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Design and testing of time-varying inductors and capcitors for an electrical speech synthesizer Wickwire, Kenneth Freeman 1967

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THE DESIGN AND TESTING OP TIME-VARYING- INDUCTORS AND CAPACITORS FOR AN ELECTRICAL SPEECH SYNTHESIZER by KENNETH FREEMAN WICKWIRE B.Eng.(Hons.), Nova S c o t i a Technical College, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l Engineering We accept t h i s t h e s i s as conforming to the required standard Research Supervisor Members of the Committee Head of the Department Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA December, 1967 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n -t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t i t y o f B r i t i s h C o l u m b i a ^ T h e Un i v e r s  t y V a n c o u v e r 8, C a n a d a ABSTRACT This thesis describes the design and testing of time-varying inductors and capacitors for use i n an e l e c t r i c a l analogue of the human vocal tr a c t . The inductors and capacitors were varied i n accordance with an external control signal by varying the value of a r e s i s t o r i n a c i r c u i t which used operational r amplifiers to simulate a variable impedance. The inductor actually tested i s not a true inductor, since i t s voltage e and current i are related by the equation e(t) = L(t) ^ r , where L(t) i s an externally controlled time function. A device for which e(t) = L(t) ^ w i l l probably be adequate for use i n a vocal tract analogue. A true inductor for which e(t) = ^ (L(t) i ( t ) ) can be realized by making a change i n the c i r c u i t tested. For the i n -ductor tested, the maximum allowable input voltage and current are + 2 volts and + 2 ma, respectively. For the capacitor, the allowable ranges are + 4 volts and + 20 ma. The inductance and capacitance can be varied over a range of 250:1 with good l i n e a r i t y with respect to external control voltage and audio frequency. The inductor's Q exceeds 50 and the capacitor's Q exceeds 200 for a l l frequencies between 200 Hz and 5 KHz. A system for routing control signals from a d i g i t a l computer to the vocal tract analogue has been devised. Each component i n the analogue, i s to be.serviced by the computer at discrete time in t e r v a l s . Between computer service times, the value of each component i s interpolated by the up-down counter-digital com-parator interpolating system described i n the thesis. TABLE OF CONTENTS Page L i s t of I l l u s t r a t i o n s v Acknowledgement v i i i 1. INTRODUCTION 1 1.1 P o t e n t i a l Uses of Speech Synthesizers 1 1.2 Previous Research on Speech Synthesis 1 1.3 Scope of t h i s Thesis 3 2. HUMAN VOCAL TRACT .. ... 4 2.1 D e s c r i p t i o n of the Vocal Mechanism ' 4 2.2 The Basic Speech Sounds 6 2.3 E l e c t r i c a l Analogue of the Vocal Mechanism 8 3. SIMULATION OF TIME-VARYING. INDUCTORS AND CAPACITORS .. . 1 3 3.1 Requirements f o r the Reactive Elements 13 3.2 The Time-varying Inductor 16 3.2.1 Steady State A n a l y s i s 16 3.2.2 Transient A n a l y s i s 19 3.3 The Time-varying Capacitor . ... 21 -4. CONTROL OF THE VOCAL TRACT ANALOGUE 24 4.1 O v e r a l l Plan f o r C o n t r o l l i n g the Inductors and Cap-a c i t o r s 24 4.2 System Components f o r Con t r o l of the Reactive Elements 26 4.3 The D i g i t a l Comparator 29 4.4 The Up-Down Counter 32 4.5 The Binary Averager 32 4.6 Routing the Control Numbers to the Reactive Elements . 35 i i i - Page 5 .' THE TIME-VARYING INDUCTOR . . . • 38 5.1 D e s c r i p t i o n of the C i r c u i t 38 5.2 Testing the Inductor 42 5.2.1 Steady State Inductor Tests 42 5.2.2 Transient Tests on the Time-Varying Inductor... 45 6. THE TIME-VARYING. CAPACITOR 5.6 6.1 D e s c r i p t i o n of the C i r c u i t 56 6.2 Testing the Capacitor 59 6.2.1 Steady State Capacitor Tests 59 6.2.2 Transient Tests on the Time-varying Capacitor . 59 REFERENCES 73 i v LIST OF ILLUSTRATIONS Figure Page 2.1 Diagram of the Human Vocal Mechanism based on a M i d s a g i t t a l S e c tion 5 2.2 Vocal Tract P r o f i l e s f o r the P r i n c i p a l Phonemes i n the E n g l i s h Language 7 2.3 E l e c t r i c a l Analogue of a Short C y l i n d e r 9 2.4 Models of Cascaded Right C i r c u l a r C y l i n d e r s 10 2.5 E l e c t r i c a l Analogue of the Human Vocal Tract .... 12 3.1 Voltage-Current R e l a t i o n f o r Time-varying Capacitors and Inductors 14 .3.2 Riordan's C i r c u i t f o r Simulation of Impedance ... 14 3.3 Simulation of an Ungrounded Impedance 15 3.4 C i r c u i t Models f o r the Ungrounded Impedance i n F i g . 3.3 18 3.5 C i r c u i t Model f o r the Ungrounded Inductor 19 3.6 C i r c u i t f o r which e(t) = L ( t ) ^  20 3.7 Time-varying Capacitor 22 3.8 C i r c u i t Model f o r Equation 3.12 .. 23 4.1 I l l u s t r a t i n g the e f f e c t of the I n t e r p o l a t i n g System 25 4.2 Block Diagram of the I n t e r p o l a t i n g System 25 4.3 Cont r o l of the Inductors and Capacitors 28 4.4 D i g i t a l Comparator ' 30 th 4.5 Truth Table f o r the k d i g i t of the D i g i t a l Comparator .' 31 4.6 3 D i g i t P o r t i o n of the Up-Down Counter 33 4-7 Binary Adder 34 th 4.8 Truth Table f o r k d i g i t of a Binary Adder 35 4.9 Computer System f o r Routing Binary Numbers to D i g i t a l Comparators and V a r i a b l e Rate Clocks .... 37 v Figure Page 5.1 Time-Varying Inductor 39 5.2 Control R e s i s t o r R 38 th 5 . 3 C i r c u i t f o r the k R e s i s t o r i n F i g . 5 .2 , 40 5.4 Control Network f o r FET Switch .' 41 5.5 Normalized Inductance L/Lp vs B f o r Lp = 100h, lOh and l.Oh 43 5.6 Contours of Inductor Q u a l i t y Factor Q.Lp = lOh . 43 5 . 7 Normalized Inductance L/Lp vs f, Lp and B ....... . 44 5 .8 C i r c u i t f o r Inductor Time-Domain Tests 45 5 . 9 Output e Q ( t ) i n F i g . 5 -8 f o r L ( t ) Constant 47 5.10 V a r i a t i o n of Binary Number B c o n t r o l l i n g the Time-varying Inductor 53 5.11 Time-varying Inductor Test Arrangement 53 5.12 C a l c u l a t e d Outputs f o r R-L C i r c u i t i n F i g . 5 .8 f o r the True Time-varying Inductance e(t) = 5.13 Calculated and Measured Outputs f o r R-L C i r c u i t i n F i g . 5 . 8 . e(t) = L ( t ) d i . (a) L = 10h/(a+bt). dt (b) L = 10h/(a-bt) . 55 6 . 1 Time-varying Capacitor 57 6.2 Control R e s i s t o r f o r the Time-varying Capacitor 56 6 . 3 FET Switch D r i v e r 58 6.4 Normalized Capacitance C/Cp vs B f o r Cp = 3750u.u.f 375(iuf and 75|-tuf.' f = 1 KHz 6-0 6 . 5 Contours of Capacitor Q u a l i t y Factor Q . . . 60 6.6 Capacitance C/Cp vs f, Cp and B • 61 6 . 7 C i r c u i t f o r Capacitor Time-Domain Tests 62 6 . 8 Output e n ( t ) f o r Constant' Capacitance 63. v i Figure Page 6.9 , Inputs to R-C C i r c u i t i n . F i g . 6.10 64 6.10 Time-varying Capacitor Test Arrangement 64 6.11 C a l c u l a t e d and Measured Oatput f o r R-C C i r c u i t i n F i g . 6.7' and Case- 1 of F i g . 6.9. (a) bRC = 1.43. Ob) bRC = 1 69 p . p 6.12 Calculated and Measured Outpu^ f o r R-C C i r c u i t i n F i g . 6.7 and Case 1 of F i g . 6.9. (a) bRC = 0.714. (b) bRC =0.5 70 6.13 Calculated and Measured Output f o r R-C C i r c u i t in. F i g . 6.7 and Case 2 of F i g . 6.9. (a) bRC^ = 0.214 (b) bRC =0.5 71 p 6.14 Calculated and Measured Output f o r R-C C i r c u i t i n F i g . 6.7 and Case 2 of F i g . 6.9. bRC = 0.714... 72 v i i ACKNOWLEDGEMENT Tie author would l i k e to thank Dr. R.W. Donaldson f o r h i s help a.id guidance throughout t h i s research p r o j e c t and Professor F.K. Bowers .for reading the t h e s i s . The author i s al s o indebted to.the graduate students who proofread the t h e s i s and e s p e c i a l l y to Mrs. M. Wein who typed i t . The author also wishes to express h i s g r a t i t u d e to the Nat i o n a l Research Council of Canada f o r a Bursary i n 1965 and to the U n i v e r s i t y of B r i t i s h Columbia f o r a Graduate Fellowship i n 1966. v i i i 1. INTRODUCTION 1.1 P o t e n t i a l Uses of Speech Synthesizers This t h e s i s describes the design and t e s t i n g of time-v a r y i n g inductors and cap a c i t o r s f o r an e l e c t r i c a l speech sy n t h e s i z e r . A speech s y n t h e s i z e r would be u s e f u l as an output f o r a computer, a language t r a n s l a t i o n system, a reading machine f o r the b l i n d , and a communication system which encodes and 1 2 transmits speech i n phonemic u n i t s . ' Such a communication system would have to transmit approximately 20 b i t s per second, as compared wi t h the 20 or more k i l o b i t s per second transmitted by conventional v o i c e communication systems. A speech s y n t h e s i z e r would be u s e f u l f o r research on l i n g u i s t i c s and as a c l i n i c a l a i d f o r speech therapy. The knowledge gained i n developing a syn t h e s i z e r would be of use to those working on automatic speech r e c o g n i t i o n . 