Classification of nonlinear elements. Abstract: Nonlinear elements Basic parameters and characteristics of nonlinear elements
Any chaotic system must have nonlinear elements or properties. In a linear system there cannot be chaotic oscillations. In a linear system, periodic external influences After the attenuation of transient processes, they cause a periodic response of the same period (Fig. 2.1). (The exception is parametric linear systems.) In mechanical systems, the following nonlinear components are possible:
1) nonlinear elastic elements;
Rice. 2.1. Scheme of possible signal transformations in linear and non-linear linear systems Oh.
2) nonlinear damping, similar to static and sliding friction;
3) backlash, clearance or bilinear springs;
4) most hydrodynamic phenomena;
5) nonlinear boundary conditions.
Nonlinear elastic effects can be associated either with the properties of substances or with geometric features. For example, the relationship between stress in a rubber sample and its deformation is nonlinear. However, although the stress-strain relationship of steel is usually linear up to the yield point, severe bending of a beam, slab, or shell may be nonlinearly related to the applied forces and moments. Such effects associated with strong displacements or rotations are usually called geometric nonlinearities in mechanics.
The nonlinear properties of electromagnetic systems are caused by the following factors:
1) nonlinear resistances, capacitances or inductive elements;
2) hysteresis in ferromagnetic materials;
3) nonlinear active elements, similar to vacuum tubes, transistors and diodes;
4) effects characteristic of moving media, for example electromotive force, where v is the speed and B is the magnetic field;
5) electromagnetic forces, for example, where J is the current, or where M is the dipole magnetic moment.
Examples of nonlinear devices are such ordinary elements electrical circuits, like diodes and transistors.
Rice. 2.2. Nonlinear problems with several equilibrium positions: a - longitudinal bending of a thin elastic rod under the action of an axial load at the end; 6 - longitudinal bending of an elastic rod by nonlinear magnetic mass forces.
Magnetic materials such as iron, nickel or ferrites are characterized by nonlinear material relationships between the magnetizing field and magnetic flux density. Using operational amplifiers and diodes, some experimenters manage to assemble negative resistances with a bilinear current-voltage characteristic (see Chapter 4).
It is not easy to identify nonlinearities in every system, firstly, because we are often trained to think in terms of linear systems, and secondly, because the main components of the system may be linear and nonlinearity is a subtle effect. For example, individual elements of a fastening truss may be linearly elastic, but they are assembled in such a way that there are gaps and nonlinear friction is present. Thus, nonlinearity may be hidden in the boundary conditions.
In the example with a curved rod, nonlinear elements are easily identified (Fig. 2.2). In any mechanical device that has more than one position of static equilibrium, play, backlash, or nonlinear stiffness are present. In the case of a rod bent by a load at the end (Fig. 2.2, a), the culprit is the geometric nonlinearity of the stiffness. In a rod bent by magnetic forces (Fig. 2.2, b), the source of chaotic behavior of the system is nonlinear magnetic forces.
Classification of nonlinear elements
Nonlinear circuits are circuits that have at least one nonlinear element. A nonlinear element is an element for which the relationship between current and voltage is not specified. linear equation.
In nonlinear circuits the principle of superposition is not followed, and therefore there are no general calculation methods. This necessitates the development of special calculation methods for each type of nonlinear elements and their operating modes.
Nonlinear elements are classified:
1) by physical nature: conductor, semiconductor, dielectric, electronic, ionic, etc.;
2) by nature they are divided into resistive, capacitive and inductive;
VAC VAC VAC
3) according to the type of characteristics, all elements are divided
For symmetrical and asymmetrical. Symmetrical are those whose characteristic is symmetrical with respect to the origin of coordinates. For asymmetrical elements, the positive direction of voltage or current is chosen once and for all, and the current-voltage characteristics are given for them in reference books. Only this direction can be used when solving problems using these current-voltage characteristics.
Unambiguous and ambiguous. Ambiguous, when several points correspond to one current or voltage value on the current-voltage characteristic;
4) inertial and non-inertial elements. Inertial elements are those elements in which nonlinearity is caused by heating of the body during the passage of current. Since the temperature cannot change arbitrarily quickly, when an alternating current passes through such an element with a sufficiently high frequency and a constant effective value, the temperature of the element remains almost constant throughout the entire period of current change. Therefore, for instantaneous values, the element turns out to be linear and is characterized by some constant value R (I,U). If the effective value of the current changes, then the temperature will change and a different resistance will be obtained, i.e. for effective values the element will become nonlinear.
5) managed and unmanaged elements. Above we talked about unmanaged elements. TO managed elements include elements with three or more terminals, in which, by changing the current or voltage at one terminal, it is possible to change the current-voltage characteristic relative to other terminals.
Parameters of nonlinear elements and some equivalent circuits
Depending on the specific task, it is convenient to use certain parameters of the elements and their total number is large, but static and differential parameters are most often used. For a resistive two-pole element, these will be static and differential resistance.
At a given point the current-voltage characteristic
At a given operating point, the current-voltage characteristic
1. Give a small voltage increment. The current increment caused by this increment is found from the current-voltage characteristic and their ratio is taken. The disadvantage of this method is that to increase the accuracy of the calculation it is necessary to reduce U and I, but it is difficult to work with the graph.
2. A tangent is drawn to a given point on the curve and then, by the geometric definition of the derivative, we get
Where the increments are taken on this tangent and can be as large as desired.
