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A sampling-type function generator and four-quadrant analog multiplier 1956

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A SAMPLING-TYPE FUNCTION GENERATOR AND FOUR-QUADRANT ANALOG MULTIPLIER BERNARD PERCY HILDEBRAND B.A.Sc. U n i v e r s i t y of B r i t i s h Columbia, 1954 A t h e s i s submitted i n p a r t i a l f u l f i l m e n t of the requirements f o r 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 t o the standard r e q u i r e d from candidates f o r the degree of MASTER OP APPLIED SCIENCE Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1956. ABSTRACT This t h e s i s describes the design. s.nd development of a sampling-type f u n c t i o n generator and a four-quadrant analog m u l t i p l i e r . The p r o j e c t i s d i v i d e d i n t o two p a r t s , the g e n e r a l arrangement and c i r c u i t r y of the f u n c t i o n generator and m u l t i p l i e r , and the t i m i n g c i r c u i t s which actuate them. This t h e s i s i s concerned w i t h the general c i r c u i t r y . The f u n c t i o n s to be generated are photographed on 35 mm. f i l m and mounted i n standard frames which a r e then fastened t o the r i m of a r o t a t i n g d i s k * An o p t i c a l system i s used to scan the fu n c t i o n s i n a tim e - s e q u e n t i a l manner. A t i m i n g system s e l e c t s the r e q u i r e d a b s c i s s a and actuates a combination of e l e c t r o n i c gates and clamping c i r c u i t s which stores the v o l t a g e , E^, re p r e s e n t i n g the o r d i n a t e , and the voltage, E^, representing t he maximum of the f u n c t i o n . These two stored v o l t a g e s , E^ and E^, are a p p l i e d t o separate sweep c i r c u i t s which produce sweep outputs of E^Ct) and Ej^T(t) r e s p e c t i v e l y . A system of comparator c i r c u i t s and gates samples the E^N(t) at the i n s t a n t a reference v o l t a g e , E, equals the sweep EgjET(t). Since the sweeps, N(t) are i d e n t i c a l , the value of EJST(t) at the i n s t a n t of sampling i s as i t i s scanned. Each successive m u l t i p l i c a t i o n i s sto r e d i n i t s own storage u n i t . A l l the c i r c u i t s are designed t o be s e l f - c a l i b r a t i n g to minimize e r r o r due to d r i f t . EE. This sequence of operations occurs f o r each f u n c t i o n i i TABLE OP CONTENTS TaDLe 03? COIl'tjeil'tjS 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 X i TableS o o o o o o o v o t o DO o o o • o o o o oo oo oo oo o o o o o o o o o o o o o o e X i i LiS"b 01* IllUStr&tXOnS 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 I V ACkH0Wl6Cig@II16irt 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 V I IntrOdUCtXOn 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 II P r i n c i p l e of Operation , 0 o o o o 0 6 o o « o o » o o o o o o o o » 3 III D e t a i l e d C i r c u i t Design o o o o o o o o o o o o . o o o o o o o o . 12 (1) Scanning U n i t » o o o o « o o o o o o o o o o » o o » » o o o o o o o 12 (2) Photo-tube A m p l i f i e r o . a o o o o o o o o . o o o o o o o o a 14 (3) Holding System of Channels 1 & 3 »»«° °»»° <> 15 (4) Holding System of Channel 2 o o o o o o o o o o o o o o 16 C 5 ) IllVGrij @X*S eoooo O O O O O O O A O O 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 20 (6 ) Gr8L"fc©S 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 oo o o o o o o 21 (Y) SW66p CirC"U.X*tS o o o o * » 0 » o © o o o o o o o o o o d * o o o * o o 24 ( 8 ) 0 QTIl J) SIX* St"t? OX* o o o o o o o o o o o o o o o o o o o o o o o e oo o o ooo 2*7 17 Accuracy Test of M u l t i p l i e r o . o o o o o o o o o o o o o o o o 3© V OOriClXlSXOIl o o o o o o o o o oo o o o o o o oooo oo oo oo oo o o o o o o 34 A^)JP^HCLl3C 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 / > 35"™3^ Sl'bllO^I^Q.piiy o o o o o o o * o o o oo o o o o o o o o o o o o o o o o o o o o 3*7 i i i TABLES 3?Q,"fol@ «L O O O O O O O O O Q O O O O O O 000000000 ooooooooo 00 000a lELbl© 2 000000 ooooooooo 000 0 0 *oooooe»a 0 0 0 0 0 0 0 0 o o o o 33 MST OP ILLUSTRATIONS Fig u r e Page 1 Symbols of computer u n i t s ..„......».... 2 2 C a l i b r a t i o n and f u n c t i o n frames .......» 4 3 Function generator assembly •........... 6 4 Four quadrant m u l t i p l i e r ...............9 5 Function storage o o o o o © . o o o o o o o o o o o o . o o . 11 6 Rota t i n g f u n c t i o n d i s k o o o o . . o . . . . « . o . o . 12 7 Scanning u n i t assembly ...... 13 8 Photo—tube a m p l i f i e r o o o . . . . . . . . . . . . . . . . 14 9 Holding system of channels 1 and 3 15 10 Voltage storage and h o l d i n g c i r c u i t ..•. 17 11 Holding c i r c u i t of channel 2 . . . . . . . . . o • 18 12 I n v e r t e r of channel 1 o o . . . . . . . . . . . . . . . . 