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Statistical watthour meter operating by polarity-coincidence correlation Warman, Lloyd Alfred 1967

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A STATISTICAL WATTHOUR METER OPERATING BY POLARITY-COINCIDENCE CORRELATION by LLOYD ALFRED WARMAN B . A . S c , U n i v e r s i t y of B r i t i s h Columbia, 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 E n g i n e e r i n g We ac c e p t t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d Members of the Department of E l e c t r i c a l E n g i n e e r i n g THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1967 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the 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 the 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 make 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 and 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 the Head 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 o f lA The 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 V a n c o u v e r 8, Canada D a t e MCUJ fr, (3G 7 ABSTRACT A d i g i t a l s t a t i s t i c a l w a t t h o u r meter which operates by p o l a r i t y - c o i n c i d e n c e c o r r e l a t i o n i s proposed. The output r e p r e s e n t s a c o n t i n u o u s l y 'up-dated', t i m e - m u l t i p l i e d e s t i m a t e of the c u r r e n t and v o l t a g e c r o s s - c o r r e l a t i o n f u n c t i o n f o r z e r o d e l a y . I n p r e v i o u s c o r r e l a t o r s of t h i s t y p e , bounded u n i f o r m l y -d i s t r i b u t e d random n o i s e has been added t o each i n p u t b e f o r e c l i p p i n g t o l i n e a r i z e the i n p u t / o u t p u t c o r r e l a t i o n f u n c t i o n r e l a t i o n s h i p . I n the work r e p o r t e d h e r e , the n o i s e sources have been r e p l a c e d by d e t e r m i n i s t i c t r i a n g u l a r wave g e n e r a t o r s . These have the advantage of b e i n g much s i m p l e r t o c o n s t r u c t and of e n a b l i n g the s t a t i s t i c a l w a t t h o u r meter t o produce an a c c e p t -a b l e e s t i m a t e of the energy ( f o r c e r t a i n t y p e s of i n p u t s ) more q u i c k l y than w i t h the former type of a u x i l i a r y f u n c t i o n genera-t o r . An attempt has been made t o p r e d i c t the performance of the d e v i c e u s i n g d e t e r m i n i s t i c a u x i l i a r y f u n c t i o n s by means of b o t h analogue and d i g i t a l computer simulations„ A p r o t o t y p e has a l s o been c o n s t r u c t e d and t e s t e d w i t h d-c and a-c i n p u t s . The r e s u l t s of the s i m u l a t i o n s and p r o t o t y p e t e s t s suggest t h a t , i f the meter e l e c t r o n i c s were r e f i n e d and con-s t r u c t e d f o r the most p a r t as an i n t e g r a t e d c i r c u i t , the s t a t i s -t i c a l w a t t h o u r meter c o u l d become a p r a c t i c a l energy-measuring d e v i c e i n d o m e s t i c , i n d u s t r i a l and l a b o r a t o r y a p p l i c a t i o n s . i i TABLE OP CONTENTS Page LIST OP ILLUSTRATIONS v i LIST OF T^ LBXlES o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 1 .5C ACE[NOWLEDGrEMENT o a o o o o 0 © 0 o o o o o O 0 0 0 < > 0 0 O O 0 o o o o 0 o 9 o o o 0 0 0 O 0 o o 0 ^ 1. INTRODUCTION . . . . 1 1.1 B r i e f H i s t o r y of Watthour Meters 1 1.2 P r e v i o u s P o l a r i t y - C o i n c i d e n c e C o r r e l a t o r s 2 1.3 The Proposed Instrument 8 2. PERFORMANCE EVALUATION OF THE PROPOSED INSTRUMENT BY COMPUTER SIMULATION 10 2.1 T h e o r e t i c a l E v a l u a t i o n 10 2.2 Analogue Computer S i m u l a t i o n 11 2.3 D i g i t a l Comput e r S i m u l a t i o n s .......••.•••••••••» 15 2.3-1 Case 1 - L e v e l I n puts and Continuous O p e r a t i o n w i t h Sawtooth A u x i l i a r y F"U_lTC "fc 1 O H S 0 0 o o o o 0 0 O 0 O 0 0 9 9 O 0 0 0 0 0 0 e e 0 0 O O o o a e 15 2„3.2 Case 2 - L e v e l I n puts and Sampling w i t h Sawtooth A u x i l i a r y F u n c t i o n s 17 2.3.3 Case 3 - L e v e l I n puts and Sampling w i t h A u x i l i a r y G e n e r a t o r s w h i c h Produce Bounded Noise w i t h a U n i f o r m Amplitude D i s t r i -TOU-"t 1 0X1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 2.3.4 Case 4 - S i n u s o i d a l Input and Continuous O p e r a t i o n w i t h Sawtooth A u x i l i a r y FU_T1C "t i OITS O O 0 0 0 0 O O 0 O 0 O O O O O O 9 O O O O O 0 O O 0 O O O O O 20 2.3.5 Case 5 - S i n u s o i d a l Input and Sampling w i t h Sawtooth A u x i l i a r y F u n c t i o n s ............. 22 2.3.6 Case 6 - S i n u s o i d a l Input and Sampling w i t h A u x i l i a r y G e nerators which Produce Bounded Noise w i t h a U n i f o r m Amplitude S "fc 31"* JL IO U-"b 1 O H O 0 O 0 O 0 0 O O O O O O 0 O O O 9 O 0 O 9 0 O O O O O O 22 i i i Page 3. RESULTS OP DIGITAL COMPUTER SIMULATIONS 23 3.1 Continuous Operation with Level Inputs 23 3.2 E f f e c t of Sampling w i t h Level Inputs 28 3.3 Comparison of the Two Types of P o l a r i t y - C o i n c i -dence C o r r e l a t o r s w i t h Level Inputs 30 3.4 Continuous Operation w i t h S i n u s o i d a l Inputs 34 3.5 E f f e c t of Sampling w i t h S i n u s o i d a l Inputs 37 3.6 Comparison of the Two Types of P o l a r i t y - C o i n c i -dence C o r r e l a t o r s w i t h S i n u s o i d a l Inputs 38 4. INSTRUMENTATION AND TESTING OP THE STATISTICAL WATT-HOUR METER 44 4.1 Implementing the C o r r e l a t o r as a Watthour Meter.. 44 4.2 Voltage and Current Measurement 46 4.2.1 Voltage Measurement 46 4.2.2 Current Measurement 48 4.3 A u x i l i a r y Function Generators 51 4.3.1 A Simple Sawtooth Generator 52 4.3.2 An Accurate Triangular-Wave Generator .... 53 4.4 Logic f o r M u l t i p l i c a t i o n of Sgn Functions ....... 55 4.5 Integrated C i r c u i t Up-Down Counter .............. 56 4.6 Sample Control Clock 59 4-6.1 Independent Clock 60 4.6.2 Sampling at Line Frequency 61 4.6.3 Sampling Synchronized to Line Frequency .. 61 4.7 Output Converter and Mechanical R e g i s t e r ........ 62 4.8 Power Supplies . . . 0 . 0 . 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.9 Test Procedure and Results 66 i v Page 4 • 9 • 1 -DC IXlJpU"fc S # o » e e i > o e » e o o « o 0 o » 9 o e « » * o « o < 9 f t o o e » 66 4 • 9 • 2 -A.C IlTp\l"t S • o e o o 9 t > o o « t » o o » o 9 o o o o o o e a e o e o o a * o 63 5. SUMMARY AND CONCLUSIONS 71 APPENDIX I - BASIC THEOREM 75 APPENDIX I I - ERROR CAUSED BY EXPONENTIAL APPROXIMATION OP LINEAR RAMP .. . . 79 REFEREE C E S 0 O O O O O 9 O « O O O 0 9 O O O O O 9 e * O t 0 O 9 9 O O O O O O O O 9 « « « O . . « 82 V LIST OP ILLUSTRATIONS F i g u r e Page 1-1 B l o c k Diagram of a P o l a r i t y - C o i n c i d e n c e C o r r e -lcl~fc 02? ( H?00 ) o o o a o o o o o a o o o o o o o o o o e o o o o o o o o o a o * * * * * ' * 3 1-2 B l o c k Diagram of PCC w i t h M o d i f i e d Input 5 I*—3 B i n E L i r y R3.-n.cl.01n Soxurc© o o o e o o t e o o o o e o o o o o o o o o o o a o o o * 7 1- 4 D i s c r e t e N o i s e G e n e r a t o r 8 2- 1 Analogue Computer 'Patch' Diagram f o r S t a t i s t i c a l Watthour Meter S i m u l a t i o n . 0 12 2-2 Analogue Computer Comparator Output 'Shaper' 13 2-3 Analogue Computer Output f o r f — 50 cps, k-^  = 0.133, k 2 = 1.0, f A 1 — 510.0 cps, f A 2 - 569.7 cps, A-j — ~ • o 0 0 0 0 0 0 0 0 0 0 0 0 0 * 0 0 O * a o 0 * o o o e * e * o o o * * * ° 14" 2-4 Waveforms I n c o r p o r a t e d i n E q u a t i o n (2-2) ......... 16 2-5 G e n e r a l Program F l o w c h a r t f o r D i g i t a l Computer S i m u l a t i o n of Continuous System w i t h L e v e l I n p u t s and Sawtooth A u x i l i a r y F u n c t i o n s 18 2- 6 D e t a i l s of L o g i c R o u t i n e Used i n Flow Chart of F 1 © 2 "~ 5 • o o s o o o e o o d e e e o e o o e A e s d a e e a e o o e e e o o f l o e e 19 3- l a E r r o r i n C o r r e l a t i o n E s t i m a t e f o r 1 ^ = 5, !X ^ 2 —- 3 0 0 0 0 0 0 0 0 . o o o o o A a e e v o a o o e e o e o e e o e o o o o o a a o a o o 25 3 - l b E r r o r i n C o r r e l a t i o n E s t i m a t e f o r 1 ^ = 57, ~ 31 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ' o e o o o 0 0 O 0 0 o o 0 0 0 0 O o o o « o o o » 26 3 - l c E r r o r i n C o r r e l a t i o n E s t i m a t e f o r 1 ^ = 283, — 15V 9 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 O O 0 O O O 0 0 O O O O O 0 O O O 0 0 O 0 0 0 3-2 Standard D e v i a t i o n , <X , of' E r r o r f o r L e v e l Inputs as a F u n c t i o n of Input L e v e l ..................... 29 3-3 E r r o r i n C o r r e l a t i o n E s t i m a t e ( f o r L e v e l I n p u t s ) as a F u n c t i o n of the Sample Number, N, f o r 1 ^ = 5, v i Figure Page 3-4 E r r o r i n C o r r e l a t i o n Estimate ( f o r Level Inputs) as a Function of the Sample Number N, Using Noise Generators to Produce the A u x i l i a r y FUJTIC "t 1 OX1S a 0 0 0 0 O 0 0 0 0 0 0 0 O 0 0 O « O O 0 0 O 0 O 0 0 e 0 0 O 0 0 O 0 0 0 « 0 0 33 3-5 E r r o r i n C o r r e l a t i o n Estimate as a Function of I n i t i a l Phase Difference Between Input Sinusoid and A u x i l i a r y Functions 35 3- 6 Standard De v i a t i o n , C, of Percent E r r o r , as a F"LlHC"fciOH Of ^/^1"^"A2 0 o » o o o o o 0 O 0 0 o o o o o 0 0 0 . » 0 O 36 3-7a-c E r r o r i n C o r r e l a t i o n Estimate as a Function of Sample Number f o r S i n u s o i d a l Input and Sawtooth A u x i l i a r y Functions 39-41 3 - 8a-c E r r o r i n C o r r e l a t i o n Estimate as a Function of Sample Number f o r S i n u s o i d a l Input and Random Noise A u x i l i a r y Functions 3 9 -41 4- 1 Block Diagram of Proposed S t a t i s t i c a l Watthour 4-2 P o s s i b l e Voltage Measuring Schemes . 47 4- 3a-c P o s s i b l e Methods of Measuring Current i n Two-Wire Systems 49 4-3d,e P o s s i b l e Methods of Measuring Current i n Three-WIX* 6 S ~fc GIGS •eaooo«ooo«««a«*»»«»»«««o«»»>»»a«»«««» 50 4-4 A Simple Sawtooth Generator 53 4-5 An Accurate Triangular-Wave Generator 54 4-6 M u l t i p l i c a t i o n Logic f o r sgn £z-^ j • sgn ........ 56 4-7 Up-Down Counter and Input C i r c u i t r y 57 4-8 Up-Down Counter Control Truth Table 58 4-9 Up-Down Counter Waveforms 58 4-10 Tuning Fork O s c i l l a t o r 60 4 —11 D©jp©n.cl©xi~fc Clock » » » e o « 0 « * o 0 e » o 9 » « » a o * * o « o o 0 * » « » « « o 61 4-12 Binary to Electromechanical Converter ............ 62 v i i Figure Page 4-13 Photograph of Electromechanical Converter ........ 63 4-14 Power Supply Locations 64 4-15 Regulators for +12V and -6V Power Supplies 65 4-16 A-C Test Configuration 69 AII-1 Error Caused by Exponential Ramp 79 v i i i I1ST OF TABLES Table Page 2- 1 Summary of D i g i t a l Computer S i m u l a t i o n s 15 3- 1 Summary of Computer R e s u l t s f o r Three Sets of Data f o r S i n u s o i d a l Inputs w i t h D e t e r m i n i s t i c A u x i l i a r y F u n c t i o n s 4- 1 R e s u l t s of l e v e l Input T e s t s (A-^l.324V., A 2=1.156V.) 68 4-2 R e s u l t s of A-C Te s t s 70 A I I - 1 Values of e(k, rj) as D e f i n e d by E q u a t i o n (AII - 4 ) .. 81 i x ACKNOWLEDGEMENT I would l i k e t o thank my s u p e r v i s o r , Dr. A. D. Moore, f o r h i s encouragement and a s s i s t a n c e throughout the course of the work and f o r h i s guidance i n the p r e p a r a t i o n of the t h e s i s . I would a l s o l i k e t o thank Dr. E. L. S i g u r d s o n f o r r e a d i n g the t h e s i s and f o r h i s h e l p f u l s u g g e s t i o n s . Many thanks t o my c o l l e a g u e s f o r u s e f u l d i s c u s s i o n s d u r i n g the pas t y e a r and f o r p r o o f r e a d i n g the f i n a l t h e s i s d r a f t . S p e c i a l thanks a re due my w i f e f o r h e r p a t i e n c e , encouragement and a s s i s t a n c e i n p r e p a r i n g the f i r s t t y p e w r i t t e n d r a f t . I a l s o a p p r e c i a t e the a s s i s t a n c e g i v e n by the B.C. Hydro and Power A u t h o r i t y i n the adjustment and c a l i b r a t i o n of a s t a n d a r d f o r the p r o t o t y p e s t a t i s t i c a l w a t t h o u r meter t e s t s . I g r a t e f u l l y acknowledge the f i n a n c i a l a s s i s t a n c e of the N a t i o n a l Research C o u n c i l t h rough i t s award of a B u r s a r y f o r 1965-66 and of a S t u d e n t s h i p f o r 1966-67. I would f u r t h e r l i k e t o acknowledge the f u n d s . p r o v i d e d by the N a t i o n a l Research C o u n c i l under NRC Grant A-3357-. x 1. INTRODUCTION 1.1 A B r i e f H i s t o r y of Watthour Meters The modern i n d u c t i o n - m o t o r w a t t h o u r meter i s the c u l m i n a t i o n of innumerable improvements and r e f i n e m e n t s t o the b a s i c i n d u c t i o n meter i n v e n t e d i n 1888 by S h a l l e n b e r g e r . P r e s e n t - d a y meters a re compensated f o r b e a r i n g and gear f r i c t i o n by a d d i n g a c o n s t a n t torque due t o an a d j u s t a b l e s h o r t - c i r c u i t e d l o o p i n the electromagnet a i r gap. B e a r i n g f r i c t i o n on the main s h a f t has been g r e a t l y reduced by u s i n g magnetic s u s p e n s i o n . Complex methods of compensation have a l s o been developed t o com-bat v a r i a b l e damping due t o changing c u r r e n t and/or v o l t a g e f l u x e s . Because the r e f e r e n c e i n an i n d u c t i o n - m o t o r watthour meter i s the permanent-magnet damping f l u x , g r e a t c a r e has been ta k e n t o make sure i t remains as c o n s t a n t as p o s s i b l e . Temperature-s e n s i t i v e a l l o y shunts a re used t