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Some new techniques for operational computers Gordon, Kenneth Ian 1960

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SOME NEW TECHNIQUES FOR OPERATIONAL COMPUTERS by KENNETH IAN GORDON BoSCo Hon„ U n i v e r s i t y of Aberdeen, 1956 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the department of E l e c t r i c a l Engineering We accept t h i s t h e s i s as conforming to the standards required from candicates f o r the degree of Master of Appl i e d Science. Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA FEBRUARY, 1960 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the 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 study. I f u r t h e r agree t h a t permission f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of 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 not be allowed without my w r i t t e n p e r m i s s i o n . The U n i v e r s i t y of B r i t i s h Columbia, Vancouver S, Canada. ABSTRACT This t h e s i s i s p r i n c i p a l l y concerned w i t h an i n v e s t i g a t i o n i n t o p o s s i b l e methods of achieving the m u l t i p l i c a t i o n of two f i v e d i g i t decimal numbers coded i n a pulse p o s i t i o n form. Several d i f f e r e n t methods are i n v e s t i g a t e d , the main e f f o r t being on devices employing a combination of d i g i t a l and analog techniques. I ^ i s seen that despite the t h e o r e t i c a l s i m p l i c i t y of these h y b r i d u n i t s , t h e i r p r a c t i c a l r e a l i z a t i o n often i n -volves considerable d i f f i c u l t y . The f i n a l m u l t i p l i e r con-f i g u r a t i o n studied o f f e r s the r e q u i r e d accuracies w i t h the use of simple c i r c u i t r y . i i TABLE OP CONTENTS Page Ab S "fc r £L C "fc e » o . o o e B O o o e O D « o o i > o o * o » o « « » o o o o o o o o f t « o o o * o » * « * X I AC^IIOWX &(l££QHldIlt e o O O O O O O O O O 0 O O O O O O O O O « e o O 9 O » e O « O Q 0 0 0 O « V l l Xo l i l t 1 * 0 d l l C t X OIX e o o t o o o o o o e o o o o o o e o o o A o o o o o o 0 0 0 0 0 0 0 0 0 1 2 0 Decimal Arithmetic Units O « o f l « 9 . » o « « 9 o « « o o o » < i « « o e c 5 2 « X • AcLcL03T o o o o o o o o o o o o o o - o o o o o o o o o t o o o o o o o w o o o * 5 2.1.1. Operation f o r 2.1.2. Operation f o r X. J Jy | 0 0 0 0 0 0 0 0 0 0 0 0 5 3&tj J y | o o o o o o o o o o o o 3 2 o 2 o Mil X t X J) X X 0 2? o o o o o o o o o a o o o o o o o o o o o o o o e o e o o o o 9 3. Trapezoidal M u l t i p l i c a t i on Methods . . . . o o . o e o . . . . . 12 3.1. Sin g l e Channel M u l t i p l i e r 14 3.2. Fi v e Channel M u l t i p l i e r 16 3.3. Disadvantages of D i r e c t Five Channel SjTStOm © © © 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 X8 3.4. Modified Five Channel M u l t i p l i e r 20 4 • CXrGXIXtry. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 23 4 « X • l i l t ©g3T£tt Or o o o o o o o o o o o o o o o o o o o o e ' o o o o o o o o o o 23 4.2. F l i p - F l o p 27 4.3* Monostable M u l t i v i b r a t o r 27 4.4. Voltage Comparator ....................... 27 4. 5.' OateS 0 » . a o » . o o o o o o o o o « o o o . o . a . . e o o o o « o . « . 30 4.6. Quant1 z er e . . . « o . o o o o » e » . . « . o » o « . . » . o o « . » « 30 5. Si n g l e Channel Test M u l t i p l i e r 35 5.1. Operatx on . o o o o o o o o o o o o . . . . . . . . . . . . . A O O . . . 36 5.3. E r r o r s and Conclusions . o o o . o o o . o . o . o o . . . . 39 6. Revised M u l t i p l i e r o . o . o . o o o . o o o o o o o . . . . . . . . . . . . . . 42 i i i Page 6.1. Generation of Tens Diagonal, Units Diagonal, and Units Off-Diagonal 42 6.2. Borrow-Carry 46 6.3. S h i f t and Transfer 46 6*4« T©s*t Unx'tf • •©«<»o<»©©»©o»«*o««©»««**«««»«» 49 6.5. Five D i g i t M u l t i p l i e r 51 6.6. Conclusions 51 Appendix 1. Decimal Counter Components 54 Appendix 2. Pulse Generator 56 Appendix 3. Three Stage A m p l i f i e r s 62 Appendix 4. T r a n s i s t o r Operational A m p l i f i e r s 66 Appendix 5. Int e g r a t o r E r r o r s 70 Bi b l i o g r a p h y 72 i v 1 LIST OP ILLUSTRATIONS Figure Page 1.1. Pulse P o s i t i o n Coding ....................... 3 2 « X 0 Add© X*i:DSl-lb"fc 2T£t C i/OI? o o v e e ' o o o o e o o o o o o o e a o c o o o o v o e 6 2.2. Pulse and Gating Waveform P o s i t i o n s 7 2.3 . D i g i t a l Decimal M u l t i p l i e r . , . , 0 , . , . , , , . , , , . . . 10 3 .1 . Waveforms f o r Trapezoidal M u l t i p l i c a t i o n 13 3.2. Single Channel M u l t i p l i e r ....».,,.»,,.,.,... 15 3.3. D i r e c t Five Channel M u l t i p l i e r .............. 17 3.4. Sample Calcu X£L°tsX OH o e o o o o o o e o o o o o o o O o o o t t O o o o o 19 3 o 5o Modi f x ©d Mu.X*bxpXx©x> 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 a 2X 4 o 1 • Ill"fc ©g3T8.."t OX" CX^CIXXi) 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 24 4o2 # A-C C o u p l ©3?0 R@s"tor©r o o o o o o o o o o o o o o o o o o * 24 4 o 3 • FX Xp-^FX Op OXX'CVLX'tif 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 28 4 .4 . Monostable M u l t i v i b r a t o r Timer C i r c u i t ...... 28 4 .5 . Voltage Comparator C X r C T X X ' t S o o o o o o o o o o o o o o o o o 29 4.6 . C i r c u i t s of Current Gates ....,..,»,,........ 31 4.7 . Block Diagram of Quantizer 33 4.8 . Waveforms of Quantizer ...................... 34 5.1. Single Channel Test M u l t i p l i e r .............. 37 5.2. S i n g l e Channel Test M u l t i p l i e r Waveforms .... 38 6.1 . M u l t i p l i c a t i o n Table . . . o , , . , . . . . . . . . . . 4 3 6.2. C i r c u i t f o r Generation of Tens Diagonal, U n i t s Diagonal, and Units Off-Diagonal 44 6.3. Input Waveforms f o r Revised M u l t i p l i e r 45 6.4. C i r c u i t f o r Barrow-Carry Operation ........... 47 6.5. C i r c u i t f o r S h i f t and Transfer Operation ..... 48 Figure Page 6.6. Test C i r c u i t f o r Transfer and S h i f t A.2.1. Pulse Generator C i r c u i t s (Counters A.2.2. Pulse Generator C i r c u i t s (Logic and BlJ •£ f. © X* 3.2jt^  ) o t > o o o o c o o o o o w o o o o o o o o o o o o o o i > o e O « o o » 60 A.2.3. Block Diagram of Pulse Generator Logic ...... 61 A.3.1. Three Stage A m p l i f i e r C i r c u i t s .............. 63 v i * ACKNOWLEDGEMENT The author i s indebted to Dr. E.V. Bohn, supervisor of the p r o j e c t , Dr. P. Noakes, grantee of the p r o j e c t , and to other s t a f f members and graduate students of the E l e c t r i c a l Engineering Department f o r t h e i r h e l p f u l a s s i s t a n c e . The p r o j e c t was supported by the Na t i o n a l Research Council Block Term Grant B.T. 68. v i i 1 SOME NEW TECHNIQUES FOB OPERATIONAL COMPUTERS. 1. INTRODUCTION There are, two basic types of computers a v a i l a b l e today; the d i g i t a l and the analog. In the d i g i t a l machine every operation performed by i t must be broken down i n t o a sequence of elementary binary steps„ In the analog machine, as i t s name i m p l i e s , an analog or s i m u l a t i o n of the given equation or system can be set up d i r e c t l y by means of p h y s i c a l elements. Where the problem concerns system s i m u l a t i o n , the analog machine o f f e r s easy se$%ip, and r e a l time s i m u l a t i o n , but only l i m i t e d aecuracyj while the d i g i t a l machine requires complex programming, gives r e a l time s i m u l a t i o n only w i t h the f a s t e s t machines, but p r a c t i c a l l y u n l i m i t e d accuracy. In an attempt to obtain easy set-up, r e a l time s i m u l a t i o n , o and high accuracy, a s e r i e s of hy b r i d computers have been de-veloped which can best be described as Operational Computers. These are s i m i l a r to the analog computers, i n that a l l the computing operations such as a d d i t i o n , m u l t i p l i c a t i o n , i n t e -g r a t i o n , e t c . , are performed i n op e r a t i o n a l blocks, but the block i t s e l f may be d i g i t a l or analog, or a combination of the two. Information t r a n s f e r between the blocks may also be i n e i t h e r d i g i t a l or analog form. The " D i g i t a l D i f f e r e n t i a l Analyzer", of which TRICE 1 i s an example, has ope r a t i o n a l blocks which are i n t e g r a t o r s operating by purely d i g i t a l means, and thus comes i n t o t h i s category of Operational Computers. The " A n a l o g - D i g i t a l 2 2 Computer" developed at the Massachusetts I n s t i t u t e of Tech-nology and described by Lee and Cox i s also i n t h i s category. This machine i s b a s i c a l l y a number of time-shared analog o p e r a t i o n a l u n i t s eontrolled'*:by a d i g i t a l computer i n such a way as to extend t h e i r dynamic range. The N a t i o n a l Bureau of Standards has developed "A Combined A n a l o g - D i g i t a l D i f f e r e n t i a l 3 Analyzer" . In i t the more s i g n i f i c a n t part of the v a r i a b l e i s represented d i g i t a l l y , and the l e a s t s i g n i f i c a n t part by an analog v o l t a g e . The usual a d d i t i o n s , m u l t i p l i c a t i o n s , and i n t e g r a t i o n s are then performed on numbers coded thus. I n v e s t i g a t i o n s i n t o Operational Computers have also been made at the Department of E l e c t r i c a l Engineering at the Uni-v e r s i t y of B r i t i s h Columbia, r e s u l t i n g i n the development of 4 a "Pulse-Position-Modulation Analog Computer" . The method used i n t h i s computer to represent the v a r i -ables i s alsor s u i t e d to d i g i t a l type decimal representation of q u a n t i t i e s . The method of coding i s shown i n P i g . 