US3564230A - Function generator for linear interpolation - Google Patents

Function generator for linear interpolation Download PDF

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Publication number
US3564230A
US3564230A US766448A US3564230DA US3564230A US 3564230 A US3564230 A US 3564230A US 766448 A US766448 A US 766448A US 3564230D A US3564230D A US 3564230DA US 3564230 A US3564230 A US 3564230A
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output
channels
function
channel
variables
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Jean-Claude Caross
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Commissariat a lEnergie Atomique et aux Energies Alternatives CEA
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Commissariat a lEnergie Atomique CEA
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR 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
    • G06G7/30—Arrangements for performing computing operations, e.g. operational amplifiers for interpolation or extrapolation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR 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
    • G06G7/26—Arbitrary function generators
    • G06G7/28—Arbitrary function generators for synthesising functions by piecewise approximation

Definitions

  • French Pat. No. 1,537,798, filed July 10, 1967 by the Commissariat a IEncrgie Atomique and delivered July 22, 1968 describes a device constituted by a function generator comprising, in combination, a number n, at least equal to three, of channels fed by voltages, variable or constant, proportional to the n input magnitudes, at least two of these channels comprising a function unit, a comparator receiving the voltages of (n 1) of these channels (voltages taken downstream or at the output of the function units which may be disposed in these channels), a storage device connected to the 12" channel, downstream or at the output of the function unit necessarily disposed in this channel, for storing the instantaneous value which is transmitted to it by this n' channel, in response to an order delivered by the comparator when this comparator finds that the algebraic sum of the voltages which arrive at it by the first (n 1) channels has reached a determined value, in particular the value zero, means for transmitting the content of the storage device and means for resetting this device and the function units to their
  • the device of the above mentioned French Pat. No. 1,537,798 comprises an integrator as function unit in the second channel, on the one hand, and the third channel, on the other hand, whereas the first channel does not comprise any function unit.
  • the present invention relates to improvements to the above mentioned device, in particular according to the particular embodiment specified above, these improvements having the object of realizing a function generator which permits an analogue magnitude of two or more than two variables to be obtained by linear interpolation, the precision of the generator being limited only by the error of interpolation of the function, having two or more than two variables, to be generated, for the error due to the device according to the above-mentioned French Pat. No. 1,537,798 is less than the interpolation error (except if the number of points of reference for the interpolation is very high, which is not very practical).
  • the generators of functions having two variables, at present available on the market are of the potpadder servomechanism type with inputs on a track at a plurality of distributed, accessible points, a cursor realizing the interpolation between two points; the pass band of such known generators is limited to a few cycles per second by the mechanical system on the input which determines the position of the cursor.
  • FIGS. 1 and 2 represent two graphs of linear interpolation
  • FIGS. 3 and 4 represent, in the form of function blocks, two ensembles used in a generator according to the invention, namely respectively an integrator with an unblocking gate and a device according to the above-mentioned French Pat. No. 1,537,798;
  • FIGS. 5 and 6 illustrate, in the form of function blocks, an interpolator respectively for one and several segments constructed from the ensembles according to FIGS. 3 and 4;
  • FIGS. 7 and 8 illustrate similarly the devices to be added to the interpolator of FIG. 6 in order to constitute a generator of functions of two variables, provided with the improvements according to this invention
  • FIGS. 9, l0 and 11 represent the curves or graphs showing the evolution of a particular function of two variables, namely of the density of carbon dioxide as a function of its temperature and of its pressure;
  • FIG. 12 illustrates the device to be added to the interpolator of FIG. 6 in order to constitute a particular function generator according to the invention adapted for the generation of the particular function mentioned above represented by the curves of FIGS. 9 to 11.
  • each integrator is constituted by an analogue amplifier mounted in parallel with a condenser F and F Means for resetting Q to zero comprise the discharge of each condenser F and F by associated transistors T, and T (FET), illustrated in FIG. 7 of this French patent.
  • the memory M is constituted by a low-loss capacitator.
  • the comparator C is an open-loop gain amplifier.
  • z be a (general) function of two variables x and y, and let there by (n +1 (particular) functions, in the plane defined by z and x, for (n +1) particular values of y: y,,, y ',...y, y,, y, ,...y,, these functions being noted:
