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A polynomial computer Ruegg, Frank Arthur 1957

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A POLYNOMIAL COMPUTER by FRANK ARTHUR RUEGG BoAoSco, University of B r i t i s h Columbia, 1955 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 this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA. May, 1957o ABSTRACT A computer for solving polynomial equations, performing Fourier synthesis, and displaying system functions is a valuable asset to an electrical laboratory•> A number of successful computers have been built for such purposes but each has its l imitat ions„ The design pf a versatile and precise instrument superior to existing computers is the purpose of this study., Using a voltage analogue of the function, this computer is designed for the solution of 20th - degree polynomials with real coefficients, 10th - degree polynomials with complex coefficients, and Fourier synthesis of either even or odd functions to the 20th harmonic or mixed functions to the 10th harmonic. Provision is made for the addition of circuits which will plot the magnitudes and angles of the polynomial or will form combinations of two lOth-degree polynomials0 The computer, using servo phase-shifters harmonically geared in pairs and excited with a two-phase, 400-cps carrier, generates a polynomial term by term in exponential formQ The coefficients and radial components of the terms are obtained by the use of precision potentiometers» The terms are summed by operational amplifiers which, mounted as a separate unit, can be removed from the computer for other analogue computations0 The radial and angular components of the independent variable, j>, of the polynomial may be constant or linear i i functions of time. The display consists of a long-persist-ence cathode-ray-tube on which the zeros of a polynomial are automatically plotted. For greater accuracy, zero positions may be determined manually by using a null indicator. Fourier synthesis can be performed with minor changes in the output circuit. The periodic function may be dis-played continuously or point by point. The major part of the computer, the phase-shifter unit, has been built and most of the other components partially assembled. Completion and testing of the computer will be part of a further project. i i i In presenting t h i s thesis i n p a r t i a l fulfilment of the requirements fo r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available f o r reference and study. I further agree that permission for extensive copying of t h i s t h e s i s for scholarly purposes may be granted by the Head of my Department or by his representative. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed, without my written permission. Department of The University of B r i t i s h Columbia, Vancouver 5\ Canada. ACKNOWLEDGEMENT The author is indebted to members of the Department of Electrical Engineering at the University of British Columbia, especially to Dr. A. D. Moore for his untiring guidance and to Dr. F. Noakes for his helpful suggestions and encouragement. This project is supported by the Defence Research Board under DRB Grant 5503-04. The author's post-graduate studies were made possible through a National Research Council Bursary which he was awarded in 1955. vi TABLE OF CONTENTS Page L i s t of I l l u s t r a t i o n s v Acknowledgement v i CHAPTER 1.0 Introduction 1 2.0 Choice of System 13 3.0 Function Generator 20 3.1 Basic System 20 3.2 Extended System 25 4.0 Detection and Output Units 31 4.1 Detection Unit 3l'. 4.2 Output Unit 32: 5.0 Components, Layout and Future Extensions 40 6.0 Preliminary Calibration 44 7.0 Manipulations and Transformations of the Polynomial 47 8.0 Operation 50 9.0 Conclusions . •. 53 Bibliography 55, i v LIST OF ILLUSTRATIONS Figure 1 ! Page 1. Direct-Coupled Phase-Shifters with Variable Excitation (System A). To follow 14 2. Seared .Phase-Shifters with Variable Excitation (System B) To follow 15 3. (reared Phase-Shifters with Constant Excitation (System C) To follow 15 4. Function Generator TO foXXOW o * o o o o « o o o o o « o 9 o o o o o « « e e o o o 20 5. Detection and Output Units TO f 0 XXOW o « o . ' o o o o o o o o o * « o o e o « « o o « o o o b 3X 6. C i r c u i t f o r Fourier Synthesis TO f 0 1XOW • t t o o o f t o f t O 0 a o o * 0 9 O o « « » « * « » o « 5X V A POLYNOMIAL COMPUTER 1 . 0 INTRODUCTION Polynomials a r i s e i n many s c i e n t i f i c studies. Often the zeros of these algebraic functions, i . e . , the roots o€ the r e s u l t i n g equations when such functions are equated to zero, must be found. A t y p i c a l occurrence i s i n the analysis of l i n e a r e l e c t r i c a l networks where polynomials of various degree must be factored. E x p l i c i t solutions for the z e r o s of a polynomial with degree up to four have been devised. However, f o r polyno-mials of higher degree, each must be treated as a special case and laborious numerical methods used. Solution by automatic analogue or d i g i t a l computation i s desirable when a high-degree polynomial must be factored. Where extreme accuracy i s not essenti a l , analogue s o l u t i o n i s preferable due to i t s s i m p l i c i t y and directness« Hence, i f f a c t o r i n g polynomials of high degree i s a frequent problem, a root-fi n d i n g analogue computer w i l l be a valuable asset* The object of the present investigation has heen the preliminary development of such an instrument. A polynomial, W(p), where the complex number, j>, i s the independent variable, may be written as k=n W(p) = 21 a. p k = U -f jV . . . ( 1 , 1 ) k=o * fc where n = degree of the polynomial, a^ = c o e f f i c i e n t of the kth power of p_, 2 and U and V are r e a l functions of j>. £ can be expanded i n the forms, P = x + jy = r e J f t = r (cos a + j s in a) , and, therefore, p k = (x + j y ) k = r k e j k a * r k ( c o s k a + j s i n ka). A l l present polynomial computers, with the exception of those using the potent ia l analogy, simulate a polynomial using one of these equivalent representations- f o r p_s Form I (cartesian form): W(P) = 21 a k ( x + i y ) k ' • ' •<1'2>; k=o Analogues of x and y are generated separately, combined to give the complex form, then squared, cubed, e t c . , to give the required terms. These terms are then weighted according to the coe f f i c i ent s and summed. Form II (exponential form): n W(p) = XI a, r k « J k a . . . . ( 1 . 3 ) k=o K Analogues of r e ^ a , r 2 * ^ 2 " , ... xnc^na are formed separately with weighting according to the coe f f i c i ent s and then summed. Form III (harmonic form): n n *(p) • 21 a i t 3 f k c o s k a + 3 ZT a k r k s i n k a - • • • ( 1.4) k=o k=o Analogues of cos k a and s i n k a are formed independently of one another for each k, or a l t e r n a t i v e l y , d i f f e r e n t i a t i o n or in tegrat ion i s used to form one member of the p a i r from the other. The two weighted sums are then computed sep-a r a t e l y . 3 Because the present study of polynomial computers i s limited to e l e c t r i c a l or electro-mechanical devices, i . e . , instruments using a voltage analogue, a subdivision of types of simulators can be mades (a) the voltage output i t s e l f represents the function (low-frequency system), or (b) the envelope of a modulated signal represents the function (carrier-frequency system). In the l a t t e r case a form of demodulation for the output signal i s required. Prom the defining forms II and I I I , i t i s apparent that there i s an intimate r e l a t i o n between polynomials and Fourier series (a f i n i t e polynomial with the modulus of the inde-pendent variable constant i s a c t u a l l y a truncated complex Fourier.Series), and, as we s h a l l show l a t e r , any instrument which simulates a polynomial i n either the exponential or harmonic form can also perform Fourier synthesis. As early as the middle of the 18th century mechanical linkages f o r finding graphically the r e a l zeros of a poly-nomial with r e a l c o e f f i c i e n t s were suggested by Rowning*. However, at that time the proposal was merely a mathematical c u r i o s i t y . J . Rowning, "Directions for Making a Machine for Finding the Roots of Equations Universally with the Manner of Using I t , " Royal Society of London Transactions„ v o l . 60 (1770), pp. 240-256. 4 2 In 1872, Kelvin developed a p r a c t i c a l linkage to predict tides by Fourier synthesis. This linkage was the forerunner of many of the l a t e r polynomial computers generating the poly-3 nomial i n harmonic form. A few years l a t e r Lucas developed and used ingenious electrolytic-tank methods to solve for both r e a l and complex zeros of a polynomial with re a l c o e f f i c i e n t s . Several accurate mechanical and e l e c t r i c a l synthesizers 4 and root finders have been b u i l t since 1900. Frame has written an excellent h i s t o r i c a l survey of polynomial solvers up to 1945. In his paper he divides the various methods into six typess 1. Graphic and v i s u a l means, 2. Kinematic linkages, 3. Dynamic Balances, 4. Hydrostatic Balances, 5. E l e c t r i c and electromagnetic methods, 6. Methods of harmonic analysis. Most of these types, especially the mechanical and f l u i d forms, were clumsy and slow to set up and are not of p r a c t i c a l i n t e r e s t here. The s i x t h method l i s t e d i s important because the modern e l e c t r i c a l harmonic systems are analogue's of this mechanical o Lord Kelvin, "The Tide Gauge, T i d a l Harmonic Analyser, and Tide Predicter," Mathematical and Physical Papers, v o l . VI, Cambridge University Press, Cambridge, 1911. 3 * * ' F. Lucas, "Resolution Immediate des Equations au Moyen de L ' ^ l e c t r i c i t e , " C.R.Acad.Sci., Paris, vol.106 (1888), pp.645-648. ^J.S.Frame, "Machines f o r Solving Algebraic Equations," Mathematical Tables and Other Aids to Computation, v o l . 9 (Jan., 1945), pp. 337-353. 5 method. The most important and successful of the purely mechan-i c a l systems f a l l i n g i n t h i s category i s B e l l Laboratories' 5 "Isograph" (the name i s now applied to a number of root-fi n d e r s ) . This machine uses high-precision gears, cams, levers and pulleys to produce sinusoidal mechanical d i s -placements up to the 10th harmonic. A stylus plots the output displacement representing W(p) from which the zeros are e a s i l y determined. Other similar linkage designs followed the B e l l Laboratories' machine. However, such mechanical computers are large and slow i n operation. Since 1945 a number of diverse electronic or electro-mechanical methods have been used with good r e s u l t s . With the a v a i l a b i l i t y of servo-mechanism components such as synchros and resolvers having very accurate sinusoidal f i e l d d i s t r i b u t i o n s , formation of harmonically related voltages becomes p r a c t i c a l , and several modern systems use these com-ponents. A few of the more interesting of the modern instruments can be grouped as follows8 R.L.Dietzold, "The Isograph, a Mechanical Root Finder," B e l l Laboratories Record, vol.16 No. 4 (Dec, 1937), pp. 130-134. So L.Brown and L.L.Wheeler, "Mechanical Method f o r Graphical Solution of Polynomials," Journal of the Franklin Institute, v o l . 