o compensate f o r changes i n the permanent-magnet damping f l u x due t o temperature f l u c t u a t i o n s and as w e l l , a d j u s t a b l e wedge-shaped p o l e p i e c e s are used t o compensate f o r a g i n g of the permanent magnet. Even w i t h these p r e c a u t i o n s , i n d u c t i o n - m o t o r watthour meters are s u s c e p t i b l e t o i n a c c u r a c i e s r e s u l t i n g from u n c o n t r o l -l a b l e e n v i r o n m e n t a l hazards such as s t r a y magnetic f i e l d s , d i r t , d u st and v i b r a t i o n . Poor meter performance c o u l d a l s o be caused by gears which a re too t i g h t , a bent s h a f t or warped d i s k , each a p o s s i b l e r e s u l t of rough h a n d l i n g , or by improper i n s t a l l a t i o n which l e a v e s the meter s l i g h t l y o f f l e v e l . Even w i t h the numerous compensations mentioned i n the 2. opening paragraph, i n d u c t i o n - m o t o r watthour meters are s t i l l r e s t r i c t e d t o s i n g l e - f r e q u e n c y s i n u s o i d a l i n p u t s , v e r y l i m i t e d v o l t a g e ranges (+10$ of t h e i r n o m i n a l V a l u e ' s ) ^ and power f a c t o r s between u n i t y and 0.5. These r e s t r i c t i o n s and the tremendous amount of i n t e r -n a l compensation i n an i n d u c t i o n - m o t o r w a t t h o u r meter c o u l d be overcome by u s i n g an e l e c t r o n i c watthour meter. T h i s i d e a becomes e s p e c i a l l y a t t r a c t i v e when one c o n s i d e r s the p r e s e n t ' s t a t e of the a r t ' i n i n t e g r a t e d and t h i n f i l m c i r c u i t r y . R e c e n t l y \ an a l l - e l e c t r o n i c w a t t h o u r meter has been c o n s t r u c t e d which sampled and d i g i t i z e d v o l t a g e s p r o p o r t i o n a l t o the i n s t a n t a n e o u s l i h e - t o - l i n e v o l t a g e and t o the l i n e c u r r e n t . The two d i g i t i z e d samples were m u l t i p l i e d t o g e t h e r by a l a r g e amount of d i g i t a l c i r c u i t r y and the r e s u l t was s t o r e d i n one of two c o u n t e r s ; one c o u n t e r handled the p o s i t i v e p r o d u c t s w h i l e a n o t h e r s t o r e d the n e g a t i v e ones. The energy consumed was then the d i f f e r e n c e i n the c o n t e n t s of the two r e g i s t e r s . I n t h i s t h e s i s , a much s i m p l e r e l e c t r o n i c w a t t h o u r meter i s proposed which t a k e s advantage of the f a c t t h a t the c o r r e l a t i o n f u n c t i o n as g i v e n by E q u a t i o n ( l - l ) , f o r T = 0, i s the average power over the p e r i o d T. 1.2 P r e v i o u s P o l a r i t y - C o i n c i d e n c e C o r r e l a t o r s 2 l e e d e f i n e s the c o r r e l a t i o n f u n c t i o n between two v a r i a b l e s f-^(t) and f ^ . t ) , whether p e r i o d i c , random or a co m b i n a t i o n of b o t h , as the time average g i v e n by 0 1 2(f) i T ^ f p f 1 ( t ) f 2 ( t + r ) d t ( i - i ) J o I n o r d e r t o use the s t a t i s t i c a l concept of i n f o r m a t i o n t o s o l v e p r a c t i c a l problems i t has become n e c e s s a r y t o d evelop i n s t r u -ments which can compute the c o r r e l a t i o n f u n c t i o n f o r a c t u a l messages and n o i s e . An e a r l y e l e c t r o n i c c o r r e l a t o r s u c c e s s f u l l y used analogue m u l t i p l i c a t i o n and i n t e g r a t i o n of i n p u t samples t o approximate 0-^ 2 (T) by N (1-2) -L >OJX C i = l where T ^ i s the s a m p l i n g p e r i o d . T h i s c o r r e l a t o r was m o d i f i e d and c o n v e r t e d t o a d i g i t a l machine so t h a t g r e a t e r a c c u r a c y and s t a b i l i t y over l o n g p e r i o d s of time c o u l d be a t t a i n e d . IN f , ( t > c l i p p e r c l i p p e r p o l a r i t y -c o i n c i d e n c e l o g i c d e l a y T sample, sum and d i v i d e %(T) F i g u r e 1-1. B l o c k Diagram of a P o l a r i t y - C o i n c i d e n c e C o r r e l a t o r (PCC) I n an e f f o r t t o s i m p l i f y the i n s t r u m e n t a t i o n r e q u i r e d t o produce a u s e f u l s t a t i s t i c a l measure of two i n p u t s i g n a l s , the p o l a r i t y - c o i n c i d e n c e c o r r e l a t o r ^ ( F i g u r e 1-1) was developed 4. Each i n p u t was c o n t i n u o u s l y q u a n t i z e d i n t o two l e v e l s (+1 or - l ) a c c o r d i n g t o i t s i n s t a n t a n e o u s p o l a r i t y . P o l a r i t y c o i n c i -dence l o g i c was then used t o m u l t i p l y the output from one c l i p p e r by the d e l a y e d output from the o t h e r . An up-down c o u n t e r was incremented ( i n c r e a s e d or decreased by one) a c c o r d i n g t o sgn jjf^( t)j . sgnp 2( t+T)J , a t s a m p l i n g i n s t a n c e s s e p a r a t e d by ^SA° ^ e s u m w a s n o r m a l i z e ( ^ a c c o r d i n g to the number of samples which had been t a k e n , t o g i v e an output N C N(T) = | y ^ s g n [ f 1 ( i T S A ) ] / s g n [ f 2 ( i T S A + T ) ] (1-3) i = l w h ich approximated the p o l a r i t y - c o i n c i d e n c e c o r r e l a t i o n (PCC) f u n c t i o n 0PCC (T) = Q£S> i \ s g n [ f 1 ( t ) J . s g n [ f 2 ( t + T ) ] d t (1-4) However, the d i s a d v a n t a g e of PCC i s t h a t a n o n l i n e a r r e l a t i o n s h i p e x i s t s between the c o r r e l a t i o n f u n c t i o n a t the i n p u t of the c o r r e l a t o r and t h a t a t the o u t p u t . g F o r t u n a t e l y , f o r bounded i n p u t s , a method was found which l i n e a r i z e d the i n p u t / o u t p u t c o r r e l a t i o n f u n c t i o n r e l a t i o n -s h i p i n p o l a r i t y - c o i n c i d e n c e c o r r e l a t o r s . I n F i g u r e 1-2 N C N ^ ) = l ^ s g n ^ d T g A ) + n i ( i T S A ) ] . s g n [ f 2 ( i TgA+T) i = l + n 2 ( i TSA+T)J (1-5) where y ( t ) i s sampled, -summed, and n o r m a l i z e d a c c o r d i n g t o the number of samples t a k e n . I f the i n p u t s are bounded a c c o r d i n g 5. t o | f . j < A . , then the a u x i l i a r y s i g n a l s n . ( t ) a re n o i s e sources whose p r o b a b i l i t y d e n s i t i e s are g i v e n by / \ A 1 p j ( n j } = 2IT ' In . I < A . and (1-6) p a ( n d ) = 0 • l n d l > A J I t can be shown (see Appendix I) t h a t , 0 ( T ) — - ^ - A ^ N ^ f o r l a r g e N. f-L(t)->i r c l i p p e r f 2 ( t ) p o l a r i t y -c o i n c i d e n c e l o g i c sample, y sum and d i v i d e d e l a y f n 2 ( t ) c l i p p e r cN(T) F i g u r e 1-2. B l o c k Diagram of PCC w i t h M o d i f i e d Input N o i s e s o u r c e s , n . ( t ) , w i t h a r e c t a n g u l a r a m p l i t u d e p r o b a b i l i t y d e n s i t y as g i v e n by E q u a t i o n s ( l - 6 ) , are the most d i f f i c u l t p a r t of these c o r r e l a t o r s t o i n s t r u m e n t . I n one i n s t r u m e n t the n . ( t ) a re co n t i n u o u s waveforms, w h i l e i n 8 9 o t h e r s ' they a re t r a i n s of d i s c r e t e l e v e l s s y n c h r o n i z e d t o the s a m p l i n g c o n t r o l . The c o n t i n u o u s n . ( t ) were t r i a n g u l a r waveforms w i t h 3 incommensurable p e r i o d s . T h i s type of waveform was c r e a t e d by i n t e g r a t i n g an output l e v e l from a random v o l t a g e s u p p l y u n t i l 6. a p r e s e t a m p l i t u d e was reached. When t h i s p r e s e t a m p l i t u d e was rea c h e d , a new random l e v e l was made a v a i l a b l e a t the i n p u t of the i n t e g r a t o r . Because the l e v e l b e i n g i n t e g r a t e d was d i f f e r e n t f o r each p e r i o d of the t r i a n g u l a r wave, and because these l e v e l s o c c u r r e d randomly, the s l o p e s of the t r i a n g u l a r waves, and hence t h e i r p e r i o d s , were incommensurable. I t i s e a s i l y r e c o g -n i z e d t h a t the t r i a n g u l a r waveform produced by t h i s d e v i c e has a r e c t a n g u l a r a m p l i t u d e p r o b a b i l i t y d e n s i t y between two p r e s e t l e v e l s . The o t h e r type of a u x i l i a r y f u n c t i o n g e n e r a t o r , which produced 2 m d i s c r e t e v o l t a g e l e v e l s , was c o n s t r u c t e d as f o l l o w s . A m p l i f i e d n o i s e of a t h y r a t r o n was compared w i t h a c e r t a i n v o l t a g e l e v e l and the p a r t s which were g r e a t e r than t h i s l e v e l were changed i n t o a t r a i n of p u l s e s . The p u l s e r a t e of such a t r a i n has a P o i s s o n d i s t r i b u t i o n . Thus, the p r o b a b i l i t y t h a t a f l i p - f l o p , r e c e i v i n g these p u l s e s , operates r times d u r i n g a time i n t e r v a l A t ^ i s g i v e n by P ( r ) = r , 1 e ^ A t l (1-7) where X i s the mean v a l u e of the number of p u l s e s p e r second. 7. n o i s e source gate : reshape f l i p f l o p NS — t G —C R — 6 PP gate p u l s e GP F i g u r e 1-3. B i n a r y Random Source T h e r e f o r e , the p r o b a b i l i t y t h a t the f l i p - f l o p o p e r ates an even number of times (P ") i s g i v e n by even ° J - \ A t , (\-At-,) 2 ( \ . A t n ) 4 " " ( l + ^ 7 ^ — + T^— + P e v e n = 7 P ( r ) = e ( l + 2! + 4 i + " • ) r=even 2 S i m i l a r l y , , '-2\At. = i d + e x ) d - 8 ) P ( r ) = e " L ( \ - A t 1 + + ... ) P o d d - [_ , x v x / - c w v ' u w l T V. r=odd -2\At-. = | (1-e X ) (1-9) I f XAt-^ i s v e r y l a r g e , t h e n ? e v e n — a n d- ^ ] i e f l i p - f l o p i s i n s t a t e 1 a f t e r time A t ^ as o f t e n as i t i s i n s t a t e 0. With m of these u n i t s f e d i n t o a D-A c o n v e r t e r as shown i n F i g u r e 1-4, the output i s g i v e n by 8. m n ( t ) = E Q \ b . ( t ) , (1-10) where P(b .=1) o± ^ and P ( b . = 0 ) ~ \. V a l u e s of n ( t ) are d i s t r i -buted between 0 and E ^ w i t h u n i f o r m p r o b a b i l i t y and take d i s c r e t e v a l u e s s e p a r a t e d by 2 - i 1 1EQ. D-A C o n v e r t e r b x ( t ) P E 1 i RS 1 i G i NS 1 GP TbT^TT pp_ z p. R S 2 G 2 a 4 NS 2 n ( t ) f b l ( t ) m P P m RS m . G m » A NS m F i g u r e 1-4. D i s c r e t e N o i s e Generator I t i s e v i d e n t from the p r e v i o u s d e s c r i p t i o n s of the g e n e r a t o r s f o r n ( t ) t h a t the e n t i r e c o r r e l a t o r would become much s i m p l e r t o i n s t r u m e n t i f these c o u l d be reduced i n comple-x i t y . 1.3 The Proposed Instrument The proposed s t a t i s t i c a l w a t t h o u r meter w i l l have the same g e n e r a l form as the c o r r e l a t o r of F i g u r e 1-2, except t h a t the d e l a y w i l l be removed and the complex random n o i s e sources r e p l a c e d by much s i m p l e r sawtooth g e n e r a t o r s . That i s , the f u n c t i o n s n . ( t ) are d e t e r m i n i s t i c and are g i v e n by 3 n . ( t ) J a - l (1-11) where T^. i s the p e r i o d of the sawtooth, x < 0 x > 0 and a. i s the i n i t i a l phase ( i n r a d i a n s ) of the sawtooth J waveform, t t M. i s an i n t e g e r g i v e n by T H — - K M . ^ m — . J x A j 3 ' I f we d e f i n e Z . ( t ) k f ( t ) + n , ( t ) j j j (1-12) we can w r i t e the n o r m a l i z e d output of the proposed i n s t r u m e n t as sgn i = l S i n c e a l l l e v e l s of a sawtooth are d i s t r i b u t e d e v e n l y i n time (are e q u a l l y l i k e l y i f chosen randomly), i t was f e l t t h a t AjA^C-^i.'f), w i t h CJJCT) g i v e n by E q u a t i o n ( l - 1 3 ) , c o u l d approximate 0(T) as d e s c r i b e d by E q u a t i o n ( l - l ) f o r p e r i o d i c (and l e v e l ) i n p u t s w i t h slow, random a m p l i t u d e f l u c t u a t i o n s . T h i s c o n j e c t u r e w i l l now be i n v e s t i g a t e d . 2. PERFORMANCE EVALUATION OF THE PROPOSED INSTRUMENT BY COMPUTER SIMULATION 2 .1 T h e o r e t i c a l E v a l u a t i o n 8 9 Ikebe, Sato and Turner have found e x p r e s s i o n s f o r the s t a n d a r d d e v i a t i o n of the c o r r e l a t i o n e s t i m a t e s f o r p o l a r i t y -c o i n c i d e n c e c o r r e l a t o r s i n whi c h the n j ( " k ) were p r o b a b i l i s t i c and the p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n s f o r the i n p u t s were known. I n an o t h e r i n s t a n c e , Knapp 1 0 made e r r o r e s t i m a t e s f o r a s i n g l e - i n p u t s i g n a l - r e c o n s t r u c t i o n system based on the p r i n c i p l e of n o i s e i n j e c t i o n , c l i p p i n g , s a m p l i n g and f i l t e r i n g . The n o i s e source i n t h i s case was narrow-band f i l t e r e d G a ussian n o i s e which r e p r e s e n t e d a p s e u d o - d e t e r m i n i s t i c s i n u s o i d . In o r d e r t o approach the problem of f i n d i n g e r r o r e s t i m a t e s f o r the proposed system, where d e t e r m i n i s t i c s i g n a l s are added t o the i n p u t s , the i n p u t s were chosen t o be as s i m p l e as p o s s i b l e . S i n c e the c o r r e l a t o r i s t o be implemented as an energy measuring d e v i c e , o n l y the case T = 0 w i l l be c o n s i d e r e d . Thus, f o r i n p u t s f-^(t) and f 2 ( t ) we have M J L 2 j ( Ti T sgn T7~ ~ 7 U < I T S A - * j V L ^ A j i = l j =I q_.=l 1 T S A : _ / 1 T S A (2-1) where —m - l < M . ^ - s . U n f o r t u n a t e l y , even f o r s i m p l e Aj J x A j i n p u t s , i t was not p o s s i b l e t o manipu l a t e t h i s e x p r e s s i o n f o r i Cjy(O) (N l a r g e ) i n t o a t r a c t a b l e form which c o u l d be r e c o g n i z e d as a r e a s o n a b l e e s t i m a t e of 0 ( 0 ) . I n f a c t , no t h e o r e t i c a l assessment c o u l d be made of how c l o s e l y C^(0) approximated 0(0). F o r t h i s r e a s o n , and the l a c k of any o t h e r approach" t o the problem, i t was d e c i d e d t o use computer s i m u l a t i o n s t o stud y the performance of the system. 