Electronic A method for accurately measuring the average value of an alternating voltage in the infrared frequency range, based on the use of a high-speed electronic digital DC voltmeter and a controlled rectifier-type converter with an averaging link. quasi-stationary mode, hardware errors, silicon diode, linear differential equation. III International Scientific And Practical Conference “Information
The methodological errors of the device were analyzed (Fig.1). To determine the errors of measuring instruments, it is necessary to conduct special studies. In this case, the following assumptions were made: -the averaging element I is ideal
RC circuit without parasitic capacitances and leaks; -the electronic digital DC voltmeter has an infinitely large input resistance, zero errors and instantaneous speed; -the control unit for the key and electronic digital voltmeter generates an ideal square-wave key control signal and a start pulse for the voltmeter at the moments of transition of the instantaneous value of the measured sinusoidal signal through the zero level; -the key K has an infinitely large resistance in the open state, infinitely small resistance in the conducting state, and the construction of sources of emf. and there is no current in the key.
Let us consider the hardware errors of the
device caused by the violation of these conditions.
The main relationship that determines the error of
the device is found when considering the process
of changing the voltage across the capacitor of the
equivalent circuit in Fig. 3, given in [
− 2 | + | 2 = = = − - period of the measured voltage;
= -is the time constant of the averaging element. The given errors are used to characterize the accuracy properties of measuring instruments only (such a concept as the reduced error of the measurement result is not used). The systematic and random components of the error of measuring instruments can be considered. These errors are usually expressed as a percentage, but they can also be expressed in relative values. − 1, (1)
measuring instruments.
Changes in the resistance R and capacitance C of the capacitor of the averaging element AND affect the measurement result, since this changes the parameter , on which the error depends1) If the values of R and C are chosen in such a way that a small value of the error is always ensured ,then even relatively large (2-3%) changes in R and C, which can actually arise as a result of aging or changes in the temperature of these elements, practically do not affect the measurement result.
In general, the partial relative error ,due to a change in the value , (due to the change in R and where C), can be found from the expression = = Taking into account that the partial derivative is The relative error after transformations has the capacitor C - is determined from the difference equation compiled from the differential equations of the circuit in Fig. 1 for cases when the key K is closed and open.
It is convenient to solve this problem by the
method of discrete Laplace transform [
In quasi-steady-state mode, the voltage across the capacitor UvvR at the moment of opening the key is determined by the expression = [1+ − 1(1+ 1⁄ 2]
, 1− − 1(1+2 1⁄ 2 =
1 2+ 12(1+ 1⁄ 2)2
; 1 =
2 1
Partial relative error н, due to the presence of R2, is expressed as =
− уст ≈ −
Taking into account (6) and the value The permissible basic and additional errors are measuring instruments and measurement methods given in the technical descriptions and forms of and increasing the number of measurements.
≈ −1 + ∙ {1 +
2 12 1 + 12 [(1 + (0,02-0,03) this error does not exceed 0.075%.
2. The final value of the input resistance of the digital voltmeter and the leakage resistance of the averaging element capacitor can also be sources of error. It is advisable to take into account the influence of these resistances according to the scheme in Fig. 1, where they are combined into one
equivalent resistance parallel with the capacitor C.
Re, connected in the expression for the relative error н will take the form discard the higher-order terms, starting from the third (since they are small compared to the terms of the first two orders), then (9) is transformed to the form (6) (7) (8) .
(9) (1 + (10) and
The dependence of н on 1 for different values of the parameter R_1 / R_2 is presented by a family of curves in Fig. 2. From these curves it can be seen that to ensure the partial relative error н no more than 0.25% at 1 less than 0.1, it is necessary that the ratio R_1 / R_2 ≤0.01.
To meet this requirement in the package of the device, the dependence of the resistance R2 on the frequency was experimentally determined (this resistance is nonlinear and frequency-dependent due to the specifics of the operation of the adopted ECV). Measurements are recommended to be carried out by gradually increasing the measured value to the limit value for the device under test, followed by a gradual decrease in it to a minimum. The increase and decrease of the measured value are carried out as many times as the measurements need to be made, each time fixing the readings of the investigated measuring device at the selected points of the scale.
According to the obtained values of R2, such values of R1 and C
were selected, at which
sufficiently small errors are combined with a
sufficiently short measurement time [
3. The influence of the key control unit and an electronic digital voltmeter on the device error can be divided into: the influence of the switching angle (or phase) of the control voltage; the influence of the cutoff angle of the control voltage and the influence of the pulse fronts that control the key.
a) The influence of the switching angle is
determined from consideration of the circuit in [
Mathematically, the
problem
is
solving
the
difference
equation
following from the linear differential equation for
the voltage u (t) on the capacitor of the circuit in
[
(11)
If we take into account the closure and opening of the circuit with the key K at the moments of time 0 and T / 2, then (11) turns into a difference equation of the form [ + 1] − [ ] −
= where ( − )(1 − e− ) (12) = = =
; 2 √1 + 2 2 = −2 − .
