The characteristics of real nonlinear elements that are usually determined by experimental studies have a complex look and are presented in the form of tables or graphs. At the same time, an analytical representation of the characteristics is needed to analyze and calculate the chains, i.e. Representation in the form of sufficiently simple functions. The process of compiling an analytical expression for the characteristics presented graphically or tables, is called approximation.
When approximated, the following problems are solved:
1. Determining the approximation area, which depends on the range of changes in the input signals.
2. Determination of the accuracy of approximation. It is clear that the approximation gives an approximate representation of the characteristics in the form of any analytical expression. Therefore, it is necessary to quantify the degree of approximation of the approximating function to an experimentally definite characteristic. Most often used:
an indicator of uniform approximation - the approximating function should not differ from the specified function by more than a number, i.e.
the average quadratic approximation rate is an approximating function should not differ from a given function in an average quadratic approximation of more than a number, i.e.
the nodal approximation (interpolation) is an approximating function must match the specified function at some selected points.
There are various ways of approximation. Most often, an approximation of a power polynomial and piecewise linear approximation is used for approximation, the approximation with the use of indicative, trigonometric or special functions (Bessel, Hermita, etc.).
7.2.1. Approximation by power polynomial
The nonlinear volt-ampere characteristic in the surroundings of the working point is a finite number of the terms of the Taylor series:
The number of members of the series is determined by the required approximation accuracy. The more members of the series, the more accurate approximation. In practice, the necessary accuracy is achieved using the approximation of the second and third polynomials. The coefficients are numbers that are simply determined from the WAT graph, which is illustrated by an example.
Example.
Approximate presented in Fig. 7.1, and Wah in the vicinity of the working point of the powerful polynomial of the second degree, i.e. polynomial type
Choose an approximation area from 0.2 V to 0.6 V. To solve the problem, it is necessary to determine the three coefficients. Therefore, we restrict ourselves to three nodal points (in the middle and at the boundaries of the selected range), for which we make a system of three equations:
Fig. 7.1. Approximation of Vakh Transistor
Solving the system of equations, determine ,. Consequently, an analytical expression describing the VH timetable has the form
Note that the approximation of the power polynomial is used mainly to describe individual characteristics fragments. With significant deviations of the input signal from the working point, the accuracy of approximation can deteriorate significantly.
If the Wah is specified not graphically, and any analytical function has the need to present it with a power polynomial, the coefficients are calculated according to the well-known formula
It is easy to see that it is a steepness of the Wah at the operating point. The steepness value significantly depends on the position of the operating point.
In some cases, it is more convenient to represent a number of McLoren
7.2.2. Piecewise linear approximation
If the input signal changes in magnitude in large limits, then you can approximate the broken line consisting of several straight segments. In fig. 7.1, B is shown by the transistor, approximated by three sections of straight lines.
Mathematical formula of approximated Wah
This type of approximation is associated with two important parameters of the nonlinear element: voltage began the characteristics and its steepness. To increase the accuracy of approximation, increase the number of lines segments. However, this complicates the mathematical formula of the Wah.
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
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 the 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
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 a rather complicated appearance. This makes it difficult to accurate mathematical description. In addition, the table form of the VAC is making 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) is the finding of intermediate values \u200b\u200bof the function according to some known values. For example, finding values \u200b\u200bat points lying between points according to known values. If a 
The same procedure is the task of extrapolation.
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 degree no more n such that 
i \u003d 0, 1, ..., n, if the values \u200b\u200bof the source function are known in fixed points, i \u003d 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 Taylor and McLoren rows known from the course of higher mathematics and lies in the decomposition of nonlinear Wah into an infinite-dimensional row, which is located at 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).
The voltage determines the position of the working point and, consequently, 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 workspace, 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
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.
For graph in Fig. 3, having accepted that the tree is formed by branches 2, 1 and 5 Answer: B \u003d solve problem 5, using relations (8) and (9). Theory / TOE / Lecture N 3. Presentation of sinusoidal values \u200b\u200busing vectors and integrated numbers. Alternating current did not find a practical ...
Second orders operating under the actions of random perturbations, and get analytical expressions for these systems, which is its dignity. In practice, use a combination of various methods. Analysis of the nonlinear mode of operation of the Chap system to determine some of the characteristics of the system, we will produce a qualitative analysis of the chap system (Fig. 1) Fig.1. Structural diagram nonlinear ...
In addition, you can create new documents in which the calculation will be calculated for other model parameters. 5.4. The performance of the program in Appendix 4 shows graphs for various parameters of the model of the reflector - modulator. According to these graphs, it can be seen that for the results calculated in Chapter 4, the consumption of results is about 20-30%, which, generally speaking, is a good result, since the conclusion ...