20 13 I n v e r t e r of channel 2 . o ^ o . . . . . . . . . . . . . . 21 14 Diode gate o o . o o o . . o e o o o o o o o o « o o . . . . o . o o 22 15 Test setup f o r the diode gate .......... 23 16 Sweep c i r c u i t of channel 1 s......... • 25 17 Sweep c i r c u i t of channel 2 •...........• 26 18 C i r c u i t diagram of m u l t i a r comparator 28 19 C i r c u i t to match time constants ........ 30 20 Test setup f o r the m u l t i p l i e r .........« 31 V ACKNOWLEDGEMENT The author wishes to express h i s a p p r e c i a t i o n f o r the assis t a n c e r e c e i v e d from Dr. E . V . Bonn, Dr. A.D. Moore and Dr. P. Noakes of the Department of E l e c t r i c a l Engineering, The U n i v e r s i t y of B r i t i s h Columbia. The author i s indebted to the Defence Research Board, Department of N a t i o n a l Defence. Canada., f o r sponsoring the research p r o j e c t under Grant Number DRB C-9931-02(550-GC) The author's post-graduate s t u d i e s were made po s s i b l e through the N a t i o n a l Research C o u n c i l o f Canada's post- graduate bursary granted i n 1954. A SAMPLING-TYPE FUNCTION GENERATOR AND FOUR-QUADRANT ANALOG MULTIPLIER 1. I . I n t r o d u c t i o n There i s a d e f i n i t e need f o r an economical e l e c t r o n i c computer of s u f f i c i e n t accuracy f o r s o l v i n g complex engineering problems. U s u a l l y the greatest expense of such a computer l i e s i n the f u n c t i o n generator and i n the p r e c i s i o n components r e q u i r e d f o r reasonable accuracy. This t h e s i s covers the b a s i c design of a sampling type f u n c t i o n generator and the complete design of a four - quadrant analog m u l t i p l i e r w i t h the exception of the tim i n g c i r c u i t s . The design of the timing c i r c u i t s was undertaken under a separate t h e s i s by J.S. F i o r e n t i n o . The e l e c t r o n i c c i r c u i t r y contains a number of gates, f l i p - f l o p s , comparators, d.c. a m p l i f i e r s , e t c . The a m p l i f i e r s are P h i l b r i c k operation p l u g - i n u n i t s . * I t was decided t h a t a l l other standard c i r c u i t s such as gates, f l i p - f l o p s and comparators would a l s o be b u i l t as p l u g - i n u n i t s . The general c i r c u i t r y i s shown i n block form f o r convenience and i s b u i l t up of t h e standard p l u g - i n u n i t s . F igure 1 shows the symbols assigned t o these u n i t s . See appendix. Philbrik operational dc amplifier Philbrick operational dc amplifier - A / W M - Inverter »> Plip-Plop Delay Plip-Plop Blanking Phantastron Triode gate conducting (closed) non-conducting (open) Diode Gate Cathode follower *- Multiar HI" Integrator Pulse Amplifier Storage Unit Pig. 1 Symbols for computer units. IV) I I P r i n c i p l e of Operation The b a s i c operation of the f u n c t i o n generator, i s t o generate a r e p e t i t i v e waveform repr e s e n t i n g a l l f u n c t i o n s to be generated and then o b t a i n the r e q u i r e d ordinate values by sampling the waveform. The simplest method of generating the r e q u i r e d wave- form i s by means o f f i l m and an o p t i c a l , scanning system. This i s the method used i n the present i n v e s t i g a t i o n . Each f u n c t i o n t o be generated i s photographed on 35 mm. f i l m and mounted i n a standard 35 mm. frame which i s fastened t o the ri m of a r o t a t i n g d i s k . The fu n c t i o n s are drawn so that they a l l have the same maximum. The f i r s t frame on the d i s k i s used f o r automatic c a l i b r a t i o n and contains the zero l e v e l and the maximum l e v e l of the f o l l o w i n g f u n c t i o n s . Figure 2 shows the c a l i b r a t i o n frame and the f i r s t f u n c t i o n frame - together w i t h the pulse sequence which operates the f o l l o w i n g c i r c u i t r y o A narrow beam of l i g h t from an o p t i c a l system i s passed through the negatives onto a photo-tube. The output of the photo-tube then represents the ordinate of a l l the fun c t i o n s i n a tim e - s e q u e n t i a l manner. The l i n e a r i t y of t h i s method depends on the u n i f o r m i t y of the p h o t o - e l e c t r i c e f f e c t over t h e surface of the photo- tube. Tests made on a commercial type 917 photo-tube show that the departure from l i n e a r i t y i s about 3#» This could be reduced by p l a c i n g a mask i n the l i g h t path. By c a r e f u l shaping of the mask the e r r o r could be made very s m a l l . Po - marker pulse P1,P3«P5 - t r i g g e r pulses obtained from f u n c t i o n frames P2 - delayed pulse d e r i v e d from PI P4 - m u l t i a r pulse from t i m i n g c i r c u i t s P6 - sampling pulse from t i m i n g c i r c u i t s P7 - delayed pulse d e r i v e d from P6 P8 - m u l t i a r pulse from m u l t i p l i e r P9 - delayed pulse d e r i v e d from P8 P i g . 