Classification of nonlinear elements

Nonlinear electric lancets

ROZDIL II. NON-LINEAR LANTZUGS

Nonlinear lanyards are all lanyards that have one nonlinear element. A nonlinear element is a whole element for which connections the flow and stress are assigned to nonlinear elements.

Non-linear lances do not follow the principle of overlap, so there are no hidden methods of decomposition. This calls for the need to develop special methods for the development of non-linear elements and the mode of their work for the skin type.

Nonlinear elements are classified:

1) beyond physical nature: conductor, overhead conductor, dielectric, electronic, ion, etc.;

2) by character divide into resistive, eminent and inductive;

VAC VAC VAC

3) by type of characteristics share all elements

On symmetrical and asymmetrical. Symmetrical - those in which the characteristic is symmetrical to the coordinates. For asymmetrical elements, once and again select a positive voltage or current and for them the current-voltage characteristic is induced at the conductors. Only in this way can one directly vikorystuvat under the hour of the highest order from the vikors of these current-voltage characteristics.

Both unambiguous and ambiguous. It is ambiguous if the current or voltage on the current-voltage characteristic shows a single point;

4) Inertial and non-inertial elements. Inertial elements are those elements whose nonlinearity is caused by heating of the body during passage of the stream. Since the temperature cannot be changed as quickly as possible, then when a changeable stream passes through such an element with high frequency and constant values, the temperature of the element is lost during a practically constant period of change no struma. Therefore, for mitten values, the element appears linear and is characterized by a constant value R(I,U). If the value of the flow changes, then the temperature changes and a different basis appears, then for the other values ​​the element becomes nonlinear.

5) keroved and non-keroved elements. Most of all we talked about non-coated elements. Before the ceramic elements, elements with three or more pins are brought in, in which, with alternating strings or voltage on one pin, you can change the I-V characteristics of other pins.

For a specific task, it is important to manually set these and other parameters of the elements, since the number of them is large, but most often the static and differential parameters are used. For a resistive bipolar element there will be a static differential support.

At the given point, the current-voltage characteristic

At the given operating point, the current-voltage characteristic

1. Give a slight increase in voltage. Find out from the current-voltage characteristics, the results of these increases, the increase in the flow and take their bearings. There are a few that require changing the layout to improve accuracy D.U.і D.I. However, it is important to work with the schedule.

2. To a given point of the curve, carry out further and then follow the geometric values

Take more money on this and can be great forever.

If the operating mode of the nonlinear element is selected, then this point is the same as its static support, as well as the voltage and strings, which can be replaced in one of 3 ways.

It is clear that during the course of time the work of the lancet strings and voltages change between the “less linear section of the current-voltage characteristic”, then this section should be described to linear levels and set to similarity valence circuit.

Linearize this plot to the equals U=a+ib.Remove your coefficient of equality.

At i=0 і U=U 0 =a,

No matter how chaotic the system is, it is to blame for its non-linear elements and power. A linear system may have chaotic oscillations. In a linear system, periodic external actions call out after the extinguishing of transient processes, periodic output of the same period (Fig. 2.1). (This includes parametric linear systems.) Mechanical systems may have the following nonlinear components:

1) non-linear spring elements;

Rice. 2.1. Scheme of possible signal transformations in linear and nonlinear systems.

2) non-linear extinction, similar to the loss of calm and forging;

3) dead travel, clearance and white springs;

4) more hydrodynamic fluids;

5) nonlinear boundary minds.

Nonlinear spring effects can be associated either with the power of speech or with geometric features. For example, the relationship between stress and deformation is nonlinear. However, if the stress and deformation of the steel varies linearly right up to the inter-plane, the strength of the beam, slab or shell can be non-linearly related to the applied forces and moments. Similar effects associated with strong displacements and rotations are called geometric nonlinearities in mechanics.

