- 28.07.2018
The figure shows a diagram of a simple and very easy-to-use thermostat, DS18B20 is used as a sensor, and the controller is controlled using a ky-040 encoder. The integral temperature sensor DS18B20 has a temperature measurement range from -55 to + 125 ° C, the temperature readings are displayed on the first line of the indicator 1602 HD44780, the controller readings are displayed on the second line of the indicator ...
- 29.09.2014
The field-effect transistor receiver receives a radio signal in the MW and LW ranges. Receiver sensitivity 1…3mV/m SW and 2…5 mV/m LW. Pout=250mW, Icon=10mA(65mA max). The radio receiver can operate with a voltage drop of up to 4 V. The receiver consists of a 3-stage HF (T1-T3), detector (D1 D2) and VLF (T4 T7). Increased sensitivity and output power achieved…
- 20.09.2014
Twice the author had to deal with the simplest, but very unpleasant malfunction of household microwave ovens: a breakdown of a protective mica plate covering the exit of the magnetron waveguide into the oven's frying chamber. Probably, the mica plate contained metal inclusions that evaporated during the operation of the furnace magnetron, which led to a breakdown of the mica. The place of breakdown was charred, and the operation of the furnace became ...
- 13.10.2014
Main technical characteristics: Rated output power at load resistance: 8Ω - 48W 4Ω - 60W Frequency response with frequency response unevenness of not more than 0.5 dB and output power of 2 W - 10 ... 200000 Hz Nonlinear distortion factor at rated power in the range of 20 ... 20000 Hz - 0.05% Rated input voltage - 0.8V Output ...
In AC circuits, the following types of resistance are distinguished.
Active. The resistance of the resistor is called active. Symbol
The unit of resistance is Ohm. The resistance of a resistor is independent of frequency.
jet. In the reactive section, three types of resistances are distinguished: inductive xL and capacitive xc and actually reactive. For the inductive reactance, the formula X L = ωL was obtained above. The unit of measure for inductive reactance is also Ohm. The value of xL depends linearly on the frequency.
For the capacitive reactance, the formula X C = 1 / ωC was obtained above. The unit of capacitance is Ohm. The value of xc depends on the frequency according to an inversely proportional law. The reactance of the circuit is simply called the value X \u003d X L - X C.
Impedance. The total resistance of the circuit is called the value
.
From this relation it follows that the resistances Z, R and X form a triangle: Z is the hypotenuse, R and X are the legs. For convenience, this triangle considers the angle φ, which is determined by the equation
φ = arctg((X L - X C) / R),
and is called the phase angle. With this in mind, additional connections can be given
The introduction of a complex representation of currents and voltages requires determining the resistance of the elements of electrical circuits in a complex form - Z.
It is well known that the resistance of a resistor is defined as the ratio of the voltage across the resistor to the current flowing through it. If voltage and current are in complex form, then
But in the previous lecture it was established that . That's why
Thus we see that the complex resistance of a resistor is expressed only by a real number. It does not introduce phase distortions between currents and voltage. To emphasize this fact, such resistance is often called active.
The complex capacitance resistance is determined by the ratio
. (3.2)
We see that the complex resistance of the capacitance to alternating current is expressed by an imaginary number. The imaginary unit -j physically determines the phase shift between current and voltage by 90o. This agrees well with its maximum value
Therefore, on the capacitance, the voltage lags behind the current by 90 °. This means that first the current flowing through the capacitor increases, then, with some delay, the charge and voltage increase.
The complex resistance of the inductance is determined by the ratio
. (3.4)
coefficient w L defines the resistance value in ohms. It is proportional to the frequency, is called inductive reactance and is denoted by XL, i.e.
To emphasize the fact that the resistances of capacitance and inductance are expressed in imaginary numbers, they are called reactances, and the capacitor and inductance are called reactive circuit elements.
Let us now determine the complex resistance of an electrical circuit containing active and reactive elements, for example, R, L and C elements connected in series (Fig. 3.1). Such a circuit is a closed
contour, so Kirchhoff's second law is valid for it
In the last expression, we will replace the symbols of instantaneous voltages and emfs with their complex images according to the rules defined in lecture 1.2. This technique is called the symbolic method. Since the current flowing through all elements of the series circuit is the same, then (3.6) comes to the form
Let's transform this expression to the form
.
