Let us consider separately the cases of connecting an external alternating current source to a resistor with resistance R, capacitor capacitance C and inductors L. In all three cases, the voltages across the resistor, capacitor and coil are equal to the voltage of the AC source.
1. Resistor in AC circuit
Resistance R is called active because a circuit with such resistance absorbs energy.
Active resistance - a device in which the energy of an electric current is irreversibly converted into other types of energy (internal, mechanical)
Let the voltage in the circuit change according to the law: u = Umcos ωt ,
then the current strength changes according to the law: i = u/R = I R cosωt
u – instantaneous voltage value;
i – instantaneous current value;
I R- amplitude of the current flowing through the resistor.
The relationship between the amplitudes of current and voltage across a resistor is expressed by the relation RI R = U R
The current fluctuations are in phase with the voltage fluctuations. (i.e. the phase shift between the current and voltage across the resistor is zero).
2. Capacitor in AC circuit
When a capacitor is connected to a DC voltage circuit, the current is zero, and when a capacitor is connected to an AC voltage circuit, the current is not zero. Therefore, a capacitor in an AC voltage circuit creates less resistance than in a DC circuit.
I C and voltage
The current is ahead of the voltage in phase by an angle of π/2.
3. Coil in AC circuit
In a coil connected to an alternating voltage circuit, the current strength is less than the current strength in a constant voltage circuit for the same coil. Consequently, the coil in an alternating voltage circuit creates more resistance than in a direct voltage circuit.
Relationship between current amplitudes I L and voltage U L:
ω LI L = U L
The current lags in phase from the voltage by an angle π/2.
Now we can construct a vector diagram for a series RLC circuit in which forced oscillations occur at frequency ω. Since the current flowing through series-connected sections of the circuit is the same, it is convenient to construct a vector diagram relative to the vector representing current oscillations in the circuit. We denote the current amplitude by I 0 . The current phase is assumed to be zero. This is quite acceptable, since it is not the absolute phase values that are of physical interest, but the relative phase shifts.
The vector diagram in the figure is constructed for the case when or In this case, the voltage of the external source is ahead in phase of the current flowing in the circuit by a certain angle φ.
Vector diagram for a serial RLC circuit
From the figure it is clear that
whence follows
From the expression for I 0 it is clear that the current amplitude takes a maximum value under the condition
The phenomenon of increasing the amplitude of current oscillations when the frequency ω of an external source coincides with the natural frequency ω 0 of the electrical circuit is called electrical resonance . At resonance
The phase shift φ between the applied voltage and current in the circuit becomes zero at resonance. Resonance in a series RLC circuit is called voltage resonance. In a similar way, using a vector diagram, you can study the phenomenon of resonance at parallel connection elements R, L And C(so-called current resonance).
At sequential resonance (ω = ω 0) amplitudes U C And U L The voltages on the capacitor and coil increase sharply:
The figure illustrates the phenomenon of resonance in a series electrical circuit. The figure graphically shows the dependence of the amplitude ratio U C voltage on the capacitor to the amplitude 0 of the source voltage from its frequency ω. The curves in the figure are called resonant curves.
At today's meeting we will talk about electricity, which has become an integral part of modern civilization. Electric power has invaded all areas of our lives. And the presence in every home of household appliances that use electricity such a natural and integral part of everyday life that we take it for granted.
So, our readers are offered basic information about electric current.
What is electric current
Electric current means directed movement of charged particles. Substances containing a sufficient number of free charges are called conductors. A collection of all devices connected to each other using wires is called an electrical circuit.
In everyday life we use electricity passing through metal conductors. The charge carriers in them are free electrons.
Usually they rush chaotically between atoms, but the electric field forces them to move in a certain direction.
How does this happen
The flow of electrons in a circuit can be compared to the flow of water falling from high level to low. The role of level in electrical circuits is played by potential.