Nonlinear power of electromagnetic systems is determined by the following factors:

1) non-linear supports, capacities and inductive elements;

2) hysteresis in ferromagnetic materials;

3) nonlinear active elements, such as vacuum tubes, transistors and diodes;

4) effects characteristic of dry media, for example, electrodestructive force, where v is fluidity, and is magnetic field;

5) electromagnetic forces, for example, de J - strum, or de M - dipole magnetic moment.

Applications of nonlinear devices include primary elements of electrical circuits, such as diodes and transistors.

Rice. 2.2. Non-linear installations with a number of alignment positions: a - late insertion of a thin spring shear under axial thrust at the end; 6 - late bending of the spring shear by nonlinear magnetic mass forces.

Such magnetic materials, such as iron, nickel or ferrite, are characterized by nonlinear material relationships between the magnetization field and the strength of the magnetic flux. For additional operational boosters and diodes, some experimenters are encouraged to select negative supports with a white current-voltage characteristic (Div. Chapter 4).

It is not easy to detect nonlinearity in any system, firstly because we often tend to dismiss linear systems, and secondly, because the main components of the system can be linear and nonlinearity has a subtle effect ectom. For example, around the fastening truss elements can be linear springs, but they can be assembled so that there are no gaps and no linear rubbing. Thus, nonlinearity can be found in marginal minds.

In a butt with a curved stem, non-linear elements are easily visible (Fig. 2.2). Any mechanical device that has more than one position of static balance has a gap, slow motion or non-linear rigidity. The type of haircut that is bent at the ends (Fig. 2.2 a) has a geometric nonlinearity of stiffness. In a stream that is driven by magnetic forces (Fig. 2.2, b), the chaotic behavior of the system is caused by nonlinear magnetic forces.

Using the method of mutual substitutions in the extracted expressions, we can extract the expressions for R ab, R bc and R ca (the same expressions for transforming the eye into the trikutnik):

R ab = R a + R b + R a R b;

Rbc = Rb + Rc + RbRc;

R ca = R c + R a + R c R a.

1.5.1. Zagalnye Vidomosti

Nonlinear electric lancet This is an electric lance that can accommodate one or a number of nonlinear elements 1 ] .

Nonlinear element the element of an electric lance, the parameters of which are based on the initial values ​​(the support of the resistive element in relation to the flow and voltage, the capacitance of the resistive element in relation to the charge and voltage, the inductance of the element in relation to the magnetic flux and tricate struma).

Thus, the volt-ampere u(i) characteristic of a resistive element, the Weber-ampere ψ(i) characteristic of an inductive element and the coulomb-voltage q(u) characteristic of an inductive element do not appear to be a straight line (as is the case with a linear element), but I am curve, which appears to be determined experimentally and does not have an exact analytical manifestation.

A non-linear electric lance has a number of significant advantages over a linear one, and specific components may be at fault.

Rice. 1.28. UGO of nonlinear resistive, inductive and amnesic elements

(for example, hysteresis), which is why the method of degrading linear lances to non-linear lances is not stagnant. It is especially important to note the non-linearity of the superposition method.

It is important to understand that the characteristics of real elements are by no means linear, but most engineering processes can, with acceptable accuracy, be considered linear.

All conductor elements (diodes, transistors, thyristors, etc.) are nonlinear elements.

Smart graphical designations of nonlinear resistive, inductive and amnesic elements are shown in Fig. 1.28. On the wine indicator you can specify a parameter that indicates nonlinearity (for example, temperature for a thermistor)

1.5.2. Parameters of nonlinear elements

Nonlinear elements are characterized by static (R st, L st, i C st) and differential (R d, L d, i C d) parameters.

Static parameters The nonlinear element is defined as the ratio of the ordinate of the selected point of the characteristic to the abscis (Fig. 1.29 ).