231 (Mar., 1941), pp.j 223-243. Simulation in form Is Bubb , g Simulation in form l i s Hart and Travis , Parker and Simulation in form III g Williams , 9 without carriers Marshall^, Lofgren"^, Simulation in form III, 12 with carriers Porte, Rose, and Willoughby , : Choudhury^j, In order to show the differences, each of these instru-ments w i l l be described briefly. Although Bubb's proposed method has not, to our knowledge, been applied to a practical computer, i t is of interest because i t is the most promising of those computers operating in form I. 7 Fo W„ Bubb, "A Circuit for Generating Polynomials and Getting their Roots," Proc. I.R.E.. vol.39 (Dec, 1951), pp. 1556-1561. g H. C. Hart and I* Travis, "Mechanical Solution of Algebraic Equations," Journa1 of the Franklin Institute, vol.225(1935), pp.63-72. g Go M. Parker and R. W. Williams, "A Magslip Isograph," Journal  of Scientific Instruments, Vcl.32 No.9 (Sept 0, 1955), pp.332-335. "^0. B. Marshall, "The Electronic Isograph for Roots of Poly-nomials," Journal of Applied Physics, vol.21 (April, 1950) pp.307-312. • ^ L i . Lofgren, "Analog Computer for Roots of Algebraic Equations,!' Proc. I.R.E.. vol.41 No.7 (July, 1953), pp.907-911. 12 Wo Go Forte, G. A. Rose, and E. 0. Willoughby, "Analogue Compu-ter to Solve Polynomial Equations of nth Degree with Real Coeffi-cients," Automatic and Manual Control, ed. Tustin, Butterworth Scientific Publications, London, 1952, pp.541-546. 13 • A. K. Choudhury, "The Isograph - an Electronic Root Finder," The Indian Journal of Physics, vol.29, No.10 (Oct., 1955)o 7 x and y_ are represented by variable gains of a pair of amp-l i f iers . Such amplifier pairs forming (— x + jy) are cas-caded to give the required power of (— x + jy). The x and y potentiometers of each pair of amplifiers are ganged and by varying each through its limit the f irst quadrant (with unit boundary) in the p_-plane is covered. Switching to -x gives the second quadrant. . Provision is made for the addition of complex coefficients on a l l terms. Although this system is theoretically sound, the large number of amplifiers required makes i t impractical for high-degree polynomial solution. Hart and Travis have produced one of the f irst practical and moderately accurate machines to solve polynomials by sim-ulation in form II. Their instrument solves polynomials up to the eighth degree. Modified fractional-horsepower motors serve as phase-shifters. . The rotors are dc-excited and are mounted on a common shaft rotating at constant angular velocity. The stators are geared to turn through angles a, 2a . . . 8a. In this system the carrier is produced through rotor rotation. Coefficient potentiometers weight the outputs and r variation is ob-tained with specially-wound, cam-operated potentiometers 2 3 8 that multiply directly by r, r , r . . . r . Zero indica-tion is provided by a meter on the output of an adding network. This computer is slow in operation but the principle can be greatly elaborated and developed to high accuracy. Parker has s u c c e s s f u l l y s o l v e d sixth-degree polynomials by s i m u l a t i o n i n form I I . Three-phase e l e c t r o m e c h a n i c a l p h a s e - s h i f t e r s geared to t u r n through angles -3a, -2a, -a, 0, o 6 a, 2a, 3a, f o r the terms p to p r e s p e c t i v e l y , generate the exponential components. The c o e f f i c i e n t s are set by po-tentiometers on the p h a s e - s h i f t e r outputs. Each term i s m u l t i p l i e d by the appropiate power of the modulus by ganged, cascaded potentiometers with b u f f e r cathode-follower stages i between u n i t s . The terms are added and f i l t e r e d and the r e s u l t a n t sum enters a meter which g i v e s a n u l l i n d i c a t i o n when the polynomial equals zero. The p h a s e - s h i f t e r s are motor-driven and the modulus u n i t s are manually adj u s t e d to l o c a t e the polynomial zeros. Although the p h a s e - s h i f t u n i t i s compact and accurate an excessive number (n, where n i s the degree of the p p l y ^ nomial) of modulus u n i t s and a s s o c i a t e d a m p l i f i e r s i s r e q u i r e d . M a r s h a l l has developed a h i g h l y s u c c e s s f u l computer us i n g form I I I without c a r r i e r f o r s o l v i n g polynomials with degree up to ten. Ten h a r m o n i c a l l y r e l a t e d square waves are generated by means of commutators of various-numbered segments on a common s h a f t . The i n p u t v o l t a g e to the commutators i s obtained from a c h a i n of ganged modulus po-tentiometers separated by b u f f e r a m p l i f i e r s such t h a t the f i r s t commutator r e c e i v e s a v o l t a g e p r o p o r t i o n a l to r , the 2 n second to T , un to a v o l t a g e p r o p o r t i o n a l to r on the l a s t term. Appropriate c o e f f i c i e n t s are set by poten-9 tiometers on the output of each commutator. The weighted square waves are f i l t e r e d to extract the fundamental-fre-quency components i n both sine and cosine terms. These terms are added separately and the two sums, representing U and V, are placed on the horizontal and v e r t i c a l deflection plates respectively of a CET. Thus a polar pl o t of V(p) i s produced. Once W(p) has been adjusted to pass through the origin, a marker pip i s gen-erated at fundamental frequency and. by phase-shift control, moved along the curve of W(p) u n t i l the o r i g i n i s reached; a i s then determined from the phase-shift. Solution of higher order polynomials using this system introduces greater commutation and f i l t e r i n g problems. Con-stant p_ cannot be generated and complex c o e f f i c i e n t s are unattainable. Lofgren has completed an elaborate computer for solving i polynomials with degree up to eight. Simulating the poly-nomial i n form III without c a r r i e r , his instrument i s an electronic counterpart of Marshall's unit. Harmonically related square waves are produced by locked o s c i l l a t e r s . By suitable choice of time-constants, eight RC-networks form exponential voltages of i n t e g r a l l y related powers. Each of these voltages, representing r , causes amplitude modulation of the output of the corresponding o s c i l l a t o r . Coefficient potentiometers weight the terms and f i l t e r s select the fundamental-frequency components of the square waves. The 10 terms are added directly and in quadrature to produce U and V, The zeros are automatically and continuously indicated on a jj-plane CRT. The £-plane locus i s obtained from the fundamental oscillator and the zero positions are located by pulsing the CRT from a crossover and gating network which is energized when U and V are simultaneously zero. Automatic normalization and root-correction circuits give high accuracy. Aside from i t s inability to handle complex coefficients, this computer has many desirable features. However, the prototype instrument contains over 200 tubes and extension of such a scheme to solve 20th-degree polynomials would require an impractical number of tubes. Forte, Rose, and Willoughby have proposed a computer for solving sixth-degree polynomials and completed one appli-cable to fourth-degree polynomials. Two-pole rotor, two-pole stator, servomechanism units are used as the harmonic generators. These units are geared to turn at harmonically related angular displacements or velocities. Modulus values are imposed by exciting the generators through ganged Variacs. Voltages representing U and V are simply obtained by connecting the direct and quadrature generator output windings in series. Demodulation i s achieved by a saturable-reactor unit. Zeros are indicated on a CRT display of W(p). Zero location i s basically the same as in Marshall's computer. Choudhury has produced a unique instrument completely electronic and yet simple. A delay line consisting of low-pass f i l t e r sections i s fed from one end by a matched frequency-sweep generator. From the theory of f i l t e r s and transmission l i n e s i t can be seen that phase s h i f t i s a l i n e a r function of frequency. Thus, the function of the phase-s h i f t e r i s replaced by an a r t i f i c i a l l i n e operating at variable frequency. None of the existing systems has a l l the q u a l i t i e s de-s i r e d i n a general-purpose instrument f o r a research labora-tory. The prime requirement i s that of solving polynomials, and i t was decided to attempt to solve polynomials with r e a l c o e f f i c i e n t s up to the 20th degree and with complex c o e f f i -cients up to the 10th degree. In addition, as secondary requirements, i t was decided that the computer shoulds (a) permit generation of Fourier series point by point or continuously with up to 21 terms, (b) permit a plot of W(p) to be viewed either point by point or continuously with a and r both varying or with either a or r constant, (c) permit generation of two r e a l - c o e f f i c i e n t polynomials of degree 10 simultaneously and enable suitable combination of the outputs, (d) permit plots of contours of constant phase angle of W(p), constant magnitude of W(p), and constant W 1 ( P ) In on the £-plane display. If a l l of the above requirements are to be met, elimin-ation of c e r t a i n forms of generation i s necessary. Simulation up to the 20th degree puts s t r i n g e n t requirements on the type of system chosen. Systems haying g e a r - t r a i n s of the type 14 used by F o r t e , Rose, Willoughby or ganged potentiometers 15 as used by Parker and Williams become e x c e s s i v e l y bulky and accumulated e r r o r and speed l i m i t a t i o n s render such systems i m p r a c t i c a l . The a b i l i t y t o perform F o u r i e r s y n t h e s i s d i c -t a t e s a system i n form I I or I I I . The b a s i c requirement of generating the polynomial f o r f i x e d £ as w e l l as f o r time-v a r y i n g £ excludes a l l types except those u s i n g a c a r r i e r s i g n a l . Hence the p o s s i b i l i t y of using any instrument employing methods such as commutation, locked o s c i l l a t o r s . o r any other device i n which harmonically r e l a t e d s i g n a l s are generated by c o n t i n u o u s l y running u n i t s , can be d i s c a r d e d . S t i p u l a t i o n t h a t complex c o e f f i c i e n t s must be p o s s i b l e a l s o leads to the same c o n c l u s i o n . These l i m i t a t i o n s leave the p o s s i b i l i t y of using a machine s i m u l a t i n g the polynomial i n e i t h e r form I I or I I I with a c a r r i e r s i g n a l . Since the d e s i r e d performance of the instrument has been s p e c i f i e d and general l i m i t a t i o n s s e t on the method of simu-l a t i o n , the range of c o n s i d e r a t i o n of instrument design i s narrowed. As a r e s u l t o f t h i s p r e l i m i n a r y i n v e s t i g a t i o n three systems were s t u d i e d i n d e t a i l . These are presented i n the f o l l o w i n g chapter. 