Properties of nonlinear elements. Linear and non-linear elements of the electrical circuit. Non-linear electrical circuits

Subject: Automatic control theory

Topic: NON-LINEAR ELEMENTS

1. Classification of nonlinear elements

Nonlinear dependences z \u003d f (x) can be classified according to various criteria:

1. By smoothness of characteristics: smooth - if at any point of the characteristic there is a derivative dz / dx, ie, the function is differentiable (Fig. 1a, b); piecewise-linear - a characteristic in which the derivatives have a discontinuity of the first (Fig.2a) or second kind (Fig.2b).

Figure: 3

By symmetry: evenly symmetric - symmetric about the ordinate, that is, z (x) \u003d z (- x) (Fig. 4a); odd symmetric - symmetric relative to the origin of coordinates, while z (x) \u003d - z (- x) (Fig. 4b); not symmetrical (Fig. 4c).

Figure: 4

2. Non-linear circuits

Circuits are called non-linear if they contain at least one non-linear element. Nonlinear elements are described by nonlinear characteristics that do not have a strict analytical expression, are determined experimentally and are set in tables or graphs.

Non-linear elements can be divided into two - and multi-pole. The latter contain three (various semiconductor and electronic triodes) or more (magnetic amplifiers, multi-winding transformers, tetrodes, pentodes, etc.) poles, with which they are connected to electrical circuit... A characteristic feature of multi-pole elements is that, in the general case, their properties are determined by a family of characteristics that represent the dependence of the output characteristics on the input variables and vice versa: the input characteristics are plotted for a number of fixed values \u200b\u200bof one of the output parameters, the output characteristics for a number of fixed values \u200b\u200bof one of the input parameters.

On another basis of classification nonlinear elements can be divided into inertial and non-inertial. Elements whose characteristics depend on the rate of change of variables are called inertial. For such elements, the static characteristics that determine the relationship between the effective values \u200b\u200bof the variables differ from the dynamic characteristics that establish the relationship between the instantaneous values \u200b\u200bof the variables. Inertialess are called elements whose characteristics do not depend on the rate of change of variables. For such elements, the static and dynamic characteristics are the same.

The concepts of inertial and non-inertial elements are relative: an element can be considered as inertial in the permissible (top-limited) frequency range, beyond which it becomes inertial.

Depending on the type of characteristics, nonlinear elements with symmetric and asymmetric characteristics are distinguished. Symmetric is a characteristic that does not depend on the direction of the values \u200b\u200bthat determine it, i.e. having symmetry about the origin of the coordinate system. For an asymmetric characteristic, this condition is not met, i.e. The presence of a symmetric characteristic of a nonlinear element makes it possible in a number of cases to simplify the analysis of the circuit, carrying it out within one quadrant.

By the type of characteristic, you can also divide all non-linear elements into elements with unambiguous and ambiguous characteristics. An unambiguous characteristic is a characteristic in which each value of x corresponds to a single value of y and vice versa. In the case of an ambiguous characteristic, some x values \u200b\u200bmay correspond to two or more y values, or vice versa. In nonlinear resistors, the ambiguity of the characteristic is usually associated with the presence of a falling section, and in nonlinear inductive and capacitive elements, with hysteresis.

Finally, all non-linear elements can be divided into managed and unmanaged. Unlike uncontrolled non-linear controlled elements (usually three- and multipoles) contain control channels, changing the voltage, current, luminous flux, etc. in which they change their main characteristics: volt-ampere, weber-ampere or coulomb-volt.

Depending on the type of constituent non-linear elements, non-linear circuits are called.

3. Gain of nonlinear element

Consider a nonlinear element (Fig. 5). Let's apply to the input of the nonlinear element a harmonic signal with an amplitude - A 0 and determine the first harmonic of the output signal.

In this case, for the input and output signals, the following ratios can be written

(1)

where: - vector module; is the vector argument.

Consider the characteristic of a nonlinear element - which is called the complex transfer coefficient of a nonlinear element. This characteristic can be plotted in the complex plane as well as the complex gain of the linear part. In this case, the characteristic - depends on the frequency of the signal and does not depend on its amplitude. Characteristic - depends on the amplitude of the input signal and does not depend on the frequency, since the nonlinear element is inertial free. For unambiguous characteristics, its values \u200b\u200bare real values, and for multivalued ones, complex ones.

Consider examples of constructing complex transfer factors for the most typical nonlinear elements -.

