Device for computing functions arcsin x, arccos x

 

(57) Abstract:

The invention relates to the field of measurement technology and can be used as functional Converter to calculate values of the function arcsin X, arccos x the Purpose of the invention is to improve the accuracy of conversion. This device uses two trigonometric Converter, each of which operates in a limited range of variation of the argument X, which reduces the requirements for each of them, and reduced the methodological and instrumental error of the conversion, the adder, two switches, a comparator, and voltage reference with corresponding connections. 1 Il.

The invention relates to measuring technique and can be used as a functional Converter that allows you to calculate values of the function arcsin X, arccos X with high accuracy when changing the argument in the range O X 1.

Known anxiously functional Converter [1] is built on an operational amplifier (op-amp), in the feedback circuit which is enabled units containing resistors, capacitor, three key scheme OR NOT and RC filter.

The drawback of the device in a low-precision is holding a reversible counter, a comparison circuit, a generator of clock pulses, two switches and block the decisive amplifiers and voltage dividers. The device performs the calculation of desired functions arcsin X, arccos X, however, the device decomposes the functions arcsin X, arccos X in a Taylor series, which complicates the device, increases the instrumental error.

It is known device [3] for inverse trigonometric transformations, containing sensors sine and cosine outputs of which are connected to the inputs of the comparator, a switching unit to switch the octant, a source of reference oscillations, shaper rectangular pulses, a phase detector, low pass filter, a second comparator, two-threshold comparator and the pulse counter output.

The device is limited by the speed and accuracy of the conversion.

It is known device [4] for the solution of trigonometric equations of the form arcsin X/Y, arccos X/y

The device comprises a generator of harmonic voltages, signals X, Y, modulator, block comparison, the shaper and the Registrar.

The instrument displays the results in the form of time intervals between the output pulses of the block comparison and shaper that shapes ieno on speed and accuracy of conversion, requires subsequent conversion time intervals.

The closest technical solution to the claimed is a device [5] contains one connected trigonometric Converter and the adder, the input device connected to the input of trigonometric Converter, and the output devices are connected to the output of the adder, the second input is connected to the voltage reference.

This device uses known trigonometric ratios

.

As trigonometric Converter is used, usually, anxiously Converter operating in the range of variation of the argument from 0 X 1, which receive the values of the function arccos X in the same range of variation of the argument. To trigonometrically Converter high demands for 0 X 1.

The aim of the invention is to improve the accuracy of the conversion.

Purpose and device for evaluation of the functions arcsin X, arccos X, containing the first trigonometric Converter, whose input is connected to the input of the adder, a first input of which is connected to the reference voltage, is achieved by the fact that istwa connected to the inputs of the second trigonometric Converter and comparator, the outputs of the first and second trigonometric transducers are connected respectively with the first and second information inputs of the first switch, the output of which is connected with the second input of the adder of the first information input of the second switch, the output of the adder is connected to the second input of the second switch, the outputs of which are the outputs of the device, the comparator output is connected to control inputs of the first and second switches.

The essence of the invention lies in the fact that to increase the accuracy of the calculation functions arcsin X, arccos X are two trigonometric Converter, each of which operates in a limited range of variation of the argument X. This allows to reduce the requirements for each of them, and reduced the methodological and instrumental error of the conversion in the range of argument 0 X 1.0.

Functional diagram of the device shown in the drawing.

The device includes: a first trigonometric Converter 1; a second trigonometric Converter 2; a first switch 3; the adder 4, and the second switch 5; a comparator 6, a source 7 of the reference voltage.

Blocks in our 6 connected together and connected to the input device. The first and second information inputs of the first switch 3 are connected respectively to the outputs of the first and second trigonometric converters 1 and 2. Source output 7 of the reference voltage connected to the first input of the adder 4. The output of the first switch 3 is connected with the second input of the adder 4 and the first information input of the second switch 5. The control inputs of the first and second switches 3, 5 combined and connected to the output of the comparator 6. The first and second outputs of the second switch 5 are connected respectively with the first and second outputs of the device.

The device operates as follows. The input device receives an input voltage signal Ubxf(X) corresponding to the values 0 X 1. This voltage is fed to the input of the comparator 6, and also to the inputs of the first and second trigonometric converters 1, 2.

