As a rule, CVC of nonlinear elements = F(u) remove experimentally, most often you can see the table or graphs . Shchob mothers on the right for analytical views , to be brought go to approximation.

Significantly given in a table or graphically CVC of nonlinear elementi = FV(u), A analytical function, I approximate given characteristics, i = F(u, a 0 , a 1 , a 2 , … , a N ). de a 0 , a 1 , … , a N – coefficients these functions, what you need to know heir to approximation.

A) Chebishev’s method coefficients a 0 , a 1 , … , a N functions F(u) know from your mind:

that stinks The process of minimizing the maximum impairment of the analytical function as specified is indicated. Here u k, k = 1, 2, ..., G - Vibration voltage values u.

At root mean square proximity coefficients a 0 , a 1 , …, a N guilty be like that To minimize the value:

, (2.6)

B) Similar functions to Taylor based on taxes functions i = F(u) Taylor series on the outskirts of the point u = U 0:

and the number of coefficients whoa unfolding. Yakshcho share the first two members of the table at the Taylor row, then we are talking about replacing the folding non-linear layout F(u) more simple linear placement . Taka replacement is called linearization of displays.

First member of the spreadsheet F(U 0) = I 0 is itself steady flow at the working point at u = U 0 A another year Lyon

differential slope of the current-voltage characteristic at the operating point , then when u = U 0 .

IN) The greatest let's expand the approach given function є interpolation(Method of picking points), at any coefficients a 0 , a 1 , …, a N approximating function F(u) be consistent with the given function F x (u) at selected points (interpolation nodes) u k = 1, 2, ..., N +1.

D) Stupina (polynomial ) approximation. I rejected this name approximation of current-voltage characteristics by static polynomials:

Inodi it is possible to perform the approximation problem manually specified characteristics on the outskirts of the point U 0 called robotnik. Todi vikorist with static polynomial

Step approximation wide vikorystvoyutsya during analysis nonlinear robots devices that serve food small external infusions that it is necessary to achieve more precise nonlinearity characteristics on the outskirts of the operating point.

E) sheet-linear approximation. In quiet times, when voltages with great amplitudes are applied to the nonlinear element, more can be allowed replacement of the characteristics of the nonlinear element is approaching і vikorystuvati more simple approximating functions . The greatest often under the hour of analyzing the work of the nonlinear element in such a regime is real characteristic is being replaced in sections of straight lines with different patches .

From a mathematical point of view, this means that on the skin part that is replaced, the characteristics are determined by the static polynomials of the first stage ( N=1 ) with different coefficient values a 0 , a 1 , … , a N.

In such a manner The specified approximation of I-V characteristics of nonlinear elements depends on the choice of the type of approximating function and the assigned coefficients one of the most important methods.

Lecture No. 16

APPROXIMATION OF CV CHARACTERISTICS OF NON-LINEAR ELEMENTS. METHODS OF ROZRAKHUNKU NON-DIY ELECTRICAL LANZYUGIV

Basic nutrition

1. Approximation of current-voltage characteristics of nonlinear elements. Polynomial approximation.

2. Shmatkovo-linear approximation.

3. Classification of methods for analyzing nonlinear Lantzugs.

4. Analytical and numerical methods for the analysis of nonlinear Lantzugs of a stationary stream.

7. Strum of a nonlinear resistor when there is a sinusoidal voltage.

8. The main transformations that occur with the help of nonlinear electric lances of the alternating stream.

1. Approximation of current-voltage characteristics of nonlinear elements

Voltage-ampere characteristics of real elements of electric coils appear in a complex form and are presented in the form of graphs or a table of experimental data. In a series of vipadkiv, the shutty of the wah, it is in the same form, and the uninterrupted to pierce the description for the pre -unit to finish the forgiveness of the anal spiwins, and the character is yak.

Replacing folding functions with nearby analytical viruses is calledapproximation .

Analytical calculations that approximate the current-voltage characteristics of nonlinear resistive elements must more accurately describe the course of real characteristics.