If the operating mode of a nonlinear element is known, then at this point its static resistance, as well as voltage and current, are known, so it can be replaced in one of 3 ways.
If it is known that during operation of the circuit, the current and voltage change within the “more or less straight section of the current-voltage characteristic,” then this section is described by a linear equation and such an equivalent circuit is associated with it.
Linearize this section with an equation of the form U=a+ib. Obtain the coefficients of the equation for it.
For i=0 and U=U 0 =a,
average value in this area.
Then, what corresponds to the following substitution scheme:
This scheme will be valid for the area limited by a wavy line.
The same expression can be written differently:
Therefore, in some problems where it is known in advance that the currents and voltages of a nonlinear element are represented as the sum of a constant component Urt, Irt and a variable component 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.
The same approach is also used in circuits with multi-pole elements, but there it is not possible to introduce only one resistance, since CPs are characterized by four coefficients of the equations. But you can find these coefficients for small alternating components of currents and voltages.
Example: Bipolar transistor (common emitter circuit).
Let it be known that u j =U p f+u kj, i j =I p f+i kj
Substitution scheme:
Let's apply differentiating parameters and get it in the form “I”.
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
These equations are written for variable components because the procedure for calculating the elements changes.
H 11 =U be /I b at I b =0, i.e. i b =I br.t.
H 12 =U be /U ke at I b =0
H 21 =I k /I b at U ke =0
H 22 =I k /U ke at I b =0, i.e. i b =I br.t.
h 12 = DU be / DU ke h 21 = Di k / Di b h 22 = Di k / DU ke,
where I, U are the increments of currents and voltages in the vicinity of the operating point.
Current-voltage characteristics of this nonlinear element.
Calculation methods nonlinear circuits direct current
There are: numerical, analytical and graphical methods.
1) Numerical are methods for numerically solving nonlinear equations. Usually a computer is used. They allow you to solve a wide range of problems, but the answer is obtained in the form of a number.
2) Analytical - these are methods based on the approximation of the current-voltage characteristic of some suitable function. If this function is nonlinear, then a nonlinear system of equations is obtained. In order for it to be solved, one has to choose the approximating function very carefully.
Content. Nonlinear elements. Saturation of magnetic materials. Ferroelectrics, varistors and posistors. Nonlinear resistors. Semiconductor diode and its current-voltage characteristic. The concept of the design of bipolar transistors and thyristors. Linear voltage stabilizer. Operating principle of field effect transistor and insulated gate bipolar transistor (IGBT).
The values of the elements R, C, L were entered as coefficients between current and voltage (R), charge and voltage (C), and magnetic flux and current (L). Further, from these relations, the generalized Ohm's law was formulated.
When considering the simplest problems, the assumption was made that these values do not depend on the electromagnetic energy flowing through these elements. And we took great pleasure in manipulating the so-called linear elements and even selecting the corresponding “linear” components.
However, linear components do not exist in nature!
They can have approximately linear parameters only in a certain range of currents and voltages. Any substance exposed to electromagnetic fields, one way or another, changes its structure and, accordingly, its physical characteristics, namely resistivity, dielectric and magnetic permeability, and even geometric shape. Therefore, the parameters of components made from these materials also change, since R=rl/s; C»es/l; L»ms/l. If these changes are not significant, then we talk about the linearity of the elements and corresponding components. Otherwise, it is necessary to take these changes into account and then we should talk about nonlinear elements and components.
UGO of nonlinear elements in equivalent circuits have the following form:
nonlinear resistor
inductor with magnetic core
nonlinear capacitor - varicap
Nonlinear elements are widely used in electrical circuits to change the shape of a signal, in other words, to excite or absorb certain harmonics that make up the signal.
From a mathematical point of view, in this case, the coefficients composed of R, C, L depend on unknown parameters (current and voltage), and the energy equations compiled according to Kirchhoff's rules become nonlinear with all the ensuing consequences for calculations.
The most common methods for solving them are:
- approximation, when the known nonlinear dependence of the element value on current or voltage is approximated by segments of linear functions and solutions of linear equations are obtained for each of them;
- graphic method when equations are solved graphically using
known nonlinear graphical dependences of the element on current or voltage;
- machine method, when the nonlinear dependence of the element value on current or voltage is approximated by a model mathematical function and integro-differential nonlinear equations are solved by numerical methods.
nonlinear inductance used in electrical engineering Weber-amp characteristics, which are similar to the BH hysteresis curves for ferromagnetic materials that physicists like to use. If on the Weber-ampere characteristic L=dY/dI, then on the HV curves m=dB/dH, but Y=NBS, and H»I/r. Sometimes they use volt-second characteristic, because Y=òUdt.
When approximating, this characteristic is usually divided into parts: before saturation it is a straight line with a slope m =dB/dH, and after saturation at Vm this is a straight line with m =1. Residual magnetization values INr and coercivity NS determine the area occupied by the hysteresis loop, i.e. active losses due to magnetization reversal. Therefore, in most cases they can be taken into account by introducing a resistive element into the circuit and excluded from the approximation of the Weber-ampere characteristic.