Static parameters are proportional to the tangent of the slope of the straight line drawn through the coordinate origin and the point at which the expansion is carried out. For example in Fig. 1.29 omitted:

F st = y A = m y tg α, x A m x

where α is the cut of the straight line drawn through the coordinate root and the working point A;

m y and m x - scales along the ordinate and abscissa axes.

Rice. 1.29. Before static and differential parameters are determined

nonlinear elements

F st = y A, F diff = dy x A dx

The static parameters of the resistive, inductive and amnesic elements look like this:

Differential parameters The nonlinear element is defined as the ratio of a small increase in the ordinate of the characteristic point to a small increase in the abscise (Fig. 1.29).

Differential parameters are proportional to the tangent of the slope, which is significant in the operating point of the characteristics and the entire abscis. For example in Fig. 1.29 omitted:

F diff = dy = m y tan β, dx m x

de β - cut-off point at the operating point B of the characteristics and the entire abscis;

m y and m x - scales along the ordinate and abscissa axes. Below are the differential parameters of resistive and inductive.

Many and omniscient elements may look like this:

1.5.3. Methods for the development of nonlinear lances

Nelinіnіinost parameter is the contributions of the rosrahunki Lantsyuga, the vibrati abbrei is the abstrail of the abstract, the close to the nyoye dile, the characteristics of the rzpyatyaty, with the allowable yak lininiyniy. Since this is impossible, because the nonlinearity of the characteristics is the reason for the choice of element (especially typical for conductor elements), then special methods of design must be used - graphical, approximation

(analytical and sheet-linear) and lower than others. Let's take a look at these methods in the report.

Graphic method

The idea of ​​the method lies in the individual characteristics of the Lanczug elements (volt-ampere u(i), Weber-ampere ψ(i) or coulomb-voltage q(u)), and then, by the way of their graphical transformation (for example, addition), the derivation of the specific characteristic for every Lanzug chi yogo dilyanka.

The graphical method of squaring is the simplest and most straightforward, ensuring the necessary accuracy for the majority of squaring, but it is difficult to stool for a small number of non-linear elements in lance. This ensures accuracy during busy times.

An example of the expansion of a nonlinear lance using a graphical method for the sequential connection of linear and nonlinear resistive elements is shown in Fig. 1.30, and for parallel - in Fig. 1.30, b.

When the sequential lancet is developed in one axes, the characteristics of all elements that need to be developed will be determined (for the analyzed application, ce u ne (i) for a nonlinear resistor R ne i u le (i) for linear R le). The nature of the change in the ignition voltage u(i) is determined by the composition of the characteristics of the nonlinear un(i) and linear ule(i) elements u(i) = un(i) + ule(i). The addition is carried out at the same values ​​of the strum (for i = i 0: u 0 = u ne 0 + u le 0, div. Fig. 1.30, a.).

The development of the parallel lancet is carried out in a similar way, only the characteristic of each lancet will be the way of folding strums, under constant stress (for u = u 0: i 0 = i ne 0 + i le 0, div. Fig. 1.30, b.).

Rice. 1.31. Active linear double-pole switch as a replacement circuit for a nonlinear element

Approximation method

The idea behind the method is to replace the experimentally determined characteristics of a nonlinear element with an analytical virus.

Disaggregate analytical approximation , when any characteristic of an element is replaced by an analytical function (for example, linear y = ax + b, ste-

som y = a th βx and inshimi) i piecewise-li-

linear, when the characteristic of an element is replaced by a set of straight lines

new cuts. The accuracy of the analytical approximation

The difference is determined by the correctness of the choice of the approximating function and the accuracy of the selection of coefficients. The advantage of the linear-linear approximation is the ease of drawing and the ability to view the element as linear.

In addition, in the enclosed range of signal changes, in which changes can be made linear (in this case small signal mode), a non-linear element, with acceptable accuracy, can be replaced with an equivalent linear active double-terminal (Fig. 1.31, the detailed double-terminal will be discussed in § 2.3.4), de strum and voltage are connected by the following:

U = E + R diff I,

where R diff is the differential support of the nonlinear element at the distance that is linearized.

