US2971698A - Function generating circuits requiring only linear elements - Google Patents

Description

Feb. 14, 1961 B. E. coWART ET AL 2,971,698

FUNCTION CENERAIINC CIRCUITS REQUIRINC ONLY LINEAR ELEMENTS Filed Maron 14, 1955 I 53,561 I ffl-E7; @ILD 2,971,698 Patented Feb. 14, 1961.

FUNCTION GENERATING CIRCUITS REQUIRING ONLY LINEAR ELEMENTS Brooks Ehrmon Cowart, Pacoima, Lloyd David Ball, Los Angeles, George Bruer Crane, Redondo Beach, and Orin Henry Knowlton, Jr., Los Angeles, Calif., assignors to Gilfillan Bros. Inc., Los Angeles, Calif., a corporation of vCalifornia Filed Mar. 14, 1955, Ser. No. 494,178

1 Claim. (Cl. 23S- 197) This invention relates -to function generating circuits requiring only linear elements and, more particularly, to such circuits for generating time-varying signals representing such functions as: hyperbolas; parabolas; and ellipses, only linear circuit elements being required, where these elements may Ialso be of a .passive nature such as inductors, capacitors, or resistors.

Many types of function generators for providing time varying functions are presently `available in the art, where nonlinear elements are employed. For example, diode limiter circuits may be utilized to provide a time varying output voltage Xo which varies in accordance with a time varying input voltage X1, in -a specified manner; the diode characteristic being selected to provide the desired functional transition frorn the input signal Xl to the output signal Xo. Suitable circuits of this type, for example, are shown on pages 273 through 278 of a bool; entitled Electronic Analog Computers by Korn and Korn published in 1952 by the McGraw-Hill Book Compan New York, Toronto and London.

Many other nonlinear elements have been employed such as, for example, thyrites' and ferrites. Moreover, vacuum tubes have been utilized to provide logarithmic functions by biasing the tube at a nonlinear characteristic portion which approximates the desired function.

It is apparent that in utilizing nonlinear elements to provide a desired time varying function, considerable care must be exercised to obtain speciic characteristics, As a result the elements specified are necessarily ycostly since they must conform with particular specifications.V

Furthermore, such carefully selected nonlinear elements may vary in characteristics due toy changes in operating conditions such as temperature or pressure. Thus an accurate function generator employing such` elements not only has an added cost factor due to the special characteristics required but also must include means for regulating such operating conditions as temperature and pressure to within certain prescribed boundaries.

A further limitation inherent in the prior art practice is the fact that frequently the nonlinear element required must be` a vacuum `tube or other active element, where the term active is utilized to indicate that the element requires activating power in order for it to function. As aA result such circuits require power to maintain the nonlinear elements in the proper operating position and are also subject to inaccuracies due to variation in the power supply utilized,

, Furthermore, it is apparent that such active elements cost considerablymore than passive elements such as resistors andcapacitors, for the same degree :of accuracy.

The present invention obviates the above and other disadvantages inherent in the prior art by providing a class of function generating circuits utilizing linear elements, which may all be of a passive nature. According to the basic concept of the invention, the particular function to be generated is first translated into a sum of vexponential functions where the exponential functions 2 may be delayed in time according to a. predetermined initial condition of the function desired.

A large class of time-varying functions may be simulated in this manner, where the degree of accuracy inherent in the simulation is a function of the number of linear elements employed providing a corresponding number of exponential Variables. The basic technique of the invention, for example, may be utilized to simulate a.

time-varying hyperbolic function such as: y=c/t. Since the simulating exponential functions cannot provide an initial condition of y=oo, the `sirmllation process must begin after a predetermined initial time interval which may be designated as ti. This hyperbolic function then may be approximated with a reasonable accuracy by. the combination of two exponential functions as follows:

Aeree-m L Bene-m gli.

von the other hand, has a small negative slope and therefore may be most accurately simulated by an exponential having a small exponential factor k2.

In this manner any function may be simulated by4 combining a plurality of exponentials, where the accuracy inherent in the simulation is a function of the4 number of points that the exponential's are selected toV pass through. The second basic step of the invention is t0 select a set of linear elements, which -rnay all bev passive, to providev the basic exponential functionerequired. Where two exponentials are speciiied, for example, two circuits including resistance and capacitance yelements may be utilized or two equivalent circuits in-v cluding resistance and linductance elements.V

After establishing the basic conguration of linear cir-V cuit elements which may provide the simulating eX- ponential functions, the transformation function for theV simulating circuit arrangement is formulated.V This ktransformation or transfer function may be obtained by wellknown Laplace-transform operational calculus.

The transformed function is then converted intoa time-varying function where the coefhcients and initial conditions speciiyingthe function remain as unl-:nowns in terms of the linear circuit elements employed; Suitable values for these linear circuit `elements are then determined by a simultaneous equation solution Vwhere-the coefcients of the desired simulating exponential function series are compared with the unknown coelicients of the time varying function. i

Many other time varying functions may be generated in this manner. A parabolic function, for example, may be generated througha combination of increasing exponential circuits where an initial shift in time may be introduced corresponding to the y=0 portion of the parabola. It should also be understood that although the actual circuit must function in a time domain, the time variable may nevertheless be considered to represent other independent variables. Thus the function y=]"(x) may be represented where x is considered to be equivalent to the independent variable of time.

lAccordingly, it is an object of the present invention to provide a class of function generating circuits which utilize only linear elements.

