Contains reports on four research projects

DeClaris, N.

Ho, Y. C.

Lee, Francis F.

Macrakis, M. S.

Scott, R. E.

Stern, T. E.

DSpace@MIT

XX. ANALOG COMPUTER RESEARCHProf. E. A. Guillemin N. DeClaris M. S. MacrakisProf. R. E. Scott Y. C. Ho T. E. SternF. F. LeeA. INTRODUCTIONThe Analog Computer group is engaged in a number of basic investigations in thefield of analog and special-purpose computers. The emphasis has been on the under-lying concepts and the introduction of new components rather than on the design, con-struction, or operation of a large computing facility.We are primarily concerned with analog computers, rather than digital computers,because more basic research seems to be needed in the analog field. There is greatneed for an increase in the speed and the accuracy of analog equipment.While some combined analog-digital equipment has been constructed, the emphasisis on special-purpose computers. The underlying philosophy of special-purpose com-puters is that they are cheap enough and simple enough to be used by the person who hasthe problem to be solved. They do not interpose the extra link of a large, trainedoperating staff between the problem and its ultimate solution.Since a great many of the special-purpose computers that have been built are directlyor indirectly related to network theory, our group has maintained an active interest innetwork theory and has done some work in this field.At the present time the research falls into three groups: (a) those problems relatingto computer systems and the basic philosophy of computers, (b) the design of new com-puting elements, and (c) applications of modern network theory.R. E. ScottB. COMPUTER SYSTEMS1. Energy and Power in Nonlinear SystemsA basic investigation is being conducted into energy and power relations in nonlinearsystems, such as those involved in computers. This investigation begins with a transi-tional stability study of second-order systems.In the design of electromechanical multipliers and function generators, variousschemes have been employed to extend their dynamic range. Among other things, satu-ration at high-power level represents a limitation to such efforts. This limitation issimilarly encountered in the related field of feedback control systems. The electro-mechanical device, which is stable in the linear sense, may exhibit relative instabilitywhen saturation occurs. Such operation involves the transition between linear and non-linear modes. A phase-plane study has been made of the transitional stability of twosecond-order electromechanical systems (Figs. XX-i, XX-2) that are frequently usedto form multipliers and function generators.-74-AMPI IFIFPFig. XX-1Excitation-limited motork k mkkT =+ f 1 + tachometer loop gain.Fig. XX-2Torque and rate-limited motorJT k kmkTam Ta = maximum acceleration =mmmaximum torqueinertia= maximum velocity.REGION OF LINEARSYNCHRONIZATION0 02 0.4 0.6 0.8 10 1.2 1.4t DAMPING FACTORFig. XX-3Transitional behavior of the system shown in Fig. XX-1.-75-rPUTMOTOR(XX. ANALOG COMPUTER RESEARCH)1412I08CU CREGION OF LINEARSYNCHRONIZATION0 02 04 06 08t DAMPING FACTORTransitional behaviorFig. XX-4of the system shown in Fig. XX-2.The result for the system of Fig. XX-1 is shown in Fig. XX-3. The system issaid to be linearly synchronized if the response to a step input of very large amplituderemains linear once it has left the saturated region. The result obtained by W. Hurewiczand N. B. Nichols (Servos with Torque Saturation: Part 1, Rad. Lab. Report 555, May1944) is indicated by a cross on the curve. The result for the system of Fig. XX-2 isshown in Fig. XX-4. A formula for the approximate number of nonlinear entries whenthe number n is large, is given by1 mn< <n+12 a T85 mF. F. Lee2. Investigations and Applications of the Theory of Signal Flow Graphsto Analog ComputersAn attempt has been made to indicate to what extent computer component errors(especially of a dynamic nature) affect the form of the equations of the original physicalproblem being solved. Instead of expressing the resultant computer errors as actualfunctions of the independent variable, the errors are expressed as modifications of theoriginal differential equations.Thus far, we have determined the theoretical limit on the number of dynamic com-ponents necessary to solve a system of simultaneous linear constant coefficient equations-76-10 12IY,(XX. ANALOG COMPUTER RESEARCH)on a computer. A systematic method of proceeding from the original equations to thefinal optimum computer connection is in the process of being completed.Given a set in the formL x 1+ L2X2 + ... + Lln x = SLn1 LnZX2 nn n 1nL x +L x+ ... + L x = Swhere L is a linear operator (that is, a polynomial in s), and S is a known functionpq qof the independent variable, a signal flow graph is drawn representing the relations asthey are written above, that is, with the unknown functions as sources and the knownfunctions as sinks. The graph then consists of open paths, each path being a singlebranch. To make this system physically realizable on a computer we must invert atleast n paths. The manner in which these paths are to be chosen depends largely uponthe characteristics