DE1802007C - Interpolating function generator - Google Patents

Description

ι 2

The invention relates to oinon interpolating radio comparators, each of which the voltage nn dem

Ion generator to generate a function ζ = z (y) output of the second channel with the voltage values

or ζ = z (x, y) of one or two variables yy _x ,,. y _n compares to a gate in one of the η cannula

or x, y. for the slope differences as a result of one of the

Means for generating functions of 5 first comparator emitted equality signal

one or more variables, in which diodes- 'open, an integrator with η I- 1 inputs, which the

Function generators are used are known. Output from the η cannula for the position differences

For the mapping of a given function of bot- tomers and the laston scoreless canal for the initial

spiolswcise a variable one divides the slope received, a storage device that the

Function in successive straight line segments, io »output variable of the integrator with the η -t- 1 input

where the slope of each section is received through recirculation cannulas and the received value

and the break points with the help of diodes under the control of the

can be set. The accuracy of this well-known parator emitted gloichheitssignal stores, and

Diode function generators are at least used for quenching devices for quenching the integrators

correct voltage ranges still unsatisfactory. 13 as a result of the transmitted by the first comparator

This is mainly due to the voltage drop to the equality signal.

together, which a diode due to its character- In a further development of the invention is a radio «

ristics caused. This voltage drop is about the generator for generating a function z— z (x, y)

0.4 to 0.7 V. In the known function generators, at least two variables x, y are characterized in this way.

with a voltage dynamics of about 100 V makes ao that for feeding the inputs the η [ 1 Ka-

the voltage drop across the diode does not differ too closely for the initial slope and the slope difference.

strongly noticeable. With function generators, their limit a system of simple function generators

Voltage range is around 10 V, the inputs are for the variable χ and one for the variable

conditions, on the other hand, are much less favorable. Output as well as differential amplifiers with two input

For the generation of functions of two as gangs and one output is provided, whereby variables with a diode function generator each differential amplifier at its two uses one the constant reference voltage of the diode ganges the output variable of two simple radio function generator by a changeable span - generation generators of successive order empnungji. The output voltage χ α of the function starts.

generator is then a function 30 for each value of y. According to a further characteristic of the invention

of two variables, ie x. \ f (x, y). contains the function generator for two variable

It is also a function generator for the BiI- also borrowed the output voltage of a divider

Calculation of functions of two variables for each differential amplifier by the difference of the

where several voltages as functions divide the variable y , whereby the differential ratio

of the one variable and between 35 stronger the difference of the variable ζ is applied.

These voltages then according to the other conditions. An arrangement can also be selected in which

variable is linearly interpolated. In the case of a differential amplifier the quotient of the

known function generator for two variables two variable y and ζ relevant differences

of this type is a servo mechanism with an inserted.

adjustable switching element with inputs on a track 40 In the invention, for the implementation of the

with several distributed, accessible points before multiplication that under the English name

see, whereby a slider indicates the interpolation between the »time-division-multiplier« known principle.

makes two points. The bandwidth of such turns.

Function generator is created by the mechanical interpolating according to the invention

System, which determines the position of the slide, on 45 the function generator possesses opposite known

limited to a few Hz. Function generators have important advantages. His

The object of the invention is to provide an accuracy only due to the interpolation error

to create interpolating function generator, co-generating function limited. He also has a

means of which an analog quantity of one or more very large time constancy and can be used for the calculation

Variables derived by linear interpolation 50 different functions can be used. for

can be, whereby the accuracy of the function- the structure of the function generator will only be

generator only from the interpolation error of the usual switching elements (amplifiers, integrators, com-

generating function should depend. parators, etc.) are required.