The operating mode of inductors with linear characteristics is selected within large values of m or L. Magnetic devices such as chokes for storing magnetic energy, transformers for transmitting power through magnetic coupling of coils, as well as electric motors operate in this area. At the same time, the effect of nonlinearity of magnetic materials is widely used to create magnetic amplifiers, ferroresonant stabilizers, and even magnetic key elements, which use magnetic materials with the so-called rectangular magnetic characteristic, where m can reach values of 50 or more. Currently, mainly 3 types of magnetic materials are used in inductors: electrical steel, amorphous iron (metaglass) And ferrites with very different hysteresis curves.
Nonlinear inductors were historically the first to be created due to the availability and low cost of magnetic materials, as well as the ease of their manufacture. They are distinguished, first of all, by their reliability, but have large weight and size characteristics, and therefore high inertia. Magnetization reversal losses and active losses due to heating of windings also represent a serious problem, especially in power electrical engineering. Therefore, at present, the use of nonlinear inductors is limited.
To represent dependency nonlinear capacitance use coulomb-volt characteristics, since C=dQ/dU.
They are similar to ferromagnetic weber-ampere characteristics, only here there is the dielectric constant e=dD/dE, where D is electrical induction or electrical displacement.
The most interesting dielectric for creating nonlinear capacitors are ferroelectrics, such as Rochelle salt (potassium sodium tartrate), barium titanate, bismuth titanate, etc. Due to the domain structure of electric dipoles, at low voltages they have a high dielectric constant with e » 1000, which decreases with increasing voltage, similar to the magnetic permeability of ferromagnets . Therefore, in foreign literature they were called ferroelectrics. These materials are widely used to create linear capacitive elements such as ceramic capacitors with high electrical energy density, where they operate in the unsaturated region of the coulomb-voltage characteristic. Nonlinearity is used to create capacitors with variable capacitance, varicondas, which have a narrow application.
In an alternating field in ferroelectrics, the direction of the electric moment of the dipoles, which are connected into large domains placed in crystal structures, changes. This leads to a change in the geometric dimensions of the crystal, the so-called effect electrostriction. There is a similar effect in magnetic materials magnetostriction, but it is difficult to use due to the presence of an external winding. In some groups of ferroelectric crystals, effects similar to electrostriction are observed. This direct piezoelectric effect – the appearance of an electric field (polarization) in a crystal during its mechanical deformation, and back– mechanical deformation when an electric field appears. These crystalline materials are called piezoelectrics, and they have received extremely great use. The direct effect is used to obtain high voltages, in primary transducers of mechanical forces (for example, microphones, sound pickups in mechanical sound recording systems), etc. The inverse effect is used in sound and ultrasonic emitters, in ultra-precise positioning systems (positioner for moving the hard disk head), etc. Both effects are used when creating resonant crystal oscillators, where the dimensions of the crystals are selected in such a way that mechanical vibrations are in resonance with electrical ones. With a very high quality factor of such a system, stability and accuracy of the generator frequency settings are ensured. Two such crystals, having a sound connection, can transmit electrical power without galvanic connection, for which they are called piezotransformers.
The domain structure of both electric and magnetic dipoles decays at a certain temperature, called the Curie point. In this case, a phase transition occurs and the conductivity of the ferroelectric changes significantly. On this basis they act posistors, in which, with additional doping of the material, a certain Curie point can be set. After reaching this temperature, the rate of increase in resistance can reach 1 kOhm/deg.
Essentially this nonlinear resistor, which has an S-shape or "key" current-voltage characteristic (volt-ampere characteristic).
That is, this element can work like an electric switch controlled by passing current or external temperature.
PTC resistors are widely used for protection against current overloads in analogue telephone networks, as well as for relieving magnetic energy from coils when they are turned off, soft starting of motors, etc. They have found a rather interesting application as adjustable fuel elements in fan heaters in which the element itself is located at almost constant temperature, and the consumed electrical power is automatically maintained equal to the removed thermal power. That is, the fan rotation speed can be controlled by the thermal power of such a heating device.
With another type of ferroelectric doping, it is possible to achieve the effect of a nonlinear dependence of its conductivity on voltage, i.e., this is actually nonlinear resistor, called varistor. This effect is due to a change in the conductivity of thin layers of matter surrounding the domains at a certain voltage. Therefore they are characterized current-voltage characteristic, where the function U(I) can be represented by a fifth-degree polynomial. It is convenient to characterize nonlinear resistors with static resistance Rst = U/I and differential resistance Rd = dU/dI. It can be seen that in the linear section Rst ~ Rd, in the nonlinear section Rst £ Rd.
Their main application is the protection of electrical circuits from switching surges of dangerous overvoltages. In a varistor, the energy of such a surge is converted into active energy and heats its mass. Therefore, varistors are distinguished by two main parameters - the voltage at which the current-voltage characteristic breaks and the energy that the element can absorb without affecting its performance.
Nonlinear resistors of various types occupy a large place in modern electrical engineering. Generally speaking, any conductor is nonlinear. If you pass current through an ordinary copper wire, then at first its resistance, as is known, will change as R0(1+αT). This dependence will persist until the wire melts and then the resistance will remain constant until the material evaporates. And in this state, the wire actually becomes an insulator.
Conductor resistance R is inversely proportional to current density, so the resistance of a bare copper conductor is considered linear up to the current density 10 A/mm2 . As heat removal from the conductor deteriorates, this value decreases. For example, in the winding of an inductor this value can be at the level of 2 A/mm2. Since when these current density values are exceeded, an increasing release of thermal energy occurs, which leads to its melting, they are considered permissible current density values and are used when choosing safe conductor cross-sections.