2.2 Analogue Computer S i m u l a t i o n S i n c e the proposed system ( w i t h o u t sampling) c o u l d be e a s i l y s i m u l a t e d on an analogue computer, t h i s method was chosen t o p r o v i d e a s i m p l e f i r s t e s t i m a t e of the performance. A PACE 231-R analogue computer was patched as shown i n F i g u r e 2-1. Three e x t e r n a l s i g n a l g e n e r a t o r s were used t o s u p p l y the system w i t h in-phase s i n u s o i d s f-^(t) and f 2 ( t ) and two independent t r i a n g u l a r waveforms n-^(t) and n 2 ( t ) . S i n c e the e l e c t r o n i c o u t puts from the comparators on the PACE s w i t c h e d between 0 v o l t s and +5 v o l t s , b i a s s i n g was n e c e s s a r y i n orde r t o s i m p l i f y the sgn m u l t i p l i c a t i o n . The b l o c k s i n F i g u r e 2-1 l a b e l l e d "SHAPE" were PACE i n t e g r a t o r s patched as shown i n F i g u r e 2-2. The mode of these i n t e g r a t o r s was c o n t r o l l e d by the comparator outputs so t h a t each produced a waveform w h i c h s w i t c h e d between two p r e s e t l e v e l s a c c o r d i n g t o the output from i t s r e s p e c t i v e comparator. F i g u r e 2-1. Analogue Computer 'Patch' Diagram f o r S t a t i s t i c a l Watthour Meter S i m u l a t i o n H IV) I.C. MODE 13-OPERATE MODE (comparator i n p u t +10mv) I.C.100K 100K Xo — j sfy-3 l i s e e l •^O.OOlni e l e c t r o n i c ^ * — s w i t c h IM 1 (comparator i n p u t -lOmv) I . C. . 100K 100K X o s/v 1 v V « v — V v 3 ^sec i ^ - ^ p OGl l Lf e l e c t r o n i c i » s w i t c h F i g u r e 2-2. Analogue Computer Comparator Output 'Shaper' The product sgn j j f ^ ( t ) - n ^ ( t ) J • sgn |f 2 ( t ) - n 2 ( t y j w a s then formed by summing a m p l i f i e r s 1-3, the p o s i t i v e a b s o l u t e -v a l u e c i r c u i t of a m p l i f i e r s 4 and 6 and the n e g a t i v e a b s o l u t e -v a l u e c i r c u i t a m p l i f i e r s 5 and 7. T h i s p r o d u c t was i n t e g r a t e d and compared t o the t r u e v a l u e of T f 1 ( t ) . f 2 ( t ) d t .0 formed by i n t e g r a t i n g the output of the q u a r t e r - s q u a r e m u l t i -p l i e r M l . F o r cases where the f r e q u e n c i e s of the i n p u t s i n u s o i d and the t r i a n g u l a r waveforms were not i n t e g e r m u l t i p l e s of each o t h e r , r e a s o n a b l e r e s u l t s were o b t a i n e d . F i g u r e 2-3 g i v e s the output from one of the s i m u l a t i o n s . I n t h i s case, the f r e -quency ( f ) of the i n p u t s i n u s o i d s was 50 cps, w h i l e t h e i r s a m p l i t u d e s were 0.133 and A 2; the a u x i l i a r y t r i a n g u l a r waves had f r e q u e n c i e s of f A-j_ — 510.0 cps and 569.7 cps w i t h e q u a l a m p l i t u d e s A-^  and A 2 r e s p e c t i v e l y . 14. ML . .AjA2 TIME (sec) F i g u r e 2-3. Analogue Computer Output f o r f a 50 cps, s k x = 0.133. k 2 = 1.0, f A 1 ~ 510 cps, f A 2 ~ 569.7 cps, A 1 = A 2 = 1 A l t h o u g h the analogue computer p r o v i d e d a q u i c k and v e r y s i m p l e way of c h e c k i n g the performance of the proposed system, i t was not p o s s i b l e t o c o n t r o l most system parameters w e l l enough t o e s t a b l i s h t h e i r e f f e c t on the outpu t . I t was f e l t t h a t d i g i t a l computer s i m u l a t i o n s would g i v e a d d i t i o n a l i n f o r m a t i o n . 15 2.3 D i g i t a l Computer S i m u l a t i o n s The s i x cases shown i n Table 2-1 were c o n s i d e r e d i n s e p a r a t e d i g i t a l computer s i m u l a t i o n s . -Inputs O p e r a t i o n A u x i l i a r y F u n c t i o n 1. l e v e l 2. l e v e l 3. l e v e l 4. s i n u s o i d a l 5. s i n u s o i d a l 6. s i n u s o i d a l c o n t i n u o u s sampled sampled c o n t i n u o u s sampled sampled sawtooth waveforms sawtooth waveforms n o i s e w i t h r e c t . prob. dens. sawtooth waveforms sawtooth waveforms n o i s e w i t h r e c t . prob. dens. Table 2-1 Summary of D i g i t a l Computer S i m u l a t i o n s A b r i e f d e s c r i p t i o n of each s i m u l a t i o n a l o n g w i t h a g e n e r a l f l o w c h a r t f o r the f i r s t program w i l l f o l l o w . Some of the computer s i m u l a t i o n r e s u l t s w i l l be g i v e n i n Chapter I I I . 2.3.1 Case 1 - L e v e l I n p u t s and Continuous O p e r a t i o n w i t h Sawtooth A u x i l i a r y F u n c t i o n s I n o r d e r t o make the e x p r e s s i o n g i v e n by E q u a t i o n (2-1) as s i m p l e as p o s s i b l e , the i n p u t s f-^(t) and f g ( t ) were chosen t o be c o n s t a n t s k-^  and k 2 r e s p e c t i v e l y . The e f f e c t of sa m p l i n g was e l i m i n a t e d by s i m u l a t i n g a c o n t i n u o u s l y o p e r a t i n g c o r r e l a t o r w i t h an output g i v e n by cT(o) = ^ T 0 j = l a. -1 TC sgn d t A. 4k.+2 3 / J T M. JL q r l U ( t - q j T A.) (2-2) S i n c e the i n t e g r a n d o f E q u a t i o n (2-2) i s a w e l l -d e f i n e d sequence of p o s i t i v e and n e g a t i v e l e v e l s of u n i t a m p l i -16. tude, the i n t e g r a l can be e v a l u a t e d by summing these l e v e l s w e ighted a c c o r d i n g t o t h e i r d u r a t i o n . The problem i s t h e r e f o r e t o f i n d t h i s sequence i n the q u i c k e s t p o s s i b l e way. Waveforms i n the system f o r an a r b i t r a r y time i n t e r v a l are shown i n F i g u r e 2-4. ^ f 1 ( t ) , n 1 ( t ) (a) A _ A f 2 ( t ) , n 2 ( t ) CI .0SS1 rTMCl sgnp-^t^n ^3 A sgn [ f ? ( t ) - n ^ ( t ) ] C R 0 S S 2 ~T2~ 4 S g n P l ] l ^gn [Z 21 , J sgnp-Jsgn p^] d t 0 -O t -> t r _T V -o t (b) (c) (d) (e) ( f ) F i g u r e 2-4. Waveforms I n c o r p o r a t e d i n E q u a t i o n (2-2) The program o u t l i n e d by the f l o w c h a r t i n F i g u r e 2-5 was used t o o b t a i n graphs of the e r r o r i n e s t i m a t i n g 0(0) by 17. Equat ion ( 2 - 2 ) as a f u n c t i o n of the i n i t i a l phase angle between the two a u x i l i a r y sawtooth waveforms. The s e c t i o n of the p r o -gram which determined the type of sequence shown i n F i g u r e 2 - 4 e i s g i v e n i n more d e t a i l i n F i g u r e 2 - 6 . This i s a f a s t l o g i c r o u t i n e c o n s i s t i n g of a d d i t i o n s , s u b t r a c t i o n s and c o n d i t i o n a l comparisons. For l e v e l i n p u t s t h i s method proved much f a s t e r than that of approximat ing the i n t e g r a l by increment ing t from 0 to T, c a l c u l a t i n g the i n t e g r a n d of Equat ion ( 2 - 2 ) and summing the r e s u l t s . 2 . 3 . 2 Case 2 - L e v e l Inputs and Sampling w i t h Sawtooth A u x i l i a r y F u n c t i o n s In an instrument which i s to operate over a l o n g p e r i o d of t i m e , f o r example a watthour meter, i t would be more convenient to sample the q u a n t i t y sgn(Z^)»sgn(Z2) and to add the r e s u l t s than to perform a continuous i n t e g r a t i o n . For t h i s reason , i t i s important to cons ider the e f f e c t s of sampling on the est imate of the c o r r e l a t i o n f u n c t i o n . Output from t h i s d e v i c e , f o r l e v e l i n p u t s , would be g i v e n by N 2 _ " i T M sgn A.<k .+2 37 3 SA T Aj U < i T S A - * j T A j + i T i = l j = l i T, ' r 1 it ( 2 - 3 ) where T, S A - 1< KL<T "Aj J -AJ This was an easy system to s imulate on the d i g i t a l computer because i t was only necessary to compare each saw-t o o t h waveform w i t h i t s r e s p e c t i v e l e v e l i n p u t at a d i s c r e t e i ' READ Al,T1,A2.T2 V MOLEV J Generate NOLRV l e v e l inputs evenly d i s t r i b u t e d from 0 to Al and store i n vector Bll(NOLEV). Repeat for l e v e l s between 0 and A2 and store i n vector B22 (MOLEV). SEE NEXT PAGE FOR DETAILS \ The estimate of the c o r r e l a t i o n function i s now given by RESULT=Al • A2 • SUM/DIV I Compute and write out error of the estimate. Store each error in array ERR0R(36,N0LEV) Set i n i t i a l phase, B, between the two sawtooth waveforms to zero and set PH=0, 11=1, 12=1 Define: CR0SS1 TMCl = T l - CR0SS1 CR0SS2 k T2-|B22(,I2)|0 - 5j TMC2 = T2 - CR0SS2 SUM =0.0 SEG1=CR0SS1 C1IECK=0.0 SEG2=PH-TMC2 Change input l e v e l s by s e t t i n g 12 = 12+1 Advance B by 10 deg. and define PH = £/360 Set 11=11+1 and 12=1 NO NO TES Compute and write out the MEAN and STANDARD DEVIATION over the 36 values of B for each set of l e v e l inputs Construct s c a t t e r p l o t s of ERROR as a function of B for each set of l e v e l inputs. (VOLEV plots/run) F i g u r e 2-5. ^STOP^ G e n e r a l Program Flowchart f o r D i g i t a l Computer Simulation of Continuous Sys-tem w i t h L e v e l I n p u t s and Sawtooth A u x i l i a r y Functions H oo F i g u r e 2-6. D e t a i l s of L o g i c Rout 7. SUM = SUM+SEGI DIV = DIV+TI CHECK-CHECK+ 1.0 IF (CHECK. EQ. T2) GO TO 300 SEG2=SEG2-SEG1 SEGI = CROSSI IF(SEG I -SEG2) 04. II. 07 2. SUM = SUM + SEGI SEG2 = SEG2-SEG1 SEG1 = TMC1 IFfSEGI - SEG2) 03. 12. 08 3. SUM = SUM —SEG1 DIV = DIV+T1 CHECK=CHECK+1.0 IF( CHECK. EO. T2) GO TO 300 SEG 2 =SEG2-SEG1 SEGI = CROSS1 IF(SEG1-SEG2) 02. 13. 06 4. SUM = SUM —SEGI SEG 2 =SEG2-SEGI SEGI = TMCI IF(SEG1-SEG2) 01. 13. 05 5. SUM = SUM + SEG 2 SEGI = SEGI -SEG2 SEG2 = CROSS 2 IFlSEGl -SEG2) 03. 12. 08 6. SUM =SUM+SEG2 SEGI =SEG1 -SEG2 SEG 2 = TMC 2 IFfSEGI -SEG2) 04. II. 07 7. SUM = SUM -SEG I SEGI =SEG1-SEG2 SEG2 = CROSS 2 IF(SEGI - SEG2) 02. 10. 06 8. SUM = SUM -SEG 2 SEGI = SEGI - SEG2 SEG 2 = TMC 2 IF(SEG1-SEG2) 01.09.05 9. SUM =SUM +• SEG2 DIV = DIV + TI CHECK = CHECK +1.0 IF (CHECK. EQ. T2) GO TO 300 SEGI= CROSS I SEG2 = CROSS 2 GO TO 06 10. SUM = SUM + SEG 2 SEGI = TMCI SEG 2 = TMC 2 GO TO OS 11. SUM = SUM - SEG 2 SEGI = TMCI SEG2 = CROSS 2 IFfSEGI -SEG2) 03. 12. 08 12. SUM = SUM-SEG2 DIV-DIV+T1 CHECK = CHECK +1.0 IFtCHECK. EQ, T2) GO TO 300 SEGI - CROSS I SEG 2= TMC 2 IF(SEG I -SEG 2)04. II. 07 i n Flow Chart of F i g u r e 2-5 20. number of p o i n t s i n t i m e , p e r f o r m the s g n ( Z ^ ) • s g n ( Z ^ ) m u l t i p l i -c a t i o n and sum the r e s u l t s . P l o t s of the e r r o r i n the e s t i m a t e of the c o r r e l a t i o n f u n c t i o n were o b t a i n e d , f o r a g i v e n s e t of a u x i l i a r y waveform f r e q u e n c i e s and i n i t i a l phases, as a f u n c t i o n of the t o t a l sample number N. 2.3.3 Case 3 - L e v e l Inputs and Sampling w i t h A u x i l i a r y G e n e r a t o r s which Produce Bounded Noise w i t h a U n i f o r m A m p l i t u d e D i s t r i b u t i o n At t h i s p o i n t , i t was of i n t e r e s t to see how the output from the proposed system would compare w i t h t h a t from e a r l i e r 7-9 c o r r e l a t o r s w hich use n o i s e w i t h a r e c t a n g u l a r p r o b a b i l i t y d e n s i t y as t h e i r a u x i l i a r y f u n c t i o n s . Even though the output from the l a t t e r i n s t r u m e n t can be e v a l u a t e d t h e o r e t i c a l l y f o r l e v e l i n p u t s ( S e c t i o n 3.3), a d i g i t a l computer program was w r i t t e n t o get p l o t s of as a f u n c t i o n of N which would co r r e s p o n d t o those of S e c t i o n 2.3.2. A pseudo-random sequence on [o,l] g i v e n by n . , , = n. ,«(27+l) + l l l l o (mod 235) ,3=1,2 (2.4) was used t o generate the n o i s e t o be added t o the i n p u t . T h i s sequence has almost the same mean and s t a n d a r d d e v i a t i o n as a bounded, p e r f e c t l y u n i f o r m d i s t r i b u t i o n . 2.3.4 Case 4 - S i n u s o i d a l Input and Continuous O p e r a t i o n w i t h Sawtooth A u x i l i a r y F u n c t i o n s Because the s i g n a l s from the f u n c t i o n g e n e r a t o r s i n the proposed system are p e r i o d i c , i n a c c u r a t e r e s u l t s c o u l d be o b t a i n e d i f the p e r i o d s of the i n p u t s i g n a l s t o be c o r r e l a t e d 21. were c l o s e l y s y n c h r o n i z e d t o t h e p e r i o d s of the a u x i l i a r y saw-t o o t h waveforms. S i n c e t h i s i n s t r u m e n t i s t o be implemented as a s t a t i s t i c a l w a t t h o u r meter, one of the most common type of i n p u t s w i l l be s i n g l e - f r e q u e n c y s i n u s o i d s . As i n the case of l e v e l i n p u t s , the e f f e c t s of s a m p l i n g were e l i m i n a t e d by simu-l a t i n g a system w i t h an output g i v e n by cT(o) = ^ sgn B. sin(wt+tf,) + A.<{2 <3 J J 0 d=l M. T L A j q.=l U ( t - q . T ) a. + - r 1 - l ! TC d t (2-5) where B. are the peak a m p l i t u d e s of the i n p u t s i n u s o i d s , J g\ are t h e i r i n i t i a l phases, and o t h e r v a r i a b l e s are as d e s c r i b e d p r e v i o u s l y . I n t h i s s i m u l a t i o n , 10,000 p o i n t s i n one c y c l e of a s i n u s o i d were c a l c u l a t e d and s t o r e d , s i n c e each v a l u e had t o be used many t i m e s . The component s u b s c r i p t s of the s t o r a g e v e c t o r were used as time markers, w h i l e the s i z e of the time u n i t was determined by the f r e q u e n c y of the s i n u s o i d . T h i s f r e q u e n c y and those of the sawtooth a u x i l i a r y f u n c t i o n s were r e a d i n as d a t a so they c o u l d be changed b e f o r e each r u n . Because the s i n u s o i d was r e p r e s e n t e d by a f i n i t e number of p o i n t s , some e r r o r was i n t r o d u c e d i n e v a l u a t i n g E q u a t i o n ( 2 - 5 ) . However, t h i s e r r o r was e s t i m a t e d t o be l e s s t h a n 0 . 