1.1. The pulse has nine p o s s i b l e uniformly spaced p o s i t i o n s r e l a t i v e -to which represents the zero p o s i t i o n . This method of r e p r e s e n t a t i o n o f f e r s several advantages f o r use i n o p e r a t i o n a l type computers, v i z . 1. A l l q u a n t i t i e s are represented i n the f a m i l i a r decimal form, thus o b v i a t i n g the need f o r any conversion between codes, (binary to decimal, ete.). ; 2. The s i n g l e pulse p o s i t i o n can r e a d i l y be converted to an analog form (k.x v o l t s ) , or d i g i t a l form (x p u l s e s ) , f o r use i n a computer. 3. Decimal counting devices ( c o l d cathode gas counter and beam switching tubes), w i t h t h e i r attendant output devices ( N i x i e 3 t «1 0 9 8 g 0 0 P o s s i b l e coding pulse p o s i t i o n s g, g o x,.10 4 x Q„10 3 4 3 X g . l O2 x ^ l O 1 x0.10 x ^ X g X g X ^ X Q i s 97,385 s i g n pulse present; number +ve si g n pulse absent; number -ve Coded pulse p o s i t i o n s and values 50 u n i t s » Sign pulse — )«— 50 u n i t s H> 50 u n i t s — 9 ^ — space f o r s i g n pulse S e r i a l p r e s e n t a t i o n of 2 numbers F i g . 1.1. Pulse P o s i t i o n Coding and T r i x i e ) are commercially a v a i l a b l e and adaptable to the processing and d i s p l a y of the d i g i t a l form of t h i s code. These devices are described i n Appendix 1. 4. The accuracy of number storage i s greater than thAt achiev-able by purely analog means. As here used i t i s one part i n 10 5. The advantages discussed f o r t h i s coding method, plus the ease of using counter tubes to perform a d d i t i o n s and sub-t r a c t i o n s on numbers so coded; make worthwhile an i n v e s t i g a t i o n of. the p o s s i b i l i t y of b u i l d i n g a complete Operational Computer based on t h i s number code. In t h i s t h e s i s , a p o s s i b l e method of performing a d d i t i o n and s u b t r a c t i o n i s discussed f i r s t , to show that these opera-t i o n s are indeed r e a d i l y achievable. A d i g i t a l method of m u l t i p l i c a t i o n i s covered next, followed by the main body of the t h e s i s which i s concerned w i t h performing the m u l t i p l i -c a t i o n operation by a combination of d i g i t a l and analog techniques. The emphasis on these h y b r i d techniques i s based on the p o s s i b i l i t y of achieving high accuracies w i t h r e l a t i v e simple, inexpensive equipment. To have a v a i l a b l e s i g n a l pulses f o r t e s t i n g the various c i r c u i t s , the generator described i n Appendix 2 was constructed. 5 2. DECIMAL ARITHMETIC UNITS 2.1. Adder A p o s s i b l e method of adding and su b t r a c t i n g numbers coded as i n P i g . 1.1. i s shown i n block diagram form i n P i g . 2.1. The f u n c t i o n of the various u n i t s i s as f o l l o w s . S.D. i s the sign detector. When both numbers have the same s i g n , i t sets the plus-minus c o n t r o l on the counter tubes so that both are added i n , i . e . a l l pulses received by the tube w i l l d r i v e i t around i n the p o s i t i v e d i r e c t i o n . When the signs d i f f e r , the operation of the counters i s reversed f o r the second number. PF1 i s a f l i p - f l o p set by g and reset by x or y. During t h i s i n t e r v a l i t c o n t r o l s "And" gate A . l . and allows the appropriate number of S.P. pulses through. These pulses occur between the cl o c k pulses (c) as seen i n F i g . 2.2. The g, x, and y pulses always coincide w i t h the cl o c k pulses. The "And" gates c o n t r o l l e d by periods GQ - - - are open during these periods and closed during the r e s t of the time. Each G^ p e r i o d l a s t s from the pulse to the g^ +^ pulse. See F i g . 2.2. C.C i s the complement c o n t r o l . Read-out i s performed by su b t r a c t i n g pulses unless the complement c o n t r o l has been a c t i v a t e d . 2.1.1. Operation f o r |x|>Jyj From the d e s c r i p t i o n of the operation of the various u n i t s , i t i s seen t h a t f o r x and y of the same s i g n , X Q ~ - - x^ and yQ — — - y^ are fed i n t o the Z Q - - - cbunters r e -s p e c t i v e l y by adding the appropriate number of pulses i n t o P i g . 2.1. Adder-Subtractor <— 9 8 7 6 5 4 3 2 1 Clock pulse p o s i t i o n s Last pulse added F i r s t pulse added S.P. pulse p o s i t i o n s g 4 « 3 g2 «1 g 0 -<*2 G 3 4 F i g . 2„2. Pulse and Gating Waveform P o s i t i o n s the r e g i s t e r . In t h i s operation x i s already contained i n the r e g i s t e r and y i s t o be added (or subtracted). C a r r i e s are produced a u t o m a t i c a l l y as a pulse appears at the 10 p i n . During read-out of the r e s u l t , the counter i s reversed and s u b t r a c t i o n terminated when a pulse i s detected at the zero p i n . For reasons of s i m p l i c i t y , the c o n t r o l s f o r t h i s are not shown. This process returns the number to the appropriate time code. When the two numbers are of opposite s i g n , the process i s s i m i l a r except that the second number i s subtracted from the f i r s t , and borrows from the next higher counter occur auto-m a t i c a l l y when a pulse i s detected at the 9 p i n . 2.1.2. Operation f o r |x|<|y| ^Addition i s as above. Su b t r a c t i o n i nvolves a d i f f e r e n t operation, and requires t a k i n g the nines complement. I t i s based on the f o l l o w i n g . During t h e , s u b t r a c t i o n i n z^ the count w i l l pass the zero p i n , which i s equivalent to borrowing a ten. Thus the r e s u l t l e f t i n the whole r e g i s t e r w i l l a c t u a l l y be 10^+ |x| - |yj i . e . (9+x 4-y 4)10 4+(9+Xg-y 3)10 3+ —+(9+x 0+l-y Q)10° z4 z 3 " Z 0 I f now we subtract 1 from the Z Q term, and then subtract each of the bracket terms from 9, ( i . e . take the nines complement of each) we get. ( y 4 - x 4 ) 1 0 4 + ( y 3 - x 3 ) 1 0 3 + — +(y 0-x p)10° This i s the d e s i r e d r e s u l t . 9 In the device, s u b t r a c t i o n of 1 from ZQ i s automatic, as i s a lso a c t i v a t i o n of the complement c o n t r o l . During read-out pulses are then added i n , and the count terminated when a pulse i s detected at the 9 p i n . This gives the d e s i r e d nines complement i n the appropriate time code. During read-out, no c a r r i e s or borrows are produced; pulses being blocked by the plus minus c o n t r o l . A r e s e t a c t i o n i s required f o r a l l f l i p - f l o p s , and to r e t u r n a l l coun-t e r s to zero before'the next c y c l e . I t i s seen then t h a t using standard components and simple techniques i t i s p o s s i b l e to b u i l d an adder-subtractor. Be-cause of i t s s i m p l i c i t y , t h i s u n i t was not constructed; e f f o r t being concentrated on the operations l i k e l y to prove d i f f i c u l t . 2.2. M u l t i p l i e r A p o s s i b l e way of r e a l i z i n g a d i g i t a l 5 place decimal m u l t i p l i e r i s shown i n F i g . 2.3. The basic u n i t s operate i n the f o l l o w i n g manner. The c i r c u l a t i n g r e g i s t e r has a pulse rate ten times that of the clock pulses, and supplies XQ - - - X^ pulses on the re s p e c t i v e l i n e s during each clock c y c l e . The "And" gates AQ - - - A^ are c o n t r o l l e d by the f l i p - f l o p F F l , and are open f o r the per i o d between the g pulse and the y pulse. Thus on l i n e (0) there w i l l appear during consecutive periods X-QYQ9 x 0 y l ' x 0 y 2 ' x 0 y 3 * a n d X 0 y 4 P u l s e s s a n d s i m i l a r l y f o r the other l i n e s . The switching matrix has the appropriate "And" gates open during the GQ - - - G^ periods. From the i n t e r -connections on the gates i t i s seen t h a t t h i s r e s u l t s i n the clock pulses 10 times " " Last pulse added A F i r s t pulse added Counters 10 times c l o c k F i g . 2.3. D i g i t a l Decimal M u l t i p l i e r pulses on each l i n e being t r a n s f e r r e d to the f o l l o w i n g counter on each consecutive p e r i o d . The counters themselves w i l l be of the high speed type, (see Appendix 1), able to accept the pulse r a t e of jthe c i r c u l a t i n g r e g i s t e r . The c a r r y hold i s a b i s t a b l e devieje which i s set by a c a r r y pulse from the pre-vious counter. i . e . by d e t e c t i o n of a pulse on the 10 p i n . The c a r r y hold i s reset and a pulse t r a n s f e r r e d to the next counter at each c l o c k pulse. I f there has been no c a r r y , there w i l l be no reset pulse and hence no t r a n s f e r . Since 9 i s the maximum number that can be added i n t o a counter du-r i n g any c l o c k p e r i o d , there i s never more than 1 c a r r y , and hence no delay i s necessary f o r c a r r y propagation. Prom the above d e s c r i p t i o n i t i s seen that the r e s u l t of the operation i s to add the appropriate x.y products i n t o the c o r r e c t counters to produce the t o t a l product d e s i r e d . This can be checked w i t h the sample c a l c u l a t i o n given i n P i g . 3.4. While the c o n t r o l operations on t h i s m u l t i p l i e r are simple, the p r e c i s i o n of timing required may be d i f f i c u l t to achieve. I f the clock frequency i s lOOKc, the ten times c l o c k frequency w i l l be lMc, and each of the gates w i l l have to open f a s t enough and p r e c i s e l y enough to pass a pulse occurring' 1 microsecond a f t e r the time when i t was due to open. Assuming the r e q u i r e d p r e c i s i o n i s achievable, there remains the problem of cost, as the system w i l l r e q u i r e a minimum of 9 high speed counting tubes, not i n c l u d i n g the c i r c u l a t i n g r e g i s t e r . 12 3. TRAPEZOIDAL MULTIPLICATION METHODS P i g . 