By definition, the expression on the right side of the last equality is nothing more than the complex resistance of the circuit in Fig. 3.1, i.e.
(3.7)
where R is the real part or active resistance of the circuit.
is the imaginary part or reactance of the circuit.
Expression (3.7) represents the complex resistance in algebraic form. The ratios between the components of the complex resistance are in full accordance with the ratios for the complex representation of the current. But for greater clarity, the concept of a triangle of resistance is introduced (Fig. 3.2).
In a triangle, the hypotenuse is determined by the modulus of the complex resistance Z, and
(3.8)
Opposite leg - reactance X, and
The angle determines the phase shift between current and voltage, which is introduced by the complex resistance of the circuit, and
Taking into account the expressions (3.8) ¸ (3.11) it is easy to switch from the algebraic to the trigonometric form of the complex resistance
a apply the Euler formula to obtain the exponential form
Now you can write Ohm's law for a circuit section without an EMF source in a complex image
(3.14)
Expression (3.14) shows that in AC circuits, the current modulus is determined by the ratio of the voltage modulus (its amplitude value) to the complex resistance modulus, and the current phase is determined by the phase difference of the voltage and complex resistance. From this follows another expression useful for practice
. (3.15)
complex conductivity
In DC circuits, the conductivity of a resistor is determined by the ratio of current to voltage:
This value is inversely proportional to the resistance.
In AC circuits, the concept of complex conductivity should be used, which is denoted by Y and, in the general case, contains the real G and the imaginary B part:
Complex conductance of the resistor
(3.17)
Complex conductance of a capacitor
. (3.18)
Complex conductance of inductance
. (3.19)
In conclusion, note that it is convenient to use complex resistance to analyze sections of an electrical circuit with a series connection of elements, and complex conductivity - for sections with parallel connection of elements.
Inductors in DC Circuits
The primary purpose of an inductor in a DC circuit is to provide resistance in the form of resistance. Inductors are usually coils of wire that create resistance. Although the resistive resistance of an inductor is usually low, the coil creates a reaction. In addition, power is dissipated by the resistance of the inductor.
Inductance effects appear when the current in a DC circuit changes. Although the current is usually a fixed amount in a running DC circuit, remember also that you still need to turn the circuit on and off. scheme. When current is initially introduced into or removed from the circuit, there is a significant change in current. This change in current causes the inductor to oppose this change. The result is an induced (induced) voltage which, as in an AC circuit, opposes the change in current.
The most significant effect is achieved when the current through the inductor is suddenly suppressed. The magnetic field around the inductor disappears, inducing a very high voltage in the coil. This voltage can even damage components in some cases. Other applications, on the other hand, take advantage of this effect to generate very high voltages to power certain special components or circuits. Examples are line-scan transformers in television receivers and ignition coils in car ignition systems.
An inductor has the ability to create a magnetic field. This property is characterized by the coil parameter - inductance (L), which depends on the number of turns, the core, the geometric dimensions of the coil.
L = ψ/I; where ψ = W F- coil flux linkage;
W- number of turns of the coil; F- magnetic flux; I is the current flowing through the coil.
In addition to inductance, a real coil has active resistance:
ρ - specific resistance of the conductor of the coil; l- conductor length;
S is the cross-sectional area of the coil conductor.
Rice. 4-1
For the convenience of analyzing the operation of a coil in an alternating current circuit, we will conventionally assume R k \u003d 0. Alternating current i = I m sin(ωt), flowing through the coil, creates a variable magnetic flux F, which, crossing the turns of the coil, induces an EMF of self-induction in them. According to Lenz's rule, the self-induction EMF, the self-induction current prevent the flow of current in the circuit, fig. 4-1.
Impedance modulus.
Rice. 4-4
Multiplying each side of the voltage triangle by the current, we get a similar power triangle (Fig. 4-4c).