Fin interpolating function generator for generating. Embodiments of the invention are shown in FIG

tion of a function ζ z (y) from at least one drawing and are shown in more detail below

Variable ^ contains according to the invention: one explained. Show it

first channel for entering the size y, a two-F i g. 1 and 2 two divided into straight sections

th channel with an integrator for generating a curve of functions ζ from a variable ^,

linear time function, η channels for an input of FIG. 3, a block diagram of an integrator with

discrete voltage values y, ... y _{n of} the variable y, 60 an unlocking gate,

η channels with one gate each for entering the slope F i g. 4 is a block diagram of a sampling multiplier

differences at each discrete point of the radio circuit,

tion, another scoreless channel for an input F i g. 5 a block diagram of an elementary inter-

the slope at the starting point, a first polisher for a single straight line section,

parator, which determines the voltage at the output of the first 63 F i g. 6 is a block diagram of an interpolating

Channel with the voltage at the output of the second function generator for several straight line sections,

Compares the channel and sends out an equality signal, FIG. 7 and 8 block diagrams of switchgear

if these two voltages are equal, η further voltages that are necessary for the formation of a function generator

(jr two variables the function generator me ϊ i a. C) are to be added,

Fig. 9. 10 and H Schaubikler the course of a "Sonderen function of two variables, and was de« ¹ volumetric mass of carbonic acid gas ils function of its temperature and its Druckesund

'f i g. 12 is a block diagram of an arrangement which the function generator according to FIG. 6 to add is a function generator for the formation of the curves in FIGS. 9 to U shown »o Functions.

Before explaining the invention in detail first the mathematical expression of a linearly interpolated function will be examined.

Let ζ be a general function of two variables χ and let y and z ₀ ... z _n η + 1 be special

Functions in the level defined by ζ and χ for

"1- I special values of y:

a) for any yt <y <, Y \ n '■

z (y) « Zi -V (y

Value of y in the range

b) at any value of y in the range

= z (-, - l (y-yt- \)

Z, - Zi-i

yi-yi-i

Area

These functions can be written as follows:

z ₀ = / (JCi y »)

Zn = fix,

If the function y is to be interpolated between the two values ^ and V ₁₊ , the variable y for a value X _{0 of} the variable χ and if z (v ) denotes the function z {x _a , y) , we get how directly from FIG. 1 shows:

expression

0 »- yt)

the change in the slope of the function i (y) to take into account, and then receive.

That means

z <- Zf-!

{y- yi) * Y {y

d) even more generally with the addition of a further segment of FIG. 1 (the segment for which y ~ yt between ₂ and ^ j- is!) Is obtained for loading \ o undesirables values of y in the range <

yt- _z <y <ytn z {y) - zi- _a + 0 '-> * <- «)

+ (y - wi) (tg / i ) (t

+ (y - wi) · (tg / i tgy) \ - (y - yt) · (tg * - tg ß) · Y {y

- y *

what can be written

(y-

Ux - ^z it «

I- {y -.V ₁ -!) ¹

Z _{1 +} IZ ₁

■ y \ _ ^z _ »- i_7 ^z '~ * · y (>» - .Vi t)

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

• iy-y,) 'ny ~ v, h

ϊ 802

.l'o-J'-i

= 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 hai, to let each yj of the observed interval correspond to a time tj given by t] = k yj (where k is a constant) and in a memory element according to FIG. 4 to sample the value of the voltage (representing z) corresponding to the time / - A y , where y is the value of the variable set during the measurement.

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

When using the name

Et =

Zi H 1 - Zi .VHi -J ¹ *

Room

(2)

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

Ei-iy-y _t )

(ν-

Ei- ' ^h -Y (IU). k

According to the invention, this is done by a device shown in FIG. 3 arrangement shown, which in series a gate or an electronic switch P and an integrator J? with the time constant ä of known design, the integrator being unlocked by the opening of the gate (which becomes permeable) from the point in time t = ti , the input of the arrangement at A being the voltage - Ei and the A 'output at S being the voltage

40

If this integrator R with the line 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 the one on the same line of the table specified voltage is supplied

50

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

A _x , Λ ₂ , A ₃ the three inputs,
L ₁ , L ₂ , L ₃ the three channels,
A ₂ ; A _{3 are} the integrators of channels L ₂ and L ₃

C is 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 Lj and L ₂ ,

Q the cancellation device for resetting the integrators A ₂ and R ₃ to zero or to the initial state, also as a result of the determination of this equality, and

K _{5 is} the output voltage of M, ie the stored quantity.