1. Non-linear element of the "amplifier with limitation" type. The link characteristics are shown in Fig. 6. Similar characteristics have different types amplifying and executive elements of automation (electronic, magnetic, pneumatic, hydraulic, etc.) in the area of \u200b\u200blarge input signals.

If the amplitude of the input action is less than a, then this is an ordinary linear non-inertia link, while the gain k is a constant value. The phase shift between input and output is zero because the characteristic of the nonlinear element is symmetrical. As the amplitude increases, the gain decreases. Some methods for studying nonlinear systems use the characteristic of the inverse complex gain of a nonlinear element (-1 /). This characteristic is shown in Fig. 6.

Since there is no phase shift between the harmonics of the input and output signals, the characteristic coincides with the real axis.

Non-linear element of the "dead band" type. The link characteristics are shown in Fig. 7. Amplifiers of various types have similar characteristics in the area of \u200b\u200bsmall input signals.

Figure: 7

If the amplitude of the input signal is within the range of ± a, then the output signal is zero, otherwise the output signal is non-zero, since the peaks of the input harmonics appear. There is no phase shift. At large amplitudes of the input signal, the gain is constant, that is, nonlinearity does not significantly affect the output signal.

3. Non-linear element of the "three-position relay without hysteresis" type. Link characteristics are shown in Fig. 8. This characteristic is inherent in closed-loop relay systems.

Since the characteristic is unambiguous, there is no phase shift. If the amplitude of the input signal® ¥, then the output signal turns into a pulse train. At small and large amplitudes, the coefficient k is small.

Figure: eight

4. Non-linear element of the "relay characteristic" type. Link characteristics are shown in (Fig. 9).

5. Non-linear element of the "backlash, gap" type. The characteristics of this

nonlinear element are shown in Fig. ten.

Non-linear element models. Models of nonlinear elements can be realized by including an operational amplifier (at the input or at feedback) nonlinear two-terminal networks. Depending on the characteristics of the two-terminal network and the method of its connection, any non-linear dependence can be realized (Fig. 11a, b, c).

Figure: eleven

Models of nonlinear links are widely used in computer simulation of automatic control systems.

Literature

1. Atabekov G.I., Timofeev A.B., Kupalyan S.D., Khukhrikov S.S. Theoretical Foundations of Electrical Engineering (TOE). Non-linear electrical circuits. Electromagnetic field. 5th ed. Publishing house: LAN, 2005. - 432p.

2. Besekersky V.A., Popov E.P. "Theory of automatic control systems". Profession, 2003 - 752s.

3. Gavrilov Nonlinear circuits in circuit simulation programs. Publishing house: SOLON-PRESS, 2002. - 368p.

4. Dorf R., Bishop R. Automation. Modern control systems. 2002 - 832s.

5. Collection of problems on the theory of automatic regulation and control / Edited by V. A. Besekersky. - M .: Science, 1978.

The characteristics of most real elements are non-linear to one degree or another. In some cases, the nonlinearity of the elements is small and when constructing a simplified model, it can be neglected, in others, nonlinearity cannot be neglected. Moreover, the functioning of most radio electronic devices is impossible without nonlinear elements (rectification, multiplication, limitation, generation, etc.).

Real nonlinear elements are subdivided into non-inertial and inertial. If the relationship between the instantaneous values \u200b\u200bof the current and voltage of the elements during periodic exposure is determined by the static volt-ampere characteristic (VAC), then the element belongs to non-inertial non-linear elements. If the static I - V characteristic and the dynamic one, taken at a frequency equal to or less than the operating frequency, do not coincide, then such an element should be considered as inertial.

Thus, the inertial nonlinear element is linear with respect to the instantaneous values \u200b\u200bof current and voltage, and the I – V characteristic connecting the effective values \u200b\u200bis nonlinear. Inertia-free elements are non-linear both in relation to instantaneous values \u200b\u200band in relation to effective and.

Depending on the number of external terminals, nonlinear bipolar elements (diodes, thermistors) and multipolar ones (transistors, triodes, pentodes) are distinguished. Volt - ampere characteristic of a nonlinear two-pole element can be symmetrical or unbalanced. The I - V characteristic of a two-terminal device with a symmetric characteristic is shown in Fig. 1. The condition is satisfied for it:

Obviously, the mode of operation of a nonlinear circuit will not change if the terminals of a nonlinear element with a symmetrical characteristic are swapped. If condition (1) is not met, the I - V characteristic is asymmetric.

The ratio of the voltage measured by the segment AB to the current measured by the segment OB (see Figure 1.) determines, on a certain scale, the static resistance R at point A.

