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Publication numberUS3500445 A
Publication typeGrant
Publication dateMar 10, 1970
Filing dateAug 27, 1965
Priority dateAug 27, 1965
Also published asDE1524322A1
Publication numberUS 3500445 A, US 3500445A, US-A-3500445, US3500445 A, US3500445A
InventorsCollings Jerry M
Original AssigneeZeltex Inc
Export CitationBiBTeX, EndNote, RefMan
External Links: USPTO, USPTO Assignment, Espacenet
Apparatus and method for producing square-law function
US 3500445 A
Abstract  available in
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Claims  available in
Description  (OCR text may contain errors)

March l0, 1970 APPARATUS AND METHODFO PRODUCING SQUARE-LAW FUNCTION Filed Aug. 27. 1965 4 Sheets-Sheet 1 Jerry M. Collings Attorneys March 10, 1970 J. M.- coLLlNGs APPARATUS AND METHOD FOR `PRODUCING SQUARE-LAW FUNCTION- Filed Aug. 2v. i965 4 Sheets-Sheet 2 2G Fig. 2d

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BY Jerry M. Collings atm, WL 0( u Attorneys March 10, 1970 .1.M. coLLlNGs 3,500,445

I APPARATUS AND METHOD FOR PRODUCING 4SQUARE-LAW FUNCTION Filed Aug. 27. 196s 4 sheets-sheet s ERROR CURVE ,107" Fig. 4a

Jerry M. Collings BY WI M ab *01km .La Attorneys J'. M, CoLLlNGs 3,500,445

APPARATUS AND METHOD FOR PRODUCING SQUARE-LAW FUNCTION Filed Aug. 27. 1965 4 sheets-sheet 4 March 10, 1970 .....udvhw www.

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United States Patent O 3,500,445 APPARATUS AND METHOD FOR PRODUCING SQUARE-LAW FUNCTION Jerry M. Collings, Concord, Calif., assignor to Zeltex, Inc., a corporation of California Filed Aug. 27, 1965, Ser. No. 483,180 Int. Cl. G06f 15/34 U.S. Cl. 23S-197 13 Claims ABSTRACT OF THE DISCLOSURE In an analogmultiplier, method and apparatus are disclosed for generating a voltage output signal having a segmental form approximating a parabolic of squared function of an input voltage signal. The input signal is shaped by a series of non-linear electrical conversion networks into a plurality of half wave triangular Voltage signals having pre-selected frequency, amplitude and symmetry with the input voltage and thereafter all of the triangular voltage signals are electrically summated to produce the desired segmental parabolic or squared output signal.

The invention relates to electronic analog multipliers of a quarter-square type and more particularly to that portion of the apparatus which performs the squaring operation. An analog device as here used represents the output as a voltage.

The quarter-square type analog multiplier has as its basis of operation the algebraic relationship:

The basic scheme of the quarter-square multiplier is shown in FIGURE l of the drawings. The principal parts of this type of multiplier are (l) one each summing and difference input networks for two-phase input signals which form the quantities a(xly) and y-a(x-y); (2) one each positive and negative absolute value networks which select only positive values of a(x-iy) and negative values of -a(xy); (3) one each positive and negative square law function generators to form the quantities b(xl-y)2 and -b(x-y)2; and (4) an output summing unit which forms the desired output b(x|y)2-b(x-y)2=cxy.

Existing quarter-square multipliers generally employ passive diode function generators to perform the squaring operation. The major disadvantages of this-method are temperature sensitivity, cumulative error caused by leakage current in the many diodes needed to'achieve high accuracy, and the ditliculty in adjusting the networks within the diode function generators to obtain the desired overall output curve characteristics because of the uncertainty of the individual diode switching characteristics.

An objectof the present invention is to provide an active-type squaring network which to a very large degree eliminates the various errors associated with the diodes in the passive type squaring network and thus to obtain greater precision, accuracy and dependability.

Another object of the present invention is to provide an apparatus and method for producing the desired square-law function having the improved results above and to do so at a reasonable cost.

A further object of the present invention is to provide an apparatus and method for producing a square-law function of the character above and which will square any signal either DC or AC and do it substantially instantaneously whereby the apparatus .will be able to constantly present the square of an input voltage which may vary rapidly with time; and which will automatically convert a negative input voltage to a positive squared output voltage in accordance with the algebraic principle that the square of a negative quantity is always a positive quantity.

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The invention possesses other objects and features of advantage, some of which of the foregoing will be set forth in the following description of the preferred form of the invention which is illustrated in the drawings accompanying and forming part of this specification. It is to be understood, however, that variations in the showing made by the said drawings and description may be adopted within the scope of the invention as set forth in the claims.

Referring to said drawings:

FIGURE 1 is a schematic block diagram of a quartersquare analog multiplier.

FIGURE 2a is a graph showing one of the half-wave rectied triangular wave functions used in the present method and apparatus.

FIGURE 2b is a graph showing another of the triangular wave functions.

FIGURE 2c is a graph of another of the triangular wave functions.

FIGURE 2d is a graph of another of the triangular wave functions.

FIGURE 2e is a graph showing a segmental approximation of a parabolic function derived by summating the functions shown in FIGURES 2a, 2b, 2c and 2d.

FIGURE 3 is a graph showing an error curve function used in the present apparatus and method.

FIGURE 4 is a schematic block diagram showing essential portions of the apparatus and method used for generating and summating the particular wave form functions used in the present invention.

FIGURE 5 is a schematic diagram of a quarter-square electronic analog multiplier constructed in accordance with the present invention.