3.1. i l l u s t r a t e s how m u l t i p l i c a t i o n of two coded numbers, x and y, can be achieved. The area of the t r a p e z o i d i n (a) i s equivalent to the product x.y, but since the area i s not completed u n t i l time x+y, t h i s i s modified to the form of (b). The i n t e g r a l of the curve i n (b), or the measure of the area i n the f i g u r e at any given time, w i l l have the shape of the curve i n ( c ) , where the maximum represents the product X.y. This can be converted back to a pulse p o s i t i o n code by decreasing the amplitude at a slope of -1, and d e t e c t i n g when i t crosses the zero a x i s . The maximum number coded i n a time i n t e r v a l i s 9, which gives a product of 81, and t h i s would have to be re-coded as a separate 8 and 1. This can be done by s u b t r a c t i n g 10 from the product each time i t reaches 10, and s t o r i n g separately the number of times t h i s i s done. To see how the above system of m u l t i p l i c a t i o n can be r e a l i z e d i n a p h y s i c a l system, l e t us consider the p r o p e r t i e s of an i d e a l i n t e g r a t o r . The voltage response to an input voltage V. i s For V\ a constant, V Q i s a l i n e a r l y r i s i n g output voltage equal to ~ _ i _ . t w i t h a slope of ~ v j I t i s also n o t i c e d that C~Tl C R° h a l v i n g the value of R doubles the slope. Thus i t is, p o s s i b l e to r e a l i z e the method of m u l t i -t 0 P i g . 3.1. Waveforms f o r Trapezoidal M u l t i p l i c a t i o n p l i c a t i o n described above, with 2 i n t e g r a t o r s plus attendant c o n t r o l c i r c u i t r y . The f i r s t i n t e g r a t o r i n t e g r a t e s a con-stant voltage up t i l l time x. The output voltage V Q ^ i s shown i n ( a ) . The second i n t e g r a t o r i n t e g r a t e s V Q ^ , doing so at double the normal rate during the i n t e r v a l g to x. This would give an output as shown i n ( c ) . A symbolic method of presentation w i l l be used to describe the operation of the systems which can be assembled using t h i s p r i n c i p l e . Actual c o n t r o l methods and c i r c u i t s w i l l be de-s c r i b e d l a t e r . 3.1. Sjngle Channel M u l t i p l i e r P i g . 3.2-. shows a system f o r the m u l t i p l i c a t i o n of s i n g l e d i g i t numbers. The f u n c t i o n of each of the u n i t s i s as f o l -lows s C^ gates the f i x e d input voltage to the f i r s t i n t e g r a t o r f o r the time i n t e r v a l g to x. I t a l s o c o n t r o l s the time i n t e r v a l during which Cg i s s c a l i n g the output of the f i r s t i n t e g r a t o r by a f a c t o r of 2. Cg determines the time i n t e r v a l g to y i n which the output of 1^ i s fed i n t o the second i n t e g r a t o r l g . C^ i s a 10 l e v e l detector. I t subtracts 10 u n i t s from l g and adds I u n i t to l g whenever the output of l g reaches 10 u n i t s . For the sake of s i m p l i c i t y , the c o n t r o l s r e q u i r e d to r e -t u r n a l l the i n t e g r a t o r outputs to zero have not been included. Neither has the l o g i c required to permit y to be l e s s than x„ and to permit e i t h e r to equal zero. M u l t i p l i c a t i o n of 3 times 5 would then be performed by the u n i t as f o l l o w s . 15 P i g . 3.2. Sin g l e Channel M u l t i p l i e r On the a r r i v a l of g, C^ gates a f i x e d voltage to I ^ . scales the output of 1^ by 2, and Cg opens to permit i t to be i n t e g r a t e d by Igo When x a r r i v e s at 3 u n i t s , the i n t e g r a -t i o n i n 1^ i s stopped, and i t s output held at 3" u n i t s , Cg scales i t s former output of 6 u n i t s down to 3 u n i t s . S h o r t l y a f t e r the 3 u n i t pulse a r r i v e s , C^ detects the output of Ig at 10 u n i t s , and p u l l s i t down to zero, while at the same time feeding 1 u n i t into, Ig. On the a r r i v a l of y at 5 u n i t s , Cg c l o s e s , holding the output of Ig at 5 u n i t s . The r e s u l t of 3 times 5, or 15 i s now stored i n Ig and Ig. These values could then be reduced at a slope of - l j s t a r t i n g at the next g pulse, and by producing a pulse when they reach zero, the r e s u l t i s returned to the required pulse p o s i t i o n code. 3.2. Five Channel M u l t i p l i e r The basic s i n g l e ehannel u n i t can be expanded d i r e c t l y i n t o a f i v e channel u n i t able to m u l t i p l y two 5 d i g i t numbers. The system i s i l l u s t r a t e d i n F i g . 3.3. Since the product can 9 be of the order of 10 , ten f i n a l i n t e g r a t o r s are needed. The outputs of the f i r s t 5 i n t e g r a t o r s are switched to the appropriate f i n a l i n t e g r a t o r s by means of the switching matrix shown. Each gate i n the matrix remains open f o r the duration of the Period GQ - - G ^ . The operation i s otherwise s i m i -l a r to that of the s i n g l e channel device. At the input i t i s noted that x i s fed i n i n s e r i e s to each channel, but y i s supplied i h p a r a l l e l at the same time as x, going 5 times to Q C Q , y^ f i v e times to ^ Cgs> etc. In combination w i t h the switching matrix, t h i s then introduces the appropriate products IT Gates open 1— 1 ^ during p e r i o d s - GQ G^ Gg Gg G^ Channel no. D i f f e r e n t i a t i o n of blocks i s by p r e - f i x i n g with channel number. Thus CA i n channel 2 i s 0C.. F i g . 3.3. D i r e c t Five Channel M u l t i p l i e r ( x0^0* x 0 ^ 1 ' x1^0' - ~ ) to the various i n t e g r a t o r s . The sample c a l c u l a t i o n of F i g . 3.4. shows that t h i s system does i n f a c t accumulate the corr e c t products at each of the i n t e -g r a t o r s . A f u r t h e r operation r e q u i r e d i s tha t of reducing the l e v e l of the f i r s t i n t e g r a t o r to zero between operations. Under the worst c o n d i t i o n s , when y i s 9, l e s s than 1 u n i t of time i s a v a i l a b l e to do t h i s . A c o n t r o l and zero detect are there-f o r e required which w i l l reduce the output of the f i r s t i n t e -g r a tor .to zero at a slope of -10, and stop i t at the zero l e v e l . . As for. the s i n g l e channel case, c o n t r o l s are also r e -quired to permit y to be l e s s than x, and to permit e i t h e r to be zero. 3.3. Disadvantages of Direct Five Channel System Although t h i s system i s the most d i r e c t f o r f i v e channel operation, i t would be very d i f f i c u l t to r e a l i z e i n p r a c t i c e . As seen from the sample c a l c u l a t i o n of F i g . 3.4., one of the i n t e g r a t o r s i s r e q u i r e d to accumulate a t o t a l of 440, perform-in g 44 c a r r i e s . The next worse case has 36 c a r r i e s . The i n t e g r a t o r must operate f o r some 50 u n i t s of time, and end up w i t h i n 0.5 u n i t s of the corr e c t l e v e l . As l i s t e d i n F i g . 3.4. o v e r a l l accuracy r e q u i r e d can then be as high as one part i n 900. Achieving the required accuracy would e n t a i l high g a i n , stable a m p l i f i e r s , B.C. products w i t h accuracies of up to one part i n .900, and tim i n g w i t h s i m i l a r accuracies. The 10 l e v e l detector would have to detect to w i t h i n l/lOO of a u n i t . 19 M u l t i p l i c a t i o n of 2 f i r e d i g i t numbers, x ^ g X g X - j X g by y ^ y ^ y , or 99,999 by 99,999. 9 8 7 6 5 4 3 2 1 0 — Integrator no. Pe r i o d Operation G 0 x 4 y 0 X 3 y 0 X 2 y 0 X l y 0 x 0 y 0 P 0 8 9 9 9 9 1 P 0 8 9 9 9 9 1 S 0 G l x 4 y l X 3 y l X 2 y l X l y l x 0 y l p l 8 9 9 9 9 1 p l 9 8 9 9 9 0 ,i S1=V P1 G 2 x 4 y 2 X 3 y 2 X 2 y 2 X l y 2 X 0 y 2 P 2 'v 8 9 9 9 9 1 P 2 9 9 8 9 9 0 S 2 = S 1 + P 2 G 3 x 4 y 3 X 3 y 3 X 2 y 3 x l y 3 P 3 8 9 9 9 9 1 P 3 9 9 9 8 9 0 S 3 = S 2 + P 3 G 4 X 4 y 4 X 3 y 4 X 2 y 4 X l y 4 x 0 y 4 P 4 8 9 9 9 9 1 P 4 9 9 9 9 8 0 9 9 9 9 8 0 0 0 0 1 -F i n a l Result The number of c a r r i e s generated on any i n t e g r a t o r i s « 0 9 18 27 36 44 35 26 17 8 Approximate accuracy requirements 3 of each u n i t are 1 part i n -20 180 300 600 700 900 P, - P a r t i a l Product S, P a r t i a l Sum F i g . 3.4. Sample C a l c u l a t i o n 20 Further, each of the f i r s t i n t e g r a t o r s i s loaded w i t h 5 outputs, which would make the accuracy required from the f i r s t i n t e g r a t o r d i f f i c u l t to achieve, since t h i s i s of the order of one i n 800 i n each case. To avoid the s t r i n g e n t requirements of t h i s system, a mo-d i f i e d method of m u l t i p l y i n g two 5 d i g i t numbers was developed. This method i s discussed next. 3.4. Modified Five Channel M u l t i p l i e r The s t r a i g h t - f o r w a r d f i v e channel m u l t i p l i e r makes poor use of the a v a i l a b l e i n t e g r a t o r s , i n that while one i n t e g r a t o r i s i n continuous^ use, the others are only used a f r a c t i o n of the time. The new system equalizes the load between the i n t e -g r a t o r s , and by c y c l i n g t h e i r use, allows time to quantize and c o r r e c t the output of each one a f t e r only one c y c l e . Since each i n t e g r a t o r only operates f o r one eycle at a time, i t never accumulates more than 99 u n i t s before being corrected (only 9 of these u n i t s would be l e f t i n the one i n t e g r a t o r ) . Thus the accuracy requirements are only s l i g h t l y more s t r i n g e n t than f o r a s i n g l e d i g i t m u l t i p l i c a t i o n . However, as w i l l be seen i n the d i s c u s s i o n , each i n t e g r a t o r i s operated i n more ways, and the a u x i l i a r y c o n t r o l equipment i s considerably i n -creased. For reasons of s i m p l i c i t y only a two d i g i t m u l t i p l i e r w i l l be discussed. ( F i g . 3.5.) Two a d d i t i o n a l numbers w i l l be used to denote the p o s i -t i o n s of the; various u n i t s . The f i r s t e x t ra number denotes the channel (0., 1, or 2), while the second denotes the period Carry c o n t r o l Quantizer P i g . 