Non-linear circuits. Approximation of characteristics. Approximation of characteristics of nonlinear elements Nonlinear elements approximation of nonlinear characteristics

Converting signals to non-linear

radio engineering chains

Most of the processes (nonlinear signal amplification, modulation,

demodulation, limiting, generation, multiplication, division and frequency transfer, etc.) associated with the transformation of the signal spectrum is carried out using nonlinear and parametric circuits. In nonlinear circuits, the parameters of the elements depend on the input actions, and the processes occurring in them are described by nonlinear differential equations. In this case, the principle of superposition is not applicable to them. These chains are very diverse and therefore there are no general methods for their analysis.

We will restrict the analysis of nonlinear circuits to considering only their certain class. These are radio engineering circuits, the analysis of which is carried out mainly using the current-voltage characteristics of nonlinear elements. An intermediate position between linear and non-linear circuits is occupied by parametric circuits, which are linear and to which the principle of superposition is applicable. However, new frequencies may appear in the spectrum of the output signal of such circuits. Parametric circuits are described by linear differential equations with variable (i.e., time-dependent) coefficients. The theory of these equations is more complicated than the theory of linear equations with constant coefficients. Some parametric circuits operate in a highly non-linear mode. This makes it possible to methodologically combine parametric circuits with nonlinear circuits, especially since the result of signal processing is associated with the transformation of its spectrum.

Approximation of the characteristics of nonlinear elements

In the general case, the analysis of the signal conversion process in nonlinear circuits is a very difficult task, which is associated with the problem of solving nonlinear differential equations. In this case, the principle of superposition is inapplicable, since the parameters of a nonlinear circuit when exposed to one source of an input signal differ from its parameters when several sources are connected. However, the study of nonlinear circuits can be carried out using relatively simple methods if the nonlinear element (NE) meets the conditions of inertia. Physically, the inertialessness of the NE means the instantaneous establishment of a response at its output following a change in the input action. Strictly speaking, inertialessness (resistive, or ohmic, i.e. only absorbing the energy of the input signal) practically does not exist. All nonlinear elements - diodes, transistors, analog and digital microcircuits - have inertial properties. At the same time, modern semiconductor devices are quite perfect in their frequency parameters and they can be idealized from the point of view of inertialessness.

Nonlinear dynamic systems are described by nonlinear differential equations, in these systems nonlinearity is necessarily present. A non-linear circuit can be determined not only by the elements included in it, but also by external signs, which include: with a harmonic input signal:

ü difference from the sinusoidal form of the output signal;

ü appearance of harmonics of the input signal in the spectrum of the output oscillation;

ü nonlinearity of the transfer amplitude characteristic;

ü dependence of the phase of the amplified signal on the amplitude.

The following methods for analyzing nonlinear circuits when deterministic signals pass through them are known and used:

Ø linearization of the characteristics of a nonlinear element (NE) at

filtering higher harmonics of the signal at the circuit output;

Ø analytical, as a rule, approximate ways of solving the system

nonlinear equations describing the operation of the device;

Ø spectral, evaluating the nonlinear properties of the circuit over the spectrum

output signal;

Ø numerical methods for solving a system of nonlinear equations with

using a computer;

The most commonly used method for analyzing nonlinear circuits is based on linearizing the characteristics of the NE when filtering the higher harmonics of the signal at the circuit output.

Linearization (from lat. linearis - linear) - method of approximate

representations of closed nonlinear systems, in which the study

a nonlinear system is replaced by an analysis of a linear system, in a sense equivalent to the original one. Linearization methods are of a limited nature, that is, the equivalence of the original nonlinear system and its linear approximation is preserved only for a certain "mode" of the system's operation, and if the system passes from one mode of operation to another, then its linearized model should be changed. At the same time, using linearization, it is possible to find out many qualitative and quantitative properties of a nonlinear system.

