Approximation of one type of standard nonlinearity
Abstract Generated abstract
This paper addresses the analytical approximation of a typical nonlinear characteristic that passes through the origin and approaches a finite saturation value. It proposes a one-parameter formula chosen to preserve these limiting features, with the parameter determined from a selected point on the curve, and examines the effect of this choice on approximation accuracy using M-20 computer calculations. The study finds that selecting a point near the middle of the most nonlinear portion gives a relative error not exceeding about 1.5 percent over the assumed operating range, and that the proposed expression yields smaller errors and a wider useful range than a previously used saturation approximation.
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APPROXIMATION OF ONE TYPE OF TYPICAL NONLINEARITY
V. N. Stanevko
(Presented by the scientific seminar of the Department of Instruments and Devices of Automation Systems)
In calculations and theoretical studies one often has to deal with nonlinear characteristics of the type shown in Fig. 1.
When carrying out studies using such a family of characteristics, it is necessary to have an analytical expression describing the given curve. This analytical expression must ensure sufficient accuracy of approximation and have a convenient form for use in analytical studies. To obtain the approximation, let us note the principal features of this curve. It passes through the origin and, as the argument tends to infinity, approaches an asymptote whose equation may be represented in the form
\[ Y = Y_{\mathrm{n}}. \]
Fig. 1
As a result of the analyses carried out, the following expression has been obtained and is proposed for approximating the curve in Fig. 1:
\[ Y(x)=\frac{Y_{\mathrm{n}}\cdot S}{Sx-1}\cdot x, \tag{1} \]
where \(S\) is the parameter of approximation of the curve; it is determined by choosing a specific point on the curve in Fig. 1 with coordinates \(x_i\) and \(y_i\).
Expression (1), as the argument varies from \(0\) to \(\infty\), satisfies the features of the dependence under consideration. Indeed, the curve described by expression (1), at \(x=0\), passes through the origin:
\[ \lim_{x\to 0} Y(x)=0; \]
as \(x\to\infty\), this curve tends to \(Y=Y_{\mathrm{n}}\):
\[ \lim_{x\to\infty} Y(x)=Y_{\mathrm{n}}; \]
moreover, it passes through one of the selected points.
Depending on the choice of the point on the curve for determining the parameter \(S\), the accuracy of the approximation is different. The dependence of the approximation accuracy on the choice of the point on the curve was investigated using an M-20 digital computer. As a result of the investigations carried out, it was found that the best approximation is obtained when the point is chosen in the middle of the interval of the most nonlinear segment of the curve. In this case the relative approximation error, taken in absolute value over the assumed operating range, does not exceed \(1.5\%\).
Expression (1) provides a better approximation than the well-known formula proposed by Robinson [1] for a similar case
\[ Y=\frac{x}{\alpha+\beta x}. \tag{2} \]
Expression (2), proposed for approximating curves with saturation, satisfies only one of the features of the indicated curve. It ensures that the curve corresponding to expression (2) passes through the origin, i.e.
\[ \lim_{x\to 0}\frac{x}{\alpha+\beta x}=0, \]
but it does not ensure its approach to the saturation value \(Y_{\text{n}}\), that is
\[ \lim_{x\to\infty}\frac{x}{\alpha+\beta x}=\frac{1}{\beta}. \tag{3} \]
Since \(\alpha\) and \(\beta\) are determined by points of approximation of the curve, only in a special case can the equality
\[ \frac{1}{\beta}=Y_{\text{n}} \]
hold.
For the stated reason, expression (2) gives a substantial error in the saturation region. On the basis of the work carried out to evaluate the accuracy of approximation by expressions (1) and (2), graphs were constructed of the dependence of the relative error in \(\%\) on the value of the argument. The magnitude of the error was determined by the formula:
Fig. 2. — 1 — error of the known approximation;
2 — error of the proposed approximation
\[ |\Delta_i\%|=\left|\frac{Y_{i\,\text{exp.}}-Y_{i\,\text{apr.}}}{Y_{i\,\text{exp.}}}\right|\cdot 100\%, \]
where \(Y_{i\,\text{exp.}}\) is the value of the function in Fig. 1 at the point \(X_i\) for the experimental curve;
\(Y_{i\,\text{apr.}}\) is the value of the function at the point \(X_i\) for the approximating curve.
From Fig. 2 it is seen that:
1) the range in which the error does not exceed \(3.5\%\) is larger for expression (1) than for expression (2);
2) the magnitudes of the errors for expression (1) are substantially smaller than for expression (2).
REFERENCES
- L. A. Bessonov. Electrical Circuits with Steel. Gosenergoizdat, 1948.