introducing transmission lines for the tube sections (Figure 30).

(L, D, N)

nonlossy

TUBE

N=1

length=10mdiam=7.4m

eardrum

ERPDRPin

IEC711

in

EAR CANAL

DRP ERP

Figure 29: A short stretch of earcanal combined with the IEC711 coupler.

Ra7

311e5*1.2

Ma7

9.838e2*0.8

Ca7

2.1p

Ra5

506e5

Ma5

94e2

Ca5

1.9p

(L, D, N)

nonlossy

TUBE

N=1

length=2.7mdiam=7.4m

table2

(L, D, N)

nonlossy

TUBE

N=1


(L, D, N)

nonlossy

TUBE

N=1


(L, D, N)

nonlossy

TUBE

N=1

length=4mdiam=7.4m

D1 D D2

(L,D,N,D1,D2)

L

HOLE

length=0.2mdiam=1m

N=19D1=7.4mD2=7.4m

IEC711

DRP ERPIN

Figure 30: The improved simulation of the IEC711 coupler.

The Head And Torso Simulator HATS comprises a Silicone rubber pinna and earcanalterminated by the eardrum impedance. This is suitable for headphone measurements.But if one wants to compare measurements with computer simulations, one needs arealistic equivalent circuit for the artificial ear. The manufacturer of the IEC711 suppliedan equivalent circuit consisting of lumped elements R, L (for constrictions) and C (forcavities). But the IEC711 consists also of several tube sections for which lumped elementsare inaccurate. Figure 30 shows an improved circuit using transmission line two-portsas described in Figure 31 on p. 33.


36/57

32 CHAPTER 4. REFERENCE POINTS FOR RESPONSE MEASUREMENTS


37/57

Appendix A

Behind The Ear Canal Entrance

A.1 Tube Transfer Function

p1 p2q2

A11 A12

A21 A22

p1 p2q2q1

q1

length [m]S [m

2]

Figure 31: The chain matrix nomenclature for a tube as a two-port.

p1q1

=

cos(k l) j Zwsin(k l)

(j/Zw) sin(k l) cos(k l)

p2q2

(A.1)

p1 = cos(k l)p2 +j Zwsin(k l) q2 (A.2)q1 = (j/Zw) sin(k l)p2 + cos(k l) q2 (A.3)

The input and output currents (Figure 31) are both to the right, as is convenient forcombining such chain matrices. These equations apply to a non-lossy tube. The para-meters are k = /c where = 2f; l = tube length; Zw = c/S, where is air densityand c the speed of sound propagation.

For a hard eardrum the transfer function of the earcanal would be p2/p1 = 1/cos(k l)(setting q2 = 0 in equ. A.2). This would show an infinite peak where cos(k l) = 0. Thefrequency 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 = /2. Since k = 2f/c, we get 2fl/c = /2 giving f = c/(4l). Settingc=340000mm/s and l=24mm we get f = 340000/(4 24) = 3500Hz .

33


38/57

34 APPENDIX A. BEHIND THE EAR CANAL ENTRANCE

A.2 Eardrum Impedance

Figure 32 shows a highly simplified equivalent circuit of the eardrum impedance usinglumped elements. Its frequency response is shown in Figure 33, where it is comparedwith that from the model of Zwislocki [3], where ossicle effects, such as the decouplingof the malleus from the eardrum, are taken into account.

Resistance_acoustic300e5

Mass_acoustic1000

Compliance_acoustic6.5p

Eardrum impedance simplified as RLC circuita

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 Zwislockis circuit [3].


39/57

A.3. LUMPED ELEMENTS 35

A.3 Lumped Elements

Figure 34 shows the equivalent circuits of acoustical constrictions, cavities, the resonantcore of a LS, and a simplified radiation impedance.

Flow velocity

profile

Equivalent

Circuit R L Cf fo

0

4

2

8

f

f

d

l

=

04

41058,7

f

f

d

l

2

2d

l

=2

53,1

d

l

f


40/57

36 APPENDIX A. BEHIND THE EAR CANAL ENTRANCE


41/57

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 etc1 cause 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 36on p. 38), as expected. IHD means Individual Harmonic Distortion referred to F1 in %.