  • this integrator R of time constant k, has not one input (as illustrated in FIG. 3), but n inputs, each unblocked by a gate at the instant indicated in the table herebelow and fed respectively by the voltage indicated on the same line of the table Instant of In ut the output voltage will be of the form this formula (3) being identical with (1), when account is taken of (2) and that the indices 1' andj are dummy indices.
  • the interpolation can thus be realized by means of a particular device according to the above mentioned French Pat.
  • R R the integrators of the channels L and L C the comparator of the outputs of the channels L and L M the device for memorizing or storing the output of the channel L when the comparator C detects the equality of the outputs of L and L Q the device for resetting to zero, or to the initial state, the integrators R and R also in response to the detection of this equality;
  • V the output voltage of M, that is to say the stored magnitude.
  • the circuit is the one shown in FIG. 5, which differs from the one shown in FIG. 4 by the use of a second comparator C comparing the output of the channel L with the output of a first supplementary channel L. having an input A,,, and of an electronic gate P,-, disposed in the channel L upstream of R and closed under the control of C when C,- detects the equality of the outputs of L and L.,, the P.- -R ensemble being the one discussed above with reference to FIG. 3.
  • FIG. 6 a linear interpolator for n straight line segments is illustrated.
  • I I"
  • the device Q for resetting to zero will be actuated by the comparator C.
  • the voltage will be In order to construct, from the interpolator of FIG. 6, a generator of a function having two variables, it is sufficient to provide means for making the input magnitudes E, of the interpolator vary as a function of X.
  • a differential amplifier G whose input +1 receives the output of H, and whose input 1 receives the output of H and which delivers E,-.
  • r f (T, p) be the volumic mass r of the carbonic gas as a function of its temperature Tand of its pressure P.
  • Tand p the input magnitudes
  • r can be calculated by putting the invention into practice with more than five samplings per second.
  • FIGS. 9 and 10 have been shown the variations of r respectively when T varies from 40 C. to 700 C. (for several values of p) and when p varies from 0 to bars (for several values of T).
  • the invention is also applicable to the realization of a generator of a function having three variables.
  • a first generator of a function having two variables (x, y) of the type described previously (constituted by the combination of the circuit of FIG. 8 and the circuit of FIG. 6 fed by the outputs of the circuit of FIG. 8) permits z (x, y, u,,) which is noted 2 to be obtained for a particular value u, of u;
  • the invention permits the realization of a generator adapted to calculate functions of two or more than two variables, which presents, with respect to the function generators already existant, numerous advantages, in particular the following:
  • the generator can be realized from standard units (amplifiers, integrators, comparators, etc.) easily available on the market.
  • a generator of a function z (x, y) of at least two variables x, y which comprises in combination:
  • a linear interpolator of a function of y for the field of values y,, y,, y y,,, ofy constituted by:
  • a first channel (l comprising an integrator and supplied with a negative supply voltage
  • a second channel (L comprising an integrator and supplied with a negative supply voltage
  • n other channels U U L ,-L,,
  • direct input channels supplied by voltages k y k y k y ,-k y, respectively:
  • n other channels L,, L",, L" ,L",,
  • indirect input channels and comprising a gate (P,, P ,-P,,)
  • each of said channels having an input and at least one outlet
  • a first comparator comparing the voltage at the output of the first channel (L,) with the voltage at the output of the second channel (L and emitting a signal of equality when these two voltages are equal;
  • n other comparators each comparing the voltage at the output of the second channel (L with the voltage at the output of one of the n direct input channels (L,, L' L' ,-L',,) to open a gate (P P P ,P,,) in one of the n indirect input channels (L" L" L” ,L",,) in response to asignal of equality emitted by itself;
  • an integrator with (n 1) inputs, receiving the outputs of(n l) indirect input channels constituted by the said n channels with gate (L",, L" L" ,L",,) and said last channel (L",,) without gate, and with one output;
  • a storage unit connected to theoutput of the integrator with (n 1) channels and storing the value received under the control of the signal of equality emitted by said first comparator;
  • a generator of a function of two variables according to claim 1, wherein said system further comprises dividers adapted to divide the output voltage of each differential amplifier by the difference concerning one of the variables, whereas the differential amplifiers are adapted to operate on the difference concerning the other variable.
  • a generator of a function of two variables according to claim 1 wherein in said s stem the differential amplifiers are adapted to operate direct y on the differences concerning the two variables.