They work on this principle fuses, the cross-section of the conductor in which corresponds to the limiting value of the current passing through it. But if a power of more than 1010 W/g is put into the wire, then the evaporation, bypassing the melting stage, will follow an adiabatic path and the pressure wave of the gas evaporating from the surface will create colossal densities of matter inside the material. In this case, it was possible to free gold atoms from their electron shell and carry out thermonuclear reactions.
At a certain voltage, sufficient for the appearance of a sufficient number of electric charge carriers in the gas, an electric current begins to flow in the gas gap. This phenomenon is called gas discharge, and the gas-discharge gap itself can be considered as a nonlinear resistance with the following current-voltage characteristic.
Gas-discharge devices have become very widespread as indicators, welding machines and melting units, electric switches and plasma-chemical reactors, etc.
In 1873, F. Guthrie discovered the effect of nonlinear conductivity in a vacuum tube with a thermionic cathode. When the cathode had a negative potential, its electrons created an electric current, and with the opposite polarity they were locked at the cathode and there were practically no carriers in the lamp. For a long time, this effect was not in demand, until in 1904 the needs of radio engineering led to the creation of a thermionic (vacuum) diode. And since in such a device the electric field is responsible for conductivity, the introduction of additional small potentials makes it possible to control the flow of electrons, that is, electric current. Thus, they were created electric field controlled nonlinear resistors (radio tubes), which replaced large, inertial and current-controlled nonlinear magnetic systems. The main disadvantages of radio tubes were the heated cathode, which requires a separate power source and appropriate cooling, as well as rather large dimensions due to the vacuum flask.
Therefore, almost simultaneously with the vacuum (thermionic) diode, a solid-state diode based on p-n transition, which is formed at the point of contact of two semiconductors with different types of conductivity. However, technological difficulties in the production of pure semiconductor materials somewhat delayed the introduction of these elements in relation to radio tubes.
When two regions with different types of conductivity come into contact, the charge carriers from them mutually penetrate (diffuse) into the neighboring region, where they are not the majority carriers. In this case, uncompensated acceptors (negative charges) remain in the p-region, and uncompensated donors (positive charges) remain in the n-region, which form space charge region(SCR) with an electric field that prevents further diffusion of charge carriers. In the zone p-transition n an equilibrium is created with a contact potential difference, which for silicon, which is widely used in semiconductor devices, amounts to 0.7 V.
When an external electric field is connected, this balance is disrupted. With forward bias (“+” in the p-type region), the SCR width decreases and the minority carrier concentration increases exponentially. They are compensated by the main carriers coming through contacts from the external circuit, which creates direct current, increasing exponentially as the forward bias voltage increases.
With reverse bias (“-” in the p-type region), the SCR width increases and the concentration of minority carriers decreases. The main carriers do not enter this zone, but the existing reverse current is caused only by the removal of minority carriers from the SCR and does not depend on the applied voltage. The forward and reverse currents can differ by 105–106 times, forming a significant nonlinearity of the current-voltage characteristic. At a certain value of the reverse voltage, charge carriers, during their free movement, can acquire energy sufficient to form new pairs of charges when they collide with neutrals, which in turn gain energy and participate in the birth of new pairs. The resulting avalanche current sweeps away all potential barriers in its path, turning the semiconductor into an ordinary conductor.
UGO semiconductor diode
Typical shape of the current-voltage characteristic of a p-n junction (diode)
An approximation of an “ideal” diode is an ideal electrical switch controlled by voltage polarity. However, this does not take into account such parameters as:
1) Forward voltage drop when direct current flows, which in many real devices is 1 -1.5 V, and this leads to active losses P = (1¸1.5)I, and, consequently, to heating of the element and limiting currents for a particular element. Solving thermal problems of cooling semiconductor devices, as well as their thermal stability, are one of the main problems in the design of electrical devices. The inversely proportional dependence of the forward voltage drop on temperature limits the use of devices with p-n junctions in parallel connections.
2) Reverse currents, which can be neglected only if they are several orders of magnitude smaller than the direct currents.
3) Avalanche breakdown voltage, which determines the operating limit of the element under reverse voltage, which you need to pay attention to, especially when working with pulsed inductive elements. However, the overall thickness of the crystal limits reverse voltages to 1 – 2 kV. A further increase in the reverse voltage is possible only with sequential assembly of elements with equalization of reverse currents.
4) Temporal characteristics in particular recovery time(the time of transition from a conducting to a non-conducting state), which is actually the time of removal of minority carriers from the SCR and its expansion. And this parameter is determined by diffuse processes with characteristic durations of 10-5 s. When modeling impulse responses in diode equivalent circuits, 2 capacitive elements are used: barrier capacitance, which is determined by the size of the SCR and the space charge (it is significant at reverse voltages), as well as diffuse capacity, which is determined by the concentration of the majority and minority carriers (it is significant for a forward voltage drop). The diffuse capacitance determines the times of accumulation and resorption of the nonequilibrium charge in the SCR and can reach a value of several tens of nanofarads. The development of technological processes in the manufacture of diodes has made it possible to significantly influence the pulse characteristics and reduce the recovery time to tens of nanoseconds in fast and ultra-fast diodes.
Therefore, the mathematical model of a real semiconductor diode developed for the Spice program and its further modifications is a rather complex mathematical expression that includes up to 30 constants set by the user to model a specific element.