(14)
As a result of determining the error according to this formula, are shown in Fig. 3 for two values = 0.1 and 0.25.
b) The influence of the cutoff angle is determined according to the same initial equation as the influence of the switching angle , with the difference that instead of the times of the key operation 0 and T / 2, it is necessary to calculate the circuit when the key is triggered at the moment ( 2 − ) and (
+ )
This somewhat complicates the compilation and solution of the difference equation, without fundamentally changing anything. Measurement errors from the influence of influencing factors are the components of the measurement errors, which are a consequence of the unaccounted influence of external factors on the measurement results. has the form
The steady-state value of the voltage across the output capacitor in the measurement interval =
The results of calculating this error for several parameter values are shown in Fig. 3. these pulses can be made small by choosing the correct key circuit, for example, with a six-diode key circuit).
Non-identical current-voltage characteristics of the key diodes, as well as non-identical shape of the front and rear the fronts of the control pulse, causes the appearance of switching bursts in the signal at the output of the switch. These bursts do not lend themselves to rigorous mathematical analysis, but are easily detected by an electronic oscilloscope. Due to the fact that switching bursts are frequency-independent, their effect is stronger at high frequencies of the measured voltage.
An experimental study of the device prototype showed that balancing the diode switch (by selecting silicon
diodes and introducing a balancing resistance) and using powerful lamps in the output cathode follower of the control unit can easily achieve a short duration (<12 meters per second) and amplitude (<1 / 4Um) of these bursts. These measures, taken in the tested prototype of the device, ensured a negligible influence of the shape of the control pulses at all frequencies below 200 Hz.
1. To consider the influence of the key operation on the error, it is convenient to divide the key operation cycle into three stages: the key conducts, the key does not conduct, and the beak is "thrown over". The third state causes the just considered commutation bursts, and in the first two states an error may arise due to the finite (and not infinitely small or large) switch resistance, as well as due to the appearance of parasitic emf. or current at the output of the key.
In the considered device, the final resistance of the conducting switch, practically when using silicon diodes, can
always be neglected in comparison with the resistance
R1 integrating link, since the first resistance is of the order of several ohms, and the second - several hundred or thousands of kilo-ohms. The influence of the final resistance of the open key Rk can be considered in a manner similar to the pleasant one when considering the influence of the resistance R2. In this case, the steady-state value of the voltage at the output capacitor in the measurement interval has the form
where = 1 1− − 0 1+ − 1 1,2 = 1,2 = − 2 ∙ − 1 √1+ 2 12,2 ± 2 1,2 ; ; 1+ − 2 1− − 0, (17) 0 = 1 + 2; 2 = 1 1 2 − 1 (1− − 1)(1+ − 2
(1− − 0)(1+ − 1. and the conversion error is expressed by the formula = − ус
It is clear from (19) that this error when using selected silicon diodes can be small, so that the resistance of the open switch does not affect the overall error of the device.
From the above, the following conclusions can be drawn. Equivalent emf a closed switch can have a significant impact, since it is directly added to the measured voltage. The task of balancing the key circuit is to get rid of this emf. Theoretically, this issue does not lend itself to accurate analysis due to the instability and nonlinearity of the current-voltage characteristics of the diodes, but an experimental study of this effect is not difficult. Experiments have shown that daily changes in the equivalent emf the key does not exceed 0.5 mV, and an increase in temperature by 40 ° C (from the value of 200 ° C) causes an emf. not exceeding 5mV. Considering that these switches are intended for use in a device in which the highest value of the measured voltage at the switch output is 10V, then it is clear that such small changes in the equivalent emf. closed key are perfectly valid.
In the open state, a possible source of instrument error is the equivalent output current of a non-conductive switch. However, it can also be brought to a negligible value (in the breadboard, the value of this equivalent current was less than 10-12A). The use of unmatched silicon diodes or diodes of other types limits the upper limit of the value of the resistance R1 of the integrating element. Verification of measuring instruments should be carried out under normal operating conditions. The operating conditions of operation must be observed in the practical use of measuring instruments. Additional errors arising when operating conditions differ from normal ones are usually expressed in fractions or multiples of the basic error. The ratios used to assess additional errors are given in the operating instructions for specific types of measuring instruments.
The carried out consideration of various sources of the device hardware errors (the errors of the electronic digital voltmeter were not considered, since their influence is clear, and they are usually very small) showed that it is easy to fulfill the conditions under which the partial components of the total device error will be small enough and even with a simple arithmetic summation, the total error will not exceed 0.5%.