An example of analytical approximation of the characteristics of a conductor diode using an additional function of the form i = a (e bu − 1) is shown in Fig. 1.32, b, piece-linear approximation - in Fig. 1.32 in the output diode characteristic is shown in Fig. 1.32, a.

Rice. 1.32. Approximation of the characteristics of the conductor diode.

a - output characteristic of the diode;

b - analytical approximation for an additional function of the form i = a (e bu - 1);

c - sheet-linear approximation.

Classification of nonlinear elements

Nonlinear lancets are those lancets that have one nonlinear element. A nonlinear element is an element for which connections, flow and stress are given to nonlinear elements.

Non-linear lances do not follow the principle of overlap, so there are no hidden methods of decomposition. This calls for the need to develop special methods for the development of non-linear elements and the mode of their work for the skin type.

Nonlinear elements are classified:

1) for physical nature: conductor, overhead conductor, dielectric, electronic, ion, etc.;

2) according to the character, divide into resistive, emmonic and inductive;

VAC VAC VAC

3) according to the characteristics, all elements are divided

On symmetrical and asymmetrical. Symmetrical - those in which the characteristic is symmetrical to the coordinates. For asymmetrical elements, once and again select a positive voltage or current and for them the current-voltage characteristic is induced at the conductors. Only in this way can one directly vikorystuvat under the hour of the highest order from the vikors of these current-voltage characteristics.

Both unambiguous and ambiguous. It is ambiguous if the current or voltage on the current-voltage characteristic shows a single point;

4) inertial and non-inertial elements. Inertial elements are those elements whose nonlinearity is caused by heating of the body during passage of the stream. Since the temperature cannot be changed as quickly as possible, then when a changeable stream passes through such an element with high frequency and constant values, the temperature of the element is lost during a practically constant period of change no struma. Therefore, for mitten values, the element appears linear and is characterized by a constant value R(I,U). If the value of the flow changes, then the temperature changes and a different basis appears, then for the other values ​​the element becomes nonlinear.

5) keroved and non-keroved elements. Most of all we talked about non-coated elements. Before the ceramic elements, elements with three or more pins are brought in, in which, with alternating strings or voltage on one pin, you can change the I-V characteristics of other pins.

Parameters of nonlinear elements and their equivalent circuits

For a specific task, it is important to manually set these and other parameters of the elements, since the number of them is large, but most often the static and differential parameters are used. For a resistive bipolar element there will be a static differential support.

At the given point, the current-voltage characteristic

At the given operating point, the current-voltage characteristic

1. Give a slight increase in voltage. Find out from the current-voltage characteristics, the results of these increases, the increase in the flow and take their bearings. There are only a few methods that require changing U and I to improve accuracy, and in this case it is important to work with the graph.

2. To a given point of the curve, carry out further and then follow the geometric values

Take more money on this and can be great forever.

If the operating mode of the nonlinear element is selected, then this point is the same as its static support, as well as the voltage and strings, which can be replaced in one of 3 ways.

It is clear that during the course of time the work of the lancet strings and voltages change between the “less linear section of the current-voltage characteristic”, then this section should be described to linear levels and set to similarity valence circuit.

Linearize this plot in the form U = a + ib. Maintain for a new coefficient of equality.

When i=0 and U=U 0 =a,

average value for this division.

What is indicative of the current substitution scheme:

This scheme will be valid for a plot surrounded by a squiggly line.

The same expression can be written differently:

Therefore, in certain situations, it is clear from afar that the currents and voltages of a nonlinear element represent the sum of a stationary warehouse Urt, Irt and a changeable warehouse u ~ , i ~ with amplitude<< чем величина постоянной составляющей, отдельно рассчитывают режим на постоянном токе (напряжении) и отдельно для переменной составляющей. Из записей видно, что двухполюсный элемент для малой переменной составляющей можно заменить просто дифференциальным сопротивлением в рабочей точке.