The output of the first trigonometric Converter 1 receives the voltage U1corresponding to the arcsine function, i.e. U1f(arcsin Ux). The output of the second trigonometric Converter 2 receives the voltage U2f(arccos Ux). These voltage U1U2coming in, respectively, the first and the indices 3 and 5. At the output of the comparator 6 receives the voltage U6control switches 3 and 5. For example, if 0 X 0,7071 at the output of the comparator 6 is a logical 0 and the output of the switch 3 receives a voltage from the first data input to the first output of the second switch 5 receives the voltage from the first data input and the second output of the second switch 5 receives the voltage from the second information inputs. For the interval 0,707 <X 1,0 voltage U6a logical 1 and the output of the switch 3 receives a voltage from the second information input to the first output of the second switch 5 receives the voltage from the second information input, and the second output of the second switch 5 receives the voltage from the first data input.

For Uxcorresponding arguments 0 X 0,707 at the output of the first switch 3 receives the voltage U3-1f(arcsin X), and for an argument 0,707 <X 1,0 receive the voltage U3-2f(arccos X).

The voltage U3(U3-1or U3-2routed to the first information input of the second switch 5 and the second input of the adder 4, the first input is from the output of the source 7 laprairie U4UopU3.

The voltage Uopchoose such size that when Ux0 (or UxUmaxvoltage at the outputs of the first and second trigonometric converters 1 and 2 were equal to the voltage U1(Ux0) Uopand U2(Ux=Umax)=UopThat is, the reference voltage corresponds to the maximum value of the angle (/2).

When you change the argument in the range 0 X 0,707 at the output of the adder 4 receives the voltage U4-1f[(/2)-arcsin X) Uop[1-(arcsin X)/(p/2)] and in the range 0,7071 <X 1,0 receive the voltage U4-2f[(p/2)-arccos X] )= Uop[1-(arccos X)/(p/2)] Voltage U4(U4-1or U4-2enters the second information input of the second switch 5, the first information input of which receives the voltage U3.

In accordance with the considered operation of the comparator and switches 3 and 5 for the argument 0 X 0,707 on the first output of the second switch 5 receives the voltage U5-1with its first data input, i.e. U5-1U3-1f(arcsin X) Uop[(arcsin X)/(p/2)] and for 0,707 <X 1,0 receive the voltage from the second information input, actalogic for argument 0 X 0,707 on the second output of the second switch 5 receives the voltage U5-2with his second information input, U5-2U4-1f[(/2) arcsin X] Uop[1-(arcsin X)/(p/2)] f(arccos X), and for changes of argument 0,707 <X 1.0 on the second output of the switch 5 receives the voltage U5-2with its first data input, i.e. U5-2U3-2f(arccos X) Uop[(arccos X)/(p/2)]

Thus, the first and second outputs of the second switch 5 get the desired function, improving the accuracy of calculation is achieved by the use of two trigonometric converters, each of which operates in a limited range of variation of the argument X.

The device can be made on the standard blocks and chips:

trigonometric converters 1 and 2 [1 of 5]

the switches 3 and 5 chip series 590 KN;

4 at the operational amplifier [6]

Sources of information

1. Auth. mon. THE USSR, N 467365, G 06 G, 7/22. 1975.

2. Auth. mon. THE USSR, N 525123, G 06 G, 7/22. 1976.

3. Auth. mon. THE USSR, N 1256051, G 06 G, 7/22. 1987.

4. Auth. mon. THE USSR, N 506027, G 06 G, 7/22. 1976.

Under the editorship of Sangolda. Handbook of nonlinear circuits. M. Mir, 1977, S. 45 47; 174 176.

6. A. G. Alexenko, that is, As the La calculation functions arcsin X, arccos X, containing the first and second trigonometric converters, voltage reference and the adder, the first input of which is connected to the reference voltage, the input of the first trigonometric Converter is an input device, characterized in that it introduced the two switches and the comparator, and the input device is connected to the inputs of the second trigonometric Converter and comparator, the outputs of the first and second trigonometric transducers are connected respectively with the first and second information inputs of the first switch, the output of which is connected to the second input of the adder and the first information input of the second switch, the output of the adder is connected to a second information input of the second switch, the outputs of which are the outputs of the device, the comparator output is connected to control inputs of the first and second switches.

 

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