(V,Hz) SineVoltHzPha

volt=1Hz=F1

deg=0 (V,Hz)

SineVoltHzPha

volt=.25Hz=(F1*Nharm)

deg=180

Battery_for_DC_value

0

sum

in

in2

SUBJECT HERE IS:

1 K2 NOT NECESSARILY UNSYMMETRICAL (Nharm=2)

2 DC NOT NECESSARILY VISIBLE IN TIME SIGNAL

.define Nharm 3.define F1 1k

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 sosure. The question of symmetry or non-symmetry in the displacement direction is onlymeaningful if the signal looks the same when the time axis is reversed. Otherwise weoptically can get a sawtooth waveform and we cannot know from its shape whether evenor odd harmonics are responsible.

1This is German nomenclature (K stands for Klirrfaktor). For example, K2 is 2nd-order harmonicdistortion, expressed in % or dB relative to the fundamental.

37


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38 APPENDIX B. HARMONIC DISTORTION

0.0m 2.0m 4.0m 6.0m 8.0m 10.0m-1.000.00

1.00

v(in) (V)t (Secs)

v(in2) (V)

0.0m 2.0m 4.0m 6.0m 8.0m 10.0m-2.00

0.00

2.00

v(sum) (V)t (Secs)

0.0K 1.0K 2.0K 3.0K 4.0K 5.0K

0.0

0.6

1.2

HARM(v(sum))F (Hz)

0.0K 1.0K 2.0K 3.0K 4.0K 5.0K0.0

22.5

45.0

IHD(harm(v(sum)),F1) (%)F (Hz)

5_SYMM_DC_K3SYMM.CIR

Figure 36: Time signal with K3 is symmetrical, as expected.

0.0m 2.0m 4.0m 6.0m 8.0m 10.0m-1.00

0.00

1.00

v(in) (V)t (Secs)

v(in2) (V)

0.0m 2.0m 4.0m 6.0m 8.0m 10.0m-2.00

0.00

2.00

v(sum) (V)

t (Secs)

0.0K 1.0K 2.0K 3.0K 4.0K 5.0K0.0

0.6

1.2

HARM(v(sum))F (Hz)

0.0K 1.0K 2.0K 3.0K 4.0K 5.0K0.00

22.50

45.00


6_SYMM_DC_K2UNSYMM.CIR

Figure 37: Time signal with K2 (phase=180 deg) is unsymmetrical, as expected.


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B.1. TWO SUPERSTITIONS EXPOSED 39

0.00m 2.00m 4.00m 6.00m 8.00m 10.00m-1.00

0.00

1.00

v(in) (V)t (Secs)

v(in2) (V)

0.00m 2.00m 4.00m 6.00m 8.00m 10.00m-2.00

0.00

2.00

v(sum) (V)t (Secs)

0.0K 1.0K 2.0K 3.0K 4.0K 5.0K0.0

0.6

1.2

HARM(v(sum))F (Hz)

0.0K 1.0K 2.0K 3.0K 4.0K 5.0K0.0

22.5

45.0


7_SYMM_DC_K2NOTREALLYUNSYMM.CIR

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 cansay the distortion mechanism had no even harmonics. But if the distorted time signalis unsymmetrical we can say there must be at least one even harmonic component(unsymmetrical distortion mechanism such as spring harder in one direction than theother). But such deductive statements can only be made if the signal is reversible: inother 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 andslower drop) no deductions of the above type can be made from a qualitative inspectionof 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 insteadof IHD, so we get the sign of each component. This can have negative values. A purereal negative value means phase=180 deg, which is one of the two possible phase valuesfor 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.


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0.0m 2.0m 4.0m 6.0m 8.0m 10.0m-1.0

-0.5

0.0

0.5

1.0

v(in) (V)t (Secs)

v(in2) (V)

0.0m 2.0m 4.0m 6.0m 8.0m 10.0m-2.0

-1.0

0.0

1.0

2.0

v(sum) (V)t (Secs)

0.0K 1.0K 2.0K 3.0K 4.0K 5.0K-300.0

-150.0

0.0

150.0

300.0

(FFT(v(sum)))F

8_SYMM_DC_K2AGAINFFTDC0.CIR

Figure 39: K2 again. Phantom DC value due to unsymmetry not really there (look atFFT: lowest curve).

