Determining the Error of Measuring Devices under Large Parameter Variations
Abstract Generated abstract
A method is presented for determining the error of measuring devices when variations in structural parameters are too large for a simple linear differential approximation to be adequate. Starting from a transfer function expressed as a ratio of adjoints, the paper derives a relative error formula that avoids repeated expansion and cancellation of common terms in the conventional computation of the total increment. The approach is illustrated for a measuring device with error separation and an external input summator, yielding an exact error expression and showing the parametric invariance tuning conditions. The example indicates that the proposed formula can give a more accurate error estimate under large parameter variations than a Taylor series approximation.
Full Text
UDC 621.372.061
DETERMINATION OF THE ERROR OF MEASURING DEVICES UNDER LARGE PARAMETER VARIATIONS
L. A. NAUMOV, E. I. TSIMBALIST, V. A. BUTENKO
(Presented by the scientific seminar of the Department of Radio Engineering)
A method is proposed for determining the error of measuring devices, using an expression for the transfer coefficient. An expression is given for the error that simplifies the computation.
Let there be a transfer function of a measuring device (MD)
\[ T_{ij}=f(x,\mathbf{D}), \]
where \(x\) is the argument \((\omega,\ldots,t)\),
\[ \mathbf{D}=\{D_1\ldots D_n\} \]
are the parameters of the structural circuit. In a number of cases, for example when evaluating MD structures, and when solving problems of optimal allocation of block errors, a linear approximation in the region of the differential of the function \(f(x,\mathbf{D})\), which in the general case has a nonlinear dependence on the errors \(\delta_i=\Delta D_i/D_i\) of the blocks, is insufficient.
Computing the total increment \(f(x,\mathbf{D}+\Delta\mathbf{D})\) by the usual method is cumbersome. The amount of computation can be reduced if the determined error is expressed in the following way:
\[ \delta=\frac{\Delta T_{ij}}{T_{ij}} =\frac{f(x,\mathbf{D}+\Delta\mathbf{D})-f(x,\mathbf{D})}{f(x,\mathbf{D})}. \tag{1} \]
Represent the function \(f(x,\mathbf{D}+\Delta\mathbf{D})\) and \(f(x,\mathbf{D})\) as a ratio of adjoints
\[ f(x,\mathbf{D}+\Delta\mathbf{D})= \frac{\Delta_{ij}+\Delta_{ij}^{\alpha}}{\Delta+\Delta^{\beta}}, \tag{2} \]
\[ f(x,\mathbf{D})=\frac{\Delta_{ij}}{\Delta}, \tag{3} \]
where \(\Delta_{ij}^{\alpha}\), \(\Delta^{\beta}\) are the terms of the numerator and denominator of (1) that have received an increment.
Substituting (2) and (3) into (1), we obtain
\[ \delta= \frac{\Delta_{ij}^{\alpha}\cdot\Delta-\Delta^{\beta}\cdot\Delta_{ij}} {(\Delta^{\beta}+\Delta)\cdot\Delta_{ij}}. \tag{4} \]
Computing \(\delta\) by formula (4) eliminates the operation of reducing like terms \(\Delta\cdot\Delta_{ij}\), unlike (1).
As an example, let us find the multiplicative error of a measuring device with error separation and an external summator at the input [1]:
\[ K=\frac{-K_0}{1-K_1\gamma+K_1K_0\beta}=\frac{\Delta_{ij}}{\Delta}. \]
By formula (4) we find
\[ \delta= \frac{ -K_0\delta_0\Delta +K_0\left(-K_1\delta_1\gamma+K_1\delta_1K_0\beta+K_1K_0\delta_0\beta+K_1\delta_1K_0\delta_0\beta\right) }{ \left[\Delta+\left(-K_1\delta_1\gamma+K_1\delta_1K_0\beta+K_1K_0\delta_0\beta+K_1\delta_1K_0\delta_0\beta\right)\right]\cdot(-K_0) } = \]
\[ = \frac{ \delta_0(1-K_1\gamma)-\delta_1K_1(K_0\beta-\gamma)-K_1K_0\delta_1\delta_0\beta }{ 1-K_1\gamma+K_1K_0\beta+\delta_1K_1(K_0\beta-\gamma)+\delta_0K_0K_1\beta+\delta_0\delta_1K_1K_0\beta }. \tag{5} \]
From expression (5) the condition of parametric invariance (tuning) is obvious: \(K_1\gamma=1\); \(K_0\beta=\gamma\), then \(K=-K_0\); \(\delta=-\dfrac{\delta_1\delta_0}{1+\delta_0+\delta_0\delta_1}\).
When finding the error by the usual method, by expanding the denominator in a Taylor series, we obtain [1]
\[ \delta=-\delta_1\cdot\delta_0. \]
Thus, the use of (4) makes it possible to calculate more accurately the error of a measuring device, including under large variations of the parameters, while eliminating a number of duplications.
REFERENCES
- M. S. Roitman, V. M. Sergeev, V. A. Butenko. Some questions of the theory and practical implementation of structural methods for increasing the accuracy of measuring amplifiers. Izvestiya TPI, vol. 270, Tomsk, Tomsk Polytechnic Institute Publishing House, 1973.