2» C a l i b r a t i o n and f u n c t i o n frames on scanning d i s k a However, the no n - l i n e a r c h a r a c t e r i s t i c of the photo- tube i s the one b i g disadvantage of the o p t i c a l system. The input i s e s s e n t i a l l y an open loop and must be c a l i b r a t e d f o r each photo-tube. I t should be p o s s i b l e t o connect the input i n t o a closed loop by u s i n g a servo to p o s i t i o n the mask au t o m a t i c a l l y , since the i n t e r n a l c i r c u i t r y i s arranged t o be s e l f - c a l i b r a t i n g and capable of p r e c i s i o n o p e r a t i o n . The output of the photo-tube i s a m p l i f i e d by a d.c. a m p l i f i e r . The zero l e v e l v oltage ( E m ) i s h e l d and f e d back to the input of the a m p l i f i e r by a system of gates and h o l d i n g c i r c u i t s . This s e t s the zero l e v e l and a u t o m a t i c a l l y c o r r e c t s f o r a m p l i f i e r d r i f t . I f the output of the photo-tube r i s e s above E m , the ouput of the a m p l i f i e r i s p o s i t i v e , and i f i t f a l l s below E m , the output i s negative. The b l o c k diagram f or the f u n c t i o n generator u n i t i s shown i n Figure"3 * The s i g n a l coming from the photo-tube a m p l i f i e r i s d i r e c t e d i n t o three channels 1, 2, 3, and the ti m i n g c i r c u i t channel D. Channel 3 i s the automatic bi a s c i r c u i t and operates as described below. When the c a l i b r a t i o n frame i s scanned, pulse P I occurs at the beginning of the frame. This pulse makes t r i o d e gate TG2 conducting and diode gate LG 15 non-conducting. When TG2 conducts,the feedback loop of the ho l d i n g c i r c u i t c l o s e s . The storage u n i t i s then able t o charge up to the voltage -E m. A second p u l s e , P2, makes TG 2 non-conducting, thus breaking the feed-back loop and l e a v i n g the storage u n i t charged t o - E T h e same pulse opens P i g . % F unction generator assembly 7. DG 15 and feeds - E m back t o the photo-tube a m p l i f i e r . The a m p l i f i e r now has an output of zero v o l t s f o r an input of E m v o l t s . The c a l i b r a t i o n occurs once i n every r e v o l u t i o n of the f u n c t i o n d i s k . This simple method solves the problem o f d r i f t and z e r o - l e v e l c a l i b r a t i o n . The purpose of channel 1 w i l l become more apparent when the m u l t i p l i e r i s discussed. I t s task i s to s t o r e the maximum p o s i t i v e value of the f u n c t i o n s and to generate a sweep which i s needed f o r the m u l t i p l i e r . As the c a l i b r a t i n g frame i s scanned by the photo-tube, the storage u n i t becomes charged t o -E m. The pulse P3 opens the second h a l f df the t r i o d e gate TG 1 a l l o w i n g the storage u n i t t o discharge to -E^ i n the case when i t s former charge of - E m was l a r g e r than -E^. The pulse P4 closes the gate l e a v i n g the storage u n i t charged to -E J J . The R ^ i combination produces t he sweep r e q u i r e d by the m u l t i p l i e r . The pulse P7 makes the gate DG 7 non-conduct- i n g , thus i n i t i a t i n g the sweep, and P8 re t u r n s i t to the ground.. The ouput of channel 1 then goes t o the m u l t i p l i e r c i r c u i t and i t s value i s -E^K^t). Channel 2 i s the f u n c t i o n c i r c u i t . Gates DG 8, 9, and 10 become conducting when the f u n c t i o n frame begins, t h a t i s a t P5« At the de s i r e d a b s c i s s a , determined by the ti m i n g c i r c u i t , P6 i s generated. This pulse makes the above mentioned gates non-conducting, thus l e a v i n g the condenser C 2 charged t o the i n v e r t e d ordinate voltage - E F . The pulse P7 makes DG 1 1 and 12 conducting, thus s t a r t i n g a sweep at B with value +E.N(t). This sweep i s i d e n t i c a l t o 8. the one In channel 1, as w i l l he shown l a t e r . Since DG 10, 9, a u i 8 aie s t i l l non-conducting the sweep a t C i s i n v e r t e d , i . e . -rE^BXt). The c i r c u i t diagram of the m u l t i p l i e r i s shown i n Figure 4. The m u l t i a r comparators (M) operate only i f the reference v o l t a g e , E, i s more p o s i t i v e than the sweep in p u t . The o r i g i n a l s t a t e of DG 16 i s non-conducting and DG 17 conducting. I f E i s negative the i n v e r t e d p o s i t i v e v o l t a g e , -E, i s a p p l i e d t o M4. Since the second input of M4 i s at ground p o t e n t i a l , H4 generates a pul s e . The pulse from M4 operates f l i p - f l o p FF8 which c l o s e s DG 16 and opens DG 17, thus applying the negative v o l t a g e , E, t o M3. The pulse from M4 also opens DG 14 and closes DG 13. These gates l