~ Distortion, Directivity and Circuit nodeling .t..-- of a Needle Array Plasma LOudapeake51

A The~i, Pre~ented to The Faculty of the College ot Engineering and Technology Ohio University

In Partial Fulfillment ot the Requirements tor the Degree

~3ter of Science

Ron Sterba ~. I r~rch, 1"91

OHIO UNIV!R$Btr1r t,~~!R~'AY TABLE OF CONTENTS

Page

Cbapter 1: Introduction 1

Chapter 2: BackgroUDd of Plasaa Sound Production

2.1 History of Plasma Sound Production

2.1.1 Experiments of Gerald Shirley .

2.1.2 Experiments of Kiichiro ~tsuzawa . 2.1.3 Other Work . 10

2.~ Plasma Physics 2. 2. 1 Cold Plasma Sound Sources...... 11

.2. 2.2 Bot Ple.~.a Sound Source5...... 12

2.3 Choice of Plasma Type 12

Chapter 3: Acoustic Wave Production

3. 1 Enerq1T Absorption of Weakly Ionized Geses . .. 13 3.2 Source Teras in the Wave Equation 16

3. 3 Interference Pa t terns of Four Point Sources.. 17 3.4 Results of Directional Pattern Tests 27

Chapter 4: Experiaental Apparatus

4. 1 Sound Source Construction...... 28 4.2 High Voltage nodulation Circuit 29 4.3 Choice of Operating Point5 32 4.1 Circuit Operation 33 i i

Chapter 5: Circuit Dodelinq 5.1 Sound Source Characteristic Curve Analysis. .. 34 5.2 Pspice Small Signal Circuit nodel 5.2.1 Vacuum Triode nodel 42 5.2.2 Sound Source nodel 46 5.2.3 Computer nodel of Co.plete Circuit 47 5.3 Frequency Response Analysis 49

Chapter 6 BaraoDic Distortion 6.1 tleasureaent of Harmonic Components 56 6.2 Nonlinear Circuit Operation 56 6.3 Comparison of Distortion tleasurements and Computer Analysis...... 58

Chapter 7: Test Results aDd Analysis 7.1. Results of Frequency Response Analysis 63 7.2. Results of Computer Distortion Analysis 63

7. 3 Ef f iciency Testing of Corona Sound Source.... 64

Chapter 8: S1IIlIl8.ry...... 66

AppendiX 1: DeriY8.tioD of the Wave Equation

tor a PlaslUl...... 68

Appendix 2: Baraon1c Distortion t ro. the SigDal

Generator...... 74

AppeDdiz 3: COJq)uter Prograa 1...... 76

Appe.Ddiz 4: COJq)uter Prograa 2...... 78 tt1

AppeDdi% 5: Coaputer Prograa 3...... 80 AppeDdi% 6: Coaputer Prograa. 4...... 82

Ref.renee. 84 iv

List of Figures Figure Page

2.1 Shirley's corona triode configuration .

2.2 nBtsuzawa's sound source and driving circuit . 5

2.3 Current levels through nBtsuzawa's sound source..... 6

2.4 Characteristic curve of natsuzawa's sound source 12...... 6

2.5 natsuzswa's electrode geo.etry _. 7

:.6 Theoretical frequency responce of nBtsuzawa's sound source 9

2.7 Corona discharge between a needle and a grid 12

3.1 Four element linear array radiating in a direction g to 8 distant point C 17

3.2 Superposition of radiated signals of amplitude a and successive phase difference 0...... 18

3.3 Relative intensity of a 2 point source array 19

3.4 Relative intensities or 4 and 8 point source arrays...... 20

3.5 Definition of angles used in plots 21

3.6 Calculated intensity interference patterns (0.5# 1 kHz) . 21 v

Figure Page

3.7 Calculated inten~ity interference patterns (2, 3, 6, 10 kHz) zz

3.8 Calculated intensity interference patterns (12, 20 kHz) 23

3.9 Directional patterns of sound pressure measured 1 .eter from the sound source at 500Hz 24

3.10 Directional patterns of sound pressure measured 1 .eter from the sound source at 1 kHz...... 24

3.11 Directional patterns of sound pressure aeasured 1 .eter trom the sound source at 3 kHz 25

3.12 Directional patterns of sound pressure measured 1 aeter from the sound source at 6 kHz 25

3.13 Directional patterns of sound pressure measured 1 aeter from the sound source at 10 kHz , 26

3.14 Directional patterns of sound pressure measured 1 aeter trom the sound source at 12 kHz...... 26

4.1 Top view of cathode assembly 28

4.2 Corona sound source 29

4.3 Sound source circuit 30

4.4 Circuit tor findinq characteristic curve of ~ound source...... 30 vi

Figure Page

4.5 Sound source characteristic curve...... 31

4.6 Circuit for testing vacuum tube characteristics..... 32

4.7 6BI4 Vacuum TUbe Characteristic Curves for V~ -.5Y,-. 75V,-1V and -1.25Y 32

5.1 Vacuua diode circuit 35

5.2 Plate voltage characteristic curve of vacuua diode...... 35

5.3 Sound Source Characteristic Curve (d = 1ea) 36

5.4 Log-Log plot of sound source characteristic curve... 37

5.5 Computer generated plot of equation 5.2 (d = 1cm) ... 36

5.6 Electrode geometry 38

5.7 Effects of vaI1~ng electrode separation d 39

5.8 Plots of Iss v» d at Vss = 4,6~6~10,12 and 14kV..... 40

5.9 Log-Log plot of Iss vs Vss at d = 8, 10 .. 12, 14 :ram. .. 41

5.10 5aall signal model of 6BK4 vacuum tube 42

5.11 Characteristic curve of Ip vs Vp at bias voltage V~ = -O.75V 43 "ii

Figure Page

5.12 Plot of plate current vs grid-cathode voltage for the 6BK4 45

5.13 Sound source circuit .odel...... 47

5.14 !xperiaental sound pressure trequency response...... 49

5.15 Siaulation frequency responce of the sound source capacitance current 50

5.16 Siaulation frequency responce of the sound source resistance current.... "...... 50

5.17 Siaulation frequency responce of the sound source plate voltage...... 51

5.18 Simplified small signal model of sound source circuit...... 51

5.19 General Form of Bode Plot of Eq. 5.23 55

6.1 Operating points of sound source 60

6.2 Calculated sound pressure haraonic levels at different dc bias points 61

6.3 ~easured haraonic sound pressure levels at different de bias points 61

A.6.1 Node numbers tor Pspice ac small signal circuit .ooel 80 Chapter 1: Introduction

The object of thi~ the~i~ i~ to explore aco~tical propertie5 ot a corona loudspeaker inclUding ha~anic distortion and directivity of sound pressure. A corona loudspeaker of the type used here consists of several .etal points in front of 8 .etal screen, biased with a high voltage to produce a corona. When the high voltage across the air gap is aodulated, sound is produced. This work examines the characteristic curve of such a device and uses it to predict distortion. The results of this analysis show the optiaum operating point to achieve low distortion ie near maximum current. Since the source has several points of acoustic energy generation, an interference pattern is set up. neasureaents were made on this in two di.ensio~ and it was compared to the theoretical interference patte~. A saall signal circuit model for the sound source and amplifier circuit was created using Pspsice. The frequency response of the sound source is compared to the frequency response of the .odel. 2

Cbapter 2: Background of PlaslI8 SoUDd Production

2.1 History of Plasma Sound Production

The following sections review previous experiaents with plas.e sound sources.

2.1.1 Experiments of Gerald Shirley

~rald Shirley pre~ented a paper on his studies of corone wind loudspeakers to the New York section of the Audio Engineering Society in June 1957. (1) Shirley described the construction of 8 corona triode in which e ring is .ounted coaxially around one of two needle electrodes which point toward each other. This corona. triode was the basis for his loudspeaker design. Shirle1' claimed that . having no aovmq parts, this speaker has definite advantages over conventional loudspeakers.

, control r"ino / . ~"'? f!llectrodes

Figure 2.1 Shirley's Corona Triode Configuration

Shlrley observed that the particles composinq the plasM. of a corona discharge have a net drift velocity which he teraed a 'corona windI. He stated that the darectron. Jaagnitude and pattern of the corona wind could be observed by injectinq a smoke streaa at low velocities into the plas.a. Fro. these test he found that that the .agnitude of this wind can be controlled by a coaxial ring set at the proper controlling potential between the ring and its concetric electrode. Shirley also noted that the electrical current can also 3 be controlled by the voltage on the ring just as the plate voltage contro15 the current through a vacuum triode. He plotted the characteri~tic curve~ for different geo.etrie~ and grid potential~ or the corona t r rode. Shirley noted that the sound produced by a single triode is very faint and described the construction of a sound source with 144 triodes operating in parallel. Needles in a grid pattern were mounted in a 6 inch square fraae and interlaced vires spaced one­ half inch apart were placed over the needles which are centered in the spaces between the wires. Shirley liaited his discussion of this sound source to the practical considerations of its operation and did not explore the theoretical means of sound production. He then I18de soae coaparasons between the corona. wind loudspeaker (CiLS) and. electrostatic speakers.

1) Botb are extended rather than point sources of sound.

2) The corona wind loudspeaker can reproduce a wider frequency range .. and should be able to create a greater aap11tude of air aotion at any trequency.

3) Electrostatic speakers require a high pover audio 5ignal while C\~S can be driven by a ~aall voltage signal which is m.odulated onto the polarizing higtl voltage circuit by 8 tube amplifier.

Shirley did not verify his second point. With regard to point number three he should add that the polarizing high voltage source supplies the operating power. Due to the absence of ]loving aechanica.l parts (and assocaeted suspensions) the overall audio frequency response curve of the ClLS was in general aucb smoother than that obtained f roa a typical cone­ type speaker. Shirley' 5 relative audio response curves deaonst.rated that the cone speaker varies in e. range of 27 dB while the ClLS 4

varied only about 7 dB. It is interesting to note 8 sharp dip in the frequency responce plot at 8Y~. He explained thi~ is due to de5tructive interference when the electrode ~epar8tion is one half the wavelenqth or the aUd10 Signal. Other considerations discussed by Shirley are: the drive require.ents for the CiLS, which include different circuits for modulating the audio signal through the triode; health hazards of the ozone and nitrous oxide genera.ted by the corona; and, further directions of research a.nd develop.ent for the elLs. He suggested researching the effects of varying humidity levels, and aethods to reduce di5tortion of the audio signal. 5

2.2.2 Experiaents of Kiichiro natsuzawa

l1at~uzawe. (2) built and tested three sound sources. The t olloY~ 15 8. s~ry ot nrs experiaental and tbeoretical work. The sound sources were built by l1atsuzawa with 8 flat brass Wire grid as the positive electrode and many steel needles a~ the negative electrode. The needles were placed equidistant to each other and distributed in a circle.

The follovinq circuit vas used to operate the sourd source.

CathodEt

681<4

10kQ 3.9kQ 11

Figure 2.2 l1atsuzawa's sound source and driving circuit

The circuit supplies a de current through the sound source~ 10 = o. 75m.A and a superimposed AC current, I (PJ'IS) . To avoid clipping, 1(~} wa5 kept le~5 than (10/2). SolVing for the peak amplitude of I gives:

I ~ 10/2, I ~ 0.375.& ~ I ~ = (~2) 0.375 = 0.53 aA ~ 6

1.3 rnA max ~_1 -m-- X t '.