For current to flow in the circuit, a constant potential difference must be maintained at its ends, i.e. voltage.
It is usually denoted by the letter U and measured in volts (B).
Due to the applied voltage, an electric field is established in the circuit, which gives the electrons directional movement. The higher the voltage, the stronger the electric field, and therefore the intensity of the flow of directionally moving electrons.
The speed of propagation of electric current is equal to the speed of establishment of an electric field in the circuit, i.e. 300,000 km/s, but the speed of electrons barely reaches only a few mm per second.
It is generally accepted that current flows from a point with a higher potential, i.e., from (+) to a point with a lower potential, i.e., to (−). The voltage in the circuit is maintained by a current source, such as a battery. The sign (+) at its end means a lack of electrons, the sign (−) means their excess, since electrons are carriers of a negative charge. As soon as the circuit with the current source becomes closed, electrons rush from the place where there is an excess of them to the positive pole of the current source. Their path runs through wires, consumers, measuring instruments and other circuit elements.
Please note that the direction of the current is opposite to the direction of movement of the electrons.
Simply, the direction of the current, by agreement of scientists, was determined before the nature of the current in metals was established.
Some quantities characterizing electric current
Current strength. The electric charge passing through the cross-section of a conductor in 1 second is called current strength. The letter I is used to designate it and is measured in amperes (A).
Resistance. The next quantity you need to know about is resistance. It arises due to collisions of directionally moving electrons with ions of the crystal lattice. As a result of such collisions, electrons transfer part of their kinetic energy to the ions. As a result, the conductor heats up and the current strength decreases. Resistance is symbolized by the letter R and is measured in ohms (ohms).
The longer the conductor and the smaller its cross-sectional area, the greater the resistance of a metal conductor. With the same length and diameter of the wire, conductors made of silver, copper, gold and aluminum have the least resistance. For obvious reasons, wires made of aluminum and copper are used in practice.
Power. When performing calculations for electrical circuits, it is sometimes necessary to determine the power consumption (P).
To do this, the current flowing through the circuit must be multiplied by the voltage.
The unit of power is the watt (W).
Direct and alternating current
The current provided by various batteries and accumulators is constant. This means that the current strength in such a circuit can only be changed in magnitude by changing different ways its resistance, and its direction remains unchanged.
But Most electrical appliances consume alternating current, that is, a current whose magnitude and direction continuously changes according to a certain law.
It is generated in power plants and then travels through high-voltage transmission lines into our homes and businesses.
In most countries, the frequency of changes in the direction of current is 50 Hz, i.e. it occurs 50 times per second. In this case, each time the current strength gradually increases, reaches a maximum, then decreases to 0. Then this process is repeated, but with the opposite direction of the current.
In the USA, all devices operate at a frequency of 60 Hz. An interesting situation has developed in Japan. There, one third of the country uses alternating current with a frequency of 60 Hz, and the rest - 50 Hz.
Caution - electricity
Electric shock can occur when using electrical appliances and from lightning strikes, since The human body is a good conductor of current. Electrical injuries are often caused by stepping on a wire lying on the ground or pushing away loose electrical wires with your hands.
Voltage above 36 V is considered dangerous to humans. If a current of only 0.05 A passes through a person’s body, it can cause involuntary muscle contraction, which will not allow the person to independently tear himself away from the source of the lesion. A current of 0.1 A is lethal.
Alternating current is even more dangerous because it has a stronger effect on humans. This friend and helper of ours in some cases turns into a merciless enemy, causing breathing problems and heart function, even to the point of complete cardiac arrest. It leaves terrible marks on the body in the form of severe burns.
How to help the victim? First of all, turn off the source of damage. And then take care of providing first aid.
Our acquaintance with electricity is coming to an end. Let’s add just a few words about sea creatures that have “electric weapons.” These are some types of fish, conger eel and stingray. The most dangerous of them is the conger eel.