Also, the specification of approximation of the current-voltage characteristic includes two independent specifications:

1) selection of the approximating function;

2) depending on the value of the constant coefficients that are included in this function, two types of approximation of I-V characteristics of nonlinear elements are most often used:

polynomial;

Shmatkovo-linear.

1.1. Polynomial approximation

The approximation by a static polynomial is calculated based on the Taylor series formula for CVC NOT:

tobto. The current-voltage characteristic in this case may be uninterrupted, unambiguous and absolutely smooth (there may be no order).

In practical applications, the current-voltage characteristics should not be differentiated, but rather calculated, for example, so that the approximating curve (16.5) passes through the given streams.

The so-called three-point method requires three points of the current-voltage characteristic:

(i 1 , u 1), (i 2 , u 2), (i 3 , u 3) corresponded to the nominal value (16.5) (Fig. 16.9).

Z Rivnyan

it's hard to know the coefficients a 0 , a 1 , a 2, parts of their system (16.6) are linear.

If the current-voltage characteristic is severely damaged and it is necessary to modify its features, it is necessary to repair more points of the current-voltage characteristic. A system of type (16.6) becomes foldable, the solution to this can be found using the Lagrange formula, which means the level of the polynomial that passes through n dot:

(16.7)

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

Butt. Let the nonlinear element produce the current-voltage characteristic, specified graphically (Fig. 16.10).

It is necessary to approximate the current-voltage characteristic IE by a static polynomial.

On the graph of the current-voltage characteristic there are four points with coordinates:

On the base of the Lagrange formula (16.7) we remove

Thus, the approximating function looks like

і ne = -6.7 i 3 + 30i 2 – 13,3i.

2. Shmatkovo-linear approximation

At sheet-linear approximation CVC is NOT approximated the totality of linear plots(shmatkiv) near possible working points.

Butt. For two plots of nonlinear current-voltage characteristic (Fig. 16.11) we can remove:

Butt. Please don’t forget to linearize the VAC plot between strumas Aі IN, which is determined by the working area and the working point R(Fig.16.12).

Thus, the alignment of the linearized plot of the current-voltage characteristic near the operating point R will

It is obvious that the analytical approximation of the current-voltage characteristic is correct only for the selected linearization section.

Often it is necessary to use analytical expressions for the volt-ampere characteristics of nonlinear elements. These expressions can only approximately represent the current-voltage characteristics; fragments of physical regularities that order the relationships between stresses and flows in nonlinear applications are not expressed analytically.

The determination of a close analytical function, specified graphically or in a table of values, between changes and arguments (independent variables) is called approximation. In this case, first, choose an approximating function, then a function that approximately represents the given location, and, in other words, choose a criterion for assessing the “closeness” of this location and approximation This is its function.

As approximating functions, most often algebraic polynomials, fractional rational, exponential and transcendental functions or a set of linear functions (in and cutting straight lines).

It is important to note that the current-voltage characteristic of the nonlinear element i= fun(u) is specified graphically, so it is indicated at every point of the interval Umin≤і≤Umax, and an unambiguous non-interruptible variable function і. Then the problem of analytical representation of the current-voltage characteristic can be viewed as a problem of approximation of a given function ξ(x) by the resulting approximating function f(x).

About approximable proximity f(x) and approximated ξ( X) function or, in other words, about the loss of approximation, call to judge the largest absolute values ​​of the difference between these functions in the approximation interval A≤ X≤ b, because of the size

Δ=max‌‌│ f(x)- ξ( x)│

Often, the mean square value of the difference between the designated functions in the approximation interval is chosen as the proximity criterion.

Inodes under the closeness of two functions f( x)і ξ( x) understand the escape from the given point

x = Ho the functions themselves P+ 1 of them.

The most extensive way to approximate an analytical function to a given one Interpolation(Selected points method), if you want to escape the function f( x)і ξ( x) at selected points (at evils of interpolation) X k , k= 0, 1, 2, ..., P.

The loss of approximation can be achieved less, the greater the number of parameters that are varied and included in the approximation function, for example, the higher the stage of the approximated polynomial or the greater the number of straight lines place an approximating linear-Laman function. The task of calculation naturally increases, both from the related tasks of approximation, and from the further analysis of the nonlinear lantzug. The simplicity of this analysis and the characteristics of the approximated function within the approximation interval serve as one of the most important criteria when choosing the type of approximated function.