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

Channels L _x , L _it L ₃ : K Ybz \ NV _{x τ} or Et -, where the comparator C is the storage of

30th = Et '* =

triggers at the time t ₀ , because at the time t − t _{0 one} has: V ₁ ζ (output variable of L _a ) = V (y) (output variable of L ₁ ). So you get

The voltage Et "is thus sampled at the point in time at which the comparator C switches over

Release time point Input voltage t = '1 -Eo
-E ₁ t - ti -Egg . '= in -E _n

= X

V (y)

= τ

Av
V ₁

one obtains a final voltage from Fnrm

-He

• YV U), (3)

where this I nrmt'l (3) is identical to the lormel (l) is, if (2) is corrected and the indices / and / mumme or luufcndc are indices.

and this voltage forming V i is stored, whereupon the erasing device ρ erases the integrators R _t 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 · 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 by using a second comparator Cu, which compares the output value of channel L with that of a "first" additional channel L _t with input 4, and distinguishes between an electronic Ί ors Pt , which is used in channel /. _a is disordered in front of R ₃ and its opening by the

t-,

Comparator Ci is controlled when it equals. The triggering of C occurs when / = U, whereby

unity of the output quantities of L ₂ and L ₄ , whereby P _{1 is} closed while the triggering of C

the arrangement P ₁ , R _3, which always occurs above with reference to FIG. _{1, if / = / 0} .

F i g. 3 is explained. Finally you get

Vs = Ei · - '° ~' * · K (ίο - H) <== * Vs ^ Ef (j- - yt) ■ Y {y - y _t ).

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

Q extinguishing device not shown, but it is io t _s of the integrator ₃₀ is A, the comparator C is naturally present in the real circuit. at the time

In Fig. 6 (in which the extinguishing device Q of the

Integrators is shown) is a linear functional. .

generator for η straight line segments shown. This t = / ₀ = T ₁ -----

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

a special channel L ₁ ', L ₈ ', L ₃ '... L _n ' compares, triggered, while the comparator C ₁ , C ₈ ... C _n to which the input voltage V (J ₁ ) = Ky ₁ , V (J ₂ ) the time t - i _lt t = I ₂ ... t = t _{n is} triggered, = Ky ₂ , V (y ₃ ) = Ky ₃ ... V (y _n receives, and the ao if it opens the associated gate P ₁ , P _z ... P _n . The associated gate P ₁ , P ₂ , P ₃ ... P _n opens, which is the extinguishing device Q through the comparator C in a channel L ₁ ", L ₈ ", L ₃ " ... L _n " is located, which actuates.

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

receives and an integrator R ₃₀ with (n + 1) input voltage (which connects the integrator R ₃ with an input 25 _u __

replaced) and at the same time a scoreless channel L ₀ " ^] Et · - '· Y (t - r <)

feeds to which the input voltage - E _{0 is} connected / <> ^r

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

V _s - T Ei · '° ~''■ ■ Y (t ₀ - ti) < == » V ₁ = ■ -' X Ei · (j. - yt) ·) '( y -y _t ).

1: Il ^{T r} 2 I -U

In order to obtain the output from the function generator of the If one over (n + 1) units of the in F i g. 7th

F i g. 6 has a function generator for two types of variable 35 shown, which £ ₀ , E ₁ , £ ₂ , E ₃ ... E _n

To construct a loan, it is sufficient to place devices in front of the so-called (n ■ {■ 1) entrances of the front

see which the input variables Et of the interpolator in the direction of FIG. 6 is finally obtained

change in 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 _. _z <+ i —Zi _ zt—} ii ^ be, directly by segment generators G ₁ die

yi +1 - ) 'i yt - yt-i' expressions

2 (

In this formula, z ₍ ^ = z (x ₀ , yt). _{Yi +1} - yt

If you change x ₀ you have zt = 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 H ₁ from one another for every £ <stronger, as in Fig. 8th

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

put. This contains: ("4 D arrangements according to I" i g. 7 forms.