As an example of nonlinear circuits, more precisely, elements, one can cite a semiconductor rectifier diode, which leaves only unipolar (positive or negative) half-sine waves from a sinusoidal signal, or a transformer whose core saturation with a magnetic field leads to "dulling" of the sinusoid vertices (and from the point of view of the frequency spectrum , this is accompanied by the appearance of harmonics of the fundamental frequency, and sometimes frequencies lower by a factor of several times the fundamental frequency - subharmonics).

When using the linearization method, signal path analysis

through a nonlinear circuit, it is relatively easy to implement if the nonlinear

the element meets the conditions of inertia. Physically, inertialessness of a nonlinear element (NE) means an instantaneous change in the response at its output following a change in the input action. Strictly speaking, inertialess (resistive, or ohmic, i.e., absorbing signal energy) NEs practically do not exist. All NEs - diodes, transistors, microcircuits, electric vacuum devices, etc. - have inertial properties. At the same time, modern semiconductor devices are quite perfect in their frequency parameters, and they can be idealized from the point of view of inertialessness.

Most nonlinear radio circuits and devices are determined by the block diagram shown in Fig. 1.

Fig. 1. Block diagram of a nonlinear device

According to this scheme, the input signal directly affects the non-linear element, to the output of which the filter is connected (linear circuit).

In these cases, the process in the radio-electronic nonlinear circuit can be characterized by two operations independent of each other.

As a result of the first operation in the inertialess nonlinear element, the shape of the input signal is transformed in such a way that new harmonic components appear in its spectrum. The second operation is performed by a filter that selects the desired spectral components of the converted input signal. By changing the parameters of the input signals and using various nonlinear elements and filters, you can carry out the required transformation of the spectrum. Many schemes of modulators, detectors, oscillators, rectifiers, multipliers, dividers and frequency converters are reduced to such a convenient theoretical model.

As a rule, nonlinear circuits are characterized by a complex relationship between the input signal and the output response, which in general can be written as follows:

In nonlinear circuits with inertialess NEs, it is most convenient to consider the input voltage as the impact, and the output current as the response, the connection between which is determined by the nonlinear functional dependence:

...................... (1)

This ratio can analytically represent the usual volt-ampere characteristic of the NE. This characteristic is also possessed by a nonlinear two-terminal device (semiconductor diode) and a nonlinear four-terminal device (transistor, op amp, digital microcircuit) operating in a nonlinear mode at different input signal amplitudes. The volt-ampere characteristics (for nonlinear elements they are obtained experimentally) of most NEs have a complex form, therefore, their representation by analytical expressions is a rather difficult task. As a rule, it makes little sense to design systems for the analysis and processing of signals using high-precision formulas if the reduction of the calculation error and the corresponding complication of the systems does not give a tangible effect in increasing the accuracy of data processing. In all these conditions, an approximation problem arises - the representation of the original complex functions by simple and convenient for practical use relatively simple functions (or a set of them) in such a way that the deviation from in the region of its assignment is the smallest according to a certain approximation criterion. The functions are called approximation functions. Finding the analytical function from the experimental current-voltage characteristic of a nonlinear element is called approximation.

In radio engineering and the theory of information transmission, several methods of approximating the characteristics of NEs are used - power-law, exponential, piecewise-linear (linear-broken line). s m polynomial and piecewise linear approximation of complex functions.

Power polynomial approximation of I - V characteristic

This type of approximation is especially effective at low amplitudes of input signals (as a rule, fractions of a volt) in those cases when the characteristic of the NE has the form of a smooth curve, i.e. the curve and its derivatives are continuous and have no jumps. Most often, when approximating, the Taylor series is used as a power polynomial:

where are constant coefficients;

- the voltage value, relative to which the expansion is carried out in a series and called operating point.