The genomes of plants caused by the FPU transformed human speech, which resonantly interacts with chromosomal DNA in vivo. This result, meaningful by us from the standpoint of a semiotico-wave component of the genetic code, has a significant methodological value for the analysis of such super-discovered objects as DNA texts, and for genome as a whole. Opened fundamentally ...
2.7.1 Nonlinear chains and approximation of the characteristics of nonlinear elements
Everything chains considered so fartreated to the class of linear systems. Elements of such chainsR, L and with are constant and do not depend on the impact.Linear chains are described by linear differential equations with constant coefficients.
If the elements of the electrical circuitR, L and with depend on impactT. the chain is described nonlineardifferential equation I.is nonlinear.For example, For oscillatoryRLC -Contraction, the resistance of which depends on the voltageu C, we get:
. (1)
Such oscillating contouris nonlinear.Electrical circuit element whose parametersdepend on the impact, called nonlinear. Distinguish resistive and reactive nonlinear elements.
For nonlinear resistive Element is characteristicnonlinear communication between current i and voltage u, i.e., nonlinear characteristici \u003d f (u). The most common resistive nonlinear elements are lamp and semiconductor devices used to enhance and convert signals. On thefigure 12.1 is given Vach type nonlinear element (semiconductor diode).
For resistive nonlinear elementsan important parameter is their resistance that unlike linear Resistors is not constant, but depends on which point is it is determined.
Figure 12.1 - WAH nonlinear element
Vakh. nonlinear elementyou can define resistanceas
(2)
where u 0 - attached to the nonlinear elementconstant pressure;
I 0 \u003d F (U 0) - chain flowingpermanent current. This is constant current resistance (or static). It depends on the applied voltage.
Let be The nonlinear element is valid Voltage U \u003d U 0 + U M COS W T, and the amplitude U M , variable component is enoughmala (Figure 12.2), so that a small section of the Wah within which the variable voltage is valid, can be considered linear. Then the current. flowing through a nonlinear element,repeats voltage: i \u003d i 0 + i m cos w t.
Determine the resistanceR DIF as aC amplitude ratioU M. aC amplitudeI M. (on the chart this is the ratio of voltage incrementD U. To the increment of currentD i):
(3)
Figure 12.2 - Impact of a small harmonic signal on a nonlinear element
it resistance is called differential (dynamic) and representsresistance to the nonlinear element of the variable current of the small amplitude.Usually go to the limit these increments and determinedifferential resistance in the formR diff \u003d du / di.
Instruments that have falling areas on the WA are called devices with negative resistance, since these areas derivativesdi / du.< 0 и du/di < 0.
Nonlinear reactive elements include non-linear capacity and nonlinear inductance. An example of a non-linear container can serve as any device with a nonlinear volt pendant characteristic.q \u003d F (U) (for example, variond and varicap). Nonlinear inductance is a coil with a ferromagnetic core, streamlined by a strong current, bringing a core to magnetic saturation.
One of the most important features of nonlinear chains is thatthey are not performedthe principle of imposition. therefore it is impossible to predict the result of the influence of the sum of the signals, if the chain reaction is known for each exposure to the effect.From what has been saidunsuitability for analyzing nonlinear chains of temporary and spectral methods, which were used in the theory of linear chains.
Indeed, let volt-ampere characteristics (VAC) of the nonlinear element is described by the expressioni \u003d a u 2. If on such element acts a complex signalu \u003d u 1 + u 2, then the response i \u003d a (u 1 + u 2) 2 \u003d a u 1 2 + a u 2 2 + 2 a u 1 u 2 differs from the amount of responses to the action of each component separately(A U 1 2 + A U 2 2) the presence of components2 A U 1 U 2, which appears only in the case of the simultaneous impact of both components.
Consider the seconddistinctive feature of nonlinear chains. Let U \u003d U 1 + U 2 \u003d U M1 COS W 0 T + U M2 COS W T,
where u m1 and u m2 - Voltage amplitudesu 1 and U 2.
Then current in nonlinear element With I \u003d A U 2 will look at:
(4)
Figure 12.3, spectra built Voltage and current. Everythingspectral components of the current turned out to be new, not contained in voltage. In this way,new spectral components arise in nonlinear circuits. In this sense, nonlinear chains have much greater possibilities than linear, and are widely used for signal transformations associated with changing their spectra.
When studying same theories of nonlinear chainsyou can not take into account the device of the nonlinear item and relying only on its external characteristics is similar to how, when studying the theory of linear chains, the device of the condenser resistors and coils does not consider and use them only by their parametersR, L and s.
Figure 12.3 - Voltage spectra and current of the quadratic nonlinear element
Stock Illustration Specified impact on the real semiconductor diode
2.7.2 Approximation of the characteristics of nonlinear elements
Usually, Wah nonlinear elementsi \u003d F (U) get experimentally Therefore, most oftenthey are specified in the form of tables or graphs. To deal with analytical expressionshave resort to approximation.