14 F o r t e , Rose, Willoughby, op. c i t . 15 Parker, W i l l i a m s , op. c i t . 2.0 CHOICE OF SYSTEM In arriving at optimum system design, serious consid-eration was given to three comparable types of instrument which satisfy the specifications set forth in the previous i chapter. It was decided to use servomeonanism components due to their simplicity and precision. This choice excludes other practical forms of instrument design that are possible, 16 e.g., Choudhury . Each of the three systems considered is a carrier system with voltage as the analogue. A computer of this type can be considered in two parts, the function generator, which forms the polynomial as modu-lation on the carrier, and the detection and output unit which removes the carrier and displays the zeros. Since the method of detection and output presentation is the same in each of the three systems, this part of the computer will not influence the choice of system and wil l be discussed later. The three systems may then be. classified according to their method of function generations A. direct-coupled phase-shifters with variable excitation, B. geared phase-shifters with variable excitation, Co geared phase-shifters with constant excitation. In each system the phase-shifters, using two-phase ro-tating fields, shift the instantaneous electrical angle of the input reference voltage by an amount equal to the angle of the phase-shifter rotor with respect to the reference Choudhury, op. c i t . voltage axis. In the following discussion a quasi-static approach w i l l be taken to simplify the mathematics. J u s t i -f i c a t i o n f o r this approach w i l l be evident i n the next chapter. System A i s shown i n Figure 1. In t h i s system the c a r r i e r r jto t i voltage, Re [e J , i s applied d i r e c t l y and i n quadrature to a chain of alternate electromechanical phase-shifters and modulus potentiometers. A l l phase-shifters turn through a common angle, a, and a l l modulus potentiometers are mechanically ganged. The output from each modulus potentiometer i s s p l i t into a two-phase voltage and applied to the following phase-shifter to produce within i t a rotating T k J ( w o * + k f l t h f i e l d . Output voltages of the form Re [ r e J are tapped off at the inputs of the phase-shifters. These voltages are passed through appropriate c o e f f i c i e n t T k i K i * + k®^1 units to obtain outputs of the form R® L&^r © J J . The c o e f f i c i e n t s , a^, are real i f potentiometers are used alone but may be made complex by the addition of passive phase-s h i f t i n g networks. When terms are added, the resultant i s k-o = U° cos w t - V° s i n w t . o o Demodulation i s accomplished by multiplying the above voltage by 2° cos u>0t and by -2»sin w ot and following each of these operations by a low-pass f i l t e r . The outputs from the two mu l t i p l i e r s are then U and V respectively. a, *2 1 I SUtojmettton 15 System B i s shown i n Figure 2. In this system, c a r r i e r voltage, Re i s applied to a chain of mechanically-ganged modulus potentiometers. Output voltages representing r°, r \ r*% ° <>. r n , drive power amplifiers. By using phase-s h i f t i n g networks, these amplifiers excite the electromechanical phase-shifters d i r e c t l y and i n quadrature to produce rotating f i e l d s . The phase-shifter rotors are geared to rotate through angles of a, 2a, 3a ... na. C o e f f i c i e n t potentio-meters weight the outputs from the phase-shifters. Voltages are then of the form Re [ a ^ r M ^ o * + k a ^ ] . Complex c o e f f i c i e n t s are attainable by allowing the phase-shifter stators to be set at angles. 6^ . This i n t r o -duces factors of the form i n the above terms so that 6^ may be considered the angle of the corresponding c o e f f i c i e n t a^ . As the voltages are i d e n t i c a l to those of system A, summation and detection are the same. System C i s shown i n Figure 3. In t h i s system the elec-tromechanical phase-shifters are excited d i r e c t l y from a common two-phase c a r r i e r source. The rotors of the units turn through i n t e g r a l l y related angles and the stators may be i n d i v i d u a l l y rotated to give angles to the c o e f f i c i e n t s as i n system B. The output voltages then represent terms of the form Re I a^e u |. These voltages enter a chain of alternate adding ampli-f i e r s and modulus potentiometers. The output of the chain can be seen to be comet /o/taae -/Hec4cu7/cad^> yanyec/ iso/lfaje o/y/bbs-s ftaO T* D F 3 ^ 0 * k jka v = Z_ Be |_e a, r e J k=o = Be , which i s the same as f o r the other two systems. A choice was not made ent i r e l y on the basis of operation as described above. Two methods of s i m p l i f i c a t i o n were studied. Both of these involve a displacement or frequency s h i f t of the carrier s i g n a l . The f i r s t s i m p l i f i c a t i o n applies only to methods B and C and allows a twofold reduction i n gearing. Instead of a l l units rotating i n the same dire c t i o n through angles 0, a, 2a • • ° na, one-half are rotated i n the opposite d i r e c t i o n . For a tenth-degree instrument) the displacements correspond-ing to p°, p \ p 2 ...p^ are then = 5 0 , -4a, -3a, -a, 0, +a, +2a o« s 5a. This, i n effect, i s a s h i f t i n the inst a n t -aneous c a r r i e r angle from (a) t) to (w t -5a). o o Demodulation using this new c a r r i e r can be obtained readily by using one extra phase-shifter positioned at -5a, through which the o r i g i n a l c a r r i e r i s passed. This c a r r i e r or spectrum displacement i s made use of by Parker and Williams The second s i m p l i f i c a t i o n i s again of the nature of a ca r r i e r s h i f t but somewhat more spectacular i n effe c t . It can be applied to a l l three systems. Consider f i r s t that a l l terms produced by the phase-Parker, Williams, op. c i t . s h i f t e r of an n^h degree instrument are i n t e g r a l units of phase ahead of the ca r r i e r at any given instant of time ( i f a i s a linear function of time then the terms can be considered integer units of frequency above the c a r r i e r ) . Thi applies to the system with no modifications. If a second set of outputs i s taken from a separate set of c o e f f i c i e n t poten-tiometers i n p a r a l l e l with the corresponding ones i n the o r i g i n a l unit, a second polynomial may be generated. This polynomial, although haying the same phase displacement per term (frequency difference) as the o r i g i n a l , can have d i f f e r -ent c o e f f i c i e n t s on any or a l l terms. If a l l terms of t h i s new polynomial are shifted through equal phase angles (equal distances up the frequency spectrum) they can be made to correspond to higher terms of the o r i g i n a l poly-nomial. Added to the o r i g i n a l polynomial, the result w i l l be a polynomial of degree 2n. This process could conceivably be continued to extend the range of the instrument any number of times i f the phase-s h i f t i n g c i r c u i t r y were s u f f i c i e n t l y l i n e a r . In our study an application of this s i m p l i f i c a t i o n only once seemed optimum. Because the f i r s t method of s i m p l i f i c a t i o n i s inde-pendent of this l a t t e r method, both can be used. The second s i m p l i f i c a t i o n , or extended system as i t i s denoted, requires n extra modulus potentiometers and t h e i r associated amplifiers and does not allow complex c o e f f i c i e n t s to be set on any of the terms when the extension i s i n use. 18 Although system A has no gearing, i t has the serious disadvantage that each simulated term of the polynomial i s e l e c t r i c a l l y dependent on the preceding term. Error-pro-ducing effects such as the nonlinear magnetizing curve of the phase-shifters become cumulative i n t h i s system. Power amplifiers are required to drive the phase-shifters and accurate phase-shifting networks must be used with each unit. These requirements make the computer design d i f f i c u l t and, considering the inherent errors, the computer using system A appears impractical. System B uses a gear t r a i n but, because the f i r s t and second simplifications can both be used, a fourfold saving i n gearing i s possible. Although the terms are formed by e l e c t r i c a l l y separate phase-shifters (the modulus factor remains cumulative), there remains the error-producing var-iable excitation of the phase-shifters. Power amplifiers and l o c a l phase-shifting networks must be used. System C, although similar i n p r i n c i p l e to B, overcomes the variable excitation on the phase-shifters by using separate addition stages with the modulus factor between stages. Although the addition process requires more com-ponents, many others have been eliminated because the phase-s h i f t e r s may be driven d i r e c t l y from the l i n e without driving amplifiers or l o c a l phase-shifting networks. This system, using both s i m p l i f i c a t i o n s , was considered the most p r a c t i c a l , and construction has been started on such a computer. The following chapters w i l l discuss the design and operation of this instrument i n d e t a i l 3o0 FUNCTION GENERATOR The design and operation of the function-generation unit of the chosen system comprises the content of t h i s chapter. The purpose of the function generator i s to generate a voltage proportional to tie terms of a polynomial k = 20 a.p o The c i r c u i t diagram, Figure 4, shows the k = 0 complete unit. This unit contains f i v e major component parts? a chain of gear-driven electro-mechanical phase-shifters to produce the required phase relationships, a set of co-e f f i c i e n t potentiometers or voltage dividers to select certain portions of signal, reversing switches for introducing negative signs, amplifiers for combining signals, and ganged potentio-meters f