You should not swim to it at a distance of less than 3 meters. His blow is not fatal, but consciousness can be lost.
If this message was useful to you, I would be glad to see you
Lecture: “Alternating current. Impedance" 1. Alternating current 2. Impedance of the alternating current circuit. Voltage resonance. 3. Impedance of body tissues. Physical foundations of rheography. 4. Regions of α-, β- and γ-impedance dispersion (independently, Remizov A. N., Maksina A. G., Potapenko A. Ya. Medical and biological physics. M., 2004, § 14. 4)
1. Alternating current In the broad sense of the word, alternating current is any current that changes over time. However, more often the term “alternating current” is applied to quasi-stationary currents that depend on time according to a harmonic law. Quasi-stationary current is a current for which the time to establish the same value throughout the entire circuit is significantly less than the oscillation period. We will assume that for quasi-stationary currents, as well as for constant ones, the current strength is simultaneously the same in any section of an unbranched conductor. Ohm's law is valid for them, but the circuit resistance depends on the frequency of the current change. We will neglect energy losses due to electromagnetic radiation of these currents. Alternating current can be considered as forced electromagnetic oscillations. Let's imagine three different circuits (Fig. 1, a - 3, a), to each of which an alternating voltage is applied
U = Um sinωt, (1) where Um is the amplitude value of the voltage, ω is the circular oscillation frequency. It can be shown that the current strength in a circuit with a resistor (Fig. 1, a) will change in phase with the applied voltage: I=Im sin ωt, (2) Fig. 1 The current strength in the circuit with the inductor (Fig. 2, a) will lag in phase with the applied voltage by π/2: I=Im sin (ωt – π/2), (3)
Rice. 2 and the current in the circuit with a capacitor (Fig. 3, a) will be ahead of the voltage in phase by π/2: I=Im sin (ωt + π/2), (4) Fig. 3
Vector diagrams corresponding to these examples are shown in Fig. 1, b – 3, b. For a circuit with a resistor we have ohmic resistance (5), for a circuit with an inductor we have inductive resistance (6) and for a circuit with a capacitor we have capacitive resistance (7)
2. AC circuit impedance. Voltage resonance. Let's consider a circuit in which a resistor, an inductor and a capacitor are connected in series (Fig. 4) Fig. 4 In the general case, the current in the circuit and the voltage do not change in the same phase, therefore I = Im sin (ωt-φ), (8) where φ is the phase difference between the voltage and current. The sum of the voltages in individual sections of the circuit is equal to the external voltage: U= IR+ IXL+ IXC = UR + UL + UC, (9)
Rice. 5 In Fig. 5, the current amplitude vector Im is directed along the current axis. Since the current amplitude is the same throughout the entire circuit, the voltage amplitudes in sections of the circuit are plotted relative to this vector: vector URm - in phase with the current strength; vector ULm – with current in phase leading by, vector UCm – with current in phase lagging by by. Using the Pythagorean theorem, we have U 2 m=U 2 Rm+(ULm-UCm)2 , (10)
Substituting into (10) the expressions for these amplitudes from (5)-(7) and taking into account Ohm’s law, we find (11) where Z is the total resistance of the alternating current circuit, called impedance. From (11) we obtain (12) The ohmic resistance R of the circuit is also called active; it determines the release of heat in the circuit in accordance with the Joule-Lenz law. The difference between inductive and capacitive reactance (XL - Xc) is called reactance. It does not cause heating of the electrical circuit elements.
From Fig. 5 we express the value of tanφ: (13) If Xc =XL, then tanϕ=0 and ϕ=0. This means that the current and the applied voltage change in one phase as if there was only ohmic resistance in the circuit; The voltages across the inductance and capacitance are equal in amplitude, but opposite in phase. This case of forced electrical oscillations is called voltage resonance and the resonant frequency is: (14) Under this condition, the total resistance Z of the circuit has the smallest value equal to R, and the current reaches its highest value. The vector diagram for voltage resonance in the circuit is shown in Fig. 6. a.