The analog multiplier is a device in which a voltage x and a voltage y are applied as inputs and from which a voltage cxy appears as output. Commonly these are volt multipliers having a scale factor c=-0.01. It is usual for each x and y to vary as functions of time; and an important characteristic of a present multiplier is its ability to multiply faithfully when the x and the y inputs change rapidly in value.

With reference to FIGURE l the two input voltages x and y are applied to summing networks 11 and 12 and difference networks 13 and 14. Voltages representing minus x and minus y must be provided from a source external to the amplifier and these are likewise applied to the networks of 12, 13 and 14 as illustrated. The outputs of these networks as noted on FIGURE l are applied to absolute value networks 16 and 17 which provide outputs of a|x+y| and -a|x-`y|. These two outputs are then fed into squaring circuits 18 and 19, the outputs from which are represented by the products b(x-ly)2 and-b(xy)2. The latter outputs are then fed into a summing network 20 the output of which is the desired product cxy in accordance with well understood algebraic addition.

The present invention is directed to the squaring networks 18 and 19. In the quarter-square analog multipliel the input voltages to these squaring networks are alx+y| and -a|x-y|. For simplicity in the description that follows only one squaring circuit will be described and the input signal will be identied as S; it being noted that reversed diodes in the lower squaring circuit 19 retain the negative sign of the signal being squared in that circuit. The generation of kS2 will now be discussed.

In a squaring network a'n input voltage is converted to an output voltage which is the square of an input voltage multiplied by an appropriate constant. Ideally the relationship starts at zero so that if the input voltage is zero the output voltage is likewise zero so that a direct reading output is obtained without adjustment. If this curve were plotted, say on the face of an oscilloscope, with Em as the abscissa and Eout as the ordinate, the

curve would appear. as a parabola. In a larger sense, therefore, the function of the present invention is to provide electronically a parabolic function; and this is accomplished in accordance with the present invention by summating a plurality of symmetrical half wave rectified triangular wave functions, shown in FIGURE 2 as S1, S2, S3 and S1, having a series relationship in which the successive functions have a frequency progression of 2j and with the base break point of each triangular function coinciding with the peak of the next higher order series function. If we let p equal the number of these triangular waves as illustrated in FIGURES 2a, 2b, 2c and 2d, the summating of these triangular waves will produce a segmental curve as seen in FIGURE 2e having 2P segments or break points. Finally in accordance with the present invention the successive peak magnitudes of the several triangular wave functions follow a relationship producing a change in slope in successive segments wherein the change in slope is a constant. The result is a segmental approximation of a parabolic function.

The foregoing may -be seen graphically by an examination of FIGURES 2a, 2b, 2c, 2d and 2e. With reference to FIGURE 2a, it will be seen that the function S4 is plotted as the ordinate and 11 as the abscissa and the latter has been subdivided for convenience into sixteen units. S1 appears at 11:1 as the beginning of an inclined ramp which extends to 11:2 and then breaks downwardly to zero at 11:3. In accordance with the rectified portion of the definition of this function, no signal appears between 11=3 to n:5. At 11:5 the function repeats as above and similarly repeats at 11:9 and 11:13. In other words if this were a full wave function there would be a negative portion of the curve appearing between 11:3-5, 11:7-9, and 11:11-13. Because of the rectification, this negative portion of the function is removed. The peak amplitude of the function S1 is set for purpose of this illustration at 1.

With reference to FIGURE 2b it will be seen that a similar symmetrical half wave rectified triangular wave function S3 starts at zero value at 11:2 and rises as a straight ramp to 11:4 then extends downwardly to zero at n=6; and again repeats at 11:10, with no signal appearing between 11:0-2, 11:6-10, and 11:14-16. S3 represents the next lower order function of the series relationship and accordingly there is a frequency progression between S3 and S4 of 2, that is in the present illustration S3 has two half cycles and S4 has four half cycles. It will also be seen that the base points viz 11:2, 6, 10 andl 14 of function S3 coincide with the peaks of the next higher order series function S1. The peak magnitudes of function S3 is set at 6.

With reference to FIGURE 2c it will be seen that the function S2 is zero from n=0 to 11:4, then rises on a ramp to 11:8 and decreases in a similar ramp to zero at n1:12 and remains at zero for the balance of the abscissa n=12 to 111:16. This function S2 follows the relationship above defined in that its frequency is one-half of the next higher order function S3; and its base break points 4 and 12 coincide with the peaks of S3. The peak of function vS2 is set at 28.

With reference to FIGURE 2d it will be noted that function S1 is zero from n:0 to 11:8 then rises as a straight ramp to n:16. If the curve were continued the ramp would then start down so that function S1 in common with functions S2, S3 and S4 is a symmetrical half wave rectified triangular wave function, only a portion of which is here used. It will also be observed that the frequency of S1 is one-half of S2 and the base break point of S1, viz., 11:8 is located at the peak of the next higher order series function S2. The peak amplitude of |51 is adjusted to 120.

