Primary and additional error of the humidity measurement channel



This error is determined by the error in measuring the frequency. The latter, in turn, consists of the quantization error, the error in determining the coefficients and during calibration and the error due to the noise of the operational amplifier.

1) The quantization error.

The digital code on the output of the ADC is:

Where T_mea = 0.2 s is the measurement time.

The relative error of quantization is:

(1)

(2)

2) Error due to an error in determining the coefficients and during calibration.

Taking into account (1) and (2), we find the absolute error in determining the frequency:

(3)

(4)

Then:

(5)

The relative error in determining the coefficient will be:

(6)

The absolute error in determining the coefficient is found from the formula:

(7)

(8)

(9)

Relative error of this coefficient:

(10)

We will find the error values due to the error in determining the coefficients separately for oil and for water.

In the first case, we denote both sides of by D. Then the absolute error Δ D is equal to:

(11)

The relative error in determining D will be:

(12)

The relative error due to the error in determining the coefficients for oil will be:

(13)

Absolute error:

(14)

(15)

Similarly, we find the value of the error for water

(16)

(17)

(18)

3) Error due to operational amplifier noise:

(19)

Where – standard deviation of operational amplifier noise

Since this error is small, it can be neglected. An additional error arises from the temperature drift of the amplifier, because of the instability of the resistor resistors that make up the conversion function, and also due to the temperature changes in the permittivity’s of water and oil.

The error is due to the instability of the resistors.

The relative error for the resistor R₁ is determined by its TCS and is then the error due to the instability of this resistor in frequency will be:

(20)

The absolute error in frequency is determined by the formula:

(21)

Let us determine the value of this error for oil and for water. For water: (21)

The error in determining the value of D:

(22)

(23)

Hence the error due to the instability of the resistance of the resistor R₁ for water will be:

(24)

A similar calculation is made to determine the error for oil.

(25)

(26)

(27)

(28)

The error for oil will be: (29)

Consider the error due to the instability of the divider resistors. In the transformation function they enter as the coefficient K, equal to: moreover .

If the resistors R₂ and R₃ are taken from the same lot, then their TCS will be approximately equal. Therefore, the effect of temperature on the coefficient K can be neglected.

4) Error from the effect of temperature changes and and

It has been experimentally established that the dielectric permeability of water is most susceptible to temperature changes: (30)

Since in pure water the value of D is equal to: then the absolute error is: (31)

This means that if the ambient temperature changes by 100 ° C, this error will be 16 %. This is one of the most significant errors of the moisture meter.

To correct this error, you can use the correction:

(32)

Where S some correction factor.

References:

1. BetaTHERM. Temperature sensors, discrete NTC thermistor elements and custom probe assemblies: Catalog. Ireland, 1996. — 33 p.
2. Grigoriev Yu. I. Geophysical studies of wells: Proc. For high schools / Pod. Red.E. V. Carusa. — Moscow: Nedra, 1980. — 398 p.
3. Kulikovsky K L, Cooper V. Ya. Methods and Means of Measurement: Proc. Manual for universities. — Moscow: Energoatomizdat, 1986. — 448 p.
4. M. A. Berliner. Moisture Measurements: Ed. 2-nd revision. And additional. — Moscow: Energia, 1973. — 400 p.