of the computer being used. We shall assume that we are dealingwith a computer containing only integrators as dynamic elements. To utilize the inver-sions to the best advantage, only n paths of inversion will be chosen. These paths inter-cept all 2n nodes, that is, no two paths of inversion will intercept the same node. Theyare also chosen so that the product of their branch transmissions is as large as possible,to reduce the order of each individual branch operator. If paths of inversion other thanthose specified above are used, the resultant flow graph will present difficulties inphysical realization. However, these difficulties can usually be overcome by performingappropriate loop inversions until the configuration reverts to the one that would haveresulted from the correct initial choice of open path inversions.The theoretical minimum number of integrators necessary to solve a given set ofequations is equal to the order of the determinant of the operational coefficients. Thisis always either equal to, or less than, the sum of the orders of the highest derivativesof all the dependent variables. Therefore, the next step in the procedure is to determinethis minimum and compare it with the number of integrations present in the invertedflow graph. If they are equal, no further advantage in this direction may be obtainedby more modifications. If they are not equal, the number of integrators can usually bereduced to the minimum by such manipulations as the introduction of additional nodes,and the like. These further modifications, which do not change the fundamental relationsbetween the variables, may also be necessary in certain cases for reducing the order ofsome branch transmissions that may still not be realizable on the computer even afterthe best possible combination of branch inversions. These modifications should also beused to make as many of the transmissions negative as possible, since electronic com-puter components always have an inherent phase inversion.-77-(XX. ANALOG COMPUTER RESEARCH)This method has been tried on several examples and seems to have some advantagesover the conventional methods of setting up the computer.Additional investigation is to be performed on modification of the flow-graph con-figurations to reduce the sensitivity of the solutions to errors in branch transmis-sions.T. E. SternC. COMPUTER ELEMENTS1. Square-Law NetworkTwo networks, shown in Figs. XX-5 and XX-6, were constructed and tested duringthe past months. Initial tests showed that the network in Fig. XX-5 is superior to thatof Fig. XX-6 in both stability and accuracy.A push-pull stage was then constructed for the second network to enable it to acceptboth plus and minus signals. At present, the device can be represented by Fig. XX-7.This device uses four straight-line segments to approximate the square-law charac-teristics. It has about 1 percent static accuracy. With ac signals the accuracy dropsto 10. 5 percent due to an equal gain in the push-pull stages. Frequency response isunknown at the present time because many potentiometers have been used in the circuitto faciliate experimental adjustments and they decrease the upper frequency range. Atpresent we need a more stable push-pull driving stage that has a maximum output of 3ma into a 10-kilohm load and a better switching arrangement for the diodes in the net-work.IN e+2V +4V +6e 2  e2 34eDC e OUTAMPle OUTFig. XX-5Square-law network usingactive elements.Fig. XX-6Square-law network usingpassive elements only.-78-RfIN e(XX. ANALOG COMPUTER RESEARCH)PUSH SQUARE DCeIN PULL LAW AMP OUTFig. XX-7Complete square-law devicee. = input voltage1neout = output voltagek= proportionality constantein = ke 12 < k < 35.in outThe principal limitation of square-law devices with germanium diodes is the finiteforward and backward resistance. It is safe to say that unless elaborate compensationis used an accuracy of 0. 5 percent is the practical upper limit for a square-law deviceof this nature. Any attempt to go above the limit will sacrifice the simplicity of the net-work.Y. C. Ho2. Delay LinesThe delay lines and their associated equipment built by Dr. C. A. Stutt are beingrepaired and put into operation for experimental investigations of Dr. M. V. Cerrillo'stheory of network synthesis in the time domain.Y. C. HoD. APPLIED NETWORK THEORY1. Potential AnalogsDuring the preparation of a technical report from the work on potential analogs thematerial has been completely revised. It is true that an equation such as Eq. 2(Quarterly Progress Report, July 15, 1953, p. 101) is quite costly to deal with, and forthis reason more examples have not been worked out. In the revised material the mainideas and results remain the same but they are presented in a more precise and satis-factory form.M. S. Macrakis2. RC Filter Frequency TransformationsIn the techniques of modern network synthesis it is customary to satisfy the require-ments for a desired filter on a normalized lowpass prototype and then, through a-79-(XX. ANALOG COMPUTER RESEARCH)frequency transformation, to obtain the correct type of filter, highpass, or bandpass.It is the purpose of this study to investigate the applicability