10 = 0.75 mA

0.0 ---+----ot------or--

O.53m~pt'ak 1.OmAp;") 0.2 m;' mln

Figure 2.3 Current Levels Through t1atsuzawa' 8 Sound Source

This leaves a .iniwel current (0.2 mA) still flowing to operate the corona. l1e.tsuzava aodeled the sound source electrically as 8 reaistence shunted by e. capacitance. The value of sound source res1stance is found from a linear curve fit of the I-V characteristic curve. Figure 2.4 is the curve for t1atsuzawa's second sound source.

18 ,..... +--....-...-~"--o+-..;.-l IJ :,__~...--.-e.~ ::- ..:Jt 16 '-' ~ ; 6- .-0- r- .. 14 +-~" ! '0 ; Slope g;ves : > "-~-!--""-+-.:!Il~"'-"--~"""""'~'R =6.0MetQ Q •0 : .. -: L 12 =' m0 -0c ::,1 10 0 00 8 0.0 OS 1D 15 Sound Sour~ Clrrent (rnA) figure 2.4 Characteristic curve of sound source 12 7

The experiJaental 'Value of shunting cepacttance C vas found to be 25pFd. na. tsuzawa claiaed that the current through the resistance of tbe sound source, R creates a proportional torce on the surroundmq air F. The I18gni tude of the RltS current I I throuqh the resis~ce is expressed as:

II = I [1 + (wRC»21-112 Eq 2.1 where w is the angular frequency and I is the ~ current through the sound source. natsuZ8VS8 then gives the resuitinq RltS force on tbe surrounding air as: F = (d/b) I' Eq 2.2 d is the distance between electrodes; b is a coefficient based on experiaental results I and approxiaates ion mobility.

Positive s_ (HV) anode ...... ~.

I··············TSepal"ation dlstanct'

He-gat;v~ side (lV) cat~

Figure 2. 5 natsuzawa I s electrode geOllctry

Using llorse an:! Ingard, (3) lfatsuzawa arrived at the folloW'iD} expression for the RltS sound pressure P for an observer far froa the source: B

1

+ (2~r)2]2 P = F 4:rC [ 1 Eq 2.3 where r is the distance fro~ the source to the observer, A is the wavelength and t i5 the sound velocity. Substituting equations 2.1 and 2. 2 into 2. 3 he obtains an expression relatinq the sound pressure to the sound source resistance, capacitance am current.

1

P = d I [1 (1- )2]"2 w 1 Eq 2.4 b411rc + 211r, [1 + {wCR)2J2

A plot of this euation 2.4 repre5ents the theoretical pressure output where R = 6 negQ and C = 25 pFd. The first part of the pressure equation, 1 2 d I A 2 b411 r c 1+ (211 r) [ ]

is assuaed to be constant at frequencies greater than 200 Hz where the observation distance, r = 100ca. The -3 dB corner frequency of euation 2.4 is qiven as:

f~ = wn/ (2n) = 1/{RC2n) = 1061 Hz

Figure 2.6 is a Bode plot of equation 2.4. 9

I ! !I!II

1Q------100---tf(Hz)---l000------1OK

Figure 2.6 Theoretical frequency responce of natsuzawa's sound source

na. t suzaee"s experimental results agreed veI17 well vi. th the theoretical frequency responce for frequencies less than 2kHz. Abo,'e 2kHz it is claiJaed the pressure is attenuated because of destructlve interference between the needle tips.

As an additional note, F. Bastien (4) suggested an oversight in l1a.tsuzawa's theoretical York. Through soae rough theoretical ca.lculati.ons Bastien 81..tggested that l1B.tsuzaw. should include the · heat.' source term in his calculations. He concludeed that at higher frequencies and greater observation distances fro. the sound source the nea t source term. makes a valid contribution in the equation for sound pressure. 10

2.1.3 Other York

F. Bastien (4) suamarized and expounded the re5ults of work done concerning sound production by electrical disCharges. Be presented and classified the work as 'true' corona or 'glow are' sound sources. He also discussed related fields such as lightning and thunder. The following paragraphs give his key points. True corona or cold plas.a loudspeakers include the devices of Shirley and. lfatsUZ8W8.. An' ionic triode speaker' sillilar to the one built by Shirley was patented by Halus and Holcomb (1957){4). nore recent developments include a corona source earphone developed by Bonder (4) and .arketed by Audioreference in 1984 and the Plasmatronic5 Loudspeaker (5) :.arketed in the U.S. by Plas.atronics Inc. in 1980. The Plas.atronic5 Loudspeaker is 8 ~drange-treble plasma speaker driven by a high voltage tube amplifier. The unit is described as a lavender triangle-sbaped cbaaber With helium bled into the speaker tro. a tank (the helium suppresses the ozone production) . A hot plasaa or glow arc discharge is another type of sound source. nost of the work done with these type speakers has been done by S. Klein (6) who first introduced his ionophone in 1946. His speaker used a 20,000 volt potential with a aodulated audio signal to drive a single electrode glow arc sound source. Bastien (4) reported tbat a version of Klein's systea has recently been co..ercialized in Europe as 8 tweeter by the nagnat corporation (The speaker produces ozone levels too high for sale in the U.S.). 11

2.2 Plas.a PhY5ic5

Sound pressure can be created by using a corona discharge in two ways: witn 8. cold pla~ or 8 hot plaSJaa.

2.2.1 Cold Plas.a Sound Sources

When a pointed electrode is put near another conductive electrode under a high voltage 8 pre-breakdown discharge occurs. The tera 'corona' desiqnates this pre-breakdown discharge. The corona is coaposed of electrons, ions, and neutral aolocules. The electron teaperature in the corona is relatively high (several eV) but the temperature of the neutral .olocules and ions is al.ost equal to the a.bient temperature. Therefore the pIas.a can be qualified as • cold' . Cold plasaa corona discharges present different features

depending on the polarity of the point J the point radius. and the gap length. Both positive and negative point polarities can be used to produce sound (4) but a negative p01nt allows higher current levels (of the order of 100~) in a range in Which a positive point produces pre-breakdown streaaers with their resulting noise. (4) Further.ore the plasBa region can be divided into two parts (7): a small one (r=200~. for a point radius of 50~) around the point at which the ionization takes place, and a larger drift region which contains only negative ions (positive ions if the point has positive polarity) . Since this region contains only one type ion it is naaed the unipolar region (see figure 2.7). In this unipolar region the ions produce a net t orce on the surrounding air molecules which is one mode of sound production of the cold plasma. The other contribution to sound production is caused by the te.perature difference of the electrons and the neutral aolocules. This teaperature difference sets a condition for energy to be exchanged betveen the electrons and the 8 toIlS which causes pressure change! in the neutral gas., leading to sound waves. 12

.....ipolar zone

'ionisation Z~

Figure 2.7 Corona discharge between a needle and a grid

2. 2. 2 Hot PlasltB. Sound Sources

In hot plaslaE1 sound sources sound is produced by a neeted plas.a of about 1000°[. Ions for this breakdown are generated by a heated electrode. The .odulation of the heat of the plaslIfl leads to ~ound wave production. nost hot plasBa sound sources subject the heated ions to a constant electric field witll a .odulated audio field. These sound sources are only efficient at frequencies above 10kHz (4) while the cold plasBa sources produce sound at much lower frequencie~ as veIl.

2. 3 Choice of PlasJIa Type

For the pre~ent york a cold pl~~mo ~ound ~ource wa~ built. The speaker is & small version of nBtsuzawals needle array - wire mesh configuration in which only four needle elecrodes were used. The needles were placed in a linear array in order to test the interference pattern5 around the speaker. One objective of the experiaent was to test the haraonrc distortion generated by the speaker in the medium and high frequency audio range. A hot plas.a speaker is aost efficiant above'10kHz (4); therefore a cold pla~ma ~peaker, which operate~ veIl at all audio frequencies, was chosen f or testing baraomc distortion. 13

Chapter 3: Acoustic ¥aTe Production

3.1 Energy Absorption of Weakly Ionized Gases

This section relates the heat energy absorbed by the electrons and ions in the plasma to the polarizing electric field in teras of .ability, drift velocity, and the electron charge e. The high voltage put across the anode and cathode of the sound source creates wha.t is known as 8. weakly ionized plaSlla in the fora of a plas.a sheath. The degree of ionization greatly influences the dynamic effects of the sheath. For this reason 'Weakly ionized gases must be considered separately when discussing acoustic effects. tlorse and Ingard (3) regard 8 'weakly ionized gas' as one with 8 low ratio of ionized particles to neutral particles. An exa.ple of a typical weakly ionized gas is a glow discharge of the following composition:

neutral gas density: 1016 cm-3 electron and ion density: 1010 to 1011 cm-3

This weakly ionized gas or plaslI8 is creeted by 8. la.rge electric field which is external to the plasma sheath. The charged particles draw energy from the external electric fields. Soae of this energy is transferred to the neutral gases which leads to sound gelleration. To study the acoustic aspects of the plasma we aust be concerned with the dynamics of the ionized gases. '!be external electric field produces forces on the ions and corresponding torces are produced by the plasma itself on the surrounding gases as 8 result of its .otion. These forces also contribute to sound generation. The weakly ionized gas in the plaslIfl sheath can be regarded as a aixture of three different coaponents: the neutral gas atoas or llo1ecule~, the ions, am the electrons. For our purposes the basic 14

difference between 8 plas.a gas and a neutral gas ~xture is that the plasBa .otion is strongly coupled to changes in the surrounding electric and .aqnetic field~ while the effect~ of the~e field5 on a neutral gas atxture is very 5:.8.11. The excited electrons in the plas.a lose their energy by several processes. In a typical weakly ionized gas created by a glow discharge as in the sound source co~idered here, the main 1055 of energy by the electro~ is due to the energy transfer to the neutral gas particles through ra.daa collisions. Other sources of losses are due to the excitation and ionization of the neutral gas particles. For a typical weakly ionized gas the difference lD the enerq}9 of the plasma components is made eVident by the difference in the particle te.peretures. The steady state electron te.perature is in the order of 104 to 105 OX while the neutral gas temperature is only around 50~. Thus the electron energy level is much higher thall the neutral gas energy level. This energy iabalance creates a constant energy flow fro. the electrons to the neutral gas particles. Since energy is supplied to the electron~ by the electric field and the electric field strength is controlled by the applied voltage the rate of energy exChange fro. the electrons to the neutral particles can be controlled by the voltage level across the sound source. If the voltage is .ade to vary with tiae the rate of energy exchange to the ions will e Leo vary with tille which will produce sound in the neutral (Jas and radiate outside the plasJaa. In the tille intervals between collisions the electrons are accelerated in the direction opposite to the electric field. During this acceleration tiRe the electron absorbs energy from the electric field corresponding to the distance the electron travels. The collisions .ake randaa the aotion of the electrons to 8 certain extent but because of the unifora electric field they have a saall net drift aotion. The rate of this motion is called the drift i~ velocity Yf and proportional to the electric field E. The constant of proportionality is called the electron .obility, be included in the drift velocity equation: 15