1.2 Previous Research on Speech Synthesis Speech synthesizers may be d i v i d e d i n t o two cat e g o r i e s : terminal analogues and v o c a l t r a c t analogues. A term i n a l analogue s y n t h e s i z e r attempts to produce an acoust i c s i g n a l whose short-time F o u r i e r transform d u p l i c a t e s that of the desired speech s i g n a l . The f i r s t s u c c e s s f u l terminal 3 4 analogue s y n t h e s i z e r was the Vocoder i n 1936. ' I t produced connected speech; however, t h i s speech lacked naturalness. Further 5 research has improved the q u a l i t y of the speech output. At • present, a computer c o n t r o l l e d t e r m i n a l analogue i s being developed at the Massachusetts I n s t i t u t e of Technology. ' A v o c a l t r a c t analogue attempts to produce speech by s i m u l a t i n g the human v o c a l t r a c e . The pressure and a i r v e l o c i t y • r e l a t i o n s h i p s of the v o c a l t r a c t can be simulated by a time-8 9 v a r y i n g non-uniform transmission lane. ' S t a t i c e l e c t r i c a l 9 10 analogues ' which produce vowels and c e r t a i n consonants have been b u i l t , but these analogues do not produce connected speech. A time-varying e l e c t r i c a l analogue which synthesized c e r t a i n vowel-consonant and consonant-vowel sequences was b u i l t by Rosenj"^" however, l i m i t a t i o n s of h i s system f o r c o n t r o l l i n g the analogue made i t d i f f i c u l t to synthesize longer phoneme sequences. 1 3 , 1 4 The work described i n t h i s t h e s i s i s the beginning of an e f f o r t to "synthesize speech by c o n t r o l l i n g the components of an e l e c t r i c a l analogue of the human v o c a l t r a c t by a high-speed d i g i t a l computer The p r i n c i p a l components of the analogue are time-varying inductors and ca p a c i t o r s whose inductance and capacitance vary i n accordance with an e x t e r n a l c o n t r o l s i g n a l . There are devices which simulate t i m e - i n v a r i a n t inductance and capacitance, 1 5 , 1 6 , 1 7 11 12 but to the author's knowledge, only Rosen ' has b u i l t i nductors and ca p a c i t o r s whose t i m e - v a r i a t i o n i s c o n t r o l l e d by an ex t e r n a l v o l t a g e . For h i s inductor Rosen used a saturable r e a c t o r For h i s c a p a c i t o r he used a v a r i a b l e - g a i n a m p l i f i e r with a ca p a c i t o r i n the feedback path. In the saturable r e a c t o r the i n -ductance i s c o n t r o l l e d by using an a u x i l i a r y winding to c o n t r o l the degree of s a t u r a t i o n i n an i r o n core. The a u x i l i a r y winding r e q u i r e s a non-linear current d r i v e r to compensate f o r the non-l i n e a r i t y of s a t u r a t i o n i n the i r o n . The ca p a c i t o r used by Rosen i s undesireable because a high supply voltage i s required to 3-prevent input voltages i n excess of one v o l t from s a t u r a t i n g the a m p l i f i e r when i t s gain i s high. 1.3 Scope of t h i s Thesis This t h e s i s describes the design and t e s t i n g of a time-v a r y i n g inductor and c a p a c i t o r . The t i m e - v a r i a t i o n depends on t.n ex t e r n a l time-varying c o n t r o l voltage. Although the inductor and c a p a c i t o r are f o r use i n an e l e c t r i c a l analogue of the human vo c a l t r a c t , they are not r e s t r i c t e d to t h i s context. The inductor a c t u a l l y t e s t e d i s not a true time-varying inductor, since the voltage e and the current i are r e l a t e d by the equation e(t) = L ( t ) ^ . As shown i n Chapter I I I , a device f o r which e(t) = L ( t ) ^ w i l l probably be adequate f o r use i n a vo c a l t r a c t analogue. A true time-varying inductor f o r which e( t ) = ^ ( L ( t ) i ( t ) ) can be r e a l i z e d by making a minor change i n the c i r c u i t t e s t e d . • The inductance and capacitance of the devices tested can be v a r i e d over a range which exceeds 250:1 w i t h good l i n e a r i t y with respect to e x t e r n a l c o n t r o l voltage and audio frequency. The maximum allowable input voltage and current i s + 2 v o l t s and + 2 ma f o r the inductor and + 4 v o l t s and + 20 ma f o r the ca p a c i t o r . The inductor's Q exceeds 50 and the c a p a c i t o r ' s Q exceeds 200 f o r a l l frequencies between 200 Hz and 5000 Hz. The most expensive elements i n the inductor and ca p a c i t o r are the op e r a t i o n a l ampli-f i e r s ; the inductor r e q u i r e s four and the c a p a c i t o r , which has one ter m i n a l grounded, re q u i r e s two. I f both c a p a c i t o r terminals are to be ungrounded, two a d d i t i o n a l o p e r a t i o n a l a m p l i f i e r s are required. 4. 2. OPERATION AND SIMULATION OP THE HUMAN VOCAL MECHANISM 2.1 D e s c r i p t i o n of the Vocal Mechanism Speech i s produced by c o n t r o l l e d movements of the vocal-mechanism. This mechanism, shown i n c r o s s - s e c t i o n i n P i g . 2.1, c o n s i s t s b a s i c a l l y of the lungs, trachea, v o c a l cords, v o c a l t r a c t , and n a s a l t r a c t . The lungs, trachea and' v o c a l cords provide an a i r v e l o c i t y source f o r the v o c a l t r a c t . The s l i t - l i k e o r i f i c e between the v o c a l cords i s the g l o t t i s . The instantaneous v e l o c i t y of the a i r passing through the g l o t t i s i s approximately p r o p o r t i o n a l to the area of the g l o t t a l opening, independent of the shape of the v o c a l and n a s a l t r a c t s . The v o c a l t r a c t i s a tube which i s non-uniform i n c r o s s -s e c t i o n a l area. I t begins at the v o c a l cords and terminates at the l i p s . The t r a c t of an adult male i s approximately 17 cm. long. The forward p o r t i o n of the t r a c t may be v a r i e d i n c r o s s - s e c t i o n a l 2 area from zero to approximately 20 cm . The v a r i a t i o n i n c r o s s -s e c t i o n a l area of the t r a c t near the g l o t t i s i s very s m a l l . In the region near the mouth, the len g t h of the t r a c t also, v a r i e s during speech production. The n a s a l t r a c t i s s i m i l a r to the v o c a l t r a c t , except that i t s c r o s s - s e c t i o n a l area i s f i x e d . For an adult male the nasal t r a c t i s approximately 12 cm. long and has a volume of approximately 60 cc. The degree of coupling between the v o c a l and nasal t r a c t s i s c o n t r o l l e d by the s i z e of the opening at the velum. 5. TONGUE \ \ LIPS \ PHARYNX / Boundaries of the vocal tract during the pro-GLOTTIS duction of the vowel [e]. A midsagittal sec-tion and two transverse sections are shown (from T. Chiba and M. Kajiyama 1 3 ). Fig.2.1. Diagram of the human vocal mechanism based on a midsagittal section. The g l o t t i s i s the primary source of e x c i t a t i o n . In voiced sounds, i t opens and closes at the r a t e of approximately 125 Hz. In unvoiced sounds, a i r i s blown through the g l o t t i s which, i n t h i s case, remains open and does not v i b r a t e . Secondary sources of e x c i t a t i o n may appear w i t h i n the v o c a l t r a c t . They may r e s u l t from turbulence created by a narrow con-s t r i c t i o n i n the t r a c t or by a closure followed by a sudden release of a i r . 19 2.2 The Basic Speech Sounds The ba s i c u n i t of speech i s the phoneme. Phonemes are c l a s s i f i e d according to t h e i r manner and place of production i n the v o c a l t r a c t . . The approximate v o c a l t r a c t c o n f i g u r a t i o n s f o r the p r i n c i p a l E n g l i s h phonemes appear i n F i g . 2.2. During the production of vowels, the v o c a l t r a c t remains i n a r e l a t i v e l y f i x e d c o n f i g u r a t i o n with no nasal coupling. A l l vowels are vo i c e d , and the a i r flow through the v o c a l t r a c t i s non-t u r b u l e n t . The consonants include a l l other phonemes except diphthongs and a f f r i c a t e s , which are combinations of vowels and consonants. F r i c a t i v e consonants are produced by a c o n s t r i c t i o n which causes a i r turbulence i n the v o c a l t r a c t . They may be e i t h e r voiced or unvoiced. Stop consonants are produced by the sudden release of pressure b u i l t up behind a complete closure at some point i n the v o c a l t r a c t . They may be e i t h e r voiced or unvoiced. f (FOR) 0 (THIN) S (SEEl 1 (EVE) I (IT) e (HATE) e(WET) S(SHE) 5 (THEN) h (HE) z(zoo) V (VOTE) 3 (AZURE) V o c a l t r a c t prof i les for the f r i t a t i v c c o n s o n a n t s of E n g l i s h . T h e short p a i r s of l ines d r a w n o n t h e t h r o a t represent v o c a l cord o p e r a t i o n [ a d a p t e d f r o m P o n t R , K o p p a n d G R E E N ) ae (AT) u (FOOT) a ( FATHER) U (BOOT) 0(ALL) A(UP) 0 (OBEY) $ (BIRD) S c h e m a t i c v o c a l t ract prof i les for the p r o d u c t i o n of E n g l i s h v o w e l s (adapted f r o m P O T T E R , . K o r p a n d G R E E N ) W (WE) j (YOU) r (READ) 1 (LET) V o c a l t ract c o a l i g u n t i o n s for the b e g i n n i n g p o s i t i o n s of the gl ides a n d s e m i v o w e l s (after P O T T E R , . K o p p a n d G R E E N ) m (ME) n (NO) T) (SING) V o c a l prof i les for the n a s a l consonants (after P O T T E R , K O P P a n d G R E E M ) P (PAY) t» (SE) t (TO) d ( DAY ) X(KEY) g (GO) A r t i c u l a t o r ) - p r o f i l e s for the E n g l i s h s t o p c o n s o n a n t s (after P O T T E R , K O P P a n d GRE F i g . 