e a d to the storage c i r c u i t s shown i n Figure 5° The sweep - E ^ t ) from channel 1 s t a r t s at P7 and i s a p p l i e d t o M3. At the i n s t a n t E - E ^ ( t ) a pulse P8 I s generated. This pulse opens DG 11 shown i n Figure 3» thus stopping the sweep -E ^ N ( t ) , l e a v i n g the storage c i r c u i t charged t c -E^H(t)o Since the sweep i s stopped at the i n s t a n t E = EjgNtt), and since the N(t) of channel 1 i s i d e n t i c a l w i t h the N(t) of EE-channel 2, i t i s obvious that the value s t o r e d i s - —S. , I f E i s p o s i t i v e , diode gate DG 13 i s open and EE- DG 14 i s cl o s e d . The voltage s t o r e d w i l l then be £ . EM From the above i t can be seen th a t four-quadrant m u l t i - p l i c a t i o n has been achieved. The pulse P9 r e s e t s the gates DG 12, 13, 14, 16, 17 as shown i n the diagram. E K2 -W P9 P(~)-N P(^)-C F F 8 P(~) P(+) A M 3 *8 M 4 Pigo 4« Pour Quadrant M u l t i p l i e r . 10. The storage system which i s shown i n Figure 5 c o n s i s t s of a bank of storage u n i t s operated by a counter. A pulse PO sets the f i r s t f l i p - f l o p to make the f i r s t storage channel open and sets a l l the other f l i p - f l o p s so that the remaining channels are c l o s e d . The pulse PO comes from a reference marker on the disko The f u n c t i o n generated from the f i r s t frame i s stored i n the f i r s t storage u n i t . The pulse P9 at the end of the m u l t i p l i c a t i o n s e t s the f i r s t storage channel i n the c l o s e d p o s i t i o n and opens the second. The o p e r a t i o n repeats u n t i l a l l the f u n c t i o n s have been processed and s t o r e d .  12,. I l l D e t a i l e d C i r c u i t Design. The c i r c u i t s are designed t o make use of standard components. The d.e. a m p l i f i e r s used are P h i l b r i c k model K2-X and K2-W o p e r a t i o n a l a m p l i f i e r s . These a m p l i f i e r s have a g a i n o f 30,000 and 15,000 r e s p e c t i v e l y . The K2-W output i s + 50 v o l t s at + 1 ma. and the K2-X maximum output i s + 100 v o l t s at i 2 ma. The a m p l i f i e r s are used w i t h 100# feed back i n p r a c t i c a l l y a l l cases. This provides maximum s t a b i l i t y . ( l ) Scanning Unit The scanning u n i t c o n s i s t s of two p a r t s j a r o t a t i n g frame holder or d i s k and an o p t i c a l system. "The d i s k c o n s i s t s o f a s i x t e e n - i n c h diameter, one-quarter i n c h t h i c k , s t e e l d i s k , with an outer r i m o f aluminum three inches wide. The heavy centre o f the d i s k a c t s as a f l y - w h e e l , smoothing out any v a r i a t i o n s i n the speed of the d r i v i n g motor. The aluminum r i m contains the c l i p s to ho l d eighteen f u n c t i o n frames. Figure 6 shows the c o n s t r u c t i o n . itn aluminum Figure 6. Rotating f u n c t i o n d i s k . 13. The d i s k i s d r i v e n by a one-tenth horsepower. 700 rpm, 110-volt d i r e c t - c u r r e n t shunt motor. The d r i v e i s transmitted through a rubber belt.which a l s o helps t o smooth out any v a r i a t i o n s i n the speed of the motor. The l i g h t source i s a 6-v o l t , d i r e c t c u r r e n t , s t r a i g h t , v e r t i c a l f i l a m e n t lamp, which gives an intense v e r t i c a l l y uniform l i g h t . The l i g h t i s * f o c u s e d by c y l i n d r i c a l l enses so that the f o c a l point of the narrow beam i s at the f i l m . The type 917 photo-tube i s s i t u a t e d on the opposite s i d e of the d i s k to the l i g h t source. Figure 7 shows the complete mechanical assembly. r o t a t i n g d i s k and f i l m holder o Figure 7. Scanning U n i t Assembly. The output of the photo-tube v a r i e s w i t h the amount of l i g h t i t r e c e i v e s and th e r e f o r e represents the ordinate of 14 o the f u n c t i o n s passing through the beam of l i g h t . That i s the output of the tube i s amplitude modulated by the l i g h t beam0 (2) Photo-tube A m p l i f i e r . The a m p l i f i e r c o n s i s t s of the two P h i l b r i c k K2-X d.c. a m p l i f i e r s i n s e r i e s . These have two i n p u t s . The p o s i t i v e i n p u t , . 1 , i s used when i n v e r s i o n i s r e q u i r e d and the negative i n p u t . .2, when i n v e r s i o n i s not r e q u i r e d . In the present a p p l i c a t i o n , i n v e r s i o n i s normally req u i r e d because negative feed-back i s used. The p o s i t i v e input i s then used f o r balance c o n t r o l . The c i r c u i t diagram of the photo-tube a m p l i f i e r appears i n Figure 8. R l , R2 0 R5, R6 - 1 Megohm R 3 , R7 - 470 Kilohms B#, R8 - 47 Kilohms R2 ,—'WWvV- R l M W A M - ~ K2-X R5- Figure 8. Photo-tube A m p l i f i e r . 