Work to reduce forward voltage drop led to the creation Schottky diodes, in which the p-n junction is replaced by a Schottky barrier formed by a metal-semiconductor pair. This made it possible to reduce the size of the SCR and approximately halve the forward voltage drop, but at the same time the permissible reverse voltage significantly decreased (< 250 В) и увеличились обратные токи. При этом улучшились импульсные характеристики, что позволило применять эти диоды при частотах до 100 кГц.
Sharp decrease dynamic resistance(Rд=dU/dIt) at reverse breakdown voltage allows the use of diodes as voltage stabilizers, like varistors. But diodes, unlike varistors, have lower dynamic resistance values. However, it should be taken into account that in the stabilization mode, the energy released in the SCR of the p-n junction is equal to P = Ul. pr×I. That's why they were created Zener diodes And avalanche diodes with heat-resistant p-n junction and based on them Zener diodes.
When a direct current passes through the SCR, charge carriers recombine with the emission of a photon, the wavelength of which is determined by the semiconductor material. By varying the composition of this material and the design of the element, it is possible to create LEDs with coherent ( laser diodes) and incoherent radiation for a very wide spectral range, from ultraviolet to infrared light.
The development of semiconductor technologies has led to the creation bipolar transistor, which consists of three layers of semiconductor material with different types of conductivity, n-p-n or p-n-p. These layers are called collector-base-emitter. Thus, we got 2 consecutive p-n junctions, but with oppositely directed conductivity. To achieve the transistor effect, it is necessary that the emitter conductivity be greater than the base conductivity, and the thickness of the base be comparable to the width of the SCR of the collector-base junction with reverse conductivity. To operate an n-p-n transistor according to a circuit with a common base, the positive pole of the source is connected to the collector, the negative pole is connected to the emitter, and the base-emitter junction is opened with an additional source. At the same time, minority carriers - electrons - will begin to flow into the thin base layer. Some of them, under the influence of the positive potential of the collector, will pass through the closed base-collector junction, causing an increase in the collector current as a reverse current through this junction. Moreover, the collector current can be several hundred times higher than the base current ( transistor effect).
Thus, a bipolar transistor can be thought of as a nonlinear resistance controlled by a base current.
UGO bipolar transistors have the following form:
I-V characteristics of a bipolar transistor or the dependence of the collector current on the collector-emitter voltage UCE(IC) for the 2N2222 transistor at different base currents.
Thus, the collector current is determined by the base current, but this dependence at low base currents is significantly nonlinear. This is the so-called active mode.
At high base currents, when the full opening of the collector-base junction is achieved, the transistor goes into saturation at a minimum collector-emitter voltage drop equal to double the contact potential difference » 1.2¸1.4 V (two open p-n junctions connected in series). We get rich mode.
This leads to 2 possibilities for using transistors - in active mode, as amplifier, and in saturated mode - like electric key.
Let's take an example of using a transistor in active mode - linear voltage stabilizer.
In this circuit, the transistor is connected according to a common collector circuit, i.e., the collector current and base current sources are connected by a common point and the control current enters the base through resistor Rv. Since the base-emitter junction is open, we can assume that the voltage drop across it does not depend on the current and is equal to the potential barrier UBE = 0.6-0.7 V. In the absence of a zener diode DZ, the output voltage according to the voltage divider rule is UOUT ~ UIN RL/RV+RL. The DZ zener diode maintains a constant voltage level based on the UZ. But then UOUT= UZ - UBE is a constant value and does not depend on the input voltage and load current. At a constant load current and, accordingly, base current, any increase in the input voltage Uin will not change the collector current, since the dynamic resistance of the collector-base junction in the active mode of the transistor is close to ¥. At the same time, a change in the load current will simply lead to a change in the base current and, accordingly, a change in the collector current.
Operation of a bipolar transistor in saturation mode requires large control currents, commensurate in magnitude and duration with the switched currents. Therefore it was proposed thyristor, consisting of 4 consecutive p-n-p-n layers.
When the control current is turned on, the first p-n junction opens (base-emitter of transistor Q1) and electrons from the emitter begin to penetrate through the second p-n junction (base-collector of transistor Q1). At the same time, the third p-n junction opens (base-emitter p-n-p transistor Q2) and, respectively, the second p-n junction (base-collector of transistor Q2). This ensures the flow of current into the first pn junction and the control current is no longer needed. The deep connection between all transitions ensures their saturation.
Thus, with a short pulse of control current, we managed to transfer the system to a saturated state with a voltage drop of about 2 V. To turn off the current in this structure, we need to reduce it to 0, and this is quite easy to achieve with a harmonic signal. As a result, we have obtained powerful semiconductor switches for alternating current networks, controlled by short pulses at the beginning of each half-cycle.
You can also change the conductivity of a semiconductor structure by applying an electric field to it, which will create additional carriers for current. These media will be main and they don't need to diffuse anywhere. This circumstance provides two advantages compared to bipolar structures.
Firstly, the times of change in conductivity are reduced, and secondly, control is carried out by a potential signal at practically zero current, that is, the main current is practically independent of the control current. And another advantage arose due to the homogeneity of the semiconductor structure, controlled by an electric field - this is a positive temperature coefficient of resistance, which made it possible to manufacture these structures using microelectronics in the form of individual microcells (up to several million per sq. cm) and, if necessary, connect them in parallel.