3.5. Modified M u l t i p l i e r d u r i n g w h i c h i t i s u s e d . Thus I -01 i s t h e s e c o n d i n t e g r a t o r 2 ( I g ) i n t h e z e r o c h a n n e l , w h i c h i s u s e d d u r i n g t h e G^ p e r i o d . The b a s i c m u l t i p l i c a t i o n o p e r a t i o n i s t h e same as b e f o r e . However, t h e f i n a l i n t e g r a t o r s ( i g - j k ) a c c u m u l a t e t h e p a r t i a l p r o d u c t s o n l y on a l t e r n a t e odd-even c y c l e s , i . e . t h e p a r t i a l p r o d u c t X g y Q i s g e n e r a t e d by t h e i n t e g r a t o r Ig-OO d u r i n g t h e GQ (even) c y c l e . Borrows and c a r r i e s a r e g e n e r a t e d by C^-OO and i n d i c a t e d b y ^10 and +1 on t h e r e s p e c t i v e l i n e s . A t t h e end o f t h i s c y c l e t h e p a r t i a l sum i s a v a i l a b l e i n t h e Ig-OO, I g - l O and Ig-^O i n t e g r a t o r s . D u r i n g t h e G-^  (odd) c y c l e , t h e p a r t i a l p r o d u c t x-^yg i s g e n e r a t e d by t h e i n t e g r a t o r I g - O l a n < ^ t h e b o r r o w s and c a r r i e s by C^-01. S i m u l t a n e o u s l y t h e o l d p a r t i a l sum i s s h i f t e d and added t o t h e new p a r t i a l p r o d u c t s t o o b t a i n a new p a r t i a l sum. T h i s i s a c c o m p l i s h e d by c l e a r -i n g t h e Ig-OO, I g - l O and ^-^O i n t e g r a t o r s . The r e s p e c t i v e sweep d u r a t i o n o f t h e s e i n t e g r a t o r s i s q u a n t i z e d i n t o i n t e g r a l c l o c k p e r i o d s by t h e r e s p e c t i v e z e r o l e v e l d e t e c t o r s D-00, D-10 and Dr-20. By q u a n t i z i n g t h e p a r t i a l sum d u r i n g e ach c y c l e t h e e f f e c t o f a c u m u l a t i v e e r r o r i s a v o i d e d . The a c c u r a c y p r o b l e m s o f t h e m o d i f i e d u n i t a r e s i m i l a r t o t h o s e o f t h e p r e v i o u s u n i t , b u t t h e r e q u i r e m e n t s a r e n o t as s t r i n g e n t ; b e i n g 1 p a r t i n 200 r a t h e r t h a n 1 p a r t i n 1000. 23 4. CIRCUITRY At the beginning of the work on the t h e s i s , various c i r -c u i t s were constructed so as to t e s t the basic t r a p e z o i d a l m u l t i p l i c a t i o n operation. This gave an i n d i c a t i o n of which operations would be c r i t i c a l , and how the various c o n t r o l operations could be r e a l i z e d i n p r a c t i c e . I t was on the ba-s i s of t h i s knowledge that the various systems were i n v e s t i -gated; i n i t i a l e f f o r t being on the s t r a i g h t - f o r w a r d f i v e channel system. The c i r c u i t s used i n the a c t u a l r e a l i z a t i o n of the system do not n e c e s s a r i l y correspond to the symbolic blocks used to e x p l a i n the operation. Thus, Cg the s c a l i n g by 2 c o n t r o l i s done i n the modified system by changing the time constant of the second i n t e g r a t o r ( l g ) by means of a current gate at the input of l g . This gate i s c o n t r o l l e d by a f l i p - f l o p which also c o n t r o l s a current gate at the input of 1^, which per-forms C^ (see P i g . 3.2.). The d e s c r i p t i o n of the Single Channel Test M u l t i p l i e r (Section 5) shows how the r e q u i r e d operations are r e a l i z e d . 4.1. Integrator The c i r c u i t used f o r the a m p l i f i e r ( P i g . 4.1.a.) i s known as a compound connection with a t h e o r e t i c a l current gain of 2 P . The v a r i a b l e 10k r e s i s t o r on the input permits a d j u s t -ment of the output l e v e l . The 10k r e s i s t o r between the f i r s t emitter and 1.5 v o l t s has the two-fold purpose of p r o v i d i n g increased temperature s t a b i l i t y , and of operating the f i r s t t r a n s i s t o r out of c u t - o f f i n a higher {3 r e g i o n . 24 (a) (b) Low g a i n a m p l i f i e r H i g h g a i n a m p l i f i e r R^ = 6•8K R^ a 22K R» = 33K R 2 = 330K R 5 = 50K T x , T g = 2N247 R 3 = 10K R 6 = 270K C± = 4000pF, f i r s t i n t e g r a t o r = 500pF, s e c o n d i n t e g r a t o r F i g . 4.1. I n t e g r a t o r C i r c u i t C 1 = l O u F R x = 0.5K D x = 1N191 F i g . 4.2. A-C C o u p l e r , D-C R e s t o r e r As shown i n Appendix 4, the i d e a l t r a n s i s t o r a m p l i f i e r f o r use i n an i n t e g r a t o r must be a high current gain device, w i t h low input impedance, high load impedance, and large value of feedback c a p a c i t o r . The 2N247 t r a n s i s t o r s a v a i l a b l e per-m i t t e d only an approximation to these requirements, the f o l -lowing compromises having to be made. The load r e s i s t o r i s undesirably low because f o r a nega-t i v e sweeping voltage i t has to supply current to the i n t e -dV" g r a t i n g c a p a c i t o r according to the r e l a t i o n i = C . 0 . C i s dt d e s i r e d as large as p o s s i b l e and i s made 4000 pF on the f i r s t i n t e g r a t o r , and 500 pF on the second which has to change at a f a s t e r r a t e of about 1 v o l t per microsec. A l a r g e value of C increases the accuracy and the holding a b i l i t y of the i n t e -g r a t o r . Due to the non-zero input impedance of the Ampli-f i e r , the input voltage v a r i e s by up to 0.1 v o l t and the d i v i d i n g chain impedance of 25k then causes an e r r o r deter-mined by; e r r o r = 1 ^ /0"1 d t . 25.10 . C J Q For t=100 microsec, the e r r o r of the f i r s t i n t e g r a t o r would be 0.1 v o l t , and t h a t of the second 0.8 v o l t . The impedance of the b i a s i n g chain can be increased to 150K to reduce t h i s e r r o r ( F i g . 4.1.b.) and increase the over-a l l current gain, but t h i s reduces the temperature s t a b i l i t y of the a m p l i f i e r . The a m p l i f i e r voltage gain at 1000 c/s w i t h the c i r c u i t shown i n F i g . 4.1.a. i s 400, dropping to 100 at 1 Mc/s. The current gain i s 600 at 1000 c/s, and can be increased to over 26 1000 by using the higher impedance b i a s i n g chain, as i n P i g . 4.1.b. No attempt was made to provide absolute temperature s t a -b i l i t y i n the a m p l i f i e r . No equipment was a v a i l a b l e to carry out proper d r i f t measurements, but i t was no t i c e d that the a m p l i f i e r output would change as room temperature changed. The d r i f t could be corrected by a d j u s t i n g the 10k p o t e n t i o -meter, and was normally slow enough not to i n t e r f e r e with any t e s t s . The i n t e g r a t o r does not use a d i f f e r e n t i a l a m p l i f i e r g i v -i n g zero output f o r zero i n p u t . Because of t h i s , and to avoid using an i n v e r t e r , a-c coupling w i t h d-c r e s t o r a t i o n (Pig.4.2.) was^ used to connect the two i n t e g r a t o r s . The voltage to which the d-c r e s t o r a t i o n i s made can be adjusted to coin c i d e w i t h the input voltage of the second i n t e g r a t o r . Because of the various disadvantages of t h i s design, the pr o p e r t i e s of 3 stage t r a n s i s t o r a m p l i f i e r s were i n v e s t i g a t e d . This work i s covered i n Appendix 3. I t was found that no improvement could be produced unless considerable e f f o r t were to be expended i n developing an automatic zero set temperature compensated o p e r a t i o n a l a m p l i f i e r . This was not attempted as being too time consuming f o r the present t h e s i s . Appendix 5 contains an a n a l y s i s of the accuracies to be expected from the present a m p l i f i e r . I t i s seen that the a v a i l a b l e accuracies are not s u f f i c i e n t f o r the second ampli-f i e r . Doubling the gain of the second a m p l i f i e r by changing the biasing- c h a i n , and i n c r e a s i n g the feed-back c a p a c i t o r to 1300 pP should make i t usable f o r products of 8 by 8, but the gates w i l l then have to handle currents of up to 1.9 ma. 4.2. F l i p - F l o p The design of the f l i p - f l o p was based on the 2N240 t r a n -s i s t o r , a high speed low power switching t r a n s i s t o r . The c i r c u i t ( F i g . 4.3.) i s of conventional design, except f o r not having the base r e s i s t o r taken to a voltage p o s i t i v e with r e -spect to the emitter. However, the s a t u r a t i o n voltage of the 2N240 t r a n s i s t o r i s low enough not to require t h i s . Since t r i g g e r i n g i s performed by p o s i t i v e pulses, base t r i g g e r i n g through -the diodes i s used. i s the i n j e c t i o n diode, while Dg prevents loading of the p o s i t i v e pulse source. The c o l -l e c t o r voltages are used to c o n t r o l diode gates, and must be able to c o n t r o l the currents being i n t e g r a t e d . 4.3. Monostable M u l t i v i b r a t o r '>• This determines the time i n t e r v a l f o r the borrow-carry operation. I t i s seen i n F i g . 4.4. T^  normally conducts. When T^ i s turned o f f r (by a p o s i t i v e pulse on the base of T^ or a negative pulse on the base of Tg) Tg w i l l conduct f o r a p e r i o d determined hy the time constant GjR, a f t e r which there i s a f a s t t r a n s i t i o n back to the o r i g i n a l s t a t e . The c o l -l e c t o r voltages are used to c o n t r o l diode gates. 4.4. Voltage Comparator I t i s necessary to have a simple voltage l e v e l detector which could be adapted to detect e i t h e r p o s i t i v e or negative going voltages at -3, -13, and -20 v o l t s . The detectors shown i n F i g . 4.5. meet the requirements since t h e i r s e n s i t i v i t y F i g . 4.3. F l i p - F l o p C i r c u i t C x m 250pF R 2 = 6.8K R 4 = 10K Tl» T 2 = 2 N 2 4 0 R x = IK R g = 2.2K R 5 = 4.7K F i g . 