As will be apparent from the foregoing, the slope of the successive curves S1-S4 decreases in the following order: 15, 7, 3 and 1. Thus viewing the slope progression in a reversed fashion, that is, reading from the higher to the lower order functions, the slope of each successive wave form increases by powers of two. The summation of functions S1 plus S2 plus S3 plus S1 appears in FIG- URE 2e. The relationship of these figures permits a graphic addition. From 11:0 to 11:1 all functions are Zero and hence the summation shown in FIGURE 2e as segment 21 is likewise Zero. From 11:1 to 11:2 the only function appearing is S4 which has a slope of 1. Hence the next segment 22 of the summated function S is 1. Between 11:2 to 11:3 S4 subtracts its slope of 1 from S3 whose slope is 3 so that the next segment 23 has a slope of 2. Between 11:3 to 11:4 only function S3 appears and accordingly the corresponding segment 24 will have a slope of 3. Between 11:4 to 11:5, function S3 subtracts from function S2 so that the resulting segment 25 has a slope of 7-3:4. From 11:5 to 11:6, function S4 is additive so that the resulting segment 26- has a slope of 7-3+1:5. From 11:6 to 11:7 function S1 subtracts from function S2 so that the resulting segment 27 has a slope of 7-1:6. From 111:7 to 11:8 only function S2 is involved so that the corresponding segment 28 will have a slope of 7. Between 11:8 and 11:9 function S2 subtracts from function S1 so that the resulting segment 29 has a slope of 15-7:8. Between 11:9 and 111:10 function S4 is added in so that the resulting segment 30 has a slope of 15-7-}1:9. Between 11:10 and 11:11 function S1 subtracts while function S3 adds so that the slope of the resulting segment 31 is 15-7-|3-1:10. Between 11:11 and 11:12 function S1 is at zero so that the resulting segment 32 has a slope of 15-7-}-3:11. Between 11:12 and 11:13 function S3 subtracts from function S1 so that the resulting segment 33 has a slope of 15-3:12. Between 11:13 and 11:14 function S4 is added so that the resulting segment 34 has a slope of 15-3-|-1:13. Between 111:14 and 11:15 Ifunction S4 is subtracted from function S1 so that the resulting segment 35 has a slope of 15-1:14. Between 11:15 and 11:16 only function S1 is involved so that the final segment 36 has a slope of 15.

The successive peaks or break points of the segments 21-36 will fall as seen in FIGURE 2e at 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 79, 91, 105, and 120.

For convenience in understanding the summation of 4a series of symmetrical half wave rectified triangular wave functions as above explained the analysis of each segment 21-36 is presented in the following table:

Summated Individual Slopes m- Peak 11 Segment S1=15 Sg=7 S3=3 S4=1 Slope Value Ko, MKO

2 2D Where the first three terms represent the amplitudes of S1020), 52(1'21), S3(j=2) respectively and the last term is a general formula defining the amplitude of the S(j-|1)th triangular function for values of and where: l

The present invention accordingly may be used whenever it is desired to generate a segmental approximation of a parabolic function having the generalized formula of AxZ-i-Bx-i-C. Where the invention is used for a quarter square multiplier a further refinement is desired viz. the elimination of the linear term Bx in the generalized formula or the teun n in the formula of the curve shown in FIGURE 2e. This may be done by shifting the base break points of the several triangular wave functions S1, S2, S3 and S1 one-half of a unit of n to the left as of their positions seen in FIGURES 2a-2e.

In FIGURES 2a-2e, the horizontal coordinate or abscissa is represented by the variable n which in the present apparatus is the scaled input voltage. Once the number of triangular wave for-ms is selected and the number of segments or break points thus established, the

units on n may -be considered as the full-scale input voltage for which the system is designed, divided by the number of segments. In the illustration of FIGURES 2a-2e, four triangular wave forms are used thus producing a segmental parabolic curve having 24 or 16 segments. It is convenient to consider nmx to be the amount of input voltage which would correspond with the upper limit of the parabolic curve and which would be the maximum input voltage for which the system is designed. Conveniently, the input voltage range may be set at 100 volts and accordingly each unit of n in the illustration given would be 100 divided by 16.

It can be shown that by the step of shifting all of the triangular waves to the left by an amount of one-half unit, by adding a constant bias of minus one-half unit to the input signal n, the function becomes S=1/2 (n2-1A as illustrated in dashed lines in FIGURE 2e.

Summarizing the foregoing and restating portions of the definition in some alternate terms for clarity it can be stated that the straight-line approximation of the parabolic function illustrated in FIGURE 2e is generated as the sum of a series of half-wave symmetrical triangular wave functions S1, S2, S3, S4, etc., having the following relationships:

(l) The number of cycles of each waveform is 2P-2 where p is the order of the waveform in the series.

(2) The horizontal shift from zero of each successive waveform, relative to the maximum span of the variable n, is 11mm/2P (3) The slope of each successive waveform from the highest order down to S1 changes according to increasing powers of 2, forming a sequence proportional to 1:3:7:15 :3l: etc. The relationship of the slopes is determined by the requirements that:

(a) the sum of the slopes at any Value of n must be equal to the slope of the curve of the desired output function at the same value of n, and

(b) the slope of the function 1/2(n2-n) increases by a constant amount for each successive unit increment of n. In the straight-line approximation of the function, the significant unit increment of n is the constant value corresponding to the distance between breakpoints along the output curve which occur at the junctions of the straight-line segments.

Most importantly, in the foregoing it will be observed that the number of segments in the parabolic function, and hence the accuracy of the system, is greatly multiplied over the number of basic circuits or triangular wave forms used. Generally for p rectifying circuits providing p number of symmetrical half-wave rectified triangular wave functions, 2p segments in the parabolic function is obtained.

As hereinabove noted, the accuracy of the square-law curve simulated by the foregoing method increases with the number of straight-line segments comprising the curve which in turn is dependent upon the number of component wave forms. Since the number of segments is equal to 2p where p is the number of component wave forms, the method and apparatus of the present invention enables the production of a curve approximating a parabola to any degree of accuracy desired. From a practical standpoint 64 segments approach an optimum result since such a curve has a fine granular error which is within the tolerances of the electrical components used to generate the wave forms. For example the theoretical error with 64 sections is approximately 0.003 volt out of volts full scale, which is 0.003 percent. In most analog devices, 0.01 percent is usually considered very good.

The generation of the triangular Wave function is not as simple as doing a similar type job in a straight passive manner because as will be more fully hereinafter shown, it requires a number of feed-forward amplifiers and sum- 'ming resistors for the several triangular wave functions.