2.2 Vocal Tract P r o f i l e s f o r the P r i n c i p l e Phonemes i n the E n g l i s h Language 20 8. Nasal consonants, which are vo i c e d , are produced when the f r o n t p o r t i o n of the v o c a l t r a c t i s closed and the velum i s opened. In t h i s case, a i r passes along the nasal t r a c t and out through the n o s t r i l s . G l i d es and semivowels are s i m i l a r to vowels and are voiced. Glides are dynamic sounds which r e s u l t from movement of the v o c a l t r a c t towards or away from a vowel c o n f i g u r a t i o n . The o r a l channel i s more c o n s t r i c t e d i n the semivowels than i n vowels, and the tongue t i p i s up. The diphthongs and a f f r i c a t e s are combination sounds. The diphthong- i s a combination of -two vowel sounds, and the a f f r i c a t e i s a combination of a stop and f r i c a t i v e consonant. 2.3 An E l e c t r i c a l Analogue of the Vocal Mechanism A l l speech sounds depend on the v o c a l t r a c t c o n f i g u r a t i o n , g l o t t a l e x c i t a t i o n and degree of coupling w i t h the nasal t r a c t . By making an e l e c t r o n i c analogue of the v o c a l t r a c t and by c o n t r o l l i n g the analogue by a d i g i t a l computer, i t should be p o s s i b l e to synthesize connected speech. The v o c a l t r a c t may be approximated as a cascade of short 8 Q ' r i g h t c i r c u l a r c y l i n d e r s ' . Two e l e c t r i c a l analogues f o r a c y l i n d e r who's leng t h $ i s much l e s s than the highest sound wave-le n g t h , appear i n Fig.'2.3. Voltage i s analogous to pressure and current to volume v e l o c i t y . Inductance L i s analogous to the inertance of the a i r mass. Capacitance C i s analogous to the com-pl i a n c e of the a i r volume. The r e s i s t a n c e R represents the power d i s s i p a t e d i n viscous f r i c t i o n at the tube w a l l , and the conductance 9. Right-Circular Cylinder ( Cross-sectionalaarea A Circumference S C 2 =r R . L 'ggco^ — G G 2 2 C =r 2 R I 2 2 L R 2 2 -TTJTiO 'VV-n section; - - 4 R = T section 0 = /" 2 \0) 2 V 0) = C = a i r density v i s c o s i t y c o e f f i c i e n t of a i r radian frequency s p e c i f i c heat of a i r at constant pressure c A \ speed of sound i n a i r adiabatic constant of a i r c o e f f i c i e n t of heat conduction Pig. 2.3 E l e c t r i c a l Analogues of a Short Cylinder G represents the power loss due to heat conduction at the tube wall. In the vocal t r a c t , R/coL and G/coC are small f o r the frequencies of interest and R and G may be neglected. An analogue of the vocal tract i s obtained by cascading T or it sections corresponding to adjacent cylinders, as shown i n F i g . 2.4. -One Section > One Sect lot) (.a) •One Section — kAu,k -One Section-fit ZKAL ns$®r> 1 =£> - = 1—r7-sdvm^ 1 I '<T30cr> kA'*iL+i < One Section ?- < One Section > F i g . 2.4 Models of Cascaded Right C i r c u l a r C y l i n d e r s (a) % s e c t i o n s ; separate and combined (b) T s e c t i o n s ; separate and combined 11. Constant k . i s an a r b i t r a r y constant which i s chosen to make the L and C correspond to p r a c t i c a l e l e c t r i c a l values. Since the c r o s s - s e c t i o n a l area of most of the nasal passage i s constant, i t i s simulated mainly by f i x e d components (see P i g . 2.5). The degree of coupling to the v o c a l t r a c t depends on the v a r i a b l e inductor and c a p a c i t o r nearest the j u n c t i o n of the nasal stub and the vo c a l t r a c t . The g l o t t i s acts as an a i r v e l o c i t y source, the v e l o c i t y being p r o p o r t i o n a l to the area of the g l o t t a l opening. In the analogue the g l o t t i s becomes a current source whose current ' i ( t ) = A(t) "f^Vso/f . C onstant Pso i s the mean value of the s u b - g l o t t a l pressure and A(t) i s the area of the g l o t t a l opening. To simulate turbulance or a sudden release of pressure which f o l l o w s a c l o s u r e , a voltage source i s placed i n s e r i e s with one of the ungrounded inductors i n the analogue. In s i m u l a t i n g turbulance the voltage source i s a white noise source. A suddenly t a p p l i e d decaying exponential of the form v ( t ) = E e t i s used to simulate closure followed by r e l e a s e . To a f i r s t approximation, the mouth and n o s t r i l s appear to the v o c a l t r a c t as plane v i b r a t i n g s u r f a c e s , a l l parts of which move i n phase. The r a d i a t i n g element i s set i n a b a f f l e that i s the head. To the analogue, the mouth and n o s t r i l s appear as a la r g e constant r e s i s t o r i n p a r a l l e l with a small inductor whose inductance i s k^o / '\fitK~. Scale f a c t o r k i s i d e n t i c a l to the scale f a c t o r i n F i g . 2.4 and A i s the c r o s s - s e c t i o n a l area of the mouth or nose opening. Re sis tor-Fig. 2.5 E l e c t r i c a l Analogue of the. Human Vocal Tract. Inductors, Capacitors and E x c i t a t i o n Voltage and Current Sources are to be C o n t r o l l e d by a D i g i t a l Computer H ro 13. 3. SIMULATION OF TIME-VARYING- INDUCTORS AND CAPACITORS 3.1 . Requirements f o r the Reactive Elements The instantaneous inductance L of a two-terminal device i s defined as the r a t i o of the f l u x l i n k a g e s X of the device to the current i passing through the device. The voltage e across the device i s equal to the time d e r i v a t i v e of the f l u x l i n k a g e s . The instantaneous capacitance C of a two t e r m i n a l device i s defined as the r a t i o of the charge q contained i n the device to the voltage e across the two t e r m i n a l s . The current passing through the device i s equal to the time d e r i v a t i v e of the stored charge. The s i m i l a r i t y between these two d e f i n i t i o n s i s i n d i c a t e d i n F i g . 3.1. The s p e c i f i c a t i o n f o r the inductor and c a p a c i t o r were decided upon a f t e r c o n s i d e r i n g a previous work by Rosen, and a f t e r d e c i d i n g that s o l i d s t a t e c i r c u i t r y would be used. As a r e s u l t of t e s t s on a s t a t i c v o c a l t r a c t analogue, Rosen,"'""'" found that the maximum le n g t h of a c y l i n d e r s e c t i o n should be 1.5 cm. S a t i s f a c t o r y vowels have been produced by using, only an area r a t i o of 29:1, but i t was decided to have a f u l l range of 250:1 i n the dynamic analogue. This means that the inductors and c a p a c i t o r s must vary over a 250:1 range. The Q ( q u a l i t y f a c t o r ) of the r e a c t i v e elements was chosen to be greater than 50. Rosen *^ showed that a Q equal to or greater than 30 was adequate. The Q can always be reduced by p l a c i n g r e s i s t o r s i n s e r i e s w i t h the inductors and i n p a r a l l e l with the c a p a c i t o r s . Because of the l a r g e dynamic s i g n a l range expected i n the v o c a l t r a c t analogue, i t was decided that the maximum allowable input voltage and current at the terminals of the r e a c t i v e 14. \ = L i q = Ce e dA _ d ( L i , dt ~ dt T d i . dL dt + 1 dt 1 = do _ d(Ce; dt - dt pd e dC 0 dt + e dt Inductance Capacitance F i g . 3-1 Voltage-Current R e l a t i o n s f o r Time-Varying Capacitors and Inductors 2* A W F i g . 3.2 Riordan's C i r c u i t f o r Simulation of Impedance .15. element's: should be as large as possible subject to the constraint' that standard s o l i d state components be used to r e a l i z e these elements. I n i t i a l experiments showed that i f 15 volt operational amplifiers wore used, the input voltage could be as large as + 2 volts for both the inductor and capacitor. The maximum input current was ± 2 ua for the inductor and ± 20 ma for the capacitor. 