Lecture No. 16.

Approximation of batteries of nonlinear elements. Methods for calculating NEDINION electrical circuits

Curriculum

1. Approximation of batteries of nonlinear elements. Polynomial approximation.

2. Partly linear approximation.

3. Classification of analysis methods is not linear chains.

4. Analytical and numerical methods for analyzing nonlinear DC circuits.

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

8. Basic transformation carried out using nonlinear electrical chains alternating current.

1. Approximation of volt-ampere characteristics of nonlinear elements

Volt-ampere characteristics of real elements of electrical circuits usually have a complex look and are presented in the form of graphs or tables of experimental data. In some cases, the immediate use of the Wahs specified in this form is inconvenient and they are striving to describe with sufficiently simple analytical relations, qualitatively reflecting the character of the Wah.

Replacing complex functions by approximate analytical expressions is calledapproximation .

Analytical expressions, approximating nonlinear resistive elements, should as accurately describe the course of real characteristics as accurately.

Consequently, the task of approximation Wah includes two independent tasks:

1) Choosing an approximating function;

2) Determining the values \u200b\u200bof the values \u200b\u200bof permanent coefficients included in this function, two types of approximation of batteries of nonlinear elements are most common:

Polynomial;

Piecewise linear.

1.1. Polynomial approximation

Approximation of power polynomials is performed on the basis of the formula of a series of Taylor for the WAT NE:

those. Wah in this case should be continuous, unambiguous and absolutely smooth (should have derivatives of any order).

In practical calculations, it is usually not differentiated, and require, for example, so that the approximating crying (16.5) passed through the specified currents.

In the so-called method of three points, it is necessary that some three points of the Wah:

(i. 1 , u. 1), (i. 2 , u. 2), (i. 3 , u. 3) - responded with rates (16.5) (Fig.16.9).

From equations

easy to find the desired coefficients a. 0 , a. 1 , a. 2, since relative to their system (16.6) linear.

If the Wah is strongly cut and it is necessary to reflect its features, it is necessary to take into account the larger number of points of the Wah. The system of type (16.6) becomes complex, but its solution can be found according to the Lagrange formula, which determines the polynomial equation 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 the VAC specified graphically (Fig.16.10).

It is required to approximate the effects of IE power polynomial.

There are four points with coordinates on the graph of the WAY:

Based on the formula Lagrange (16.7) we get

Thus, the approximating function has the form

and NE \u003d -6,7 i. 3 + 30i. 2 – 13,3i..

2. piecewise linear approximation

For piecewise linearapproximation of Wah Na approximated a combination of linear sites(pieces) near Possible working points.

Example. For two sections of nonlinear Wah (Fig.16.11) we get:

Example. Let it be required to linearize the section of the Wah between currents BUTand INused as a working area near the working point R(Fig.16.12).

Then the equation of the linearized area of \u200b\u200bWah near the working point R will be

It is obvious that the analytical approximation of the Wah is true only for the selected linearization site.

Academy of Russia

Department of Physics

Abstract on the topic:

"Approximation of the characteristics of nonlinear elements and analysis of chains in harmonic effects"

Curriculum

1. Approximation of the characteristics of nonlinear elements

2. Grafo-analytical and analytical methods of analysis

3. Analysis of the circuits by the cut-off angle

4. The impact of two harmonic oscillations on rampant

nonlinear element

Literature

Introduction

For all previously discussed linear chains, the principle of superposition, from which it follows a simple and important consequence: a harmonic signal, passing through the linear stationary system, remains unchanged in form, acquiring only other amplitudes and the initial phase. That is why the linear stationary chain is not able to enrich the spectral composition of the input oscillation.

A feature of the NE, compared with linear, is the dependence of the NE parameters from the values \u200b\u200bof the applied voltage or the flow of the flowing current. Therefore, in practice, when analyzing complex nonlinear circuits, various approximate methods are used (for example, a nonlinear circuit linear in the area of \u200b\u200bsmall changes in the input signal is replaced and linear analysis methods are used) or are limited to high-quality conclusions.

An important property of nonlinear electrical circuits is the ability to enrich the spectrum of the output signal. This important feature is used in constructing modulators, frequency converters, detectors, etc.

The solution of many tasks associated with the analysis and synthesis of radiotechnical devices and chains requires knowledge of processes occurring while simultaneously exposed to a nonlinear element of two harmonic signals. This is due to the need to multiply two signals when implementing devices such as frequency converters, modulators, demodulators, etc. It is natural that the spectral composition of the output current of the NE with bicaronic effects will be much richer than with monogarmonic.