Q L- the reactive power of the coil is used to create a magnetic field. Reactive power measurement unit: var - volt - reactive ampere;
R- the active power of the circuit is converted into heat. Unit W;
S is the total power of the circuit, the unit of measurement is VA - volt-ampere.
S= P + jQ- complex value of total power.
Full power module.
Power factor showing what part of the electrical power supplied to the circuit S, turns into useful power R.
To calculate voltages and currents through the elements of an electrical circuit, you need to know their total resistance. Energy sources are divided into two types:
- direct current(batteries, rectifiers, accumulators), the electromotive force (EMF) of which does not change with time;
- alternating current(domestic and industrial networks), the EMF of which changes according to a sinusoidal law with a certain frequency.
Active and reactances
Load resistance is either active or reactive. Active resistance(R) does not depend on the mains frequency. This means that the current in it changes synchronously with the voltage. This is the resistance that we measure with a multimeter or tester.
Reactance is divided into two types:
— inductive(transformers, chokes);
— capacitive(capacitors).
A distinctive feature of a reactive load is the presence of an advance or lag of the current from the voltage. In a capacitive load, the current leads the voltage, and in an inductive load it lags behind it. Physically, it looks like this: if a discharged capacitor is connected to a DC source, then at the moment of switching on, the current through it is maximum, and the voltage is minimum. Over time, the current decreases and the voltage increases until the capacitor is charged. If you connect a capacitor to an alternating current source, then it will constantly recharge with the mains frequency, and the current will increase before the voltage.
By connecting an inductance to a DC source, we get the opposite result: the current through it will increase for some time after the voltage is connected.
The amount of reactance depends on the frequency. Capacitance:
Corner frequency related to mains frequency f formula:
As can be seen from the formula, as the frequency increases, the capacitance decreases.
AC circuit impedance
In the AC network, there is no load only active or only reactive. The heating element, in addition to the active one, contains an inductive resistance; in an electric motor, the inductive resistance prevails over the active one.
The value of impedance, taking into account all the active and reactive components of the electrical circuit, is calculated by the formula:
Calculation of the equivalent resistance of circuit elements
Several resistors can be connected to one power supply. To calculate the source load current, the equivalent load resistance is calculated. Depending on the connection of the elements to each other, two methods are used.
Series connection of resistances.
In this case, their values add up:
The more resistances connected in series, the greater the equivalent resistance of that circuit. A household example: if the contact in the plug deteriorates, this is tantamount to connecting additional resistance in series with the load. The equivalent resistance of the load will increase, and the current through it will decrease.
Parallel connection of resistors.
The calculation formula looks much more complicated:
The case of applying this formula for two resistors connected in parallel:
Connection Case n same resistance R:
The more resistances connected in parallel, the lower the total resistance of the circuit. We observe this in everyday life: the more consumers are connected to the network, the less equivalent resistance and the greater the load current.
In this way, calculation of the impedance of an electrical circuit happens in stages:
- An equivalent circuit is drawn containing active and reactive resistances.
- Equivalent resistances are calculated separately for the active, inductive and capacitive components of the load.
- The impedance of the electrical circuit is calculated
- The currents and voltages in the power supply circuit are calculated.
Explains that if a potential difference is applied at the ends of a section of the circuit, then under its action an electric current will flow, the strength of which depends on the resistance of the medium.
AC voltage sources create a current in the circuit connected to them, which can follow the shape of the sinusoid of the source or be shifted in angle from it forward or backward.
If the electrical circuit does not change the direction of current flow and its phase vector coincides completely with the applied voltage, then such a section has a pure active resistance. When there is a difference in the rotation of the vectors, then they talk about the reactive nature of the resistance.
Various electrical elements have an unequal ability to deflect the direction of the current flowing through them and change its magnitude.
Coil reactance
Take a source of stabilized alternating voltage and a piece of long insulated wire. First, we connect the generator to the entire straightened wire, and then to its own, but wound in rings around, which is used to improve the passage of magnetic fluxes.
Accurately measuring the current in both cases, it can be seen that in the second experiment a significant decrease in its value and a phase lag by a certain angle will be noticed.
