DE1802007B2 - Interpolating function generator - Google Patents

Description

The invention relates to an interpolating function generator for generating a function ζ = z (y) or ζ = z (x, y) from one or two variables y or x, y.

Devices for generating functions of one or more variables, in which diode function generators are known. For mapping a given function of, for example of a variable one divides the function into successive straight line segments, the inclination of each section by means of resistors and the kink points by means of diodes can be set. The accuracy of these known diode function generators is at least for certain Voltage ranges still unsatisfactory. This is mainly related to the voltage drop, caused by a diode due to its characteristics. This voltage drop is approximately 0.4 to 0.7 V. In the case of the known function generators with a voltage dynamics of around 100 V it makes the voltage drop across the diode is not too noticeable. For function generators whose Voltage range is around 10 V, the conditions are much less favorable.

To generate functions of two variables with a diode function generator, the constant reference voltage of the diode function generator is replaced by a variable voltage y. The output voltage χα of the function generator is then for each value of y a function of two variables, i.e. Xa = f (x, y)

It is also a function generator for education of functions of two variables known, in which several voltages as functions of the one variable and then between these voltages according to the other variable is linearly interpolated. In a known function generator for two variables of this type is a servomechanism with an adjustable switching element with inputs on a track provided with several distributed, accessible points, with a slider interpolating between makes two points. The bandwidth of such a function generator is determined by the mechanical The system that determines the position of the slide is limited to a few Hz.

The invention is based on the object of creating an interpolating function generator by means of its an analog quantity derived from one or more variables by linear interpolation can be, the accuracy of the function generator only from the interpolation error of the to generating function should depend.

An interpolating function generator for generating a function ζ = z (y) of at least one variable y contains according to the invention: a first channel for the input of the variable y, a second channel with an integrator for generating a linear time function, η channels for an input of the discrete voltage _{values J 1} ... y _{n of} the variables y, η channels with one gate each for entering the gradient differences at each discrete point of the function, another channelless channel for entering the gradient at the starting point, a first comparator that allows the Compares the voltage at the output of the first channel with the voltage at the output of the second channel and sends out an equality signal if these two voltages are equal, η further comparators, each of which the voltage at the output of the second channel with the voltage values J ₁ . .. y _n compares to a goal in one of the η channels for the slope differences as a result of one of the first comparator to open the equality signal sent out, an integrator with η + 1 inputs that receive the output variables of the η channels for the slope differences and the last scoreless channel for the initial slope, a storage device that receives the ίο output variable of the integrator with the η + 1 input channels and the received value under the control of the equality signal sent out by the first comparator, and canceling means for canceling the integrators as a result of the equality signal sent out by the first comparator.

In a further embodiment of the invention is a function generator for generating a Funktionz = z (x, j>) x of at least two variables, y formed so as to feed the inputs of the η + 1 channels for the initial slope and the slope differences simple, a system of Function generators with an input for the variable χ and an output as well as differential amplifiers with two inputs and an output are provided, each differential amplifier receiving the output of two simple function generators of consecutive order at its two inputs.

According to a further characteristic of the invention, the function generator for two variables also contains dividers which divide the output voltage of each differential amplifier by the difference between the variables y , the difference between the variables ζ being applied to the differential amplifiers. An arrangement can also be selected in which the quotient of the differences relating to the two variables y and ζ is applied to the differential amplifiers.

In the invention, for carrying out the multiplication, the term »Time-division-multiplier« principle known - - applied.

A trained according to the invention interpolating function generator has compared to known Function generators have important advantages. Its accuracy is only due to the interpolation error of the too generating function limited. It also has a very high time constancy and can be used for the calculation various functions can be used. Only Usual switching elements (amplifiers, integrators, comparators, etc.) are required.

Embodiments of the invention are shown in the drawing and are described in more detail below explained. Show it

F i g. 1 and 2 two curves of functions ζ of a variable y, F i g, divided into straight line segments. 3 is a block diagram of an integrator with an unlocking gate,

F i g. 4 is a block diagram of a scan multiplier circuit;

F i g. 5 is a block diagram of an elementary interpolator for a single straight line section,
F i g. 6 shows a block diagram of an interpolating function generator for several straight line sections, FIG. 7 and 8 are block diagrams of circuit arrangements which are used to form a function generator

for two variables the function generator according to FIG. 6 are to be added,

9, 10 and 11 are diagrams showing the course of a special function of two variables, namely the volumetric mass of the carbonic acid gas as a function of its temperature and pressure and

Fig. 12 is a block diagram of an arrangement which the function generator according to FIG. 6 is to be added to a function generator for the formation of the by the curves in FIGS. 9 to 11 to obtain the functions shown.