Another object of the invention is to provide a linear 'between the hyperbolic obviating the special charac- -teristics -arid accurately regulated temperature and pressure conditions.

A further object is to provide. a class'of function generating circuits which may utilize passive elements obviating the necessity for active elements providing the desired function which require power for activation, the power being necessarily accurately regulated to provide the desired function. l

Still another object of the invention is to provide a function generator which may utilize passive elements avoiding the inaccuracies resulting 1n operation where a well regulated power supply is required for activating active elements.

Yet a further obiec't is to provide an economical .class of function generating circuits where special nonlinear elements are not required nor particular operating conditions such as temperature or pressure.

A more specific object of the invention is to provide a class of functiongenerating circuits where a desired function is simulated by exponential functions, each exponential function being provided by a corresponding set of linear elements which may be passive elements such as inductors, capacitors or resistors. A

'Ihe novel features which are believed to be .characteristic of the invention, both as to its organization and method of operation, together with further objects and advantages thereof, will be better understood from the following description considered in connection with the accompanying drawings. -It is to be expressly understood, however, that the drawings are for the purpose of illustration and description only, and are not intended as a definition of Vthe limits of the invention.

Fig. l is a block diagram of a basic embodiment of the invention;

Fig. 2 is a schematic diagram of one species of the invention for providing av signal simulating a rectangular hyperbolic function;

Fig. 2a is a graph indicating the correspondence function to be simulated and the exponential functions utilized according to thepresent invention; and

Fig. v2b is an equivalent circuit for the embodiment of Fig. 2.

Reference is now made `to Fig. embodiment of the invention is illustrated in block diagram form. As indicated in Fig. l the basic embodiment comprises a plurality of exponential circuits indicated as providing the functions:

AgkiU-ti); Balou-ti); NekMt-ti) Each of the exponential circuits receives certain initial amplitude conditions, reprcsentedaby corresponding signails, and also initial time conditions in order to establish in the interval 5 seotSO sec.

Since many curve fitting techniques may be utilized to approximate this function in the interval between tL--S and t=50 it is not deemed necessary to show the specific l wherein the basicV 4 manner of deriving a simulating function. A suitable function is found to be:

It will be noted that this function is expressed in a form where the initial time condition t1=5 is effectively provided by failing to utilize the iirst 5-second portion of the function which starts from an amplitude 65. 'Ibis function may `also be expressed as follows:

where tS for simulation. 'I'lhis function may be achieved by introducing an actual time delay in initiating the operation of the exponential circuits.

This exponential simulating function Ymay be provided by the double RC exponential circuit shown in Fig. 2.

Referring now to Fig. 2, it is noted that the embodiment includes a first RC exponential circuit including a capacitor C1 :and a resistor R1. The capacitor C1 is initially charged to the voltage E applied thereto through normally closed relay contact Rta 1 forming part of a relay Rta. Y

Ari output signal is derived from the junction of C1 and R1 and -is applied to a second RC exponential circuit, including resistor R2 and capacitor C2, having one end connected to resistor R2 and the other end connected to ground. Capacitor C2 receives an initial charging voltage of 0 volt through a normally closed second contactl Rta 2 `of relay Rta. It will alsov be noted that a second relay Rtx is included, having 4a normally closed contact Rtx 1 completing connection between capacitor C1 and ascisse-621+.scie-wm) g resistor R-l in the first exponential circuit. When relayv will be understood that the particular potentials illus-- trated in Fig. 2 lare by no means intended as a limitation of the invention.

The operation of the circuit begins at time ta when relay Rta is actuated opening associated contacts Rt 1 and Rtarz. In the particular operation to be discussed below, it will be assumed that potential E is 65 volts and that vat time fa occurs at zero time as indicated in Fig. 2a. It will be understood, however, that other suitable times ta may be selected such as ta: l0 seconds 4at which timey the potentiall across capacitor C1 should be selected to be 17.5 volts as indicated in"Fig. 2a. 'Ilhe time condition ta then, corresponds to the general condition ti indicated in Fig; l and discussed-above as covering any of the possible initial time conditions. Y

After signal ta actuates relay Rta, opening the associated contacts, capacitor C1 begins to discharge through the parallel path of the exponential circuit including the resistor. R2 and capacitor C2 coupled in series and through resistor R1. -It will be Vshown in the detailed discussion which follows that the values of the components of these exponential circuits may be selected to simulate the desired hyperbolic function with a high degrec of accuracy during a considerable portion of the time interval. Thugs, in Fig. 2a, it will be noted that the function y=/ t is accurately simulatedby the summation of .two exponentials: y=56.l6ert/595 and y=8.84el=/621 in the interval Istarting from t approximately equal to 8 seconds through the time t=60 seconds.