[q 3.1

The electron .obility is in units of ca2 I Volt-sec. Since E is in units of Volt I CIl thi5 gives us V~in CIl I sec. In natsuzawa's (2) sound source experiaents he arrived at a value of be: 2.2 c~/Volt-sec experiaentally. Like the electrons the ions are also colliding with ataas and have a similar drift velocity: Vi = - bi E, 'Where b; is ion ..ohility and Vi is ion velocity. The average rate of energy absorption by the electrons per unit voluae, ~ , is expresaed by (3):

Eq 3.2 vhere ~ is the electron density and e is the .agnitude of the electron charge. There is a similar relation for the ion energy tr~fer. Since the number of electrons, ~ is nearly equal to the number of ions, ~, the difference in the energy contributed by the electrons and the ions is related by their respective mobilities be & bi· Since the energy absorption of the electrons is dependent on the ti.e between collisions so.e esti.ate of the tiae delay of this process can be :.ade. In a glow discharge as used in a sound source the average electron collision time is around 10-8 sec (3) while the collision ti.e required to produce high audio frequencies is 6x10-5sec. Since the collision tiae is auch s~ller than the period tor high audio frequencies we can regard the tr~fer of energy fro. the electrons to the neutral atoas as instantaneous. We can then consider ~ = e~[E(t)]2 as the tiae-dependent net heat transfer equation. The next section deals with the conversion of this heat tr~feI te~ to sound. 16

3. 2 Source Teras in The Wave Eauatiop

Appendix 1 developes the theory whiCh expre5~e5 the 1nteraction ot electrons and ions With the neutral gas in teras ot sources in the wave equation. The result (Eq A 1.14) is the following wave equation:

Eq 3.3 where H is the heat energy added to the plas.a per unit voluae from the field, )1 is the ratio of spec1f1c heats of the plas-a and C is the speed ot sound.

F(rl t ) is the force per unit voruae of plasJla. caused by the field. This is related to the force term that natsuzawa uses to relate the current through his cold plas.a sound source to the resulting sound pressure by the equation: t = I(d/b) where d is the d1stance between electrode5 and b corresponds to the ion .ability. The reJDaininq term:

C')' -1) oH c'') at is referred to as the 'heat term'. It doainates the wave equation for hot plasma sound sources such as the glow arc sound source built by Klien (6). 17

3.3 Interference Patterns of Four Point Sources

The corona loudspeaker .ade f or this study can be matheJaatically .od.eled as e. linear arranqeaent ot tour point sources. This is valid at great distances fro. the sound source. The sources cause a directional pressure pe t tern around the speaker which is a function of the wavelength of the tr8Dsaitted frequency, the spacing of the point sources, and the position of the listener with respect to the orientation of the needle array. A s.all four eleaent _odel is the basis of the following analysis (8). The ele.ents are separated by a dis~ce d with the total array diaension L given by: L = 3(d). Referring to figure 3.1, The phase difference between the signals tro. two successive sources is given by:

{'- = 2n d o sin Q Eq 3.4 )..

~/' ·· t-·-...",;: . d I I...! ..-. (N-t)d I · ~ I ~ (N-t)d SlnB 1- --_...... \ ~ ~ ~dsinB

Figure 3.1 Four ele.ent linear array radiatin1 in a direction s to a distant point C

l If all of these radiation vectors operate at a magnitude la and to a point C which is a t an angle ctI fro. the perpen:1icular to the array the result of the superposition is the radiation vector R which can be found fro. the folloviD1 figure. 18

') . li a= ..rm- '"~

H(' s'in N8/2 R=2rsln_o = a --- 2 sin 812

90- NS 2

A a B Figure 3.2 Superposition of radiated signals of aaplitude a and successive phaae diff erence ~

The intensity, I, of the ~ound signal at the point C is: I = R2. Substituting the expression for R we have:

~ = I sin-ON ndsin B]/)..) 2 s sin ( Indsan B] I A)

Where 13 =[n.dsin g] I).. Eq 3.5

and Is is the sound intensity f roll. each source

Whenever the sin teras of the nuaerator and denoatnator both equal zero constructive interference of the wave patterns occurs. This occurs in the nuaerator when NO = Oft,N1t,2Nn... and coincidentally in the denoainator when D = O,n,2n.. Those points where D is 8 aultiple of 11. occur when: d sing = n la.bda. At the points of constructive interference the relative intensity can be calculated by: 19

2 2 2 I = I sin (Nf» -+1 N I3 -+ If s 2 s 132 Is sin (13)

Between the principle .axima there are N-1 points where the intens i ty I I: O. this occurs when the nuaerator stn2NJ3 = 0 but the denoainator 5iri2~ is still finite. This occurs when dsin9/(laabda) = alN where _ I: 1,2, ... N-i. This also sets the condit ron to define the wid th of the principle ISB.xiJD.um.. Froll the center of the principle .axiaua to the first null is called the balt-width principal .axiaUl\, or HWPl1. The following intensity plots show relative intensity (N2 tera divided out) of linear arrays containing 2,4 and 8 sources. Note the horizontal axis are in units of SJ.D9 and the principle Il8.xaUJllS occur when sana = n{laJ1bda)Id. The crests of the principle .axilluas can be nUltbered in unit s of d sin9!laabda. Also note as N becoaes large the BJPn becoaes s.aller corresponding to it·s definition as: HiPn=lambda/Nd.

? .. d Slrl.8· -1 -.,. -)-.--+ 1 (w " 1.0 Is ( T 0.5 Is I \ o sinH --+ N = 2

Figure 3.3 Relative intensities of a 2 point source array 20

-2 .. d s1n.8' ... -1 0 1 ") ).. " 1.0 Is

4 i i 0.5 Is I

0

N = 4 sinB ---+

u d SlnS -2 +- -- -+ -1 0 1 2 A 1.0 I!=

~2*HiPn r 0.5 Is I

0 sinB --+ N = 8

Figure 3.4 Relative intensities of 4 and 8 point source arrays

Between each principle maxiaua there are N-2 secondary maxiaa whiCh have lower intensities. Since our sound source consists of four point sources we wish to look at the polar plots of the relative intensities when N=4. A prograa (See Appendix 3) was used to generate the polar plots of the sound intensity at different angles around the speaker where the JlaxaUl1. intensity has been nonaalized to be Imax = Is = 1.0.. (see figure 3.6, 3.7 and 3.8). The needles point in the direction of 180°. At the angles where destructive interference brought the signal to a null the prograa draws an asyaptotic line. These nulls are at the BIPl1 angle~. 21

wire mesh ~ ¥'riICJ 1w/)~ ~

1 rMt.,.

figure 3.5 Definition of angles used in plots ( overhead view )

,..,..-' 2100 / /' I I .5 1" .5 ,.. 500 Hz / 1000 Hz I WL: 66.2 ell I iL: 33.1 em. 180V----+----+----+----- .5 .5 .5 .5

\ .5 .5 \ \ / ,-/ ~ / ~

Figure 3.6 Calculated intensity interference patterns 9 = 00 along riaht hand horiz axis Radius represents relative sound intensity. 22

I 2000 Hz 3000 Hz ¥L: 16.5ea TiL: ii.0ca HWPl1: 900

1

I I 6000 Hz 10000 Hz ¥L: 5.5C1\ WL: 3.3 0 ell HiPl1: 30 HWPn: 17.5 0

I:' • ...J .5

Figure 3.7 Calculated intensity interference patterns o = 00 along right hand horiz axis Needle array parallel to vertical axis. Radius represents relative sound intensity. 23

ft', I 12000 Hz 20000 Hz iL: 2.75 iL: 1.6cll. c. 0 HWPI1: 14.480 HVPI1: 8.63

.5

Figure 3.8 Calculated intensity interference patterns e = 00 along right hand horiz axis Needle array parallel to vertical axis. Radius represents relative sound intensity.

The needle separation of the source model is 2.76 em. This sets the needle ~eperation di~~ce, d, to one half the wavelength at 6kHz. This results in a balf width principle aaxt.a at 300 from the relationship: 5in~ = la.nda/Nd where N = 4~ d = la~a/2 (see figure 3.7). ¥hen the needle separation equals the wavelength at 12 kHz the constructive interference at 9 = 900 is pronounced (see figure 3.8). The following plots in figures 3.8 - 3.13 represent the experiaental aeasureaents of the directional patterns of sound pressure 1 .eter frCDl. the 50und source. The sound pressure wa~ aeasured with a sound pressure aeter connected to a epectrua analyser. The um.t,s of relative sound pressure are in dB. Tests were conducted at various frequencies. The needle separation dis~ce d is 2. 76ca in all cases. The needles point in the directian ot 1800 (see figure 3.5) 24

500Hz 90

180...------+- 0

270 Figure 3.9 Directional patterns of sound pressure aeasured 1 aeter fro. the sound source at 500Hz Radius represents relative sound intensity in dB.

1kHz 90

o

270

Figure 3.10 Directional patterns of sound pressure aeasured 1 aeter fro. the sound source at 1kHZ Radius represents relative sound intensity in dB. 25

Figure 3.11 Directional patte~ of ~ound pre5sure .ea5ured 1 aeter tra. the sound source at 3 kHZ Radius represents relative sound intensity in dB.

~ 90

270

Figure 3.12 Directional patterns of sound pressure measured 1 aeter fro. the sound source at 6 kHZ Radius represents relative sound intensity in dB. 20

10kHz

90

o

270 Figure 3.13 Directional patterns of sound pressure measured 1 meter frOll the sound source at 12 kHz Radius represents relative sound intensity in dB.