3.2 The Time-Varying Inductor 3.2.1 Steady State Analysis Many c i r c u i t s for simulating time-invariant inductors have been considered, 15,16,17 ^ u t n o n e 0 f these could be extended to the simulation of time-varying inductors. The required range of inductance or the required Q could not be real i z e d , or the maximum input voltage and current were too small. The basic c i r c u i t 21 f i n a l l y chosen was one used by Riordan to produce an adjust-able time-invariant inductor. Experimentation showed that the i n -ductance range of 250:1 could be realized by varying only one re s i s t o r . Riordan's c i r c u i t appears i n Pig. 3-2. Assuming that the current into the amplifier inputs i s neg l i g i b l e , and that A^ >>1 and A2>>1, i t can be shown that The input current I i s the current i n Z^, so that E. Z-, Z^,ZC h ' - I - ^ f <3.3) If either or Z^ i s a capacitor and i f the remaining Z's are r e s i s t o r s , then the input impedance i s inductive. •16 k This c i r c u i t , as i t stands, i s not suitable for use i n a vocal tract analogue because one input terminal i s grounded. To obtain" an ungrounded inductor, two i d e n t i c a l c i r c u i t s are placed i n the back-to-back configuration of F i g . 3 . 3 . • 2 5 F i g . 3 .3 Simulation of an Ungrounded Impedance Analysis of the c i r c u i t i n F i g . 3«3 i s sim i l a r to that of the grounded inductor, i n Fig. 3«2 17. and Let and E, = E, = E 1 3 u E., = E-, --- E, 1 3 d E4 = E u " Ift <Eu - V 1 3 and I = 7 % % (E - E, ) u Tj^TjJLr- u d' 1 3 5 i t Z Z S i m i l a r l y I = * (B - E ) 1 3 5 Z 1 Z 3 Z 5  Z 2 Z 4 Z 1 Z 3 Z 5  Z 2 Z 4 (E - E,) Then l u = u z d (3.5) (E ' - E,) and • i d = u z , d (3.5) Equations 3-5 and 3.6 suggest the c i r c u i t model of P i g . 3.4a. Note that current A l = I - I , r e s u l t s from a mismatch between Z u d and Z*. I f Y = \ and Y" = \j ,.then F i g . 3.4b becomes the Norton Z x - Z equivalent of F i g . 3-4a. Let Y = Y' + AY Then YE, = Y'E, + AYE,,, : d d d 18. (cO • W W >— 7 - Z- Z.? Zs - > — Id _>— • Y ( |)YEct Y'EM(T) <Y' Y -U In AY= Y-Y' > A Y Aid" Ed 1>YEJ ?Y' © r t d (|)Y'E, fy' AT I--O f i d ^ = AY 1A_ Ed Fig. 3.4 C i r c u i t Models for the Ungrounded Impedance in Fig. 3.3 1 9 . The c i r c u i t model appears as i n F i g . 3 . 4 c . Further rearrangements r e s u l t i n the model i n F i g . 3 - 4 d , where AZ = AY Therefore AZ = Z 3 Z 3 Z 4 Z 4 1 Z 2 Z 3 Z 4 Z 5 Z 2 Z 3 Z 4 Z 5 (3 .7) Equation 3 . 7 shows that the l a r g e r the mismatch between Z^Z^Z^Z^, and ZIZ^Z^Z,-, the smaller the magnitude of AZ. Therefore, Z^ i s chosen to be the v a r i a b l e r e s i s t o r , and Z„Z'Z.Z' i s matched as 2 3 4 5 c l o s e l y as p o s s i b l e to Z^Z^Z^Z,-. I f Z^ and Z£ i n F i g . 3 - 3 are cap a c i t o r s and a l l other impedances are r e s i s t o r s , then the r e s u l t -i n g ungrounded inductor can be modelled as i n F i g . 3 . 5 . Eu Ed Ju >-L. L C2 R, R3 Si AL . Rafts CzCz F i g . 3 - 5 C i r c u i t Model f o r the Ungrounded Inductor 3 . 2 . 2 Transient A n a l y s i s I f i n F i g . 3 - 1 i s a time-varying r e s i s t o r , R-^(t), and Z^ a c a p a c i t o r , the c i r c u i t does not behave as a true time-v a r y i n g inductor. 20. R5 C R R Pig. 3.6 C i r c u i t for which e(t)=L(t) .L(t)=R, (t) 2r? ^ ax JL i t ^ If the c i r c u i t i n Pig. 3..6 i s at rest for t = - <x=> , then the input current i i s -t r e. RTTtT d t C0R„R,- ; LV-. 2 3 5 J_ ^  1 D i f f e r e n t i a t i o n and rearrangement gives e ± = L(t)-'|| , (3.10) C R R where L(t) =R,(t) 2 ^ (3.11) 4 Equations 3.10 and 3.11 also'apply to the c i r c u i t i n Pig. 3.3, provided Z± = R±(t), Z 2 = Z£ = C 2, Z = Z^ = R Z^= = R 4 and 21. and Z c = Z» = R c 5 5 5 A true ungrounded time-varying inductor i s r e a l i z e d i f e i t h e r R„ and Rl or R,_ and R' , r a t h e r than Rn , are v a r i e d 3 3 5 5 1 simultaneously. More c i r c u i t r y i s required to vary R^ and R^ . or R^ and R^ L, and considerable mismatch caused, by l o s s i n synchroniz-a t i o n between r e s i s t o r p a i r s could r e s u l t . In eqn. 3.10 the term i ^ i s absent. In the v o c a l t r a c t analogue, t h i s term i s p r o p o r t i o n a l to the r a t e of change of A/£ The term L ^ depends on the r a t e of change of inductor current.' I f . i ( t ) = 1 cos (wt + 0 ) , then ' |l |^ | ' = | Iw^oi /kA.|, . dL 1 dt k ^ A ) d t , ana i d t |/|L ^ dt / I dt LOA dt In the human v o c a l t r a c t , the term wA~ ^ d t ^ l << 1> e x c e p t when the area A changes suddenly. A sudden change occurs, f o r example, i n moving from a vowel to a stop consonant. Such changes occur between phonemes, and are present f o r only a few m i l l i s e c o n d s . I t i s b e l i e v e d that the i n t e l l i g i b i l i t y of synthesized speech w i l l not be n o t i c a b l y impaired by using inductors f o r which e = L(t) | | . 3-3 The V a r i a b l e C a p a c i t o r The v a r i a b l e c a p a c i t o r was made from the same basic c i r c u i t as the i n d u c t o r . The c a p a c i t o r has the advantage that i n the analogue one t e r m i n a l i s grounded. As before, the equation f o r the input impedance i s eqn. 3.3. Laboratory experiments showed that i t was best to choose Z^ as the c a p a c i t o r , as the v a r i a b l e R-^(t), and the remaining impedances as f i x e d r e s i s t o r s , as shown i n F i g . 3.7. 22. F i g . 3.7 Time-Varying Capacitor I f R-^  i s a f u n c t i o n of time, the c a p a c i t o r behaves as a true time-varying c a p a c i t o r . In a n a l y z i n g the c i r c u i t , consider e4 the r a t i o ^ when the c a p a c i t o r C^ i s removed. Because the components are a l l r e s i s t o r s , eqn. 3.2 can be used d i r e c t l y to give R R e4 = e i ( 1 - R ^ T t l R ^ ( 3 ' 1 2 ) A model f o r eqn. 3.12 i s shown i n F i g . 3-8, where d(Ke.) d(Ce.) i = Cr 1 1 5 dt - dt C R R 0(t) = K(t)C 5 = g ^ f g A . ( 3 . 1 3 ) 23-i >-r" F i g . 3-8 C i r c u i t Model f o r Equation 3-12 The equations f o r the v o c a l t r a c t analogue i n F i g . 2.3 show that the inductance i s i n v e r s e l y p r o p o r t i o n a l to k/JL and the capacitance i s d i r e c t l y p r o p o r t i o n a l to k£ . I f i s i n v e r s e l y p r o p o r t i o n a l to k/i then L = K T C0R„R,-_1_ 2 3 5 A// R/i K f o r R. = L 1 " A// where i s a constant. I f C i s p r o p o r t i o n a l to A I C = °5R2 R4 A £ R 3 K f o r R-, = C 1 - ki where i s constant. I t w i l l be seen i n Chapter 4 that because C i s p r o p o r t i o n a l to A, and L i s i n v e r s e l y p r o p o r t i o n a l to A, the c o n t r o l of the v o c a l t r a c t analogue i s s i m p l i f i e d , when the lengths £ of adjacent sections i n F i g . 2.4 are constant and equal, 24. 4. CONTROL OP THE VOCAL TRACT ANALOGUE 4.1 O v e r a l l Plan f o r C o n t r o l l i n g the Inductors and Capacitors The v o c a l t r a c t analogue i s to be a cascade of L-C sec t i o n s which simulate c y l i n d e r s of leng th 2. and time-varying c r o s s - s e c t i o n a l area A ( t ) , as shown i n P i g . 2.5- I t v f o l l o w s from P i g . 2.4 that f o r each s e c t i o n the product LC = / /c must be maintained. During speech production the c r o s s - s e c t i o n a l area at each point along the v o c a l t r a c t changes w i t h time. In the e l e c t r i c a l analogue, the c o n t r o l of the components i s to be by d i g i t a l computer. I t i s d e s i r a b l e to minimize the number of times per second that the computer s e r v i c e s the analogue. The computer w i l l then be able to devote more of i t s time to c a l c u l a t i n g the required parameter s e t t i n g s from stored phoneme sequences. To reduce the number of computer s e r v i c e c a l l s r e q u i r e d , an i n t e r -p o l a t i n g system was b u i l t . Between s e r v i c e c a l l s , t h i s i n t e r -p o l a t i n g system causes each r e a c t i v e element to vary with time i n a stepwise f a s h i o n , as shown i n F i g . 4.1 f o r a ca p a c i t o r . The i n t e r p o l a t i n g system makes stepwise approximations of s t r a i g h t l i n e s which are used f o r piecewise l i n e a r approximations of curves. I t i s de s i r e a b l e to use the analogue as a s t a t i c analogue as w e l l as a dynamic one. For t h i s reason the i n t e r -p o l a t i n g system was designed to make a r e a c t i v e element move t o -wards a s p e c i f i e d value and then remain there u n t i l a new value was s p e c i f i e d . 25. P i g . 4.1 I l l u s t r a t i n g the e f f e c t of the i n t e r p o l a t i n g system Up-Down Count Control • '• ->» , P i g . 4.2 Block Diagram of the I n t e r p o l a t i n g System During each s e r v i c e c a l l , the computer w i l l d e l i v e r to the element s e r v i c e d an eighteen b i t binary number. Eight b i t s of t h i s number w i l l s p e c i f y the new value towards which the element i s to proceed. Ten b i t s • J i l l s p e c i f y the rat e l / A t at which the element' i s to proceed to i t s new value. For sect i o n s where both A and i vary, the computer w i l l c o n t r o l the inductors and cap a c i t o r s separately. For sections where $ i s constant and only A v a r i e s i t w i l l be shown that the c o n t r o l can be s i m p l i f i e d . 