Before explaining the invention in detail, the mathematical expression of a linear interpolated function can be examined.

Let ζ be a general function of two variables χ and y and z ₀ .. ■ '. z _n η + 1 special functions in the plane defined by ζ and χ for η + 1 special values of y:

JWi

-i, yuyt + i

These functions can be written as follows:

Z ₀ = fix, JO)

Zi = f (x, Ji)
Z _n = f {x, y _n ).

25th

If the function y is to be interpolated between the two values j «and yt + _{1 of} the variable y for a value x _{0 of} the variable χ and if the function z (x ₀ , y) is denoted by z (y) , one obtains as immediately from Fig. 1 shows:

a) for any value of y in the range yt <y < Yi + i ■

z (y) = Zi + (y- yi) tg ot,

z (y) = Zi + Cv - yi) ^Z -p-- ~ - ■ -; yi yt

b) at any value of y in the range

ζω =

tv - yt-ύ

c) more generally, around z (y) using the same formula for any values of y in the range

to express, one uses Heaviside's unit function Y (y - yi) as a factor with the expression

to account for the change in the slope of the function z (y) , and then get:

z {y) = Zi- _x

That means

- ji-j) tgjff

z (y) = zi-i + (y - yt-ύ

Zi - Zi-L

(y-yi) * Y (y-y _t );

d) even more generally with the addition of a further segment of FIG. 1 (of the segment for which y lies between yi- ₂ and j> i_ _x ) is obtained for

what can be written

z (y) = Zi-2 + tv - yi-2) random values of y in the range yt-2 <y <yi + i-

ζCv) = Zi- ₂ + (y- yi-i) tgy + Xy - yt-ύ · (tg ß - tg γ) ■ Y (y -yt + Cv- yd · (tg «- tg / S) · Y (y - yd,

+ ■ (y-yi-ύ

Zi - Zi-!

Zj-L - Zj-2

yt-i - yt-2

e) in the most general way, one obtains by extending the range of the variable y to the complete range under consideration, i.e. for all values of y in the range y ₀ < y < y _n :

Ji ~ JO - - {y-yi) -Y (y — yii-

In the special case in which the function z (y) only defines values of y in the range 0 < y < y _n and z ₀ = 0 (Fig. 2), the following can be written:

Z ₀ - z- ₁

= 0

is set.

In order to carry out a linear interpolation of z (y) it is sufficient to generate a function over time which shows the appearance of the broken line with successive slopes of the F i g. 2 has to allow a time tj given by tj = ky ₃ - (where k is a constant) to correspond to each y $ of the observed interval and in a memory element according to FIG. 4 to sample the value of the voltage (representing z ) corresponding to the time t = ky, where y is the value of the variable set during the measurement.

It is sufficient to repeat the process over time, in which case the output variable V _{s represents} z (y) = z (x ₀ , y) .

When using the name

Egg =

- Zj

Zj - Zj-x

j «-ji-i

(2)

it is a question of generating a sequence of slopes such that one obtains

Ei · (y - J «) · F (j - y _t ) <== Φ E ₁ -

Y {t-n).

According to the invention, this is done by a device shown in FIG. 3 arrangement shown, which in series contains a gate or an electronic switch P and an integrator R with the time constant k of known type, the integrator being unlocked by the opening of the gate (which becomes permeable) from the point in time t = U , the The input of the arrangement at A is the voltage - Ei and the output at S is the voltage

The interpolation can therefore be carried out by means of a special device with two integrators of which one is in the second channel and the other in the third channel, namely of the type "sampling multiplier" which is shown in FIG. 4 is shown. In this figure denote

A ₁ , A ₂ , A ₃ the three inputs, L ₁ , L ₂ , L ₃ the three channels,

-R ₂ , R ₃ the integrators of channels L ₂ and L ₃ C the comparator of the output variables of channels L ₁ and L ₂ ,

M the storage device for storing the output variable of the channel L ₃ , when the comparator C determines the equality of the output variables of L ₁ and L ₂ ,

Q the erasing device for resetting the integrators -R ₂ and R ₃ to zero or to the initial state, also as a result of the establishment of this equality, and Vs the output voltage of M, ie the stored variable.