The constant coefficients of the Taylor series are determined by the well-known formula

. .................. (3)

The optimal number of terms of the series is taken depending on the required accuracy of the approximation. The more members of the series are selected, the more accurate the approximation. Typically, it is possible to approximate the characteristics with sufficient accuracy by a polynomial not higher than the second or third degree. To find the unknown coefficients of series (2), it is necessary to specify a range, several possible voltage values \u200b\u200band the position of the operating point in this range. If it is required to determine the coefficients of a series, then on a given characteristic, points with their coordinates are selected. To simplify calculations, one point is combined with a working point that has coordinates; two more points are selected at the boundaries of the range and. The rest of the points are placed arbitrarily, but taking into account the importance of the approximated section of the I - V characteristic. Substituting the coordinates of the selected points into formula (2), a system of equations is formed, which is solved with respect to the known coefficients of the Taylor series.

LECTURE number 16

APPROXIMATION OF YOUR NONLINEAR ELEMENTS. METHODS FOR CALCULATION OF NON-LINEAR ELECTRIC CIRCUITS

Study questions

1. Approximation of I - V characteristics of nonlinear elements. Polynomial approximation.

2. Piecewise linear approximation.

3. Classification of methods for the analysis of nonlinear circuits.

4. Analytical and numerical methods for the analysis of nonlinear DC circuits.

7. Current in a nonlinear resistor when exposed to a sinusoidal voltage.

8. Basic transformations carried out by means of nonlinear electrical circuits of alternating current.

1. Approximation of current-voltage characteristics of nonlinear elements

Current-voltage characteristics of real elements of electrical circuits usually have a complex form and are presented in the form of graphs or tables of experimental data. In a number of cases, the direct application of the I – V characteristics specified in this form turns out to be inconvenient and they tend to be described using rather simple analytical relations that qualitatively reflect the nature of the I – V characteristics under consideration.

Replacing complex functions with approximate analytical expressions is calledapproximation .

Analytical expressions that approximate the I - V characteristics of nonlinear resistive elements should describe the course of real characteristics as accurately as possible.

Consequently, the problem of approximating the I - V characteristic includes two independent problems:

1) the choice of the approximating function;

2) determination of the values \u200b\u200bof the constant coefficients included in this function, two types of approximation of the I - V characteristic of nonlinear elements are most often used:

Polynomial;

Piecewise linear.

1.1. Polynomial approximation

Approximation by a power polynomial is performed based on the Taylor series formula for the I - V characteristic of NE:

those. The I - V characteristic in this case must be continuous, unambiguous and absolutely smooth (must have derivatives of any order).

In practical calculations, the I – V characteristics are usually not differentiated, but require, for example, that the approximating curve (16.5) passes through the given currents.

In the so-called three-point method, it is necessary that some three points of the I - V characteristic:

(i 1 , u 1), (i 2 , u 2), (i 3 , u 3) - corresponded to the face value (16.5) (Figure 16.9).

From equations

it is easy to find the required coefficients a 0 , a 1 , a 2, since system (16.6) is linear with respect to them.

If the I - V characteristic is strongly indented and it is required to reflect its features, it is necessary to take into account a greater number of I - V characteristic points. A system like (16.6) becomes complicated, but its solution can be found by the Lagrange formula, which determines the equation of the polynomial passing through n points:

(16.7)

where A k ( u) = (u – u 1) ... (u – u k-1) ( u – u k + 1) ... ( u – u n).

Example... Let the nonlinear element have a VAC set graphically (Figure 16.10).

It is required to approximate the I - V characteristic of IE by a power polynomial.

Four points with coordinates are highlighted on the I - V characteristic:

Based on Lagrange's formula (16.7), we obtain

Thus, the approximating function has the form

and ne \u003d -6.7 i 3 + 30i 2 – 13,3i.

2. Piecewise linear approximation

When piecewise linearapproximation of the I - V characteristic of the NE is approximated a set of linear sections(pieces) near possible operating points.

Example... For two sections of the nonlinear I - V characteristic (Fig. 16.11) we get:

Example... Let it be required to linearize the I - V characteristic section between currents ANDand ATwhich is used as a work area near the work point R(fig. 16.12).

Then the equation of the linearized section of the I - V characteristic near the operating point is R will be

It is obvious that the analytical approximation of the I – V characteristic is correct only for the selected linearization section.