12kHz

90

270 Figure 3.14 Directional patterns of sound pressure .easured 1 aeter fro. the sound source at 12 kHz Radius represents relative sound intensity in dB. 27

3.4 Result5 of Directional Pattern Tests

The polar plots of sound pressure qenerally correspond to the caaputer generated predictions. The large side lobes predicted at 12kHz are easily seen in the experi.ental plot. One difference is that the sound pressure behind the grid (fraa 90° to 270°) is slightly greater than behind the needles. The sound source projected the sound aore in the directian of the needles. If each needle tip was a true point source the patterns should have been symaetrical about the 900-2700 axis. The wire grid .ay have reflected and diffused the sound on the back side of speaker. At higher frequencies the pressure aaplitudes were widely varied and often hard to aeasure. The use of an anechoic room would have given aore accurate results at the5e higher frequencies. 26

Chapter 4: Experiaent81 Apparatus

4.1 Sound Source Construction

The speaker is constructed of four steel needles aounted perpendicular to the brass wire cloth. The anode of the speaker is the brass cloth and the needles are ..ultiple cathodes. The JD.omlting structures for both the needles and the vire cloth are .ade of steel and are supported above the table by steel posts. The bases of these .ountiDg posts are each bolted to 8 separate plexiglass insulating structure which insulates the speaker parts froa the table surface. The cathode asseJlbly (figure 4.1) is constructed to vary the needle geoaetry in two ways. First, the dis~ce between adjacent needles can be adjusted by sliding each individual needle mount along tracks which run throuah all f our of the mounts.

steel treeks /

Fiqure 4. i Top view of cathode asseJably

Second, the needles can be positioned closer or farther from the plane of the brass wire grid by a linear aicropositioner which supports the cathode insulating structure. 29

.assernb~ - t -, I w. cloth InO\Ilti\gframe

llnO\1lt post - -

-st..l mount post eflon sheet -. 0.125" T PlexiglAssfr~, / -Pie xig~s frMne C1 table surface Figure 4.2 Corona sound source

4.2 High Voltage nodulation Circuit

Figure 4.3 shows the circuit ~ed to operate the sound 50urce. The high voltage needed to ionize the air and create the corona is supplied by a 20kV-10laA de voltage supply. A 6BK4 vacuus tube is used to aaplity the audio signal froa the signal generator to the high voltage signal needed to drive the speaker. The ca thode resistor Rkath is used to bias the de operating point of the tube and provide negative teedbacK. 30

+ Vss -

anode cathode \I platE' 20kY plate­ OD47UF de Yoltigf 68K4 ._••-t----.....-CTM ...... 1---.. supply y+ + + - gk + Yg

"insulators - ll~th

Figure 4.3 Sound source circuit

Since no current enters the grid of the tube we can S8.y tbe.t Iss = IRkath· With no ac signal across the 10kQ resistor (Vg = OY) we can write the following voltage loop equations:

Vkath = IRkath Racath = -V~. Also: VHY = Vplate + Vss . We need to know the characteristics of the vacuum tube and the sound source to find values tor VH\I and P"'ath to properly bias the circuit. The circuit in figure 4.4 was used to find the characteristic curve of the sound source. + Vss - Iss VHV= Vss anode cathode ----+

20kV + p~l de volt. 5.6Y ~t~f supply Zene-r dJodE. (protects amrnet.,.)

------insulators -- -- figure 4.4 Circuit for finding characteristic curve of sound source 31

By varyiD;r the de vol taqe (Vss ) and aeasuring the current through the sound source (Iss) the nonlinear curve in figure 4.5 was plotted.

0.0006

0.0005 t-----4---bias current =400p.A --...... ,....--1 : : ; : 0.0004

~ -< '-' 0.0003 ~~~~~~~~~~,~~~_.~~~~C13jY ~'" 0.0002 .·t 0.0001 ; :;rr : :

0.0000 t----~.....-oio-.._..-...._.p_~...._..__"t_.,...._c 2000 4000 6000 8000 10000 12000 14000 16000 'Iss (V)

Figure 4.5 Sound source characteristic curve

~ find the characteristic curves of the 6BK4 vacuum tube the circuit in figure 4.6 was used. Note: VH\I = Vplate. By setting the grid voltage to a constant value, varying the plate voltage and recording the corresponding plate current, Ip~te, curves for different values of Vgk were .ade. l1a.ny aeasureaents were Bade but only four points on each curve are displayed in figure 4.7. The curve-fitting program used was Cricket Graph. 32

lplit~

20kY plate de vo1t~ 681<4 ••••-+----.. ~la:I +

Figure 4.6 Circuit for testing vacuum tube characteristics

v~ =-.5V / "'fit< =-.75 Y 1~~~~~~~~~~~~~~~~V~=-1DY

1100 ~...... -+-...... -+-...... -...... ;....,---...... -.....~~.. - yN =-1.25 \I 1000 t-+-.....-..~.~~....-~....-;.-+-...... ~~.,t ':1" 900 ~ ~ ~ ~ 800 ~ ~~t ~ 400pA' ~ i....~~...... ~~... 700 ~ 600 +tn-I ttltH+~ : ,-,' ~• 50CI ...... -!.....-!--!-+.~..-;,."+-!i~~-...-.-...~---, ~ 400 ~-.....-.....-...... ~."".;, .,.~~ -+--.-...-",

300 ...... -!.....~~~~....,~.-- ;.....,.~-.;-...;-+~ 200 t--!rq....;~~..".a:..~,...;-....-.....;..., 100 O~ i-t- -.... -;- ...... 2.02.53.03.54.04.55.05.56.06.57.0 7.58.0 Vplat~kY) Figure 4.7 6BK4 Vacuum Tube Characteristic Curves for Vgk = -.5Y.. -. 75Y,-1V and -1.25Y

4.3 Choice of Operating Points

ibile takinq the sound source characteristic curve aeaeureaenta. arciDJ between the needles and the brass cloth would occur at de currents higher than 600~. Therefore bias points of 33

400~A or lower were chosen to reduce the chances of arcing. A bias point for Vgrid vas chosen to be -0.75V. Froll the 6BK4 Iplat~ characteristic curve of Ygk = -0. 75V a t I: 400JLA the pIate voltage is 4. 25kV. The voltage across the sound source, Vss = 13.1kY at Iss = 400.uA (from Figure 4.5). This gives:

YHY = Vplate + Yss = 4. 25kV + 13.1kY = 17. 35kY

The final consideration is the value for Racath. Since we know

Vkath = -Vgk I: -0. 75V and lRkath = Iss = 400J,lA,

RK~th = Vk~th / lRkath = O. 75 l O. 0004 = 1875 Q

A decade power resistor was used f or ~ath. The resistance was tested and vas found to be accurate and non-inductive.

4.4 Circuit Operation

When the de voltage V~, is raised to its value of 17. 35kV, the aaaeter in series with RKath will read 400.u.A. ¥hen the voltage aJlPlitude of signal generator,Vsignal , is applied the grid voltage is no longer zero (as in the de case) but, Vg = A sin ¥t. The voltage loop around the grid and cathode loop is now: Vg = Vgk + Ykath. The variation in V~ is small, so

so that the sound source current 15 closely controlled by the grid voltace. IRbth is observed on the oscilloscope as the voltaqe wavefoI'll across Rtcath. If the aaplitude of Vsignal is too high elippinfJ in Iss will occur. 34

Chapter 5 : Circuit llodeliDg

5.1 Sound Source Cbaracteri~tic Curve Analysis

The corona sound source with the needles configured as the cathode and the bra~5 wire grid as the anode has characteristics siIilar to e. high voltage vaCUUlD. diode. By studying a diode I 5 Characteristics we can establish a basis for the sound source distortion analysis. The vacuua diode operates on the principle of unilateral conductivity. See figure 5.1. Source A supplies the current to heat the f ilaaentary ca thode. By the process of theraionic emission electrons are eaitted troll am surround the cathode. When switch S is closed e. potential between anode and cathode is supplied by source B. Thi5 potential draws the electrons to the anode. The J88gnitUde at the resulting current depems on:

1) the total number of electrons emitted 2) the Jlaqnitude of anode-cathode potential 3) the negative space Charge

Since so.e electrons are emitted at low velocities they do not travel far frOll the cathode. The result is a region of high electran density around the cathode. This high electron density may depress the potential in the space beloY this electron cloud at the cathode surface, and thus has a current limiting effect at low plate potentials. The application of high plate potentials suppresses or overco.es the effect of the negative space charge region by dravinq aore electrons towards the anode and lowering the density near the cathode surface. 35

Fig 5.1 Vacuum diode circuit

The plate characteristics are aeasured by holding the cathode heater source Va at a rated value and increasing the plate or anode voltage. Fig 5.2 identifies the different regions of operation of a vaCUUID. diode.

...c •"­ ~I .,~ o -0• CUTM\t limited o ..C by emission

o ~ voltage, eb

Figure 5.2 Plate voltage characteristic curve of vacuum diode

Over the first part of the curve fro. 0 to a the current rises slowly as a result of the retarding influence of the negative space charge. .As the electric tield between anode and cathode overcoaes the negative space charge, 8.5 seen in the region froll. a to b the curve becoaes aore linear. Finally when the plate voltage beco.es high enough all the ent ted electrons reach the anode and the curve becoaes aore horizontal as seen in segllent b-e. 36

The characteristic curve can be described in the region fro. 0 to b for e. parallel plate diode by the followiD;J equation credited to Child (9):

Eq 5.1

where i b = total pIate current eb = ptete voltage d = distance between electrodes A = constant depelXliDJ on the geoaetry of the electrodes

The characteristic curve of the open - air corona sound source was plotted experaentally and is presented in figure 5.3. The curve is seen to be siailar to that of the vacuum diode except the sound source curve never saturates. Above 150~A the curve is fairly linear. When the device vas operating: in the most linear region (around 600~) the electric field would break down the air and a violent arc would occur.

0.0006

0.0005

0.0004

""-c[ '-' 0.0003 ~" 0.0002

0.0001

OO ...... ilii--r-w--r--t--....-i--....-t--..--+--....-tסס.0 2 4 6 8 10 12 14 16 Yss(kY) Figure 5.3 Sound Source Characteristic Curve (d = 1ea) 37

To fit the ~ound source date to the diode equation we used the 2 z equation: i I: (A/d ) Vss and evaluated the exponent z. Since our electrodes are not operating in a vacuum and the geoaetry is different this tera .ay be different than the 1.5 constant for the vacuum diode. In fact, the slope of the log-log plot of the curve in figure 5.4 gives us z = 2.76 Iss= Ie x Yss 2.76

To find A, the bias point of 400~ and 13.1kV when d = lea was used:

A = (400%10-6)x(1)2 I (13100 2.76) = 1.68x1o-15 (A cm2)/V

The characteristic function for the sound source is:

Iss = (1. 68x1o-15 X Vss2.76)/dn (A) Eq 5.2 with n = 2. Plugging this into equation 5.1 and plotting this with the aeasured data for d = 1clD. on e. linear scale gives the plot of fiqure 5.5. 38

800 ···········r····· ···,·~·~~r~~ .. ·A····! ~ 600 ...... L 1.•.•.•....••L j

iss (UA) I I I/ I 400 ···········"1"···········1···········"1"·· ········1

200 ...... j , -r ,

o 4 B 12 16 Vss (kY) Figure 5.5 Coaputer generated plot of equation 5.2 (d = 1ca)

The anl}· reJlaininq term to consider ill equation 5. 1 is the effect of the electrode separation distance d shown in the following figure.

high voltage ~ anode --- ..__ ....~.

1-··-..-..-_..-Tseparat'ion dlstanct'

ca~ lowvo1t~s.

Figure 5.6 Electrode geoaetry

In a separate experiment aeasureaents of Iss and Vss were taken at separation dis~ces of 8, 10, 12 and 14... During these ezperaaents d vas held constant while Vss was varied and recorded with Iss. The t olloviDg plot of experaental data shows the effeet 39 of varyiIllJ d on the characteristic curve. This date. was plotted am curve-fit using Cricket Graph software.

d=1cm d=1.2cm

~ ~ ~ 200 --t..~~--;.~~ ...... ~~..-..-..~.- ~ ~ I-----..-.-.~.--~r".--o!--.-._--..--: L,. 100 "~"'------..Ii~...... :-t t ~ t~~~~~~:-...;~-·..e.------~4 ; ; ! ~ o ...... -"""""------~------.--~ ...... 4 6 8 10 12 14 16 18 Yss (kV) Figure 5.7 Effects of varying electrode separation d

Since the geparation distance where all of the audio testiIllJ took place vas at 1ea we ~ consider this as the reference distance. The inverse square relationship prove5 to be accurate at d=. Scm.. At 6kY the 46.uA reading divided by (. 8)2 qives us 71. 8.uA where the actual reading is 72}lA. At longer distances the currents are higher than predicted by the equa.tion. The 46}lA reading at d=1 i~ divided by (1.4)2 to give U~ 23.4?}lA when the act~l current i~ 26}lA. Errors becoae larger for distances qreater than lea at higher voltages. These errors could be due to the unusual geo.etry of the sound source, which is different fro. that of a typical vacUUll diode. To further test the relationship of equation 5.2 a plot was aade of the log of the sound source current VB the log of the separation distance d (see figure 5.8). Taking the log of both sides of equation 5.2: 40

log(Iss) = loq(1.6Bx1cr15 ) + 2.76 log(Vss ) -n log(d)

At constant values of Vss the slope of the plot5 in fiqure 5.8 should be -2 103 --_._-- _. .