In the tasks of approximating the current-voltage characteristics of electronic and conductor devices, their creation, as a rule, is not necessary through a significant distribution of the characteristics of devices from expression to expression influx of factors on them that destabilize, for example, the temperature in the conductor devices. It is most often necessary to do it “correctly” to create the ignorant, averaging nature of longevity i= f(u) at the boundaries of the working interval. In order to be able to analytically analyze lanyards with nonlinear elements, it is necessary to use mathematical expressions for the characteristics of the elements. These demonstrations themselves are called experimental, then. are removed as a result of the modification of the corresponding elements, and then on this basis the sub-type (type) data are formed. The procedure of mathematical description of any given function of mathematics is called approximation of this function. It is based on low types of approximation: behind fixed points, according to Taylor, according to Chebishev and others. It is necessary to remove the mathematical expression from which the output and approximation functions are satisfied with the given tasks.

Let's look at the simplest method: the method of identifying points or nodes of interpolation by a static polynomial.

It is necessary to calculate the coefficients of the polynomial. For whom to choose (n+1) point on a given function and a system of equations is formed:

The coefficients are determined from this system a 0, a 1, a 2, ..., a n.

At selected points, the approximating function is consistent with the output, at other points it varies (strongly lies in the static polynomial).

You can use the exponential polynomial:

Another method: Taylor approximation method . In this case, one point is selected, where the output function will be separated from the approximable one, and then additionally put in mind, so that this point will be avoided even further.

Butterworth approximation: the simplest polynomial is selected:

For which condition can be determined the maximum vitality ε at the edges of the range.

Approximation according to Chebishev: It is static, there is a bias established at a number of points and the maximum variation of the approximating function as an output is minimized. Theoretically, the approximation of the function is carried out to the maximum of the absolute value of the polynomial f(x)step P as a non-stop function ξ( X) will be minimally possible, as in intervals of proximity A≤ X≤ b sacristy

f( x) - ξ( X) no less, no less n + 2 once again accepts its boundaries, which are consistently drawn out, the greatest f(x) - ξ( X) = L> 0 the lowest f(x) - ξ( X) = -L value (Chebishev criterion).

In many applied problems, it is known that the polynomial approximation is based on the mean-square proximity criterion, if the parameters of the approximated function f(x) are chosen deliberately to a minimum in the approximation interval A≤ X≤ b square deviating functions f(x) according to the given non-interruptible function ξ( X), then think about it:

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

Obviously, before the rules for finding extrema, the problem is reduced to the development of a system of linear levels, which is created as a result of leveling the first private similar functions to zero. Λ according to the skin of the identified coefficients a k approximating polynomial f(x), totto rivnyan

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

It has been proven that this system of ranks has a single solution. In the simplest types, it is analytical, and in the most advanced form, it is numerical.

Chebishev established that it is possible for maximum inspiration to overcome jealousy:

In engineering practice, vikoryst is also called sheet-linear approximation- This is a description of a given curve in sections of straight lines.

Between the skin and linearized sections, the current-voltage characteristics are consistent with all methods of analysis of colivania in linear electric lancets. It is clear that the more linearized plots are divided into a given current-voltage characteristic, the more it can be approximated and the more difficult it is to calculate the amount of heat in lancus.

In many applied problems of vibration analysis in nonlinear resistive lanyards, the volt-ampere characteristic in the approximation interval is represented with sufficient accuracy by two or three straight lines.

Such an approximation of the volt-ampere characteristics gives, in most cases, the results of the analysis of vibrations in a nonlinear resistive lance with “small” influxes on the nonlinear element in terms of accuracy, so that Mitte values ​​of strums for a nonlinear element change I= 0 to I = I swing

As stated earlier, the main characteristics of nonlinear elements are not the connection, but the current-voltage characteristic of the active support
or else
, or staleness
- for nonlinear inductance (amperweb characteristic) or storage capacity q(u) – for nonlinear capacitance (voltcoulomb characteristic) (Fig. 3.8).