Figure 6.3

The first family of characteristics in (6.1) is called input characteristics, the second - output characteristics (it is assumed that pole 1 acts as the input of a nonlinear element, and pole 2 as an output). A general view of the input characteristics of the transistor is shown in Figure 6.3, b, the output characteristics are shown in Figure 6.3, c. Since the third family in (6.2) characterizes the effect of the output voltage on the input voltage, it is called the voltage feedback characteristic. The fourth family is the direct current transfer or throughput characteristics.

Like nonlinear two-pole networks, three-pole elements in the “small” signal mode are well described by differential parameters, which can be determined by differentiating the static characteristics. So, from the first family, the parameter can be found

which is called differential input impedance. Family 2 finds differential output conductance

With the help of nonlinear circuits, a number of problems that are very important for practice are solved. Let's note some of them.

1. Converting AC to DC. Devices that implement such a conversion are called rectifiers.

2. Converting DC to AC. It is produced using devices that are called autogenerators in radio engineering, and inverters in industrial electronics.

3. Frequency multiplication, that is, obtaining a voltage at the output of the device, the frequency of which is several times higher than the frequency of the input signal. This function is implemented in frequency multipliers.

4. Frequency converters - changing the frequency of the carrier wave without changing the type and nature of modulation.

5. Implementation of various types of modulation; devices that allow modulation are called modulators.

6. Demodulation of signals, that is, the extraction of a low-frequency control signal from a high-frequency oscillation; devices that perform demodulation are called demodulators or detectors.

7. Stabilization of voltage or current, that is, obtaining a voltage or current at the output of the device, which practically does not change in magnitude when the input voltage and load resistance vary in a wide range.

8. Waveform conversion; for example, sinusoidal to rectangular voltages.

9. Increased signal strength.

10. Conversion and storage of discrete signals.

Approximation of nonlinear characteristics

As noted in the previous section, the analytical form for representing the static characteristics of nonlinear elements is the most convenient for practical use. To obtain an analytical description of the characteristics, one of two approaches is used, as a rule. The first involves the analysis of the physical processes taking place in the element under consideration, the drawing up of equations describing these processes, and then the search for an analytical expression for the static characteristic by solving the equations. The advantage of this approach is that the resulting relations are characterized by parameters that have a specific physical meaning. However, this approach also has significant drawbacks. Firstly, sufficiently reliable information about the physical processes taking place in the element is required. Secondly, the equations describing internal processes in real elements are usually quite complex, their analytical solution is possible only with the introduction of significant simplifying assumptions. As a result, the obtained analytical expression may very little reflect the real static characteristic.

The second approach is based on the approximation of the characteristics of nonlinear elements found experimentally.

The operating modes of the elements can be different. In some modes, the currents and voltages of an element change only in a small vicinity of a certain point of rest, in other modes, the region of change in currents and voltages covers the entire characteristic or most of it. In accordance with this, the function approximating this characteristic should reproduce the working area with the greatest accuracy. The smaller the working section of the curve, the simpler the function that approximates this section of the characteristic can be chosen.

There are various ways to approximate:

1) linear;

2) nonlinear;

3) piecewise linear;

4) piecewise nonlinear.

Linear approximation is used when a nonlinear element operates in a small signal mode. Approximation of a nonlinear function in this case is carried out, as a rule, by a tangent drawn or calculated at the point of the characteristic, in the vicinity of which changes in currents and voltages occur. In the case of a nonlinear resistive two-terminal network, such an approximation can be interpreted as a replacement in the calculation of nonlinear resistance by a linear one, numerically equal to the differential resistance. The advantage of linear approximation is the ability to move from nonlinear circuit analysis to linear (linearized) circuit analysis, which is much simpler. The disadvantage is that the accuracy of such an approximation is low, and even in the mode of a small signal, the calculation error can be significant.

In nonlinear approximation, various power series are used most often.