1 % of C T(0) 22. 2.3.5 Case 5 - S i n u s o i d a l Input and Sampling w i t h Sawtooth Waveform A u x i l i a r y F u n c t i o n s The s i n u s o i d and a u x i l i a r y f u n c t i o n s were e s t a b l i s h e d by the method mentioned i n S e c t i o n 2.3.4. Output from t h i s s i m u l a t i o n i s g i v e n by N 2 sgn s i n ( t o i T g A + y ) i = l 0=1 " i T M SA T ^ r 1 U ( i T S A - * J T A j | ] + ^ " ^ (2-6) where a l l q u a n t i t i e s have p r e v i o u s l y been d e f i n e d . S i n c e t h i s program was w r i t t e n t o s i m u l a t e a s a m p l i n g system the f a c t t h a t the s i n u s o i d was s t o r e d as a f i n i t e number of p o i n t s d i d not cause any e r r o r i n e v a l u a t i n g E q u a t i o n ( 2 - 6 ) , as i t d i d i n Case 4. 2.3.6 Case 6 - S i n u s o i d a l Input and Sampling w i t h A u x i l i a r y G e n e r a t o r s which Produce Bounded Noise w i t h a U n i f o r m Amplitude D i s t r i b u t i o n A g a i n , t o compare the output from'the proposed system w i t h t h a t which would be o b t a i n e d from systems u s i n g n o i s e w i t h a r e c t a n g u l a r p r o b a b i l i t y d e n s i t y f o r the a u x i l i a r y f u n c t i o n s , the l a t t e r type of c o r r e l a t o r was a l s o s i m u l a t e d . The pseudorandom sequence of E q u a t i o n (2-4) was used t o s i m u l a t e the output from the a u x i l i a r y n o i s e s o u r c e s and the o t h e r parameters were kept the same as those used i n Case 5. 3. RESULTS OP DIGITAL COMPUTER SIMULATIONS 3.1 Continuous O p e r a t i o n w i t h L e v e l I n p u t s Parameters which c o u l d he v a r i e d i n the e v a l u a t i o n of 0^(0) f o r l e v e l i n p u t s , were: k., A., T , <x.(j=l , 2 ) and T, the l e n g t h of the i n t e g r a t i o n i n t e r v a l . A f t e r s t u d y i n g the r e s u l t s of v a r y i n g each of the parameters mentioned, i t was f e l t t h a t the most i n f o r m a t i v e way t o d i s p l a y the s i m u l a t i o n r e s u l t s was t o p l o t e T = ^CT - gSjversus 0 ! <x2 - 0^ f o r v a r i o u s v a l u e s of f ^ t ) = f 2 ( t ) = kA. T h i s graph appears i n F i g u r e 3-l a - c f o r t h r e e v a l u e s of the sawtooth f r e q u e n c y r a t i o f A i / f A 2 ' The r e a s o n T i s not i n d i c a t e d on these graphs w i l l be e x p l a i n e d l a t e r . When t h i s method of e s t i m a t i n g 0(0) i s implemented as a w a t t h o u r meter, and oc2 w i l l o c cur a t random w i t h each v a l u e of ct^ (and hence (3) b e i n g e q u a l l y p r o b a b l e . T h e r e f o r e , the t h r e e graphs of F i g u r e s 3-2a-c are g i v e n . Each d i s p l a y s the mean and s t a n d a r d d e v i a t i o n of e T f o r a s p e c i f i c r a t i o of f A 1 / f A 2 . R a t i o s of f A 1 / f A 2 i n F i g u r e s 3-2a-c cor r e s p o n d t o those used i n F i g u r e s 3 - l a - c . As an a i d t o i n t e r p r e t i n g the r e s u l t s of these simu-l a t i o n s , q u a n t i t i e s 1^. and I ^ 2 s h o u l d be d e f i n e d . I f the sawtooth f r e q u e n c y r a t i o ^ 1 / ^ 2 ' o r i s same, the p e r i o d r a t i o T A 2 ^ T A 1 ' i s w r i ' t ; " f c e n a s a n integer r a t i o w i t h no common f a c t o r s , then 1^. and I ^ 2 are the i n t e g e r s , where I > I 1 A 1 A2" 24. f T I . a X M XA2 XA2 /_ -v i . e . -s— = m— = j — ( 3 - D XA2 XA1 XA1 Now, because the i n p u t s a re l e v e l s and the n - j ( t ) a r e p e r i o d i c , the i n t e g r a n d of E q u a t i o n (2-2) w i l l be p e r i o d i c . W i th l i t t l e e f f o r t the p e r i o d of sgn J z i ( t ) J . sgn | z 2 (t ) J can be seen t o be T^ = ^^j^^ = ^A2^M s e c o n c * s * S i n c e the i n t e g r a n d of E q u a t i o n (2-2) i s p e r i o d i c , w i l l be the same f o r a l l T=pT^, where p i s any p o s i t i v e i n t e g e r . T h e r e f o r e , w i l l approach t h i s v a l u e f o r T » T ^ , and i t i s p o s s i b l e t o f o r e c a s t the l o n g - t e r m a c c u r a c y e x a c t l y , by l o o k i n g a t the v a l u e of at T=T A. In p l o t t i n g Erp v e r s u s |3 i t i s o n l y n e c e s s a r y t o c o n s i -2 J t der the range 0 ^ 3 < - r — s i n c e a l l p o s s i b l e v a l u e s of 0™ appear XA2 1 i n t h i s i n t e r v a l and are r e p e a t e d I ^ 2 times i n the i n t e r v a l 0^(3<2TC. D e t a i l s of the r e s u l t s d i s p l a y e d i n F i g u r e s 3-1 and 3-2 not r e l e v a n t t o the p o s s i b l e c o n s t r u c t i o n of a p r a c t i c a l w a t t h o u r meter w i l l not be d i s c u s s e d . From the s i m u l a t i o n r e s u l t s of F i g u r e 3-1, i t can be seen t h a t f o r T ^ T ^ , the e r r o r tends t o a v a l u e no l a r g e r than A - ^ A 2 / I A - ^ I A 2 , and the s t a n d a r d d e v i a t i o n t o a much s m a l l e r v a l u e . F o r example, i f we r e q u i r e an e s t i m a t e of 0 t o w i t h i n one p e r -c e n t , i t i s t h e o r e t i c a l l y p o s s i b l e t o a c h i e v e t h i s , or b e t t e r , i f f A 1 and f ^ 2 are chosen so t h a t ^ 1 ^ 2 ^^ iP ' w^-ere ^ e expected v a l u e of 0 i s g i v e n by KA-^A2. A l t h o u g h the r e s u l t s shown i n F i g u r e 3-1 and 3-2 use prime-number f r e q u e n c y r a t i o s f o r the sawtooth waveforms, s i m u l a t i o n s were a l s o r u n f o r non-prime r a t i o s and the l i m i t s on the e s t i m a t e e r r o r as g i v e n above, a p p l y f o r any f r e q u e n c y r a t i o . 0-06 r ~°'°6 INITIAL PHASE DIFFERENCE (DEGREES) F i g u r e 3 - l b . E r r o r i n C o r r e l a t i o n E s t i m a t e f o r 1^ = 57 and I ^ 2 = 31 ro INITIAL PHASE DIFFERENCE (DEGREES) F i g u r e 3 - l c . E r r o r i n C o r r e l a t i o n E s t i m a t e f o r I.-. = 283 and I.p = 157 28. I t was p o s s i b l e t o determine a few g e n e r a l c h a r a c t e r -i s t i c s a p p l i c a b l e f o r a l l f r e q u e n c y r a t i o s from graphs of the type shown i n F i g u r e 3-2a ( i . e . complete c u r v e s f o r s m a l l f r e -quency r a t i o s ) . F o r 1 ^ and I ^ both odd, t h e r e are 1 ^ + I ^ z e r o s between +A i n t h i s type of p l o t . The maximum s t a n d a r d A, A„ d e v i a t i o n o c c u r s f o r z e r o i n p u t and i s g i v e n by ^— n r, n T — f — . When e i t h e r 1 ^ or I ^ i s even, the curve has 1 ^ + I ^ ~ 1 z e r o s and the maxima occur on the end l o b e s of the symmetric p l o t . 3.2 E f f e c t of Sampling w i t h L e v e l Inputs I n o r d e r t o check t h a t a 'sample and add' d e v i c e has an a c c u r a c y which tends t o t h a t of c o n t i n u o u s i n t e g r a t i o n , i t i s n e c e s s a r y t o d e f i n e q u a n t i t i f e s I g ^ and 1^. R e p e t i t i o n p e r i o d o f the two sawtooth waveforms was shown t o be T^ = -^1^2 I ^ T ' A I ' L e t the s a m p l i n g p e r i o d be g i v e n by Tg^; then the p e r i o d of sgn ^ ( i T g ^ j J » s g n [ f 2 ( i T S A ) j i s g i v e n by T A I g A = "'"SA^A ^ ''"ASA w ^ e r e ^ S A ^ A 1 S r e ^ u c e c ^ ^° a n i n t e g e r r a t i o w i t h no common f a c t o r s and i s g i v e n by ^ g ^ ^ A ' From t h i s , i t i s seen t h a t i s the same f o r a l l N = p l ^ , where p i s any p o s i t i v e r e a l i n t e g e r . I t i s expected then t h a t £^ a t N - 1^ would be a p p r o x i m a t e l y e q u a l t o a t T = T^. The p l o t s i n F i g u r e 3-3 show t h i s t o be the case. The envelope of the e r r o r , e^, i n F i g u r e 3-3 was found t o be I , o H + B ,• (1-2) eN max N ° d > OJ 0-2 0-4 0-6 0-8 1-0 INPUT LEVEL (a) 0-04Y o 0 - 0 3 | ct ki 0-02 TA2'31 UJ J-m 1_ _1 L J i 0-0 0-2 0-4 0-6 INPUT LEVEL 0'8 1-0 (b) cc o 0'0015 cc cc 0-0010] o 0-0005| ^o-oooo\ 'AJ-2'3 IA2 = 1 5 7 O'O 0-2 0-4 0-6 INPUT LEVEL 0-8 1-0 F i g u r e 3-2. Standard D e v i a t i o n / ^ , of E r r o r f o r L e v e l I n p u t s as a F u n c t i o n of Input L e v e l 30. where B i s e q u a l t o f o r T = T^ and H i s a c o n s t a n t <|-( A-^A2~B)J T and u s u a l l y v e r y much l e s s than t h i s amount. 3.3 Comparison of the Two Types of P o l a r i t y - C o i n c i d e n c e Cor- r e l a t o r s w i t h l e v e l I n p u t s The output expected from the proposed i n s t r u m e n t w i l l now be compared w i t h t h a t expected from i n s t r u m e n t s which use the more complex a u x i l i a r y n o i s e g e n e r a t o r s . Output from the l a t t e r d e v i c e can be t h e o r e t i c a l l y c a l c u l a t e d , or determined by computer s i m u l a t i o n . I n o r d e r t o determine t h e o r e t i c a l l y an e x p r e s s i o n f o r the s t a n d a r d d e v i a t i o n of the s t a t i s t i c a l e s t i m a t e of 0, i t i s con v e n i e n t t o b e g i n w i t h Var C ( T ) = <C2(T)> - <C(f)> 2 R e c a l l t h a t , f o r the e s t i m a t e of an a u t o c o r r e l a t i o n f u n c t i o n , N - l 2 s g n [ f ( i T S A + f j ) + n d ( i T c(f) i A\(X) = SA 1=0 j = l where f ,• = 0 , j = 1 The e x p r e s s i o n f o r Var C(T) i s t h e r e f o r e N - l N - l 0(f) m=0 p=0 <Q X (mT g A) Q 2 (mT s A + T) Q±(pTgA) Q 2 ( PTg A + T)> N - l N <Q1(mTgA)Q2(mTSA+r)> m=0 31. (c) F i g u r e 3-3. E r r o r i n C o r r e l a t i o n E s t i m a t e ( f o r L e v e l I n p u t s ) as a F u n c t i o n of the Sample Number,N, f o r I A , = 5, I A 2 = 3 and T A S A = 5 0 0 T g A 32. where Qj(mT S A) = sgn [f (mT S A) + ( m T g A ) ] . A p p l y i n g the Theorem of Appendix I , we get ( i ) <Q± (mT S A) Q2 (mT s A + T) Q]_ (P^g A ) Q2 (P?sA+T)> f o r p ^ m = 1, and = 1 , f or p = m ( i i ) <Q1(mTSA)Q2(mTSA+r)> = i 2 < f i ( m T S A ) f 2 ( m T S A + T ) > F o r f ^ = f 2 = kA, i t i s c l e a r t h a t the e x p r e s s i o n g i v e n by ( i ) i s e q u a l t o k 4 f o r p ^  m and 1 f o r p = m. S i m i l a r l y , ( i i ) i s p e q u a l t o k . We have then Var C(f) = p [ l + ( N - l ) k 4 ] N - ( N k 2 ) ] 2 or and Var C(f) = $- [ l - k 4 | , - l < k ^ l (3-3) o - g ( T ) ] = ^ ( l - k ^ (3-4) T h i s r a t e of convergence was v e r i f i e d by computer s i m u l a t i o n . P l o t s of the type shown i n F i g u r e s 3-4a-c i n d i c a t e t h a t the e s t i m a t e , of 0 does converge as l/ \ | N as expected. The s t a n d a r d d e v i a t i o n , as g i v e n by E q u a t i o n (3-4). i s a l s o shown on these p l o t s . By comparing p l o t s l i k e those i n F i g u r e s 3-3a-c w i t h the type shown i n F i g u r e s 3-4a-c, i t i s r e a s o n a b l e t o assume t h a t f o r the cases where the i n p u t s f l u c t u a t e s l o w l y w i t h r e s p e c t t o the i n t e r n a l sawtooth f r e q u e n c i e s or are s m a l l compared t o the a u x i l i a r y f u n c t i o n , the c o r r e l a t o r u s i n g d e t e r m i n i s t i c •I i . . , . . . . , . . . _ , ; , • I I U_U_i r  ur*e"3-4~. E r r o r i n " C o r r e l a t i o n E s t i m a t e T f o r L e v e l Inputs as a F u n c t i o n of the Sample Number N, U s i n g N o i s e Generators t o Produce the A u x i l i a r y -F u n c t i o n s 34. sawtooth g e n e r a t o r s w i l l g i v e a good e s t i m a t e of the c o r r e l a t i o n f u n c t i o n much more q u i c k l y t h a n the c o r r e l a t o r which uses a u x i l i a r y random n o i s e g e n e r a t o r s . T h i s i s easy t o see when we c o n s i d e r t h a t i n the f i r s t case the e r r o r i s n e a r l y p e r i o d i c ; t h u s , when i t has been n o r m a l i z e d w i t h r e s p e c t t o the number of samples which have been t a k e n i t must, p e r i o d by p e r i o d , be reduced by l / N . However, when a n o i s e source produces the a u x i l i a r y f u n c t i o n , the s t a n d a r d d e v i a t i o n of the e r r o r b e f o r e n o r m a l i z a t i o n grows as OP and t h e r e f o r e the n o r m a l i z e d e r r o r o n l y d e c r e a s e s as 1/<V1P . 3.4 Continuous O p e r a t i o n w i t h S i n u s o i d a l Inputs I n o r d e r t o s t u d y the worst case, the sawtooth wave-forms of the a u x i l i a r y f u n c t i o n g e n e r a t o r s were chosen so t h a t ^ A l a n <^ "^A2 w e r e 0 < ^ a n c ^ P» "kke i n i t i a l a n g l e between the a u x i l i a r y waveforms,! was z e r o ( c f . , c o n d i t i o n s on a u x i l i a r y sawtooth waveforms f o r maximum a b s o l u t e e r r o r and maximum s t a n -dard d e v i a t i o n i n the l e v e l i n p u t case of S e c t i o n 3«l). For a g i v e n s e t of sawtooth f r e q u e n c i e s and i n p u t s i n u s o i d f r e q u e n c y , •the parameter which had the most e f f e c t on the e r r o r i n the c o r r e l a t i o n f u n c t i o n e s t i m a t e was the i n i t i a l phase r e l a t i o n s h i p , o, between the i n p u t and the i n t e r n a l a u x i l i a r y f u n c t i o n s . A g a i n , s i n c e a l l the waveforms we: are d e a l i n g w i t h are c o n s i d e r e d t o be e x a c t l y p e r i o d i c , the e s t i m a t e of the c o r r e l a t i o n f u n c t i o n f o r T—»co tends t o the v a l u e of the e s t i -mate f o r T. = T.I = I 'T . Here, 'T i s the p e r i o d of the As A s A s s * 1 i i n p u t s i n u s o i d and T /T. = I / l A , where I / l A i s an i n t e g e r r S A s A S A 0 30 • 20 • cc o 10 » a: cc UJ 0 -Uj -10 • o Ct Uj 0. -20 • -30 • 2.0 • Ct O 1.0 • CC CC Ui 0.0- • Ui -1.0 . o cc Uj CL -2.0 . 