Transistors created on this principle are called field(in foreign literature FET or Field emission transistor). Currently, a large number of different designs of such devices have been developed. Consider a field-effect transistor with an insulated gate, in which the control electrode ( gate), separated from the semiconductor by an insulating layer, usually aluminum oxide. This design is called MOS (metal-oxide-semiconductor) or MOS (metal-oxide-semiconductor). The space of a semiconductor where additional carriers are formed under the influence of an electric field is called channel, the entrance and exit to which, respectively, are called source And drain. Depending on the manufacturing technology, channels can be induced (p-conductivity is created in the n-material or vice versa) or built-in (space with p-conductivity is created in the n-material or vice versa). The figure shows a typical horizontal design of an MOS transistor with an induced and a built-in p-channel.
UGO MOS transistor
 |
Here are the transfer characteristics of the BUZ11 transistor, namely the dependence of the drain current and drain-source voltage on the gate voltage. It can be seen that the opening of the transistor begins with a certain value of Uthr and quite quickly it enters saturation.
Here is the static characteristic of the BUZ11 transistor, namely the dependence of the drain current on the drain-source voltage. Markers mark the points of transition to saturation mode
The resistance of field-effect transistors to current overloads, high input resistance, which can significantly reduce control losses, high switching speed, positive temperature coefficient of resistance - all this allowed devices with field control not only to practically replace bipolar devices, but also to create a new direction in electrical engineering - intelligent power electronics, where control of energy flows of almost any power is carried out with clock frequencies of the order of tens of kilohertz, i.e., virtually in real time.
However, at high currents, field-effect transistors are inferior to bipolar transistors in terms of direct losses. If in a bipolar transistor, provided it is saturated, the losses are determined by P = IКUpr, where Upr is practically independent of the current and is approximately equal to the height of the potential barrier at two open p-n junctions, then in field-effect transistors P = IС2 Rpr, where Rpr is mainly the resistance of a homogeneous channel .
A solution to this problem was found by combining field control with a bipolar transistor. This insulated gate bipolar transistor is better known by its trade name IGBT (Insulation Gate Bipolar Transistor).
UGO for IGBT
As you can see, here a p+ layer was added to the vertical structure of the field-effect transistor as a substrate, and a bipolar pnp transistor was formed between the emitter E and the collector K. Under the influence of a positive potential at gate G, a conducting channel appears in the p-region, which opens junction J1. At the same time, injection of minority carriers begins deep into the low-resistance n-layer, layer J2 opens slightly and a current begins to flow between the collector and emitter, supported by carriers in the p-layer, which keep the p-n junction J1 in the open state. The voltage drop across the JGBT is determined by the voltage drop across the open p-n junctions J1 and J2, just like in a conventional bipolar transistor. The turn-off times of JGBT are determined by the times of resorption of minority carriers from these junctions. That is, the device turns on as a field-effect transistor, and turns off as a bipolar one, as can be seen in the example of switching the GA100T560U_IR device.
This structure can be thought of as a combination of a field-effect control transistor and a bipolar main transistor.
The temperature dependence of the voltage drop across the JGBT is determined by the negative coefficient on the J2 junction and the positive coefficient on the p-layer channel, as well as the n-layer. As a result, the developers managed to make the positive temperature coefficient prevail, which opened the way for parallel connection of these semiconductor structures and made it possible to create devices for practically unlimited currents.
Assembly on IGBT for switching
voltages up to 3300 V and currents
Subject: Theory of automatic control
Topic: NONLINEAR ELEMENTS
1. Classification of nonlinear elements
Nonlinear dependencies z = f(x) can be classified according to various criteria:
1. According to the smoothness of the characteristics: smooth - if at any point of the characteristic there is a derivative dz/dx, i.e. 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).
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By symmetry: even-symmetric - symmetrical with respect to the ordinate axis, i.e. z(x) = z (- x) (Fig. 4a); odd-symmetric - symmetric about the origin, with z (x) = - z (- x) (Fig. 4b); not symmetrical (Fig. 4c).
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2. Nonlinear circuits
Nonlinear circuits are those that contain at least one nonlinear element. Nonlinear elements are described by nonlinear characteristics that do not have a strict analytical expression, are determined experimentally and are given in tables or graphs.
Nonlinear 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 the help of which they are connected to the electrical circuit. A characteristic feature of multi-pole elements is that, in the general case, their properties are determined by a family of characteristics representing the dependence of the output characteristics on the input variables and vice versa: the input characteristics are built for a number of fixed values of one of the output parameters, the output ones - for a number of fixed values of one of the input parameters.
According to another classification criterion, nonlinear elements can be divided into inertial and non-inertial. Inertial elements are elements whose characteristics depend on the rate of change of variables. For such elements, the static characteristics that determine the relationship between the current values of the variables differ from the dynamic characteristics that establish the relationship between the instantaneous values of the variables. Inertia-free elements are those 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 inertial-free elements are relative: an element can be considered as inertial-free in the permissible (limited from above) frequency range, beyond which it becomes inertial.
Depending on the type of characteristics, nonlinear elements with symmetrical and asymmetrical characteristics are distinguished. A characteristic that does not depend on the direction of the quantities that determine it is called symmetric, i.e. having symmetry with respect to the origin of the coordinate system. For an asymmetric characteristic, this condition is not met, i.e. The presence of a symmetrical characteristic of a nonlinear element allows, in a number of cases, to simplify the analysis of the circuit, carrying it out within one quadrant.