. : ...

101 10 d(mm)

Figure 5.6 Plots of Iss vs d at Vss = 4,6,8,10,12 and 14kV

Computer generated curve fits from the plots of figure 5.8 show the slope to be -2.065 at Yss = 4kV. The slopes decrease as the bias of Vss increases and therefore the lidO relationship of equation 5.2 changes. The rele.tionship is dependent on the geoaetry of the electrodes and the plas.a. Since the bias voltage Vss changes the geoaetryof the plas.a, (seen in figure 2.2) it also causes n to change. The following table shows values of n obtained fro. the slope of the curve fits of figure 6.8.

Table 5.1 Yss (xY) n.. 4 2.065 6 1.810 10 1.639 12 1.577 14 1.292 41

Since the n changes at higher voltages ve find separate equations for the sound source at fixed di5~ce5 of d = 8, 10, 12 and 14... To tim the exponents tor these equations lOCJ-lor;] curves of Iss vs Vss were plotted in figure 5.9.

: : ::: ; : : : : ~~1Omrn Iss = A I Yss~.9431

____~__.cr~~~··....".~.~~~~~t2nvnIss =A* 'Ys;"2:9is~ :: ..-----~--..-; :-f-+-+ .. ~. ~ .i .! : ~ :: ~~·~14mmIss= A" YS5"3.0335 : Illit ! ~ ~ :1111

figure 5.9 Log-Log plot 01 Iss vs Yss at. d = 6 .. 10, 12 .. 14 JDIl.

Using the exponents found from the curve fits we can solve for the cons~t te~ of each characteristic equation.

@ d = 81m: Iss = 5. 776x1o- 16 x Vss2.9379 (A) Eq 5.3 @ d = 101uD.: Iss = 3. 39x10-16 X Vss2.9431 (A) Eq5.4 @ d = 121ml: Iss = 2. 885x1o-16 X Vss2.9351 (A) Eq5.5 @ d = 14Dl.: Iss = 9. 759x10-17 X Vss3 .0335 (A.) Eq 5.6 where Vss is specified in volt.s. 42

5.2 Pspice S.all Signal Circuit nodel

A working Pspice computer model of the corona sound source circuit was aade to try to better understand the circuit operation. The coaputer .odel is also used to see the current and voltage wavefora5 which are hard to .easure in the lab due to the high de voltages in the circuit.

5.2.1 Vacuum Triode nodel

The first step in aodelinq the circuit is to construct an accurate aoc1el tor the 6BK4 vaCUUIl tube. The s.a.ll signal model for the tube (10) is given in figure 5.10.

CGP

grid p1at~

+

Figure 5.10 5118.11 signal .odel of 6BK4 vacuua tube

The plate resistance is .odeled as a noruinear controlled current source where the ac plate current is a tunction of the ac plate voltage:

The Characteristic curve of de plate current, Ip V5 plate voltage, Vp at the bias point of V~ = -0.75V is shown in figure 5.11. 43

00010 ...... -.-.-...-~~~...... ~o:----:--~ . . : : : ~ : ~ ~, : :

: : : . . : 1000 2000 3000 4000 5000 6000 7000 Yp (Y) Figure 5.11 Characteristic curve of Ip VS Vp at bias voltage Vf/t:. = -0.75V

The curve-fit gives us the following de characteristic equation:

Eq. 5.7

Since we are deriving an ec small signal m.odel of the vacuum tube we need to describe the ac plate current function, ip(vp) at its de bias points. Fro. section 4.3 we know the tube is biased at the quiescent values of:

IpQ = 400JiA and VpQ = 4.25kY and the sinusional voltage at the plate is vp. To find the s]I8.11 signal plate current function ip(vp) we can take a Taylor expansion of the characteri5tic plate current function, Eq. 5.7, about the quiescent point. 44

This gives us,

Ip(Vp} = 4.0x10-4 + 1. 67x1o-7 (vp) + 1. 3x1o- (vp}2 Eq. 5.8 "

Note that the e~sion gives us the quiescent current of 400~ tor the de bias current. Since we are .odeling a s.all signal ac circuit we subtract the de bias current fraa both sides and it becoaes:

The small signal pIate current function f or the aodel is thus in the t OIllL: ip(vp) = (gp)vp where gp = 1. 67x10-7 + 1.3x10-11 (vp), and the plate resistance where vp ~ 0 is:

Ip = 1lcJp = 1 / 1. 67x10-7 = 6.0 l1egQ

Next the transconductance of the tube is .odeled as a nonlinear controlled current source where the ae current, im, is 8. function of the controlling voltage f roa grid to ca thode, vgk:

To find the de transconductance curve we hold the plate voltage constant at the bias point of 4.25kV and vary the grid-cathode voltage while recording the plate current to get the plot in tiqure 5.12 45

O.OOQ6. ...-~ ~-.--.....--r--.....-~~ -t ! 0.0005 -+--!-,-..----

0.0004 r- ~ 4: ._-- ..+...--._:._. '-" ~ ~ ~ ~ ~ 0.0003 -. .----.-..-,+-....-.~.,..r;.;--!---__!_---4___t

~ 0.0002 .~...-.l-L-~;-.~-.D-t--....~-:- ~ ; ~ ~ ~ ~

"'------...... -..-.-.-..------..-..~--'" OOסס.0 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 V gk(Y) Figure 5.12 Plot of plate current vs grid-cathode voltage for the 6BK4

The curve-fit program gives the following expression for Ip(V~):

Eq.5.9

From. figure 5.10 we can see that at the bias point of Ipo = 400.uA, V~ = -0.75. Taking the Taylor expansion of equation 5.9 at the quiescent point we find the ac current function im(v~):

Im{Vgk) = 4.0x1(14 + 2.83x1(14(vgk) + 4.16x1o-5 (vgk)2 Eq 5.10

Subtracting the 400~A de bias current gives us the ac current:

The slI8.11 signal current function f or SPICE is in the f ora:

im(Vm) = (gm)vQk where gp = 2.83x10-4 + 4. 16x1o-5 (v Qk ) .

When Vole #11 0, qm • 2.83x1~ aho 46

5.2.2 Sound Source nodel

The sound source is Ilodeled as a nonlinear current generator shunted by 8. capacitance . Css. Equation 5. 2 relat.es the sound source current, Iss to the speaker voltage, Vss when the separation distance, d = 1cm.. Iss = 1.68*10-15 V 2.76

Taking the Taylor e~sion of the sound source characteri5tic equation at the bias points of,

IssQ = 400JiA and ,TSsQ = 13. ftV and keeping only the first few te~5, we get the following ac expression for Iss:

Subtracting the 400~ de bias current we get:

Eq 5.11

The s.all sianal current function tor the .odel is in the fora:

8 When vss ItS 0, Iss = 1/gss = 1/8. 44xlo- = 11.85 l1eqQ 47

5.2.3 Co.outer noQel ot COlDlete Circuit

The diagrc.. rcprcocnting the coaputcr aodcl of the coaplcte ec circuit i~ seen in fiqure 5.13. The cathode reaistence. ~ith, i:s 1875 Q (see section 4.2).

CGP

plat~

Css

Figure 5.13 Sound source circuit .odel

All eIeaente in the circuit are known values except the ca.pacitance of the sound source .. Css. We can estiJlate this value by using tie.t5uzawa· 5 (2 ) equation f or sound pressure: 2]~ p= dI 1+(.1.) w 1 b 411r Vs[ 211 r [1 + (wCR)2]2

C is the pa~llel coabination of the sound source capacitance and the capacitance of the tube fro. plate to cathode. R is the parallel combination of the resistance of the sound source and the effective output resistance of the tube at the 400~ bias point. r is the observer's distance froa the speaker and Vs is the sound velocity. 48

The center t era including the wavelength (l8.Jada) is approxi.atly equal to 1 for frequencies over 200Hz. The te~s: dI/(b4nrVs ) are constant for a constant ac current through the sound source. The equation reduces to a constant tiaes a frequency dependent part:

P = const * w [1 + (vRC) )2]-1/2

Sound pressure aeaSUIeaents were taken a distance r = SOc. for different values of audible frequency and the results are plotted in figure 5.12. The lower 3dB cutoff point of the frequency response

plot occurs at f3dB II: wn/21t z: 1/ (2nRC) .

The effective output resistance of the tube aaplifier is given a~ (i1) :

where: grrl and I p are given in section 5. 2 as 2. 8257x10-4llhos and 6. OI1()6 Q respectively.

Therefore: rout = I p + (gm1p+1 )~ath = 6.0x1()6+(2.83x10-4x6.0x1()6+1)x1875 = 9.18 ttegQ

The total resistance fro. plate to ground (R) is approxiaately:

The f o110winq plot is the experiaental frequency response when the ac current through the cathode resistor was held constant at 150.uA peak. 49

-35 ------...... ------...... ----.-.-...... 101 102 103 fr~qu«ICtj (Hz) Figure 5. 14 Experi.ental sound pressure frequency response ( ac plate current constant with frequency )

The plot gives a -3dB frequency of 420Hz. Solving for the total capacitance value, we get:

C = 1/ (f3dB2n.R) = 73.3 pFd

The capacitance f roa the pIate to the cathode was ..easured to be 2.8pFd. Solving for the sound source capacitance we get:

Css = 73.3pFd - 2.8pFd = 70.5pFd

See appendix 5 for the Pspice input listing. Note the polynomials used for qplate, gm and gS5. Their values are from. the results in section 5.2.2.

5.3 Frequency Response Analysis

The plots of figures 5.15, 5.16 and 5.17 shows the Pspice saulation data including frequency response of the current through 50 the sound source capacitance, the current through the sound source resistance and the ac voltage across the tube whiCh is alaost the ~e as the sound source voltage.

II Ill!

100

o·~i 10 1 102 103 frequency (Hz)

Figure 5.15 Simulation frequency respo~e of the sound source capacitance current

8O..,...-o:--~ ...... ~--~--- ...... ,....------

60

~ ~ 'D •~ -c 40 :1- '-".,.. Va g; 20 IiIii:;

11III11

Figure 5. 16 SiJauls.tion f requency response of the sound source resistance current 51

1000

800

'""~ .. 600 •0- 1.1111 ....W. '0 :>- 400 '-' Q. > 200

0 101

Figure 5.17 Siaulation frequency response of the 8C plate voltage (saae as the sound source voltage)

The 5.all 5ignal circuit of the tube amplifier and the sound source can be simplified to the parallel circuit of figure 5.16.