Assuming it is desired to generate the 64-section parabola, it would take six of these associated circuits. The last two or three of these circuits have quite a number of input resistors because all of the previous units sum into the later sections. Accordingly, it is a feature of the present invention that the last several triangular wave functions are replaced by a single special error correction wave function which may be obtained by a passive diode network. The present apparatus and method thus combines a basic output curve constructed of a relatively low number of triangular wave forms with an error correction curve to produce a final output curve consisting of a relatively large number of segments.

Examination of wave functions S1, S2, S3 and S4 in FIG- URES 2a-2d shows that each successive wave form provides a correction to the function approximation generated by the sum of preceding wave forms. The true correction curve at any stage of the progression is equal to the difference of the continuous square-law function and the sum of the progression of triangular wave forms. This difference is itself a repetitive parabolic function.

As a feature of the present invetnion I have found that a satisfactory error curve may be generated by the fourth triangular Wave function S4. To do so, wave form 54 is generated as a full wave as shown in solid line 42, in FIG- URE 3 half Wave rather than half wave, as is fed into a passive diode network to obtain the parabolic error curve 41.

Method of generating component triangular wave forms A series of half-Wave triangular wave forms which meets the requirements of the discussion relating to FIGURE 2 can be generated by the method illustrated in FIGURE 4. The first wave form of the series, S01 is generated by a signal summing-inverting-rectifying device N1. This device produces an output signal which is proportional to the input signal Sin, representing the variable n. N1 may be defined as a non-linear function generating electric circuit having input and output relationships as follows:

if input S 0, output=CS if input SSO, output=0 where C is a constant. A horizontal offset from the original signal shown in the box N1, is depicted in FIGURE 4a and is determined by the value of input bias signal SB1. This bias signal (opposite in polarity to Sm) holds the output of the device N1 at zero -because of the rectifying characteristic until the absolute value of input signal exceeds it. The slope of the inverted output wave form S01 is controlled by the proportionality or gain characteristic of the device N1.

A second summing-rectifying-inverting device N2 is identical to N1 except that (l) the bias signal SBZ is half the value of SBI, (2) the slope of the output wave form S02 is set at half the value of the slope of S01, and (3) the inverted output S01 is added to the summing input 43 of device N2. The resultant wave form is shown in FIG- URE 4b.

With reference to FIGURE 4, it will be seen that signal S111 is fed to both the summing inputs 44 and 43 for circuits N1 and N2; and the output S01 is fed forwardly by conductor 46 to summing input 43 for circuit N2. Thus summing input 44 for the rst circuit N1 has fed to it only the input signal S1n and the bias signal SBI. On the other hand summing input 43 for the second device N2 has fed to it not only the input signal S111 and its bias SB2, but also the output S01 of device N1. As a result the output wave form, shown in FIGURE 4b, has a horizontal offset controlled by bias SB2 and thereafter increases proportionately to the input S111 at one-half the rate of slope of the output S01 up to the value represented by bias SB1, at which point it decreases at the same rate because of the subtraction of output S01 at the input 43.

The next wave form S03 shown in FIGURE 4c is generated by device N3, and it will be observed that the summing input 47 for device N3 is connected to all of the preceding signals viz. Sin, S01, and S02 as Well as its own bias signal SB3. The derivation of output signal S04 shown in FIGURE 4d is obtained in a similar manner from device N4. The summing input 48 for device. N4 is connected to all of the preceding signals, viz. S111, S01, S02, and S03 as Well as its own bias signal SB4.

As will be observed from the foregoing there is thus derived a series of symmetrical half-wave rectified triangular wave functions having the series relationship hereinabove defined. With reference to FIGURES 2a-e, it will be noted that the base break points of the several wave functions progress l, 2, 4, 8, and that this indexing of the several functions is controlled by the feed forward and bias method depicted in FIGURE 4. The outputs S01, S02, S03 and S04 of the several devices, N1, N2, N3 and N4 may be fed to a summing output 57 to provide a segmental approximation of a parabolic (square-law) function; or device N4 may be adjusted to provide a full-wave output which then may be converted into an error curve function and then summated with outputs S01, S02 and S03 as hereinabove described.

In accordance with the present invention the several devices N1, N2, N3 and N4 each inclu-de an electronic network centering about an operational amplifier (see operational amplifiers 1, 2, 3, 4, 5, 6, 7 and 8 in FIGUR-E An operational amplifier is a direct coupled high gain amplifier of huge negative gain as indicated by the minus infinity sign inside the standard block envelope for the amplifier. Also characteristic of the use of such an amplifier because of its infinite gain, is of the large amount of feedback from output to input which normally is so large that the behavior of the circuit is described by the components feeding the amplifier and placed around it to provide the feedback. With reference to amplifier 1 in FIG- URE 5 it will be noted that resistors R1, R2 and R3 are connected in the input to the amplifier and resistor R25 is connected in the feedback path. Also characteristic of the operational amplifier, the input is held at a virtual ground since even a very small input voltage will drive the amplifier to saturation. Accordingly, the gain is a function of the ratio of the input and feedback resistors. If these are made equal the gain is minus l and if the ratio is changed to say 10, the. gain would be minus 10. While the ideal operational amplifier has a gain of minus infinity, a commercial operational amplifier may have a gain of about 100,000 which for practical purposes is equivalent of infinite. For example, in such an amplifier one millivolt input will cause a full swing volt output. Since operational amplifiers are well understood in the art further vdetails of construction are not required to be given here and the usual block diagram as used in FIGURE 5 will sufiice.

summarizing in part the foregoing and with reference also to FIGURES 4 and 5 the input signal S111 for amplifier 1, see FIGURE 5, is appiied to resistor R1 and to the input terminal S6. The output terminal 57 of the amplifier is connected to the input terminal 56 by a feedback loop including diode CR1 and resistor 2S, the diode CR1 being connected to pass current only when the amplifier output is positive. Importantly also there is connected between the input and output terminals 56 and 57 a second feedback loop including diode CR2 which is connected to pass current only when the output terminal 57 is negative and block feedback when the output voltage at 57 is positive. The output terminal for the network is taken at point 58 in the first mentioned feedback loop between diode CR1 and resistor R25. Output voltage S01 (with reference also to FIGURE 4) appears at this point.