This approach should be used in circuits with multi-pole elements, otherwise it is not possible to introduce only one element, i.e. The emergency situations are characterized by the same coefficients of the equals. Alternatively, it is possible to calculate the coefficients for small exchangeable storage pipes and voltage.

Butt: Bipolar transistor (circuit with a carbon emitter).

Let us know that u j =U p f+u kj , i j =I p f+i kj

Substitution circuit:

There are definitely differentiating parameters that are taken away from the “I” form.

u bk = h 21 i b + h 12 u ke

i ke = h 21 i b + h 22 u ke

U be =H 11 I b +H 21 U ke

Prices should be written for changing warehouses, since the procedure for unpacking items changes.

H 11 =U b /I b at I b =0, then. i b = br.t.

H 12 =U be /U ke at I b =0

H 21 =I to /I b at U ke =0

H 22 =I to /U ke at I b =0, then. i b = br.t.

h 12 =ДU be /ДУ ke h 21 =Ді to /Ді b h 22 =Ді to /Дu ke,

where I, U is an increase in flow and voltage around the operating point.

Current-voltage power of this nonlinear element.

Methods for the development of nonlinear lancets of a stationary stream

Disaggregate: numerical, analytical and graphical methods.

1) Numericals are methods for numerically unraveling nonlinear equations. Start vikorist EOM. The stench can be spread widely around the area, but it is consistent with the appearance of the number.

2) Analytical methods, which are based on the approximation of the current-voltage characteristic of any type of function. Since this function is nonlinear, the result is a nonlinear system of levels. In order for this to be possible, it is necessary to carefully select the approximating function.

Yakshcho staleness U(I) or I(U linear ta yogo opir R steadily ( R =c onst ) , then like this element call linear (LE) , and the electric lance that develops only from linear elements - linear electric lancet .

CVC of a linear element symmetricalі є straight line, which passes through the coordinate root (Fig. 16, curve 1). Well, in linear electric lances, Ohm’s law applies.

Yakshcho staleness U(I) or I(U) any element of the electric Lanzug Not linear, and yogo opir lie in the stream in new chi voltage on yogo vysnovki ( R ≠с onst ) , then like this element call Not linear (NOT) , and the electric lance for obviousness I'd like one nonlinear element - nonlinear electric lancet .

CVC of nonlinear elements non-straightforward, and the inodes may be asymmetrical, for example, the conductor fittings (Fig. 16, curves 2, 3, 4). Thus, in non-linear electric lancets there is a deposit between the flow and the voltage doesn't conform Ohm's law.

Rice. 16. Current-voltage characteristics of linear and nonlinear elements:

curve 1- VAH LE (resistor); curve 2- VAH NOT (lighting lamps with metal thread); curve 3- VAH NOT (lighting lamps with carbon thread;

curve 4- VAC NOT (supply diode)

Butt linear element resistor.

butts nonlinear elements serve: frying lamps, thermistors, conductor diodes, transistors, gas-discharge lamps, etc. Umovne assigned NOT shown in fig. 17.

For example, with an increased flow through which a metal thread flows through the heating of an electric lamp, its heating increases, and therefore its operation increases. In such a manner, the heating lamp remains unstable.

Let's take a look at the offensive butt. A table has been provided with the values ​​of element supports at different values ​​of flow and stress. Which table represents a linear element and a nonlinear element?

Table 3

Table 4

Give your answer to the question, on which of the image graphs is Ohm's law? What element does this graph represent?

1 2 3 4

What can you say about graphics 1,2 and 4? What elements characterize these graphs?

A nonlinear element at any point of the current-voltage characteristic is characterized by a static support, which is the same as the voltage to the flow, which is indicated by these points (Fig. 18). For example, for a point A :

.