15. The ouput of the photo-tube v a r i e s from zero t o approximately 1 v o l t as the f u n c t i o n v a r i e s between i t s negative and p o s i t i v e extremes. The output of the a m p l i f i e r should represent the f u n c t i o n , therefore a feedback system was designed t o give zero output f o r zero input l e v e l , Em<> The output of each a m p l i f i e r i s f e d back t o the input to give a g a i n of 10. The two i n s e r i e s then produce a ga i n of 100. The gain of 100 was chosen so t h a t the maximum photo-tube s i g n a l , approximately 1 v o l t , would not overload the a m p l i f i e r s which have a maximum output range of -50 t o + 50 v o l t s . (3) Holding System of Channels 1 and 3. Channels 1 and 3 operate p r i n c i p a l l y as v o l t a g e - clamping c i r c u i t s . The block diagram of the h o l d i n g system i s shown i n Figure 9« Figure 9« Holding system of Channel 1 and 3» 16. When a voltage e^ i s a p p l i e d as shown, and the t r i o d e gate i s made conducting, the condenser C charges up t o t h i s v o l t a g e . The a m p l i f i e r s are connected so as to provide i n v e r s i o n at the output of the hold i n g a m p l i f i e r . That i s , to obtain negative feed-back between the input of K2=-W and the output of the h o l d i n g a m p l i f i e r , the p o s i t i v e (+) input of K2 - W and the negative (<=>) input o f the h o l d i n g a m p l i f i e r must be used. I f the gate i s made non-conducting the feed-back loop i s broken and the condenser C i s l e f t charged t o =e^0 Any v a r i a t i o n i n the input no lo n g e r a f f e c t s the output. I f the charge begins to le a k from the condenser, the change i n p o t e n t i a l i s f e d back to the input of the h o l d i n g a m p l i f i e r . This tends to make the ouput r e t u r n to i t s o r i g i n a l s t a t e . The d e t a i l e d c i r c u i t diagram o f the h o l d i n g system appears i n Figure 10. (4) Holding system of Channel 2. The h o l d i n g c i r c u i t used i n channel 2 must be one which f o l l o w s the v a r y i n g f u n c t i o n very r a p i d l y . I t need not hold the f u n c t i o n f o r as long as the h o l d i n g c i r c u i t s i n channels 1 and 3. E s s e n t i a l l y , a condesner i s charged through a d.e. a m p l i f i e r . However, the condenser must be sma l l i n order to charge r a p i d l y . At f i r s t a P h i l b r i c k K2-W a m p l i f i e r was used to charge the condenser, but t h i s proved to be too slow because of i t s low current output which i s l i m i t e d t o one ma. R 2 ! V +300 Main Chassis Bus R i R, B O—V W v — • — W W loOM 0.1# 10Z P o t , - M a l l o r y loO -~10# 220K 10$ 2.2M 10$ 470K 10$ 68K> 10$ 120K 10# 1 watt 2.7M 10# R ^ s 4.7M 10$ R ^ s 1500X110^ l13' 33K 15K 10% 10$ A l l r e s i s t o r s are watt unless otherwise noted. K2-V -̂ )—I S *- REPRESENTATION 0 0.5 MFD Polystyrene (Low Leakage) 5 50 MMFD 600 VDC _ ^, _ , , 0 „ m For b i d i r e c t i o n a l c o n t r o l ( p o s i t i v e 1 10 MMPD 600 TO and negative v o l t a g e storage) both 1 470 MMPD 600 VDC c o n t r o l g r i d s of TG are connected i n s e r i e s . Figure 10g VOLTAGE STORAGE AND HOLDING CIRCUIT. 18. Suppose the value of the storage condenser I s .001 microfarads. I f the condenser i s charged by the Z2-W a m p l i f i e r , the time of charging i s as follows s 0 = q/v = A~£_± v Therefore t = C v / i I f V = 50 v o l t s and i = 1 ma. —"5 t = .001 x 50 x 10 - 50 micro-seconds. Since t h i s time i s too long the c i r c u i t i n Figure 11 was developed. I n t h i s c i r c u i t a cathode f o l l o w e r i s used i n s i d e the feed-back loop. Since the 12AU7 tube i s capable of c a r r y i n g 10 ma. average current and supplying up t o 50 ma. current pulses, i t was found to be s a t i s f a c t o r y as a cathode f o l l o w e r . The b i a s t o the K2-W a m p l i f i e r . i s adjusted so that zero input g i v e s zero output of the cathode f o l l o w e r . Figure 11. Holding c i r c u i t of Channel 2. 19< Suppose a negative step of 50 v o l t s i s a p p l i e d at (a ) . The cathode f o l l o w e r i s then made t o conduct very s t r o n g l y , say about 50 ma. Then t = £S = .001 x 10°^ = 1 micro-second, i I f a p o s i t i v e step i s , a p p l i e d t o ( a ) ? the cathode f o l l o w e r i s cut o f f and the-'condenser i s charged from the -300 v o l t source through the 30Z r e s i s t o r , . The time constant, Ô*,,, o f ^ e c i r c u i " ^ i s 30 microseconds. Let the voltage across C 2 be e 2 and the -300 v o l t supply voltage be e^„ The charging time of the condenser may be found as f o l l o w s s - 6 2 ( p ) ^ R 2 e^TpT 1 + pR 20- 2 e 2 = 50 v o l t s maximum e 1 = - 3 0 0 v o l t s e 2 ( t ) = e ^ t ) (1 - e" > t/ R2 C2) e l t = R 2C 2 I n 5/6 Therefore t = 5.088 x 10 ' = 5 micro-seconds <> Since the f u n c t i o n i n v o l v e d w i l l probably never be a step type, the charging time w i l l be l e s s than 5 micro- seconds and may be considered t o be n e g l i g i b l e . 