4.4. Monostable M u l t i v i b r a t o r Timer C i r c u i t 29 Pig. 4.5. Voltage Comparator C i r c u i t s can always be increased by adding another stage of gain. In operation, i s normally c u t - o f f . When the input voltage goes more negative than -20 (or -13) v o l t s ( P i g . 4.5.a.) or more p o s i t i v e than -3 v o l t s ( P i g . 4.5.b„), conducts, the base voltage f o l l o w s the input, and an a m p l i f i e d voltage i s produced at the c o l l e c t o r which t r i g g e r s the f l i p - f l o p . Under normal c o n d i t i o n s , as soon as the l e v e l i s detected the r i s e or f a l l i s stopped. 4.5. Gates F i g . 4.6.a. shows a conventional current gate which i s c o n t r o l l e d by t r a n s i s t o r f l i p - f l o p s of the type discussed i n s e c t i o n 4.2. I t supplies a constant current t o the f i r s t i n t e g r a t o r f o r a time i n t e r v a l determined by the f l i p - f l o p . This produces a l i n e a r r i s e or f a l l i n the i n t e g r a t o r . F i g . 4.6.b. shows the gate c o n t r o l l i n g the input to the second i n t e g r a t o r . S c a l i n g by two i s performed by a l l o w i n g current to flow through 2 i d e n t i c a l r e s i s t o r paths from the f i r s t to the second integrator.. A l t e r n a t i v e l y i t can be allowed to flow through only one path? g i v i n g s c a l i n g by 1, or stopped a l t o g e t h e r . The remaining two paths produce sweep downs and 10 subtract operations. The value of the r e s i s t o r s are determined by the RC pro-ducts required f o r a given operation of the i n t e g r a t o r . 4.6. Quantizer Pig.4.7. shows 2 p o s s i b l e quantizer c o n f i g u r a t i o n s . C i r c u i t (a) supplies a quantized r e s u l t to another i n t e g r a t o r , To f l i p - f l o p s D l •w-D 4 31 -25v R, To input of f i r s t R-i i n t e g r a t o r +25v R-, 82K D 2,D 3 - 1N100 VltV4- 1N191 To a-e coupler To scale by Z f l i p - f l o p To c u t - o f f f l i p - f l o p (a) R-, B l . u2 ' D l - KJ- 1 -25V To 10 subtract f l i p - f l o p To sweep-down f l i p - f l o p - 2 5 v - — v w v - i — K l 1 3h To input of •** second i n t e g r a t o r R x m 56K R 2 = 39K R 3 = 560K D1 = 1N191 D 2 = 1N100 (b) P i g . 4.6. C i r c u i t s of Current Gates 32 while (b) codes the contents of the i n t e g r a t o r i n t o a quan-t i z e d pulse p o s i t i o n form. Operation of the quantizer requires that c l o c k pulses (c) be a v a i l a b l e on one l i n e . A pulse g occurring a h a l f u n i t of time before the g pulse i s a l s o required on another l i n e . Because these pulses were not r e a d i l y a v a i l a b l e from the s i g n a l source, and because the system i s i n h e r e n t l y simple, i t was not incorporated i n t o the f i n a l u n i t . Operation f o r F i g . 4.7.a. i s as f o l l o w s . F l i p - f l o p FF1 i s set by g , c l o s i n g "And" gate A^, and s t a r t i n g a sweep down of l g at a slope of -1. FF2 i s set by G^, thus adding a slope of +1 to the r e s u l t being accumulated i n l g . A c t i v a t i o n of the zero detector resets FF1, stopping s u b t r a c t i o n from l g and opening A^. The next clock pulse a r r i v i n g at A^ passes through the gate and resets FF2, which stops the r i s e i n l g . The operation f o r F i g . 4.7.b. i s s i m i l a r except that FF2 i s not required, and the f i r s t pulse to pass through A^ i n the c o r r e c t G period i s used as the pulse p o s i t i o n code of the contents of l g * As seen i n the diagram of F i g . 4.8., t h i s c i r c u i t w i l l c o r r e c t any value i n I ? which i s out by +0.5 u n i t s . 33 Corrected sweep up Sweep down Sweep down PP2 0 det 2 / > FF1 T (a) I* - I 0 det PP1 to store T I c (b) P i g . 4.7. Block Diagram of Quantizer 34 F i g . 4.8. Waveforms of Quantizer 5. SINGLE CHANNEL TEST MULTIPLIER 35 The u n i t to be described i s the f i n a l one b u i l t . I t i s based on knowledge obtained from the c o n s t r u c t i o n of another s i m i l a r u n i t which incorporated an i n v e r t e r w i t h a s c a l i n g by 2 c o n t r o l ; and d-c coupling of a l l u n i t s . The f i r s t u n i t gave an i n d i c a t i o n of where the performance would be c r i t i c a l , and l e d to mod i f i c a t i o n s i n the second u n i t . Thus, cumulative d-c d r i f t s , and the d i f f i c u l t y of a l i g n i n g non-zero input and out-put voltages r e s u l t e d i n abandoning the i n v e r t e r and using a-c coupling d i r e c t l y between the f i r s t and second i n t e g r a t o r s . Since the input p o t e n t i a l of the i n t e g r a t o r s v a r i e d by up to 0.1 v o l t s , i t was de s i r e d to have the d r i v i n g voltages at l e a s t 20 v o l t s (1/200 accuracy). They were made 25 v o l t s , t h i s be-in g the supply voltage f o r the a m p l i f i e r s , and the output of the f i r s t i n t e g r a t o r was made to have a slope of +2 v o l t s per cloc k p e r i o d . Thus, on the f i r s t i n t e g r a t o r , 2 v o l t s repre-sents 1 u n i t . This l a r g e r output voltage of the f i r s t i n t e -g r a t o r also reduced the loading on i t by the i n t e g r a t i n g r e -s i s t o r of the second stage. Since the amount of droop of the output, and change of the input are dependent on the magnitude of the output, the second i n t e g r a t o r was set to have a slope of -1 v o l t per clo c k p e r i o d . The droop was not as c r i t i c a l on the f i r s t stage due to the l a r g e r value of i n t e g r a t i n g ca-p a c i t o r . To make sure the current gates are completely on or o f f , the f l i p - f l o p c o l l e c t o r s are made to operate between about 36 +1.5 v o l t s , and -2.5 v o l t s f o r an i n t e g r a t o r input voltage of about -0.4 v o l t s . The method of p r e s e n t a t i o n i s now changed from the sym-b o l i c one used p r e v i o u s l y , to one where each of the blocks r e -presents one of the u n i t s described. The complete operation i s then given f o r a sample c a l c u l a t i o n of 3 times 5. F i g . 5 . 1 . shows the interconnections of the various c i r c u i t s , while F i g . 5.2. gives the waveforms. i 5 . 1 . Operation In the quiescent s t a t e , a l l i n t e g r a t o r s are at zero l e v e l , and a l l f l i p - f l o p s are reset w i t h t h e i r g side negative. The g Q pulse sets f l i p - f l o p s FF1 and FF2, which by t h e i r c o n t r o l of gates T^ and Tg cause a l i n e a r r i s e w i t h a slope of 2 i n i n t e g r a t o r I ^ j operate the s c a l i n g by 2 c o n t r o l ; and open the gate p e r m i t t i n g 1^ to feed current to Ig. The output of Ig then r i s e s p a r a b o l i c a l l y as shown i n F i g . 5.2. Pulse x 0 occurs at t=3, r e s e t t i n g FF1, which stops the r i s e i n 1^ and changes the s c a l i n g c o n t r o l so t h a t i t now scales by 1 . Ig now r i s e s l i n e a r l y with a slope of 1, reach-i n g -13 v o l t s (which represents the number 10) at t=3j|. The 10 l e v e l detector generates a pulse which sets the borrow-c a r r y m u l t i v i b r a t o r . A c t i n g through Tg, the m u l t i v i b r a t o r causes a s u b t r a c t i o n of 10 u n i t s over a p e r i o d of 5 microsec, l e a v i n g some 1.5 u n i t s (-4.5 v o l t s ) i n Ig at the end of the operation. While the borrow-carry m u l t i v i b r a t o r was i n opera-t i o n i t would a l s o have been feeding 1 u n i t i n t o Ig (carry operation) by c o n t r o l l i n g T„. I p then continues to r i s e 37 PP n Output goes +ve f o r pulse on same side of f l i p - f l o p . T Diode gate Source i s c u t - o f f f o r b of opposite p o l a r i t y to source; cut-on i f b i s of same p o l a r i t y . +source F i g . 5.1. S i n g l e Channel Test M u l t i p l i e r 38 © W> «5 +» r-t o • a ft - p o U O + > (30 o> -P a i -14v -20v -13v -3v -4v •3v 8 FF3 PF4 FF5 1 2 3 4 5 6 7 8 9 g l 1 2 3 4 5 6 7 8 9 'I I I I I I I I I I I ' I I I • I I I u n i t s o f t i m e i n terms o f c l o c k p u l s e s JZL F l i p - f l o p waveforms i n d i c a t e p e r i o d o f change o f s t a t e F i g . 5.2. S i n g l e C h a n n e l T e s t M u l t i p l i e r Waveforms l i n e a r l y to 5 u n i t s , at which time pulse y^ resets PP2, and cuts o f f l g . Obtaining the r e s u l t i n the proper pulse p o s i t i o n code i s performed i n the f o l l o w i n g p e r i o d 0^. 1^ i s al s o returned to zero i n t h i s time p e r i o d . The pulse g^ sets PP3, PP4, and PP5, sweeping 1^, I^, and l g down at slope of -1 u n t i l they each reach zero l e v e l . At that l e v e l a pulse i s generated by the zero detector, which r e s e t s the corresponding f l i p - f l o p and the sweep down stops. The zero detector pulses from l g and l g represent the coded r e s u l t of the computation. I f i t were d e s i r e d to quantize and c o r r e c t the r e s u l t i n l g and l g , the method described i n Sub-section 4.6. would have been used i n s t e a d . This q u a n t i z i n g procedure would be essen-t i a l i n the s h i f t and read-out operations of a l a r g e r system f o r the m u l t i p l i c a t i o n of 2 f i v e d i g i t numbers. 