4.2 System Components f o r Control of the Inductors and Capacitors An inductor or c a p a c i t o r 'whose instantaneous value depends on a binary number can be made to move towards a s p e c i f i e d value such as C(t^) i n F i g . 4.1 with the a i d of the d i g i t a l comparator and an up-down counter. In F i g . 4-2, the r e a c t i v e element's value depends on the binary number i n the counter. The same binary number i s al s o stored i n one side of the d i g i t a l comparator. The new binary number towards which the inductor or c a p a c i t o r i s to proceed i s placed i n the memory of the other side of the com-parator. The d i g i t a l comparator compares the new number with the number i n the up-down counter. I f the new number i s l a r g e r , then -the . d i g i t a l comparator t e l l s the up-down counter to count up, u n t i l the number i n the up-down counter equals the new number. I f the new number i s smaller, the up-down counter counts down. The ra.te at which the counter counts depends on the computer-con-t r o l l e d c l ock r a t e . When the lengths and $± + \ °^ "^wo adjacent sections i n F i g . 2.4a are constant and equal, the combined capacitance C 21, between these sections i s , f) A. + A. -, « ~ f> C 2 2 Although the inductors and Lj_ +-j_ must be c o n t r o l l e d independently, )ap-the c o n s t r a i n t LC = j? /C. i s maintained by c o n t r o l l i n g the CE a c i t o r C between the two adjacent sections w i t h the average of the instantaneous value of the binary number f o r the two adjacent i n d u c t o r s , as shown i n P i g . 4.3a. When and i n Pi g - 2.4b are constant and equal, the combined inductance between these sections i s T _ 1(2- . 1 ) ^TT 2 lA. + A. / - [2A.A. n/(A. +A. n )] l i + l L l i + l ' l i + l ^ Although the cap a c i t o r s and must be c o n t r o l l e d i n -2 2 dependently, the c o n s t r a i n t LC = jl /C i s maintained by con-t r o l l i n g the inductor between two adjacent sections with the average of the instantaneous value of the binary number f o r the two adjacent c a p a c i t o r s as shown i n P i g . 4.3b. The c o n t r o l number f o r the combined inductance should be p r o p o r t i o n a l to 2k.k. n/(A.+A. , ) . A device which averages A. * * i i + l ' i i + l y TO I and A. , w i l l approximate the c o n t r o l number with (A.+ A. ,)/2. i + l * r l i + l ' For c o n t r o l numbers d i f f e r i n g by small r a t i o s , the two q u a n t i t i e s are almost equal. The e r r o r i s 12.5% f o r an area r a t i o A. -,/A. of 2. The c o n t r o l number f o r the combined capacitance which i s p r o p o r t i o n a l to (A^+A..+-^)/2 i s r e a l i z e d e x a c t l y using an averager. For t h i s reason the scheme i n F i g . 4.3a w i l l be used i n preference to the one i n F i g . 4.3b. O n s TTSection n Up-Dovsn Counter PatP thctt gijoil Com ^orator Element | Number " T Number ' From Computer iRote Number One TTSkr.fi on PzrOTC^ -j Bihar y Averctger- Up-DowrX Count 4 ConTrvl'-I Up-Dov,n 'Counter ( Pigir&l Comporotcr J Variable. Element Number (a) ion Number From Computer iiRate Number -One 7TSe-ction-Bittary Averager Up-Down 'Counter ) ( Pigi'tci! Comparator One T Section One T Sect ion Section Number )\ Fro m Computer Element Number (b) Section Number From Computer CO Fig. 4-3. Control of the Inductors and Capacitors (a) it sections (b) T sections • 29. 4.3 The D i g i t a l Comparator A block diagram of the d i g i t a l comparator designed and teste d i s shown i n F i g . 4.4- F a i r c h i l d uL-923 J-K f l i p - f l o p s and F a i r c h i l d u.L-914 dual two-input NOR gates were used i n the a c t u a l c i r c u i t . Each d i g i t Xj\ of the binary number X = \ 2 i s .k=o compared w i t h i t s corresponding d i g i t Y^ . of the binary number Y = \ 2 Y, . A d e c i s i o n i s f i r s t made as to whether X , i s k=0 . . . • . l a r g e r than, smaller than, or equal to Y- I f X_ / Y o • > • n-1 . n-1 ^ -n-1 then the up-down counter i s i n s t r u c t e d to count i n the d i r e c t i o n to make X , = Y . When X , = Y , X „ and Y „ are com-n-1 n-1 n-1 n-1' n-2 - n-2 pared and the r e s u l t determines whether the counter counts up, down, or remains unchanged. This process i s repeated u n t i l X=Y. The way i n which the d i g i t a l comparator was designed can th be understood w i t h the help of F i g . 4.5. The k d i g i t must have t h the t r u t h t a b l e of F i g . 4.5, where and D^ are the k d i g i t s ' d e c i s i o n that the counter must count up. or down r e s p e c t i v e l y . C ontrol d i g i t Z^._-^  prevents comparison of l e s s s i g n i f i c a n t d i g i t s from i n f l u e n c i n g the counter u n t i l X^ . '= Y^ .. The t r u t h t a b l e of F i g . 4.5 may be expressed i n boolean form as f o l l o w s : IL. = Z, X, % k k k k D, = Z, X. Y, k k k k Z, n = Z. U~ D~ k-1 k k k 31. Module 2fe Xk Vk DK - l 0 0 0 0 0 0 o o 0 0 0 o 1 0 0 0 0 o 1 1 0 0 0 1 0 0 0 0 1 1 0 \ 0 1 0 I 1 0 1 0 0 1 1 1 0 0 1 F i g . 4.5 Truth table for the k d i g i t of the d i g i t a l comparator It follows that u k = z k + x k + Y k D k = zk + x k + Y k -1 - z k + u k + D k . 32. Pig. 4.4 i s a r e a l i z a t i o n of the truth table i n F i g . 4 .5 . 4.4 The Up-Down Counter F i g . 4.6 shows the schematic for three d i g i t s of an up-down counter. The counter was designed by Austin and i s described 22 in d e t a i l i n his thesis. It i s b a s i c a l l y a row of J-K f l i p - f l o p s , each f l i p - f l o p being toggled by the same clock pulse. When U=0 and D=l, the counter counts up; when U=l and D=0, the counter counts down. When U=D=1, the counter does not change. The state of each f l i p - f l o p . i s controlled by 3 NOR gates. In counting up, these gates do not permit a f l i p - f l o p to change to the ONE state unless a l l the lesser s i g n i f i c a n t d i g i t s are i n the ONE state. In counting down, the NOR gates prohibit a f l i p - f l o p from changing to the ZERO state unless a l l the lesser s i g n i f i c a n t d i g i t s are i n the ZERO state. When the counter i s to remain un-changed, then the J-K r a i l s are both made po s i t i v e . 4.5 Binary Averager . A binary averager may be rea l i z e d by s h i f t i n g the output of a binary adder one d i g i t towards the least s i g n i f i c a n t b i t . th Fi g . ,4.7 shows the k d i g i t component of a binary adder which was th b u i l t and tested. In Fi g . 4-7, X, and Y, are the k d i g i t s of n-1 n-1 X = ^  I 2 ^ X k a n (^ Y = ^ t ^ ^ k ' •w^ 1-0''1 a r e ^ e ^ w 0 n u m h e r s being k=0 k,-0 • added, and j _ s the k^*1 d i g i t of the sum.' Dig i t Z^ . i s the carry from the sum of the less s i g n i f i c a n t d i g i t s . F i g . 4.8 shows the th truth table for the k d i g i t . •'own Rail Var i a bid Rale Clock Pulse Fig. 4.6 3 D i g i t Portion of the Up-Down Counter F i g . 4.7 Binary Adder. S = X + Y 35. Yk 2k SK 0 0 0 O 0 o i 0 1 \ \ 0 1 D O 1 0 0 1 0 0 1 I 0 0 1 1 1 0 1 1 1 1 0 0 1 1 1 1 F i g . 4.8 Truth Table for k d i g i t of a Binary Adder Analysis of F i g . 4.7 show that Z. , = X, Y, + Y, Z, + Z, Y, k+l k k k k k k and S, = (Z, + X,Y, + X.Y, )Z, n+ X,Y, Z, k k k k k k k+l k k k Since d i g i t s and S^ . have the same truth table as F i g . 4.8, the c i r c u i t i n F i g . 4.7 i s a binary adder. 4.6 Routing the Control Numbers to the Reactive Elements The instantaneous value of each reactive element i n the vocal tract analogue depends on the binary number i n the J-K f l f l o p memories of the d i g i t a l comparator and the variable rate clock. In the f i n a l version of the analogue, these J-K f l i p -flops memories are to be arranged i n matrix form as shown i n . . . • 36. th t h F i g . 4.9- The i row corresponds to the i r e a c t i v e element. The j " ^ 1 column corresponds to the j ^ * 1 binary d i g i t of the binary t h number that goes to the i element. The J-K terminals i n the th 3 column are common, and are connected through b u f f e r s to the output terminals of the j ^  f l i p - f l o p i n the computer in-out th r e g i s t e r . Tht, j d i g i t w i l l be t r a n s f e r r e d from the computer to th any toggled J-K f l i p - f l o p i n the j column. By t o g g l i n g a l l th J-K f l i p - f l o p s i n the. i row, the device s e l e c t o r t r a n s f e r s the binary number i n the computer to the J-K memories of the t h d i g i t a l comparator and the clock of the i element. The parameters of the analogue's e x c i t a t i o n sources w i l l be c o n t r o l l e d i n the same manner. Column O boo J T fc± bi o J T K • Di'gi'tc>l Comparator > j •< Column I Column 8 I Column 9 Bo! J T K T ? 1 ^ 1 r fc>05 J T K £——t——5-A I r\ Us J T K j r < J T K VoiKaWc Clock R a f e Column -1 WW J T « Diq hiq J T K 1 T K; FF 9 C amput&r- Ou tput F i g . 