o r introducing the r a d i a l v a r i a t i o n . The function-generator operation can be considered i n two separate stagess the formation of the polynomial terms from p^ to p"^ (the basic system) and the formation of the poly-11 20 nomial terms from p to p (the extended system), 3.1 Basic System A two-phase, 400-cycle voltage of approximately 10 volts peak i s applied to the stator of each phase-shifter. These machines have 2-phase, 2-pole windings on both rotor and stator with t h e i r rotors gear-driven and t h e i r stators rotatable through an angle of 1 8 0 ° . From noise considerations, i t i s found best to keep the magnetizing current out of the s l i p rings. One rotor winding i s not used and i s loaded with a resistance equal to the load on the other rotor winding (approximately 11.4 kilohms) to minimize d i s t o r t i o n of the f i e l d d i s t r i b -ution within each phase-shifter. The output rotor winding of each phase-shifter drives a pair of precision c o e f f i c i e n t potentiometers, only one of which i s used i n the basic system, through a r e s i s t i v e network to attenuate the output to approx-imately 1.8 v o l t s . This degree of attenuation i s dictated by the necessity of keeping the adding amplifiers from saturating under the most adverse conditions. The numbering on the c o e f f i c i e n t potentiometers refers to the terms i n the polynomial to be generated. Consider f i r s t only the generation of terms a Qp^ to a^p*^. The outputs from the c o e f f i c i e n t potentiometers [ a ^ t o \&^Q\ enter i d e n t i c a l reversing switches. The signal from a reversing switch i s connected to an adding amplifier where the c o e f f i c i e n t output i s added to or subtracted from the signal representing the sum of a l l preceding stages. These adding amplifiers are cascaded through modulus or r potentiometers which are mechanically ganged and motor driven. The voltage outputs from the phase-shifters, after atten-uation, are denoted V q to v^o0 ^ c a n readily be seen that the output from the l a s t adder i s . 10 v = 2Z - J a k | r k v k . . . . (3.1) Note that r i s a real positive number but unlike | a^| , which has a maximum of unity, an e f f e c t i v e maximum of 1 02 has been chosen f o r r since i t i s desirable to keep the unit c i r c l e within the range of the instrument,, Consider the voltage v^. It i s a well-known fa c t that i f a pair of voltages of frequency f ,90 degrees apart i n phase, are supplied to two pairsof pole windings, one p a i r being 90 mechanical degrees from the other, a rotating f i e l d i s set up with angular v e l o c i t y OJ^ = 2%t& radians per seconds Such i s the case within the phase-shifters, where f Q = 400 cps. Denote the rotor angle by and the stator angle by B_k, measured i n opposite and l i k e sense, respectively, to that of f i e l d rotation, re l a t i v e to a fixed axis of r e f e r -ence. Then the voltage induced i n the rotor winding due to the rotating f i e l d can be written v, = v k max Due to the gearing a simple correspondence can be seen between the values of a^. This r e l a t i o n i s = (k -5)a^. For si m p l i c i t y the subscript on i s dropped i n a l l further references and equation (3.2) becomes v = v k max B . [ I J K * * f c + <* -«>«>] . . „ ( 3 . 3 ) . This equation applies only when a and are independent of time. If a i s allowed to increase at constant angular ve l o c i t y , u>r, so that a *s a + (*>rt, the induced voltage i n the rotor winding i s increased or decreased i n magnitude i n pro-portion to the difference between co, and u> . Thus the out-o r put voltage becomes r. , c V / -i _ r j [w t + 8. + (k~5)(a+u> t ) J ] o o o • ( 3 o 4 ) © The factor [ l + (k-5) w r/u oj is the commonly encountered s l i p factor i n induction motors. Since equal voltages are applied to a l l stators, v.., i s the same for a l l v, «. Equation (3.1) can now be written v T - « I L w w v « J . i . [ * J h w < w ' 1 , i P k . t M A J k ' ] k— 0 o o o o (3e 5 ) In the computer as b u i l t , e ^ k „ i j a ^ l c a n then be denoted a general c o e f f i c i e n t a ^ that can take on any value on or within the unit c i r c l e in-be complex-number plane. It w i l l be noticed that signal i s applied to the negative input terminal of an amplifier when the corresponding reversing switch i s i n the positive position. This i s done so that the loading effect of the v a r i ation of the modulus potentiometers w i l l not be re f l e c t e d i n the accuracy of the c o e f f i c i e n t potentiometers for the case of r e a l positive c o e f f i c i e n t s . In order that t h i s procedure be v a l i d i t must be understood that the reference direction f o r i s that which gives the maximum posit i v e output signal contribution at t = 0 at the end of the summing chain f o r the positive settings and f o r a = 0 at t = 0. Two conditions exists ( i ) when the phase-shifter rotors are s t a t i c , and ( i i ) when the rotors are rotating at constant angular v e l o c i t y . If there i s no rotation so that a = a Q , the s l i p factor i s unity and equation (3.5) reduces to 10 v = v Z Be [yK*-S«> S ^ k - J . . . ( 3 > 6 ) o  x k=0 . Referring to chapter 1.0, i t i s apparent that a, r k e j k a n * i s a t y p i c a l term i n the polynomial W(p) = 5~~ a-p^ where re d = p 9 and that the output voltage i s k=0 It i s not d i f f i c u l t to demodulate this signal to produce U and V g however, this s t a t i c situation i s only a special case, because generally a w i l l be a function of time 0 In that case, the s l i p terms introduce unwanted factors. The error so produced cannot be tolerated for accurate solution and steps must be taken to compensate for i t . Analysis of equation (3.5) shows that i t i s represented by a lin e spectrum with components lying between ( f Q + 5 f r ) and ( f Q - 5 f ^ where f = cor/2tt. Each l i n e i s associated with one term i n the sum. The r e l a t i v e amplitude of any one l i n e i s l i n e a r l y proportional to i t s frequency due to the s l i p factor, i . e . , the kth term has a frequency oj = 00 + (k-5)u> and an amplitude proportional to 1 + (k=5 ) a) /oo = w/co . r o o In order that the spectral l i n e s have the same r e l a t i v e amplitudes as when u r = 0, the kth term must be corrected by multiplying by a>o/a>o Three p o s s i b i l i t i e s were studied for this correction: ( i ) i n d ividual scale factors introduced at each phase-s h i f t e r output, ( i i ) adjustments on the modulus potentiometers to give approx-imate corrections based on the binomial expansion, ( i i i ) an integrating network at the end of the summing s t r i p . The f i r s t two methods are applicable for only one velocity of angular rotation whereas the t h i r d method i s an exact correction for any speed,provided the modulus variation i s slow i n comparison. The t h i r d method was chosen not only for t his advantage but also for i t s s i m p l i c i t y . An active i n -tegrating c i r c u i t i s used sas large attenuation from a passive integrator cannot be allowed at t h i s point. Since the band of frequencies i s centred compactly around i$Qt the integrating network need only have the desired response i n t h i s region and the pole at the ori g i n may be removed. The phase and amplitude errors caused by amplifier d r i f t correction and the o r i g i n pole removal are small. Letting K be the integrator transmission function magnitude at u> = UQ9 the output from t h i s unit i s r j(w t~5a-V2) -i v = K v m a x Re [e 0 W(p)J . . . . ( 3 . 8 ) This i s the form required for demodulation. Notice that K can be made adjustable. This i s desirable to keep the peak value of v within a range of optimum voltage for the m u l t i p l i e r regardless of what degree polynomial i s being solved. 3.2 Extended System Referring to the lower half of Figure 4. i . e . . terms p*^ to p ^ , notice that there i s a chain of summing amplifiers i d e n t i c a l to those f o r the p^ to p ^ terms. The c o e f f i c i e n t potentiometers feeding these are i d e n t i f i e d as ) a i j j to i a20^ ^ al()) * s n o ^ n o r m a H y u s e d i n t h i s s e t ) . Similar to the out-26 put from the f i r s t 11 terms, the output from the 11th to 20th terms i s v« = 21 * l ak+lC-' ^ k 0 .o.(3.9) k=l Because the voltages v^ 9 Vg o o . V ^ Q above are those i n the 0 , 10 , . . , p to p terms, the output becomes 10 v'=vmax Z : [ l + ( k - 5 ) W r / c o o ] . R e [ e ^ V " 5 a ) e J P k ± r V * ] k.=l ..o(3.10)» These terms are passed as a sum into two band-shift phase-s h i f t e r s , which are d i r e c t l y coupled e l e c t r i c a l l y , and driven at a speed of 5u r, thus acting as one unit turning at 10o)r. The reason for using two phase-shifters rather than a single one turning at twice the angular v e l o c i t y i s to achieve a reduction i n gear speed. The output voltage, v, i s s p l i t into two components at right angles by a phase-shifting net-work and applied to the input of the units . This network has not been designed as part of this study B Although t h e o r e t i c a l l y i t i s impossible to achieve the same phase-shift for a l l f r e -quency components while retaining a constant magnitude response, a good approximation can be expected, as the spectrum i s compact. Analysing the output term by term, i t can be seen that i f r j(w t~5a) jfl. . . -I V = v m a x B e L e ° e • 1 k * l 0 l r 6° J — f 3 ' 1 1 ) i s applied to the input, the output from the f i r s t band-shift unit i s V = * k + 1 0 l ^ < k + 5 ) " ] . . . (3.1 2, when the transmission factor of the units i s adjusted to unity. The output from the second unit i s V k = Vmax R e L e 6 " - l ak+10l r B J ...(3.13)* However, i f a i s a function of time, a = a +co t 9 and i f the ' — . y o r y variation of r_ with respect to time i s neg l i g i b l e , there occurs a s l i p factor for each u n i t . The factor f o r the f i r s t unit i s 1 + 5w/[co + (k-5)co land f o r the second unit i s 1 + 5w /[(a +ku)l. r u o r r ' - o r J Therefore the overall factor f o r the kth term i s 1 + 10(o r/[u o+(k-5)u r] . Introducing the above changes i n equation (3.10), the result i s i ° r / v / -i r ^ K t + ( k + 5 ) ( a o + U ) r t l ] + l I j P k kl v =vmaxZ: [ 1 + ( k + 5 V W o ^ R e L e * O ± l a k + l o | e r J k=l ... (3.14). By passing t h i s voltage through the ten addition stages of the f i r s t section of the function generator the result i s a mul t i p l i c a t i o n by r ^ . The output due to the extended system voltage i s then 10 ttti v =v max r -1 r j [w t+ (k+5)(a -5-00 t)] , . , k . i n1 [l+(k+5)o»r/a.J.Be[. L o 0 1 J.