5.3. E r r o r s and Conclusions The accuracy of adjustment required i s p r o p o r t i o n a l to the magnitude of the product to be obtained. Thus f o r a pro-duct of 5 times 2, the accuracy required i s 1 part i n 20, while f o r a product of 8 times 8 i t i s 1 part i n 128. In a d j u s t i n g the R of the RC product to t h i s degree of accuracy, the forward impedance of the diodes must be allowed f o r . Since i f we neglect temperature e f f e c t s , t h i s impedance w i l l be constant f o r a constant current, i t i s p o s s i b l e to allow f o r t h i s e x tra impedance i n a l l gates except the one con-t r o l l i n g the current path between the f i r s t and second i n t e -g r a t o r s . Here the current v a r i e s w i t h the magnitude of the out-put of 1^, and i s at a l l times s m a l l ; being from zero to 300 microamps. Tests on a s i l i c o n and germanium diode i n s e r i e s showed tha t t h e i r impedance i s 75k at 8.5 microamp, and 12k at 58 microamp. The e f f e c t of operating the diodes i n t h e i r t r a n -s i t i o n region i s that the m u l t i p l i e r cannot be adjusted to give a r e s u l t of 9 f o r both 1 times 9 and 3 times 3 since the RC product of Ig d i f f e r s by n e a r l y 100$ between the two cases. A d d i t i o n a l i n a c c u r a c i e s are produced by having 2 diodes i n one path and 1 diode i n the other path of the s c a l e by 2 c o n t r o l , when both are r e q u i r e d to have i d e n t i c a l impedances. Within the accuracies of reading the o s c i l l o s c o p e , i t was p o s s i b l e to adjust 1^ to give m u l t i p l e s of 2 v o l t s per u n i t of time w i t h no v i s i b l e e r r o r . For a set d r i v i n g v o l t a g e , t h i s only required adjustment of R, the a m p l i f i e r gain being s u f f i -c i e n t to produce l i n e a r i n t e g r a t i o n . A s l i g h t droop i n the h e l d voltage was encountered due to the b i a s i n g chain e f f e c t (see Sub-section 4.2.). Adjustment of I0 i s mqre complex, due to the f i n a l r e s u l t being determined by s e v e r a l v a r i a b l e s . Thus err o r s can be produced by i n c o r r e c t r e s i s t o r values i n the 2 current paths of the scale by two c o n t r o l , by an i n c o r r e c t RC product f o r the 10 subtract, i n c o r r e c t d u r a t i o n of the 10 s u b t r a c t , and i n c o r r e c t 10 l e v e l d e t e c t i o n . In a d d i t i o n the a m p l i f i e r gain was shown to be i n s u f f i c i e n t to provide accurate i n t e g r a t i o n . (See Appendix 5). The higher gain a m p l i f i e r was not used 41 b e c a u s e o f t h e i n c r e a s e d t e m p e r a t u r e d r i f t t h a t i t p r o d u c e s . A h i g h e r v a l u e o f i n t e g r a t i n g c a p a c i t o r was n o t u s e d b e c a u s e o f t h e i n c r e a s e d l o a d i n g i t w o u l d p r o d u c e on t h e f i r s t i n t e g r a t o r , and on t h e f l i p - f l o p s c o n t r o l l i n g t h e g a t e s . W i t h o n l y t h e o s c i l l o s c o p e as a m e a s u r i n g i n s t r u m e n t , i t was n o t p o s s i b l e t o a d j u s t a l l t h e v a r i a b l e s t o g i v e t h e c o r r e c t r e s u l t o v e r a wide r a n g e . A t t h e l o w e r end, a c c u r a c y i s l i m -i t e d b y t h e e f f e c t o f t h e d i o d e impedance, and a t t h e u p p e r end, by t h e a m p l i f i e r g a i n , and d i f f i c u l t y o f a d j u s t m e n t . I t i s s e e n t h e n t h a t t h e r e a r e 3 p r i n c i p a l f e a t u r e s o f t h e p r e s e n t method w h i c h i t w o u l d be d e s i r a b l e t o a v o i d . 1. The 17 o p e r a t i o n a l a m p l i f i e r s w h i c h w o u l d be r e q u i r e d f o r a 5 d i g i t m u l t i p l i e r , w i t h t h e i r a t t e n d a n t p r o b l e m s o f r e l i a -b i l i t y , c o m p l e x i t y , d i f f i c u l t y o f a d j u s t m e n t , e t c . 2. The l a r g e number o f 10 s u b t r a c t o p e r a t i o n s w i t h t h e i r a t t e n d -a n t a c c u r a c y r e q u i r e m e n t s . ! 3. The d o u b l e i n t e g r a t i o n p r o c e s s w h i c h p r o d u c e s s m a l l c u r r e n t s t h r o u g h one o f t h e d i o d e g a t e s , and c o n s e q u e n t n o n - l i n e -a r i t i e s . To g e t a r o u n d t h e s e r e q u i r e m e n t s i t i s n e c e s s a r y t o a ban-don t h e t r a p e z o i d a l method o f m u l t i p l i c a t i o n i n f a v o r o f t h e s y s t e m t o be d i s c u s s e d n e x t . 42 6. REVISED MULTIPLIER Consider the m u l t i p l i c a t i o n t a b l e of 2 numbers shown i n P i g * 6.1. When supplied with both numbers at the same time, the f a s t e s t method of l o o k i n g f o r a product i s to f o l l o w the diagonal down u n t i l meeting the f i r s t number on the v e r t i c a l column at the l e f t , and then h o r i z o n t a l l y u n t i l at the column of the second number. We w i l l then have the d e s i r e d r e s u l t . This i s b a s i c a l l y the method which w i l l be used to ob-t a i n a product, but since generation of the complete t a b l e would be r a t h e r complex, the values i n the diagonal alone are generated, and the o f f - d i a g o n a l values are then obtained by adding to these values. The t o t a l number r e q u i r i n g to be added would then never exceed 20. As before, analog methods are used to perform the a d d i -t i o n s , but as a l l the operations are now l i n e a r , a constant current source i s used i n s t e a d of an o p e r a t i o n a l a m p l i f i e r . From the d e s c r i p t i o n of the c i r c u i t s i t w i l l be seen how t h i s p r i n c i p l e i s to be used i n assembling a m u l t i p l i e r . 6.1. Generation of Tens Diagonal, U n i t s Diagonal, and U n i t s  Off-Diagonal The proposed c i r c u i t f o r performing the m u l t i p l i c a t i o n i s shown i n F i g . 6.2. The r e q u i r e d waveforms are given i n F i g . 6.3. Consider X as being the i n t e r v a l between the g and x pulses, and T the i n t e r v a l between the g and y pulses. Then the c o l l e c t o r of T, i s p o s i t i v e f o r ( X . I ) , the time on the 43 Maximum Number number of added C a r r i e s 1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 9 8 0 2 4 6 8 10 12 14 16 18 14 i 3 9 12 15 18 21 24 27 18 2 4 16 20 24 28 32 36 20 2 5 25 30 35 40 45 20 2 6 36 42 48 54 18 2 7 49 56 63 14 2 8 64 72 8 1 9 81 0 0 Values i n diagonal 1 4 9 6 5 6 9 4 1 -Units 0 0 0 1 2 3 4 6 8 -Tens F i g . 6.1. M u l t i p l i c a t i o n Table 44 X k — A / W S / - I r3 K l — J - V V W -V 2 ^ W S A Or 1 T ko X k - \ A A / V -H h V V W H 10^'k+l o r -TV 'kd F i g . 6.2. C i r c u i t f o r Generation of Tens Diagonal, U n i t s Diagonal, and Un i t s Off-Diagonal i 45 0 1 2 3 _ l I L 4 5 6 7 8 9 J L I I I L Clock 0 pulse p o s i t i o n s +100v +100v - V 10-1 4 r _5_r e r 8 6 K 6  1 5 if  4 P i g . 6.3. Input Waveforms f o r Revised M u l t i p l i e r diagonal, and negative the r e s t of the time. The c o l l e c t o r of Tg i s p o s i t i v e f o r (X+Y).(X.I) the time on the o f f - d i a g o n a l . During the i n t e r v a l (X.Y), samples the u n i t s diagonal value as given by V^, ^ developes the tens diagonal by means of l i n e a r sweeps supplied by V^ and Vg, while C^Q samples Vg. At the end of (X.Y), t r a n s f e r s i t s charge to through Tgj the sweep on i s stopped, and the a m p l i f i e d voltage of C^Q i s a p p l i e d to to produce the o f f - d i a g o n a l values. During the p e r i o d of the o f f - d i a g o n a l i . e . (X+I).(X.Y), as the l i n e a r sweeps are o c c u r r i n g i n C^, 10 subtract and c a r r y opera-t i o n s i n t o 9 k + i m a v b e produced. At the end of the o f f - d i a g o n a l p e r i o d , the operation stops and the contents of C^ and C^-^ are t r a n s f e r r e d to s t o r -age capacitors f o r the s h i f t operation. The borrow-carry and s h i f t operations are described next. 6.2. Borrow-Carry The borrow-carry c i r c u i t F i g . 6.4. i s s i m i l a r to t h a t used i n the previous u n i t . When V^ the voltage on C^ reaches zero, the schmitt t r i g -ger changes s t a t e and opens gates long enough to subtract 10 v o l t s from and add 1 v o l t to ^. i 6.3. S h i f t and Transfer The proposed c i r c u i t i s shown i n F i g . 6.5. The contents of are t r a n s f e r r e d to C j t g by a p p l i c i a t i o n of a - V ^ Q voltage to the base of T^ which i s normally c u t - o f f . This leaves C^ f r e e f o r re-use during the next time p e r i o d while the contents of are being quantized and s h i f t e d t o 4 7 -V 40- S.T. T 10 i - T i o + V i • V M — + V S.T. - Schmitt Trigger (Timer) P i g . 6 . 4 . C i r c u i t f o r Borrow-Carry Operation 48 P i g . 6.5, C i r c u i t f o r S h i f t and Transfer Operation C^_^. The s h i f t i s i n i t i a t e d by a g pulse s e t t i n g FF1, thus s t a r t i n g a sweep down of C j t g at a slope of -1, and a r i s e of C j ^ ^ at a slope of +1. When a clock pulse passes through the comparator, FP1 i s r e s e t , r e s u l t i n g i n quantizing the contents of C k g i n t o C k ^ . 6.4. Test Unit The t r a n s f e r and s h i f t operations were t e s t e d by means of i the c i r c u i t shown i n P i g . 6.6. Since the low l e v e l f l i p - f l o p s were used to provide the timing i n t e r v a l s (see Sub-sections 4.2. and 4.3.), the voltages had to be changed by means of T^, Tg, and Tg to the new l e v e l s r e q u i r e d to operate the gates. In operation the charge equivalent to 2.5 v o l t s on was t r a n s f e r r e d i n 3 microsec to C k g by c o n t r o l of the base of the 2N404 t r a n s i s t o r (T^). Subsequently the voltage on the ^ks t r a n s i s t o r was swept down l i n e a r l y to -5 v o l t s while that on Cjj. was being swept up. The r i s e was stopped by a pulse passing through the comparator when V k s reached -5 v o l t s . The t e s t s showed that no d i f f i c u l t y would be encountered i n a c c u rately t r a n s f e r r i n g charge between capacitors or i n holding t h e i r output l e v e l s . Due to the capacitors being close-l y matched, there was no detectable d i f f e r e n c e i h the voltage l e v e l s reached. The f i n a l comparator c i r c u i t may d i f f e r from the one used here, but again there should be l i t t l e d i f f i c u l t y i n achieving a s u i t a b l e system f o r q u a n t i z i n g and s h i f t i n g v o l t a g e s . 