4.9 Computer System f o r Routing Binary Numbers to D i g i t a l Comparators and V a r i a b l e Rate Clocks. Each column corresponds to a separate, b i n a r y d i g i t ; 5. THE TIME-VARYING INDUCTOR 38. 5.1 D e s c r i p t i o n of the C i r c u i t The v a r i a b l e inductor to be used i n the analogue i s based on the c i r c u i t i n F i g . 3.3. F i g . 5.1 shows the c i r c u i t f o r n-1 k L = lOh/B, where B n b^ .2 and the b^'s are binary d i g i t s . k=0 The inductance L i s p r o p o r t i o n a l to R-^ = R/B, where R i s constan F i g . 5.2 shows the c o n t r o l r e s i s t o r R^. R, = B."' B R AX R I 2 X k, ft If X grv. 1 X^b« \ Binary Control AJumber F i g . 5.2 C o n t r o l R e s i s t o r R^ th t h If b^ .= 0, the k switch i s open; i f b^= 1 the k switch Is closed. Thus, n-1 !_ _ 1 R± ~ R 2k b _ B k - R (5.1) k=0 th F i g . 5.3 shows the a c t u a l c i r c u i t f o r the k r e s i s t o r i n W V ; -15 +15 — A A A F i g . 5.1 Time-Varying Inductor 40. F i g . 5.2. . The maximum voltage allowable at terminals T and T-, t J ° u d i s + 10 v o l t s . Terminals T and T-, are connected to the inputs >-D , &Nf303s ' R k Rp£ S ON¥303 fj , ^ l - -f-V\A t-r^ *• +ie CN -QOrF t h P i g . 5-3. C i r c u i t f o r the k R e s i s t o r i n P i g . 5.2 of the d i f f e r e n t o p e r a t i o n a l a m p l i f i e r s , i n F i g . 5.1. Two FET's are needed f o r each switch to prevent the sw i t c h i n g mechanism from c o n t r i b u t i n g any input current to these a m p l i f i e r s , which are s e n s i t i v e to very small input c u r r e n t s . When i n F i g . 5-3 i s equal to + 12 v o l t s , t r a n s i s t o r Q i s OFF. Diodes D-^  and are back-biased, and there i s no voltage drop, across the r e s i s t o r s r ^ and . Thus VQ.Q2_= ^Gg2 = ^ a n d ^ e FET's. are ON. The only path that current can take i s through the two FETs, v a r i a b l e pot Rp^-j and r e s i s t o r R-^.. By v a r y i n g the value of pot R , , the t o t a l r e s i s t a n c e R-, = R/ + R , + Rn,T1 (R n, T 1 i s the * pk' k k pk ONk ONk t o t a l ON r e s i s t a n c e of the two FET's) can be set ac c u r a t e l y . • 41. When V i = -6 v o l t s the four diodes are forward biased and ck the t r a n s i s t o r Q i s ON. Gates G^  and G^  of the FETs are at -6 v o l t s and sources and are approximately at ground. As a r e s u l t , both PETs are OFF. R e s i s t o r s r ^ and cannot be too l a r g e because they serve to discharge the capacitance of each FET. The network i n F i g . 5.4 was used to convert the binary d i g i t b, i n t o the c o n t r o l voltage V , and at the same time turn to k to ck on the lamp when = + 12. The outputs of a m p l i f i e r s A^ ' and A^ i n F i g . 5.1 depend only on the magnitudes of E^, E^ and the c i r c u i t elements R^, and •H2v - 6 v. F i g . 5.4 C o n t r o l Network f o r FET Switch (Eqn. 3-4). Capacitors C^ and C^ are matched to prevent one input a m p l i f i e r from s a t u r a t i n g when the other does not. Large r e s i s t o r s R2 and Ri, prevent the cap a c i t o r s from charging up to the supply voltage when the input i s grounded. The v a r i a b l e pots i n s e r i e s with R^ and R i enable each inductor " h a l f " to be set to an exact 3 3 / " 42. , inductance value. To prevent the a m p l i f i e r outputs from s a t u r a t i n g f o r l a r g e B, the input voltage E i s l i m i t e d to + 2v. The input current i s l i m i t e d to + 10v/R^ -y ± 2ma; l a r g e r currents causa the a m p l i f i e r s and A l to s a t u r a t e . I f i s too s m a l l , the i n -ductor c i r c u i t o s c i l l a t e s , since R,- provides p o s i t i v e feedback path f o r a m p l i f i e r A^. The inductor i s c a l i b r a t e d by grounding points E and E^ i n F i g . 5.1.and a d j u s t i n g the 100K trimming r e s i s t o r s on . a m p l i f i e r s A^ and A^ u n t i l f i r s t E^ and then E^ are zero. The impedance bridge i s then connected to E^ and Ej^, B i s set to u n i t y and R^ i s adjusted to make the inductance equal to the req u i r e d value. The procedure i s repeated f o r the other h a l f of the inductor. 5.2 Testing the Inductor 5.2.1 Steady State Inductor Tests The inductor was tested i n the time and frequency domain. The frequency domain t e s t s y i e l d measures of the inductor's q u a l i t y f a c t o r Q and l i n e a r i t y w i t h respect to c o n t r o l number B and frequency f. The time domain t e s t s i n d i c a t e d how w e l l the inductor would perform as the inductance changed w i t h time. Let L be the inductance f o r B=l. Inductance L i s changed P P & by changing c a p a c i t o r s and i n F i g . 5.1 F i g . 5-5 shows L/L vs B f o r f = 1 KHz and L equal to lOOh, lOh and l.Oh. The P I> corresponding values of and CJ, are 0.1, 0.01 and 0.001 uuf, r e s p e c t i v e l y . 43. 0-& 0-6 O.V ^ O.04 N. 0.02 ^ O.0I ^ O.008 O-0O<5 0.0D) L " b o 0 VO CQ p C\J ^ 10 o $ g 3 - B>inary Control Number ^ 1 .1 i i ... ! i l_ 1 i i . | \\ a ! i !' i i i | : i ! •! \ j j | id-Lp :-'£>&>>! i | j | \ j | jb-l-4 ,<A| lij- I 1 \ | :C-L-p- 1.0hi i ; | i i : i i \ i i M l I i i 1 i I. - i i M l I X M j j ; 1 ! i I • \ i i i i " i i : i 1 i I ! 1 j ! ! i I c J p 1 I i i . i i i | Pig. 5.5 Normalized Inductance L/L vs B for L P P l.Oh, f = 1 KHz = lOOh, lOh and loo 60 W H o oh 0.2 Of 1 M l 1 i 1 1 i i 10 h j 1 • \ ! i i -J— i\ ! ; i I i i i \ i i : 1 I I • i -—-j — r i M / \ /• 1 i I ... J I i : •i / ! / l l 1 , 7 ^ 1 , ' I I I \ Q>\i.Oo • i i ! 1 i I i i j 1 ! i i i i ! 1 j j | I 1 •! j i i i 1 Region of Interest for Vocal Tract Analogue CVI *• IS ~a> or " S X § l %• o o o U <B O F i g . 5.6 Contours of Quality Factor Lp = lOh 44. I — B ~- ic • o o o o to (a) o frequency f (Hz) 03 o - y / ^vN6"^~irJ' o o o (to vj •frequency f (Hz) 8 Q o Region o f Intore.st for Vocotl Tract Analogue ft-e •FCHz) F i g . 5.7 ^ Normalized Inductance L-/L vs f,. and B, i s the inductance f o r B = 1 measured at f = 1 KHz (a) L p = lOOh, (b) = lOh and (c) L p ='l.Oh .45. P i g . 5.6 shows Q vs f and B. In the range required f o r use i n the vo c a l t r a c t analogue, the Q i s f a r i n excess of the required minimum of 50. P i g . 5.7 shows normalized inductance vs f f o r B = 1, 10, 100 and 255 and f o r = lOOh, lOh and 1.Oh.The range of operation needed f o r use i n the v o c a l t r a c t analogue i s shown. 5.2.2 Transient Tests on the Time-Varying Inductor The R-L c i r c u i t i n F i g . 5.8 was used f o r the time-domain t e s t s . With e. - E U _ n ( t ) , the output e n ( t ) was measured on an eiU) Id) -v R -AAAA ecd) F i g . 5.8 C i r c u i t f o r Inductor Time-domain Tests o s c i l l o s c o p e and compared with the c a l c u l a t e d output. The step f u n c t i o n fO, t < 0 u At) = \ x | i , t > o . The c i r c u i t was at r e s t f o r t < 0. Thus, i ( t ) = e( t ) = 0 f o r t < 0. For t > 0, e^Ct) can be c a l c u l a t e d by t a k i n g the time d e r i v a t i v e of the f l u x l i n k a g e s \=Li. Thus e n ( t ) = 44> where dX R , dt + LTtT" x e. 1 46. (5.1) Eqn. 5.1 i s of the form dt The s o l u t i o n of eqn. 5.2 + P(t)y =Q(t) (5.2) 23 i s P dt P dt Q e dt (5.3) The f i r s t time-domain t e s t was made w i t h L ( t ) = L, where L i s a constant. For e^(t) = E U _ ^ ( t ) , e 0 ( t ) = E e Rt L F i g . 5.9 shows a graph of the measured and c a l c u l a t e d outputs e Q ( t ) f o r L = lOh, l h , and O.lh, E = 2 v o l t s and R = lOK/i . The c a l c u l a t e d and measured values coincided almost e x a c t l y . Rounding at the top of the output at t=0 occurred f o r L = O.lh. This rounding was caused by the f i n i t e slew r a t e of the o p e r a t i o n a l a m p l i f i e r s i n F i g . 5.1. The output was measured using an o s c i l l o -scope with a v a r i a b l e time base. In the experiment was equal to lOh. Consider eQ(t) i n F i g . 5.8 f o r L ( t ) = L / B ( t ) , where .B(t) = a + bt, and a, b and _L^,are p o s i t i v e constants. Equation 5.1 becomes f | + R ( V b t ) X = E U (t) P (5.4) From eqn. 5.2 i t f o l l o w s that f o r t > 0 L t/Jiv lOli lh OA h 0-5 msec /ct'w 5Vsac/J iV Fig. 5.9 Output e n ( t ) i n Fig. 5.8 for L(t) Constant 48. R / , b .2-— ( a t + ^ "t . R + ^ / * b 2 \ t — (ax + ^ x ) P \ ( t ) - E e dx. (5.5) J 0 The output voltage i s the time derivative of equation 5.5 R t b ^ r t R /•__.- b „2 e n ( t ) = E ( l - f - ( a + bt)e p L (at + | V '0 L (ax + 2 x ) dx). 0 (5.6) Now l e t B(t) be changed by unity at times t=nT (n=0,l,...k...N) where T = ^ . Thus, and B(t) = a ± h L L(t) a + n nT ^ t ^ (n+l)T (5.7) nT ^ t ^ (n+l)T. (5.8) Eqn. 5.1 becomes for nT ^ t ^ (n+l)T f| + |- (a + n)\ = E U _ 1 ( t ) . (5.9) P Por nT ^ t ^ (n+l)T the solution i s n-1 - F(n-k-l)(a±(^)) \ n P P 1 " ( a i n ) ) , 3 (i-e 'n+k-e Q ( t ) = E RT L (a+k) k=0 x e - |-(a+n)(t-nT) P (5.10) The d e t a i l s of the solution are omitted, as they are similar to those to be presented i n conjunction with eqn. 5.19-49. I n P i g . 