20. The diode gates i n the c i r c u i t are used to i s o l a t e the condenser and leave i t charged to t he vo l t a g e representing the ordinate at the i n s t a n t of sampling. The purpose of the p a r a l l e l feed-hack i s t o allo w the a m p l i f i e r t o he used as the i n v e r t e r i n Channel 2 when charging has been completed,, ( 6 ) I n v e r t e r s , The i n v e r t e r s c o n s i s t of a K2-W a m p l i f i e r w i t h 100$ feed-back g i v i n g a g a i n of one. The i n v e r t e r i n Channel 1 i s shown i n Figure 12. I R2 b i a s Figure 12. I n v e r t e r of Channel 1, As mentioned e a r l i e r , the i n v e r t e r i n Channel 2 employs the same a m p l i f i e r that i s used to charge the h o l d i n g condenser. This can be done because the h o l d i n g a m p l i f i e r i s i n o p e r a t i v e during the time t h a t the i n v e r t e r i s needed,, The c i r c u i t diagram f o r the Channel 2 i n v e r t e r i s shown i n Figure 13. The gates are non-conducting and therefore the K2-W a m p l i f i e r and the cathode f o l l o w e r a ct as an i n v e r t e r . 21. - W V v W RloR2-200K R3 - 400K input output Figure 13. I n v e r t e r of Channel 2. (7) Gateso Two types of gates were i n v e s t i g a t e d ; the t r i o d e p gate and the six - d i o d e bridge-type gate D The gates must be b i - d i r e c t i o n a l and i n most cases be capable of passing voltages w i t h great accuracy. That i s 9 the g a i n through the gate should be as close as p o s s i b l e to u n i t y . A double-triode can be used as a b i - d i r e c t i o n a l gate, but i t i s extremely d i f f i c u l t , i f not i m p o s s i b l e 9 t o o b t a i n a c o n s i s t e n t l y accurate gain l e v e l through the gate. The only circumstances under which t r i o d e gates can be used are i n s i d e a feed-back loop, where accuracy i s not necessary. These were used i n the h o l d i n g systems of Channels 1 and 2 as mentioned p r e v i o u s l y . I n most cases diode gates are a n e c e s s i t y because a voltage must be passed a c c u r a t e l y . The c i r c u i t diagram of R2 -wvvw- £ 6AL5 •I L £ 6AL5 * 6AL5 R4 i n R l , R2, R3 - 1M R4 - IK DG F i g . 14. Diode gate -§ out i 6AL5 £ 6AL5 B - 300 v o l t s C - 60 v o l t s K 60 v o l t s 23. the diode gate appears i n Figure 14• The gate i s designed to pass i 50 v o l t s maximum? s i n c e t h i s i s the highest voltage passed by the d.c. a m p l i f i e r s . The c o n t r o l voltages are ± 60 v o l t s . The tubes used are three 6AL5's. The accuracy o f the gate was t e s t e d as shown i n Figure 15. The r e s u l t s are given i n Table 1. conducting Figure 15. Test Setup f o r the Diode Gates. 24. Table 1. E i n (volts) Ein-Eo (volts) % r e l a t i v e error .3.00 .01774 .59# 5.00 .03898 .78# 10.0 .05255 .53# 20.0 .07909 .39# 30.0 .10720 ,35# 40.0 .13425 .33# 50.0 .16320 .33# 60.0 .19157 .32?S -60i>0 .1608 .27# -50.0 .12615 .259& -40.0 .09119 .23# -30.0 .05690 .19# -20*0 .02824 .14# -10.0 .00079 .008# The greatest r e l a t i v e error occurs at an input of 5 v o l t s . The gain at t h i s point i s s G = fa = 5 . 0 0 - .03898 = , 9 9 2 Bin 5.00 (8) Sweep C i r c u i t s . The sweep c i r c u i t s of Channel 1 and 2 are d i f f e r e n t , but the sweeps are i d e n t i c a l . The sweep c i r c u i t of Channel 1 i s an RG network with a diode gate from the output to ground. Figure 16 shows the c i r c u i t . The holding c i r c u i t prevents the input to the sweep from changing as the sweep progresses. The equation of the sweep i s e Q ( t ) = E ^ l - e" t / / Rl Cl)„ 25 R l BM • rtwvw* From ho l d i n g c i r c u i t R l - 1M CI - .001 microfarads Figure 16. Sweep c i r c u i t of Channel 1* The time constant, R^C^, i s chosen as one m i l l i - s e c o n d because of the speed of scanning. The frames are eight inches from the centre of the d i s k so th a t at 700 rpm the l i n e a r speed of the frame i s approximately 560 inches per second. The frame i s about l£ inches wide, th e r e f o r e the t i m e r e q u i r e d f o r one f u n c t i o n to be scanned i s 2.5 m i l l i - s e c o n d s . The time between frames i s 1.5 m i l l i - s e c o n d s . I f the f u n c t i o n i s sampled near t h e end of the frame, there i s 1.5 m i l l i - seconds f o r the sweep and comparators t o a c t before the next f u n c t i o n begins. Therefore,the sweep time-constant was chosen to be one m i l l i - s e c o n d . The gate s t a r t s the sweep when i t i s made non-conducting and stops the sweep when i t i s made conducting. The sweep c i r c u i t of Channel 2 cannot