(2)

The limit of the ratio of the voltage increment in the circuit section to the current increment in it or the derivative of the voltage by current on the same scale determines the differential resistance:

Distinguish between nonlinear elements with monotonic and non-monotonic IVC. For monotonic IVCs, or always greater than zero.

Non-monotonic characteristics are divided into N and S types... For elements with an N-shaped characteristic (Fig. 2.a), several different voltages can correspond to the same current value. For an S-shaped I-V characteristic, several currents can correspond to one voltage value (Fig. 2.b).

Fig. 2. I - V characteristics of various nonlinear elements

a) non-monotonic N-type; b) non-monotonic S - type;

c) I - V characteristic of a non-electrically controlled two-terminal device - a thermistor.

The type of the I - V characteristic of a nonlinear element may depend on a certain value that is not related to the currents and voltages of the circuit in which the element is connected, in particular, on temperature (Fig.2c), illumination, pressure, etc. Such elements belong to non-electrically controlled two-pole networks .

Fig. 3. Electrically controlled element

Non-linear elements are all semiconductor and electronic devices operating with signals whose instantaneous values \u200b\u200bvary within a fairly wide range. For concreteness, we will consider nonlinear two-terminal networks, when the input signal is voltage, and the output signal is current
in him. All methods and results can be carried over to the case of a nonlinear four-port network, for example, a transistor operating in a nonlinear mode at large values \u200b\u200bof the input signal amplitude. Here the output circuit is represented by a current source controlled by the input voltage. Characteristicnonlinear element establishes a functional nonlinear relationship between voltage
and amperage
in him:

(2.1)

IN inertial elementinstantaneous current value
depends not only on the voltage value
at the same time , but also on the values \u200b\u200bof this voltage at previous times. Inertial elements, strictly speaking, do not exist. Inertia-free condition is performed approximately if the characteristic time of change of the input signal is much longer than the process settling time inside the nonlinear element itself. The time to establish a stationary state in semiconductor devices is
from.

The inertia of devices can be associated with the inertia of current carriers. With an increase in the vibration frequency, it begins to manifest itself when the transit time of carriers through the device becomes commensurate with the vibration period. Such inertia manifests itself in the appearance of a phase lag (shift) of the output current relative to the input voltage, a change in the active input and output resistances and their transformation into complex ones, etc. As a result, the amplification factors of amplifiers and the output powers of the generators usually decrease. A characteristic type of inertia is thermal inertia in temperature changes, and hence the resistance of thermistors. Only at a sufficiently low vibration frequency does its element temperature manage to follow the instantaneous voltage values. For example, already at a frequency
Hz, the resistance of the filament of the incandescent lamp has practically no time to change, which ensures uniform illumination. Such inertial elements are used in harmonic oscillators to improve their characteristics.

The calculation of a nonlinear inertial device can be simplified if it is possible to represent it by combining two simpler devices: a nonlinear inertial device and a linear inertial device (filter). This approach can be used, for example, to calculate a resonant or bandpass amplifier at large input signal amplitudes. Let the active element of the amplifier (transistor or electronic tube) can be represented as a non-inertial nonlinear device, and nonlinear distortions in its passive load (oscillatory circuit or system of coupled circuits) can be neglected. The load containing the reactive elements is approximated by a linear inertial device.

Nonlinear dependences z \u003d f (x) can be classified according to various criteria:

1. By smoothness of characteristics: smooth - if at any point of the characteristic there is a derivative dz / dx, ie, the function is differentiable (Fig. 1a, b); piecewise-linear - a characteristic in which the derivatives have a discontinuity of the first (Fig.2a) or second kind (Fig.2b).

By unambiguity: unambiguous - in which each value of the input quantity corresponds to one value of the output quantity (Fig. 3a); multivalued - in which each value of the input quantity x corresponds to several values \u200b\u200bof the output quantity z (Fig. 3b, c, d).

By symmetry: evenly symmetric - symmetric about the ordinate, that is, z (x) \u003d z (- x) (Fig. 4a); odd symmetric - symmetric relative to the origin of coordinates, while z (x) \u003d - z (- x) (Fig. 4b); not symmetrical (Fig. 4c).

Non-linear circuits

Circuits are called non-linear if they contain at least one non-linear element. Nonlinear elements are described by nonlinear characteristics that do not have a strict analytical expression, are determined experimentally and are set in tables or graphs.