Let us assume that some constant influence is applied to the nonlinear two-terminal device, which determines its initial operating mode. This effect will be called “displacement”. In this case, the value of the function at the starting point. If the initial action is changed by some amount, then, representing the new value of the function in the form of a Taylor series, we get

where are the values \u200b\u200bof the derivatives of the function f (x) at the point.

Since, instead of (6.3) we can write

The last relation is an expansion of the function f (x) in a Taylor series in the vicinity of a point and is an analytical description of the characteristic of an element. The resulting formula is a power series. The more members of the series are taken into account, the more accurately the actual characteristic will be expressed. Leaving the terms in the expansion, we get a polynomial of degree. Thus, the approximation of the characteristics by polynomials leads to the following equations:

a) if, then; (6.4)

b) if, then. (6.5)

The coefficients must be selected in such a way that the approximating equation describes the working section of the characteristic with acceptable accuracy. In order not to complicate the calculations, the number of terms of the approximating equations (6.4) and (6.5) try to limit as small as possible.

Along with power polynomials, other types of functions (exponential, trigonometric, etc.) can be used for nonlinear approximation. The advantages of this approach to obtaining an analytical description of nonlinear characteristics are, firstly, in the possibility of finding an arbitrarily exact expression and, secondly, in the absence of the need for knowledge about the principle of action of the element under consideration. Disadvantage - the coefficients of the approximating expressions have no physical meaning, their numerical values \u200b\u200bcannot be estimated and corrected from the general, theoretical provisions. A slight change in the course of the characteristic or consideration of the approximated area can lead to significant changes in the numerical values \u200b\u200bof the coefficients,.

In the practice of radio engineering calculations, the method of piecewise linear approximation is widely used. In this case, the characteristic of the nonlinear element is replaced by a set of straight line segments that coincide with the real curve with satisfactory accuracy. An example of a piecewise linear approximation of an N-shaped I – V characteristic is shown in Figure 6. 4. It is obvious that the approximating relations for each section will be different.

Figure 6.4

This method, while preserving the advantages of linear approximation, allows, in comparison with it, to significantly increase the accuracy of describing the characteristics and, at the same time, significantly simplifies the approximation process itself in comparison with nonlinear approximation.

The disadvantage of piecewise linear approximation is the complication of the algorithm for calculating the electrical circuit due to the need for constant control of the values \u200b\u200bof the variables. This procedure does not create difficulties if there is only one element in the analyzed circuit for which the piecewise linear approximation is used, but it may turn out to be excessively laborious with an increase in the number of such elements.

Piecewise nonlinear approximation is used in cases where none of the three considered approximation methods gives a satisfactory result either because of low accuracy or because of the complexity of the relations obtained (an excessively large number of terms in approximation by power polynomials, a very large number of segments for piecewise -linear approximation). Sometimes the piecewise nonlinear approximation is resorted to in cases when, as a result of the analysis of physical processes in the element, a relation is obtained that well describes a significant section of the static characteristic, but is poorly acceptable for any qualitative change in the operating mode of the nonlinear element (for example, the phenomenon of breakdown of an electron-hole transition in semiconductor devices). Quite often, such an approximation makes it possible to describe the characteristic with the required accuracy with a relatively small number of sections described by different ratios (as a rule, 2 - 3 sections).

When studying the properties of electrical circuits, the phenomenon of hysteresis, as a rule, can be neglected. Only when studying circuits based on this phenomenon (for example, the operation of magnetic storage devices with a rectangular hysteresis loop), hysteresis must be taken into account.

In fig. 15.11, a shows a typical symmetric characteristic y \u003d f (x).

For a nonlinear inductance, the role of x is played by the instantaneous value of induction, the role of y is the instantaneous value of the field strength H. For a nonlinear capacitor, y is the voltage - charge q. For nonlinear resistors (for example, tirite resistances), the role of x is played by the voltage, y is the current.

