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

 

(57) Abstract:

The invention relates to measuring technique and can be used in trigonometric converters to obtain values funktsii1= arcsin x2=arccos x, and various analog computing devices. The purpose of the invention to enhance the functionality of the device through the implementation of consistent transformation of the argument x, which is the ratio of input voltage to the amplitude of the harmonic voltage in a time interval whose duration is proportional to , and then into a voltage, proportional to the duration of this time interval, which is then converted into the desired functions1and2. The device comprises a block of comparison, the harmonic generator voltage, the time-amplitude Converter, the comparator character driven key, the adder and the controlled amplifier. 2 C. p. F.-ly, 2 Il.

The invention relates to measuring technique and can be used in various trigonometric functional converters, allowing to obtain information about the values of F1arc sin X and F2arc cos X. the device pred.
arc sin 0,707 X 7,0744 X/(10,03164 X2)

if 0 X 1.

The disadvantage of this device is that it only performs conversion arc sin X and in a limited range of variation of the argument from 0 to 45about.

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 device converts the functions sin X and os X arcsin, arccos X, forming approximating functions, however, the accuracy of the transformation is insufficient, because the device performs the expansion of the function arcsin X, arccos X in a Taylor series with different coefficients, which complicates the design.

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, pay the comparator and the pulse counter, the number of pulses which is proportional to the measured angle.

The device is difficult by design, has an output signal in the form of code, it is not possible to determine the arcsin and arccos simultaneously, as it is necessary to specify different initial conditions on the two inputs of the device.

The closest technical solution to the invention is a device for solving trigonometric equations of the form:

F1arcsin X/Y, F2arccos X/Y

The device comprises a generator of harmonic voltages, source X, source Y, modulator, block comparison, the shaper and the Registrar, other output of the signal source connected to the first input of the modulator, the output of the signal source connected to the first input of the comparison, the output of the generator is connected to the input of the shaper and to the second input of the modulator, the output of which is connected to the second input of the comparison, outputs a block comparison and shaper connected to two inputs of the recorder.

The device gives the solutions of the equation F1arcsin 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 e.g. icepay possible areas of change in the argument either positive or negative values, as in the time interval does not contain the information about the sign of the argument, moreover, the subsequent transformation of time intervals in the electrical analogue to pair your device with other analog devices.

The aim of the invention is to expand the functional capabilities of the device.

The purpose of the device to calculate the inverse trigonometric functions arcsin X and arccos X, containing the harmonic generator voltage, the unit of comparison and the input signal, the output of which is connected to the first input of the comparison is achieved by the fact that it introduced the time-amplitude Converter, the comparator mark key and the two adder, the second input unit of comparison is connected to the output of the harmonic generator voltage, the output of the Comparer connected to the input of the time-amplitude Converter, the output of which is connected to the first inputs of the first and second adders to the second input of the first adder connected to the output of the key, the first information input of which is connected to the bus reference potential of the device, the second information input of the key is connected to a source of reference voltage and to the second I is and connected to the control inputs of the key and the second adder, the outputs of the adders are the outputs of the device.

The first adder includes inverting adder and amplifier, the first and second inputs of the first adder are respectively the first inverting input of the adder and the input of the amplifier, the amplifier output is connected to the second input inverting adder whose output is the output of the first adder.

The second adder contains inverting adder and managed amplifier, the first and the second input of the second adder are respectively the same inverting inputs of the adder, the output of which is connected to the input of the control amplifier, the control input and output of which are respectively managing input and output of the second adder.

In Fig.1 presents a functional diagram of the device.

The device comprises a generator 1 harmonic voltage, the unit of comparison 2, the time-amplitude Converter 3; a comparator mark 4; key 5; the first adder 6, the second adder 7.

Blocks in the device are interconnected as follows.

To the first input unit of comparison 2 the connected input source (in the drawing, the terminal UI), to second his wholetone Converter 3, the output of which is connected to the first inputs of the first adder 6 and the second adder 7. To the second input of the first adder 6 is connected to the output of the key 5, the second input of the second adder 7 is connected to the reference voltage (indicated by terminal Uop), to the same source connected to the first information input key 5, the second information input of which is connected to the bus reference potential of the device. The comparator input of the mark 4 is connected to the source signal UIthe comparator output of the sign is connected to the control inputs of the key 5 and the second adder 7. The outputs of the adders 6 and 7 are the two outputs of the device.