350 F i g u r e 3-5. 6 (DEG.) (c) E r r o r i n C o r r e l a t i o n E s t i m a t e as a F u n c t i o n of I n i t i a l Phase D i f f e r e n c e Between Input S i n u s o i d and A u x i l i a r y F u n c t i o n s 36, r a t i o w i t h no common f a c t o r s . P l o t s of 6 v e r s u s e , the e r r o r i n r e s t i m a t i n g 0 a t T=T^ are g i v e n i n F i g u r e 3 - 5 , where <S i s the i n i t i a l phase an g l e between the i n p u t and the a u x i l i a r y i n t e r n a l f u n c t i o n . A method f o r p r e d i c t i n g the magnitude and shape of these curves c o u l d not be found. However, f o r I.,»I A O/T even, A l A2 s i t was p o s s i b l e t o r e l a t e the t r e n d i n the s t a n d a r d d e v i a t i o n of the p e r c e n t e r r o r t o the product ^ ^-^^ and the the i n t e g e r 1^ which r e l a t e s the p e r i o d of the i n p u t s i n u s o i d t o the sawtooth p e r i o d s . E q u a t i o n ( 3 - 5 ) , o b t a i n e d by r o u g h l y f i t t i n g c urves t o s i m u l a t i o n r e s u l t s , g i v e s t h i s e m p i r i c a l r e l a t i o n s h i p . (3 -5) T h i s curve i s p l o t t e d i n F i g u r e 3-6 i n two s e c t i o n s ; c T ^ 6 5 ( l A ) ° ' 2 9 , • N - 0 . 5 8 ( IA1 1A2) one f o r c o n s t a n t ^A2 a n c i ^ e °^eT ^ 0 T c o n s t a n t 1^. R e s u l t s from a l l computer s i m u l a t i o n s of t h i s type a r e a l s o shown i n F i g u r e 3 - 6 . More computer runs were not made because the r e m a i n i n g ones (those f o r l a r g e r v a l u e s of ^^2.^ls2 a n c i ^h) w 0 1^^ have r e q u i r e d a l o n g e r e x e c u t i o n time t h a n was p r a c t i c a l . . 1 5 6 -5" - f Q/<3 -2 -1 ^ 1.0 . 1 1 I \ i 0 -o.s ( o 3 ' ° F i g u r e 3 - 6 ( a , b ) . 1 Standard D e v i a t i o n , ^ , of P e r c e n t E r r o r , as a F u n c t i o n of ^^j^^ a n < 1 ^A 37. These r e s u l t s a re o n l y i n t e n d e d t o show t h a t i n t h i s c a s e , the e r r o r i n e s t i m a t i n g the c o r r e l a t i o n f u n c t i o n decreases as ^hj_-tfc2 i n c r e a s e s , and i n c r e a s e s as the f r e q u e n c y of the s i n u s o i d i n c r e a s e s r e l a t i v e t o the f r e q u e n c i e s of the a u x i l i a r y sawtooth waveforms. 3.5 E f f e c t of Sampling w i t h S i n u s o i d a l I n puts F o r s i n u s o i d a l i n p u t s , as i n the case of l e v e l i n p u t s , we would expect the sampled output t o :approach p e r i o d i c a l l y the output which was r e a l i z e d f o r c o n t i n u o u s o p e r a t i o n . The p e r i o d i n t h i s case i s g i v e n by T A S g A = ^ g 1 Q& = ^ S A ^AS W H E R E T S A I S the s a m p l i n g p e r i o d , T g ^ / T A S = I S . A / I A S a n d "'"SA^AS I S A N I N _ T E S E R r a t i o w i t h no common f a c t o r s . F o r purposes of comparison w i t h p r e v i o u s r e s u l t s , the p l o t s i n F i g u r e s 3-7a-c are n o r m a l i z e d w i t h r e s p e c t t o time and t h e r e f o r e show e r r o r i n average power, as a percentage of f u l l s c a l e , as a f u n c t i o n of the number of samples which have been t a k e n . A g a i n we expect t h a t , p e r i o d by p e r i o d , the e r r o r w i l l d ecrease as l / N towards the e r r o r of the c o n t i n u o u s case f o r T = T^g. From the p l o t s i n F i g u r e s 3-7a-c we can see t h a t the e r r o r tends t o £ ( ^ A g ) a p p r o x i m a t e l y as l / N a t i n t e r v a l s much l e s s than ^ A g g ^ - T h i s occurs because w i t h i n T^gg^ t h e r e e x i s t q u a s i p e r i o d i c s t r u c t u r e s i n the unn o r m a l i z e d e r r o r . Thus, when these are n o r m a l i z e d w i t h r e s p e c t t o N, t h e i r a m p l i t u d e s decrease as l / N . S i n c e we can choose sawtooth p e r i o d s which would 38. a l l o w l o n g - t e r m r e s u l t s of any d e s i r e d a c c u r a c y , i t i s of i n t e r e s t t o note how many samples would have t o be tak e n b e f o r e the average power (and hence, the energy)' measured by t h i s d e v i c e , would f a l l w i t h i n a g i v e n range of a c c u r a c y . Table 3-1, based on the r e s u l t s of F i g u r e s 3-7a-c, r e l a t e s the number of samples t a k e n t o the expected a c c u r a c y . Sinewave Frequency 60 cps C I^g =-9J'4'O^0G'9 Sawtooth F r e q u e n c i e s ~ 6 1 cps, ^ 6 7 cps I 1 ^ = 9836, = 8955 Sampling Frequency ~98.75 sa/sec I g A = 6077 F r a c t i o n of yiax. V o l t a g e F r a c t i o n of Max. C u r r e n t Number o f Samples ( x l O 3 ) Expected A c c u r a c y (% of F.S.) E l a p s e d Time (min.) a t 100 s a / s e c . 1.000 1.00 45 270 540 1.2 0.2 0.1 7.5 45.0 90.0 0.885 0.10 20 100 200 1.0 0.2 0.1 3.3 16.6 33.3 0.770 0.01 75 150 300 0.2 ' 0.1 0.05 12.5 25.0 50.0 Table 3-1 Summary of Computer R e s u l t s f o r Three Sets of Data f o r S i n u s o i d a l Input's w i t h D e t e r m i n i s t i c A u x i l i a r y Func-t i o n s and Sampling 3.6 Comparison of the Two Types of P o l a r i t y - C o i n c i d e n c e  C o r r e l a t i o n w i t h S i n u s o i d a l I n p u t s An e x p r e s s i o n f o r the s t a n d a r d d e v i a t i o n of the o u t -put from a s a m p l i n g p o l a r i t y - c o i n c i d e n c e c o r r e l a t o r which uses n o i s e w i t h a r e c t a n g u l a r p r o b a b i l i t y d e n s i t y f o r the a u x i l i a r y 39. 0.2[ 0.0 g -0.21 ct ct "•-OK 2 -0.6 o Ui Q> -0.8 •1.0 -1.2 NO. OF SAMPLES WOK 200'i 300K f 400/T fio = Ao Figure 3-7a. ..Error i n Correlation Estimate as a Function of Sample Number for Sinusoidal Input and Sawtooth « r l A u x i l i a r y Functions F i g u r e 3-8a. E r r o r i n C o r r e l a t i o n E s t i m a t e as a F u n c t i o n of Sample Number f o r S i n u s o i d a l Input and Random Noise A u x i l i a r y F u n c t i o n s 0.2 0.0 Oz o g -0.2 UJ I-» ^ -0.4 Uj O Q: S -0.6 •0.8 4 0. NO. OF SAMPLES WOK 20 OK 300 K 40P* BJ=0.8BSAJ BfO. 100 A 2 F i g u r e 3-7P. E r r o r i n C o r r e l a t i o n E s t i m a t e as a F u n c t i o n of Sample Number f o r S i n u s o i d a l Input and Sawtooth A u x i l i a r y F u n c t i o n s Bj=0.885A j B2=0.100A2 F i g u r e 3-8b. E r r o r i n C o r r e l a t i o n E s t i m a t e as a F u n c t i o n of Sample Number f o r S i n u s o i d a l Input and Random No i s e A u x i l i a r y F u n c t i o n s 0.8 ct o ct ct Uj Uj O Ct U] 0. 0.6" 0.4 • 0.2 0.0 NO. OF SAMPLES •m * ' • * 100K 20 OK 300K Bj = 0.77Af B2=0.01A2 400K 41. F i g u r e 3-7c. ( E r r o r i n C o r r e l a t i o n E s t i m a t e as a F u n c t i o n of • Sample Number f o r S i n u s o i d a l Input and Sawtooth A u x i l i a r y F u n c t i o n s BJ=0.77AJ 0.4 0.2 0.0 ct o Ui 5 °-4 £ * OS 0.8 1.0 Bf0.01A2 NO. OF SAMPLES fOOK 20 OK 300K 40 OK F i g u r e 3-8c. E r r o r i n C o r r e l a t i o n E s t i m a t e as a F u n c t i o n of Sample Number f o r S i n u s o i d a l Input and Random Noise A u x i l i a r y F u n c t i o n s 42.. functions can be determined t h e o r e t i c a l l y f o r p e r i o d i c input s i g n a l s . In cases where the input i s not c l o s e l y synchronized o. w i t h the sampling period, i t was found that oo .4 Var C(T)= t - h ) ( a ' h + ^ h ) 2 h=l (3-6) CO CO b ) ) ( a ^ 1 ) ( a ^ 2 + b ^ ) ( l + c o s ^ T ) h 1 = l h 2=l h ^ where the a^ and b^ are F o u r i e r c o e f f i c i e n t s given by T/2 a h = f ( f ( t ) c o s ( h w t ) d t -T/2 and b h = |j f (t)sin ( h c o t)dt -T/2 and the input f ( t ) , of period X, i s expressed as oo f ( t ) =X 1 j a ^ cos (hoot) + b^ sin (hojtTj , OJ = ~ h=l For a s i n g l e frequency s i n u s o i d a l input given by f ( t ) = kA s i n cot, Equation (3-6) gives Var 0 ( f ) - |1 ( i - |) .41 i J/2 or cr[c(T)]- A 2 ( | [l - I])' (3-7) 43. Equation (3-7) for the standard deviation of the correlation estimate i s indicated on the plots of Figure 3-8a-c. These plots were obtained from the simulation mentioned i n Section 2-6. For purposes of comparison, corresponding graphs i n Figures 3-7 and 3-8 are shown above one another; the former show the results of using deterministic a u x i l i a r y functions -the l a t t e r the results of using random functions, with the same input i n both cases. By comparing these two sets of graphs, i t can be seen that the correlator using deterministic a u x i l i a r y functions w i l l give a reasonable estimate of the correlation function f o r this type of input more rapidly than that using noise for the a u x i l i a r y functions. This i s especially true, as expected, for small inputs. Because the simulations of Section 3-5 each required a r e l a t i v e l y large amount of computer time, only the results shown were obtained. It was f e l t that additional information on the f e a s i b i l i t y of the proposed s t a t i s t i c a l watthour meter could be obtained by constructing a prototype and testing i t under various load conditions. 4. INSTRUMENTATION AND TESTING OP THE STATISTICAL WATTHOUR METER 4.1 Implementing the C o r r e l a t o r as a Watthour Meter The c o r r e l a t i o n f u n c t i o n g i v e n by 0 1 2 ( T ) i T i ^ i r T f i ( ^ f 2 ^ d t Jo can be t r a n s f o r m e d t o a measure of average power over a f i n i t e i n t e r v a l by l e t t i n g r = o , T -»oo ~ T—•T.. , l a r g e ' f 2 ( t ) ~ v ( t ) , ( 4 _ 2 ) and f 2 ( t ) ^  i ( t ) . Energy i s then average power m u l t i p l i e d by t i m e , or E ( v , i , T ) = T 0 =1 v ( t ) i ( t ) d t ^ 0 F o r l o n g p e r i o d s of o p e r a t i o n , t h i s i s most e a s i l y i n s t r u m e n t e d by u s i n g the e s t i m a t e of the c o r r e l a t i o n f u n c t i o n g i v e n by C^. That i s , the energy w i l l be g i v e n by the s t a t i s t i c a l e s t i m a t e d e f i n e d by E ( v , i , T ) ~ E ( v , i , N ) k ^r- A n A ? C M (4-3) I t i s now p o s s i b l e t o f i n d a c o n s t a n t which w i l l r e l a t e the 'count', c, i n the output up-down c o u n t e r t o the energy consumed. S i n c e C^ = c/N, we have A A c E ( v , i , T ) ^ ^ - 2 - (4-4) X S A Ll N L2 POWER SUPPLY FUNCTION GENERATOR COMPARATOR 2 FUNCTION EXCLUSIVE GENERATOR OR H W v,(t) MECHANICAL REGISTER LOGIC AND CURRENT DRIVERS UP-DOWN COUNTER SAMPLE CONTROL F i g u r e 4-1. B l o c k Diagram of Proposed S t a t i s t i c a l Watthour Meter 46. T h i s i s v e r y u s e f u l because i t e l i m i n a t e s the n e c e s s i t y of k e e p i n g t r a c k of the sample count f o r the purpose of ' n o r m a l i z i n g ' the output a c c o r d i n g t o sample s i z e . I t a l s o p r o v i d e s a v e r y s i m p l e means by which the wattho u r meter can be a d j u s t e d . A b l o c k diagram of the meter w i l l t ake the form shown i n F i g u r e 4-1. Each b l o c k w i l l now be d i s c u s s e d . 4.2 V o l t a g e and C u r r e n t Measurement I f s i g n a l s f ^ ( t ) and f ^ C t ) , p r o p o r t i o n a l r e s p e c t i v e l y to the i n s t a n t a n e o u s l i n e - t o - l i n e v o l t a g e and t o the sum of the l i n e c u r r e n t s , were a v a i l a b l e , the c o r r e l a t o r d e s c r i b e d e a r l i e r c o u l d be used as a wattho u r meter. I n o r d e r t o keep t h i s s e c t i o n b r i e f , o n l y methods f o r measuring energy i n two-wire d-c and i n two- or t h r e e - w i r e s i n g l e - p h a s e a-c power systems w i l l be c o n s i d e r e d . 4.2.1 V o l t a g e Measurement The o b j e c t of the v o l t a g e measuring schemes shown i n F i g u r e s 4-2a-c i s t o o b t a i n a v o l t a g e p r o p o r t i o n a l t o the l i n e v o l t a g e , w i t h v L ( t ) = 0 ^ v y ( t ) = V R/2 v L ^ m i n ~ V t ) > 0 (4-5) v L ^ m a x - V ^ < V R a n d | v L ^ | m a x " V * ) - ^ ^ ' where V R i s the r e f e r e n c e v o l t a g e f o r the a u x i l i a r y - f u n c t i o n g e n e r a t o r . Each method of measuring v o l t a g e s h o u l d shunt l e s s 47. line load TO COMPARATOR line vL(t) load I Invert r •Ar— TO COMPARATOR (b) line v,(t) GJUUUJ pnnnnrL load (cj TO COMPARATOR F i g u r e 4-2. P o s s i b l e V o l t a g e M easuring Schemes 48. than 0.1$ of the minimum c u r r e n t f o r which the w a t t h o u r meter i s r a t e d . The t h r e e - w i r e systems shown i n F i g u r e s 4-2 b,c are f e d by a c e n t r e - t a p p e d t r a n s f o r m e r secondary. A l t h o u g h v r ( t ) 2 ± - v r ( t ) , i t i s assumed t h a t a measure of the l i n e - t o -L l L 2 . l i n e v o l t a g e o? o f both l i n e - t o - g r o u n d v o l t a g e s i s needed t o o b t a i n an a c c u r a t e energy e s t i m a t e . S i n c e the s i g n a l s ^ ( t ) and V j - ( t ) , r e p r e s e n t i n g l i n e v o l t a g e and c u r r e n t r e s p e c t i v e l y , must have the same phase d i f f e r e n c e as the v o l t a g e and c u r r e n t they r e p r e s e n t , the elements i n F i g u r e s 4-2 w h i c h c o u l d be r e s p o n s i b l e f o r some phase s h i f t s h o u l d be mentioned. The r e s i s t i v e v o l t a g e d i v i d e r f e e d i n g i n t o an o p e r a t i o n a l a m p l i f i e r s h o u l d cause no concern. However, the p o t e n t i a l t r a n s f o r m e r o p e r a t i n g a t 50 t o 60 cps c o u l d i n t r o d u c e a s l i g h t amount of phase s h i f t . To a v o i d a l a r g e d e p a r t u r e of the v o l t a g e r a t i o from the t u r n s r a t i o , 12 w i t h consequent phase s h i f t s , i t can be shown t h a t the w i n d i n g r e s i s t a n c e s and l e a k a g e i n d u c t a n c e s must be kept low. 4.2.2 C u r r e n t Measurement The c u r r e n t measuring systems shown i n F i g u r e 4-3a-e s h o u l d produce a v o l t a g e p r o p o r t i o n a l t o the l i n e c u r r e n t ( s ) , w i t h i L ( t ) = 0 ~ v x ( t ) = V R/2 H ^ m i n ~ ^ ( t ) > 0 (4-6) - l / ^ m a x ~ - ! ^ ) < \ a n d l ^ l m a T V I ( t ) ^ V 2 ± V 2 ' 50. 