Fig.3.8. Types of characteristics of nonlinear elements

However, the graphical form of the characteristics of nonlinear elements (Fig. 3.8.) does not allow for the selection of the location (3.1-3.15), for the folding of the robot circuits with nonlinear elements. Therefore, one of the most important tasks that arises when analyzing the dynamics of circuits that accommodate nonlinear elements lies in the approximation of nonlinear characteristics. The greatest expansion of the approximation of nonlinear characteristics is polynomial and piece-linear, as well as approximation using various types of transcendental functions.

When analyzing nonlinear circuits, the ability to obtain the correct result depends primarily on the correct choice of the approximation method and on the determination of the approximated function of the nonlinear element. The point is that the more precise the approximation of the nonlinear element is, the easier it is to determine the necessary analytical characteristics of the nonlinear element. In addition, it will be easier to solve the nonlinear equation that describes the dynamics in such a nonlinear system, using the chosen form of the approximating function. Therefore, the correct choice of approximation of a nonlinear characteristic allows one to completely simplify the process of solving a nonlinear equation. In addition, it is necessary to note that often one and the same characteristic of a nonlinear element must be approximated in different ways, depending on the minds in which the nonlinear element is processed and what nutrition is responsible for the investigation. Therefore, methods of approximation are selected according to the specific skin conditions of the coliform in circuits with various nonlinear elements.

Let's look at methods of approximation of various functions of nonlinear elements. The most advanced methods of approximation of nonlinear elements include the following:

polynomial approximation ─ submission of a nonlinear characteristic to an additional static series,

line-linear approximation ─ representation of the approximated function in sections of straight lines,

approximation using various types of transcendental functions

Polynomial approximation. If the non-linear characteristics are specified by an analytical model, then on the outside of the working point the function can be represented by the expansion of the Taylor series (
on the outside of the point x 0)

, (3.16)

where R is the excess in the Taylor series, which is not necessary during approximation.

If the characteristic is specified graphically (Fig. 3.9), then the approximation can be done by a shortened linear series (polynomial), intervening it with another - the fifth step

3.9. Graphic manifestation of nonlinear characteristics

To determine the coefficients a k, it is possible that with the values ​​of the variable x k left side of the polynomial (3.17) the values ​​of the function y k are obtained.

Let's create a system of rankings:

, de
. (3.18)

This system has equal values ​​y n , 0 x n x 0 – depending on the quantity, so this system can be calculated using the Cramer method, using coefficients a k .

If x = x 0 + S (x 0 is constant, and S is a small signal), then

de - differential parameter of the nonlinear element. Thus, it can be understood that the first coefficient a 1 of the polynomial approximation of the nonlinear characteristic (3.17) is avoided by the differential parameter of the nonlinear element. In addition, it is significant that x = 0 lies in the middle of the interval (x 5 - x 1) of approximation of the nonlinear characteristic by a polynomial, then the coefficient a 0 means the value of the function on the grain of coordinates (as we see As a nonlinear characteristic i=φ(u), then the coefficient and 0 =i(0) is calculated as the value of the strum at u=0.

Shmatkovo-linear approximation. Linear approximation is based on replacing the real characteristics of a nonlinear element with parallel sections, which are replaced by sections of straight lines (Fig. 3.10).

3.10. Linear approximation of a nonlinear element

The accuracy of the sheet-linear approximation depends on the number of intervals, which are replaced by straight cuts at a given interval and vikoristan of the sheet-linear approximation. The greater the number of direct splits the interval for which we have a stagnant line-linear proximity, the greater the accuracy of the run with the real nonlinear characteristic, but at the same time it becomes more complicated analysis of colivan in such a system. To simplify the expansion, it is necessary to use a minimum number of straight cuts that replace the nonlinear characteristic. For example, the dynamic flow characteristic of a triode (Fig. 3.10) can be approximated with a sufficient level of accuracy using only three straight line segments:

. (3.20)

Replacing non-linear sections of parameters of non-linear elements with sections of straight ones, allows the influence of linear power itself, and this means that all methods of the linear theory of lancets are now at a standstill iv. By pulling linear sections, nonlinear elements are replaced with linear ones, with characteristics equal to their differential values.