+

,..,v Vss,... p1at~

Figure 5.18 Simplified 5Il8.11 signal m.odel of sound source circuit

G is the sua of the conduc~ce of the sound source resistance and the effective output conductance of the tube. C is the combination of the sound source capacitance and the capacitance of the tube f roll. plate to cathode. The voltage across the parallel circuit can be expressed in teras of the ~t sinusoidal current fro. the tube. 52

Vss = II (G + jwC) Eq. 5.14

To relate the sound pressure response of the speaker to the electrical frequency response of the circuit we look at the wave equation (Eq 3.3) which relates the sound pressure to the heat and force source te~.

Bast1en (4) reduces the wave equation to the the following equation relating the div F(r,t) force te~ to sound pressure:

K Eq 5.15 41Jrc where r~ the distance troa the source, c, the speed of sound, and k, a constant dependent on position are all a5sumed to be constant. F i5 the total force applied on the unipolar zone by the ionization zone and is roughly proportional to the electric field intensity at the limit of the two zones, ~ which is DrODortional to the voltaae across the sound source, vss . In the t requency do.a.in Eq 5. 15 becomes: Eq. 5.16

Bastien also evaluates the sound pressure due to the heat term in the wave eque.tion:

y -1 aw P(r) = Eq. 5.17 H 411r c 2 at 53 where Y as the ratio of specific heats of the plaSJla (assuaed constant) and ¥ is the total energy injected fro. electrons in the ionization zone and i~ given 8.5 (3):

If :::: IJJ H dVo

From equation 3.2 we can see that H is proportional to [2, therefore we can assuae : W= kE2 Eq 5.18

\'here E can be expreseed as a 5teady state, Eo value and an ec component, aE:

Substituting this expression for E into Eq. 5.18 we get:

iT :::: k(Eo + clE)2 :::: kEQ2 + kEQaE + kar2 and

aW/at:::: kEQ[a(aE)/atj + k[d(oE)2;dt] Eq. 5.19

The second tera: k[a(aE)2/at], generates distortion of the sound pressure. Assuae expression 5.19 can be approxiaated by:

Eq. 5.20

The frequency daaain of equasion 5.17 is then:

Where dE is proportional to the ac voltage across the sound source # Vss. This gives us: 54

Eq. 5.21

To find the frequency dependent sound pressure equation for the speaker we coabine Eq. 5.16 and Eq. 5.21:

Psp(v) = Pr(w) + PH(V) = Vss[ k: 1j w + )(2 + K3jV] = vss[ k2 + jW(k1+K3 ) ] = X2Vss[ 1 + j (w/wn)] Eq. 5.22

Where wn = K2/{Kl+k3)· Substituting Eq. 5.14 for vss we get:

Psp(W) = K21[1 + j(W/wn)]/(G + jwC) = X2IR[1 + j(v/Wn)]/(1 + jwRC) Eq. 5.23

where R ItS i/G and I is the ac input current f rom. the tube which is constant. Fro. Eq. 5.22 we can see that the -3dB cutoff frequency occurs when w = 1/(CtR) as we assumed before. Note that the heat term pressure, PH(w) contributes the term k3jw to the output pressure equation (Eq. 5. 22). ltatsuzawa (2) neglects the heat tern in his pressure equation (Eq.2.4). Bastien (4) does soae rough calculations and finds the heat term aight contribute to the sound pressure at frequencies higher than 360Hz. 5S

! I I I T 11 ·1 1" 1" ·1 1 1Scm l : ! llHl··· i f/II. ~+-W. ~ wn 1 ~ ~ T :. .:- .:- ...i ... .~ ... ~ -3<18 +3c13 ~yrl I ll"1" 1OdS ! I I 211RC f/ItS. 420Hz . ~.

5dB r I i

.1.1 .:. T T ..

10 100 f (Hz) 1k 1Ok

Figure 5.19 General fora of Bode plot of equasion 5.23

This Bode plot is somewhat similar to the experimental pressure frequency respance of figure 5.14. 56

Chapter 6: Baraonic Distortion

6.1 neasure.ent of Haraonic Distortion

Operation along the non-linear sound source characteristic curve induces haraonic coaponents of the driving frequency w, the tundaaenta; frequency. The second harmonic has a frequency of 2w

and the third, 3w 1 and so on. The corresponding aaplitudes for the tundeaental and its baraonrcs are A1 .. A2 .. .Ao and so on. The percentage of distortion represented by each har.onic is defined as (10):

D2 = A2/ A1 D3 = 13/ .&1 D4 = A4 / Al

The total ras haraonic distortion is defined as:

+ .... Eq 6.1

6.2 Nonlinear Circuit Operation

To evaluate the distortion caused by nonlinearity, consider a general function:

The bias points of the circuit are a t quiescent values of I;Q and VoQ. To evaluate the harm-onic coaponents we can take the Taylor expansion of the function about the quiescent point (10). 57

v y + (dVo ) (I; -I ) + o) (I, -I iQ)2 + '" = iO (lV o oQ er dI.2 1 I =I 1 1 .=Ii n 2' Eq. 6. 2 i iO 1 ~

This can be expressed as:

Vo = VoQ + 81 (1;-110) + a2(I; - 110)2 + ... Eq 6.3

¥hen the interval of operation does not include clipping (and the resultant sharp changes in the derivative), operation can be adequately represented by the first three or four te~ in the series. Constants a 1 ~ a2, 63, and 8.4 can be found from differentiation of the polynomial curve fit function evaluated at the quiescent point. The sinusoidal current input fro. the vacuum tube 8.Jlplifier is in the rora:

." 11 - Ii0 = r, cos vt

Substitution of this current function into the Taylor series and using trigonoaic identities for cos2wt, cos3wt, etc. yields the following haraonic series for the signal component of output voltage: 1 A2 3 /\.4 ".: VoQ = (2'8 Ii + 6 8 1 + ... ) 2 4 i

/\. ~ A 3 + (e. I. + - a I. + ... ) cos wt 1 1 4 3 1

. 1 r-.2 1 A 4 + (2' 8 r, +"'2 8 r, + ... ) cos 2vt 2 4 1 ""3 5 1"5 + ( - 8. I. + - a I. + ... ) cos 3vt 4 3 1 16 5 1 1 r-. 4 ) +( - a I· + ... cos 4wt + ... 8 4 1 58

The baraoruc series can be vrit ten as:

Yo - YoQ :: Ao + Al cos wt + A2 cos 2wt + ... Eq 6.4

where .62, 43, .64 ... are the a.plitudes of the bar.onic coaponents of voltage. Their Jaagnitudes depend on I; and the nonlinee.rity of the curve. Note that Ao represents a de shift in output voltage under signal conditions. Using these aaplitudes we can solve tor the percentage teras of the various baracm.c coaponents.

1 ,,2 1 " 4 1 -aI·+-aI.+ A? 2 2 1 2 4 1 f, = --=- = ~ I. (2) 2 A A 3 A3 1 2a 1 a I. + - e. I. + 1 1 1 4 3 1

1 "'3 5 "'5 -aI·+-aI·+ 4 :; 1 16 5 1 ~ I~(~) »; ~ A :t: , 4a a I .• '::""a I."'. 1 1 1 4 3 1

1 /'.4 - a I. + 8 4 1 ~ I~(~)' A ~ A 3 1 8a a I. + ~ a I. + 1 1 1 4 3 1

6.3 Comparison of Distortion nea3ur~ents and Cgmputer Analysis

The distortion approxt.ation technique described in section 6.2 was written in a Pascal prograll seen in Appendix 5. The program evaluates the sound source cbaracteristic function of Eq 5.2 in the followiDtJ fora:

Vss = 2. 254x105 IssO•36

It reads in 'Values of bias current and the Ili.n.aUll and I18xaua current leve15 of the sinusoidal signal. The routine then gives the 59 values of 81, 6.2, 63, am 8.4, am the naraonrc atap11tUl1e values ~, ~ and D4 in percent and dB. The vacma tube 8.J\Plifier used to drive the sound source is an alaost linear current source input to the speaker. Since speaker voltage is proportional to the force tera (F) of the output sound pressure equation (see Eq 3.2) we assuae a direct relationship between the speaker voltage distortion and the experimental sound pressure distortion. The sound pressure output of the speaker was ~pled by a Quest Electronics .odel 216 sound level aeter at 26ca fro. the needle tips. A Hewlet-Packard .odel 13561A Spectrua Analyser was used to display the Ilicrophone's signal in the frequency doaain. A tundaaental frequency of 2kHz was used because this places the tundaaental and its haraonics in a relatively flat region of the frequency response curve (see figure 5.12). Ha~onic5 were generated at 4, 6, and 8kHz. All sound pressure readings were noraalized to correspond to 6 tundaaental sound level of 0 dB. Since the dc current that flows through the sound source is the same as the current that flows through the vacuum tube the points of operation are dete~ed by observing the voltage across the C8thode resistor. The corresponding current and Yoltage operating points can then be read from the characteristic curve of the sound source. 60

. VIi -+--;" 152kY 0.0004 /' ,...,. ~- -c 4- "'-' 0.0003 --[-l-~ - ... ---j ~ ~ ~ 7f-; -1- ~ 13.3kY j/' 0.0002 ---t. v-/T ; - 0.0001 /; 10.9kV ~ OOסס.0 I J I • J I 2 4 6 8 10 12 14 16 Yss (kV) Figure 6.1 Operating points of sound source

Distortion .easurements were taken in the lab for de biases of 300J,lA., 350J,lA., 400.uA, and 450.uA with ac signal aIlplitude of 150.uA peak in all cases. The following tables and graphs show this experimental data as well as results of the computer siaulation.

Coaputed distortion values at different de bias points

Bias Point: 300uA 360uA 400uA 450uA 2nd ha raome : -22 dB -23 dB -25 dB -26 dB 3rd naraonic : -39 dB -42 dB -44 dB -46 dB 4th baraomc : -54 dB -59 dB -62 dB -65 dB

neasured baraonrc distortion at diff erent de bias points

Bias Point: 300l1A 350l1A 400uA 450uA 2nd baraonic : -15 dB -16 dB -18 dB -20 dB 3rd baraonac: -24 dB -29 dB -29 dB -29 dB 4th baraonic: -29 dB -21 dB -39 dB -29 dB 61

-20 r------

-30 prfldictftd distortion JI7JlIoe------I ~b ~ trio pmnb: ~ -40 • ....~ !!. -so •~

-70 2nd harmonic 3rd harmonic 4th harmonic

Figure 6.2 Calculated sound pressure ha~onic levels at different de bias points (nonaalized to the funda..ental pressure = 0 dB)

-10 ...------..