±K+10|r k + 1 0J o o o (3 o 15) • i f i t i s assumed that 8^=0. A s i m p l i f i c a t i o n i s produced i f k+10 i n equation (3.15) i s replaced by k, with l i m i t s correspondingly altered. Then 20 II 11 v =v max J2 r _ r j fu t+(k-5)(a +u t)l . , .-1 2Z [ M W V - J .Be [i 1 ° ° 1 • ^ a k | r k J k=ll 0 . 0 ( 3 .16 ). This i s recognizable as a continuation of the same series generated by the f i r s t section of the function generator, equation ( 3 . 5 ) o Complex c o e f f i c i e n t s cannot be set when the extended terms are used, i . e . , a l l c o e f f i c i e n t s of the poly-nomial must be r e a l . The t o t a l output from both parts before integration i s . 20 <r-r , . -j r J k t + ( k - 5 ) ( a + u t ) ] -k=0 ...(3.17), and after integration i t becomes i ? r j[« t+(k-5)(a +a t)-*/2] . - i k=0 2 0 r i kW2-5(a +u t)] _ =KvmaxZ! B e f S ° W ( P ) - I 0 0 0 ( 3 o l 8 ) , i k=0 To determine the numerical values i n the detection and root-indicating units, i t i s necessary to f i r s t choose the rates of scanning the p-plane. The polar angle and the modulus r can be s t a t i c or func-tions of time. When they are made functions of time, i . e . , when the units are motor-driven, there exist three c r i t e r i a which determine t h e i r approximate speeds § 1. the d e f i n i t i o n of the p_-plane trace, 2. the mechanical l i m i t a t i o n s of gears, resolvers, and potentiometers, 3. the bandwidth of the function-generator spectrum. It was found through experiment that a repetition rate as low as 2 cycles per minute allowed a trace to be viewed 29 continuously on a P-7 phosphor CRT screen. Also, considering the spot size of the CRT beam, i t was estimated that 10 lines per inch gives adequate d e f i n i t i o n on a 12-inch screen for approximate root p o s i t i o n i n g . Then, because the p_-plane trace i s a c l o s e l y woven s p i r a l (r increasing slowly as a rotates very rapidly)$ and because the rep e t i t i o n rate of r i s set at a minimum of approximately 2 rpm. the c y c l i c rate of a must be 10 li n e s / i n c h x 6 inches x 2 cycles/min. = 120 rpm. Considering the mechanical l i m i t a t i o n s , this angular vel o c i t y of 120 rpm gives a maximum gear speed of 5 x 120 = 600 rpm. This i s allowable noisewise and i s well below the l i m i t f or the gears and resolvers. Assuming a l i f e of 10 revolutions, a speed of two rpm for the r potentiometers gives an expected l i f e of more than 8000 hours, which should be adequate. The frequency bandwidth of the function generator f o r a varying at 2 cps i s from 390 to 430 cps. This i s a compact spectrum and produces l i t t l e phase s h i f t error i n the i n t e -grators and phase-shift network of* the band-shift u n i t . The two motor drives f o r r and a are not geared or t i e d together i n any way; the absence of interconnection allows random i n t e r l a c i n g of the s p i r a l s , giving more e f f e c t i v e coverage of the p_=plane. If higher d e f i n i t i o n i s desired, the r potentiometers can be slowed down to give a more det a i l e d coverage of the p_-plane. However, i n that case the roots w i l l not be con-tinuously illuminated. Construction of the phase-shifter, c o e f f i c i e n t , adding, and modulus units i s complete except f o r interconnecting wir-ing. 31 4.0 DETECTION AND OUTPUT UNITS Figure 5 shows the detection and output units i n d e t a i l ; these w i l l be described separately. 4.1 Detection Unit The output voltage of the function generator after integration i s r i(w t-5a»n/2) , VT - K vmax B e L6 T <P)J — where K and v are constants and max ¥(p) =51 a,p k = U + jV, k i a To locate the zeros of the polynomial, W(p) must be sep-arated from the above expression. Equation (4.1) can be written v T = Kv m a x E e |[cos(a)ot-5a-Ti/2)+ j sin (uot-5a-*/2)] . [u+jv]j = Kv [ucos(u t-5a-ir/2)-V sin(co t-5a-n/2 )1 ...(4.2). max L o o J Because |w(p)| = *^ U^  "+ V*% equation (4.2) can be written V T = K vmax | ¥ ( P ) | C O S (0 + u *-5o-ii/2) ... (4.3), where 0 = tan""'- ^ . By r e c t i f y i n g and f i l t e r i n g , |w(p)| i s obtained. However, as w i l l be seen l a t e r , the components U and V are desired f o r Fourier synthesis and Nyquist plots. These are obtained i n the following manner. The function generator output enters two separate m u l t i p l i e r s , one multiplying v^ , by ^cos(a) ot-5a-n/2) and the other multiplying i t by ~ g sin(w ot-5a-7t/2). —<- Ale J Hf 32 M u l t i p l i e r #1 yi e l d s • m l= v m a x [ U + V 2 cos j&-2. (wot-5a-n/2)]J ...(4.4). Similarly multiplier #2 y i e l d s vm2 = vmax [ V ~ s i n fa2• (wQt-5o-i./2)}] ...(4.5). The frequency components of U and V range from 0 to 40 cps whereas the components of the second terms i n equations (4.4) and (4.5) range from 780 to 860 cps. Therefore, the second terms are eas i l y removed by f i l t e r s placed after the multi-p l i e r s . The f i l t e r s are twin-T units designed with peak attenuation at 800 cps. The f i n a l outputs aft e r f i l t e r i n g are then Tml' = Vmax • U ... (4.6), a n d vm 2 ' = % a x ' V ••• ( 4 ' 7 ) ° 4.2 The Output Unit. To locate the zeros, i . e . , to determine the values of r and a where W(p) = 0, the polar plot of W(p) can be produced by placing U and V on the horizontal and v e r t i c a l d e f l e c t i o n plates respectively of a CRT. When W(p) passes through zero the values of r and a can be read either on a p_-plane CRT or from calibrated d i a l s on the r and a units. However, for a f i r s t approximation involving many zeros th i s w i l l be a tedious process. Although provision has been made for th i s form of presentation, the computer was also designed to indicate the root positions on a map of the p_-plane presented on a long-persistence CRT screen. The jj-plane trace i s generated by a separate phase-s h i f t e r geared to have an instantaneous angular position of a. One rotor winding i s excited by a voltage proportional to cos w Qt; the other winding i s l e f t open. Separate out-puts are taken from each of the stator windings. For wr= 0, these outputs are of the forms it n 7 v = v cos w t.cosa . . . 14.8). c p o v = v cos ca t.sina . . . (4.9), s p o . * where v " i s constant. Each output i s attenuated to keep the voltage l e v e l low i n the following components, the choppers, because t h e i r delicate contacts work best under low voltage and current conditions. The choppers are polarized, double-throw, single-pole vibrators having armatures energized from the 400-cps supply through a suitable phase-compensating network. The arma-tures are adjusted to swing i n phase with the voltage, Vp" cos w Qt, so that the choppers act as almost i d e a l , un-f i l t e r e d , synchronous demodulators with full-wave output. The output from each chopper i s passed through a shunt m-derived, low-pass f i l t e r having cutoff at 640 cps and high attenuation at 800 cps. The resultant voltages are then v = v cosa . . . (4.10), c p and v = v sina , . . (4.11), s p where v ' i s proportional to v P P Each of these voltages i s fed into an e l e c t r i c a l l y sep-arate modulus potentiometer which i s mechanically ganged with the modulus potentiometers of the function generator. Hence, since the potentiometer r a t i o , r / l . 2 varies between 0 and 1, the output voltages are v • • °.r coao/l.2 = v '.x/l.2 • . . (4.12), * j? and v g = v ' »r sina/l.2 = T 1 .y/l..2 . . . (4.13), where x and y_ are the instantaneous cartesian coordinates of the point p_ defined by the computer polar coordinates r and a . The voltages v and v are applied to the horizontal and c s v e r t i c a l deflection plates respectively of the p-plane CRT, applying suitable amplification to exch to give the d esired p_-plane size. Regardless of the size, however, the l i m i t of x, £ and r corresponds to a numerical value of 1.20. When a i s a linear function of time, a = a + a> t, the — ' o r above voltage expressions are not exact. Like the function-generator phase-shifters, the j>-plane phase-shifter or re-solver, as i t should be more c o r r e c t l y termed, generates a " s l i p " voltage. The error so produced can be analyzed as follows. Consider a pulsating single-phase f i e l d within the resolver proportional to Re [e J . This f i e l d can be thought of as composed of two equal but oppositely rotating f i e l d s with amplitudes one-half that of the o r i g i n a l and wi th ±u>_ as t h e i r angular v e l o c i t i e s . Then equation (4.8) must be replaced by n r N , i r 3 (w„t+a +o) t) , j (-01 t+a +u> t ) n yc = L v p /2jRe[(l +a) o/u) r)e J 0 0 r +(l-a>0/u>r)« 0 0 r ] . . .(4.14) • As i n the case of the function generator, i t i s possible to remove the error due to the s l i p voltage by integrating. However, the maximum error without correction i s only 0.5$ to the sine term. The CRT beam i s normally biased to cut-off during scanning of the p_-plane. Ideally, when the value of W(p) goes through zero, a pulse i s formed and placed on the control grid of the CRT. This illuminates the screen and indicates the zero po-s i t i o n s . To generate such a pulse, a device i s required which "triggers" when W(p) = 0. However, a device cannot p r a c t i c a l l y 18 be b u i l t to "trigger" at exactly that moment. Lofgren , i n designing his computer, analyzed three d i f f e r e n t approximations which were independently considered for t h i s computer. These are intended to indicate a root when8 of radius 6 , (b) |U| + |V|^  6; the zero range i n the W(p)-plane i s a square of the peak voltage because ( ~ l s 0.005. Such a small * °'max error does not j u s t i f y correction. The same argument applies (a) $ the zero range i n the W(p)-plane i s a c i r c l e of sides 6 with i t s corners on the U and V axes, (c) simultaneously; the zero range i n the W(p) plane i s a square of sides 6 p a r a l l e l to ihe U and V axes. Lofgren, op.cit. Note that 6 must be small for accurate solutions. Method (a) can be applied easily as |w(p)| i s obtainable by r e c t i f i c a t i o n and f i l t e r i n g . Cases (b) and (c) appear to have equal merit and were studied i n d e t a i l . Although many minor items were considered, the only deciding factor appeared to be the number of amplifier stages required and hence, of these two cases, (b) was chosen. . Since U and V are available, method (b)is probably most e a s i l y applied, but method (a) has the advantage that i t i s not subject to the multiplier errors. To make the instrument as ve r s a t i l e as possible, both methods are available, and the choice of one or the other involves only a switching c i r c u i t . When method (a) i s used the output from the |w(p)| f i l t e r enters a dc amplifier which raises the peak output