Functional diagram of the first adder 6 and the second adder 7 is shown in Fig.2. The first adder 6 is an amplifier 8 with the gear ratio TO8-1 and inverting adder 9, the first input connected to the first input of the first adder 6, and the second input is connected to the second input of the first adder 6 via the amplifier 8. The second adder 7 includes inverting adder 10 and the controlled amplifier 11 with transfer factor11+1, two input inverting adder 10 is connected to the same two inputs of the second adder 7, and the output of inverting summarizedin between them.

Inverting adders of the drawing shows schematically in the form of an operational amplifier without feedback resistors.

The device operates as follows.

The first input unit of comparison 2 receives an input voltage signal UIand on his second input signal harmonic voltage U U1sin t with amplitude U1, which is chosen equal to the maximum value of the input signal UI. The unit of comparison 2 generates voltage pulses, the duration t is proportional to the value 2 arccos l UI/U1l, then the ratio UI/U1we will call the argument X.

Under voltage U1mean amplitude of these harmonics; the value of U1choose equal to the maximum value of the input signal UI. Therefore, U1this is an absolute value, so the sign of the argument X UI/U1depends only on the sign of the signal UI.

The time-amplitude Converter 3 converts the pulse duration t in the DC voltage U3proportional to this length, its conversion coefficient K3choose equal TO3-(U1/P), and the voltage e of the spine from the editing area of argument X. In the first operation mode when 0 X 1 comparator mark 4 produces a signal of logical "1", which closes the 5 key on the terminal with zero potential and sets the gain of the controlled amplifier 11 is equal to K11-1. For the two inputs of the first adder 6 receives signals:

at the first sign U3_[U1/(P/2] arccos l X l; the second input of U30, these signals are summed inverting adder 9 and output it, and accordingly the output of the first adder 6 is formed of the voltage signal:

U6[U1/(P/2)] arccos l X l

The inputs of the second adder 7 receives signals at the first sign

U3-[U1/(P/2)] arccos l X l; the second input of UopU1these signals are summed inverting adder 10 and the output it generates a voltage signal that can be represented as follows:

U10-{-[U1/(P/2)] arccos l X l + U1}

[U1/(P/2)] [arccos l X l ( P/2)]

[U1/(P/2)] arcsin l X l, which is inverted by the amplifier 11 and the output of the second amplifier 7, a signal is generated

U7[U1/(P/2)] arcsin l X l

In the second mode of operation when -1 X 0 comparator mark 4 produces a signal of logical "0", which closes the 5 key on the terminal voltage Uop3[U1/(P/2)] arccos l X l; the second input of U5UopU1. The voltage signal U5is inverted by the amplifier 8 and the inverting adder 9 adds the signals U3and-U1. At its output and the output of the first adder 6 is formed of the voltage signal:

U6-{-[U1/(P/2)] arccos l X l - U1}

[U1/(P/2)][arccos l X l + (P/2)]

In the second adder 7 inverting adder 10, as in the first operation mode, and generates the output signal U10:

U10-[U1/(P/2)]arcsin l X l, which is in the second mode of operation is not inverted and directly on the output of the second adder 7, where U7-[U1/(P/2)] arcsin l X l

Thus, when changing the argument as in the field of positive and negative values, the device on its two outputs simultaneously generates signals in analog form, is proportional to the value:

if 0 X 1 arccos l arcsin X l l X l

when -1 X < 0 arccos l X l + (P/2),

-arcsin l X l

The device can be made on the standard blocks.

1. The DEVICE FOR calculating the INVERSE TRIGONOMETRIC FUNCTIONS ARCSIN X AND ARCCOS X, containing the harmonic generator voltage, the unit of comparison and the input signal, the output of which is connected to the first the character, key and two adder, the second input unit of comparison is connected to the output of the harmonic generator voltage, the output of the Comparer to the entrance FremantleMedia Converter, the output of which is connected to the first inputs of the first and second adders to the second input of the first adder connected to the output of the key, the first information input of which is connected to the bus reference potential of the device, the second information input to the reference voltage and the second input of the second adder, comparator input of the sign is connected to the input source, output to the control input key and the second adder, the outputs of the adders are the outputs of the device.

2. The device under item 1, characterized in that the first adder includes inverting adder and amplifier, the first and second inputs of the first adder are respectively the first inverting input of the adder and the input of the amplifier, the amplifier output is connected to the second input inverting adder whose output is the output of the first adder.

3. The device under item 1, characterized in that the second adder contains inverting adder and managed amplifier, the first and second inputs of the second adder are sootla, managing input and output of which are respectively managing input and output of the second adder.

 

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