By type of characteristic, you can also divide all nonlinear elements into elements with unambiguous and ambiguous characteristics. A characteristic is called unambiguous 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 may correspond to two or more y values or vice versa. For nonlinear resistors, the ambiguity of the characteristic is usually associated with the presence of a falling section, and for nonlinear inductive and capacitive elements - with hysteresis.
Finally, all nonlinear elements can be divided into controlled and uncontrolled. Unlike uncontrolled, controlled nonlinear elements (usually three- and multi-terminal networks) contain control channels, changing voltage, current, luminous flux, etc. in which their main characteristics change: volt-ampere, Weber-ampere or coulomb-voltage.
Depending on the type of constituent nonlinear elements, they are called nonlinear circuits.
3. Gain of a nonlinear element
Let's consider a nonlinear element (Fig. 5). Let us apply a harmonic signal with amplitude – A 0 to the input of the nonlinear element and determine the first harmonic of the output signal.
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In this case, the following relations can be written for the input and output signals:
(1)
where: - vector module; 
- vector argument.
Let us consider the characteristic of a nonlinear element -, which is called the complex transmission coefficient of the nonlinear element. This characteristic can be constructed in the complex plane in the same way as the complex transmission coefficient 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 frequency, since the nonlinear element is inertia-free. For single-valued characteristics, its values are real quantities, and for multi-valued ones, they are complex.
Let's consider examples of constructing complex transmission coefficients for the most typical nonlinear elements - .
1. Nonlinear element of the “limited amplifier” type. The characteristics of the link are shown in Fig. 6. Various types of amplifying and actuating elements of automation (electronic, magnetic, pneumatic, hydraulic, etc.) in the area of large input signals have similar characteristics.
If the amplitude of the input action is less than a, then this is an ordinary linear inertia-free link, and the gain k is a constant value. The phase shift between input and output is zero since 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 transfer coefficient of the 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.
Nonlinear element of the "dead zone" type. The characteristics of the link are shown in Fig. 7. Various types of amplifiers in the field of small input signals have similar characteristics.
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If the amplitude of the input signal is within the range ± a, then the output signal is zero, otherwise the output signal is not zero, since the peaks of the input harmonic appear. There is no phase shift. At large amplitudes of the input signal, the gain has a constant value, i.e., nonlinearity does not have a significant effect on the output signal.
3. Nonlinear element of the “three-position relay without hysteresis” type. The characteristics of the link are shown in Fig. 8. This characteristic is inherent in relay systems with feedback.
Since the characteristic is unambiguous, there is no phase shift. If the amplitude of the input signal is ®¥, then the output signal turns into a sequence of pulses. For small and large amplitudes, the coefficient k is small.
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4. Nonlinear element of the "relay characteristic" type. The characteristics of the link are shown in (Fig. 9).
5. Nonlinear element of the “backlash, clearance” type. Characteristics of this
nonlinear element are shown in Fig. 10.
Models of nonlinear elements. Models of nonlinear elements can be implemented by including nonlinear two-terminal networks in the operational amplifier circuit (input or feedback). Depending on the characteristics of the two-terminal network and the method of its connection, any nonlinear dependence can be realized (Fig. 11a, b, c).
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Models of nonlinear links are widely used in modeling automatic control systems on a computer.
Literature
1. Atabekov G.I., Timofeev A.B., Kupalyan S.D., Khukhrikov S.S. Theoretical foundations of electrical engineering (TOE). Nonlinear electrical circuits. Electromagnetic field. 5th ed. Publishing house: LAN, 2005. – 432 p.
2. Besekersky V.A., Popov E.P. "Theory of automatic control systems." Profession, 2003 - 752 p.
3. Gavrilov Nonlinear circuits in circuit modeling programs. Publishing house: SOLON-PRESS, 2002. – 368 p.
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.
When a circuit is subject to constant EMF and voltages, the value of the direct current in it is determined by the resistances and conductivities G circuit elements, i.e. these parameters are basic. As for capacitance and inductance, in the case of nonlinear DC circuits they play a role only when deciding the issue sustainability regime in such a circuit. But even in an alternating current circuit, for many nonlinear elements, their resistance and conductivity are of primary importance. In this regard, we will consider such nonlinear elements and their characteristics, the main parameters of which are resistance and conductivity.
For an element characterized by constant resistance, the current-voltage characteristic is a straight line (Fig. 9.1).
The characteristics of nonlinear elements are usually divided into static and dynamic. Static refers to characteristics obtained with a very slow (infinitely slow) change in current or voltage. Dynamic characteristics give the relationship between voltage and current when they change rapidly. These characteristics may differ from static ones, for example due to thermal inertia.
There are concepts of static and dynamic resistance, as well as static and dynamic conductivity. Under static resistance ( R c) for a given current, the ratio of the voltage corresponding to the specified current according to the static characteristic to the value of this current is called (Fig. 9.2). The reciprocal of static resistance is called static conductivity.
, υ, a – voltage and current scales.
Under dynamic resistance ( R e) at a given point of the dynamic characteristic is called derivative voltage versus current at a specified point in the dynamic characteristic. The reciprocal of dynamic resistance is called dynamic conductivity ( G d). Let the dynamic characteristic coincide with the static one. Then the dynamic resistance can be determined from the given static characteristic as follows:
Where β – angle of inclination of the tangent to the dynamic characteristic to the abscissa axis.