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  • Physics & Mathematics (AREA)
  • Engineering & Computer Science (AREA)
  • Mathematical Physics (AREA)
  • Theoretical Computer Science (AREA)
  • Software Systems (AREA)
  • Computer Hardware Design (AREA)
  • General Physics & Mathematics (AREA)
  • Control Of Velocity Or Acceleration (AREA)
  • Measurement And Recording Of Electrical Phenomena And Electrical Characteristics Of The Living Body (AREA)
  • Feedback Control In General (AREA)
  • Complex Calculations (AREA)
US766448A 1967-10-12 1968-10-10 Function generator for linear interpolation Expired - Lifetime US3564230A (en)

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
FR8757A FR93224E (fr) 1967-07-10 1967-10-12 Perfectionnements aux calculatrices analogiques.

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US3564230A true US3564230A (en) 1971-02-16

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US (1) US3564230A (fr)
BE (1) BE722119A (fr)
CH (1) CH496996A (fr)
DE (1) DE1802007B2 (fr)
ES (1) ES358876A1 (fr)
FR (1) FR93224E (fr)
LU (1) LU57041A1 (fr)

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US3748447A (en) * 1971-11-18 1973-07-24 Sperry Rand Corp Apparatus for performing a linear interpolation algorithm
US3996456A (en) * 1975-02-13 1976-12-07 Armco Steel Corporation Recursive interpolation
US4188627A (en) * 1969-05-14 1980-02-12 Elliott Brothers (London) Limited Display apparatus

Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US3043516A (en) * 1959-10-01 1962-07-10 Gen Electric Time summing device for division, multiplication, root taking and interpolation
US3231729A (en) * 1961-03-31 1966-01-25 Systems Inc Comp Dynamic storage analog computer
US3333092A (en) * 1964-07-14 1967-07-25 Sperry Rand Corp Alternating current integrators
US3383501A (en) * 1964-10-27 1968-05-14 Honeywell Inc Arithmetic circuit for multiplying and dividing
US3412240A (en) * 1963-02-21 1968-11-19 Gen Precision Systems Inc Linear interpolater

Patent Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US3043516A (en) * 1959-10-01 1962-07-10 Gen Electric Time summing device for division, multiplication, root taking and interpolation
US3231729A (en) * 1961-03-31 1966-01-25 Systems Inc Comp Dynamic storage analog computer
US3412240A (en) * 1963-02-21 1968-11-19 Gen Precision Systems Inc Linear interpolater
US3333092A (en) * 1964-07-14 1967-07-25 Sperry Rand Corp Alternating current integrators
US3383501A (en) * 1964-10-27 1968-05-14 Honeywell Inc Arithmetic circuit for multiplying and dividing

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US4188627A (en) * 1969-05-14 1980-02-12 Elliott Brothers (London) Limited Display apparatus
US3748447A (en) * 1971-11-18 1973-07-24 Sperry Rand Corp Apparatus for performing a linear interpolation algorithm
US3996456A (en) * 1975-02-13 1976-12-07 Armco Steel Corporation Recursive interpolation

Also Published As

Publication number Publication date
BE722119A (fr) 1969-03-14
DE1802007A1 (de) 1969-06-26
LU57041A1 (fr) 1969-01-20
ES358876A1 (es) 1970-06-16
CH496996A (fr) 1970-09-30
DE1802007B2 (de) 1970-11-12
FR93224E (fr) 1969-02-28

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