50 P i g . 6.6. Test C i r c u i t f o r Transfer and S h i f t Operations 51 6 . 5 . Five D i g i t M u l t i p l i e r Five u n i t s of the type shown i n F i g . 6 . 2 . , w i t h t h e i r attendant borrow-^carry and s h i f t c o n t r o l s would be r e q u i r e d . The waveforms would be supplied to a l l of the u n i t s i n p a r a l -l e l , but each would have i t s own separate X and T p e r i o d con-t r o l s . Each of the capacitors i n the u n i t would be used i n s e v e r a l ways. Thus during one p e r i o d the c a p a c i t o r would develop the tens diagonal from the previous stage, accept c a r r i e s from the previous stage, accept the s h i f t e d and quan-t i z e d value of the previous p a r t i a l sum, accept the t r a n s f e r from arid develop the o f f - d i a g o n a l sweeps. I t would also be swept down by i t s own c a r r y - s u b t r a c t operations. 6 . 6 . Conclusions This f i n a l r e v i s e d system s t i l l has i t s disadvantages, but i t appears that these w i l l be e a s i e r to overcome than those of the previous systems. The need f o r o p e r a t i o n a l a m p l i f i e r s i s overcome by use of s p e c i a l waveforms and con-stant current supplies to the c a p a c i t o r s . * The number of carry-subtract operations i s reduced to a maximum of 4 as com-pared w i t h the previous 9. Two of the 4 c a r r i e s are produced by the o f f - d i a g o n a l sweep, and the other 2 by the s h i f t opera-t i o n , and the c a r r i e s and tens diagonal of the previous stage. The problem of small diode currents i s only p a r t l y overcome i n that there i s s t i l l a change by a f a c t o r of 10 during the o f f -diagonal sweeps. In a d d i t i o n the low d r i v i n g voltages ( f o r low values of Vg) may not be able to produce s u f f i c i e n t l y l i n e a r sweeps i n C,. This could be overcome by use of t r a n -5 2 s i s t o r s to t r a n s f e r the required Vg voltage as many times as needed, rather than using sweeps. An added advantage would then be the e l i m i n a t i o n of the d-c a m p l i f i e r . Due to the new method of obta i n i n g the product, the over-a l l accuracy requirements are also considerably reduced. With a maximum of 4 car r y operations, the accuracy required of the 10 subtract operation i s 1 part i n 80. This can aJLs,p be taken as the accuracy requirements of the supply waveforms Vg and V^, although i n general t h i s accuracy i s only required when 4 car-r i e s a c t u a l l y occur. The problems i n t h i s system w i l l be concerned w i t h the 10 subtract operation, and w i t h obtaining capacitors matched and s t a b l e to the req u i r e d degrees of accuracy. Since the t r a n s f e r operations of the whole system depend on charge t r a n s -f e r from one ca p a c i t o r to another, the capacitors must be matched i f the t r a n s f e r i s also to in v o l v e the same voltage l e v e l s . The r e s u l t s of t h i s i n v e s t i g a t i o n i n t o p o s s i b l e m u l t i p l i e r c o n f i g u r a t i o n s are that the decimal m u l t i p l i e r would prove too expensive due to i t s need f o r high speed beam switching tubes. The t r a p e z o i d a l method of m u l t i p l i c a t i o n , while appearing to be t h e o r e t i c a l l y rather simple, leads to e x c e s s i v e l y high accu-racy requirements which are d i f f i c u l t to r e a l i z e i n p r a c t i c e , and which would re q u i r e the use of elaborate and expensive oper a t i o n a l a m p l i f i e r s . The f i n a l r e v i s e d system i s l e s s s e l f - c o n t a i n e d than the t r a p e z o i d a l system, r e q u i r i n g the supply of various e x t e r n a l waveforms. However, no complex c i r c u i t r y i s needed, the accuracy requirements are low, and 53 r e t a i n i n g the q u a n t i z i n g operation developed f o r the modified t r a p e z o i d a l system should make these accuracies easy to achieve* 54 APPENDIX 1 DECIMAL COUNTER COMPONENTS Gas Counter Tube (Dekatron). S y l v a n i a type 6910. These tubes are b i - d i r e c t i o n a l c o l d cathode decade counter and s e l e c t o r tubes. The count may be determined v i s i b l y by noti n g the p o s i t i o n of the orange-blue glow spot on any of the 10 cathodes r a d i a l l y spaced around a c e n t r a l anode d i s k . S u i t -able p u l s i n g of the 2 g r i d s causes the discharge t o move from one cathode to the next. Output connections may be derived from a l l 10 cathodes, making a v a i l a b l e across 10 cathode r e -s i s t o r s 10 sequential gating type waveforms. Currents drawn are of the order of 0.6 ma, and the top counting r a t e i s about 100 kc i n e i t h e r d i r e c t i o n . Beam Switching Tube. Borroughs type BD 301. Their operation depends on the switching of a t r o c h o i d a l e l e c t r o n beam which i s c o n t r o l l e d by orthogonal magnetic and e l e c t r i c f i e l d s . Switching between output electrodes i s done by p u l s i n g the anode and 2 g r i d groups. The p o s i t i o n of the beam can be detected by 10 output electrodes i n s e r t e d through the anode, g i v i n g 10 sequential gating type waveforms. There i s no v i s i b l e i n d i c a t i o n of the beam p o s i t i o n . Tube currents are of the order of 10 ma, and top counting speeds are .about 1 Mc. N i x i e . Borroughs type 6844A. This i s a s p e c i a l purpose gas f i l l e d c o l d cathode v i s u a l d i s p l a y read out tube. The cathodes, placed one behind the 55 other, are formed i n the shape of the numerals which are to be dis p l a y e d , and each i s connected independently to an output p i n . Applying a voltage to one of these output pins w i l l then cause the appropriate cathode to have an orange blue glow; making the number v i s i b l e . I t can be d i r e c t l y connected to the beam switching tube to provide a d i g i t a l d i s p l a y of the count. An am p l i f y i n g stage i s req u i r e d when i t i s used with the Dekatron. T r i x i e . S y l v a n i a 1750 t r a n s i s t o r s plus N i x i e . This i s the same tube as the N i x i e i n d i s p l a y method, but i s intended f o r use with low voltage and low power t r a n s i s t o r c i r c u i t s . 5 6 APPENDIX 2 PULSE GENERATOR Int r o d u c t i o n . In a computer u t i l i z i n g the type of pulse code mentioned p r e v i o u s l y , a s u i t a b l e means must be a v a i l a b l e to generate the code from manually operated s e l e c t o r switches. A generator f o r t h i s pulse code and necessary s e l e c t o r gates was con-s t r u c t e d . This generator was designed to have the f o l l o w i n g outputs. 1 . Output on 1 l i n e , 5 i n i t i a t i n g pulses (gg - - - g±)> ten u n i t s apart, w i t h a delay of 5 0 u n i t s before repeating. 2 . Output on 1 l i n e , a pulse (gg), delayed 1 0 u n i t s behind g^. 3 . Output i n s e r i a l on 1 l i n e , 5 independently adjustable pulses ( X Q - - - x^) delayed 0 to 9 u n i t s behind the i n i -t i a t i n g p ulses. 4 . Output i n p a r a l l e l on 5 l i n e s , 5 sets of pulses (yQ, y^, Y 2 ' Y 3 ' Y 4 ^ ' e a c h equally and adjus t a b l y delayed 0 to 9 u n i t s behind the i n i t i a t i n g pulses, i . e . each y^ pulse i s to occur 5 times w i t h the same delay. The x and y terms are interchangeable, depending on which must be supplied i n s e r i e s and which i n p a r a l l e l . C i r c u i t . The design i s based on 2 S y l v a n i a 6 9 1 0 c o l d cathode counter tubes. One i s d r i v e n so as to change stat e every u n i t of time, and the other every 1 0 u n i t s . By s u i t a b l y gat-in g the voltages at the cathodes of the 2 tubes, the required outputs can be obtained. 57 j The d r i v i n g c i r c u i t f o r the 2 counter tubes i s shown i n P i g . A.2.1. I t i s a m o d i f i c a t i o n of that recommended by S y l -v a n i a . Ten 12 p o s i t i o n switches have 10 of t h e i r contacts connected to the cathodes of the f i r s t tube. The wipers of these switches can then independently make contact with any one of the cathodes. The method of d r i v i n g used causes the d i s -charge to change from p i n 10 (P^g) °* ^ e * ub e» ^° ^° etc . The r e s t of the c i r c u i t r y can be broken down i n t o the sub-u n i t s shown i n F i g . A.2.2. These are then interconnected as shown i n the l o g i c schematic of F i g . A.2.3. Because the counter tubes are high impedance devices, cathode f o l l o w e r s are used to d r i v e the "And" gates. The clamping l e v e l of the "And" gates i s adjusted to remove spurious low l e v e l pulses from the output. Due to the high impedance of the gates, f u r t h e r cathode f o l l o w e r s are r e q u i r e d to supply a low impedance output s i g -n a l . A-c coupling, w i t h d-c r e s t o r a t i o n i s used on a l l out-puts so as to have the output pulses going p o s i t i v e from zero v o l t s . L o g i c . The period GQ i s obtained from p i n 10 of the second counter tube, and i s designated ^2^10° ^9 S i v e s &i» e t c . The wipers on the switches S ^ to S ^ Q can be designated to give the pulses X Q to x^ and y^ to y 4 r e s p e c t i v e l y . Operation on these pulse sources by the "And" - "Or" gates connected as shown i n the schematic of F i g . A.2.3. w i l l then give the r e -58 quired outputs. The x and y pulses taken from the high speed counter can only occur once during each G p e r i o d . Thus the l o g i c equation Y Q ( G o + G l + G 2 + G 3 + G 4 ^ m e a n s that y Q occurs i n every one of the G periods since none overlap. Due to the f i n i t e t r a n s f e r time of the tubes, the pulse on T^P^Q which i n i t i a t e s the t r a n s f e r i n the second tube, has passed before T 2 has changed s t a t e . ^1^9 * s thus used f o r the zero p o s i t i o n . The n i n t h pulse now occurs during the change-over, but a pulse of about l / 2 the c o r r e c t magnitude can get through the "And" gates i n the c o r r e c t time i n t e r v a l . However, when the switches are set to s e l e c t the n i n t h pulse, the a d d i -t i o n a l loading of T^P^Q u s u a l l y makes i t incapable of d r i v i n g T g o The c i r c u i t was not modified during the c o n s t r u c t i o n to c o r r e c t t h i s d e f i c i e n c y as i t was f e l t t hat 8 d i g i t s would be enough f o r the i n i t i a l t e s t s . The pulse generator was normally operated so as to give a c l o c k p e r i o d of 14,3 microsec; the i n t e g r a t o r time constants being c a l c u l a t e d f o r t h i s value. To supply the s i n g l e channel t e s t m u l t i p l i e r , the c i r c u i t was l a t e r modified so as to give only the gg, X Q , y^, and g^ p u l s e s j a l l on separate l i n e s . This was done by removing the connections on TgPg to T g P g and attaching them t o p o i n t (a) (see P i g , A.2.1.). The connection to TgPg was then changed to T g P g o As seen i n P i g . A.2,3., t h i s serves to suppress the G^ to G^ periods c o n t r o l l i n g the x and y pulses, and to replace the G,- p e r i o d by the G, p e r i o d . 