All specified parameters R st, R d, change from point to point, i.e. depend on current. For passive elements, i.e. containing no energy sources, always R c > 0, G c > 0 but R d , G d are positive only for points lying on the ascending part of the characteristic and negative for points of the falling part (Fig. 9.3).
9.2. Current-voltage characteristics of some nonlinear elements
1. Semiconductor diode, its ampere-voltage characteristic is shown in Fig. 9.4.
2. In technology high voltage Tirite nonlinear elements made of ceramic material – tirite – are used. The characteristic of tirite is as follows (Fig. 9.5)
The resistance of tirite decreases with increasing voltage, i.e. conductivity increases. This dependence of conductivity on voltage allows the use of thirite elements to protect high-voltage installations of power plants, substation transformers, etc. from overvoltage. Install so-called thirite spark gaps (T) (Fig. 9.6), connected across the spark gap and in parallel with the protected installation ( N) between the high voltage alternating current (HV) line and ground.
At the rated voltage, the spark gap is not broken and no current passes through the spark gap. When the rated voltage is exceeded, the spark gap breaks through and a large current passes through the spark gap, i.e. As the voltage increases, its resistance drops sharply. As a result, the line (VN) is discharged into the thirite spark gap (T) and the voltage on the line drops. In this case, the resistance of the spark gap increases, and the current through it decreases. A sharp decrease in current leads to the cessation of the discharge in the spark gap and, consequently, to the cessation of current in the spark gap circuit.
3. The electric arc, which is a nonlinear element of electrical circuits, is of great practical importance. In Fig. Figure 9.7 schematically shows an electric arc burning in air at atmospheric pressure between carbon electrodes.
The active part of the cathode (K), emitting electrons ē , has a temperature of ~ 3000°C. Part A of the anode bombarded with electrons ē , has a temperature of ~ 4000°C. Between the active parts K and A there is the arc D itself, the temperature of which is ~ 5000°C. In the arc region, the gas is in an ionized state; the main current carriers are ē .
Currently, the electric arc is used as a light source in spotlights and projection devices. In metallurgy, powerful arcs are used in arc furnaces. Electric arc welding has become widespread.
The electric arc has a nonlinear characteristic, which is shown in Fig. 9.8.
It can be seen that as the current increases, the arc voltage drops.
9.3. Calculation of simple circuits with passive nonlinear
elements
Graphical method calculation.
a) Series connection of nonlinear elements.
Characteristics of nonlinear elements are given in the form of graphs in Fig. 9.10.
In this case, according to Kirchhoff’s laws, we can write , .
Therefore, by adding the ordinates of the characteristics and , we find the characteristic. Having this characteristic, it is not difficult to find the current i,u 1 ,u 2 for any mode .
For example, for voltage u=u*(Fig. 9.10).
This method can be extended to the case of any number of nonlinear and linear elements connected in series.
b) The case of parallel connection of nonlinear elements (Fig. 9.11).
Characteristics of nonlinear elements u 1 =F 1 (i 1), u 2 =F 2 (i 2) are shown in Fig. 9.12.
In this case, in accordance with Kirchhoff's laws, we have i=i 1 +i 2 ,
Therefore, by adding the abscissas of the curves and , we obtain the characteristic.
c) Consider a mixed compound (Fig. 9.13).
The characteristics of nonlinear elements are known (Fig. 9.14). The current-voltage characteristic of linear resistance is written as follows: .
According to Kirchhoff's laws, we have the equations , , .
First we add the ordinates of the curves. We get a curve.
Then adding the abscissas of the curves and , we obtain the dependence . Having the given curves, you can find all the voltages and currents if one of these voltages or one of these currents is given.
G). Calculation of simple nonlinear circuits containing EMF sources (Fig. 9.15).
The characteristic, magnitude and direction of the EMF are specified. e”.
According to Kirchhoff’s second law, taking into account the direction of traversal, we have:
Let e bc>0. Then we have the case shown in Fig. 9.16. The case when e bc<0, соответствует рис. 9.17.
That. EMF can be taken into account by correspondingly shifting the characteristics of a nonlinear element connected in series with an EMF source. Therefore, the calculation of nonlinear circuits containing EMF sources is carried out using the same methods as the calculation of passive nonlinear circuits.
Known; magnitude and direction of EMF e 1 > 0, e 2 > 0 (Fig. 9.19, 9.20).
We set the direction of the currents in all branches. Let's build higher in the specified way the resulting characteristics of all branches.
(Fig. 9.19), (Fig. 9.20).
We add up the abscissas of the curves , , and find (Fig. 9.21).
9.4. Calculation of simple nonlinear DC circuits using the iterative method
The term “iteration” comes from the Latin word and means “repetition.”
To calculate circuits with nonlinear elements, the iterative method of solving nonlinear algebraic equations is often used.
To understand the essence of the method, consider an equivalent circuit in which the EMF source E and resistance r in(Fig. 9.22) represent an arbitrary linear part of the original circuit, i.e. represent some equivalent source.
Let external characteristic equivalent source coincides with straight line 1 (Fig. 9.23), and the characteristic of the nonlinear element is given by curve 2.
If the solution is made geometrically, then the point “ A” the intersection of characteristics determines the mode of the circuit, i.e. voltage and current in this mode.
If this task To solve numerically, for example, using the iterative method, you must proceed as follows:
1. We perform the so-called zero approximation. To do this, set the voltage U 0 equal to, for example, E and using curve 2 we find the current I 0 .