of |w(p) | to 48 v o l t s . For method (b), the outputs U and V enter separate absolute-value c i r c u i t s , each of which consists of a dc amplifier with feedback and a pair of diodes. The output voltage of these units, l i k e that of the JW(p) | amplifier, w i l l always be p o s i t i v e . The voltages representing [U| and |V/ are added i n a DC amplifier giving a peak output voltage of 48 v o l t s f o r the signal |U| + |v| . For either method, the output enters a high-gain dc amp-l i f i e r without feedback which w i l l be saturated by an input of the order of i 0.01 v o l t . Because the input to thi s stage has a peak value of 48 v o l t s , the amplifier w i l l swing to saturation on x 100$ = 0.02$. For case (a), the numerical peak of |w(p)|is 1.2, and hence the value of 6 i s 0 o 0 2 1 x 0 1 ° 2 = 0.00024. For ease (b), the numerical peak of )U(+|V| i s 2.4 and, therefore, the value of 6 i s 0 O 0 ^ Q Q 2 O 4 = 3 7 0.00048. Either of these i s more than adequate. W(p), U and V are not constant (unless u>r = 0) but vary p e r i o d i c a l l y at least with the frequency of wr, i . e . , i f any of these functions pass through zero at a l l , they must do so at least every 0.5 second ( f y = 2 cps). Considering a very extreme case, t h a t when the time taken to swing from 0 to .01 volts i s 0.5 seconds, then the saturating amplifier swings through approximately 100 v o l t s i n 0.5 seconds. The average rate of change of the output i s 200 volts per second. At some time there must be a rate of change equal to or greater than this average value. Using an operational amplifier as a d i f f e r e n t i a t o r , pulses are formed each time the saturating amplifier changes state. Both positive and negative pulses are formed as the polynomial passes through zero and both pulses are passed to the CRT g r i d although only the p o s i t i v e pulse which precedes the zero i s effective i n indicating the zero position. The d i f f e r e n t i a t o r design i s set by the minimum rate of change of voltage (200 v/sec) from the saturating stage and the desired minimum pulse peak (22.5 v o l t s ) . To prevent more than 22.5 v o l t s from developing with higher rates of change of W(p) and hence giving unequal brightness to the zero location on the CRT, a biased diode l i m i t e r i s shunted across the output. It should be noted that the zero-detection unit may not pulse the CRT (and hence indicate zeros) i f the j>-plane i s being swept very slowly. However, once the region of the zero i s e s t a b l i s h e d approximately, a q u a s i - s t a t i c i n d i c a t i o n can be obtained by manually moving a or r r a p i d l y through a small r e g i o n i n the neighborhood of the zero. The r e g i o n where a zero w i l l be i n d i c a t e d i n the j>-plane w i l l not n e c e s s a r i l y have the same numerical l i m i t s as the corresponding r e g i o n i n the W(p)»plane. The l i m i t s i n the W(p)-plane have been chosen as the s i d e s of a square o f l e n g t h 6. . The mapping of t h i s i n t h e j>-plane f o r s i n g l e - o r d e r zeros i s an approximate square, d i f f e r e n t l y o r i e n t e d and depending i n s i z e upon the extent to which the zeros c l u s t e r . For m u l t i p l e zeros, the slope and area w i l l be d i f f e r e n t ; however, as i n the s i n g l e - o r d e r case, o n l y the slope can .be determined whereas the area, i n general, may only be s t a t e d to v a r y dependent upon the pr o x i m i t y of other z e r o s . The t h e o r e t i c a l d i f f i c u l t y of determining the zero area i n the £-plane need not be of concern because, f o r most c o n d i -t i o n s , the l i m i t i n g s i z e of t h i s area w i l l be determined by the pulse lengths from the d i f f e r e n t i a t o r , the "spot" s i z e of the CRT beam, and the d e f i n i t i o n of the scanning t r a c e . The c a l i b r a t i o n u n i t c o n s i s t s o f two separate c i r c u i t s : the a - p o s i t i o n c i r c u i t and the rebalance c i r c u i t . An e l e c t r o -mechanical p h a s e - s h i f t e r i d e n t i c a l to those i n the phase-s h i f t e r u n i t i s connected, when c a l i b r a t i o n i s to be done, i n t o a synchro-type c i r c u i t together with the p h a s e - s h i f t e r t h a t p o s i t i o n s a t a. T h i s c a l i b r a t i n g u n i t w i l l synchronize i t s s h a f t to a and from a c a l i b r a t e d d i a l on the s h a f t the angle a can be read d i r e c t l y . A potentiometer i d e n t i c a l to those i n lie modulus unit i s connected to n u l l the voltage from the p-plane r potentiometer. Sensitive balance i s ob-tained by the use of an open-loop operational amplifier with neon glow-lamps on i t s output. A d i a l on the potentiometer calibrated from 0 to 1.2 i s manually adjusted u n t i l a n u l l i s reached. The zeros of a polynomial can be viewed continuously on the CRT display or, f o r higher d e f i n i t i o n , can be observed at a reduced rate by slowing the modulus drive motor. For maximum accuracy, before any non-essential manipulations are used, the computer motors can be stopped ( u ) r = 0, r s t a t i c ) at a zero and the modulus and angle can be obtained from the c a l i b r a t i o n u n i t . 40 5.0 COMPONENTS, LAYOUT AND FUTURE EXTENSIONS The complete computer, with the exception of the power supply and CRT display, i s mounted i n two 42-inch relay racks. The computer can be thought of as a number of physically separate parts, the phase-shifter unit, the c o e f f i c i e n t units, the adding units, the modulus unit, the multiplying unit, the output unit, and the c a l i b r a t i n g u n i t . The phase-shifter unit, the largest single section of the computer, occupies the top half of one rack. Autosyn 400-cycle resolvers were chosen for the phase-shifters. These are physically small, highly accurate, and require only a two-phase source to produce a rotating f i e l d . They are driven by a 500-rpm, 24-volt dc motor through Boston 32-pitch spur gears. The gears are arranged on a v e r t i c a l main shaft which turns at 3 u>r and the resolvers are paired o f f along th i s shaft (the largest gear r a t i o i s 3 s i ) . The additional phase-shifters for multiplying, s h i f t i n g , and p_-plane pl o t t i n g are also mounted along t h i s shaft. A c o e f f i c i e n t angle, 6 k, i s set by rotating a calibrated disc that i s mounted d i r e c t l y on the resolver stator. To avoid excessive twisting of leads, the physical l i m i t of i s 180°. Larger angles for the Oth to 10th terms and negative c o e f f i c i e n t s f o r the 11th to 20th terms are obtainable through reversing switches mounted below the c o e f f i c i e n t potentiomet-ers. 41 The c o e f f i c i e n t potentiometers are mounted i n the other rack with j a j to \&^Q\ i a one row and with | a ^ | tojaggl i n the other row (the second c o e f f i c i e n t I&JQI * s u s e ( * only when two polynomials are simultaneously generated). These compo-nents are 2500-ohm, 10-turn, 0.2 5#-linearity, miniature Helipots which can be set without a vernier to three s i g n i -f i c a n t figures. The adjusting and attenuating re s i s t o r s f o r the c o e f f i c i e n t c i r c u i t s are mounted on either side of the phase-shifter unit . The adding amplifiers are P h i l b r i c k model K2-X operational amplifiers, self contained (except for plate supply) i n two 10-unit manifolds i n the c o e f f i c i e n t rack. The eleventh amplifier i n each set i s separately mounted at the side. Jacks to match those on the manifolds are mounted d i r e c t l y below i n a fixed panel. A mating plug-board connects each unit to the corresponding fi x e d jacks. Using this form of construction, the amplifiers can be completely disconnected from the computer for other uses. The modulus unit i s mounted at the rear of the c o e f f i c i e n t potentiometers and contains two rows o f 11 precision 10,000-ohm continuous potentiometers. These rows are geared together and coupled to a gear-reduced, 500-rpm, 24-volt dc motor so that the potentiometers turn at a maximum speed of 2 rpm. The associated network of r e s i s t o r s f o r the adding and modulus units i s mounted i n two parts: one group i s mounted on the amplifier plug-boards and the other group i s compactly wired between -foe panel jacks and c o e f f i c i e n t switches. 42 Precision wire-wound r e s i s t o r s with 0.25$ tolerance are used throughout. The two mul t i p l i e r s are self-contained as a unit (Philbrick model M U / D V ) similar to the adding manifolds. This unit i s mounted near the bottom of the c o e f f i c i e n t rack. Like the adding manifolds i t i s available for other uses. Output f i l t e r s for the mul t i p l i e r s are located at the rear of the same rack. The output u n i t consists of two panels on the lower half of the phase-shifter rack. One contains f i v e P h i l b r i c k operational amplifiers and associated networks for zero-crossover detection and pulse formation. The other panel contains the two s i x - v o l t choppers, the m-derived f i l t e r s and associated r e s i s t i v e networks. The c a l i b r a t i o n unit occupies the panel d i r e c t l y below the phase-shifter u n i t . It contains one Autosyn resolver i d e n t i c a l to those i n the phase-shifter unit but with shaft position indicated on a d i a l calibrated i n degrees, and one modulus potentiometer of the same type as used i n the modulus unit but l i n e a r l y calibrated from r = 0 to 1.2. The lowest panel on the c o e f f i c i e n t rack is a "patch board" containing banana jacks to which a l l major outputs are connected. By suitably interconnecting these, the i n -strument can be set f o r any one of i t s various functions. The main power-supply unit i s not mounted on the racks. It consists of one 300-ma, i 300-volt plate supply, one 24-vo l t , 5-amp dc supply and a 10-volt, 2-phase, 25-va, 400-cps supply. The phase-shifter unit, coefficient unit, adding and modulus units have been partially completed. Some of the other units have been mounted in the racks but interconnecting wiring has not been done. Design and construction of a phase-shifting network for the extended function-generator terms and the CRT display unit, not being essential components,, a re considered part of a further project. A number of other interesting and useful extensions are possible, three of which are listed below. 1. By the use of an amplitude-sorting circuit, lines of constant magnitude of W(p), U and V could be plotted in the j>-plane. A comparable device for sorting angles could similarly map lines of constant angle of the above functions. 