0.0m 2.0m 4.0m 6.0m 8.0m 10.0m-1.0

-0.5

0.0

0.5

1.0

v(in) (V)t (Secs)

v(in2) (V)

0.0m 2.0m 4.0m 6.0m 8.0m 10.0m-2.0

-1.00.0

1.0

2.0

v(sum) (V)t (Secs)

0.0K 1.0K 2.0K 3.0K 4.0K 5.0K-300.0

-150.0

0.0

150.0

300.0

(FFT(v(sum)))F

9_SYMM_DC_K2AGAINFFTDCMP5.CIR

Figure 40: With real DC value (-0.5V) in battery it is no longer a phantom.


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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 followedfor the other causes of non-linearity:

non-linear membrane stiffness (F vs x)non-flat Bl profile (B l v s x).

For a Bl profile combining two Fermi functions (a concept used in semiconductor- andsolid-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 Cms is the compliance. Note the opposite sign ofthe charges for the charge sensor and the place where the Cms capacitor was, before non-linear elements were introduced. The charge sensor is essentially a piece of wire, whichdoes not disturb the circuit. The shift in charge to the right represents displacementof 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 outputof the two-port NL Compliance, works against the restoring force of the spring.

Figure 41 shows how all three mechanisms can be combined in a single circuit.

X Y+

Y-

A

B C

D

Pin_for_Bl

XY+

Y-

X Y+

Y-

Rdc8

Mms+

charge sensor

(V,Hz)

volt=0.3

Hz=F1

N

mech

Ac1

IDEALTRA_nonlin

X Y+

Y-NL Area

coeff1=-1ucoeff2=0

area0=1e-4

coeff3=0

mechelectr

++

(Tm)

A

B C

D

Pin_for_Bl

Bl_NONLIN

XY+

Y-NL Bl Fermi

ShiftCoilOut=0.1mBlvalue_at_rest=0.5

X Y+

Y-

NL Compliance

coeff2=-1coeff3=0

Cms=Cms

(m3/140000)

Pa_Cavity

Displacement



.define Cms 3m

Mms=93.816u

.define F1 30

Area

(1cm2)

+ -

Force

A capacitor Cms

would be here

for linear case

Bl_value

.define m3 50u

Figure 41: Combining 3 distortion mechanisms in a circuit for a simplified headphone.


46/57



47/57

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-linearsymmetrical 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 massplays no role (Figure 44 on p. 45). As expected the coil displacement clips where the fieldbecomes weak. As also expected the distortion for sound at a distance is much worsethan that of displacement (displacement x being differentiated twice for acceleration,which determines SP L).

Rdc8

Mms

acoust.1

N

mech

1e-4

+

charge sensor

NL Bl Fermi

ShiftCoilOut=0

Blvalue_at_rest=0.5

mechelectr

++

(Tm)

Bl_NONLIN

Rmech

40mCms

(AREAm2,DISTm)

DISTm=2

(V, Hz, Deg)

volt=4

Hz=F1

volts

SPL

displacement

Bl_value



.define Cms 3mMms=93.816u

.define F1 30

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 thedistortion 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. TheBl profile is not a mechanical impedance. It cannot properly be classed with suchbuilding blocks as spring compliances which definitely distort if stretched beyond thelinear limit.

43


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44 APPENDIX C. SOME CURIOUS OBSERVATIONS

10 100 1K 10K10.0

20.0

30.0

40.050.0

60.0

70.0

dB(v(SPL))+94F (Hz)

10 100 1K 10K0.0m

1.0m

2.0m

3.0m

4.0m

5.0m

v(displacement) (V)F (Hz)

2BLAC.CIR

Figure 43: AC analysis of Figure 42.

What we see here is an irregularity in the force generator, which need not necessarilybe revealed in the reaction of the mechanical system, if the latter has a high qualityfactor.

A pendulum clock shows a very similar behaviour. It receives an impulse every timeit goes through the mid point. And yet its oscillation is perfectly sinusoidal. The LSsimulated here does the same because the damping resistance Rms=40m is not veryhigh. But, as seen in Figure 43 the Q value is fairly typical of an average LS.


49/57

C.1. BL DISTORTION IN RESONANCE 45

0.0m 20.0m 40.0m 60.0m 80.0m 100.0m-5.0

0.0

5.0

v(volts) (V)T (Secs)

0.0m 20.0m 40.0m 60.0m 80.0m 100.0m-1.5m

0.0m

1.5m

v(displacement) (V)T (Secs)

-1.0m

0.0m

1.0m

v(SPL) (V)

-2.0m -1.0m 0.0m 1.0m 2.0m 3.0m0.0m

250.0m

500.0m

v(Bl_value) (V)v(displacement) (V)

3BL30HZ.CIR

Figure 44: Transient analysis of Figure 42 at 30Hz.