The operational amplifier and its network thus has (a) a conducting mode; and (b) a non-conducting mode. In the conducting mode:

In the non-conducting mode:

Accordingly, when the input voltage S1n increases negatively, the output voltage S01 appears as a fiat ramp as seen in FIGURES 4a and 2d; and when the input voltage Sm is positive, the output voltage S01 is zero, thus accomplishing the half-wave rectification hereinabove discussed.

With reference to FIGURE 5 and to operational amplifier 1, the linear ramp output S01 is shifted by reference voltage Sbl. As will 4be observed a constant positive 100 volt reference voltage is provided at the terminal 61 of a voltage divider circuit including series connected resistors R19, R21, R22, R23, R24 and ground 62. For purposes of present illustration the bias voltage SBI is set at +32 volts as derived from terminal 63 in the voltage divider circuit and resistor R2. Consequently, the diodes in the first circuit constrain the output S01 to zero except when the input signal is greater than 32 volts. In other words so long as the input signal Sin is positive no output voltage will appear. As S111 goes negative no output voltage will appear until S111 reaches and exceeds minus 32 volts. At that point the lineal ramp output S01 of FIGURES 2d and 4a will commence. As an important feature of the present invention this break point or corner is very sharp-much sharper than can be obtained with passive diode networks.

Operational amplifier 2 and its surrounding network is essentially similar to that described in connection with operational amplifier 1 with the following important changes:

(a) The bias voltage SBZ is reduced one-half so that the first break point of the triangular wave to be formed will be located at 11:4 instead of 11:8 as illustrated in FIGURES 2c and 2d; and

(b) The signal input and feedback resistors R4 and R26 and the feed forward input resistor R5 are adjusted to provide relative slopes of S1n=+l and S01=-2.

As a result of the foregoing adjustments output signal S07I will remain at zero as the input signal goes negative to 16 volts and then will increase as a lineal ramp until the input signal reaches minus 32 volts. At that point the output signal S01 of the first circuit appears also at the input terminal 64 of the second circuit by reason of a forward connection made by conductor 66 which leads from output terminal 58 of the first circuit forwardly through resistor R to the input 64 of the second circuit. Resistor R5 adjusts the gain of output S01 to two times the gain of the second circuit. From this point on the voltage fed forward from the first circuit will increase at twice the rate as the voltage S111 applied to the second circuit and at opposite polarity because of the reversal in the polarity produced lby operational amplifier 1. Accordingly, as the signal reaches minus 32 volts, corresponding with 11:8 in FIGURE 2, the ramp turns down as seen at 67 in FIGURE 2c. When the output S02 has been driven down to zero by the forward fed signal S01, diode CR3 holds the output at zero to the end of the scale, that is, between 11:12 and 11:16 as shown in FIGURES 2c.

Operational amplifier 3 and its network again is essentially similar to that shown in circuits 1 and 2 with the following modication: the bias voltage SB3 is again reduced one-half to eight volts so that the ramp of the output signal S02 of circuit number 3 will commence at 11:2 instead of 11:4 as shown in FIGURES 2b and 2c.

The output signals S01 and S02 of lthe first two circuits are both fed forward by conductors A66 and 68 and are applied through resistors R9 and R10 respectively to the input terminal 69 of operational amplifier 3.

As a result of the foregoing signal S03 will be zero as input signal S111 changes from zero to minus 8 (corresponding with n=2 in FIGURE 2b) and S03 will start up` as a ramp as the input signal S111 goes from minus 8 to minus 16 (corresponding with 11:4 in FIGURE 2b). At this point the output signal S02 from the second circuit is received at the input 69 of the third circuit and, as above explained, signal S02 increases at twice the rate and at reverse polarity as compared to the signal S111 of the third circuit. Accordingly, when the input signal S reaches minus 16 volts (the bias of the second circuit and corresponding with 11:4) the output ramp S03 turns downwardly as seen at 71 in FIGURE 2b until it is driven to zero by signal S02 and it will remain at zero as signal S02 continues to increase to a point corresponding with 11:8. At this point, signal S01 is impressed at the input terminal 69 of the third circuit through R9 with a relative slope of +2 and signal S02 reverses direction to a relative slope of 2, so that the net slope of these two feedforward signals is zero. The slope of the output signal S03 at point 11:8, therefore, is determined solely by the input signal S111 with a relative slope of -1. The output S02 remains clamped at zero until the net sum of all three input voltages reaches zero potential at n=10. At this point, the output becomes positive and begins to rise positively from zero to a second peak value at 11:12 as S111 increases. At 11:12, the feedforward voltage S02 reaches Zero so that the net input voltage, consisting of S111 with a relative slope of -l and the feedforward voltage S01 with a relative slope of +2, causes the output voltage to reverse direction with a negative slope of 1. Voltage S03 decreases until it reaches a value of zero at 11:14 and remains clamped at zero even though both S01 and S111 continue to increase to 11:16.