There are a large number of different analytical expressions, in one way or another, suitable for the analytical description of the characteristics of nonlinear elements. When choosing the most suitable analytical expression for the function y \u003d f (x), one proceeds not only from the fact that the curve described by the analytical expression should be sufficiently close with all its points to the experimentally obtained curve in the expected range of displacements of the working point on it, but take into account and the possibilities that the chosen analytical expression gives in the analysis of the properties of electrical circuits.

Further, for the analytical description of symmetric characteristics of the type in Fig. 15.11, but we will use the hyperbolic sine:

In this expression - numerical coefficients; a is expressed in those units that - in units inverse to units so that the product is a dimensionless quantity. To determine the unknown coefficients, one should arbitrarily choose the two most characteristic points through which the analytical curve should pass on the experimentally obtained dependence y \u003d f (x) in the assumed operating range, substitute the coordinates of these points into equation (15.1) and then solve the system of two equations with two unknowns.

Let the coordinates of these points (Fig. 15.11, a). Then

Attitude

The transcendental equation (15.2) is used to determine the coefficient. Consequently,

Example 147. The magnetization curve of transformer steel is shown in Fig. 15.11, b. Find the coefficients a and.

Decision. Select two points on the curve:

According to equation (15.2), we have We set arbitrary values \u200b\u200band make calculations:

Based on the results of the calculations, we build a curve and find it from it. Next, we define

The dotted line in Fig. 15.11, b is built according to the equation. § 15.14. The concept of Bessel functions. When analyzing nonlinear circuits, Bessel functions are widely used, which are a solution to the Bessel equation

Bessel functions are expressed by power series and tables are compiled for them. The Bessel function from the argument is denoted, where is the order of the Bessel function. The general expression for in the form of a power series can be written as follows:

Table 15.1

It is often necessary to have analytical expressions for the current - voltage characteristics of nonlinear elements. These expressions can only approximately represent the I – V characteristic, since the physical laws that govern the relationship between voltages and currents in nonlinear devices are not expressed analytically.

The problem of an approximate analytical representation of a function, given graphically or by a table of values, within a given range of variation of its argument (independent variable) is called approximation. In this case, firstly, a choice is made of an approximating function, that is, a function with which a given dependence is approximately represented, and, secondly, a criterion for assessing the "proximity" of this dependence and its approximating function is selected.

Most often, algebraic polynomials, some fractional rational, exponential and transcendental functions, or a set of linear functions (straight line segments) are used as approximating functions.

We will assume that the I - V characteristic of a nonlinear element i= fun (u)set graphically, i.e. defined at each point of the interval U min≤ and≤ U max,and is a single-valued continuous function of the variable and.Then the problem of analytical representation of the current-voltage characteristic can be considered as the problem of approximating a given function ξ (x) by a chosen approximating function f(x).

On the proximity of the approximating f(x) and approximated ξ ( x) functions or, in other words, the approximation error, is usually judged by the largest absolute value of the difference between these functions in the approximation interval and≤ x≤ b,that is, the largest

Δ \u003d max\u200c\u200c│ f(x)- ξ( x)│

The closeness criterion is often the mean square value of the difference between the specified functions in the approximation interval.

Sometimes, under the proximity of two functions f ( x) and ξ ( x) understand the coincidence at a given point

x \u003d Hothe functions themselves and p+ 1 of their derivatives.

The most common way to approximate an analytical function to a given one is interpolation(the method of selected points), when the functions f ( x) and ξ ( x) at the selected points (y evils of interpolation) X k, k= 0, 1, 2, ..., p.

The approximation error can be achieved the smaller, the more the number of varied parameters is included in the approximating function, i.e., for example, the higher the degree of the approximating polynomial or the greater the number of line segments contains the approximating polygonal line function. At the same time, of course, the volume of computations grows, both in solving the approximation problem and in the subsequent analysis of a nonlinear circuit. The simplicity of this analysis, along with the features of the approximated function within the approximation interval, is one of the most important criteria when choosing the type of the approximating function.