be a simple RC network. The reason i s th a t ordinate voltage i s h e l d on a r e l a t i v e l y small condenser. When the sweep s t a r t s , the charge 26. on the condenser i s drained o f f , thereby reducing E^. Hence some means of c o r r e c t i n g f o r the voltage drop must be found. The s o l u t i o n i s t o use a d i r e c t - c u r r e n t a m p l i f i e r w i t h c a p a c i t a t i v e feed-back. The c i r u i t appears i n Figure 17. e. =b C2 P5-C P6-N P5-C P8-N Figure 17. Sweep C i r c u i t of Channel 2. An a n a l y s i s o f the c i r c u i t w i t h gate DG 11 conducting and DG 10 and DG 12 non-conducting, gives the f o l l o w i n g s Since E i s e s s e n t i a l l y zero because of the l a r g e g a i n 27< The c a p a c i t o r discharges through Ego Hence, i f B i s the o r i g i n a l v oltage on C^, e K t ) - B e " t / E 2 ° 2 and e-]_(p) E P + 1 *2°Z Therefore .2(p) = - = - ̂ _ ( ) '22 Taking the inverse Laplace Transforms 0 Ct) = «E ( ! _ e-VVfe) °3 I f 0, = C, 2 3, e 2 ( t ) = - E ( l - e R2°^2 ) which i s the same as the sweep used i n Channel 1 except f o r the si g n . was determined by the requirements of the ho l d i n g c i r c u i t and was set at .001 microfarad. I n a preceding paragraph i t was found t h a t the sweep time- constant was one m i l l i - s e c o n d . This r e q u i r e s t h a t R2 must be 1 megohm i n both sweeps. (8) Comparator. Comparators of the m u l t i a r type were chosen because of t h e i r inherent s i m p l i c i t y and accuracy. The c i r c u i t diagram appears i n Figure 18. The negative reference voltage on the p l a t e s -of VI keeps the diode from conducting. The tube V2 i s normally conducting s t r o n g l y and V3 i s c l o s e t o c u t o f f . When the 176 AW2F or 134 BW2F 381 R l - 100Z 4 R2 - 4o7M I R3.R8 - 53K 1 R4,R9 - 56Z 1 W v R5 - 100K R6 - 33K R7 - 1M C1,G3 -'.001 >ufd 02,04 - . l>ufd 05 -.01y*fd F ig» 18. C i r c u i t diagram of m u l t i a r comparator 29* f a l l i n g sweep reaches e q u a l i t y w i t h the reference v o l t a g e 9 VI conducts, thus completing the r e g e r a t i v e loop. The regeneration d r i v e s the g r i d of V2 to c u t - o f f very r a p i d l y and as a r e s u l t we get a l a r g e p o s i t i v e pulse at the p l a t e of V2. The tube V3 i s used to i n v e r t the puise^although a pulse transformer could a l s o be used f o r pulse i n v e r s i o n . The charge on CI discharges through R2 and makes V2 conducting again. This again closes the r e g e n e r a t i v e loop and causes another pulse. A t r a i n of pulses occurs u n t i l the sweep r e t u r n s to a l e s s negative value than the reference v o l t a g e . I n a c t u a l p r a c t i c e the pulse i s a p p l i e d to a f l i p - f l o p which turns on the gate DG- 7, thus stopping the sweep and r e t u r n i n g i t to ground. Therefore only the i n i t i a l pulse i s generated since VI cuts o f f as soon as DG 7 conducts. The r i s e time of the pulse i s of the order of 1 t o 2 microseconds and occurs again as soon as the sweep voltage i s equal t o the reference voltage. The pulse height i s 230 v o l t s . 30. 17 Accuracy Test of Multiplier. The circuit as a whole was not tested because no precision components or power supplies were available. However, a test procedure i s desirable for future testing. The test set-up i s shown i n Figure 20. Aa was shown earlier the equation of the sweep i n channel 1 i s e ^ t ) = B (1 - e - t / Rl°l) The equation of the sweep i n channel 2 i s e 2(t) - • U < i - e-*/*2*2) ! 3 If the sweeps are to be equal, EgGg must equal R 1C 1 and Og equal Cj • RgCg was matched to R^Pl a s i s s n o w n i n Figure 19 < 01 Rl r—-WVVVV-Cvv3lvvuL /V) iooo cps R2 oscilloscope bias Figure 19. Circuit used to match Time Constants.  3 2 . was adjusted u n t i l the o s c i l l o s c o p e showed no 1000 cy c l e output. Therefore = R 2 C2° T h e K 2 ™ ^ a m p l i f i e r s introduce l i t t l e e r r o r since t h e i r gains are so l a r g e . I t would he extremely d i f f i c u l t t o match C 2 t o C^. Therefore a method to overcome t h i s was devised. The input to the R 2 C 2 network must be p o s i t i v e i n order to produce a negative sweep. Therefore, an i n v e r t e r i s introduced as shown i n Figure 20. The input to the R 2 C 2 r 0 sweep i s then E — . Hence the output of the sweep i s r l ~ E ^ ^ ( l - e - t / R 2 ° 2 ) . r l ° 3 I f r 2 C 2 = r ^ , the sweep w i l l be - E ( l - e~X'*2GZ) which i s the same as the R^C-L sweep. The r 2 C 2 , r ^ C ^ time constants were matched i n the same way as i s shown i n Figure 19. The m u l t i p l i c a t i o n m u l t i a r i s KL, and M2 i s the peak d e t e c t i n g m u l t i a r . The pulse generator produces a pulse P I which operates the gates as shown i n Figure 20, thereby i n i t i a t i n g both sweeps. The negative reference voltage of KL i s s e t at the d e s i r e d l e v e l of m u l t i p l i c a t i o n , E^. When the R - ^ l s w e e P reaches e q u a l i t y w i t h E.