Often there is a situation when one of the two signals acting on the NE is small in amplitude. Analysis in this case is greatly simplified. We can assume that with respect to the small NE signal is linear, but with a variable parameter (in this case, the steepness of the WAH). Such a mode of operation of the NE is called parametric.

1. Approximation of the characteristics of nonlinear elements

When analyzing nonlinear chains (NCs), the processes occurring inside the elements constituting this chain are not considered, and are limited only by external characteristics. This is usually the dependence of the output current from the applied input voltage

, (1)

which is customary to be called a volt-ampere characteristic (VAC).

The simplest is to use the existing table form of the Wah for numerical calculations. If the analysis of the chain should be carried out by analytical methods, the task of the selection of such a mathematical expression arises, which would reflect all the most important features of experimentally removed characteristics.

This is nothing more than the task of approximation. In this case, the choice of the approximating expression is defined both the character of nonlinearity and the calculated methods used.

Real characteristics have enough sophisticated view. It makes them accurate mathematical description. Moreover, table form VAH representations makes discrete characteristics. In the intervals between these points, the values \u200b\u200bof the Wah are unknown. Before switching to approximation, it is necessary to somehow decide on unknown values \u200b\u200bof the Wah, make it continuous. There is an interpolation task (from Lat. inter. - between, polio. - Smooth) - This is the finding of the intermediate values \u200b\u200bof the function according to some known values. For example, finding values

At points lying between points according to known values. If, a similar procedure is an extrapolation problem.

It is usually approximated by only the part of the characteristic that is a working area, i.e. within the limits of changing the amplitude of the input signal.

When approximating the Volt-ampere characteristics, two tasks must be solved: select a specific approximating function and determine the corresponding coefficients. The function should be simple and at the same time enough to transmit the approximated characteristic. The determination of the coefficients of the approximating functions is carried out by interpolation, the standard or uniform approximation, which are considered in mathematics.

Mathematically, the setting of the problem of interpolation can be formulated as follows.

Find a polynomial

degrees no more n. such that i. = 0, 1, …, n.if the initial function values \u200b\u200bare known in fixed points, i. = 0, 1, …, n.. It is proved that there is always only one interpolation polynomial, which can be represented in various forms, for example in the form of Lagrange or Newton. (Consider independently on self-preparation according to the recommended literature).

Approximation by power polynomials and piecewise linear

It is based on the use of a series of Taylor and Maclogen, well-known from the course of the highest mathematics and lies in the decomposition of nonlinear Wah

In an infinite-dimensional row, which is in some surroundings of the working point. Since such a number are not physically implemented, it is necessary to limit the number of members of the number, based on the required accuracy. The power approximation is used at a relatively small change in the amplitude of the impact relative.

Consider a typical form of any NE (Fig. 1).

Voltage

Defines the position of the working point and, therefore, the static mode of operation of the NE.

Fig. 1. An example of typical Wah Na

It is usually approximated by all the characteristics of the NE, but only the working area, the size of which is determined by the amplitude of the input signal, and the position on the characteristic - the amount of constant displacement

. Approximating polynomial is written in the form, (2)

where coefficients are

Defined expressions.

Approximation of power polynomial is to find the coefficients of a number

. For a given form, these coefficients are significantly dependent on the choice of the working point, as well as the width of the characteristic site used. In this regard, it is advisable to consider some of the most typical and important cases for practice.

1. The operating point is located in the middle of the linear section (Fig. 2).

Fig. 2. Working point Wah - in the middle of the linear section

Plot on the characteristic where the law of change of current is close to linear, relatively non-sequins, therefore the amplitude of the input voltage

should not go beyond this area. In this case, you can write :, (3) - the current of rest; ; - Differential steepness characteristics.

This case is applicable only at a weak signal.

Usually, Wah nonlinear elementsI \u003d F (U) get experimentally Therefore, most often they are specified in the form of tables or graphs . To deal with analytical expressions have resort to approximation.

Designated table or graphic Wah nonlinear elementi \u003d F V (U), but analytical function, approximatingspecified characteristic i \u003d f (u, a 0, a 1, a 2, ..., a n ). Where a 0, a 1, ..., a n – factors this function, which need to find As a result of approximation.

A) in the Chebyshev method Factors a. 0 , a. 1 , … , a. N Functions F (U) are located from the condition:

i.e. they determined in the process of minimizing the maximum evasion of the analytical function from the specified one. Here u k, k \u003d 1, 2, ..., g - selected voltage values u.