If it is assumed that the two integrators R ₂ and R _{3 have} the same time constant τ and the voltages V (y) = Ky or - V ₁ or - Ei are applied _{to the three inputs ^ 1} , A ₂ , A ₃ (where K is a constant), the outputs are the

Channels L ₁ , L ₂ , L ₃ : KY or V ₁ - or Ei -, with the comparator C storing

t - n

Y (t-ti)

Unlock time Input voltage i = 0 -E ₀
-E ₁ t = ti -Egg t = tn - E _n

an output voltage is obtained from the form

(3)

1 = 0

V (y)

triggers at the time t ₀ , since at the time t = t _{0 one} has: V ₁ - (output variable of L ₂ ) = V (y) (output variable of L ₁ ). So you get

EiKy

If this integrator R with the time constant k does not have an input (as shown in Fig. 3), but η inputs, each of which is unlocked by a gate at the time indicated in the table below and with that on the same line of the table specified voltage is supplied Vs =

V ₁

The voltage Ei - is thus sampled at the point in time at which the comparator C tilts

v (y)

V ₁

V ₁

this formula (3) being identical to formula (1) when (2) is taken into account and the indices z and j are silent or running indices.

and this voltage forming V _s is stored, whereupon the erasing device Q erases the integrators R ₂ and R ₃ , so that the working cycle can begin again.

To produce an interpolator, the scheme in FIG. 4 taking into account the assembly of the F i g. 3 can be modified somewhat.

First, an elementary interpolator with a single straight line section is to be produced. The assembly is then that of FIG. 5, which differs from that of FIG. 4 through the use of a second comparator Q, which compares the output variable ^J of channel L ₂ with that of a "first" additional channel L ₄ with input A ₁ , and an electronic gate Pi , which is arranged in channel L ₃ before R ₃ and its opening through the

7 8

Comparator C <is controlled when this is equal to The triggering of C ₁ occurs when t = U, whereby

determines the unity of the output quantities of L ₂ and L ₄ , whereby Pi is closed, while the triggering of C

the arrangement P ₁ , R _3, which always occurs above with reference to FIG. 11, when t = i ₀ .

F i g. 3 is explained. Finally, one obtains ·.

V _s = Ei -AHA .. Y ( _to - α) Φ = ^ V _s - E ₁ · (y - y _{ ) ■ Y (y - y _t ).

To simplify the FIG. 5 is in FIG. 4 the If T _{1 is} the time constant of the integrator R ₂ and

Quenching device Q is not shown, but it is ίο τ ₂ that of the integrator R ₃₀ , the comparator C is of course present in the real circuit. at the time

In Fig. 6 (in which the extinguishing device β the
Integrators is shown) a linear function generator for η straight line segments is shown. This _t - _t - _τ W
Function generator contains η arrangements with a 15 V ₁
Comparator and a gate, each comparator. C ₁ , C ₂ , C ₃ ... C _n the output variable of L ₂ with the

of a special channel L ₁ ', L ₂ ', L ₃ '... L _n compares, triggered, while the comparator C ₁ , C ₂ ... C _n to which the input voltage V (J ₁ ) = Ky ₁ , F (J ₂ ) the time t - t _lf t - t ₂ ... t = t _{n is} triggered, ■ = Ky ₂ , V (y _s ) = Ky ₃ ... V (y _n receives, and the 20 if he opens the associated gate P ₁ , P ₂ ... P _n . The associated gate P ₁ , P ₂ , P ₃ ... P _n opens, which extinguishing device Q is through the comparator C in a channel L ₁ ", L ₂ ", L ₃ " ... L ₇₁ " is located, which actuates.

the input voltage -E ₁ , - E ₂ , ■ - E ₃ ... - E _n At the output of the integrator R ₃₀ one receives the

receives and an integrator R ₃₀ with («+ 1) one voltage
(which the integrator R ₃ with an input 25

replaced) and at the same time a goalless channel L ₀ "* Vüi · ^t ~ ^t% · Y (t U)

feeds to which the input voltage - E _{0 is} connected - i ^ o ^r

will lay. It is assumed that V ₁ and V (yi) das
have the same characters. and thus

I = O

In order to use the output from the function generator of the If one uses («+ 1) units of the in F i g. 7th

F i g. 6 has a function generator for two variable 35 shown type, which E ₀ , E ₁ , E ₂ , E ₃ ... E _n borrowed to construct, it is sufficient to place devices in front of the so-called (n + 1) inputs of the Provide which the input variables Et of the interpolator in the direction of the F i g. 6, one finally obtains change as a function of χ . a generator according to the invention for generating

But now there is a function ζ of two variables x, y.