59 P i g . A.2.1. Pulse Generator C i r c u i t s (Counters) i t I ..... 0 0 Clamp Adjust o 1M Rg = 27K R 5 = 25K Cg = 2 5 ^ T± = 124T7 = 22K R 4 = 100K C 1 = 01 uF J>1 = 1N191 F i g . A.2.2. Pulse Generator C i r c u i t s (Logic and B u f f e r i n g ) S, X 1 o S2 X l 8 3 X2 S4 X 3 5 4 CFl CFl * A c CF1 CF1 | — ^ A * CF1 s6 y 0 s8 y 2 S 9 y 3 S 1 0 y 4 -?[°CFT CFl T F T CFl A e CFl G 0 T 2 P 1 0 CFl G l T 2 P 9 [My- G 2 T 2 P 8 L V 3 ° R C F T > — G 3 T 2P 7rpU C F 1 > — G 4 T 2 P 6 )R-> CF2 DCR X G G 0 + X 1 G 1 + X 2 G 2 + X 3 G 3 + X 4 G 4 OR CFl — 2 A JIT * A # A A *| CF2 I »} DCR | ? y 0 ( G 0 + G l + G 2 + G 3 + G 4 CF2 1—*\ DCR | * y x ( * CF2 — * DCR • • CF2 DCR CF2 |—j DCR |—* y 4 ( J E T -H CF2 |—»1 DCR | > g ( A K- CF1 G 5 T 2 P 5 CF2| H DCR | > g c F i g . A.2.3. Block Diagram of Pulse Generator Logic 62 APPENDIX 3 THREE STAGE AMPLIFIERS In an attempt to overcome the accuracy l i m i t a t i o n s of the compound a m p l i f i e r , the p r o p e r t i e s of 3 stage a m p l i f i e r s were i n v e s t i g a t e d . As shown i n F i g . A.3.1, a-c coupling was used so as to i s o l a t e temperature d r i f t s between stages, and make i t easy to set output l e v e l s . However, a-c coupling cannot be used i n an a m p l i f i e r to be used i n an i n t e g r a t o r , unless some method i s found of a c c u r a t e l y d-c r e s t o r i n g the i n t e r -stage c a p a c i t o r s . (Voltages develop across the capacitors which are dependent on how much the i n t e g r a t o r i s used i n a given p e r i o d of time.) I t proved q u i t e easy to b u i l d a m p l i f i e r s with open loop gains of 8000 at lOkc, but phase s h i f t s w i t h i n the device pro-duced o s c i l l a t i o n s when the i n t e g r a t i n g c a p a c i t o r was connected. The frequency of o s c i l l a t i o n v a r i e d from 1 to 7 megacycles, de-pending on the c o n f i g u r a t i o n , and the o s c i l l a t i o n s could only be stopped when the open loop gain was reduced. The c i r c u i t c o n f i g u r a t i o n s of F i g . A.3.1. show 2 p o s s i b l e methods of s t a -b i l i z a t i o n . In the f i r s t ; gain was reduced by use of 470k r e s i s t o r s i n s e r i e s w i t h the emitter de-coupling c a p a c i t o r s . A 500pF capacitor between the f i r s t emitter and the second c o l -l e c t o r would then stop the o s c i l l a t i o n s . Open loop gain was around 600 at lOkc and 100 kc. The second c i r c u i t w i t h the more complex s t a b i l i z a t i o n shown, had a voltage gain of 2000 at l k c , and 1000 at lOkc, but only 60 at lOOkc. 63 -16'5v R± = IK R 4 = 10K R ? = 330K = lOuF C 4 = 500pF R 2 m 15K R 5 = 4«7K R g = 82K Cg = 20uF C g = 'lOuF R 3 = -47K R 6 = 150K RQ = 0»1K Cg = 15uF Cg = .05uF T, - 2N247 C 7 = 1 5 0 p F F i g . A.3.1. Three Stage A m p l i f i e r C i r c u i t s 64 The problems encountered i n developing these i n t e g r a t o r a m p l i f i e r s d i f f e r from the normal ones i n that the device i s requ i r e d to have a considerable gain at frequencies where s t r a y c a p a c i t i e s and t r a n s i s t o r c u t - o f f a f f e c t the performance. For the 2N247 t r a n s i s t o r , the common emitter c u t - o f f frequency i s f n T C • =500kc. Since the a m p l i f i e r output can be r e q u i r e d to P produce a t r i a n g u l a r waveform w i t h a frequency of about lOOkc, there i s not much range f o r changing the attenuation-frequency c h a r a c t e r i s t i c of the a m p l i f i e r without e x c e s s i v e l y reducing the high frequency g a i n . Most of the high frequency attenuation c i r c u i t s t r i e d would reduce the frequency of o s c i l l a t i o n without a f f e c t i n g the amplitude. A l t e r n a t i v e l y the o s c i l l a t i o n would be very much attenuated, but not completely removed. (The s i n g l e capa-c i t o r used may have had t h i s e f f e c t . ) A t h i r d c o n d i t i o n en-countered was a so r t of c o n d i t i o n a l s t a b i l i t y ; high frequency o s c i l l a t i o n s o ccurring only at c e r t a i n output l e v e l s . No method was found which could give s t a b i l i t y f o r open loop gains appreciably greater than those a v a i l a b l e from the simple compound a m p l i f i e r . The only improvement would have been i n a reduced input impedance. When the 3 stage a m p l i f i e r was converted to d-c coupling, i t proved extremely d i f f i c u l t ; due to the high d-c gain, to adjust the output l e v e l by changing the input b i a s chain s e t t i n g . Temperature d r i f t was continuous and d i f f i c u l t to adjust f o r by changing the input s e t t i n g . From t h i s work i t was r e a l i z e d that any improved ampli-f i e r would have to incl u d e automatic zero set, automatic 65 temperature compensation, and high gain over a broader band-width than had been obtained i n the a m p l i f i e r s above. Since the development of such an a m p l i f i e r would be a p r o j e c t i n i t s e l f , f u r t h e r work on i t was stopped. APPENDIX 4 TRANSISTOR OPERATIONAL AMPLIFIERS Consider the current r e l a t i o n s h i p s I = A l . sc 1 Where A i s the short c i r c u i t current gain of a m p l i f i e r . h - h+h S u b s t i t u t e ( l ) and (2) i n (3) I. sc - V^sc x s c A Consider the input 67 Z^ - Feed-forward impedance - Feed-back impedance - Input impedance of a m p l i f i e r I. = 1 V.-V, x b V b - h Z i (6) (7) S u b s t i t u t e from (2) f o r I. t I V, = Z. . _ S £ b x A (8) S u b s t i t u t e (8) i n (6) V. Z.' I I. = _ A _ _JL x Z. Z. sc A (9) S u b s t i t u t e from (5) f o r I. Z i sc t- t Z. r:+ 1 X u AB or I = ~ A sc Z i 1-AB+ Zi Z. J x (10) (11) Consider the output o 68 B, - Load impedance Z q - Output impedance of a m p l i f i e r * ^sc ~ "V S u b s t i t u t e from (2) f o r I P S u b s t i t u t e from (11) f o r I sc P 0 1 4 - 1 V. 1 - AB + Z i Z7 i (12) (13) (14) Convert to a form containing yr— • V. ti. x o i 1 " + £ 1 (15) (15) reduces to V m • V . o Z^ x Z. AB + ™ 1 - AB + ^ x (16) To f i n d p X8 = Z ^ l V o ~ V b (17) 69 S u b s t i t u t e f o r YQ and V b from (12) and (7) sc R i Z i A (18) Now B sc S u b s t i t u t e f o r I„ and I P sc from (18) z. 1 A k + \\ 1 + Z k K 0 1 1 1*1 1 + £ T 0 ' i + z k i A 0 For 1 k 1 i n t e g r a t i o n , Z^ = R, ^ = ^ • = gf (16) gives the t r a n s f e r f u n c t i o n (19) (20) (21) For or T = 1 AB + s c Z i Sf ° z! 1 - AB + j i (22) (23) (24) To minimize the e r r o r term, AB must be maximized. From (21), t h i s means we de s i r e Z. -l 0, R. I T oo» APPENDIX 5 INTEGRATOR ERRORS Prom Appendix 4 equation (24), we have the r e l a t i o n 1 V , TT " ST 1 + AB S u b s t i t u t e f o r 8 from equation (21) of Appendix 4 V , V. ST 1 + 1 — + "~t* 1 RT Z / I L o ' A - z i 1 1 1 R L + Z i ^ 0 T SC Assume z'=»=»RT • For R. = 6'8 K, and Z.' = 15 K O h h 1 A- zi(ij; + r)= A - 3 - A The e r r o r term i n (2) now becomes e J, _ t - s f l [x + R^SC]• v i - TA f V* + CTHJ/7 ^ Jo L o -'o F i r s t Integrator (1^) -9 C = 4.10 .F -6 t = 129.10 sec V. = 25v l -6 T = 179.10 sec e = 0*03 ( 1 + 2°37) = 0'113 v. Second Integrator (Ig) -9 -6 C = 0*5.10 P V i = 9 v T s= 14*3.10 sec -6 t = 129.10 sec e = 0*135 ( 1 + 19J (9) = 2'7 v (10) To reduce the er r o r of Ig to 0*5 f o r a maximum product of 8 by 8 requires -9 -6 C = 1*3.10 F V. = 8 v T = 14*3.10 sec -6 T = 114.10 sec Then e = 0*064 (1 + 6*4) ( l l ) = 0*475 v. (12) 72 BIBLIOGRAPHY 1. M i t c h e l l , J.M., and Rushman, S., "The TRICE, A High Speed Incremental Computer," I.R.E. N a t i o n a l Convention  Record, part 4, 1958, pp. 206-209. 2. Lee, R.C., and Cox, F.B., "A High Speed A n a l o g - D i g i t a l Computer f o r Simulation," I.R.E. Transactions on  E l e c t r o n i c Computers, v o l . EC-8, no. 2, June 1959, pp. 186-196. 3. Skranstad, H.J., "A Combined A n a l o g - D i g i t a l D i f f e r e n t i a l Analyzer," Datamation, v o l . 5, no. 6, Nov. -Dec. 1959, p. 41. 4. Boulding, J.D.R., "An Analog Method of Function Generation Using a Magnetic Drum," M.A.Sc. Thesis, U n i v e r s i t y of B r i t i s h Columbia ( E l e c t r i c a l Engineering), A p r i l 1959. 5. Blecher, F.H., " T r a n s i s t o r C i r c u i t s f o r Analog and D i g i t a l Systems," B e l l System Technical J o u r n a l , no. 35, 1956, pp. 295-332. 

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