Under the rms approximation Factors a. 0 , a. 1 , …, a. N. should be such to minimize the amount:

, (2.6)

B) approximation of the function by Taylor based on the presentation Functions i \u003d f (u) near Taylor in the neighborhood of Point \u003d U 0:

and determining coefficients of this decomposition. If a restrict ourselves to the first two member decomposition In a series of Taylor, then we will talk about the replacement of a complex nonlinear dependence F (U) more simple linear addiction . Such substitution is called linearization of characteristics.

First Member of decomposition F (u 0) \u003d i 0 represents d.C. at work point for u \u003d u 0, but second C.linen

differential Volt-Amplist Speech Speed \u200b\u200bat Workpoint , i.e. with u \u003d U 0 .

IN) Most common way of approximation specified function is interpolation (method of selected points), with which Factors a. 0 , a. 1 , …, a. N approximating function F (U) are from the equality of this function and the specified F X (U) in selected points (interpolation nodes) u k \u003d 1, 2, ..., n + 1.

E) Power (polynomial ) Approximation. This name received approximation of Wah power polynomials:

Sometimes it can be convenient to solve the task of approximation specified characteristic in the neighborhood of Point 0 , called workers. Then use power polynomials

Power Approximation wide used when analyzing Nonlinear works devices to which are applied relatively small external influences , so it requires quite accurate reproduction of nonlinearity characteristics. in the surroundings of the working point.

E) piecewise linear approximation. In cases where the nonlinear element affects voltages with large amplitudes, you can allow more approximate replacement of the characteristics of the nonlinear element and use more simple approximating functions . Most often When analyzing the work of the nonlinear element in this mode Real characteristic is replaced segments of straight lines with different inclons .

From a mathematical point of view, this means that in each replaceable area, the characteristics are used by the 14-degree polynomials ( N \u003d 1. ) with different values \u200b\u200bof coefficients a. 0 , a. 1 , … , a. N.

In this way, the task of approximation of the batteries of nonlinear elements is to choose a type of approximating function and determining its coefficients One of the above methods.

As a rule, the batteries of nonlinear elements are obtained experimentally; It is less likely to find them from theoretical analysis. To study, it is necessary to choose the function of approximation such, which, being quite simple, would reflect all possible features of the experimental feature with a sufficient degree of accuracy. The most often use the following methods of approximation of the volt-amps characteristics of two-pole: piecewise linear, power, indicative approximation.

Piecewise linear approximation

Such an approximation is usually used in calculating processes in nonlinear equations in the case of large amplitudes of external influences. This method Based on the approximation of the characteristics of nonlinear elements, i.e. On an approximate replacement of the real characteristics of straight lines with different inclons. The figure shows the input characteristic of the real transistor, approximated by two sections of direct.

Approximation is determined by two parameters - voltage of the beginning of the characteristics of the UAN and the steepness S. The mathematical form of the approximated Wahova is:

The voltage of the beginning of the input characteristics of bipolar transistors has an order of 0.2-0.8 V: the steepness of the current characteristics of the base of the base of the iB (UBE) is about 10mA / V. Crudyness of the characteristic of the IK (UBE) of the collector current depending on the voltage base-emitter, then the value of 10mA / V must be multiplied by the H21E - the gain of the base current of the base. Since H21E \u003d 100-200, the specified steepness has the order of several amp to volts.

Power Approximation

The power approximation is widely used when analyzing the operation of nonlinear devices to which relatively small external influences are applied. This method is based on the decomposition of the nonlinear volt-amps characteristic I (U) in a series of Taylor, which moves in the vicinity of the working point U0.

the number of decomposition members depends on the specified accuracy. Consider example:

Input characteristic of the transistor. Working point U0 \u003d 0.7V. We select 0.5 as nodes of approximation of the point 0.5; 0.7 and 0.9 V.

It is necessary to solve the system of equations:

Spectral composition of current in nonlinear element with external harmonic effects

Consider a circuit consisting of a sequential connection of the source of the harmonic signal Us (T) \u003d Coswt, the source of constant voltage of the displacement U0 and an irrigate nonlinear element. To do this, consider the drawing.

The current in the chain has a sinusoidal shape.

The current and voltage form are different.

The cause of the current curve is simple: unequal increments of the voltage correspond to the same voltage increments, because , and the differential steepness of the Wah in different sections is different.

Consider the task analytically.

Let us know the nonlinear function I (U) \u003d I (UC, U0). The nonlinear element acts the voltage of the signal UC (T) \u003d UMCOS (WT + J).

The dimensionless value x \u003d wt + j, then i (x) \u003d i (umcosx, u0) is a periodic function relative to the argument x with a period of 2t. Imagine her next to Fourier With coefficients .

Function I (x) is even, so the Fourier series will contain only cosine components: .