0.0m 20.0m 40.0m 60.0m 80.0m 100.0m-5.0

0.0

5.0


0.0m 20.0m 40.0m 60.0m 80.0m 100.0m-1.5m

0.0m

1.5m


-50.0m

0.0m

50.0m

v(SPL) (V)

-2.0m -1.0m 0.0m 1.0m 2.0m 3.0m0.0m

250.0m

500.0m


4BL300HZ.CIR

Figure 45: Transient analysis of Figure 42 at 300Hz.


50/57


Figure 46 shows the frequency responses of THD etc. And finally Figure 47 shows thetime signals of Figure 44 on p. 45 with coil shifted out of the magnet gap by 0.3mm.

30 100 4000.0

4.0

8.0

12.0

Y_Level(THD(HARM(v(Pa)),F1),1,1,11*F1)

F1

Y_Level(IHD(HARM(v(Pa)),F1),1,1,2*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)

30 100 400-120.0

-100.0

-80.0

-60.0

-40.0

-20.0

Y_Level(dB(HARM(v(Pa))),1,1,F1)

F1

Y_Level(dB(HARM(v(Pa))),1,1,2*F1) Y_Level(dB(HARM(v(Pa))),1,1,3*F1)Y_Level(dB(HARM(v(Pa))),1,1,4*F1) Y_Level(dB(HARM(v(Pa))),1,1,5*F1)

THD AND K2 TILL K5

Figure 46: Quasi-AC analysis of same LS. Lower figure black: linear SP L component.

0.0m 20.0m 40.0m 60.0m 80.0m 100.0m-5.0

0.0

5.0


0.0m 20.0m 40.0m 60.0m 80.0m 100.0m-1.5m

0.0m

1.5m


-1.0m

0.0m

1.0m

v(SPL) (V)

-2.0m -1.0m 0.0m 1.0m 2.0m 3.0m0.0m

250.0m

500.0m


6BL30Hzcoiloutby300u.CIR

Figure 47: As in Figure 44, but coil shifted out of the magnet gap by 0.3mm.


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C.2. THE EGG DOES COME BEFORE THE CHICKEN. 47

C.2 The Egg Does Come Before the Chicken.

Does voltage U across a resistor produce current I or does the current produce thevoltage drop? And in the mechanical world: Does force F cause velocity v or the otherway round? Like the question What Comes Firstthe Chicken or the Egg?, thereseems to be no unique answer. But that is only because we were dealing with a two-pole. If we consider a four-pole (or two-port) and put the chicken and egg on oppositesides: one at the input, one at the output, we get a clear answer:

The motional parameter on one side causes the potential parameter on the other.

C.2.1 Potential and Motional Parameters

In the above question one parameter is a potential and the other is a motional parameter.Electrical current is a motional parameter because it consists of a flow of charge. Even

WORLD POTENTIAL MOTIONAL

electrical voltage U current I, charge Qmechanical force F velocity v, position x, acceleration a

mechanical (acoust.) pressure p volume flow q[m3/s]human wish (intention) action

Figure 48: Examples of potential and motional parameters.

the position x of a membrane, as in a condenser microphone, belongs to the motionalcategory, although we do not normally associate position with movement. This is be-cause we can choose to be concerned exclusively with frequency responses of amplitudeand phase of signals, which implies sinusoidal movement. Then we can always expressposition x in terms of velocity by writing v = jx, where = 2f. And of courseacceleration a = 2x.

C.2.2 Does Intention Produce Action?

In Figure 48 the last line seems to contradict what we are saying here. Most of us feelthat a wish produces an action. Nevertheless brain research has come up with apparentlyunquestionable evidence that it is the other way around (Roth [6]). Whether or not welike this view, is not the issue here. And I am not qualified to find a way of harmonisingthis so-called fact with free-will ideology. Radical materialists (an expression coinedby Zoltan Torey [7]) say it is not possible. But Torey seems to succeed in reinstatingfree-will without invoking the concept of spirit (his pseudo-free-will model).