40 If the function ζ is fixed, it can be interesting

E ₁ ~ ^{Zi + 1} ~ ^Zi __ ^Zi ~ ^Zi ~ ¹ _. be, directly through segment generators G _z the

yi yi - yi-i ' expressions

In this formula ζ «= z (x ₀ , j«). y _{i +} - j _{

If you change X ₀ you have zj = z {x, yi), and 45

Et is a function of x. and to generate them in pairs in differential

In the most general case, one can subtract more H _z from each other for each Et , as in FIG. 8

a generator of the in FIG. 7, which is an embodiment modification of

place. This contains: («+1) arrangements according to FIG. 7 forms.

three segment function generators G ₁ , G ₂ and G ₃ , 5 ° This solution was used in the special radio switch Zi ₊₁ , Zi and Zi- _x from the common generation generator, which (as a function) generate the input variable χ; volumetric mass of the carbonic acid gas in radio two differential amplifiers D ₁ , D ₂ , which are supposed to generate the input marked with the temperature and the pressure to which it is made - »- 1« by the one marked with »+ 1«, and which is hereinafter referred to as subtract the marked input, the first example of application of the invention being referenced by the output variable applied to "+1" from FIG. 9 to 12 is explained.
G ₁ and the output variable of Es applied to "-!" Let r =. / (T, p) be the volumetric mass r of G ₂ and the second by the carbonic acid gas applied to "+1" as a function of its temperature T output variable of G ₂ and the at "- 1" and its pressure p. When the input variables Γ put the output variable is applied; 60 and ρ change slowly over time (so that, for example, the greatest two divisors or quotient meters H ₁ , H ₂ , which temperature gradient reaches 500 ⁰ C per minute), the output variables (differences with regard to x) as in one through the circulation of Divide carbonic acid gas into cores cooled by D ₁ and D ₂ by the corresponding differences in a closed flow circuit with regard to y ; reactor, r, by the application of the invention a differential amplifier G, the input of which 65 loading with more than five samples per second "+ 1", the output of H ₁ and its inputs are calculated.

gear »- 1« receives the output quantity of H ₂ In Fig. 9 and 10 show the changes in r

and which egg delivers. when T changes from 40 ⁰ C to 700 ⁰ C (for several

Values of ρ) changes and if /? changes from 0 to 80 bar (for several values of ' T) .

Such functions r (T, ρΐ) were chosen such that in a useful range of 200 ° C <T <700 ⁰ C, the linear interpolation error exceeds both the variables T as well as for the interpolation between pi and pt + i does not exceed 0.2% . This condition determines five functions r (T, ρΐ), namely for

P ₁ = 7Bar,; ./;
p ₂ = 19 bar,
p _s = 35 bar,
Pi = 55 bar ,,,
P ₆ = 78 bar, -;

which by the five curves of FIG. 9 shown each of which is actually formed by 18 straight line segments:

Five function generators of the type G _z of FIG. 8 immediately result in the ¹ expressions of the type z (x, y, U ₁ ), which is denoted by Z ₁ for u = U ₁ , z (x, y, M ₂ ), which is denoted _{by z 2} for u = U ₂ is,

** \ ^ »* ■ r% £ t / y ^^ ^^ ^^ AA ^ AV AJ * t Ä l · · « W Vf * 3 ⁴ ^ ^ * ÄJ * ^ ASi * ^ AV J.AA. ** »* A

ζ (χ, j, m _w ), which denotes with z _n for u = u _{n i}

Pi + i - Pi

which go into the formula of linear interpolation and the changes thereof are shown in Fig. 11 (for clarity of this figure only the both limit curves completely drawn).