In the problems of approximating the current-voltage characteristics of electronic and semiconductor devices, as a rule, there is no need to strive for high accuracy of their reproduction due to the significant scatter of the characteristics of devices from sample to sample and the significant influence of destabilizing factors on them, for example, temperature in semiconductor devices. In most cases, it is enough to "correctly" reproduce the general average character of the dependence i= f(u) within its working range. In order to be able to analytically calculate circuits with nonlinear elements, it is necessary to have mathematical expressions for the characteristics of the elements. These characteristics themselves are usually experimental, i.e. obtained as a result of measurements of the corresponding elements, and then on this basis reference (typical) data are formed. The procedure for the mathematical description of a given function in mathematics is called the approximation of this function. There are a number of types of approximation: by selected points, by Taylor, by Chebyshev, etc. Ultimately, it is necessary to obtain a mathematical expression that satisfies the original approximating function with some given requirements.

Consider the simplest way: the method of selected points or nodes of interpolation by a power polynomial.

It is necessary to determine the coefficients of the polynomial. To do this, select (n + 1)points on a given function and a system of equations is drawn up:

From this system, the coefficients are found a 0, a 1, a 2, ..., a n.

At the selected points, the approximating function will coincide with the original one, at other points it will differ (strongly or not - depends on the power polynomial).

An exponential polynomial can be used:

Second method: taylor approximation method ... In this case, one point is selected, where the original function will coincide with the approximating one, but in addition, a condition is set so that the derivatives also coincide at this point.

Butterworth approximation: the simplest polynomial is selected:

In this case, you can determine the maximum deviation ε at the edges of the range.

Chebyshev approximation: is a power law, there is a coincidence at several points and the maximum deviation of the approximating function from the original is minimized. In the theory of approximation of functions, it is proved that the largest deviation of the polynomial in absolute value f(x) degree pon the continuous function ξ ( x) will be minimal if in the approximation interval and≤ x≤ bdifference

f ( x) - ξ( x) is not less than n + 2times takes its successively alternating limiting maximum f(x) - ξ( x) = L\u003e0 and the smallest f(x) - ξ( x) = -L values \u200b\u200b(Chebyshev test).

In many applied problems, polynomial approximation by the mean-square criterion of proximity is used, when the parameters of the approximating function f(x) are selected from the condition of minimizing in the approximation interval and≤ x≤ bsquared function deviation f(x) on a given continuous function ξ ( x), i.e., from the condition:

Λ= 1 / b-a∫ a [ f(x)- ξ( x)] 2 dx \u003d min. (7)

In accordance with the rules for finding extrema, the solution to the problem is reduced to solving a system of linear equations, which is formed as a result of equating to zero the first partial derivatives of the function Λ for each of the desired coefficients a kapproximating polynomial f(x), i.e., the equations

dΛ ∕ yes 0=0; dΛ ∕ yes 1=0; dΛ ∕ yes 2=0, . . . , dΛ ∕ yes n=0. (8)

It is proved that this system of equations also has a unique solution. In the simplest cases, it is found analytically, and in the general case - numerically.

Chebyshev established that the equality should be fulfilled for maximum deviations:

In engineering practice, the so-called piecewise linear approximation Is a description of a given curve by straight line segments.

Within each of the linearized sections of the volt-ampere characteristic, all methods of analyzing oscillations in linear electrical circuits are applicable. It is clear that the more linearized sections a given current-voltage characteristic is divided into, the more accurately it can be approximated and the greater the amount of calculations in the course of analyzing oscillations in the circuit.

In many applied problems of the analysis of oscillations in nonlinear resistive circuits, the approximated volt-ampere characteristic in the approximation interval is represented with sufficient accuracy by two or three straight line segments.

Such an approximation of the current - voltage characteristics in most cases gives quite satisfactory accuracy results for the analysis of oscillations in a nonlinear resistive circuit with "small" impacts on the nonlinear element, that is, when the instantaneous values \u200b\u200bof currents in the nonlinear element vary within the maximum permissible limits from I \u003d 0 to I = I max