1 O ' vWv 2 TO COMPARATOR (d) sA/ TO COMPARATOR (e) F i g u r e 4-5d,e. P o s s i b l e Methods of C u r r e n t Measurement i n Three-Wire Systems 51. The c u r r e n t - m e a s u r i n g d e v i c e s h o u l d not i n t r o d u c e more t h a n 0.1$ drop i n the l i n e v o l t a g e . I n F i g u r e s 4-3a,d, the l o a d r e s i s t o r on the secondary s i d e of the c u r r e n t t r a n s f o r m e r must be v e r y s m a l l compared t o the i n p u t impedance of the comparator so as t o p r o v i d e a c o n s t a n t l o a d . At the same time , t h i s l o a d r e s i s t a n c e s h o u l d be l a r g e enough t o p r o v i d e a v o l t a g e v-j-(t) adequate f o r the comparator i n p u t . The scheme shown i n F i g u r e 4-3c can o n l y be used i f t h e r e are no grounds t o the r i g h t of the c u r r e n t - m e a s u r i n g d e v i c e . I f the s m a l l r e s i s t o r i n the l i n e were p l a c e d on the 'hot' s i d e of the l o a d , the d-c d i f f e r e n t i a l a m p l i f i e r shown i n t h i s f i g u r e would need an i m p r a c t i c a l l y h i g h i n p u t common-mode r e j e c t i o n r a t i o i n o r d e r t o a c c u r a t e l y measure c u r r e n t s over a 1 0 0 - t o - l range. I n F i g u r e s 4-3a,b,d,e i t 1 , i s d e s i r a b l e t h a t the t r a n s -formers i n t r o d u c e n e g l i g i b l e phase s h i f t . That i s , f o r p o t e n t i a l t r a n s f o r m e r s , the le a k a g e impedance must be kept s m a l l , and f o r 12 c u r r e n t t r a n s f o r m e r s the Q of the secondary c i r c u i t must be h i g h . 4.3 A u x i l i a r y - F u n c t i o n Generators The a u x i l i a r y f u n c t i o n g e n e r a t o r s are b a s i c t o the o p e r a t i o n of the system s i n c e t h e i r a c c u r a c y d i r e c t l y a f f e c t s the o v e r a l l i n s t r u m e n t performarjce. From Appendix I I , i t can be seen t h a t the e x p o n e n t i a l sawtooth g e n e r a t o r w i l l cause n e g l i g i b l e e r r o r as l o n g as M/A>100. The most c r i t i c a l p r o p e r t i e s of the 52. sawtooth waveform are i t s upper and l o w e r l i m i t s and i t s p r e c i s e l o c a t i o n about the symmetry a x i s of the i n p u t s i n u s o i d s . Two g e n e r a t o r s w i l l be suggested. The f i r s t i s a s i m p l e sawtooth g e n e r a t o r which l e a d s t o a c e r t a i n amount of i n s t r u m e n t a d j u s t -ment. A l s o suggested i s a second, more e x p e n s i v e , g e n e r a t o r which produces a t r i a n g u l a r waveform between two w e l l - d e f i n e d l e v e l s . 13 4.3.1 A Simple Sawtooth G e n e r a t o r T h i s g e n e r a t o r produces a sawtooth by c h a r g i n g a c a p a c i t o r from a c o n s t a n t - c u r r e n t source produced by a b o o t -s t r a p c i r c u i t . The c a p a c i t o r d i s c h a r g e i s c o n t r o l l e d by a u n i -j u n c t i o n t r a n s i s t o r s w i t c h . More s p e c i f i c a l l y , b o o t s t r a p p i n g m a i n t a i n s a c o n s t a n t v o l t a g e a c r o s s R and thus a c o n s t a n t c u r r e n t i n t o the c a p a c i t o r . When the v o l t a g e on t h i s c a p a c i t o r reaches the upper t r i p p o i n t of the u n i j u n c t i o n t r a n s i s t o r , the s w i t c h i s c l o s e d and the c a p a c i t o r r a p i d l y d i s c h a r g e s . As the v o l t a g e f a l l s t h r o ugh the l o w e r t r i p p o i n t , the s w i t c h i s opened and the c a p a c i t o r i s a g a i n charged by the c o n s t a n t - c u r r e n t s o u r c e . L i n e a r i t y i s improved by s p l i t t i n g the c a p a c i t o r C i n t o and and p r o v i d i n g a feedback r e s i s t o r By employing t h i s method of feedback, i t was found p o s s i b l e t o c o n s t r u c t a sawtooth g e n e r a t o r c o n t r i b u t i n g l e s s t h a n 0.5% e r r o r t o the o v e r a l l w a t t h o u r meter. The main d i s -advantage of t h i s t y pe of g e n e r a t o r i s the u n c e r t a i n t y of the l o w e r t r i p p o i n t of the u n i j u n c t i o n t r a n s i s t o r . I n o r d e r t o 53. < OUT F i g u r e 4-4. A Simple Sawtooth Generator e s t a b l i s h the upper and l o w e r bounds on the ramp more a c c u r a t e l y , the t r i a n g u l a r wave g e n e r a t o r covered i n the f o l l o w i n g s e c t i o n was developed. 4.3.2 An A c c u r a t e Triangular-Wave Generator The concept of r e p l a c i n g the n o i s e sources by d e t e r -m i n i s t i c ramp g e n e r a t o r s a p p l i e s t o sawtooth and t r i a n g u l a r waveforms. The s i m u l a t i o n r e s u l t s shown i n the p r e v i o u s c h a p t e r , i n g e n e r a l , a p p l y e q u a l l y w e l l t o t r i a n g u l a r a u x i l i a r y f u n c t i o n s . F o r t h i s a p p l i c a t i o n , i t i s more c o n v e n i e n t t o produce waveforms w i t h n e a r l y e q u a l p o s i t i v e and n e g a t i v e s l o p e s than i t i s t o generate sawtooth waveforms. I n t h i s ramp g e n e r a t o r , the upper and l o w e r bounds are e s t a b l i s h e d t o w i t h i n 1 m i l l i v o l t , i . e . , t o b e t t e r than 0.1$ of the peak a m p l i t u d e of the t r i a n g u l a r waveform produced. A l t h o u g h the c i r c u i t shown i n F i g u r e 4.5 c o n t a i n s a few more 220K -6V" F i g u r e 4-5. An A c c u r a t e Triangular-Wave Generator 55. components than the p r e v i o u s sawtooth g e n e r a t o r , the p r i n c i p l e of o p e r a t i o n i s e x t r e m e l y s i m p l e . Assume FEQ^ (an N-channel d e p l e t i o n mode FET) i s ON; then the p o s i t i v e i n p u t t o the comparator i s a t V R v o l t s ( t h e n e g a t i v e i n p u t i s assumed t o be a t a l e v e l < v o l t s ) which c r e a t e s a p o s i t i v e output from the comparator. We see t h a t i n ON, Q2 i s ON, i s OFF, and i s ON, whi c h means F E ^ i s ON, FET 2 i s OFF and the M i l l e r i n t e g r a t o r i s b e i n g charged from a n e g a t i v e s o u r c e , c r e a t i n g a p o s i t i v e - g o i n g ramp. When t h i s ramp reaches V^, the output of the comparator becomes s l i g h t l y n e g a t i v e so t h a t Q^ , Q2, and r e v e r s e t h e i r s t a t e s which causes F E ^ t o be 'pinched' OFF, FET 2 t o come ON and a n e g a t i v e -g o i n g ramp t o appear a t the output of the M i l l e r i n t e g r a t o r . T h i s ramp c o n t i n u e s u n t i l i t reaches the ground l e v e l a t the p o s i t i v e i n p u t of the comparator, a t which time the p r o c e s s i s r e p e a t e d . The F a i r c h i l d ^A710 comparator used i n t h i s g e n e r a t o r has an i n p u t h y s t e r e s i s of 1 m i l l i v o l t w h i l e the |iA702 a m p l i f i e r used i n the i n t e g r a t o r i s r e s p o n s i b l e f o r a s l o p e e r r o r , e g , of 0.5%, which a c c o r d i n g t o Appendix I I c o u l d cause a 0.2% e r r o r i n the watthour-meter o u t p u t . 4.4 L o g i c f o r M u l t i p l i c a t i o n of Sgn F u n c t i o n s R e f e r r i n g t o F i g u r e 4-1 and E q u a t i o n s (1-12) and (1-13) we see t h a t each increment of the sum i s determined by the com-p a r a t o r o u tputs and a l o g i c c i r c u i t f o r m u l t i p l y i n g the sgn f u n c t i o n s . Two comparators, one g i v i n g an output r e p r e s e n t i n g 56. sgn |j£]_"~n3_J a n ( i " t n e ot,her r e p r e s e n t i n g sgn J ^ - 1 ^ ] A R E U S E ( ^ ^ ° c o n t r o l an up-down c o u n t e r . The comparator outputs a re l o g i c a l 1 and 0 r e p r e s e n t i n g sgn ( Z . ) e q u a l t o 1 and - 1 , r e s p e c t i v e l y . I n o r d e r t o m u l t i p l y the two sgn f u n c t i o n s , we s i m p l y p e r f o r m the B oolean o p e r a t i o n d = ab + a'b' (4-11) That i s , d=l o n l y when bo t h a and b are 1 or 0. T h i s r e p r e s e n t s a complemented e x c l u s i v e OR and can be r e a l i z e d by NOR l o g i c as shown i n F i g u r e 4-6. F i g u r e 4-6. M u l t i p l i c a t i o n L o g i c f o r sgnp 1 J.sgn p 2 J 4.5 I n t e g r a t e d - C i r c u i t Up-Down Counter The up-down c o u n t e r , a p a r t from two t r a n s i s t o r c u r r e n t d r i v e r s , was c o n s t r u c t e d from s t a n d a r d F a i r c h i l d (J.L923 f l i p -f l o p s and RTL i n t e g r a t e d c i r c u i t r y . The b l o c k s i n F i g u r e 4-7 l a b e l l e d 'GATE' are i n v e r t e r s (-?\iL914 w i t h one i n p u t grounded), w i t h the e m i t t e r connected t o the 'up-down' c o n t r o l l i n e . F i g u r e 4-7. Up-Down Counter and Input C i r c u i t r y /°/p C (indep) Oo/p C T0 On dep) T L L H H H H . H i L H H H L © 1 • L H H H L L H ' H I H L © L H H i • L PERMISSIBLE TRANSITION • UNWANTED TRANSITION F i g u r e 4-8. Up-Down Counter C o n t r o l T r u t h Table / 2 3 4 lunnninmnnnnnnnnn n n on nn JI J] OlO D J I L 1 JUL F i g u r e 4-9. Up-Down Counter Waveforms 59. Since, i n t h i s a p p l i c a t i o n , the up-down counter i s only required to operate at frequencies below 10 kcps, a s e r i a l counter was used. This avoids any chance of the p o s s i b l e timing problems associated w i t h p a r a l l e l counters. To guarantee that the b i t s i n the counter do not change when the clock pulse i s absent, the f l i p f l o p s (except f o r the f i r s t one) are d i s a b l e d by applying a p o s i t i v e l e v e l to both the S and R inputs. However, during the clock pulse the counter f l i p - f l o p s can receive a f a l s e t r i g g e r i f the 'up-down' c o n t r o l l i n e s change and the gates are i n a p a r t i c u l a r s t a t e . This can be seen by l o o k i n g at the t r u t h t a b l e of Figure 4-8. In order to prevent t h i s from oc c u r r i n g the c o n f i g u r a -t i o n bounded by the dotted l i n e i n Figure 4^5 was used. The 3pX914 NOR-gates and p,L900 b u f f e r are connected as 2 S-R f l i p -f l o p s w i t h an o v e r a l l a b i l i t y to hold the c o n t r o l i n one s t a t e during the clock pulse. The l e v e l of the c o n t r o l held throughout the clock pulse i s that which e x i s t s at the l e a d i n g edge of the clock pulse. To ensure the c o n t r o l l i n e s have s e t t l e d before the toggle i s a p p l i e d to the f i r s t f l i p - f l o p i n the counter, the toggle i s delayed and appears 100 psec a f t e r the l e a d i n g edge of the clock pulse. 4.6 Sample Con t r o l Clock From Equation ( 4 - 4 ) , i t i s apparent that the accuracy of the watthour meter i s d i r e c t l y a f f e c t e d by the accuracy of the clock which toggles the counter. I f the 60. watthour meter i s to operate to a p r e c i s i o n determined "by the instrumentation and not by the p r i n c i p l e , i t i s almost impera-t i v e that an independent c l o c k , or one of the type mentioned i n Section 4.6.3, be used. That i s , the clock should not be synchronized w i t h the input unless the input waveforms have a very d e f i n i t e shape. This i s mentioned i n Section 4.6.2. 4.6.1 Independent Clock' Clocks e x i s t which operate to an accuracy of the 12 order of 1 part i n 10 . However, since we are attempting to propose a p r a c t i c a l watthour meter, we must s e t t l e f o r a clock which i s inexpensive and yet does not d e t e r i o r a t e the p e r f o r -mance of the meter. A simple clock using a tuning-fork reference meets t h i s requirement, as i t can probably be constructed f o r a reasonable cost and s t i l l have an accuracy of b e t t e r than 1 part i n 10 4. A simple c o n f i g u r a t i o n used by a well-known watch manufacturer and claimed to have an accuracy of 1 part i n 43,000, appears i n Figure 4-10. magnet phase d r i v e d r i v e sensing c o i l c o i l c o i l no. 1 no. 2 tuning f o r k (the c o i l s a c t u a l l y enclose the permanent* magnets on the tuning fork) Figure 4-10. Tuning Fork O s c i l l a t o r 61. Many o t h e r more expensive schemes e x i s t but w i l l not be con-s i d e r e d h e r e . 4.6.2 Sampling a t L i n e Frequency I t i s p o s s i b l e t o a v o i d . t h e d i f f i c u l t i e s of p r o d u c i n g a p r e c i s i o n c l o c k t o c o n t r o l s a m p l i n g r a t e i f the harmonic con-t e n t of l i n e v o l t a g e and c u r r e n t can be c o n s i d e r e d n e g l i g i b l e (a s i t u a t i o n which may seldom e x i s t ) , i n which case s a m p l i n g s h o u l d o c c u r a t each p o s i t i v e peak of l i n e v o l t a g e . The d i g i t a l w a t t h o u r meter w i l l t hen g i v e an i n d i c a t i o n w h i c h i s t w i c e the metered energy. I n the p r e s e n t i n s t r u m e n t , the s a m p l i n g c l o c k may be produced by the c i r c u i t shown i n F i g u r e 4-11 and c o r r e c t i o n f o r the s c a l e f a c t o r of two can be made by b y - p a s s i n g one stage i n the c o u n t e r . T 0.02^f ={= IK compara-t o r -{> c l o c k output F i g u r e 4-11. Dependent C l o c k 4.6.3 Sampling S y n c h r o n i z e d t o the L i n e Frequency Another p o s s i b i l i t y i s t h a t of g e n e r a t i n g harmonics of the l i n e f r e q u e n c y , f i l t e r i n g one out and c o u n t i n g i t down 62. w i t h a b i n a r y c o u n t e r . T h i s p r o c e s s c o u l d be r e p e a t e d as many times as i s p r a c t i c a l u n t i l a d e s i r a b l e s a m p l i n g r a t e i s ob-t a i n e d . 