Operational amplifier 4 and its network, which produce the full-wave triangular output used to drive the errorcorrection diode network, is similar to the circuits described for amplifiers 1 through 3 with the following eX- ceptions:

(a) The signal input and feedback resistors R13 and R28 and the feedforward input resistors R14, R15, and R16 (for S01, S02, and S03, respectively( are adjusted to produce relative slopes of S111:1, 803:-2, S02=-2, 801:-2; and v (b) the rectifying circuitry, consisting of the clamping diodes in the feedback path of the amplifier, is eliminated so that a full wave output is produced; and

(c) the bias voltage SB4 is again reduced, but this time to a smaller value than the four volts which might be expected for the fourth component waveform. Since the fourth amplifier generates the driving waveform for the error correction curve in order to represent, in this example, a six-component simulation of the parabolic curve, or 26:64 segment curve, the initial bias of the fourth waveform would then be the same as for a hypothetical sixth waveform, or one volt (%4 of the total range of 64 volts assumed for S111 in the present illustration).

As a result of the foregoing, the output signal S04 will be zero for values of S111 between zero and minus one volt (corresponding to 11:1/4 That is, bias voltage SB4 is selected to adjust S04 to approach a full waveform shown by solid line 42 in FIGURE 3, for the generation of parabolic error curve 41. When S111:minus 1 volt, the negative input signal overtakes the positive one volt bias and the output S04 begins to increase lineally with a relative slope of +1 as S111 is further increased. When S111 equals -8 volts at 11:2, the output voltage S03 which commences at that point, is fed forward through conductor 73 and resistor R16 with a relative slope of 2, causing the output S04 to change directions and decrease with a relative slope of minus 1. S04 has a value of Zero volts at n=4. At this point, the feedforward voltage S02 changes to a relative slope of +2 but its effect is cancelled out by the feedforward voltage S02 which commences at that point `with a slope of 2. As a result, output S04 is determined only by the input signal S111 which causes it to Bias correction Up to this point in the discussion of the generation of the triangular waveforms S01, S02, S03 and S04, the special bias of one-half unit necessary to convert the general parabolic curve F(n):1/2(112-n) to the square-law function F(n+1/2):1/2 (m2-1A) has been disregarded. Since it is desired to shift the overall curve one-half unit to the left to accomplish the desired transformation, this may be done by shifting each component waveform one-half unit to the left by reducing the positive lbias. In the 64- segment approximation, with a 64 volt range of input, as in the present example, a unit is defined as 64 volts total range of input divided by 64 segments, or one volt, which is equal to 11:11 with reference to FIGURES 2 and 3. Therefore, a negative bias of n:1/2 %:1/s must be added to each waveform so that SB1:77/s units (+311/2 volts), SB2:3% units (151/2 volts), SB3:17; units (+71/z volts), and SB4:1; unit (+1z volt).

The error correction curve, FIGURE 3, is generated by a conventional function generator of the passive diode network type, represented by blocks +D and -D in FIGURE 5. With reference to block +D, it will be seen that the output S04 of circuit number 4 is applied to the input terminal of the passive diode network made up of seven sections 77, 78, 79, 80, 81, 82 and 83, each consisting of a resistor and diode as illustrated. These sections are connected at spaced voltage points to a voltage divider 84 connected at one end to input terminal 76 and at its other end to a reference voltage as for example minus volts as here shown. The several sections 77-83 are connected to a common output terminal 86.

As an important feature of the present passive diode network, an eight segment square-law wave form is generated by a seven-branch network for each half cycle or full excursion of the input triangular wave formi. This is true since the first diode path does not conduct at the zero input level. Since the passive network provides an eight section segmental approximation of a parabolic function,

11 it serves in effect to add three additional circuits of the type of circuits 1, 2 and 3 hereinabove described. The result is the provision of a segmental approximation of a parabolic function having `64 sections which approaches the optimum square-law function.

It is possible to make a fairly accurate sixteen section passive network. With such a sixteen section network it would not be necessary to come down to the fourth amplifier. One could use the third amplifier converted to full wave as above discussed and drive the sixteen section passive network with two peaks; and summate the result with the first two amplifiers in the manner herein descri-bed, and still get a 64 section segmental approximation of the desired parabolic function.

Description of overall apparatus The overall schematic diagram of the quarter-square electronic analog multiplier is illustrated in FIGURE 5. The primary input circuitry is the same as that for existing quarter-square multipliers. The two input signals X and Y are applied simultaneously as two-phase signals ('-j-X and -X, -i-Y and --Y) to the positive input summing and absolute value network y(Block I-j-A) and a negative input summing and absolute value network (Block A). The output of the network I-I-A is a positive quantity X -Y regardless of the polarities of the individual quantities X and Y. Similarly, the output of network -A is always a negative quantity (X-j-Y).

These inputs are applied to an active positive or negative squaring network (Block `-l-B or -B). Each active network functions electrically in accordance with the description applicable to FIGURE 4. With reference to the positive squaring network (Block `-l-B), the junction of resistors R1, R2, and R3 is equivalent to summing point S1 in FIGURE 4. The negative signal (X-l-Y) from input lblock -A is equivalent to the common input signal Sin and is applied to all four stages of the active network through R1, R4, R8, and R13. The bias signals corresponding to SB1, SBZ, etc. In FIGURE 4 are derived from a positive voltage divider (R19 through R24) and are applied to all four stages of the active squaring network through R2, R6, R11, and R17. The feedforward connection from the output of one stage to the input of each successive stage is accomplished from stage 1 through RS, R9, and R14, from stage 2 through R10 and R15, and from stage 3 through R16.

An additional input is applied to each stage through R3, R7, R12 and R18 for diode compensation as discussed below.

