The device for calculating the inverse trigonometric functions arcsin x and arccos x

 

(57) Abstract:

Usage: as a functional Converter for computing values of functions arcsin or arccos. The invention is: to improve the accuracy of conversion to calculate the inverse trigonometric functions arcsin or arccos are two trigonometric Converter, one of which works in the range of change of argument of 0 < x < 0,707, and the other in the range 0,707 < x < 1.0 in. Work in a limited range of variation of the argument x can reduce the error of each of the trigonometric converters, which determine the accuracy of the transformation in the entire range of variation of the argument x from 0 to 1.0. The device contains two trigonometric Converter, an adder, a comparator, and a switch. 1 Il.

The invention relates to measuring technique and can be used in various functional devices, allowing to measure the value of the function arc sin X arc cos x, the device must meet the requirements of high accuracy conversion when changing the argument in the region of 0 <X<1.<P> Famous anxiously functional Converter [1] built on operating the J. OR-NOT and RC filter. The device provides an approximation of the function dependence

arc sin 0,707 X 7,0744 X/(10,03164-X2) when 0 < X < 1.

The disadvantage of this device is a low-precision conversion, and that it only performs conversion function arc sin X in a limited range of variation of the argument from 0 to 0,707.

A device containing a reversible counter, a comparison circuit, a generator of clock pulses, two switches and blocks the decisive amplifiers and voltage dividers.

The receiving device performs the desired function arc sin X, arccos X, forming approximating functions, however, the accuracy of the transformation is insufficient, while it can be difficult to design because the device performs complex decomposition functions arcsin X, arccos X in a Taylor series.

A device that performs inverse trigonometric transformations, containing sensors sine and cosine outputs 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, the number of pulses which the e code that's not always convenient.

A device 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 device gives the solutions of the equation F1=arcsin X/y: F2arccos X/Y in the form of time intervals between pulses of the block comparison and shaper which generates pulses at the moments of transition harmonic voltages across zeros or maximums.

The device is limited by the accuracy of the transformation, the representation of the unknown functions in the form of time intervals requires subsequent conversion time intervals in the electrical analogue for pairing with other devices.

The closest technical solution of the invention is a device that contains one United 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.

In the known device uses what about the Converter usually use anxiously Converter, operating in the range of change of argument of 0 < X <1, which receive the values of the function arccos X in the same range of variation of the argument.

This implementation allows you to define the desired function, and the error is determined mainly by the accuracy used trigonometric Converter. Error analysis trigonometric Converter operating with arguments 0 <X<1, shows that providing high precision measurements difficult task.

The purpose of the invention increase the accuracy of the transformation.

For this purpose, the device for calculating the inverse trigonometric functions arcsin X MLM 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 a source of reference voltage, inputs of the second trigonometric Converter, the switch and the comparator, the inputs of the comparator and the second trigonometric Converter connected to the input device, the output of the first trigonometric Converter connected to the first input of the switch, the output of the second switch is connected to the second input of the adder, the output of the switch, the output which is the output device.

The drawing shows a functional diagram of the device.

It contains the first trigonometric Converter 1, the second trigonometric Converter 2, an adder 3, a switch 4, a comparator 5, a voltage reference.

Blocks in the device are interconnected as follows.

Inputs trigonometric transformers 1,2 and comparator 5 is interconnected and connected to the input device. The first input of switch 4 is connected to the output of the first trigonometric Converter 1. The second input of switch 4 is connected to the output of the adder 3, the first input of which is connected to the reference voltage, and the second to the output of the second trigonometric Converter 2. The output of the comparator 5 is connected to the third control input of switch 4, the output of which is connected to the output device.

The device operates as follows.

The input device receives an input voltage signal UIf(X) corresponding to the values 0<X<1. This voltage is fed to the input of the comparator 5 and to the inputs of the first and second trigonometric preobjective Converter 1 receives the voltage U1corresponding to the arcsine function, i.e., U1=f(arcsinUx). The output of the second trigonometric Converter 2 are respectively the voltage U2= f(arccosUx). The voltage U2is supplied to the adder 3, the output of which receives the voltage U3equal to U3=Uop-U2. The voltage Uopchoose such size that when Ux=0 and UxUmaxthe voltages at the outputs of the first and second trigonometric converters 3 and 5 were equal respectively to the voltages U1(Ux+0)= Uopand U2(Ux=Umax)=Uop, i.e., the reference voltage corresponds to the desired corner /2 90about.

The voltage U1arrives at the first input of switch 4, to the second input of which receives the voltage U3.

At the output of the comparator 5 receives the logical signal voltage U5for controlling the operation of the switch 4. For example, when 0<X<0,7071 at the output of the comparator 5 logic "0" and the output of the switch 4 receives the voltage from the first input, and when 0,7071<X<1.0, a logical "1" and the output of the switch 4 receives the signal from your second entry.

For Uxthe corresponding argument of 0<X<0,7071 on you the progress of the adder 3 U3=f(90o-arccosX)f(arcsinX).

Therefore, for an argument of 0 < X < <0,7071 at the output of the switch 4 receives the voltage from the first input U4-1f(arcsinX) Uop(arcsinX/90o), and to change the argument 0,7071 < X < 1.0 to receive the voltage from the second input U4-2f(90o-arccosX) Uop(1-arccosX/90o) Uop(arcsinX/90o).

To obtain at the output of the output voltage corresponding to the arc cosine function, it is enough to swap the first and second trigonometric converters. Then the output of the first trigonometric Converter 1 receives the voltage U1corresponding to the arc cosine function, i.e., U1f(arccosX). The output of the second trigonometric Converter 2 are respectively the voltage U2=f(arcsinX). In this case, for an argument of 0<X<0,7071 at the output of the switch 4 receives the voltage U4-1f(arccosX) Uop(arccosX/90o), and to change the argument 0,7071 < X < 1.0 to receive the voltage U4-2f(90oarcsinX)= Uop(1-arcsinX/90o) Uop(arccosX/90o).

Improving the accuracy of calculation of the inverse trigonometric functions arcsinX or arccosX is achieved through the use of two trigonome which allows you to reduce the error of each of the trigonometric converters, the error which determines the accuracy of the transformation in the range of variation of the argument X from 0 to 1.0.

The device can be made on the standard blocks.

The device for calculating the inverse trigonometric functions arcsinx and arccosx, containing the first trigonometric Converter, voltage reference and the adder, a first input connected to the output of the reference voltage, the input device is the input of the first trigonometric Converter, characterized in that it introduced a switch, a comparator, and a second trigonometric Converter, whose input is connected to the inputs of the comparator and the device, the output of the second trigonometric Converter connected to the second input of the adder, the output of which is connected to the first information input of the switch, the second information and the control inputs of which are connected respectively to the outputs of the first trigonometric Converter and comparator, the output of the switch is the output device.

 

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