4.7 Output C o n v e r t e r and M e c h a n i c a l R e g i s t e r l o n g - t e r m output from the wattho u r meter i s s t o r e d i n a 5-place d e c i m a l m e c h a n i c a l r e g i s t e r . A m e c h a n i c a l r e g i s t e r has the advantage t h a t the count i s not d e s t r o y e d i n the event of a power f a i l u r e . I t i s a l s o more c o n v e n i e n t t o re a d t h a n a count s t o r e d e l e c t r o n i c a l l y . To c o n v e r t a b i n a r y count i n the f l i p - f l o p c o u n t e r t o a d e c i m a l count i n the m e c h a n i c a l r e g i s t e r , the e l e c t r o - m e c h a n i c a l c o n v e r t e r shown i n F i g u r e s 4-12 and•4-13 was c o n s t r u c t e d . -10-L i r 0 11 0 10' 1.8K JB ) 7 V -1.8K B ) s / V -L I S \M£SU&; o -6V - \ f l M M O / 0 -6V *B 1.8K r i f r J ^ w -TjiaaaflfliL'—0 -6V 1.8K ry^ B ) yv— -ddUUijm—© -6V F i g u r e 4-12. B i n a r y t o E l e c t r o m e c h a n i c a l C o n v e r t e r Each electromagnet c o i l i n t h i s d e v i c e i s s p l i t i n two so t h a t at any one t i m e , h a l f of a c o i l i s e n e r g i z e d and h a l f of the d i a m e t r i c a l l y o p p o s i t e c o i l i s e n e r g i z e d . T h i s p r o v i d e s a 63. symmetric f o r c e on the r o t a t i n g permanent magnet. The c o i l s a re connected so t h a t as a 6-volt l e v e l i s a p p l i e d t o the f o u r c o i l s i n the sequence ...2,3,4,1,2,3...., the m e c h a n i c a l c o u n t e r i n c r e a s e s i t s count and, of c o u r s e , d e c r e a s e s i t s count when t h i s sequence i s r e v e r s e d . The f o u r i n p u t s t o the c o n v e r t e r are o b t a i n e d from the b i n a r y c o u n t e r as shown i n F i g u r e 4-7. F i g u r e 4.13. Photo of E l e c t r o m e c h a n i c a l C o n v e r t e r 4.8 Power S u p p l i e s The comparators used i n the meter r e q u i r e +12V and -6V w h i l e the l o g i c c i r c u i t r y r e q u i r e s a p p r o x i m a t e l y +3.6V. In o r d e r t o keep the power s u p p l y as s i m p l e as p o s s i b l e , i t was d e c i d e d t o use the +12V" s u p p l y as a r e f e r e n c e f o r the sawtooth g e n e r a t o r s . T h i s meant t h a t as s m a l l a c u r r e n t as p o s s i b l e s h o u l d be drawn from t h i s s u p p l y , which made i t n e c e s s a r y t o use the -6V s u p p l y t o d r i v e the l o g i c and the e l e c t r o m e c h a n i c a l c o n v e r t e r . F i g u r e 4-14 shows the arrangement used. 64. +12 v o l t s u p p l y 55mA i 2 , x 420mA_ l F u n c t i o n G e n e r a t o r s and Summing A m p l i f i e r s 40mA Y L o g i c & Counter E l e c t r o -; mechanical? c o n v e r t e r F i g u r e 4-14. Power Supply L o c a t i o n s From F i g u r e 4-14, i t can be seen t h a t the n e g a t i v e v o l t a g e source must be capable of s u p p l y i n g 515 mA a t -6.0 +0.1V, w h i l e the p o s i t i v e v o l t a g e source must be a b l e t o s u p p l y 95 mA a t +12.0 +0.01V. S e r i e s - r e g u l a t e d s u p p l i e s were chosen i n each ca s e , w i t h more c a r e f u l temperature compensation used on the +12V r e f e r e n c e s u p p l y . Both s u p p l i e s were de s i g n e d so t h a t the complete r e g u l a t o r , w i t h the e x c e p t i o n of the s e r i e s con-t r o l element, c o u l d be f a b r i c a t e d as an i n t e g r a t e d c i r c u i t and e n c a p s u l a t e d w i t h i t s own t e m p e r a t u r e - s t a b a l i z i n g element, a t e c h n i q u e which can be expected t o be v e r y common i n the f u t u r e . A t r a n s f o r m e r w i t h a two-winding secondary was r e q u i r e d t o p r o v i d e the power f o r the meter. Because of the s m a l l r e s i s t a n c e t o ground seen by the f i l t e r s , i t was n e c e s s a r y f o r these secondary w i n d i n g s t o p r o v i d e v o l t a g e s as h i g h as 9V and 18V rms r e s p e c t i v e l y f o r the -6V and +12V d - c s u p p l i e s . rectifier diodes are 1N4001 F i g u r e 4-15. R e g u l a t o r s f o r +12v and - 6 V Power S u p p l i e s 66 . Even w i t h t h e s e v o l t a g e s a v a i l a b l e , f u l l - w a v e r e c t i f i c a t i o n and 1000-ij.Fd f i l t e r c a p a c i t o r s were r e q u i r e d i n o r d e r t o have enough v o l t a g e t o be a b l e t o p r o v i d e p r o p e r r e g u l a t i o n . When the e f f i c i e n c i e s of the r e c t i f i e r , f i l t e r and r e g u l a t o r are t a k e n i n t o a c c o u n t , the 9V rms w i n d i n g must be a b l e t o s u p p l y 115 mA, w h i l e the 18V rms w i n d i n g must be a b l e t o p r o v i d e 610 mA. Thus a. 10 watt t r a n s f o r m e r w i t h a 115v. rms p r i m a r y w i n d i n g was r e q u i r e d . 4.9 Test Procedure and R e s u l t s A s t a t i s t i c a l w a t t h o u r meter based on the f i r s t e i g h t s e c t i o n s of t h i s c h a p t e r was c o n s t r u c t e d and t e s t e d . T h i s i n s t r u m e n t was not i n t e n d e d t o be a p r o d u c t i o n p r o t o t y p e , but s i m p l y a n o t h e r means of g e t t i n g i n f o r m a t i o n on the performance and f e a s i b i l i t y of t h i s type of meter. 4.9.1 DC I n p u t s T e s t s were r u n w i t h l e v e l i n p u t s p r o v i d e d by two r e s i s t i v e v o l t a g e d i v i d e r s a c r o s s a zener d i o d e . The i n p u t s were b u f f e r e d by F a i r c h i l d jiA702A IC a m p l i f i e r s b e f o r e e n t e r i n g the comparator of F i g u r e 4-1. A d-c d i f f e r e n t i a l v o l t m e t e r was used t o check the l e v e l s b e f o r e and a f t e r each t e s t . The output from, each r u n was n o r m a l i z e d w i t h r e s p e c t t o time and a c o n s t a n t of p r o p o r t i o n a l i t y between the product of the i n p u t l e v e l s and the n o r m a l i z e d output was e s t a b l i s h e d . T h i s a l l o w e d v a l u e s f o r the expected output t o be c a l c u l a t e d . The r e s u l t s of these t e s t s are compared t o the expected output 67. i n Table 4-1. The l a r g e r e a d i n g e r r o r w h i c h appears i n Te s t s 6 and 11 i s a t t r i b u t a b l e t o the f a c t t h a t the z e r o i n p u t f o r each ch a n n e l was not l o c a t e d p r e c i s e l y a t the m i d - p o i n t of the t r i a n g u l a r wave a u x i l i a r y f u n c t i o n s . As a r e s u l t , a b i a s p r o d u c t had t o be removed from the output which was, p a r t i c u -l a r l y i n Test 6, of almost the same magnitude as the outpu t . The d i f f e r e n c e , which r e p r e s e n t e d the expected o u t p u t , was t h e r e f o r e a poor e s t i m a t e . I n a l l c a s e s , the r e a d i n g e r r o r , w h i c h r e p r e s e n t e d the l i m i t a t i o n s i n r e a d i n g b o t h i n p u t a m p l i -tudes and the wattho u r meter output as w e l l as the a c c u r a c y t o which the o f f s e t was known, w i l l be reduced i n the f u t u r e by t a k i n g l o n g e r t e s t runs and by e l i m i n a t i n g the o f f s e t b e f o r e t e s t i n g . Ten of the t h i r t e e n t e s t r e s u l t s f e l l w e l l w i t h i n the maximum r e a d i n g e r r o r . The p o o r e s t r e s u l t was t h a t of Test 11 where the i n p u t p r o d u c t was o n l y 0.7% of f u l l s c a l e . T h i s was p r o b a b l y due t o the s l i g h t amount of 10 Mcps o s c i l l a -t i o n which o c c u r r e d d u r i n g the comparator s w i t c h i n g i n t e r v a l . F o r l e v e l i n p u t s , the output of the i n s t r u m e n t i s d i r e c t l y r e l a t e d t o At . = t-, . - t . ( i = l , 2 ) , where t , . i s the time d u r i n g t h w hich the j comparator output i s i n the 1 - s t a t e and t . i s the time i n the 0 - s t a t e . S i n c e the s w i t c h i n g i n t e r v a l i s a p p r o x i m a t e l y c o n s t a n t f o r a l l i n p u t l e v e l s , the p e r i o d of o s c i l l a t i o n r e p r e s e n t s an i n c r e a s i n g p r o p o r t i o n of A t . as the i n p u t tends t o z e r o . U n t i l t h i s i s improved, i t w i l l cause a p p r e c i a b l e e r r o r i n the r e s u l t s f o r i n p u t s as s m a l l as t h a t of of Test 11. 68. Test k l A l (V) ^2^2 (v) k l k 2 Test Duration (hours) A b s o l u t e % E r r o r % E r r o r of F . S . Max. % Reading E r r o r 1 1.324 1.105 0.96 0.85 0.61 0.58 1.56 2 1.324 0.7682 0.66 0.83 0.39 0.25 1.79 3 1.324 0.3208 0.28 2.3 0.30 0.08 2.13 4 1.324 0.3273 0.28 4.5 -1.13 - 0 . 3 1 4.68 5 1.324 0.0000 0.00 18.1 -0.14 0.00" 2.65 6 1.324 0.0162 0.014 5.3 -71.5 0.00" 1820 7 1.324 0.0788 0.068 4.2 -7.5 - 0 . 4 1 4.46 8 0.000 1.156 0.00 10.3 -0.16 0.00" 5.21 •9 1.010 1.027 0.68 5.3 -0.01 0.00" 1 . 6 3 10 1.010 0.6220 0.41 17.8 -0.35 -0.14 1.13 11 1.009 0.0108 0.007 17.0 104 0.39 14.1 12 0.4915 0.2472 0.17 6.9 6.1 0.95 1.25 13 1.324 1.156 1.00 2.3 2.7 2.7 2.77 Table 4-1 R e s u l t s of L e v e l Input T e s t s (A,=1.324V., A 9=1.156V.) 4.9.2 AC In p u t s I n t h i s case, the output of the s t a t i s t i c a l w atthour meter was compared t o t h a t of a c a r e f u l l y a d j u s t e d and c a l i b r a t e d s t a n d a r d s i n g l e - p h a s e i n d u c t i o n w a t t h o u r meter. A c o n s t a n t of p r o p o r t i o n a l i t y was e s t a b l i s h e d between the two meters a f t e r t e s t s had been made a t both 100% and 50$ power f a c t o r . The r e l a t i v e p o s i t i o n s of the two meters i n the t e s t c i r c u i t i s shown i n F i g u r e 4-16 and the r e s u l t s of the t e s t s are g i v e n i n Table 4-2. 69. c u r r e n t c o i l Q o f s t a n d a r d 100% pf 50%pf r 3 3 0 i i voltage c o i l of s t a n d a r d £> v y ( t ) F i g u r e 4-16. A-C Test C o n f i g u r a t i o n A l l t e s t s were made a t room temperature w i t h c o n s t a n t l o a d . The l e n g t h of the t e s t s ranged from 6.7 t o 41.7 hours as i n d i c a t e d i n Table 4-2. No attempt was made t o e s t a b l i s h con-vergence t i m e s f o r the i n s t r u m e n t d u r i n g these t e s t s . Some of the a-c t e s t r e s u l t s f e l l s l i g h t l y o u t s i d e of the maximum r e a d i n g e r r o r , but not f a r enough t o v o i d the f e a s i b i l i t y of the i n s t r u m e n t o r t o d i s c o u r a g e f u r t h e r t e s t i n g . I t i s f e l t t h a t w i t h a few r e f i n e m e n t s i n the i n s t r u m e n t a t i o n , more e x t e n s i v e t e s t s would i n d i c a t e t h a t t h i s type of s t a t i s t i -c a l w a t t h o u r meter c o u l d r e l i a b l y meet the Canadian F e d e r a l Government a c c u r a c y s t a n d a r d s f o r domestic w a t t h o u r meters. *+2% a t both l i g h t and f u l l l o a d f o r 100% power f a c t o r and +3% f u l l l o a d f o r 50% p f , where: f u l l l o a d = 50% of r a t e d c u r r e n t and l i g h t l o a d = 2T% of r a t e d c u r r e n t . 70. Test C u r r e n t % of F.S. Test D u r a t i o n (Hours) Power F a c t o r A b s o l u t e % E r r o r % E r r o r of F.S. Max. % Reading E r r o r 1 83.0 9.0 100% -1.52 -1.35 1.11 2 42.1 15.4 100% -0.38 -0.18 1.08 3 41.8 6.7 100% 0.27 0.13 1.28 4 22.0 40.3 100% 0.36 0.10 0.97 5 15.4 28.5 100% -1.13 -0.23 0.94 6 11.2 28.6 100% - r . 71 -0.29 1.00 7 4.1 41.7 100% 0.74 0.06 0.84 8 84.7 16.8 50% -3.29 -1.54 1.22 9 42.4 7.5 50% 2.22 0.57 1.40 10 22.0 27.5 50% -0.88 -0.14 0.94 11 10.3 21.8 50% 0.94 0.10 1.08 V o l t a g e was 11% of F.S. f o r a l l t e s t s . Table 4-2 R e s u l t s of A-C T e s t s 5. SUMMARY AND CONCLUSIONS A d i g i t a l s t a t i s t i c a l w atthour meter has been proposed w h i c h o p e r a t e s by p o l a r i t y - c o i n c i d e n c e c o r r e l a t i o n . The output r e p r e s e n t s a t i m e - m u l t i p l i e d e s t i m a t e of the c u r r e n t and v o l t a g e c r o s s - c o r r e l a t i o n f u n c t i o n f o r z e r o d e l a y ; the r e l a t i o n s h i p between the t r u e c o r r e l a t i o n f u n c t i o n , the l i n e a r i z e d e s t i m a t e of the c o r r e l a t i o n f u n c t i o n and the output of the proposed i n s t r u m e n t can be d e r i v e d from E q u a t i o n s ( l - l ) , ( l - 1 3 ) and ( 4 - 4 ) , The s o u r c e s of bounded, u n i f o r m l y - d i s t r i b u t e d random o n n o i s e i n the c o r r e l a t o r s of Ikebe and Sato , J e s p e r s et a l q and Turner , have been r e p l a c e d by d e t e r m i n i s t i c t r i a n g u l a r -wave g e n e r a t o r s . As i n the former c o r r e l a t o r s , these a u x i l i a r y f u n c t i o n s , when added t o the i n p u t b e f o r e c l i p p i n g , l i n e a r i z e the i n p u t / o u t p u t c o r r e l a t i o n f u n c t i o n r e l a t i o n s h i p . Because the e x p r e s s i o n f o r E ( v , i , t ) c o u l d not be mani-p u l a t e d , f o r l a r g e N, i n t o a t r a c t a b l e form which would a l l o w the q u a l i t y of the energy e s t i m a t e t o be a s s e s s e d , the system was s i m u l a t e d on an analogue computer. A l t h o u g h t h i s p r o v i d e d a q u i c k check of the performance of the proposed i n s t r u m e n t , i t was not p o s s i b l e t o c o n t r o l most system parameters w e l l enough t o e s t a b l i s h t h e i r e f f e c t on the o u t p u t . The proposed system was t h e n s i m u l a t e d on a g e n e r a l -purpose d i g i t a l computer under f o u r c o n d i t i o n s - l e v e l or s i n u s o i d a l i n p u t s , w i t h c o n t i n u o u s or sampled out p u t . From these s i m u l a t i o n s , i t was p o s s i b l e t o e s t a b l i s h some a c c u r a c y l i m i t a t i o n s and convergence r a t e s . 