Summary of function of the active squaring network stage With reference to FIGURE 5, amplifier No. 1 with the immediately connected passive components, functions as the generator N1 of triangular waveform S01 in the description applicable to FIGURE 4. Amplifier 1 characteristically inverts the output signal polarity -with respect to the input polarity. Rectifier CR'Z holds the output signal level at zero as long as the sum of input signals has a positive polarity. Rectifier CR1 switches in the amplifier feedback component R to initiate the normal summing function whenever the sum of input signals has a negative polarity. The action of these two elements provide the rectifying and biasing requirements for the stage. The gain of the stage, which determines the slope of the output signal relative to the input signal is controlled by the basic relationship of DC amplifier operation: gain: the ratio of feedback resistance to the individual input resistance. Two separate requirements for the slope of the individual waveforms were previously described: (l) to generate a series of triangular waveforms, the iirst waveforms in the series must be fed-forward to the successive waveform-generating units with a specific slope relationship, and (2) the complete series of individual waveforms must be given a different slope relationship and added together to produce the overall function curve.

Requirement (l) is fulfilled by the amplifier gain determined by the input resistors of amplifiers 1 through 4 (Block .-l-B) or amplifiers 5 through 8 (Block -B). Requirement (2) is met by the gain determined by the input resistors of the output summing amplifier 9.

Diode compensation at the input One source of error in the conventional input network (Block A or A) is the finite impedance of the diodes in the absolute value circuit and their non-linear operating characteristics. The voltage drop across these diodes may be treated as an error signal opposite in polarity t0 the desired input signal. To compensate for this undesirable signal, a voltage of equal amplitude is developed across the diode CR3 in the diode compensation network (Block C or -C). This voltage is added into the summing junction of the following amplifiers to cancel out the error voltage created across the diodes in Block A or A.

The output terminals 58, 89, and 86 of the first three operational amplifier stages and the passive squaring network stage are connected through resistors R31, R32, R33 and R34 to a common output terminal 92. In a like manner the outputs of operational amplifiers 5, 6 and 7 shown in Block -B and the output of passive squaring network No. II shown in Block -D are connected to a common output terminal 93. The two outputs of the two systems are connected from terminals 92 and 93 to the input terminal 94 of operational amplifier 9. The product KXY, K being a constant, appears at the out-put 96 of operational amplifier 9.

I claim:

1. The method of generating from a time variable input signal an output signal having 2P linear segments defining a segmental approximation of a parabolic function of the input signal which comprises, forming the input signal into a plurality (p) of symmetrical half wave rectified triangular wave signals having a series relationship in which successive triangular signals have a frequency progression of 2j where j represents the series 0, 1, 2, 3 p-l and with the base break points of each triangular signal coinciding with the peaks of the next higher order frequency signal, and summing said triangular signals and adjusting the peak magnitudes thereof to selected values producing said segmental output signal having a constant change in slope between successive segments.

2. The method of squaring a time variable quantity which comprises, representing the quantity to be squared as the magnitude of a time variable electrical input signal, forming the input signal into a series of symmetrical half wave rectified triangular wave signals having a frequency progression of 2j where y' represents the series 0, 1, 2, 3 and with the base break points coinciding with the peaks of the next higher order frequency signal, summating said triangular signals to produce a segmental wave form, adjusting the successive peak magnitudes of said triangular wave signals to produce a change in slope in successive segments of said segmental wave form wherein sa1d change in slope is a constant approximating a parabolic wave form having the general formula Ax2 plus Bx plus C, and biasing the base break points of said triangular wave signals with respect to the input signal to eliminate the factor Bx.

3. The method of squaring a time variable quantity which comprises, representing the quantity to be squared as a time variable voltage, impressing said voltage as an input to an electronic network having ya plurality (p) of voltage outputs all being functions of said input voltage and being a series of symmetrical half wave rectified triangular signals having a frequency progression of 2j and with the base break points of each triangular signal coinciding with the peaks of the next higher order frequency signal, summating said outputs, and adjusting the successive peak magnitudes of said triangular signals to where the first three terms of the series represent the amplitudes ofthe triangular signals for j equals 0, i equals l, j equals 2 respectively and last term is a general formula defining the amplitude of the fm triangular signal for values ljjmx-l.

and where:

K equals an `arbitrary constant describing the peak."

amplitude of the triangular signal for j equals 0, jmx equals the total number of triangular signals used.

and with the base break points of each triangular signals coinciding with the peaks of the next higher order frequency triangular signal and summating said triangular signals.

5. An apparatus for squaring a time variable quantity represented 'by a time variable input voltage signal comprising, a non-linear electronic network having an input adapted to receive such voltage signal and having a plurality (p) of outputs producing as a function of the input voltage signal a series of symmetrical half wave rectified triangular wave signals having a frequency progression of 25 where j represents the series 0, l, 2, 3 and with the base breakpoints of each triangular signal coincidingv with` the peaks of the next higher order frequency triangular signal and means connected to said outputs summing said triangular signals and adjusting the successive peak magnitudes of said triangular signals to produce a segmental parabolic function having 2p segments having a constant change in slope between successive segments.

6. An apparatus as defined in claim 5 wherein said electronic network includes a plurality of operational amplifiers having feedback diodes connected in the circuit thereof providing said non-linear electrical characteristics of said network.

7. An apparatus for squaring a time variable quantity represented by the amplitude of a time variable input voltage signal comprising; an electronic network having an 'input adapted for connection to the voltage signal, a plurality of operational amplifiers each having a resistive diode feedback circuit providing an output, and input resistors connected to receive the input signal and being connected to the outputs of certain other said amplifiers providing at said outputs a series of symmetrical half wave rectified triangular wave signals having a frequency progression of where j represents the series 0, 1, 2, 3 and with the base break points of each triangular signal coinciding with the peaks of the next higher frequency signal; a passive diode squaring network h aving an output; and means summating the signals appearing at said squaring network output and said electronic network outputs except the highest frequency signal output thereof and adjusting the magnitudes of the summated output signals to produce a segmental parabolic function having a constant change in slope between successive segments.