Approximation of nonlinear characteristics using additional transcendental functions. Some characteristics of nonlinear elements are approximated by transcendental functions in Fig. 3.11. As approximating transcendental functions, the exponents of their sums, trigonometric, reversal trigonometric, hyperbolic and other functions are defined. For example,

or else
. (3.21)

Fig.3.11. Applications of approximation of nonlinear characteristics

transcendental functions

As a rule, the current-voltage characteristics of nonlinear elements are determined experimentally; It is better to know them through theoretical analysis. To carry out the investigation, it is necessary to select an approximation function such that, being idle, it represents all possible features of the experimental characteristics with a sufficient level of accuracy. The most commonly used methods for approximating the current-voltage characteristics of bipolar networks are: piece-linear, static, display approximation.

Shmatkovo-linear approximation

Such an approximation may stagnate when the processes of nonlinear processes develop due to large amplitudes of external inflows. This method is based on the approximation of parameters of nonlinear elements, then. In the near future, the real characteristics will be replaced by sections of straight lines with different patches. The little picture shows the input characteristic of a real transistor, approximated by two straight lines.

The approximation is determined by two parameters – the stress of the characteristic U and the slope S. The mathematical form of the approximated current-voltage characteristic is as follows:

The voltage of the input parameters of bipolar transistors is on the order of 0.2-0.8 V: the slope of the base line parameter ib(Ube) is close to 10 mA/V. The slope of the characteristic ik(Ube) of the collector flow in relation to the base-emitter voltage, then the value of 10 mA/V may be multiplied by h21e - the coefficient of strengthening the base flow. The fragments h21e = 100-200, the indicated steepness is on the order of many amperes per volt.

Step approximation

Step approximation is widely used when analyzing the operation of nonlinear devices to which very small external inflows are supplied. This method is based on the expansion of the nonlinear current-voltage characteristic i(u) into a Taylor series, which converges on the outside of the operating point U0.

A number of folding members must be kept within the specified accuracy. Let's take a look butt:

Input characteristics of the transistor. Operating point U0 = 0.7V. We select the nodes of approximation of the point 0.5; 0.7 and 0.9 St.

It is necessary to adjust the ranking system:

Spectral storage of the stream in a nonlinear element with an external harmonic influx

Let's take a look at the lance that consists of the serial connection of the harmonic signal element Uс(t) = coswt, the constant displacement voltage element U0 and the inertia-free nonlinear element. For this purpose, let's look at the drawing.

The strum of the Lancjugu has a sinusoidal shape.

The shape of the stream and the voltages appear different.

The reason for the creation of a crooked struma is simple: however, increased stress is indicated by different increases in the struma, because , And the differential steepness of the current-voltage characteristic on different plots varies.

Let's take an analytical look at the plant.

Let us see the nonlinear function i (u) = i (Uc, U0). The voltage applied to the nonlinear element is Uc(t)=Umcos(wt+j).

The dimensionless quantity x=wt+j then I(x)=I(Umcosx, U0) is a periodic function for the argument x with a period of 2T. Let us imagine the Four'e order with coefficients .

The function i(x) is paired, so the Four's series is more suitable than cosine warehouses: .

Amplitude coefficients of harmony

The remaining two formulas provide a simple solution to the problem of the spectrum of the stream in a nonlinear element with a harmonious external influx:

tobto. Strum, in addition to the stationary warehouse I0, maintains an endless continuity of harmony with the amplitudes of In. Harmony amplitudes lie in the parameters Um and U0, as well as in the form of the approximating function.

Let's look at how to look at the approximating function.

Shmatkovo-lineyna

i(U)=

Supply voltage u(t)=U0+Umcoswt.

The flow chart looks like a cosine wave of impulses with multiples. Where the number of impulses in the stream is calculated according to the equation:

U0+Umcosq=Uн Þ .

Step approximation.

On the outskirts of the operating point U0, the current-voltage characteristic of the nonlinear element