72. Programs were a l s o w r i t t e n t o s i m u l a t e the e a r l i e r s a m p l i n g c o r r e l a t o r which used n o i s y a u x i l i a r y f u n c t i o n s . These r e s u l t s were compared w i t h those from the p r e v i o u s l y mentioned s i m u l a t i o n s , f o r both l e v e l and s i n u s o i d a l i n p u t s . The comparisons i n d i c a t e d t h a t f o r p r e d o m i n a n t l y s i n g l e -f r e q u e n c y ( o r l e v e l ) i n p u t s w i t h slow random f l u c t u a t i o n s , the i n s t r u m e n t w i t h the d e t e r m i n i s t i c a u x i l i a r y - f u n c t i o n g e n e r a t o r s would g i v e an a c c e p t a b l e c o r r e l a t i o n e s t i m a t e more q u i c k l y than i t s p r e d e c e s s o r . T h i s would be p a r t i c u l a r l y t r u e f o r s m a l l i n p u t s . A p r o t o t y p e of the proposed s t a t i s t i c a l w atthour meter was c o n s t r u c t e d and t e s t e d w i t h d-c and a-c i n p u t s . The s t a n d a r d used f o r the a-c t e s t s was a c a r e f u l l y a d j u s t e d and c a l i b r a t e d i n d u c t i o n - m o t o r w a t t h o u r meter. Short-^erm (6.7-41.7 hours) t e s t s were performed a t room temperature w i t h c o n s t a n t l o a d s a t b o t h 100% and 50% power f a c t o r . A l l t e s t r e s u l t s are g i v e n i n the t h e s i s . The e r r o r s w h i c h appeared i n these t e s t s were a l l s m a l l e r than 3.5%. S i n c e the maximum r e a d i n g e r r o r s were of the same o r d e r of magnitude, i t was f e l t t h a t some r e f i n e m e n t s i n i n s t r u m e n t a t i o n and more e x t e n s i v e t e s t s would y i e l d much b e t t e r r e s u l t s . A l t h o u g h the convergence r a t e s were not t a b u l a t e d d u r i n g the t e s t s , the s i m u l a t i o n r e s u l t s i n d i c a t e d t h a t these a c c u r a c i e s c o u l d have been o b t a i n e d i n l e s s t h a n an hour. I f the i n p u t and comparison c i r c u i t r y of t h i s i n s t r u -ment were r e f i n e d and r e f e r e n c e c e l l s were used t o d e f i n e the am p l i t u d e bounds of the a u x i l i a r y f u n c t i o n s , i t i s c o n c e i v a b l e 73. t h a t t h i s d e v i c e c o u l d a l s o be used as a p r e c i s i o n l a b o r a t o r y i n s t r u m e n t , or as a meter f o r h i g h energy systems. A more d e f i n i t e d i s c u s s i o n of these p o s s i b l e a p p l i c a t i o n s c o u l d be made o n l y a f t e r s p e c i f i c r e q u i r e m e n t s were known and e x t e n s i v e performance t e s t s were c a r r i e d out. The economic f e a s i b i l i t y of an a l l - e l e c t r o n i c w a t t -hour meter has not been c o n s i d e r e d . However, because of the r a p i d p r o g r e s s b e i n g made i n the i n t e g r a t e d - c i r c u i t and t h i n -f i l m f i e l d s , i t i s f e l t t h a t i n the near f u t u r e , the e l e c t r o n i c s of an i n s t r u m e n t as s i m p l e as t h i s one c o u l d be e c o n o m i c a l l y mass-produced and e n c a p s u l a t e d i n s m a l l , i n e x p e n s i v e , c o n s t a n t -temperature ovens. T h i s would e l i m i n a t e the need f o r temp e r a t u r e -compensated components and p r o b a b l y i n c r e a s e the " l i f e - t i m e " of the i n s t r u m e n t . Another f e a t u r e w h i c h makes the d i g i t a l e l e c t r o n i c w a t t h o u r meter p r o m i s i n g i s t h a t i t c o n t a i n s an e l e c t r o n i c c o u n t e r . T h i s appears v e r y u s e f u l i f one c o n s i d e r s the i d e a of 15 u s i n g a tele p h o n e s e r v i c e t o re a d r e s i d e n t i a l meters r e m o t e l y , because i t e l i m i n a t e s the need f o r complex s h a f t encoders. That i s , the m e c h a n i c a l r e g i s t e r and a s s o c i a t e d encoders would be e l i m i n a t e d and a ' n o n - v o l a t i l e ' core a r r a y would be used t o s t o r e the s i g n i f i c a n t b i t s i n the c o u n t e r . I n c o n c l u s i o n , b o t h the r e s u l t s from the s i m u l a t i o n s and those from the p r o t o t y p e t e s t s show t h a t the s t a t i s t i c a l w a t t h o u r meter i s p o t e n t i a l l y c a pable of measuring e l e c t r i c a l energy w i t h an a c c u r a c y adequate f o r domestic m e t e r i n g purposes. C o n s i d e r e d i n c o n j u n c t i o n w i t h the d e c r e a s i n g c o s t of i n t e g r a t e d c i r c u i t r y and the growing t r e n d t o a u t o m a t i c d a t a c o l l e c t i o n and p r o c e s s i n g , t h i s r e s e a r c h i n d i c a t e s t h a t f u r t h e r d e v e l o p -ment of the s t a t i s t i c a l w atthour meter i s m e r i t e d . APPENDIX I 75. BASIC THEOREM A rough o u t l i n e of the p r o o f of the theorem t o f o l l o w was f i r s t proposed by Veltmann and Kwakernaak ^ and l a t e r r e f i n e d by J e s p e r s , Chu and P e t t w e i s . Theorem L e t f ^ and ^ e ^ w 0 d i f f e r e n t c o n t i n u o u s random v a r i a b l e s w i t h upper and l o w e r bounds A. and -A. r e s p e c t i v e l y . Now, c o n s i d e r two a d d i t i o n a l c o n t i n u o u s m u t u a l l y independent random v a r i a b l e s n-^  and n^f which are a l s o independent of f-^ and ± 2 and have r e c t a n g u l a r p r o b a b i l i t y d e n s i t i e s g i v e n by P.(n.) = ( J (AI-1) J J We have then \ n i \ > k i f l f 2 = A 1 A 2 S g n Z ' (AI-2) where z = z ? , A _ z . = f . -n ., J 0 3 /+1 f o r z >0 " I" 1 sgn z _ <^_± f o r z < 0 , and the h o r i z o n t a l b a r s denote the s t a t i s t i c a l averages of the q u a n t i t i e s i n v o l v e d . P r o o f S i n c e the p r o b a b i l i t y t h a t sgn z = 1 i s P ( z > 0 ) and 76, t h a t sgn z = -1 i s P ( z < 0 ) , we have sgn z = P ( z > 0 ) - P ( z < 0 ) or sgn z = 2 P ( z > 0 ) - l Thus, t o v e r i f y E q u a t i o n (AI-2) we must f i n d P ( z > 0 ). Now, P ( z > 0 ) = P ( z x > 0 , z 2 > 0 ) + P ( z 1 < 0 , z 2 < 0 ) where P(a,b). i s the p r o b a b i l i t y t h a t b oth events a and b occur s i m u l -t a n e o u s l y . The j o i n t p r o b a b i l i t y d e n s i t y p(f-^. ni» n 2 ^ a ^ o v s us t o express P ( z > 0 ) as P ( z > 0 ) =y) ' p ( f 1 ? f 2 ' n l ' n 2 ' ) d f l d f 2 d n l d n 2 i = l D i (AI-3) where, s i n c e we have d e f i n e d p ( f ^ , f 2 , n ^ , n 2 ) = 0 i f the modulus of a t l e a s t one of f ^ , f 2 , n ^ . , or n 2 exceeds the c o r r e s p o n d i n g A^, we can d e f i n e the domains as -A.<f. <A. A and -A.<n.<f. X ^ 1 x -A.< f. < A. x ^ X ^ - X - f . < n. < A. x^~ x ^  x D 1 (AI-4) £ D 2 ( A I - 5 ) R e c a l l t h a t the n^ are m u t u a l l y independent as w e l l as independent of the f ^ . T h e r e f o r e , we can w r i t e p ( f 1 , f 2 , n 1 , n 2 ) = p ( f - L , f 2 ) p ( n 1 ) p ( n 2 ) Prom E q u a t i o n s ( A I - l ) and (AI-4) we have t h e n 77, P ( z 1 > 0 , z 2 >0) ""^l ^2 ^1 ^2 J — ^ d f , f p ( f , , f 0 ) d f 0 C dn n C dn. ~ 4A,A 0 ) " x l ^ ^ 1 , - 2 / ^ 2 ^ ^ 2 1 - A - A - A - A 1 2 1 2 "^1 ^2 = 4^A7) dflf ( A 1 + f 1 ) ( A 2 + f 2 ) p ( f 1 , f 2 ) d f 2 (AI-6) and from E q u a t i o n s ( A I - l ) and (AI-5) we get P ( z 1 < 0 , z 2 < 0 ) A l A 2 - A - A A l A 2 S u b s t i t u t i n g E q u a t i o n s (AI-6) and (AI-7) i n t o E q u a t i o n (AI-3) g i v e s A-^  A 2 2 P ( z > 0 ) = f d f x C (l4^ ) p ( fl' f2 ) d f2 - A ^ - A 1 A 2 Now, J d f - ^ p ( f 1 , f 2 ) d f 2 = l _ A - A A l A 2 and A-^  A 2 \ f 1 f 2 p ( f 1 ^ 2 ) d f 2 = f ^ ~ L A J A -A -Rl A 2 78. T h e r e f o r e , f f 2P(z>0) = 1 + A1 A2 and ^1^2 = 1^^ "2 s ^ n z a s r e c l u i r e ( 1 ' S i n c e C^CT) i n F i g u r e 1-2 approximates sgn z, we have 0(T) — \A2 C N ( / f ) 79. APPENDIX I I ERROR CAUSED BY EXPONENTIAL APPROXIMATION OE LINEAR RAMP The e r r o r i n d e t e r m i n i n g s g n ( f - n ) i s d i r e c t l y r e f l e c t e d i n t he e s t i m a t e of the c o r r e l a t i o n f u n c t i o n . T h i s e r r o r , due to the s l i g h t l y non-uniform d i s t r i b u t i o n of the e x p o n e n t i a l ramp l e v e l s w i t h t i m e , can be seen by l o o k i n g a t F i g u r e A I I - 1 . F i g u r e A I I - 1 . E r r o r Caused by E x p o n e n t i a l Ramp 80. F o r O ^ t ^ T , we have _ t n ( t ) = M (1 - [ ^ ] " T ) (AII-1) and p ( t ) = & (AII-2) We can now d e f i n e the e r r o r i n d e t e r m i n i n g sgn [f ( t ) - n ( t ) ] by Ac* ^ the q u a n t i t y e = ^ , or E = k - ^ (AII-3) By s u b s t i t u t i n g E q u a t i o n s (AII-1).. and (AII-2) i n t o E q u a t i o n ( A I I - 3 ) w i t h n ( t n ) = p ( t ) = kA, we get e = k - i n L ^ r-J/ l n.L"M-J A A I f we l e t n^  = ^, th e n e(k, ^ = k - j ^ j i l f f i (AII-4) A few v a l u e s of e(k,n v) are shown i n Table A I I - 1 . Once the r e q u i r e d i s chosen, the o t h e r more common measures of sweep e r r o r can be e a s i l y found and a s a t i s f a c t o r y sweep c i r c u i t con-s t r u c t e d . 0.500 0.200 0.100 0.050 0.020 0.010 0.005 0.002 0.001 1.000 5.OOxlO" 1 2.OOxlO" 1 1.OOxlO - 1 5.OOxlO" 2 2.OOxlO" 2 1.OOxlO" 2 5.OOxlO - 3 2.OOxlO" 3 1.OOxlO" 3 0.500 8„50xlO" 2 4 . 8 0 x l 0 - 2 2.60xlO" 2 1 . 3 5 x l O - 2 5.50xlO~ 3 2.77xlO" 3 1.39xlO" 3 5 . 5 7 x l O - 4 2 . 7 8 x l O - 4 0.200 2 . 7 8 x l 0 " 2 1.71xlO~ 2 9»46xlO" 3 4.96xlO" 3 2.04xlO~ 3 1.03xlO" 3 5 . l 6 x l O - 4 2.07xlO" 4 1 . 0 4 x l O " 4 0.100 1 . 3 2 x l 0 ~ 2 8 . 2 5 x l O - 3 4 o 6 l x l O " 3 2.42xlO" 3 9.99xlO~ 4 5 o 0 4 x l O " 4 2 . 5 3 x l O - 4 1.02xlO" 4 5.08xlO" 5 0.050 6 . 4 1 x l O - 3 4 . 0 6 x l O - 3 2 . 2 8 x l O - 3 1.20xlO" 3 4.94xlO~ 4 2 , 5 0 x l O - 4 1 . 2 5 x l O - 4 5.03xlO" 5 2 . 5 2 x l O - 5 0.020 2 . 5 3 x l O - 3 1 . 6 l x l O - 3 9 . 0 4 x l O - 4 4 . 7 7 x l 0 " 4 1.96xlO~ 4 9 . 9 2 x l O - 5 4.98xlO" 5 1.99xlO" 5 9.79xlO" 6 D.010 1.26xlO" 3 8„02xl0 - 4 4.51xlO" 4 2 . 3 8 x l 0 " 4 9 .79xlO" 5 4 .94xlO" 5 2.48xlO" 5 9.52xlO" 6 4 . 39x lO - 6 0.005 6 . 2 6 x l O - 4 4.OOxlO - 4 2.25xlO" 4 1 . 1 8 x l 0 " 4 4.87xlO~ 5 2 . 4 6 x l O - 5 I . l 6 x l 0 ~ 5 3.77xlO" 6 1 . 1 5 x l O - 6 Table A I I - 1 Val u e s of e(k,rj_) as D e f i n e d by E q u a t i o n ( A I I - 4 ) 82. REFERENCES 1. S a c e r d o t i , G., M.V. S c a g l i o t t i , E. Z a p p i t e l l i and N.B. Crema, " P r o p o s a l f o r a Completely E l e c t r o n i c Sampling Meter of the D i g i t a l Type f o r R e a l Power", A l t a Frequenza, V o l . 35, No. 4, pp. 282-295, A p r i l , 1966. 2. Lee, Y.W., S t a t i s t i c a l Theory of Communication. New York, London; John W i l e y and Sons I n c . , I960. 3. Lee, Y.W., T.P. Cheatham, J r . , and J.B. Weisner, " A p p l i c a -t i o n of C o r r e l a t i o n A n a l y s i s t o the D e t e c t i o n of P e r i o d i c S i g n a l s i n N o i s e " , P r o c . IRE. V o l . 38, No. 10, pp. 1165-1171, October, 1950. 4. S h i n g l e t o n , H.E., "A D i g i t a l E l e c t r o n i c C o r r e l a t o r " , P r o c . IRE . V o l . 38, No. 12, p p . . 1422-1428, December, 1950. 5. F a r a n , J . J . , J r . and R. H i l l s , J r . , " C o r r e l a t o r f o r S i g n a l R e c e p t i o n " , A c o u s t i c s Research Lab., H a r v a r d U n i v e r s i t y , Cambridge, Mass., Tech. Memo. No. 27, September 15, 1952. 6. Veltmann, B.P. and H. Kwakernaak, " T h e o r i e und Technik der P o l a r i t a t s k o r r e l a t i o n f u r d i e dynamische A n a l y s e n i e d e r f r e q u e n t e r S i g n a l e und Systeme, R e g e l u n g s t e c h n i c k , V o l . 9, No. 9, pp. 357-364, September, 1961. 7. J e s p e r s , P., P.T. Chu and A. F e t t w e i s , "A New Method f o r Computing C o r r e l a t i o n F u n c t i o n s " , Paper p r e s e n t e d a t the I n t e r n a t i o n a l Symposium on I n f o r m a t i o n Theory, B r u s s e l s , September, 3-7, 1962. 8. Ikebe, J . and T. S a t o , "A New I n t e g r a t o r U s i n g Random V o l t a g e " , ETJ of Japan. V o l . 7, No. 2, 1962. 9. T u rner, R., "A D i g i t a l C o r r e l a t o r f o r Low Frequency S i g -n a l s " , M.A.Sc. T h e s i s , Dept. of E l e c t r i c a l E n g i n e e r i n g , F a c u l t y of A p p l i e d S c i e n c e , U n i v e r s i t y of B r i t i s h Columbia, December, 1964. 10. Knapp, C.H., "A C a r r i e r I n j e c t i o n Method f o r R e c o v e r i n g the Waveshape of C l i p p e d S i g n a l s " , G e n e r a l Dynamics, E l e c t r i c Boat D i v i s i o n , T e c h n i c a l Report No. U417-65-048. December 24, 1965. 11. H u l l , T.E. and A.R. D o b e l l , "Mixed C o n g r u e n t i a l Random Number Gene r a t o r s f o r B i n a r y Machines", A.CM. J o u r n a l , V o l . 11, No. 1, January, 1964. 12. MacFadyen, K.A., S m a l l Transformers and I n d u c t o r s , Chapman and H a l l L t d . , London, 1953. 83. 13. Millman, J. and H. Taub, Pulse. Digital.and Switching  Waveforms. McGraw-Hill Book Co., New York, 1965. 14. Walston, J.A. and J.R. M i l l e r (Ed.), Transistor C i r c u i t  Design. McGraw-Hill Book Company Inc., New York, 1963. 15. Brothman A. et a l , "Automatic Remote Reading of Residential Meters". '.I.E.E.E. Trans, on Comm. Tech.. (USA), Vol. COM-13, No. 2, pp. 219-232, June, 1965. 

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