Nevertheless there is one observation we can definitely make, without becoming esoteric:We need no theory, hypothesis or model whatsoever, to observe that the results of brainresearch are analogous to transduction behaviour. The same idea is expressed bythe famous saying: The road to hell is paved with good intentions (potential not ac-tualised). It would go too far here, to speculate further about the completeness of the

analogy. But we can say this much: To be effective, the action needed seems to be inthe material world with which we are familiar. If the analogy works, the result mustbe a potential. If so, to which world does it belong? Was the category human worldchosen in Figure 48 too narrow? Maybe Bennett has an answer [8].


52/57


C.2.3 Passive Transducers

A two-port connects input and output. If input and output are in different worlds we

usually call the two-port a transducer (LS or Microphone). If they are in the same worldwe get such devices as electrical transformers. This category is not the subject here.

A microphone detects sound pressure, pressing against the membrane. But it is not theresulting force which gives the voltage at the electrical terminals. If the membrane isvery heavy there will be no electrical signal. It is the movement in the mechanical worldwhich is needed to give an electrical signal, voltage or potential, in the electrical world.

Similarly a loudspeaker needs a non-zero motional parameter (current I) at the elec-trical input to give a sound at the output. Applying a voltage will not do anything ifthe leads to the coil are broken (open circuit). We can formulate this as follows:

DEVICE CAUSE RESULT FORMULAdynamic LS current I Force F IBl = F

dynamic microphone coil velocity v voltage U v Bl = U

Figure 49: Electro-dynamic transduction parameters.

C.2.4 Pure Transduction Two-ports

Figure 50 shows the pure dynamic (gyrator) and electrostatic transduction two-ports.

They are pure in that their parameters do not involve any impedances from either

a

b c

d

(Tm)

mechelectr

++

a

b c

d

Figure 50: Pure transduction two-ports (w means = 2f).

of the worlds to be connected. As an interface the job of a transduction two-port is tomake the connection and no more. Bl is the product of magnetic field and wire length.The negative sign for one of the Bls expresses the anti-reciprocity of the gyrator.

Cem is the electro-mechanical compliance of the electrostatic membrane. This is notan obvious concept and deserves elucidation. An electrical condenser C connects twoelectrical quantities: the potential U and the charge Q, which is a motional parameter.The electro-mechanical condenser Cem relates electrical potential U with mechanical


53/57

C.2. THE EGG DOES COME BEFORE THE CHICKEN. 49

displacement x thus: U Cem = x (microphone). It may seem an amazing coincidencethat going the opposite direction involves exactly the same constant: Force FCem = Q.Well thats how things are. Why complain if theyre simpler than you expected!

The chain matrices1 for two transduction two-ports are as follows:

Dynamic transduction:

UI

=

0 Bl

1/Bl 0

Fv

(C.1)

Electrostatic transduction:UI

=

0 1/jCem

jCem 0

Fv

(C.2)

C.2.5 Piezoelectric Transducer Two-port

The concepts transduction two-port and transducer two-port are different. Sometimesbits of the mechanical world are included (Zwicker and Zollner [4]), as in Equation C.3.

Piezoelectric transducer ( = eS/x):

UI

=

1/ 0

0

Fv

(C.3)

Here = eS/x. S is the area and e = E, which contains the elasticity modulus Eof the crystal. Though convenient and correct, this is no more a transduction two-portthan the circuit of Figure 51, which is a transducer two-port.

(AREAm2,DISTm,deg)

deg

RadiatorBaffledPascal

AREAm2=0.8e-4

DISTm=1

angledegrees=0

13SPEAKER1100Hz08ALU

out

in

Figure 51: A transducer two-portyes, a LS is indeed a transducer. But this is not atransduction two-port.

Equation C.3 might lead one to think the force F itself produces an electrical voltage U,both potential parameters. Though this equation relates these two parameters correctlyand is useful, it does not reflect physically what is going onNo microphone potential Uwould be produced if the crystal did not deform (motional parameter).

1The I and v directions are defined here symmetrically: both pointing inwards at the two-port (notsuitable for matrix multiplication).


54/57


C.2.6 There Is No Magnetic World

On p. 47 we acknowledged two worlds: (1) Electrical and (2) Mechanical. One might betempted to extend this to four worlds:(1) Electrical, (2) Mechanical, (3) Acoustical, (4) Magnetic.

But the last two are lower down in the hierarchy. As a physical phenomenon Acousticsbelongs to the mechanical world. It can be defined as The Mechanics of GaseousMaterial. It is convenient to separate the mechanical and acoustical worlds, but onlybecause the dimensions are different: Force F[N] is replaced by pressure p[N/m2], andvelocity v[m/s] by volume current q[m3/s]. But that has nothing to do with physics.