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3. Impedance of body tissues. Physical basis of rheography Body tissues conduct not only direct, but also alternating current. The body does not have systems similar to inductors, so its inductance is close to zero. Biological membranes (and therefore the entire organism) have capacitive properties. Therefore, the impedance of body tissues is determined only by ohmic and capacitive resistance. The ohmic and capacitive properties of biological tissues are modeled using equivalent electrical circuits. Let's look at some of them (Fig. 7).
Rice. 7 For the circuit shown in Fig. 7. a, the frequency dependence of the impedance can be obtained from (12) at L = 0: (15)
In accordance with formula (15), the impedance decreases with increasing frequency, but there is a contradiction with experiment: when ω→ 0 Z→∞. The latter means an infinitely large resistance at constant current, which contradicts experience. For the circuit shown in Fig. 7, b, at ω→∞ Z→ 0, which does not correspond to experiment. The most successful diagram is Fig. 7, c. There are no contradictions with experience characteristic of the two previous schemes. It is this combination of resistors and capacitor that can be taken as the equivalent electrical circuit of body tissues. The frequency dependence of the impedance of the equivalent electrical circuit corresponds to the general course of the experimental dependence of the impedance on frequency.
The frequency dependence of impedance allows you to assess the viability of body tissues, which is important to know for transplantation of tissues and organs. A difference in the frequency dependences of the impedance is obtained in the cases of healthy and diseased tissue. The impedance of tissues and organs also depends on their physiological state. Thus, when blood vessels are filled, the impedance changes depending on the state of cardiovascular activity. A diagnostic method based on recording changes in tissue impedance during cardiac activity is called rheography (impedance plethysmography). Using this method, rheograms of the brain (rheoencephalogram), heart (rheocardiogram), great vessels, lungs, liver and limbs are obtained. Measurements are usually carried out at a frequency of 30 kHz.
1. Alternating current and alternating voltage. The resistance of a circuit section when alternating current flows.
2. Flow of alternating current through a resistor. Resistor resistance, effective current and voltage values.
3. Capacitor in an alternating current circuit, capacitance.
4. Flow of alternating current through an ideal inductor, inductive reactance.
5. Flow of alternating current through an RLC circuit, impedance. Voltage resonance. RCR chain.
6. Impedance of body tissues. Equivalent electrical diagram fabrics. Rheography.
7. Basic concepts and formulas.
8. Tasks.
15.1. Alternating current and alternating voltage. Resistance of a circuit section when alternating current flows
In a broad sense "variables" Any current that changes over time in magnitude and direction is called. In engineering, a variable is a current that changes over time. according to harmonic law. This is the current we will consider:
Alternating current is forced electromagnetic oscillations that occur when any device is connected to an alternating voltage network:
Typically, the beginning of the time count is chosen so that for the voltage of the electrical network the initial phase is equal to zero. Therefore, in formula (15.2) there is no term φ 0.
In a chain permanent current, the ratio of voltage to current is called the resistance of the circuit section (R = U/I). Similarly, the concept of resistance is introduced for the circuit variable current Its value is indicated by the letter X.
Resistance section of a circuit in an AC network is equal to the ratio of the amplitude value of the alternating voltage in this section to the amplitude value of the current in it:
The maximum value of alternating current (I max) and its initial phase (φ 0) depend on the properties of the elements included in the electrical circuit of the device. Let us consider the flow of alternating current through such elements.
15.2. Flow of alternating current through a resistor. Resistor resistance, effective current and voltage values
Resistor called a conductor that does not have inductance and capacitance.
For all frequencies of alternating current, which is used in technology, the resistance of the resistor (X R) remains constant and coincides with its resistance in the direct current circuit:
A resistor is the only element for which the current and voltage are in phase. In order to show the phase difference between current and voltage in the general case, a vector diagram is used, in which the vector representing the amplitude voltage (U max) is located at an angle to current axes. The angle that the vector U max forms with the current axis shows how much the voltage phase advances the current phase.