Non-linear elements can be divided into two - and multi-pole. The latter contain three (various semiconductor and electronic triodes) and more (magnetic amplifiers, multi-winding transformers, tetrodes, pentodes, etc.) poles, with which they are connected to an electrical circuit. A characteristic feature of multi-pole elements is that, in the general case, their properties are determined by a family of characteristics that represent the dependence of the output characteristics on the input variables and vice versa: the input characteristics are plotted for a number of fixed values \u200b\u200bof one of the output parameters, the output characteristics for a number of fixed values \u200b\u200bof one of the input parameters.

On another basis of classification, nonlinear elements can be divided into inertial and inertial ones. Elements whose characteristics depend on the rate of change of variables are called inertial. For such elements, the static characteristics that determine the relationship between the effective values \u200b\u200bof the variables differ from the dynamic characteristics that establish the relationship between the instantaneous values \u200b\u200bof the variables. Inertialess are called elements whose characteristics do not depend on the rate of change of variables. For such elements, the static and dynamic characteristics are the same.

The concepts of inertial and non-inertial elements are relative: an element can be considered as inertial in the permissible (top-limited) frequency range, beyond which it becomes inertial.

Depending on the type of characteristics, nonlinear elements with symmetric and asymmetric characteristics are distinguished. Symmetric is a characteristic that does not depend on the direction of the values \u200b\u200bthat determine it, i.e. having symmetry about the origin of the coordinate system. For an asymmetric characteristic, this condition is not met, i.e. The presence of a symmetric characteristic of a nonlinear element makes it possible in a number of cases to simplify the analysis of the circuit, carrying it out within one quadrant.

By the type of characteristic, you can also divide all non-linear elements into elements with unambiguous and ambiguous characteristics. An unambiguous characteristic is a characteristic in which each value of x corresponds to a single value of y and vice versa. In the case of an ambiguous characteristic, some x values \u200b\u200bmay correspond to two or more y values, or vice versa. In nonlinear resistors, the ambiguity of the characteristic is usually associated with the presence of a falling section, and in nonlinear inductive and capacitive elements, with hysteresis.

Finally, all non-linear elements can be divided into managed and unmanaged. Unlike uncontrolled non-linear controlled elements (usually three- and multipoles) contain control channels, changing the voltage, current, luminous flux, etc. in which they change their main characteristics: volt-ampere, weber-ampere or coulomb-volt.

Depending on the type of constituent non-linear elements, non-linear circuits are called.

Non-linear elements can be divided into three groups: non-linear active resistance r, non-linear inductance Land nonlinear capacitances C. Examples of nonlinear active resistances are vacuum and semiconductor diodes and triodes, nonlinear inductances - inductive coils and transformers with a magnetic circuit, nonlinear capacitors - capacitors with a ferroelectric dielectric.

In each of these groups, nonlinear elements, in turn, can be divided into two classes: uncontrolled and controlled nonlinear elements.

Uncontrolled nonlinear elements can always be represented as a two-port network. The current of this two-pole network depends only on the voltage applied to its terminals. Such a non-linear element is characterized by one current-voltage characteristic. An example of an uncontrolled nonlinear resistance is a vacuum or semiconductor diode.

Controlled non-linear elements are usually multipole. The current in the main circuit of such an element depends not only on the voltage applied to the main circuit, but also on other parameters (control factors). Control factors can be electrical or non-electrical. Examples of controlled nonlinear elements with an electrical control factor are multi-electrode vacuum tubes and magnet

amplifiers. An example of a controlled nonlinear resistance with a non-electrical control factor is a photoresistor, the current through which depends on the amount of illumination.

Uncontrolled nonlinear active resistances, according to the principle of thermal inertia, can be divided into two groups: inertial and inertial.

Examples of inertial resistance are incandescent lamps and thermistors. For these elements, the dependence is essentially nonlinear only between the effective or amplitude values \u200b\u200bof currents and voltages. Due to the thermal inertia during the period of the sinusoidal current, the resistance of these elements changes insignificantly. Therefore, with sufficient accuracy for practice, we can assume that the relationship between the instantaneous values \u200b\u200bof current and voltage within one period is linear.

An example of inertialess resistances are lamp and semiconductor diodes and triodes at not very high frequencies. Here, the characteristics are nonlinear for both the effective and instantaneous values \u200b\u200bof currents and voltages.

It should be noted that all real elements of electrical circuits have some nonlinearity. Therefore, the division of electrical circuits into linear and nonlinear is conditional. A circuit element can be considered linear or non-linear, depending on the degree of non-linearity and the problem that is posed when considering this circuit.