Similarly magnetism belongs properly to the electrical world. A dynamic loudspeakertransduces from the electric to the mechanical world. There is no need to speak of amagnetic world (even if the LS does contain a magnet). Magnetism can be entirelyaccounted for by combining Coulombs Law of electrostatic attraction with special rel-ativity (Figure 52). It is convenient to look at the problem from the point of view ofthe ionic charges (red) because they do not move. Focussing our attention on the redcharges in the left wire, we shall ask what force they feel due to the right wire. Whenno current flows, as in (a) the forces cancel.

When an electron current flows in both wires (b), from the point of view of the positivecharges (red) in the left wire, the electrons in the right wire seem closer together dueto relativistic length contraction. This is felt as an apparent increase in negative (blue)charge density, whereas the charge density of the positive (red) charges remains as before,

because they have the same reference system as the red charges in the left wire. Theresult is that the right wire appears to be negatively charged. This gives an attractionbetween the wires, which we have come to call magnetism. But this is inherently anelectrostatic phenomenon, and we need invoke no new concept to explain it.

Force

(b) Currents of electrons I(a) No currents

LEFT WIREATTRACTEDBYEXCESSNEGATIVECHARGE

Figure 52: Electrostatic attraction due to relativistic length contraction.


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C.3. PHASE DELAY IN A PRESSURE CHAMBER 51

C.3 Phase Delay in a Pressure Chamber

The speed of sound (c = 340m/s) only applies for free-field propagation or in a tubewithout reflections. See Figures 5354. Phase delay is defined as /. It vanishes forhard termination. The pressure rises and falls as in a pressure chamber without anygradient for tube length less than /4 (below 800Hz here), as if the speed were infinite.Phase alternates between 0o and 180o for rising frequency.

(L, D, N)

nonlossy

TUBElength=lengthdiam=diamN=1

50e6

(L, D, N)

nonlossy

TUBElength=lengthdiam=diamN=1

Zw(V,Hz)

volt=1

(V,Hz)volt=1

out2

.define diam 20m

.define area (pi*(diam/2)**2)

.define Zw (rho*speed/area)

in2

.define length 0.1

out1in1

(1) NO REFLECTIONS

(2) HARD WALL AT FAR END (right end closed)

Zw=1.288MEG

practically infinite

wave impedance (purely real)

Figure 53: Circuit for non-lossy tube: closed and with non-reflecting termination.

10 100 1K 5K-5.0

5.0

15.0

db(v(out1)/v(in1))F (Hz)

db(v(out2)/v(in2))

10 100 1K 5K0.0

180.0

360.0

-ph(v(out1)/v(in1)) (Degrees)F (Hz)

-ph(v(out2)/v(in2)) (Degrees)

10 100 1K 5K0.0

200.0

400.0

600.0

length/(-(pi/180)*ph(v(out1)/v(in1))/ww)F (Hz)

length/(-(pi/180)*ph(v(out2)/v(in2))/ww)

SOUND_IN_CLOSED_TUBE_ANALYSIS.CIR

TRANSFER FUNCTION in dB

-PHASE IN DEGREES

APPARENT SPEED OF SOUNDACCORDING TO PHASE SHIFT

no reflections

+ means later no reflections

no reflections

Figure 54: Phase delay for Figure 53. red: closed; blue: terminated with the waveimpedance ZW = c/S which is pure real. (ww= .)


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57/57

Bibliography

[1] M R Avis and L J Kelly. Principles of Headphone DesignA Tutorial, Audio atHomeAES 21st UK Conference 2006, pp17-1 till 17-10. (Cited on p. 1).

[2] J Borwick. Loudspeaker and Headphone Handbook 3rd edition 2001,ISBN 0 240 51578 1. (Cited on p. 5, 19).

[3] J Zwislocki. Analysis of the Middle-Ear Function, Part I: Input Impedance,J. Acoust. Soc. Am. Vol. 34 (1962) p.1513-1523 (Cited on p. 34).

[4] E Zwicker, M Zollner. Elektroakustik ISBN 3-540-18236-5 (Cited on p. 49).

[5] G Theile. Study on the standardisation of studio headphones, reprinted from EBUReview - Technical, No.197 (Feb 1983). Published by the Technical Centre ofthe European Broadcasting Union, Avenue

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Aes Tutorial 48 Headphone Fundamentals