of frequency transformationin the case of a transfer function of RC character.a. Lowpass to highpass transformationConsider a transfer function Z 1 2 (s) which satisfies all the RC requirementsh (s 2 + A.s B i )Z12(s ) = C i=1 mr (s + ci)i=1n- (s + Pi)(S + i )(s + a )i=lwhere n = 2m as the upper limit and a. is real and positive.1To make the lowpass-to-highpass transformation let s - k/s. The new transferfunction isnS(k + pis)(k + Pi s )i=lZ 2z(s) = Z1L2 C, n = 2m(k + a.s)1nsP 1 (k +i:1m11 (kPis)(k + Pis)C, 2m = n + p+ ais)1sP (s + k/i)(s + k/P i )i=1(s + k/ai)] i 2Ci=mni ai=1On examining this equation it is seen that the poles and zeros of the new function areinverted about a circle of radius k. (See Fig. XX-8.) Since the RC character of thetransfer function is preserved, that is, all poles are simple and on the real axis, andn 4 2m, the transformation is still applicable.b. Lowpass to bandpass transformationFor this type of frequency transformation letsk s(-Yo-80-z12(s) =(XX. ANALOG COMPUTER RESEARCH)jq t/ \1\ /pp / ALOWPSS HIGHPASS LOWPASS BANDPASSFig. XX-8 Fig. XX-9where wo is the center frequency desired, and k is a bandwidth factor.width product is constant.We observe that the poles of the new transfer functionThe gain band-sW swill be solely determined by the transformation of the factor(s+ 0i) - s 21 W 0S(o 2Sa.k s+wi k 0oHowever, since the degree of the numerator is smaller than or equal to the degreeof the denominator it is sufficient to consider only the factor( 2 WO+ ..i T s + W = (s+y)(s+6)The necessary condition for y and 6 to be positive is2 2-L. W Ia.o1- 1 o2k - 2+4 k2 2iLo 21/22a.1k>12kSince k determines the bandwidth, it is clear that there is a low limit in thebandwidth& =of the bandpass network. This limit is determined by the pole a.i which is closest to the1-81-\iZ(B)s) = z12 " 12% j(XX. ANALOG COMPUTER RESEARCH)Fig. XX-10origin in the lowpass structure. This constitutes the only restriction on the transfor-mation from a lowpass RC structure to another bandpass. It is obvious that forZ 1 2 (0) = z2s = +j 0 , therefore wo is the center frequency and forZ 2 (+ j) = z (B)s = + +-k - o z2k/Hence the bandwidth AW isI1AW = okThe minimum Q of the RC network therefore becomesAW 1 Cmink- 2If the roots of the equation are examined it is seen2C1L2k2k2oor2y- 6 =oTherefore yi and 6 i constitute inverse pairs about the periphery of a circle from the-82-W --[k( o s(XX. ANALOG COMPUTER RESEARCH)origin with radius ow (See Fig. XX-9.) From Fig. XX-9 the following relationZ1 (s) = Z12(s) Z Z(s)is obvious for the case of o = k = 1. This suggests an alternative procedure of super-imposing the magnitude characteristic of a lowpass and a properly derived highpassfilter as in Fig. XX-10.N. DeClaris3. Duality of Ideal TransformersGiven an electrical circuit containing R's, L's, C's, and sources, there is a tech-nique that enables one to interchange voltage with current and junction points withmeshes by merely topological considerations. The following theorem extends the appli-cation of this technique to circuits containing ideal transformers. The significance ofthe theorem is obvious, since one has to recognize the fact that ideal transformers arevery frequently used in circuit analysis.Theorem: The dual of an n-winding ideal transformer with Nk turns in each windingconsists of n two-winding ideal transformers of 1:N k turns ratio each, and with thesecondary windings connected in series; the secondary voltages are taken in the sameclockwise (or counterclockwise) direction.Proof: Consider an ideal transformer with n-windings. (See Fig. XX-11ii.) Eachof the windings has N k number of turns. Let the voltage polarities and current flowdirections be assumed to be those shown in Fig. XX- 11. Since the system is lossless,the instantaneous power is zero.nSVkIk = 0 (1)k=0Furthermore, by definitionV 1 V 2  V k1 = constant (2)N1 N2 N kDivide Eq. 1 by Eq. 2.nV Z V k 0k Vk k =k=0SinceN- = constantVknZ NkIk = 0 (3)k=0It is obvious that Eqs. 2 and 3 describe the system completely.-83-(XX. ANALOG COMPUTER RESEARCH)*MkE k,0-Fig. XX-11In order to obtain this dual setThus we haveI'N1Fig. XX-12of equations, it is sufficient to interchange Ik and V k.2 kN N - constantN2 N knNkV k = 0(4)(5)Obviously the power equation has not been alterednZ Vk k = 0k=0Therefore the system defined by Eqs. 4 and 5 constitutes the dual of the circuit shownin Fig. XX-12. In order to recognize the system more easily, let1Nk M kEquations 4 and 5 are rewritten asMII = M2 = MkI k = constantnVkZ0 kk=0(6)(7)The circuit of Fig. XX-12 corresponds to Eqs. 6 and 7, as can be easily verified.For a general orientation on this topic the reader is referred to E. A. Guillemin,Communication Networks, Vol. II, chap. V, John Wiley and Sons, Inc., New York,1935 and E. C. Cherry, "The Duality Between Interlinked Electric and Magnetic Circuitsand the Formation of Transformer Equivalent Circuits," Proc. Phys. Soc. (London)B 62, 101, 1949.N. DeClaris-84-NN22 21 LV2 JNkAd I I

Analog Computer Research

http://dspace.mit.edu/bitstream/1721.1/51163/1/RLE_QPR_032_XX.pdf

Analog Computer Research

Abstract

Similar works

Full text

Available Versions

DSpace@MIT