Amplitude coefficients of harmony

Two recent formulas give common decision The tasks about the current spectrum in the nonlinear element with harmonic external exposure:

those. The current, except for the constant component I0, contains an infinite sequence of harmony with amplitudes in. Harmony amplitudes depend on the parameters UM and U0, as well as on the type of approximating function.

Consider how it depends on the type of approximating function.

Piecewise linear

i (U) \u003d

The voltage u (t) \u003d u0 + umcoswt is supplied.

The current schedule has a form of cosine pulses with a cut-off. The cut-off angle of current pulses is determined from the equality:

U0 + umcosq \u003d UAN þ .

Power approximation.

Let in the vicinity of the working point U0 Wah nonlinear element

Many of the most important processes (non-linear amplification, modulation, detection, generation, multiplication, division and frequency conversion) is carried out in radio electronic devices Using nonlinear and parametric circuits.

In general, the analysis of the signal conversion process in nonlinear circuits is a very complex task, which is associated with the problem of solving nonlinear differential equations. In this case, the principle of the superposition is not applicable, since the parameters of the nonlinear chain when exposed to one source of the input signal differ from its parameters when several sources are connected. However, the study of nonlinear chains can be carried out relatively simple methodsIf the nonlinear element meets the conditions of rapidness. Physically the raylessness of the nonlinear element (NE) means an instantaneous response to its output after changing the input effects. If you say strictly, then the idlenetic nonlinear elements practically does not exist. All nonlinear elements - diodes, transistors, analog and digital chips 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 their idleness.

Most nonlinear radio engineering chains and devices are determined by the structural circuit shown in Fig. 2.1. According to this scheme, the input signal directly affects the nonlinear element, the filter (linear chain) is connected to the output.

Picture. 2.1. Structural scheme nonlinear device.

In these cases, the process in the radio electronic nonlinear chain can be characterized by two operations independently from each other. As a result of the first operation in an idle nonlinear element, there is such a conversion of the input signal form, in which new harmonic components appear in its spectrum. The second operation is performed by a filter, highlighting 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, autogenerators, rectifiers, multipliers, divisors and frequency converters are reduced to such a convenient theoretical model.

As a rule, nonlinear chains are characterized by complex dependence between the input signal and the output reaction that general You can write like this:

U out (t) \u003d f

In nonlinear chains with non-irresolute NE, it is most convenient as an impact to consider the input voltage U Vx (T), and the response - the output current I (T), the relationship between which is determined by nonlinear functional dependence:

i out (t) \u003d f

This ratio can analytically constitute a conventional volt-ampere characteristic of the NE. This characteristic has a nonlinear two-pole (transistor, OU, a digital chip), operating in nonlinear mode with different input amplitudes. Volt-ampere characteristics (for nonlinear elements they are obtained experimentally0 most nonlinear elements have a complex view, so the representation of their analytical expressions is a rather difficult task. In electronic devices, analytical methods of representation are widely used. nonlinear characteristics Different devices with relatively simple features (or their set) approximately reflecting real characteristics. Finding an analytical function on the experimental characteristic of the nonlinear element is called approximation. There are several ways to approximate characteristics - power, indicative, piecewise linear (linearly broken approximation). The highest distribution was obtained by approximation by power polynomial and piecewise linear approximation.

Approximation by power polynomial.This type of approximation is particularly effective at low amplitudes (as a rule, the proportion of Volta) input signals in cases where the characteristic of the NE has a form of a smooth curve, i.e. The curve and its derivatives are continuous and have no jumps. Most often, during approximation, a series of Taylor is used as a power polynomial

i (u) \u003d a o + a 1 (u-u o) + a 2 (u-u o) 2 + ... + a n (u-u o o) n, (2.1)

where a o, a 1, ... a n is constant coefficients; U O - the value of the voltage U, relative to which there is a decomposition in a row and called working point.Note that here and then the argument for the functions of the current and voltage to simplify is omitted. The constant coefficients of the Taylor series are determined by the known formula

The optimal number of row members is taken depending on the pipe accuracy of the approximation. The more chosen members of the number, the more accurate approximation. Approximation of the characteristics is usually possible to accurately accomplish the polynomial not higher than the second one - the third degree. To find unknown row factors, it is necessary to set the range U 1, U 2 of several possible values \u200b\u200bof the voltage U and the position of the working point Uau in this range. If it requires to define N coefficients, then N + 1 points with its coordinates are selected on a given characteristic (I N, U N). To simplify the calculations, one point is combined with the operating point U o, which has coordinates (I o, U o); Two two points are selected at the boundaries of the range U \u003d U 1 and U \u003d U 2. The remaining points are arbitrarily arbitrarily, but taking into account the importance of the approximated section of the Wah. Substituting the coordinates of the selected points in formula (2.1), they make up the system of N + 1 equations, which is solved relative to unknown coefficients A N of a series of Taylor.

