US3500445A - Apparatus and method for producing square-law function - Google Patents

Apparatus and method for producing square-law function Download PDF

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US3500445A
US3500445A US483180A US3500445DA US3500445A US 3500445 A US3500445 A US 3500445A US 483180 A US483180 A US 483180A US 3500445D A US3500445D A US 3500445DA US 3500445 A US3500445 A US 3500445A
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voltage
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Jerry M Collings
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Zeltex Inc
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    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06G—ANALOGUE COMPUTERS
    • G06G7/00—Devices in which the computing operation is performed by varying electric or magnetic quantities
    • G06G7/12—Arrangements for performing computing operations, e.g. operational amplifiers
    • G06G7/16—Arrangements for performing computing operations, e.g. operational amplifiers for multiplication or division
    • G06G7/164—Arrangements for performing computing operations, e.g. operational amplifiers for multiplication or division using means for evaluating powers, e.g. quarter square multiplier

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  • 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:
  • FIGURE l of the drawings 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.
  • 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.
  • 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 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
  • 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.
  • the input voltages to these squaring networks are alx+y
  • S the input signal
  • reversed diodes in the lower squaring circuit 19 retain the negative sign of the signal being squared in that circuit.
  • kS2 The generation of kS2 will now be discussed.
  • 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.
  • 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.
  • FIGURES 2a, 2b, 2c, 2d and 2e The foregoing may -be seen graphically by an examination of FIGURES 2a, 2b, 2c, 2d and 2e.
  • 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.
  • the function repeats as above and similarly repeats at 11:9 and 11:13.
  • 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.
  • S3 has two half cycles and S4 has four half cycles.
  • 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.
  • 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.
  • 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.
  • 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.
  • function S2 subtracts from function S1 so that the resulting segment 29 has a slope of 15-7:8.
  • function S4 is added in so that the resulting segment 30 has a slope of 15-7- ⁇ 1:9.
  • function S1 subtracts while function S3 adds so that the slope of the resulting segment 31 is 15-7-
  • function S1 is at zero so that the resulting segment 32 has a slope of 15-7- ⁇ -3:11.
  • function S3 subtracts from function S1 so that the resulting segment 33 has a slope of 15-3:12.
  • function S4 is added so that the resulting segment 34 has a slope of 15-3-
  • 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.
  • 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.
  • the horizontal coordinate or abscissa is represented by the variable n which in the present apparatus is the scaled input voltage.
  • n may -be considered as the full-scale input voltage for which the system is designed, divided by the number of segments.
  • four triangular wave forms are used thus producing a segmental parabolic curve having 24 or 16 segments.
  • nmx 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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:
  • 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.
  • 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.
  • summing input 44 for the rst circuit N1 has fed to it only the input signal S1n and the bias signal SBI.
  • 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.
  • 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.
  • 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.
  • 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.
  • resistors R1, R2 and R3 are connected in the input to the amplifier and resistor R25 is connected in the feedback path.
  • 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.
  • 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.
  • 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.
  • a conducting mode In the conducting mode:
  • the linear ramp output S01 is shifted by reference voltage Sbl.
  • 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.
  • 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:
  • 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;
  • 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.
  • 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.
  • 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.
  • 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.
  • 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 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:
  • 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).
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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,
  • 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.
  • the output of network -A is always a negative quantity (X-j-Y).
  • each active network functions electrically in accordance with the description applicable to FIGURE 4.
  • 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.
  • 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.
  • amplifier No. 1 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.
  • gain the ratio of feedback resistance to the individual input resistance.
  • 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.
  • 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 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.
  • 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.
  • 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.
  • K an ⁇ arbitrary constant describing the peak
  • amplitude of the triangular signal for j 0
  • jmx the total number of triangular signals used.
  • 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.
  • 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.
  • 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.
  • 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:
  • 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.
  • 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:
  • 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
  • the method of generating an electrical output signal having a parabolic relationship with a time variable electrical input signal 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.
  • 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.

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US483180A 1965-08-27 1965-08-27 Apparatus and method for producing square-law function Expired - Lifetime US3500445A (en)

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Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US3573451A (en) * 1965-10-13 1971-04-06 Monsanto Co Function generator for producing square and ramp wave pulses
US4514820A (en) * 1982-09-30 1985-04-30 Honeywell Information Systems Inc. Apparatus for generating trapezoidal signals over a single conductor coaxial bus
CN110109508A (zh) * 2019-06-11 2019-08-09 宁波新策电子科技有限公司 指数函数信号发生电路

Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US2900137A (en) * 1955-02-21 1959-08-18 Research Corp Electronic multiplier
SU151873A1 (ru) * 1961-10-30 1961-11-30 Ю.А. Тарасов Квадратор дл аналоговых множительных устройств
US3191017A (en) * 1962-09-11 1965-06-22 Hitachi Ltd Analog multiplier
US3253135A (en) * 1962-02-20 1966-05-24 Systron Donner Corp Quarter square analog multiplier

Patent Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US2900137A (en) * 1955-02-21 1959-08-18 Research Corp Electronic multiplier
SU151873A1 (ru) * 1961-10-30 1961-11-30 Ю.А. Тарасов Квадратор дл аналоговых множительных устройств
US3253135A (en) * 1962-02-20 1966-05-24 Systron Donner Corp Quarter square analog multiplier
US3191017A (en) * 1962-09-11 1965-06-22 Hitachi Ltd Analog multiplier

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US3573451A (en) * 1965-10-13 1971-04-06 Monsanto Co Function generator for producing square and ramp wave pulses
US4514820A (en) * 1982-09-30 1985-04-30 Honeywell Information Systems Inc. Apparatus for generating trapezoidal signals over a single conductor coaxial bus
CN110109508A (zh) * 2019-06-11 2019-08-09 宁波新策电子科技有限公司 指数函数信号发生电路

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GB1147806A (en) 1969-04-10
DE1524322A1 (de) 1970-08-13

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