In view of the continuous computation of the time varying function according the invention, the desired linal time condition may be selected at any point following the lirst point of reasonable accuracy. Thus, relay Rtx may be actuated at any time following to equal to approxi-V an output signal indicationrfor the value of the function at that time.

A more detailed explanation of a particular operation of the inventionas employed in Ian initial velocity computing circuit is found in copending application Variable Range Signal Generating Circuit With Means for Computing Initial Velocity by Lloyd David Ball et al., tiled March 7, 1955, Serial No. 492,482, now Patent No. 2,832,537; and Velocity Tracking System For Increasing The Range yof Acquisition of. Moving Targets by Ball et al., filed March 7, 1955, Serial No. 492,627. The present invention is shown in means 200 of Fig. 2` in the The Kirchoi law equations for the equivalent circuit shown in Fig. Zbmay be expressed as follows:

These equations may then be transformed according to..

EVS 0 Tin) range signal generating circuit application, Serial No. 492,482 and in Fig. 6 in means 610 `of the velocity tracking system covered in application Serial No. 492,627. In these applications, the present invention is employed to compute initial velocity and is actu-ated at time to to generate a hyperbolically varying signal defined as where Ad is the range difference between two points through which a target passes over the time interval At.

In `employing the present invention in the above particular application, relays Rto are actuated at the time the target whose initial velocity is to be computed passes a first range point. The circuit values of the hyperbolic function generating circuit of the invention are selected so that the output signal at all times represents r-ro Thus, as soon as the target is detected passing through a known range dilference point constituting the factor Ad relay Rrx of the present disclosure is actuated so that the output signal is directly equal to tx-to where tx-to is equal to At.

In this manner the invention provides a simple means of continuously generating a variable signal representing the initial Velocity of a target according to a hyperbolic function so that the instant the target passes through a second known range point an accurate signal may be read out.

In each of the above-ident-iied applications, it will be noted that the capacitor C1 is also employed as a feedback device for a chopper-stabilizer D.C. amplifier. In this manner, not only is the initial velocity computation erformed in :accordance with the desired hyperbolic function provided by the present invention, but also the final signal developed across capacitor C1 is incorporated into the integrating feedback path as an initial condition for a subsequent integration to form a varying velocity signal.

The equivalent circuit for the embodiment of Fig. 2 is shown in Fig. 2b, where it is assumed that the relay Ro is actuated, initiating the hyperbolic function generating operation, and that relay Rrx #has not been actuated.

The transform for the voltage ecl 4across C1 may then be expressed as follows:

` 7 (eel) :E tran-k, kras-162]? Performing thezinverse trans-form the genepaliform of" time functionvisfound to b e:

Suitable circuit constants their .bedeterrmined "byl comparing the simulatingfunction desired withv this general time function; for example:

l l l 1 l I-t will be noted that, since only three equations are provided :and there are four unknowns in terms of circuit elements, the solution which -is satisfactory is not unique. Therefore, `although precise solutions may be determined mathematical-1y, it is considered satisfactory for the purpose of the invention to establish the basic conguran'on of elements and to approximate reasonable circuit values therefor, precise values being then determined experi satisfactory'for providing the simulating ,exponentialy functions above: R1=6.8 megohms C1`=2 microfarads R2'=6.8 megohms' C2=4 microfarads The general technique of derivation in accordance with fthe present invention may now be summarized as follows.

(1) The Afunction to be simulated 'as a continuous time function is approximated by series of exponential which 'may be expressed 'as follows:

(2) A plurality of sets of linear elements are then provided, one set being included for each of the terms in the exponential series.

(3) The. basic circuit configuration is then transformed in terms of unknown element values to provide the transformation function:

A U(s) AL(s) where T(Eo) represents the transformed output signal, AU(s) represents the upper term in the transformation function and AL(s) represents the lower term, or root function which is equated to zero to determine the roots k1' and k2, etc. In the example above:

(5) Finally the values of the various circuit constants are obtained by equating the unknown coetlicients A and B, etc., to the corresponding terms derived from the transformation providing the relationship:

A Uta) TUT) From the foregoing description, it is apparent that the present invention provides ya class of functioning generating circuits requiring only linear elements, Where these elements may :also be of a passive nature Vsuch as inductors, capacitors and resistors.

While the invention has been speciically illustrated with respect to a resistor-capacitor exponential circuit, it will be understood that other exponential circuits may be utilized. Furthermore, it may be understood that, while the specific example illustrated relates to simulation of a hyperbola, other functions such as parabolas and ellipses may be simulated.

What is claimed is:

A hyperbolic function generator comprising: a iirst circuit including direct-current source of potential, a switch, and a resistor connected in series; a second circuit including a first capacitor connected in parallel with said resistor; a third circuit including a resistor and a second capacitor connected in series, said third circuit also being connected in parallel with said resistor; a switch connected in parallel with said second capacitor; and means for opening said switches simultaneously.

References Cited in the tile of this patent UNITED STATES PATENTS OTHER REFERENCES The Radio Amateurs Handbook, 1951 edition, published by the American Radio Relay League, West Hartford, Conn.` Page 247 relied on.

Electronic Analogue Integration and Differentiation (Brandon), Electronic Engineering, November 1953, pages 476 and 477.

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