-20 ,..,.."...._.. '. measured distortion m 1eve15 at bias po'ints : ~ • II ~...... ~ -30 II 35OJ1A If II 400J1A ..~ ~ 45OJ.lA -40

-so 2nd harmonic 3rd harmonic 4th ~ic

Figure 6.3 ~easured harmonic sound pressure levels at different de bias points (nonmlized to the fundamental pressure = 0 dB)

lteasured ~lues of the second baraomcs decrease as the de bias point increases as predicted by the proqr8.l1.. ltore varied readiD}s were observed f or the third and fourth baraomcs. '!he cause of these ~ried readings is discussed later. 62

Another theoretical test was done to study the effect of changing the gap distance on the haraonic distortion levels. The charactaristic equations for the sound source at gap ~eparation distances of d = 8, 10, 12 and 14 .. are given in equations 5.3, 5.4, 5.5 and 5.6 respectively. The distortion levels were evaluated by the proqraa in appemix 6 f or signals biased at 400j1A vith aaplitude~ of 150j1A.

Caaputed distortion values at different gap distances

Gap distance: d=8D d=1Omm d=12U d=14mm 2nd haraomc : -23.9 dB -23.9 dB -23.9 dB -23.8 dB 3rd baraonrc: -43.3 dB -43.3 dB -43.3 dB -43.1 dB 4th haraonac : -62.1 dB -62.1 dB -62.1 dB -61.9 dB

These result s show that varying the gap distance should have little effect on distortion levels. 63

Chapter 7: Test Results aDd ADalysis

7.1 Results of Frequency Response Analysi8

The ec s.all signal analy:sis of the sound source am high voltage amplifier circuit shoved that the _odel systea could be reduced to e. parallel RC circuit f eel by en ideal current generator where the resistance is a parallel caabination ot the output resistance of the aaplifier and the resistance of the sound source. The capacitance of the aodel is the co-.bination of the eapacatance of the sound source and the plate to cathode capacitance of the vacuua tube. The sound pressure frequency response of the speaker is related to the electrical frequency response by showing the frequency dependent relationship between sound pressure and the maqnitude of the force term of the wave equation and the magni. tude of the heat energy injected in the plasma. In the linearized model, since the torce tera magnitude and the average injected energy are both proportional to the sound source voltage the -3dB cutoff frequency of the electrical circuit should be the saae as the -3dB cutoff frequency of the sound pressure response. Using the experi.ental frequency response of the sound pressure we found a theoretical circuit capacitance value of 73pFd. Experiaental aeasureaents in the lab wi th no high voltaqe qives 23pFd. This measure.ent was taken with only a 6Vdc bias voltaue on the circuit. The actual value during circuit operation :aay be dif f erent because the high de bias voltage ionizes the air and thus affects the circuit capacitance.

7.2 Results of Coaputer Distortion Analysis

Both the experiaental results and the computer predictions show that the 2nd haraonic distortion levels increase as the de bias current throuah the sound source is decreased. The experiaental 64 distortion levels were higher than the coaputer predictians. Thi5 could have been due to distortion in the high voltage aaplifier which was not taken into account in the coaputer prediction. A Jaajor source of second haraonic distortion Bay arise because H is proportional to [2. Further.ore~ the needle and grid structure is not a parallel plate capacitance.. so we cannot even say Vss is proportional to E. There may be some nonlinearity in this function too, which effects both the r tera and the H tera in Eq. 3.3. The co~ination of all the~e nonlinearities together could in the end be a .ore serious source of distortion than the characteristic curve of the sound source.

7.3 Efficiencvof the Corona Sound Source

A test was .ade to co.pare the efficiency of the corona speaker to e. conventional cone speaker. Both were used to create the same sound pressure at the saae frequency at a fixed distance of one meter. Table 7.1 shows the voltage, current and power driving each speaker to produce the same 54 dB signal on the sound pressure meter at 2 kHZ when the aeter vas directly in front of the sound source.

Table 7.1 Power dissipated in corona speaker and cone speaker

rllS yo!tape rJlls current power corona speaker 4.24 xv 297 )11. 1.269 W conventional speaker 34 mV 4.25 mA. 0.1445 aW The table shows that the conventional cone speaker is 8700 tiaes aore efficient than the corona speaker. Shirley (1) suggests .ounting the speaker in a baffle enclosure to iaprove the low frequency efficiency. A higher efficiency could also be obtained if the aeter had been aoved to 8 point of .axiaua constructive interference. 66

Chapter 8: SUIUUlry

The object of thi5 work is to explore acoustical properties of a corona loudspeaker, especially har.onic distortion and directivity of sound pressure. A corona speaker was built to do this investigation. It vas constructed of four steel needles .ounted perpendicularly towards a wire aesh grid. By putting a high voltage between the needles and the grid the air in the gap i5 ionized, ereating a corone. . By .odulatil'KJ an audio signal onto the high voltage across the sound source the Bystea .aXes the corona undergo an acoustic vibration. The acoustic signal is created by means of two processes. The first is the heating and cooling of the ionized gas inside the plasa8 sheath th~ creatil'KJ pressure changes in the surroundil'KJ air. The ~econd i~ a net force on the surrounding air .olecules due to collisions with the ions which comprise the plasma sheath. This is shown by the wave equa. tion which has 8. the hee. t source term and 8. force source te~. The force source term dominates the wave equation for corona sound sources. The characteristic curve of the sound source resembles the curve of 8 vacuua diode. Because of the non-linear curve baraonic distortion is created. '!he second harm.onic distortion was predicted by the 5hape of the characteristic curve. Results of experimental measurements and predicted distortion levels showed that haraonic distortion is reduced at higher values of de bias current. tleasured distortion values were .uch higher than those predicted by the shape of the characteristic curve alone. '!his may be due to nonlinearities in the wave equation - particularly, in the relationships between the applied voltage, the field in the plas.a, and the heat generated by the field. Since the sound source is comprised of four point sources of fixed position, constructive or destructive interference between adjacent sources results in interference patterns around the speaker. These patterns 'Were predicted fro. the speaker geo.etry 67 and the frequency of the sound. neasureaents shoved soae aqreeaent with the~e calculations. Efficiency tests showed that the present corona speaker is .any tiaes less efficient than conventional cone speakers. Efficiency could be improved by using a baffle enclosure. 68

Appendix 1: DeriYlltioD of the ¥aye Equation for a Plas:aa

This derivation of the wave equation follows norse and !ngard (3) and starts by relating the cbanges in pressure ot the qasses in the plasma to changes in temperature. denstty .. entropy, and heet capacities of the fluid. Inside the pla~.a region the aotions of the fluid are too great tor the linear equations ot wave aotion to hold. Therefore all nonlinear teras which usually can be neglected when deriviDJ the linear wave equation .ust be included. l1odulation of the electric field ionizes aore or less of the surrounding air aolecules according to its intensity. This change in the aaount of ionized ga.s contained in the plasae sheath can be thotl]ht of as 8. net M.SS flow into and out of the ionized area. The equation of continuity for ma~s flow is (3):

3P + div (p u ) = 0 at Eq A.i.1 where p ~s the .ass density of the fluid and u is the vector representing the fluid velocity. Since a sound source operates using high ionizing voltages with a coaparatively small modulated voltage SWing of the input audio 5ignal we can separate the variables into an equilibrium. value pl~ a small acoustic part:

Pressure: P + P Temperature :T + l'

lfaS5 Density: p + d Entropy: S + (J per unit .ass

When relating the changes in temperature~ mass density, pressure and entropy norse and lnaard (3) worked with the followina measurable propertie~ of the ionized gas: 69

Cp Heat capacity per unit -.ass at canst pressure

Cv fleat capacity per unit 1l8.5S at canst volUJle t; Miab8. tic coapressibility

K. Isothe~l coapressibility B Coefficient of ther.al expansion Y Ratio of specific heat8

The r efat.i.onsmps involving the heat capacities are as foll095:

r ~ = C K "P S V t J

The isothe~l compressibility relates the change in voluae of the gas to the pressure changes as the teaperature is held constant:

K = -.!.(av) t V dP T Eq A. 1.2

The coefficient of theraal expansion, B represents the change in voluae as the temperature changes, with the pressure held constant.

Eq A..l.3

Finally, the increase of pressure with teaperature at a constant Toluae i5:

Eq .6..1.4 first we solve for the acoustic change in mass density in terms of the measurable quantities of the gas. This acoustic density tera is give as: 70

Eq A.i.6 but:

Substituting into A.1.4 gives us:

Eq A.1.6

It can be shown (3) the term for the acoustic change in entropy,~ is:

I \ 0' = -r + p = ~ II r _ 1 p II (as) (as) y - Eq A.i.? ar p ap TTl YC( I \ J

Using A.l.5 we can get the expression relating the change in mass density to the change in pressure and temperature.

dP _ oc a --- Q:: VPk:s - (p- o.r ) dt - dt dt Eq A.i.8

Now we can relate the change in entropy to the heat energy introduced into the plasma per unit mass.

Eq A.1.9

6(t) represents the heat enerqy per unit mass and the entropy ter. is given in equation A.i.? Substituting gives: 71

i , ~ ,I Y- 1 ' € (t) -11'- PI "" ()t I )lex. I c, \ J o PP e (t) -(1'--+- ) "" at (X. (X. )J c,

a p ~ 6(t) -(O:1'-P+-) ,.""., ot y c,

0 1 oP (1 6 (t) -(p-O:1') ~ -- - Eq A.i.10 at ' )J dt Cp

Where the left hand side is given in equation A.i.8.

And reaembering from equation A 1.1,

op =_ mv (p u) ~t we find:

1 ap = _1_ (_ mv (PU») )J pK, at )J pKs

1 aD (X, 6 (t) ,.""., -- - - )J at c,

Where the last line is the right side of equation A.i.iD. Rearranging this expression gives: 72

oP « ett) pK, - + otv (p u) = -C- Ypt; Eq A.1.11 at p

The la~t te~ in A 1.11 can alternatively be written as follows:

where H is the heat energy added to the plaSWi per unit voluae and is given es H =: pe (t). For any aa.terial:

)J -1 =

We define~

1 pt;

From the wave equation developed below, we see that c represents the speed of sound in the plasl'lfl.. The last te~ of equation A.1.11 can now be written as:

BH =(Y-l) Ht;p Cp 13T

And for a perfect gas~ 1 13:: - T 73

Substituting this into 1.1.11 gives us:

1 3P -- ... div(pU) = Eq A.1.12 c2 ~t The equation of continuity for ao.entum flow is given as:

Eq A.i.i3

Where F (r,t) is the torce per unit voluae acting on the ions and molecules at position r. Taking the partial derivative nth respect to tiae of equation A 1.12 and noting that from equation A 1.13:

gives us the wave equation:

Eq A 1.14 74

~peDdi% 2: Ha~onic Distortion fro. the Signal Genera. tor.