̂ , KL produces a pulse which stops the sweeps. Some way of measuring the value of the R 2 C 2 sweep had to be found, since no storage c i r c u i t was a v a i l a b l e . A second m u l t i a r , M2, i s introduced.and i t s reference voltage 33. i s r a i s e d from a l a r g e negative value u n t i l i t generates a pulse which i s seen on the scope. At t h i s point E^ i s equal to the peak of the sweep. The voltages Eg and E^ are then compared on a simple bridge as shown i n Figure 2 0 and the voltage d i f f e r e n c e i s measured. I f the m u l t i p l i e r i s ab s o l u t e l y accurate E 1 = Eg. The r e s u l t s of a t y p i c a l set of readings are shown below. Table 2 . E l *2~*1 E 2 ° E 1 (Approaching from (Approaching from below the peak) above the peak) 5 5 . 0 6 5 . 0 7 1 0 1 0 . 2 5 1 0 o 2 6 1 5 1 5 . 2 5 1 5 . 2 9 2 0 2 0 . 3 5 2 0 . 4 0 2 5 2 5 . 4 7 2 5 . 5 3 The above t e s t i s designed to measure the accuracy of the m u l t i p l i e r . The way i n which the m u t l i p l i e r has been set up i n t h i s case introduces more e r r o r than would the a c t u a l m u l t i p l i e r . In ihe example given above the f l i p - f l o p i s loaded by three gates i n s t e a d of one and, i n additions, the RgCg sweep i s stopped by making DG 1 2 conducting instead of stopping i t by making DG 1 1 non-conducting. The diode gates have a delay of 1 0 micro-seconds when made conducting, but only 1 to 2 micro-seconds when made non-conducting. Therefore . the readings above do not i n d i c a t e t he accuracy which i s obtainable, but they do show that the t e s t i n g method i s s a t i s f a c t o r y . 34 7 Conclusion. The prototype scanning system t h a t was b u i l t has i n d i c a t e d that the i d e a i s p r a c t i c a b l e . Further refinements w i l l be made as work progresses on the analog computer. The m u l t i p l i e r and f u n c t i o n generator could not be f u l l y t e s t e d because the tuning c i r c u i t r y was not completed, and p r e c i s i o n r e s i s t o r s and cmdensers were not a v a i l a b l e f o r the sweep and i n v e r s i o n c i r c u i t s . Another f a c t o r was the u n a v a i l a b i l i t y of w e l l regulated power s u p p l i e s . The.tests on the m u l t i p l i e r d i d i n d i c a t e that good accuracy can be obtained. (5) + 300 IN/ 5 »1 • .0 150K 10$ t E2 e 9 150K 10# 1 E 5 0 O 470K 10$ * R4 0 o 1.0M 10$ B 5 • 0 180K 10$ * E 6 s 68K- 10$ * *7 I 2.2K 10$ * E 8 ft o 82K 10$ * 6 watt watt watt watt watt watt HEATERS A 6.3V (3)- 300 R9 ^ 0 R R 11 12° i 220K : 10.0M . 1.5 M 4.7M 10$ 1096 10$ £ watt £ watt £ watt i watt I N > + K2-X OUT SYMBOL O5 8 A P P E N D I X : ^ 15 MMPD 0.005 MPD 7.5 MMPD MODEL K2-X OPERATIONAL AMPLIFIER George A. P h i l b r i c k Researches, Inc. (GAP/R) \J1 2 IN 7 H 8 H 6.3V :E2 /\—i\ 01 H K 12AX7 *R6 ±02 R8 R9 1 3 4 6 + -300 GND OUT IN VDC (REP) R l - .22M 5% R2 .22M 5# R3 - 1M RIO R4 - 2.2M R5 - ..47M 5% NE2 R6 - 10K NE2 R7 _ .22M 5% R8 - .22M 5# R9 - •27M 5% R l l RIO R l l ,68M 4.7M 01 - 7.5 C2 - 500 C3 — 7.5 5 +300 VDC Philbrick Model K2-W Operational Amplifier R - 10K R R l - 1M +300 VDC OUT R e s i s t i v e method of b i a s GAIN: 15,000 DC, open-loop POWER REQUIREMENTS: 4.5 ma at +300 VDC 4.5 ma at -300 VDC 0.6 Amperes at 6.3V TUBE COMPLEMENT 2 12AX7 Base Connections General S p e c i f i c a t i o n s INPUT IMPEDANCE: Above 100 Megohms OUTPUT IMPEDANCE: Less than 1 K open-loop below 1 ohm f u l l y f e d back DRIFT RATE: 5 mv per day, r e f e r r e d to the input VOLTAGE RANGE: -50 VDC to +50 VDC,at ouput & both inputs INPUT CURRENT: Less than 0.1 Microamp f o r e i t h e r input OUTPUT CURRENT: -1 ma t o +1 ma over f u l l voltage range. Appendix 37. B i b l i o g r a p h y Fiorentino,• J.S., -"Timing and Coincidence C i r c u i t r y f o r a Time-Sharing Analog Function Generator" - M.A.Sc Thesis. U n i v e r s i t y of B r i t i s h Columbia, 1956*. Millman and Puckett - "Accurate L i n e a r B i - d i r e c t i o n a l Diode Gates" - Proceedings of the I.R.E., January 1955, pp. 29-37. Freeman and Parsons 7 "A Time-Sharing Analog M u l t i p l i e r " Transactions of the I.R.E., P r o f e s s i o n a l Group on E l e c t r o n i c Computers, March 1954. Korn and Korn - " E l e c t r o n i c Analog Computers", McGraw H i l l , 1952. Williams and Moody - "Ranging C i r c u i t s , L i n e a r Time- Base Generators:and Associated C i r c u i t s " - I n s t i t u t e of E l e c t r i c a l Engineers J o u r n a l . Volume I I I A , Radio L o c a t i o n , 1946, pp.1188-1198.

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