5.1, e ^ t ) = L ( t ) . In t h i s case eqn. 5-1 becomes L || + Ri = E U^Ct) . I f L ( t ) = L / B ( t ) , where B(t) = a + bt as defined f o r eqn. 5.4, then ff + R ( V 1 = ^ L M ) U - l ( t ) ( 5 - l l } P P which i s of the same form as eqn. 5.2. The s o l u t i o n of eqn. 5-11 f o r t > 0 i s fcrf±#t ) t(t)= e r Therefore e^Ct) = E e up R(ax±bxz) r & f " ' ^ - (5.13) When L ( t ) i s changed at times t = nT and B(t) = a + n as i n eqn. 6.7, the s o l u t i o n of eqn. 5.2 i s more d i f f i c u l t . Eqn. 5.1 becomes + R- ( a ± n ) i = M a i s l u ( t ) ( 5 > 1 4 ) . P P d i + nT ^ t ^ (n+l)T Equation 5.14 i s of the form of eqn. 5.2, w i t h P(x) = Y~ (a + n) nT ± x * (n+l)T L P The i n t e g r a t i n g f a c t o r i n eqn. 5.2 becomes 50. p£x)dx R[(a±nYt-nT) + (a ± fn-,)) nT Jo • e Lp L z The solution of eqn. 5.14 for nT < t ~= (n+l)T i s 6ft) = e (a±n)(t~nT) + fa±(n^j))nT E r a * to e JR' (a±h)(x~kT)-^(a ±(^-j))kT' 2 dx R=0 JnT Efa±n) e R C a t n W - n T ) - i - f a ± m - i ) ) n T ' 2 (5.15) Integrating eqn. 5.15 gives • / i l P L (a±n)(t-nT) + f a ± ln~i))nT n - i L P L £_ e R 2 Lp J=p e 2 -ix=t (5.16) 51. Rearranging eqn. 5.16 and substituting the l i m i t s gives _R [ Ca±r)Yt-r)T) tW = £ e L p 1 • R (a ± (n-t) )nT ' , 2 R (a ±ffr-n)kT R (a zizk)T _R_ (ditnriOnT R ,a±n)Ct-nT) Lp 2 ( e L P _ ,) Rearranging gives R(azn)(t~nT) n-i e (a±rn-j))n - (a^ih^.O)k] RT(a±k) _R (a±n)(i~r)T) 2 J(e L' -,)e~L? The desired output eQ(t) = e^ - Ri(t) i s e0(t)= E n - i RT( n_^-))fa±(n + ^ )) _RT(a±k) 2 ( i - e " L " ) RfcfinXt-nT) fc = 0 (5.17) nT ^ t ^ (n + 1)T 52. To compare the measured output e ^ t ) of. the c i r c u i t i n Pig. 5-8 against the calculated output of eqns. 5..6, 5.10, 5-13, and 5.17, the number B(t) was varied as i n Fig. 5.10 by the i n t e r -polating system described i n Chapter 4. (See Fig.5.1l). For the i n t e r v a l 0 - t - Tv = NT of F i g . 5.10, eqns. 5.6, 5.10, 5.13, and 5.17 were used to calculate the outputs. For the i n t e r v a l t x^Tv, the outputs are as given by eqns. 5.10 and 5.17 with n=N. For eqns. 5.6 and 5.13, "the outputs are decaying exponential of the _ t form e = E e 'L" where T = L /R(a + bTv) and E i s the value of c p' — • c e^(t) i n eqns. 5.6 or 5.13 at t=Tv. The outputs were calculated and plotted by an IBM 7040 computer for b = 4/3 x 10 5/sec, N = 40, Tv = 0.3 msec and for a=4 and a=44. (See Figs.5.12 and 5.13). The discrepancy between eg(t) for the continuous L(t) and the corr-esponding stepwise approximation was so small that only eQ("t) calculated from eqns. 5.10 and 5.13 are shown. The calculated results i n F i g . 5.12 for the true inductor do not agree very well with the calculated and measured results i n F i g . 5.13- The d i s -crepancy i s noticable because the rate of change of inductance i s of the same order of magnitude as the rate of change of current i ( t ) . Fortunately, this problem w i l l not occur i n the vocal tract analogue. The close agreement between the calculated and measured values for the device for which e(t) = L(t) | | indicates i t s accuracy and the effectiveness of the interpolating system. 53. ,B(t) Case I o V Time t Tv a B(t) K Case 2 4>t Time t Tv F i g . 5.10 V a r i a t i o n of Binary Number B c o n t r o l l i n g the Time-Varying Inductor >-R AAA/v-Up-Down Control Signa/ Binary Number To V/liich T h d u c t o r Value Is To P r o c e e d F i g . 5.11 Time-Varying Inductor Test Arrangement 54. 1 1 sr -1 - --— ----— 1 -- \ - - -- 1 -— 1 i • - -— -- -- --- ~i 2 - - - - ----- - - ; - - - - -- - - -\ -*% V L - -J -- K i - - - - - -- ! - -0 3 -— -0 V --— 0 o / a | -0 M. 0 HZ ---- - 1 L a — - -- . . . 71 rx -A j _ -, i i 1 " ! I (a) - -- -- --> - - . . . -- -2 - -- - - -- c ~ -— --— -if -/ -- • - *• - - -0 5 - - Q o 0 1 -0. - - -- o .5 r -{ - --- 1 fZ £ n is —-- - ! iz i i i ~\~r (b) Pig. 5.12 Calculated Outputs for R-L C i r c u i t s i n F i g . 5.8 for the True Time-varying Inductance g ^ _ d(Li) where L = 10h/(a+bt) for 0 * t ^ and b = | x 10 5/sec (a) L = 10h/(a+bt); a = 4. (b) L ='10h/(a-bt); a = 44-(b) Calculated and Measured Output f o r R-L C i r c u i t i n Pig. 5.8, e Q ( t ) = L ( t ) d i . Upper Photographs: V e r t i c a l Scale; 1 v o l t / d i v dt Horizontal Scale; 0.1 msec/div. Lower Graphs: Calculated Outputs (a) L = 10h/(a+bt) where a=4 and b = 4/3 x 105/sec. (b) L = 10h/(a-bt) where a=44 and b= 4/3 xloVsec' 56. 6. THE TIME-VARYING CAPACITOR .6.1 D e s c i i p t i o n o f the C i r c u i t The time-varying c a p a c i t o r i n P i g . 6.1 i s based on the c i r c u i t i n P i g . 3-2. F i g . 6.1 shows the c i r c u i t f o r C = (0.0037)B uf. The capacitance .is v a r i e d i n the same manner as the inductance, namely, by v a r y i n g the c o n t r o l r e s i s t o r R-, = R/B, where R i s v a r y i n g the r e s i s t o r R-^  i n F i g . 5-2 i s used here. The a c t u a l c i r c u i t i s simpler, however, since the c a p a c i t o r i n the v o c a l t r a c t analogue i s grounded. F i g . 6.2 shows the v a r i a b l e r e s i s t o r n-1 constant The same general technique used f o r k=0 R. T V 6 1 4 F i g . 6.2 Control R e s i s t o r R, f o r the Time-Varying Capacitor 57. Fig . 6.1 Time-Varying Capacitor 58, When V , =0, the k t h PET i s ON. When V , = -6 v o l t s , the k t h PET i s OFF and i , = 0. FETs r a t h e r than t r a n s i s t o r s are used f o r k switches since -lOv - V-^ g - lOv. The r e s i s t a n c e R^+ R .^ + RQJ^ i s adjusted by the v a r i a b l e r e s i s t o r R^., where i s the ON th r e s i s t a n c e of the k FET. The c i r c u i t i n F i g . 6.3 converts the binary d i g i t b^ . i n t o the c o n t r o l voltage and at the same time turns on the lamp t h when the k FET i s ON. r 2N3SWO -3.6N o (lamp) 6>rS F i g . 6.3 FET Switch D r i v e r The booster c i r c u i t which terminates the o p e r a t i o n a l ampli-f i e r i n F i g . 6.1 i s a totem pole emitter f o l l o w e r capable of d e l i v e r i n g + 20 ma. I t s purpose i s to provide more output current than the inadequate amount supplied by the o p e r a t i o n a l a m p l i f i e r . The input voltage E^ to the v a r i a b l e c a p a c i t o r i s l i m i t e d to approximately + 4 v o l t s . .A higher input voltage w i l l saturate 59. a m p l i f i e r . A , when B i s l a r g e . The input current i s l i m i t e d to the maximum output of the booster, ± 20 ma. 6.2 Testing the Capacitor 6.2.1 Steady State Capacitor Tests The time and frequency domain t e s t s f o r the ca p a c i t o r were s i m i l a r to those f o r the inductor. Let 0^ be the capacitance f o r B=l, f = 1 KHz. Capacitance 0^ i s changed by changing c a p a c i t o r C^ i n P i g . 6.1 P i g . 6.4 shows C/Cp vs B f o r C p = 3750 (iuf, 375 u.u.f and 75 \i\if. The c o r r -esponding values of C^ are 5|if» 0.5u.f and O.lu-f. F i g . 6.5 shows Q as a f u n c t i o n of frequency and B. In the range required f o r use i n the v o c a l t r a c t analogue, the Q i s much i n excess of the required minimum of 50. F i g . 6.6 shows C/C^ vs frequency f o r B=l, 10, 100 and 255 and f o r C^ = 3750 \i\if, 375 |i|if and 75 u.u.f. The range of operation needed.for use i n the vo c a l tract.analogue i s shown. For each set of curves, the non- l i n e a r behaviour of C/C vs f f o r B=l i s due P mainly to s t r a y capacitance i n p a r a l l e l with the c o n t r o l r e s i s t o r . This s t r a y capacitance w i l l be reduced, when the c o n t r o l r e s i s t o r i s mounted on a p r i n t e d c i r c u i t board. 6.2.2 Transient Tests on the Time-varying Capacitor To make time domain t e s t s of the ca p a c i t o r the R-C c i r c u i t i n F i g . 6.7 was used. With e.(t) = E U , ( t ) , the output e n ( t ) was 60. F i g . 6.5 Contours of Capacitor Q u a l i t y Factor Q 61. 0.6 \ £ 1-2 5!) 0.6 .'V! 0 o f f eguency f (tt?) 7 IOO&255 — (b) frequency f cHz) Region of Interest for Voce// Tract Analogue > f fWz) F i g . 6.6 Capacitance C/Cp vs f, C p and B. C p i s the capacitance f o r B = 1 measured at f•= 1 KHz 62. measured on an o s c i l l o s c o p e and compared wi t h the c a l c u l a t e d output. R =F CCt) P i g . 6.7 C i r c u i t f o r Capacitor Time Domain Tests The c i r c u i t was at r e s t f o r t < 0. Thus, i ( t ) = e(t) = 0 f o r t < 0. For t > 0, eQ(t) i s best c a l c u l a t e d by f i n d i n g the charge q(t) stored on the c a p a c i t o r and then d i v i d i n g by the capacitance. Thus e Q ( t ) = q ( t ) / C ( t ) , and d e. dt + Rcfry = -R . ( 6 - 1 } Eqn. 6.1 i s of the form of eqn. 5-2. • The f i r s t time-domain t e s t was made wit h C(t) = C, where C i s a constant. For a step input t _ RC e Q ( t ) = E ( l - e ) U _ 1 ( t ) F i g . 