This happens due to the emergence of opposing forces of induction, which manifest themselves under the action of Lenz's law.
In the figure, the passage of the primary current is shown by red arrows, and the magnetic field it creates is shown by blue. The direction of its movement is determined by the rule of the right hand. It also crosses all adjacent turns inside the winding and induces in them the current shown by the green arrows, which weakens the magnitude of the applied primary current, while simultaneously shifting its direction with respect to the applied EMF.
The greater the number of turns wound on the coil, the stronger the inductive reactance X L is created, which reduces the primary current.
Its value depends on the frequency f, inductance L, is calculated by the formula:
X L = 2 π fL = ω L
Due to overcoming the forces of inductance, the current on the coil lags behind the voltage by 90 degrees.
Transformer reactance
This device has two or more windings on a common magnetic circuit. One of them receives electricity from an external source, and it is transferred to the other according to the principle of transformation.
The primary current passing through the power coil induces a magnetic flux in the magnetic circuit and around it, which crosses the turns of the secondary winding and forms a secondary current in it.
Since it is impossible to create ideally, part of the magnetic flux will be dissipated into the environment and create losses. They are called leakage flux and affect the amount of leakage reactance.
The active component of the resistance of each winding is added to them. The resulting total value is called the electrical impedance of the transformer or its Z, which creates voltage drops on all windings.
To mathematically express the relationships within the transformer, the active resistance of the windings (usually made of copper) is denoted by the indices "R1" and "R2", and the inductive - "X1" and "X2".
The impedance in each winding is:
Z1=R1+jX1;
Z2=R1+jX2.
In this expression, the index "j" denotes the imaginary unit located on the vertical axis of the complex plane.
The most critical mode in relation to inductive resistance and the occurrence of a reactive power component is created when transformers are connected in parallel to work.
Capacitor reactance
Structurally, it consists of two or more conductive plates separated by a layer of material with dielectric properties. Due to this separation, direct current cannot pass through the capacitor, while alternating current can, but with a deviation from the original value.
Its change is explained by the principle of operation of reactive - capacitive resistance.
Under the action of the applied alternating voltage, which changes in a sinusoidal form, a surge occurs on the plates, the accumulation of electric energy charges of opposite signs. Their total number is limited by the dimensions of the device and is characterized by capacity. The larger it is, the longer the charge takes.
During the next half-cycle of oscillation, the polarity of the voltage across the capacitor plates reverses. Under its influence, there is a change of potentials, recharging of the formed charges of the plates. In this way, the flow of the primary current is created and opposed to its passage when it decreases in magnitude and shifts in angle.
Electricians have a joke on this issue. The direct current on the graph is represented by a straight line, and when it goes through the wire, the electric charge, having reached the capacitor plate, rests against the dielectric, getting into a dead end. This barrier does not allow him to pass.
The sinusoidal harmonic, on the other hand, goes over obstacles and the charge, freely rolling over the painted plates, loses a small part of the energy that has caught on the plates.
This joke has a hidden meaning: when a constant or rectified pulsating voltage is applied to the plates, a strictly constant potential difference is created between the plates due to the accumulation of electric charges, which smooths out all the jumps in the supply circuit. This property of the increased capacity capacitor is used in DC voltage stabilizers.
In general, the capacitance Xc, or opposition to the passage of an alternating current through it, depends on the design of the capacitor, which determines the capacitance "C", and is expressed by the formula:
Xc = 1/2π fC = 1/ω C
By recharging the plates, the current through the capacitor leads the voltage by 90 degrees.
Power line reactance
Any transmission line is created to transmit electrical energy. It is customary to represent it as sections with equivalent circuits that have distributed parameters of active r, reactive (inductive) x resistance and conductivity g, referred to a unit of length, usually one kilometer.
If we neglect the influence of capacitance and conductivity, then we can use a simplified line equivalent circuit with lumped parameters.
Overhead power line
The transmission of electricity through uninsulated wires located in the open air requires a significant distance between them and from the ground.
In this case, the inductive resistance of one kilometer of the wire of a three-phase line can be represented by the expression X0. It depends on:
average distance between wire axes asr;
outer diameter of the phase conductors d;
relative magnetic permeability of the material µ;
external inductive resistance line X0';
internal inductive resistance of the X0 '' line.