Besides a static support, a non-linear element is characterized by a differential support, which means the relationship between an infinitely small or even small increase in stress ∆U to a significant increase ∆I (Fig. 18). For example, for a point A The current-voltage characteristic can be written

de β - Cut the nahila, drawn through the point A .

These formulas form the basis of the analytical method for the analysis of the simplest nonlinear Lantzugs.

Let's take a look. Since the static support of a nonlinear element with a voltage U 1 = 20 is greater than 5 Ohms, then the force of the flow I 1 is warehoused ...

The static support of the nonlinear element at a flow of 2 A becomes...

Visnovok from the third food: separate linear and nonlinear elements of the electric lance. Ohm's law does not apply to nonlinear elements. Nonlinear elements are characterized by static and differential support at the skin point of the I-V characteristic. All conductor devices, gas discharge lamps and frying lamps are connected to non-linear elements.

Nutrition No. 4. Graphical method of developing non-linear

electric lantsyugs (15th century)

For the decomposition of nonlinear electric coils, graphical and analytical methods of decomposition are used. The graphical method is simple and will be discussed later in the report.

Let Dzherelo ERS E With internal support r 0 live two sequentially connected nonlinear elements or supports NS1 і NS2 . Vidomi E , r 0 , current-voltage characteristic 1 NS1 and current-voltage characteristics 2 NS2. It is necessary to measure the strum of Lanzygu I n

Initially, we will show the current-voltage characteristics of the linear element r 0 . This is straight, so it goes through the coordinates. The voltage U that falls on the supports of the circuit is indicated by the

To encourage storage U = f ( I ) , it is necessary to graphically plot the current-voltage characteristic 0, 1 і 2 , subsumable ordinates that correspond to one abscise, then another, etc. Let's eliminate the curve 3 , which is the current-voltage characteristic of all Lanzug. Vikoristuyu Qiu CVC, we know the strum of Lanzygu I n , which indicates voltage U = E . Then, Vikorista found the value of the struma for the current-voltage characteristic 0, 1 і 2 we know the voltage U 0 , U 1 , U 2 (Fig. 19).

Let Dzherelo ERS E With internal support r 0 to live two parallel connected nonlinear elements or supports NS1 і NS2 , current-voltage characteristics. It is necessary to measure the strum at the necks of Lanzug I 1 і I 2 , the voltage drop on the internal support occurred on the nonlinear elements.

There will be a current-voltage characteristic I n = f ( U ab ) . For which the current-voltage characteristic is formed graphically 1 і 2 , subsumable abscise that indicates one ordinate, then another ordinate, and so on. There will be a current-voltage characteristic of the entire Lanzug (curve 0,1,2 ). For which the current-voltage characteristic is formed graphically 0 і 1,2 , subsumable ordinates, which represent the abscissas

Vikoristuyu Qiu CVC, we know the strum of Lanzygu I n , which indicates voltage U = E .

Vikorist current-voltage characteristic 1,2 , which means voltage U ab , which is indicative of the detected struma I n , that internal voltage drop U 0 , which is indicative of this struma. Then, vikorista and CVC 1 і 2 we know shukani strumi I 1 , I 2 What do voltage measurements indicate? U ab (Fig. 20).

Let's take a look at these examples.

When nonlinear supports are connected in series with characteristics R 1 and R 2, the characteristic of the equivalent support R E ...

pass below the characteristics of R 1

passed the characteristics of R 1

pass, according to the characteristics of R 1

pass below the characteristics of R 2

When linear and nonlinear supports are connected in series with the characteristics of a and b the characteristic of an equivalent support.

pass below characteristics a

go through more characteristics

pass, showing the characteristics of a

pass below characteristics b

Visnovok from the fourth food: Non-linear electric stakes of a steady stream form the basis of electronic lances. There are two methods for their development: analytical and graphic. The graphical design method makes it easier to determine all the necessary parameters of a nonlinear lancet.