;8. An apparatus for generating an Output signal as a segmental approximated parabolic function of a time variable input signal comprising, an electric summing circuit each having a plurality of inputs and an output equal to the summation of said inputs, a plurality of non-linear function generating electric circuits each having input and output'relationships as follows:

if input S 0, output equals -CS if input SO, output equals 0 where S is an electrical signal and C is a constant, the output of'each summating circuit being individually connected to the input of a separate non-linear circuit providing a'plurality of associated circuits, said associated circuits being arranged to provide an inter-connected series thereof in which the non-linear circuit output of each associated circuit is connected to one of the summing circuit inputs of all succeeding associated circuits of the series, another of the inputs of each summing .circuit being connected to receive the input signal, biasing means being connected one to an input of each of said summing circuits providing biasing signals thereto progressively decreasing in magnitude by a factor of l in each successive associated circuit of the series, and a summing circuit connected to the outputs of all of the non-linear circuits, whereby said circuits co-function with the aforesaid constant C providing the output signal as a plurality of continuous electrical segments having a constant change in slope between successive segments.

9. An apparatus as defined in claim 8 wherein each of said associated circuits comprise, an operational amplifier having a feedback network consisting of a serially connected resistor and diode providing a first feedback path and a diode oppositely poled with respect to said first named diode providing a second feedback path.

10. An apparatus for generating a segmental voltage signal approximation of a parabolic function in response to a time variable input voltage signal comprising, a plurality of electric summing circuits each having a plurality of inputs and an output equal to the summation of said inputs, a plurality of non-linear function generating electric circuits each having input and output relationships as follows:

if input S 0, output equals -CS if input S50, output equals 0 where S is a voltage signal and C is a constant, the output of each summing circuit being individually connected to the input of a separate non-linear circuit providing a plurality of n associated circuits, said n associated circuits being arranged to provide an interconnected series thereof defined by the non-linear circuit output of each associated circuit being connected to one of the summing circuit inputs of all succeeding associated circuits of the series, one of the inputs of each summing circuit being connected to receive the input signal, a plurality of fixed voltage biasing means being connected one to an input of each said summing circuits except that of the nih associated circuit of the series, a passive diode squaring network having an input and an output with the input connected to the nonlinear circuit output of the nth associated circuit, and a summing circuit connected to the output of said passive network and the non-linear circuit outputs of all of the associated circuits except that of said nth associated circuit, whereby said circuits co-function with the aforesaid constant C to provide the segmental voltage signal consisting of a plurality of continuous electrical segments having a constant change in slope between successive segments.

11. The method of generating an electrical output signal having a parabolic relationship with a time variable electrical input signal which comprises, shaping the input signal into a plurality (p) of symmetrical half wave rectified triangular wave signals having a series relationship in which the successive signals have a frequency progression of 2j where j represents the series 0, l, 2, 3 p-l and with the base break points of each triangular signal coinciding with the peaks of the next higher frequency signal and summating the triangular signals with successive peak magnitudes following a relationship defining a segmental parabolic electrical signal having a constant change in slope between successive segments and generating a segmental electrical signal approximating the error deviation of said segmental parabolic electrical signal from a parabola, and summating said error signal and said segmental parabolic signal to produce the output signal.

12. The method defined in claim 11 wherein a series of n triangular signals are provided and the nth signal is shaped to provide said segmental error signal.

13. The method of squaring a time variable quantity which comprises, representing the quantity to Abe squared as the amplitude of a time variable and voltage signal and impressing said voltage signal as an input to an electronic network having a plurality (p) of voltage outputs all being functions of said input voltage and being a series of symmetrical half wave rectied triangular signals having a frequency progression of 2J and with the base break points of each triangular signal coinciding with the peaks of the next higher frequency triangular signal and summating said outputs except the highest order signal output and adjusting the successive peak magnitudes of the suiimated triangular signals to produce a segmental parabolic voltage function having 2P-1 segments, applying said highest order frequency signal output to a passive diode squaring network to generate an error wave voltage approximating the error deviation of said segmental parabolic function from a parabola, and summating said error Wave voltage and said segmental parabolic function.

References Cited UNITED STATES PATENTS 2,900,137 8/1959 Giser 23S-194 3,191,017 6/1965 Miura et al 235--194 3,253,135 5/1966 Collings et al 235--194 FOREIGN PATENTS 151,873 10/ 1961 U.S.S.R.

MALCOLM A. ,MORRISON Primary Examiner R. W. WEIG, Assistant Examiner

Patent Citations
Cited PatentFiling datePublication dateApplicantTitle
US2900137 *Feb 21, 1955Aug 18, 1959Research CorpElectronic multiplier
US3191017 *Sep 11, 1962Jun 22, 1965Hitachi LtdAnalog multiplier
US3253135 *Feb 20, 1962May 24, 1966Systron Donner CorpQuarter square analog multiplier
SU151873A * Title not available
Referenced by
Citing PatentFiling datePublication dateApplicantTitle
US3573451 *Aug 25, 1969Apr 6, 1971Monsanto CoFunction generator for producing square and ramp wave pulses
US4514820 *Sep 30, 1982Apr 30, 1985Honeywell Information Systems Inc.Apparatus for generating trapezoidal signals over a single conductor coaxial bus
Classifications
U.S. Classification708/846, 708/808, 708/837, 708/852
International ClassificationG06G7/164
Cooperative ClassificationG06G7/164
European ClassificationG06G7/164