For reference: the inductive resistance of 1 km of an overhead line made of non-ferrous metal is about 0.33 ÷ 0.42 Ohm / km.
Cable power line
A power line using a high-voltage cable is structurally different from an overhead line. It has a significantly reduced distance between the phases of the wires and is determined by the thickness of the internal insulation layer.
Such a three-core cable can be represented as a capacitor with three linings of cores stretched over a long distance. With an increase in its length, the capacitance increases, the capacitive resistance decreases, and the capacitive current that closes along the cable increases.
In cable lines, under the influence of capacitive currents, single-phase earth faults most often occur. To compensate for them in 6÷35 kV networks, arc-suppressing reactors (DGR) are used, which are connected through the grounded neutral of the network. Their parameters are selected by complex methods of theoretical calculations.
Old DGRs did not always work effectively due to poor tuning quality and design imperfections. They were created for the average calculated fault currents, which often differed from the real values.
Now new developments of the DGR are being introduced, which are capable of automatically monitoring emergency situations, quickly measuring their main parameters and adjusting for reliable damping of ground fault currents with an accuracy of 2%. Due to this, the efficiency of the DGR immediately increased by 50%.
The principle of compensation of the reactive component of power by capacitor units
Electric networks transmit high-voltage electricity over long distances. Most of its consumers are electric motors with inductive resistance and resistive elements. The total power sent to consumers consists of the active component P, spent on useful work, and the reactive component Q, which causes heating of the windings of transformers and electric motors.
The reactive component Q, arising on inductive resistances, reduces the quality of electricity. To eliminate its harmful effects in the eighties of the last century, the USSR power system used a compensation scheme by connecting capacitor banks with capacitive resistance, which reduced φ.
They were installed at substations that directly supply problematic consumers. This ensured local regulation of the quality of electricity.
In this way, it is possible to significantly reduce the load on the equipment by reducing the reactive component when transmitting the same active power. This method is considered the most effective method of energy saving not only in industrial enterprises, but also in housing and communal services. Its proper use can significantly improve the reliability of the operation of power systems.
a) Active resistance R, r is an idealized circuit element in which irreversible transformations of electrical energy into thermal energy take place:
BUT.
b) Inductance L is an idealized circuit element, which is characterized by the ability to accumulate the energy of a magnetic field. The inductance is numerically equal to the ratio of the flux linkage to the current by which this flux linkage is due:
,
(3.6)
where 
is the coupling flux of the inductor,
N is the number of turns of the coil,
F- magnetic flux.
.
c) Capacitance C is an idealized element of an electric circuit, which is characterized by the ability to accumulate the energy of an electric field.
,
(3.7)
where 
- the charge on the plates or plates of the capacitor,
is the potential difference between the capacitor plates.
Capacitance C - does not depend on 
, but is determined by the size, shape of the capacitor, as well as the dielectric properties of the medium located between the capacitor plates.
.
RMS AC
Oscillations that occur under the influence of an external periodically changing EMF are called forced electromagnetic oscillations. Steady-state forced electromagnetic oscillations can be considered as the flow of alternating current in a circuit containing a resistor, an inductor and a capacitor.
On fig. 3.5 is a graph of an alternating sinusoidal current.
Rice. 3.5. AC Graph
The effective value of the alternating current is equal to such a value of the direct current, which, in a time equal to the period of the alternating current, releases in the same resistance the same amount of heat as the given current. Determined by formula 3.8.
. (3.8)
Active, reactive and impedance in AC circuits
Current in active resistance
,
(3.9)
where I r , U r- effective values of current and voltage on the active resistance R.
The phase shift between current and voltage across the resistor is zero (see Fig. 3.6).
Rice. 3.6. Vector diagram of current and voltage across a resistor
Current in inductor
,
(3.10)
where I L , U L- effective values of current and voltage on inductive resistance X L .
,
(3.11)
where ω – the cyclic frequency is zero, so at constant current the inductor has no resistance.
Loading...