three segment function generators C ₁ , G, and G ₃ , 50 This solution was used in the special radio which z <4 "Zi and Zi-" 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 , / J ₂ , which are supposed to generate the input marked with the temperature and the pressure to which it is made of- »- 1« by the one marked with »+ 1«, and which is below al <subtract the marked input, the former 55 application example of the invention with reference by the output variable applied to ") 1" from FIG. 9 to 12 is explained. G ₁ and the output variable of Let r / (T, />) be the volumetric dimension r of G ₈ and the second output variable of G, and as a function of its temperature the at »- 1« and its pressure / ». When the input variables are set to 1 output variable is applied; 60 and ρ change over time (so that e.g. the largest two divisors or quotient meters H _x , W ₁ , which temperature gradient reaches 500 "C per minute) the output variables (differences with regard to x) as in one through the circulation of carbonic acid gas ir of /), and D _t divide a closed flow circuit cooled core 7cn with respect to y ; reactor by the corresponding differences, r can, through the use of the difference, be a differential crystal (i, the input 65 of which with more than five samples per second he β ) 1 «the output variable of //, and its factored in.

gong "- 1" the output of //, receives In F i g. 9 and 10 show the changes in r

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

1 ΟΠΟ ΠΠ7 χ yj \ j Iu \ j \ j ι

10

Values of ρ) changes and if ρ changes from 0 to 80 bar (for several values of T).

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

P ₁ = Ί bar,
p ₂ = 19 bar,
p ₃ = 35 bar,
Pn = 55 bar,
p _s = 78 bar,

which by the five curves of FIG. 9, each of which is actually represented by 18 straight line sections is formed.

Five function generators of type d from FIG. 8 directly result in the expressions of the type

- η

which go into the formula of the linear interpolation and their changes are shown in FIG (to clarify this figure, only the two limit curves are fully drawn).

The five expressions of the type are obtained from the above expressions

η _ η - n- ₁

Pt Pt-Pt-i

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 «= 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 by the combination the circuit of FIG. H and the circuit of FIG. 6 fed by the output variables of the circuit of FIG. 8) the generation of

ζ «(*, y, U ₀ ).

This value is denoted by Z ₀ for a particular value W ₀ of u.

Further generators for generating a function of two variables (χ, ^) of the same type (Fig. 8 I F '' i g. 6) allow the creation of

z (x, y, U ₁ ), which is denoted by Z ₁ for 11 = M ₁ , z (x, y, M ₂ ), which is denoted _{by z 2} for μ = M _2,

z (x, y, Un), which is denoted by z “for” = M _n. 5

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 seldom happens that a function has a large one Number of points must be interpolated for all variables. It is therefore arranged in such a way that the am Interpolation requiring the least number of points is carried out last.

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 (A ₂ ) 2: to generate a linear time function, η channels (L ₁ ', L ₂ ', L ₃ '... L _n ') for an input of the discrete voltage _{values ^ 1} ... y _{n of} the variables y, η channels (L ₁ ", L ₂ ", L ₃ "... L _n ") each with one gate (P ₁ , P ₂ , P ₃ ... P _n ) for entering the gradient differences at each discrete point of the function, and 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 emits an equality signal when these two voltages are equal, η further comparators (C ₁ , C ₂ , C ₃ ... C _n ), each of which determines the voltage at the output of the second K. compares anals (L ₂ ) with the voltage values y _x ... y _{n in} order to establish a gate (P ₁ , P ₂ , P ₃ ... P _n ) in one of the η channels (L ₁ ", L _a ", L ₃ "... L _n ") to open an integrator (R ₃ ,) with η + 1 inputs, which 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 received, a storage device (M), which the output of the integrator (A ₃₀ ) with the η + 1 input channels and stores the received value under the control of the equality signal sent by the first comparator (C), and canceling devices (Q) for canceling the integrators (R _t , R _M ) as a result of that from the first comparator (C ) sent equality signal. 2. Function generator according to claim 1 for generating a function ζ => z (x, y) of at least? "" Two variables * and y, characterized in that for feeding the inputs of the η f-1 channels (L ₀ ", L ₁ ", L _a ", L, "... L _n ") a system of simple function generator ((? ,, (ft, G ₃ ; G,) with an input for the variable * and one output as well as differential powder strengthen (D 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 , / Z ₂ ), 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 pertaining to the two variables ζ and y is _{applied to the differential amplifier (W 2} ) (F i g. 8). For this purpose 3 sheets of drawings