6.8 shows the measured and c a l c u l a t e d outputs ^ ( t ) f o r 63. C = 0.006 | i f , 0.06 (if, 0.3 iif and E = 2 v o l t s . The lowest value of capacitance was not used because P i g . 6.6 shows that f o r B ^ 1 the capacitance i s a h i g h l y n o n - l i n e a r f u n c t i o n of frequency. This n o n l i n e a r i t y causes a l a r g e r i n g i n g e f f e c t i n the ourput. E - 2 volts ec(t) Fig-. 6.8 Output eQ(t) f o r Constant Capacitance The output was measured using an o s c i l l o s c o p e with a v a r i a b l e time base. In the experiment C^ was equal to 1.5 nf. When C(t) = C B(t) was time-varying, B(t) and e.(t) were as shown i n F i g . 6.9. The number B(t) was v a r i e d by using the i n t e r p o l a t i n g system f o r the vo c a l t r a c t analogue, as shown i n F i g . 6.10. Input 64 Bft)' >1 C c i s e / a —, C o s e ? BftJ X I S •Tv- !y > ^ F i g . 6.9 Inputs to R-C Network i n Fig . 6.10 i R eiti) * w w (— e0(t) U ^ - b o w h C o n t r o l Signn Binary Number To W h i c h Copeic'iior Vol'je I?-To Proceed 5^ i ' i D i g i to/ Composer Up-Down A Variei We Rate Clock F i g . 6.10 Time-Varying Capacitor Test Arrangement 65, If e ± ( t ) i s as shown i n Fig. 6.9, and i f C(t) = C B(.t) and B(t) = a + bt where C , a, and b are positive, then eqn. 6.1 becomes for 0 < t < Tv da JL I TT ( F ) dt + RC (a + bt) = R u - l l t ; P ~ (6.2) The exponent P dt of the integrating factor becomes r r P dt = dt RC (a + bt) P In I .a +_ bt + b RC The solution of eqn. 6.2 i s ln(a+bt) q(t) = e ±bRC p _ln( a+bx) +bRC 0 J E R e P dx C E J2 (1+bRC (a+bt) - a a a+bt ± bRC P bRC £ 1 P a+bt > 0 Since e Q ( t ) = q(t)/C(t) , e Q ( t ) bRC + 1 __J2T bRC _J Tl±bRC P 1 a±bt / P bRC p £ 1 (6.5) a+bt > 0 For'Tv - t - Tv + Tc, capacitance C^(a + b Tv) remains constant, and the output goes exponentially to a value of EQ at the end of the f i r s t half of the square wave. Tc E w h e r e E ' = (1+bRC ) P 1 - a+bT v bRC +1 66 bRC P ^ . E  (1 + bRC ) P i f a+bTv / bRC + 1 V bRC P < < 1 During the second half cycle of the square wave, d t - + RC (a' + bt) = °' ( 6 ' 5 ) P where time t i s measured from the s t a r t of the negative going step i n the square wave. The i n i t i a l charge on the capacitor at t = 0 i s QA = a'C E~. Let B(t) = a' + bt, where a' and b 0 p 0 — ' are p o s i t i v e . Rearranging eqn. 6.5 and integrating gives 1 + * ( t ) - a ' c P Eo (r±btj • 0 - t - T v Since e Q ( t ) = q ( t ) / C ( t ) , bRC P bRC + 1 P ~ / \ bRC (6.6) During the time Tv - t - Tv + Tc, (t i s again measured from the sta r t of the negative going step i n the square wave), the capacitance remains constant at C = C p(a' + b Tv) and e^(t) 67. decays exponentially to zero (t - Tv) RC (a'+bTvJ e Q ( t ) = E Q e P where bRC + 1 V , . T? _ a j bRC h0 = ^0 a'+bTv p (6.7) Let B(t) be incremented at times t = nT (n=0,l,...k,...N) as-xin Pig. 6.9. Thus C(t) = C p(a + n) for nT - t - (n+l)T. Eqn*. 6.1 becomes for the f i r s t half of the square wave input da, dt + RC (a+n) ~ R " - l v P ~ = 1 U -.(t) n T < t ^ (n+l)T The solution for nT - t - (n+l)T and t > 0 corresponding to eqn-. 6. 3 i s ecCt)--t n - i _j_r,j_ (can) T i - e RCo(a±k) K - n T ) Rt>fa±n) (6.8) During the second half of the square wave input eqn. 6.1 becomes da. dt + RC (a'+n] P = 0 0 £ t ^ Tv and the solution for nT - t - (n+l)T corresponding to eqn. 6.7 i s n-i - p r 1 0 - t ± Tv (6.9) a 'in 68. The d e t a i l s of the solution are omitted, as they are similar to those presented i n conjunction with eqn. 5-17. Eqns. 6.3 and 6.6 show that the general form of the out-put i s greatly affected by the parameter • Measured and calculated outputs were obtained for six different values of th i s parameter, the values being selected to show the various kinds of behaviour i n eg(t). The outputs were calculated and plotted by an IBM 7040 computer for a and a' either 15 or 207, b = 192/msec, Tv = 1 msec and Tc = 1.5 msec, (See Pig. 6.11). The discrepancy i n e ^ t ) for C(t) a continuous function and C(t) a stepwise approximation did not show on any of the plotted outputs. For this reason the calculated output eQ(t) i s shown only for C(t) continuous. The results i n Figures 6.11 to 6.14 show photographs of the measured outputs and the corresponding calculated outputs. The close agreement between the calculated and measured results indicates the accuracy of the time-varying capacitor. (a) ( b) P i g . 6.11 C a l c u l a t e d and Measured Output f o r R-C C i r c u i t i n P i g . 6.7 and Case 1 of P i g 6 9 Upper Photographs: V e r t i c a l Scale; 1 v o l t / d i v . H o r i z o n t a l Scale; 0.5 msec/div! ' Lower Graphs: C a l c u l a t e d Outputs, (a) bRCp= 1.43 (b) bRC p - 1. ^ (a) (b) 6.12 Calculated and Measured Output f o r R-C C i r c u i t i n Fig. 6.7 and Case 1 of F i g . 6.9. Upper Photographs: V e r t i c a l Scale; 1 v o l t / d i v , Horizontal Scale; 0.5 msec/div. Lower Graphs: Calculated Outputs, (a) bRC p = 0.714 (b) bRCp =0.5. -j (a) (b) F i g . 6.13 Calculated and Measured Output f o r R-C C i r c u i t i n Fig. 6.7 and Case 2 of F i g 6.9. Upper Photographs: V e r t i c a l Scale; 1 v o l t / d i v , Horizontal Scale; 0.5 msec/div Lower Graphs: Calculated Outputs. (a) bRC - 0.214 (b) bRC = 0 5 P P <] 72. F i g . 6.14 Ca l c u l a t e d and Measured Output f o r R-C C i r c u i t i n F i g . 6.7 and Case 2 of F i g . 6.9. b R C p = 0.714. (a) Measured Outpu' V e r t i c a l Scale; 1 v o l t / d i v , H o r i z o n t a l s c a l e ; 0.5 msec/div. (b) C a l c u l a t e d Output. 73. REFERENCES 1. H.F. Olson, H. Belar and E.S. Rogers, Speech Processing Techniques and A p p l i c a t i o n s , IEEE Trans, on Audio and E l e c t r o -a c o u s t i c s , V o l . AU-15, No. 3, Sept. 1967, pp. 120-126. 2. B. Gold, CM. Rader, Systems f o r Compressing the Bandwidth of Speech, IEEE Trans, on Audio and E l e c t r o a c o u s t i c s , V o l . AU-15, No. 3, Sept. 1967, pp. 131-136. 3. H.W. Dudley, Synthesizing Speech, B e l l Laos. Record, V o l . 15, No. 4, Dec. 1936, pp. 98-102. 4. H.W. Dudley, The C a r r i e r Nature of Speech, B e l l System Tech. J . , V o l . • 19, No. 4, Oct. 1940, pp. 495-515. 5. M.R. Schroeder, Vocoders: A n a l y s i s and Synthesis of Speech, Proc. IEEE V o l . 54, No. 5, May 1966, pp. 720-734. 6. R.S. Tomlinson, The Design of a D i g i t a l l y C o n t r o l l e d Attenuator, Quarterly Progress Report Ho. 74, Research Lab. of E l e c t r o n i c s , M.I.T.., J u l y 15, 1964, pp. 185-190. 7. R.S. Tomlinson, SPASS - An Improved Terminal Analog .Speech Synthesizer Quarterly Progress Report No. 80, Research Lab. of E l e c t r o n i c s , M.I.T., Jan. 15, 1966, pp. 198-205. 8. J.L. Flanagan, Speech A n a l y s i s Synthesis and Perception, (New York: Academic Press Inc., 1965) pp. 21-74. 9. K.N. Stevens, S. Kasowski and C.G.M. Fant, An E l e c t r i c a l Analog of the Vocal Tract, J . Acoust. Soc. of Am., V o l . 25, No. 4, J u l y 1953, pp. 734-742. 10. H.K. Dunn, The C a l c u l a t i o n - of Vowel Resonances, and an E l e c t r i c a l Vocal Tract, J . Acoust. Soc. of Am., V o l . 22, N.6, Nov. 1950, pp. 740-753-11. G. Rosen, Dynamic Analog Speech Synthesizer, Technical Report 353, Research Lab. of E l e c t r o n i c s , M.I.T., Feb. 10, I960. 12. G. Rosen, Dynamic Analog Speech Synthesizer, J. Acoust. Soc. of Am., Vol . 30, No. 3, Mar. 1958, pp. 201-209. 13. M.H.L. Hecker, A.S. House and K.N. Stevens, Performance of the A r t i c u l a t o r y Analog of the Speech Mechanism: A Report on the Status of Research, Quarterly Progress Report No. 64, Research Lab. of E l e c t r o n i c s , M.I.T., Jan. 15, 1962, pp. 191-198. 14- J.B. Dennis, Speech Synthesis, Quarterly Progress Report No. 67, Research Lab. of E l e c t r o n i c s , M.I.T., Oct. 15, 1962, pp. 157-162. 74. •15." J.E. Fulenwider, High Q Inductance Simulation, Proc. IRE, Vol. 48, May I960, pp. 954-955-16. Y. Kanai, On the Inductive Part i n the A.C. Characteristics of the Semiconductor Diodes, J. Phys. S o c , Japan, Vol. 10, 1955, pp. 719-720. 17. I. Ladany and R.J. Kearney, A High-Q tuned C i r c u i t using a Solid-State Inductance, J. Electronics and Control, Vol. 10, Mar. 1961, pp. 241-243-18. T. Chiba and M. Kajiyama, The Vowel, Its Nature and Structure, Phonetic Society of Japan, Tokyo, 1950, p. 121. 19. Flanagan, op. c i t . , pp. 14-21. 20. R.K. Potter, G.A. Kopp and H.C. Green, V i s i b l e Speech, D. Van Nostrand Company, Inc., 1947. 21. R.H.S. Riordan, Simulated Inductors using D i f f e r e n t i a l Ampli-f i e r s , Electronics Letters, Vol. 3, No. 2, Feb. 1967, pp. 50-51. 22. G. Austin, Design and Construction of an Opaque Optical Contour Tracer for Character Recognition Research, M.A.Sc. Thesis, Department of E l e c t r i c a l Engineering, U.B.C, 1968. 23- E.D. R a i n v i l l e , A Short Course i n D i f f e r e n t i a l Equations, (New York: Macmillan Comp., 2nd Ed., 196l). 

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