Fig.2.2. Approximation of the characteristics of the transistor power polynomial.

Example 2.1. In fig. 2.2 The dash line is represented input characteristic i B \u003d F (U BE) of the CT601A transistor. Approximate the predetermined characteristic of the transistor in the range of 0.4 ... 0.8 in the polynomial of Taylor second degree i B \u003d AO + A 1 (U BE -U O) + A 2 (U BE -U O) 2 relative to the operating point U o \u003d 0 , 6 V.

Decision. To simplify the calculations as points of approximation, select the voltage values \u200b\u200bat the boundaries of the range and at the operating point, i.e. 0.4; 0.6 I.

0.8 V. Since the selected points correspond to currents 0.1; 0.5 and 1.5 mA, then for a given polynomial, we obtain the following system of equations:

0.1 \u003d A O + A 1 (0.4-0.6) + A 2 (0.4-0.6) 2 \u003d A O --0.2A 1 +0.04 A 2

0.5 \u003d a o + a 1 (0.6-0.6) + a 2 (0.6-0.6) 2 \u003d a o

1.5 \u003d a O + A 1 (0.8-0.6) + a 2 (0.8-0.6) 2 \u003d a o + 0.2a 1 +0.04 A 2

The solution of this system of equations gives the values \u200b\u200bof the coefficients A o \u003d 0.5 mA, a 1 \u003d 3.5 mA / B, a 2 \u003d 7.5 mA / in 2. Substituting them in formula (2.1), we find an approximating function (its schedule is shown in the figure with a solid line): i b \u003d 0.5 + 3.5 (U b -0.6) +7.5 (U b -0.6) 2.

Piecewise linear approximation. In most practical cases, when the input signal of the radio electronic chain is affected by the input signal. Significant amplitude, the real volt-ampere characteristic of the nonlinear element can be approximated by a piecewise linear line consisting of several segments of direct with different angles of inclination to the abscissa axis. This approximation is associated directly with two important parameters the nonlinear element - the voltage of the beginning of the characteristics E H and its steepness S. In the general case, the differential steepness of the characteristics at the operating point is determined by the ratio of the increment of the current to the increment of the voltage, and with their small values, we have

The equation of a straight line with piecewise linear approximation The characteristic is written in the form:

i \u003d (0, u

i \u003d (s (u-e n), u\u003e e n (2.4)

In many radiotechnical devices, the characteristic of the nonlinear item to which the signal of a large amplitude is supplied with an acceptable accuracy to approximate only two sections of straight lines.

Example 2.2. An experimentally removed input characteristic I B \u003d F (U BE) of the CT601A transistor is represented in Fig. 2.3. Strike line. Perform a piecewise linear approximation of this characteristic in the vicinity of the working point U o \u003d 0.6 V.

Decision. In accordance with a given voltampear characteristic of the transistor, we find that the value of the current at the operating point I o \u003d 0.5 mA. The steepness of the characteristics at the operating point is calculated approximately by formula (2.3). Setting the linear increment of the voltage ΔU BE \u003d 0.8 - 0.6 \u003d 0.2 B, we find the increment of current Δi B \u003d

1.5-0.5 \u003d 1 mA. Then s \u003d δi b / ΔU b \u003d 1 / 0.2 \u003d 5 mA / c.

Fig.2.3. Partly linear approximation of the characteristics of the transistor.

As a result of an approximation, the characteristics of the transistor base current in the area of \u200b\u200bthe working point with coordinates О \u003d 0.5 mA, U o \u003d 0.6 V. Determined as: i B \u003d 0.5 + 5 (U BE -0.6) \u003d 5 (U BE -0.5).

From this formula it follows that when U BE<0,5 В ток базы транзистора должен принимать отрицательные значения, что не отражается заданной характеристикой. Значит, полученная функция будет аппроксимировать заданную зависимость только при амплитуде входного напряжения u бэ >0.5 V. If the input voltage U BE<0,5 В, то можно принять i б =0. Таким образом, аппроксимирующая функция (сплошная линия на рисунке), отражающая характеристику транзистора, запишется в следующем виде:

i \u003d (0, u be<0,5

i \u003d (5 (U BE -0.5), u be\u003e 0.5

Improving the accuracy of approximation of the characteristics of nonlinear elements is achieved by increasing the number of lines segments. However, this complicates the analytical expression of the approximating function.

Lecture number 9.

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