Denote set table or graphic Wah nonlinear elementi \u003d f v (u), and analytical function, but peproximatingthe specified characteristic, i \u003d f (u, a 0, a 1, a 2, ..., a n). where a 0, a 1, ..., a n - the coefficients of this function, which need to find As a result of approximation.
A) in the Chebyshev method The coefficients a 0, a 1, ..., a n function f (u) are from the condition:
, (5)
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 valuesu.
Under the rms approximation coefficients a 0, a 1, ..., a n should be so to minimize the amount
(6)
B) approximation of the function by Taylorbased on the presentation Functions i \u003d F (U) near Taylor in the vicinity of the pointu \u003d u 0:
(7)
and determining coefficients 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 dependenceF (U) more simple linear addiction. Such substitution is called linearization of characteristics.
First Member of decompositionF (u 0) \u003d i 0 representspermanent current at the working pointat u \u003d u 0, and the second h Len
- (8)
differential Volt-Amplist Speech Speed \u200b\u200bat Workpoint, i.e. at u \u003d u 0.
C) the most common way of approximation specified functionis interpolation (method of selected points),at which the ons of 0, a 1, ..., a n Approximating functionF (U) are from the equality of this function and the specifiedF X (U) in the selected points (interpolation nodes)u k \u003d 1, 2, ..., n + 1.
E) power (polynomial ) Approximation. This name receivedapproximation of Wah power polynomials:
(9)
Sometimes it can be convenient to solve the task of approximation specified characteristicin the surrounding pointU 0, called working. Then use power polynomials
(10)
Power Approximation wide used when analyzing Nonlinear worksdevices to which are applied relativelysmall external influences, so it requires quite accurate reproduction of nonlinearity characteristics. in the surroundings of the working point.
E) piecewise linear approximation. In cases wherethe nonlinear element affects voltages with large amplitudes,you can allow moreapproximate replacement of the characteristics of the nonlinear element and use more simple approximating functions. Most often When analyzing the work of the nonlinear elementin such a real mode characteristic is replacedsegments 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 coefficientsa 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.
The impact of the harmonic signal on the chain with a 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 circuits can be carried out relatively simple methods, if 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 diagram of 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 a complex dependence between the input signal and the output reaction, which in the general form can be written as:
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 are obtained, so the representation of their analytical expressions is a rather difficult task. In electronic devices, analytical methods of representing the nonlinear characteristics of various devices relatively simple functions (or their set) are widely used, approximately Relative characteristics reflecting the analytical function according to the experimental characteristic of the nonlinear element is called approximation. There are several ways to approximate the characteristics - a power, indicative, piecewise linear (linear-broken approximation). The highest distribution was obtained by approximation of 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 of the nonlinear element - voltage of the beginning of the characteristics E H and its steepness S. In the general case, the differential spinning characteristics at the operating point is determined by the ratio of the current increment 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. Similar information. In accordance with the definition of this method, the calculation of a nonlinear circuit with its use includes in the general case the following main steps: 1. The initial characteristic of the nonlinear element is replaced by a broken line with a finite number of rectilinear segments. 2. For each section of the broken, the equivalent linear parameters of the nonlinear element are determined and the corresponding linear replacement circuits of the original chain are drawn. 3. The linear task is solved for each segment separately. 4. Based on the boundary conditions, the time intervals of the image of the depicting point for each rectilinear section (the boundaries of the existence of individual solutions) are determined. Let the volt-ampere characteristics (WAH) of the nonlinear resistor have the shape shown in Fig. 1. Replacing it with a broken line 4-3- 0- 1- 2-5, we obtain the given in Table. 1 Estimated equivalent substitution schemes and linear ratios corresponding to them. Calculation of each of the resulting linear stages of substitution in the presence of one nonlinear element in the circuit and an arbitrary number linear is not possible. In this case, on the basis of the theorem on the active two-pole, the initial nonlinear circuit is first reduced to a diagram containing an equivalent generator with some linear internal resistance and a non-linear element is connected with it, after which it is calculated. If there is an energy source in the circuit in the circuit, the working (depicting) point will constantly slide according to the approximating characteristic, turning through the breakpoints. The transition through such points corresponds to the instantaneous change in the substitution scheme. Therefore, the task of determining the desired variable is reduced not only to the calculation of the substitution schemes, but also by determining the "switching" moments between them, i.e. Finding the boundary conditions in time. The analysis is significantly complicated if there are several nonlinear elements in the chain. The main difficulty in this case is related to the fact that the combination of linear sections is not known in advance, corresponding to the input voltage (current). The desired combination of linear sections of all nonlinear elements is determined by the prosperity of their possible combinations. For any adopted