2. Various combinations of two polynomials, such as In ^ 1 ^ ^ could be formed. 3. A more elaborate extension that could be added is a servo tracking mechanism used to map a predesignated locus in one plane into the other. Probably the only major change in the present instrument would be to replace the £ and a dc drive motors by servomotors and amplifiers. Although the computer has been designed with the optimum placing of present components, possible extensions have been considered and allowances for such additions have been made. 6.0 PRELIMINARY CALIBRATION The preliminary c a l i b r a t i o n involves adjustments on almost every part of the computers the phase-shifter unit, the c o e f f i c i e n t units, the adding amplifiers, the modulus unit, and the amplifiers and potentiometers i n the output c i r c u i t s . With the exception of the output unit, the ad-justments are performed i n four stages. They w i l l be des-cribed i n the order i n which they must be made. The f i r s t stage i s the mechanical alignment, of the modulus potentiometers. The bodies of these potentiometers can be rotated i n d i v i d u a l l y by loosening the appropriate clamping screws. The potentiometer must be so aligned that a l l tie wipers are simultaneously on the same f r a c t i o n of the t o t a l winding. With the motor drive clamped and the gear back-lash taken up, the bodies of the potentiometers are adjusted so that the wiper just touches the start of the winding i n each potentiometer. The second stage is the setting of the biases and gains of the adding amplifiers. The biases on a l l amplifiers are adjusted to give zero when both inputs are grounded. The adding amplifiers have an o v e r a l l gain of 1.33. However, as a gain of 1.20 i s required on the output of the modulus potentiometers, rheostats are placed i n series with the modulus potentiometers. These must be accurately adjusted. During adjustment, the ganged modulus potentio-meters are locked at p r e c i s e l y l / l . 2 r a t i o . This f r a c t i o n i s accurately determined by a voltage comparison method using a ten-turn Helipot as the reference. During this and the succeeding adjustments a l l c o e f f i c i e n t p o l a r i t y switches should be placed on p o s i t i v e . A 1.8-volt, 400-cps supply i s then connected to the tap point on the second c o e f f i c i e n t potentiometer, ja^| , and the associated amplifier rheostat i s s et so that the output from t h i s amplifier read on a pre-c i s i o n voltmeter i s 1.8 v o l t s . With the output measured at the same place, the 1.8—volt, 400-cps supply i s connected i n turn to a l l c o e f f i c i e n t potentiometers along the chain and each i s adjusted to give 1.8 volts output. The t h i r d stage of the adjustments i s the setting of the i phase-shifter rotor angles. The rotors of the 11 phase-s h i f t e r s can be independently locked at any angle. These rotors must be set so that they are at a, 2a, 3a ... with respect to the rotor supplying the ( a Q| c o e f f i c i e n t potentio-meter. For this adjustment, the stator windings are excited with rated voltage and the c o e f f i c i e n t potentiometers and the stator angles, 8^, are set at zero. The main shaft i s clamped i n position and the horizontal and v e r t i c a l inputs of a CRO are connected to one phase of the phase-shifter supply and to the output of the adding-amplifier chain respectively. Each of the f i r s t set of c o e f f i c i e n t potentiometers i n turn i s set to unity and the associated rotors adjusted with respect to the main siaft to produce a l i n e with p o s i t i v e slope on the CRO. The l a s t stage of the adjustments involves the voltage l e v e l of the c o e f f i c i e n t u n i t s . A 2000-ohm rheostat i n series 4 6 w i t h each p a i r of c o e f f i c i e n t potentiometers sets t h i s v o l t a g e l e v e l . During the adjustment, the modulus u n i t remains clamped a t l / l . 2 r a t i o and the phase s h i f t e r s are normally e x c i t e d . A l l c o e f f i c i e n t potentiometers are - i n i t i a l l y set to zero and a p r e c i s i o n voltmeter r e p l a c e s the CBO on the a m p l i f i e r output. By s e t t i n g each c o e f f i c i e n t potentiometer i n t u r n t o u n i t y the r h e o s t a t s can be a d j u s t e d to g i v e the r e q u i r e d 1.80 v o l t s on the voltmeter f o r each u n i t . The output u n i t s c o n t a i n a few a m p l i f i e r s whose b i a s e s must be adjusted, three p h a s e - s h i f t e r s which must be a l i g n e d s i m i l a r l y to those i n the f u n c t i o n generator, and v o l t a g e -l e v e l - a d j u s t i n g r h e o s t a t s i n the p_-plane u n i t * Most o f these adjustments are s t r a i g h t f o r w a r d . However, to compensate f o r the 90-degree phase l a g of the i n t e g r a t i n g network of the f u n c t i o n generator, i t i s necessary to use the o t h e r phase of the 2-phase supply as r e f e r e n c e when s e t t i n g the r o t o r angles of these p h a s e - s h i f t e r s . Although the p r e l i m i n a r y adjustments are numerous and time-consuming, they are not i n t e r a c t i n g and once a d j u s t e d no f u r t h e r a t t e n t i o n should be r e q u i r e d . Since the o v e r a l l p r e c i s i o n of the computer depends p r i m a r i l y upon the accuracy w i t h which these adjustments are made, i t i s worthwhile to e x e r c i s e c o n s i d e r a b l e care a t t h i s stage of assembly. 47 7.0 MANIPULATIONS AND TRANSFORMATIONS OF THE POLYNOMIAL It i s essential to prepare a l l polynomials for the com-puter. Four basic operations, two necessary for set t ing up the computer and two he lpfu l i n obtaining higher accuracy, are important. They ares "equalizing" coeff ic ients , normalizing coef f ic ients , the p""^  transformation, and s h i f t i n g the o r i g i n . These w i l l be discussed i n d e t a i l i n the above order. In some cases there i s a considerable range between co-e f f i c i e n t values . Accuracy i n se t t ing coe f f i c i en t s i s great-est when a l l are of the same order of magnitude. This can general ly be arranged, p a r t i c u l a r l y when there i s a tendency for the coe f f i c i ent s to increase or decrease with powers of £ , by using a scale change i n the var iab le p_. The ef fect of t h i s change i s that the zeros, are arranged as near as possible to the u n i t c i r c l e . In some systems, a modif icat ion of th i s method i s used to ensure that every zero l i e s w i th in the unit c i r c l e • Af ter coe f f i c i en t equa l i za t ion , i t i s necessary to nor-malize c o e f f i c i e n t s . The maximum value of any coe f f i c i en t has been chosen as uni ty (for convenience i n c a l i b r a t i n g the potentiometer d ia l s o n l y ) . Therefore, the largest coe f f i c i en t must be div ided into every coe f f i c i en t i n the polynomial . This computer has been designed to generate W(p) and d i sp lay i t s zeros ins ide the c i r c l e of radius r = 1.20. In general there i s no assurance that a l l z e r o s of a polynomial w i l l l i e within this rad ius . Rather than make a change of 48 scale and thus s a c r i f i c e accuracy (unless such a process i s carried out a great number of times), a better procedure i s to make an inversion of the £-plane, through use of the p~* transformation, to cover the whole p-plane from zero to i n f i n i t y . This transformation i s easily carried out by a trans-position of the order of the co e f f i c i e n t s i n the polynomial. The zeros found f o r this polynomial from zero to the unit c i r c l e are the reciprocals of the zeros of the o r i g i n a l poly-nomial from the unit c i r c l e to i n f i n i t y . It i s advisable to use the transformation p~* instead of 1.2(p"H because there i s the p o s s i b i l i t y of amplifier saturation occurring near r = 1.20. To increase the accuracy of a p a r t i c u l a r zero an " o r i g i n -s h i f t " with scale change can be made. Consider the following example. If the indicated zero of some polynomial i s A + jB where the true zero i s (A + a ) + j (B + b), then a and b are the error components. By s h i f t i n g the o r i g i n of the j>-plane to the indicated zero, i . e . , by making the substitution p = p ( + A + jB, and reforming the resultant expression to a normalized polynomial (in which c o e f f i c i e n t s may be complex), a true zero w i l l be a + jb. Since the magnitude of a + jb w i l l be less than the maxi-mum r e l a t i v e error of the computer (say 0.05), the scale of the shifted p_-plane may be changed by the reciprocal, of t h i s maximum error and s t i l l retain the zero i n question within the unit c i r c l e . Thus the o r i g i n a l error may be found with accuracy equivalent to or better than the o r i g i n a l zero. By repeating the process, the zero can be found to any desired accuracy. However, more than one or two stages of this pro-cess would be used only i n extreme cases as the calculations involved are lengthy. Careful manipulation of a polynomial into optimum form for setting w i l l decrease overa l l solving time and give high first-determination accuracy. One or two stages of " o r i g i n -s h i f t i n g " should give three—significant-figure accuracy which i s normally adequate f o r most e l e c t r i c a l network studies. 