For this test the fundamental is chosen at 3kHz with correspondilvJ ha~onics generated at 6, 9, and 12kHZ. The 3kHz input signal fro.. the signal generator vas set at 4.3 Vp-p and was connected to the input of the tube. The 3kHz unclipped fundaaental input signal generated a -21dB sound pressure output signal ~pled by a aicrophone transducer at 25cm trom the needle tips. The second, third, and fourth haraonics generated pressure levels of -38, -52, and -62dB respectively. When the signal generator was connected directly to the spectrum analyser at the saae voltage level the analyzer registered the rundaaenteI as 8 +3.5dB value. The baraonics at 6, 9 and 12kHz registered -63dB or less. The ac voltage signal (x) required to produce -63dB ~ be found by the equation:

alog (-63) 20 x = 4.3 -----+""l 5' alog ( ~) 20

This gives us, x = 2.03 aillivolts peak to peak. This was verified experimentally by feeding a ~V signal from 6 to 12 kHz into the system.. The spect.rua analyser always aeasured this as less than -63dB. Finally the baraonic output of the signal generator must be coapared to the signal needed to generate the ha.raonic outputs of the sound. source. The -38dB haraonic output into the detector a.t 6kHz ¥a5 duplice.ted U5i.ng a 6kHz fund8.ll.ental frequency. .An input sigllfll of 40o..V generated this auch output. This is ZOO tiJaes the .aqnitme of the 2 .V of the hanaonic due to the signal generator. Adjusting the signal generator to its miniaua output of 22ail1ivolts peak to peak at 6, 9 and 12kHz resulted in output sound signals of 75

-70, -80 and -80 dB respectively, tar below the haraonic outputs of -38, -52 and -62 dB at the ~e frequencies when the fundaaental of 3kHz vas applied. Therefore it can be concluded that the haraonic distortion of the sound source is al.ost entirely due to the inherent nonlinearities of the source and the high voltage a.plifier circuit and that the haraonic output of the signal generator has 8. negligible effect on the distortion aeasure.ents. 76

AppeDdix 3: Co.puter Proqraa -1 program -1;

{This program displa~ theintensity ofinterference patterns for 1; near} {arraysofsound sources and disp18l)S the relative intensity vertical1y } {vnerethe horizontal units are ofs;n(trete). Theta is theangle from the} {perpendicular to the linear array. Variables include: n = number of}

{sources J f = frequency oftransmission .. d = seperation bet-ween } {sources (em), numunits =number ofpnncl ple maxi mas to be displayed} {c =speed of sound through the medi urn (set at 33100 em/s)} {The program also calculates end displays the h81f-'width ofthe} {pri nciple maxi mum (HWPM) given ~: HWPM = \a'8Velength/nd} var

n.i , numumts J unttvidtn : integer;

theta, thetarad .. d, f} Is, Is,c, Iarrde, BJ unit, HWPM, center: real; const pi =3.141593; begin {program statements} movetot D, 200); line(O .. -5); moveto( 1OO} 200) ; 11 ne( 0, - 5) ; moveto( 200 I 200) ;

11 ne( 0 J- 5) ; moveto{ 300, 200); line(O, -5); moveto(400,200); 11 ne( O} - 5) ; moveto( 0 I 200); 11 ne (400, 0) ; n:= 4; c:=33100; f:= 12000; d := 2.75833; lamd8 := elf; Is := 100; numumte := 4; unit'width := round( 400 I numumts) ; unit ~= 0; i := 0; HWPM := unitvidth I N; center := unitvidth * numumte I 2; 77

'While i < 100 do begln B := pi * d * unit Ilamda; ;f (si n( B» <> 0 then 18:= Is * (sin(N * B) * sin{N * B) I (s10(B) * sln(B) * N I- N»; 1;neto( i of 4, 200 - round( te) ; unit := unit + (numuntts I 100) ; i :=; + 1; end; penpat(grey) ;

moveto( round{center - HWPM) J 200) ; ljne(O, -100); moveto (rourdteenter + HWPM) .. 200) ; l1ne(0, -100); end. 78

AppeDdix 4: Co.pater Prograa '2

program -2;

{This program ceculstes the inteference patterns of8 four element} {sound source and graphicallydisplays8 polar plot ofthe relative sound} {i ntensttu 81 all angles aroundthesound source. The program 81 so} {finds the null angle3 'Where destructive; nterference bri ng$ the} {1 ntensity tozero anddrevs essumtotic 11 res et these angles} {The variables are: source separation:d (em) } {traremtssion frequency: f (Hz) } {Number ofelements: n (for my model 0=4)} {HWPM: h81f-widthofthe principal maximum} {Is: represents maximum redi us ofinterference pattern} var n. i .. Is : integer;

theta,tnetersd, d, t, Is, CJ x, y, lamda, BJ pifact : real; HWPM .. sintreta. costheta : real; const pi =3.141593; begin {Sound Disperticn 81 6000Hz 'with needle seperation of 2.7583cm} n := 4; c := 33100; f:- 6000; d := 2.7583333; lemde := elf; Is := 100; S1 ntheta := lamda I (n .. d); fOOveto(80 .. 120); 1;neto{ 320, 120); moveto( 200 .. 240) ;

11 neto( 200 J 0); if (1 - (81 ntheta + 81 ntreta) > 0) then begin costheta := sqrt( 1 - (81 ntheta .. sintrete) : HWPM := erctentstnthete I costnete): . vriteln(HWPM : 6 : 3) ; end; theta := 0; i := 0; -while i <202 do begin tretared := theta ~ 2 .. pi I 360; B:= pi '* d '* sin(thetarad) 11amda; if (S1 n( B» (> 0 then la:= Is ~ (sin(N ~ B) ~ sin(N * B) / (sin(B) * sln(B) '* N of N»; x := cos(theterad); y := sin(thetarad); 79

;f; = 1 then moveto( round{ 18 .. x) + 200, 120 - round{ Ie " y» ; ,ftheta> 3.6 then 11 neto( round{ 18 * x) + 200 J 120 - round( 18 " y» ; theta := theta + 3.6 I 2; i := 1 + 1; end· mov~to( 200, 120); if {1 - (Sl nthete ... Sl ntheta) ) 0) then beQlo 11 Oeto( Is .. round(cos( HWPM» ... 200} round( 120 - Is .. 81 n(HWPM») ; moveto(ZOO,120); lineto( -(Is" round(cos{HWPM») + 200} round( 120 - Is" s;n(HWPM»); moveto(200, 120); 1;neto( Is .. round(cos( HWPM» + 200} round( 120 + Is -f Sl n(HWPM») ; moveto{200, 120).: 11 neto( - (Is * rouoo( cos ( HWPM) » + 200, round( 120 + Is + s;n(HWPM) )); end; end. 50

Appendix 5: Co.puter Prograa 13

CGP

grid S platE' '3 5

Reap

Css

Figure A.6.1 Node nuabers for Pspice ac small signal circuit aodel

*RON STERBA *PSPICE CIRCUIT THAT SIMULATES THE SMAll SIGNAL MODEL OF THE *SOUND SOURCE AND AMPLIFIER CIRCUIT.

*SIGNAL INPUT A= 1.OV AT NODE 1* VIN 1 OAC 1.0~·

*CI RCUIT ELEMEtlTS* RS 1 2 600 CIN 2 30.047u RIN 3 00 10K CGK 3 42.6PF RKATH 4 0 1875 CGP 3 5 O.03PF CPK 45 2.8PF CSS 0 7 70.5Pf 81

-vccs THAT MODELS THE PLATE RESISTANCE" GPlATE 5 4 POLY 1 (5,4) 0 1.665[-7 1.3E-11

-vccs THAT MODELS THE TUBE TRANSCONDUCTANCE*

GM 5 4 POLY 1 (314) 0 2.8257E-4 4.162E-5

-vccs THAT MODELS THE SOUND SOURCE CONOUCTANCE* GSS 6 0 POLY 1 (6,0) 0 6.4367E-8 5.6674E-12

"AN ADDITIONAL RESISTOR MUST BE ADDED TO AllOW A DC PATH TO GND fROM NODE 5* RSS 5 0 1OOOMEG

*ADDITIONAL RESISTORS fOR CURRENT MEASUREMENTS Rl 651 RCAP 7 5 1

.AC DEC 10 10 1OK .PlOT Ae I(Rl) .PLOT AC I(RCAP) .PlOT I.e I(RKATH) .PlOT AC V5 .END 52

AppeDdix 6: Coaputer Prograa -4

proqrem #3;

{Thi. program prediot. dietortion 1•...,.1.of the MCOnd, third end } {fourth h8rmon1cs. The program reads 1n parameters of} { the characteristic opereti ng function: VS8 =cons*lss**expo} { 8S ..-ell as theoperati ng poi ntsofthe Cl rcuit 1ncl 001 ng: } { tmex .. Imi n, end 10. Distortion levels ere presented 8S} { 8 percent ofthefundamental and 1ndB 'Where the fundamental } {ie normalized tozerodB} tupe list =erreul0..501 of real; ver i .. j .. 1hold: 1nteger ; n.. x, Y.. P.. q.. co os : real;

Ide J expo, Imex J ImtnJ Ihat.. VQ .. IQ .. f : real; D} e : list;

function feet (n : reel) : reel: begin f:= 1; i hold := i: ; := 1; 'While 1 < n + 1 do begin f := f .. t: 1 := i + 1; end; fect := f; 1 := ihold; end;

function pover (P .. Q: real) : real; begin if p = 0 then pover := 0 else if p > 0 then po-wer := exp(q of In(p)) else if (p < 0) and (round(q) mod 2 = 0) then pover := exptq of 1n( - p») else if (p < 0) end (round(q) mod 2 = 1) then J)O\'Ier := -exp(q *' 1n(- p» end;

{turetten SUbprogram that calculates the log (base 10) of 8 number} function log (t : reel) : reel; begin ift (> 0 then 83

log :=1n(t) 11n( 10) ; ift =0 then log := 0; end; beQin writel n{ 'Whatis the current bias?') ; reedln( tq) ; vritel n('What ts the mi ni mum current sew; ng ?') ; reedl n( Imin) ; vritel n{ What is the maxi mum curentsvi ng ?') ; re8d1 n(Imex); vrttel n('What is the exponent (x) ?'); reedlntexpo) ; vrttel n(What is the val ue oftheconst ?I) ; readln(cons) ; for i := 1 to 4 do begin x := feet (i ) ; aI1J := co ns i- expo i- pove r{iq.. (expo - 1») i x ; cons := cons" expo: expo 0= expo - 1.; ife(i] < 0 then a[i] :=a[1J ~ (-1); 'w'ri te1n( '8 [ ', i : 1} '] =·} 8 [ i] : 28 : 26) ; end; lhat := (Imex - Imin) I 2; D[ 2] := Ihat * 8(21 I (2 * e( 1]); D(3] :=pove r(Ihat} 2) * 8[ 3] I (4 * 8[ 1]) ; o[41 := po¥er ( Ihet} 3) * 8[ 4] I (8 * 8 [ 1] ) ; Ide:= 0.5 * 8(2] T Ihat T Ihat + 0.75 * 8[4] ;: povert lhet .. 4); v ritel n( •0(2] =', 0(2) : 10 : 8) ; v rt teln( I D[ :3 J=', D[3) : 10 : 8) ; 'vir;te1n(·DI4] »', D[ 4] : 10 : 8) ;

0[2] := 20 *log(D[2]).; D[3] := 20 * log(O[3]); D[4] := 20 * log( D[4]) ;

I vrtteln( 'D(2] =' I D[2] : 4: 1.. 'dB ); vrtteln{ 'D[ 3] «, D[3] : 4 : 1} IdB') ; I vrtteln( 'D(4] =1 I D[ 4] : 4 : 1.. 'dB ); vrttel n{ II) ; {...rtteln( 'The de voltage bias is:', Ide: 8 : 7} , A') ;} end. References

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32-37 J Aug., 1980

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