combination, the schema parameters are known, and, therefore, voltages and currents can be determined for all elements. If they lie within the corresponding linear sections, the adopted combination gives a correct result. If at least one nonlinear element variables go beyond the boundaries of the linear portion under consideration, then go to another combination. Table 1. Piecewise linear approximation of a nonlinear resistor It should be noted that there is always the only combination of linear sections of the characteristics of nonlinear elements corresponding to the change in the input signal in some limits. As an example, we will define the voltage in the chain in Fig. 2 in which 1. In accordance with the specified WHA, a nonlinear resistor on a plot 1- 2 is replaced with a linear resistor with resistance on a plot 2-3-source current with current 2. Based on this equivalent replacement for current on a plot of 1- 2, you can write: When moving the depicting point by section 2-3, we have when moving in a portion of 1- 4 cars 3. Determine the intervals of the movement of the depicting point in separate areas of the Wah. For the point of breakfast 1 on the base (1) the equation is valid From here we obtain two values \u200b\u200bof the instantaneous phase of the supply voltage on one period, corresponding point 1 :. The first value determines the transition of the depicting point from section 4-1 1 to the section 1-2, the second from the section 2-1 to the section 1-4. Similarly, write to the point 2 breaks where (the value corresponding to the transition from section 1- 2 to section 2-3) and (the value corresponding to the transition from the portion 3- 2 to the portion 2-11). Thus, we obtain for one period of the supply voltage In accordance with the frequency of the sinusoidal function, these solutions are repeated after 360 ° N. In fig. 4 shows a graph of the desired value. Method of harmonic balance The use of an analytical expression for approximation The characteristics of the nonlinear element allows the least laborious calculation when the law of changes in time by one of the variables that determine the operation of the nonlinear element (current or voltage for the resistor, streaming or current for the inductance coil, charge or voltage for the condenser), is set or It follows from the preliminary analysis of the physical conditions of the proceeding process, which took place when solving previous tasks of this section. If such a certainty is absent, then the task in the general case can only be solved approximately. One of these methods most widely applicable in practice is the method of harmonic balance. The method is based on the decomposition of periodic functions in a Fourier series. In general, the desired variables in the nonlinear electrical circuit are unsuccessful and contain an infinite spectrum of harmonics. The expected decision can be represented as the sum of the main and several higher harmonics, which are unknown amplitudes and initial phases. Substituting this amount into a nonlinear differential equation recorded for the desired value, and equating in the resulting terms of the coefficients before harmonics (sinusoidal and cosine functions) of the same frequencies in its left and right parts, come to the system of 2N algebraic equations, where N-number of accusative harmonics . It should be noted that the exact solution requires the consideration of an infinite number of harmonics, which is impossible to implement practically. As a result of limiting the number of the number of the harmonics under consideration, the exact balance is violated, and the solution becomes approximate. The method of calculating the nonlinear circuit in this method includes in the general case the following main steps: 1. The equations of the state of the chain for instant values \u200b\u200bare recorded. 2. The expression of analytic approximation of a given nonlinearity is selected. 3. Based on the preliminary analysis of the chain and nonlinear characteristics, the expression of the desired value in the form of a finite range of harmonics with unknown at this stage amplitudes and initial phases is given. 4. The functions as defined in paragraphs 2 and 3 are substituted in the state equation, followed by the implementation of the necessary trigonometric transformations to isolate the sinus and cosine components of the harmonic. 5. A grouping of members in the obtained equations on individual harmonics is performed, and on the basis of equating coefficients for single-order harmonics in their left and right-wing parts (separately for sinus and cosine components) a system of nonlinear algebraic (or transcendental) equations relative to the desired amplitudes and initial phases is recorded. The decomposition functions of the determined value. 6. The solution is carried out (in the general case with numerical methods on the computer) of the obtained system of equations relatively and. A special occasion of the method of harmonic balance is Method for calculating the first harmonics Nonnisoidal values \u200b\u200b( method of harmonic linearization) When the highest harmonics of the desired variables, as well as input influences neglected. When analyzing, the characteristic of the nonlinear element according to the first harmonics is used, for which the analytical expression of the nonlinear characteristic for instantaneous values \u200b\u200bis substituted with the first harmonic of one of two variables that define this characteristic, and is a non-linear connection between the amplitudes of the first harmonics of these variables. The calculation stages correspond to the outlined harmonic balance method. At the same time, due to the fact that the final system of nonlinear equations has the second order, in some cases the possibility of their analytical solution appears. In addition, since only the first harmonics of non-censoroidal values \u200b\u200bare considered, a symbolic method can be used in the calculation.
. The battery of the nonlinear resistor is shown in Fig. 3, where.
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and on a plot 4- 1- current source with current 
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