From the above expressions we get the five expressions of the type _v,,

Pi + i-Pi

η - n- _x

Pi-Pi-i

35

by means of the in F i g. 12 with four differential amplifiers H _z , the five inputs of which are acted upon by five function generators of the type Gz (FIG. 8) and the five outputs of which are an interpolator of the type shown in FIG. 6, in which the index η is 4.

The combination of these five generators of the circuit of FIG. 12, the five inputs of which are each connected to the output of one of these generators, and the circuit of FIG. 6, if η = 4, forms the overall generator for generating the function r = / (T, p) for the carbonic acid gas.

The previous statements concerned the generation of a function from only two variables. However, the invention also relates to the manufacture of a generator for producing a function of three Changeable applicable.

Namely, if z. B. a function z (x, y, u) of three variables x, y, u is to be generated, allows a first generator to generate a function of two variables (x, y) of the type described above (which is made by combining the Circuit of FIG. 8 and the circuit of FIG. 6 fed by the output variables of the circuit of FIG. 8) the generation of

■ ζ = (x, y, U ₀ ).

This value is denoted by z ₀ for a particular value M ₀ of u. ·

Further generators for generating a function of two variables (x, y) of the same type (FIG. 8 + FIG. 6) allow the generation of

One obtains z (x, y, u) on an interpolator as shown in FIG. 8, which performs the interpolation on the variable μ changing _{between M 0} and u _n.

The number needed to produce the complete generator to produce the function of three variables required circuits can be very large, especially when between a very large number must be interpolated from points. However, it rarely happens that a function has a large one Number of points must be interpolated for all variables. It is therefore arranged so that the am Interpolation requiring the least number of points is carried out last.

20th

Claims (4)

Patent claims: 1. Interpolating function generator for generating a function ζ = z (y) from at least one variable y, characterized by a first channel (L ₁ ) for the input of the variable y, a second channel (L ₂ ) with an integrator (R ₂ ) for generating a linear time function, η channels (L ₁ ', L ₂ ', L ₃ '... L _n ') for an input of the discrete voltage _{values J 1} ... y _{n of} the variables y, η channels (L ₁ " , L ₂ ", L ₃ " ... L _n ") each with a gate (P ₁ , P ₂ , P ₃ ... P _n ) for entering the slope differences at each discrete point of the function, another scoreless channel (L ₀ ") for an input of the slope at the starting point, a first comparator (C) which _{compares the voltage at the output of the first channel (L 1} ) with the voltage at the output of the second channel (L ₁ ) and an equality signal sends out, if these two voltages are equal, η further comparators (C ₁ , C ₂ , C ₃ ... C _n ), each of which the voltage at the output of the second Ka compares nals (L ₂ ) with the voltage values y ₁ ... J _{n in} order to generate a gate (P ₁ , P ₂ , P ₃ ... P _n ) in one of the η channels (L ₁ ", L ₂ ", L ₃ "... L _n ") to open an integrator (i? ₃₀ ) with η + 1 inputs that receive the output variables of the η channels (L ₁ ", L ₂ ", L ₃ "... L _n ") for the slope differences and the last scoreless channel (L ₀ ") for the initial slope , a storage device (M) which receives the output of the integrator (R ₃₀ ) with the η + 1 input channels and stores the received value under the control of the equality signal sent from the first comparator (C), and cancellers (Q) for cancellation of the integrators (R ₂ , R ₃₀ ) as a result of the equality signal sent out by the first comparator (C). 2. Function generator according to claim 1 for generating a function ζ = ζ (χ, y) of at least two variables χ and y, characterized in that for feeding the inputs of the ■ m + 1 channels (L ₀ ", L ₁ ", L ₂ ", L ₃ " ... L _n ") a system of simple function generators (G ₁ , G ₂ , G ₃ ; Gz) with an input for the variable χ and an output as well as differential strengthen (JD ₁ , D ₂ ; H _z ) is provided with two inputs and one output, each differential amplifier receiving the output of two simple function generators of successive order at its two inputs. 3. Function generator according to claim 2, characterized in that the system also contains dividers (H ₁ , H ₂ ) which the output voltage of a Divide each differential amplifier (D ₁ , D ₂ ) by the difference in the variable y , while the difference in the variable ζ is applied to the differential amplifier (FIG. 7). 4. Function generator according to claim 2, characterized in that the quotient of the differences between the two variables ζ and y relating to the differential amplifier (Hz) is applied (Fig. 8). For this purpose 3 sheets of drawings