50 8.0 OPERATION 8.1 Polynomial Solving Polynomial solving on the computer can be carried out systematically as follows. The normalized c o e f f i c i e n t s of the polynomial (and the angles of these i f applicable) are set. The appropriate "plug-board" connection and integrator gain are chosen. The r and a motors are then turned on at maximum speed. The computer w i l l scan values of the variable j> and zeros w i l l be continuously illuminated on the CRT display. If higher accuracy of zero location i s desired, the r and a motors are stopped. Bias on the CRT i s decreased u n t i l the trace i s just perceptible and the r and a manual controls adjusted u n t i l the CRT "spot" i s i n the neighbourhood of the zero to be determined. With a headset connected to the un f i l t e r e d output of the magnitude c i r c u i t , r and a are varied u n t i l a n u l l i n the 400-cps tone i s reached. This indicates the location of the zeros. With the zero " s t a t i c a l l y " located, the c a l i b r a t i n g r and a units can be adjusted to give a very accurate zero location. For each zero that i s desired to a high accuracy the above process should be followed. For even greater accuracy a number of stages of " o r i g i n -s h i f t " can be used with t s t a t i c " zero location a f t e r each. 8.2 Fourier Synthesis Continuous or point-by-point 21-term Fourier synthesis 51 can be performed d i r e c t l y on the computer. Either a sine series or a cosine series up to the 20th harmonic or a mixed series of sines and cosines up to the 10th harmonic can be generated and viewed on a CRO. Figure 6 outlines the connections required for the l a t t e r s e r i e s . A lternatively, a generali zed cosine series may be set up to the 10th harmonic. The operation may be explained as follows. The modulus potentiometers are set to unity and the c o e f f i c i e n t angles are set to zero unless the generalized cosine i s being used. The normalized c o e f f i c i e n t s of the harmonics are set i n order on the c o e f f i c i e n t potentiometers. The technique for Fourier synthesis follows from equation (1.4), 20 W(p) = Z l * , r k (cos k a + j sin k a) = U + jV, k=0 K k where i n this case r = 1 Then depending upon which m u l t i p l i e r i s used either 20 y~ a, sin k a = V k=0 K 20 or 2Z. a i cos k a = U k=0 * i s formed. The sine-series plug-board connection uses the V-multiplier and the cosine-series connection uses the U-multiplier. For a mixed series of sines and cosines, the cosines are set on the (a^l to I^ QI potentiometers and the sines are set on the ia^^ l to lag^l potentiometers. The l a t t e r set of terms 4^  vet-f CosisiecT^ort ^?^eo^s&/-/^s status?) a/j cos R« fe may be interpreted as a k + i o = ^k where b k i s the general c o e f f i c i e n t of the sine se r i e s . The mixed-series plug-board connection d i r e c t s the terms \B.Q\ to I^ QI to the U-multiplier only and the terms | b . | to / b 1 0 | to the V-m u l t i p l i e r only (the band-shift unit i s not used) and also adds, i n a separate amplifier, the U and V outputs. The result i s The r e s u l t i s v a l i d whether a i s a linear function of time or not and thus t h i s output can be displayed on a CRO with a linear time base or can be measured by a meter. As the operating adjustments on the computer are st r a i g h t -forward and the components l o g i c a l l y placed, only a short time should be required f o r anyone with basic knowledge of the problem involved to learn to operate the instrument. 1 0 1 0 b, s i n k a. 9.0 CONCLUSIONS The particular use to which polynomial computers may be put vary somewhat. Therefore, i t is difficult to compare their value on an absolute scale of merit. The prime purpose of a computer of this type is to find the zeros of a polynomial as quickly and accurately as possible. As a result of this investigation, the system has been designed so that the zeros of a polynomial with complex coefficients up to the tenth degree and with real coefficients up tot he twentieth degree can be visually located with two settings of coefficients on the computer. It is expected that each polynomial s etting will take less than 5 minutes and that each visual display will be complete within 30 seconds. Overall time to search the p_-plane will be approximately 10 minutes. Overall accuracy with automatic viewing should be better than 5$ and with manual detection better than 2$. Other forms of possible output such as (a) generation of Fourier series with up to 21 terms, (b) plots of W(p), |w(p).|, and the angle of W(p) with either or both of the polar coordinates of p_ varying, (c) generation of two tenth-degree real-coefficient polynomials simultaneously and provision for suitably combining and plotting their outputs, make this computer more versatile than comparable instru-ments. The formation of 21 terms with only 13 phase-shifters by a band-shift (the extended system) i s an o r i g i n a l c i r c u i t i n t h i s computer and gives i t almost twice the number of terms of any machine of equivalent size and complexity. The adding process, although not o r i g i n a l i n design, i s an important feature of the present system i n that i t reduces the number of adding stages and modulus potentiometers. The use of the m u l t i p l i e r s allows d i r e c t plots of W(p), i t s r e a l and imaginary parts, magnitude and angle. The generation of the modulus i n an active c i r c u i t allows cover-age of an area i n the j)-plane with a radius s l i g h t l y greater than unity. This makes possible d e f i n i t e locations of zeros lying on or near the unit c i r c l e . In the course of t h i s investigation, the system design has been f i n a l i z e d , the function T- generator mechanism b u i l t , and most of the essential equipment mounted i n the racks. F i n a l assembly, testing and extensions w i l l be considered part of a further project. It i s believed that this instrument w i l l prove extreme-ly useful for solution of polynomials and for Fourier synthesis, and with extensions, w i l l provide good v i s u a l displays for demonstration and study of system functions and various complex-plane problems. 55 BIBLIOGRAPHY Atkinson, C , "Polynomial Root Solving on the Electronic D i f f e r e n t i a l Analyser," Mathematical Tables and  Other Aids to Computation,, vol., 9 (Oct., 1955) 0 pp. 139-^143. Azaroff, L. V., "A One Dimensional Fourier Analog Computer," Review of S c i e n t i f i c Instruments, v o l . 25, No. 3 (May 1954), pp. 4 Y 1 - 4 7 Y . BeeverSo CA., "A Machine f o r the Rapid Summation of Fourier Series," Proc. Physics Society, vol.51 (1939), p. 660. Beevers, Q.A. and Macewan, D., "A Machine for the Rapid Summation of Fourier Series," Journal of S c i e n t i f i c  Instruments, v o l . 19 (1942), p. 150. Blakesley, T. H., "A Kinematic Method of Finding the Roots . Roots which are Less than Unity of a Rational Integral Equation of Any Degree Followed by a More General Method of Finding the Roots of Any Value," Philoso-phical Magazine, v o l . 23 (1912), pp.892-900. Borselino, A., "Un metodo e l e c t t r i c o por l a determinazione approssimata d e l l e r a d i c i r e a l i o complease die una equazine algebraica," Nuovo Cimento, (Feb., 1948), P«23. Brown, S.L., "A Mechanical Harmonic Synthesizer-Analyzer," Journal of the Franklin Institute, v o l . 228 (1939), pp. 675-694. Brown, S.L., and Wheeler, L.L., "Mechanical Method f o r Graphical Solution of Polynomials." Journal of the  Franklin Institute, vol.231 (1941), pp.223-243. Brown, S.L., and Packard, R.C, "Graphical Solution for the Real and Complex Roots of Polynomials with a Mechanical Harmonic Synthesizer," Physical Review, v o l . 73 (1948), p.648. Bubb, F. W., "A C i r c u i t f o r Generating Polynomials and Getting t h e i r Roots," Proc. I.R.E., v o l . 39 (Dec, 1951), pp.1556-1561. Buttler, C.C, and Rymer, T.B., "An E l e c t r i c C i r c u i t for Harmonic Analysis and Other Calculations," Philo- sophical Magazine, vol.35 (1944) pp.606-616. Calvert, J.F,, Johnston, H„Ro, and Singer, G.H., "Root- °° solver for lOth-Degree Algebraic Equations," Proc. N.E.C,, vol.4 (1948), p.254. Choudhury, A.K., "The Isograph - an Electronic Root Finder," The Indian Journal of Physics, vol.29 No. 10, (Oct., 1955), pp.468-473. Dietzold, R.L., "The Isograph, a Mechanical Root Finder," B e l l Laboratories' Record, vol.16 No.4 (Dec.1937) pp.130-134. Diprose, K.V., "Polynomial Equation Solver Isograph," Automatic and Manual Control, ed. Tustin, Butterworth S c i e n t i f i c Publications, London, 1952, p.541. Ferrara, G., and Nadeau, R.L., "A Complex Wave Synthesizer" E l e c t r i c a l Engineering, vol.70 (July,1951), p.585. Forte, W.G., Rose, G.A., and Willoughby, E.O., "Analogue Computer to Solve Polynomial Equations of n^h Degree with Real Coefficients, "Automatic and Manual Con- t r o l , ed. Tustin, Butterworth S c i e n t i f i c Publications, London, 1952, pp.541-546. Fox, T. S. "A Waveform Synthesizer," Engineering Digest. vol.3 (Mar., 1956), pp.44-47. Frame, J.S., "Machines for Solving Algebraic Equations," Mathematical Tables and Other Aids to Computation, vol.9 (Jan., 1945), pp.337 -3536 ! Furth, R., and Pringle, R.W., "A New Photoelectric Method of Fourier Synthesis and Analysis," Philosophical  Magazine, vol.35 (1944), pp.643-656. Hagg, Go, and Laurent, T., "A Machine for the Summation of Fourier Series," Journal of S c i e n t i f i c Instruments, vol.23 (1946), pp.155-158. Hart, H.C., and Travis, I., "Mechanical Solution of Algebraic Equations." Journal of the Franklin Institute, v o l . 225 (1935), pp.63-72. ~~ [ ~ Kelvin, "The Tide Gauge, T i d a l Harmonic Analy zer, and Tide Predicter," Mathematical and Physical Papers, vol.VT, Cambridge, Cambridge University Press, 1911. Kempe, A.B., "On the Solution of Equations by Mechanical Means," The Messenger of Mathematics, vol.2 (1873), pp.51-52. Kempner, A.J., "On the Separation and Computation of Com-plex Roots of Algebraic Equations," Univ. Colo. Studies, vol.16 (1927-29) pp. 75-87. 57 Krantz, F.W., "A Mechanical Synthesizer and Analyser," Journal of the Franklin Institute. v o l . 204 (1927), pp. 245-262. Layton, J.M., "A Polynomial Computer," Automatic and Man- ual Control.»d. Tustin, Butterworth S c i e n t i f i c Publications, London, 1952, p.546. Lofgren, L., "Analog Computer f o r Roots of Algebraic Equations," Proc. I.R.E. vol.41 No.7 (July, 1953), pp.907-911. Lucas, F., "Resolution Immediate des Equations au Moyen de L ' ^ l e c t r i c i t e , " C.R.Acad.Sci.. Paris, vol.106 (1888), pp.645-648. Lukaszewicz, L., "A Simplified Solution and New Applica-t i o n of an Analyzer of Algebraic Polynomials," B u l l .  Acad. Polon.Sci., vol.1 (1953), pp.103-107. Marshall, O.B., "The Electronic Isograph for Roots of Polynomials," Journal of Applied Physics,vol.21 ( A p r i l , 1950), pp.307-312. M i l l e r , D. C , "A 32-element Harmonic Synthesizer," Journal of the Franklin Institute,vol.181.(1916), pp.51-81. Murray, F.T. The Theory of Mathematical Machines, New Tork, King's Crown Press, 1948. Parker, G.M., and Williams, R.W., "A Magslip Isograph," Journal of S c i e n t i f i c Instruments,vol.32 No.9 (Sept.1955) pp.332-335. ; Robertson, J.H., "A Simple Harmonic Continuous Calculating Machine," Philosophical Magazine, vol.13(1932),p.413. Robertson, J.H., "A Simple Machine Capable of Fourier Synthesis Calculation," Journal of S c i e n t i f i c Instru- ments, vol.27 (1950), pp.276-278. Schooley, A.H., "An Electro-mechanical Method for Solving Equations," R.C.A.Review, vol.3 (1938), pp.86-96. Sommerville, J.M., "Harmonic Synthesizer for Demonstrating and Studying Complex Wave Forms." Journal of Scien-t i f i c Instruments, vol.21 (1944), p.174. 58 Soroka, ¥,W,, Analog Methods i n Computation and Simu-la t i o n , New York and London, McGraw-Hill, 1.954. Tait , P.G., and Thomson, W., Treatise on Natural Philosophy, Cambridge, England, The University Press, 1879, pp. 379-482. 

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