The circuit with resistor R and the corresponding vector diagram are shown in Fig. 15.1.
Rice. 15.1. AC circuit with resistor and its vector diagram
Since current and voltage change in the same phase, vectors U max and I max are plotted along one straight line in one direction.
In principle, any alternating current is accompanied by electromagnetic radiation. However, for alternating current frequencies used in industry, the intensity of such radiation is negligible, and energy losses due to electromagnetic radiation are neglected. Therefore, the work done by an alternating current flowing through a resistor is completely turns into his internal energy. In this regard, the resistance of the resistor is called active.
Calculations show that the average power released in a resistor during the flow of alternating (harmonic) current is calculated using the formulas
The values of alternating current and voltage determined by formula (15.7) are called valid. There is an agreement
that by default the actual values are indicated for the AC circuit. For example, the household AC voltage is 220 V. The specified value of 220 V is current voltage value.
15.3. Capacitor in AC circuit,
capacitance
Let's connect a capacitor with capacitance C to the alternating voltage circuit (15.2). Along with the change in voltage, the charge of the capacitor will also change, and a current will arise in the supply wires. The charge of the capacitor is related to the voltage in the circuit by the relation (see formula 10.16)
The resistance of a capacitor in an alternating current circuit is called capacitive reactance. We find its value using formulas (15.3, 15.9):
The circuit with a capacitor and its corresponding vector diagram are shown in Fig. 15.2.
Rice. 15.2. AC circuit with capacitor and its phasor diagram
Because the voltage lagging behind in phase with the current by π/2, the vector U max is rotated relative to the current axis clockwise arrow (in mathematics this direction is considered negative).
15.4. Flow of alternating current through an ideal inductor, inductive reactance
Let us include in the alternating voltage circuit (15.2) a coil with inductance L, the active resistance of which can be neglected. This coil is called perfect. Due to self-induction, an emf will arise in it, preventing a change in the current in the circuit.
Since we neglect the active resistance of the coil, the emf. and voltage are the same: ε = U. Using formula (10.15) for the emf. self-induction, we obtain a differential equation for the current
The circuit with coil L and the corresponding vector diagram are shown in Fig. 15.3.
Rice. 15.3. AC circuit with coil and its vector diagram
Because the voltage ahead in phase the current is by π/2, then the vector U max is rotated relative to the current axis counter-clockwise arrows (in mathematics this direction is considered positive).
When alternating current flows through a capacitor and an ideal inductor no energy loss occurs. These elements take energy from the network for half a period and convert it into electric and magnetic field energy, respectively. During the second half of the period, the field energy returns to the network, maintaining the current. Due to the absence of energy losses, capacitive and inductive reactances are called reactive.
15.5. Flow of alternating current through an RLC circuit, impedance. Voltage resonance
Consider a circuit consisting of a resistor R, an inductor L, and a capacitor C connected in series (Fig. 15.4). If an alternating voltage is applied to it (15.2), then the current in the circuit will be out of phase with the voltage by a certain angle φ:
Such a circuit has both active and reactive resistance. Therefore, its resistance is called impedance and is denoted Z.
Impedance equal to the ratio of the amplitude value of the alternating voltage at the ends of the circuit to the amplitude value of the current in it:
Z = U max /I max .
Rice. 15.4. RLC circuit in an alternating current network and its corresponding vector diagram
The RLC circuit and its corresponding vector diagram are shown in Fig. 15.4.
Elements of the RLC chain are connected sequentially. Therefore, the same current flows through them, and the applied voltage U(t) is the sum of the voltages in individual sections of the circuit:
Voltage resonance
If the values of L, C and ω are selected in such a way that X c = X l, then the impedance Z (formula 15.16) has the minimum possible value equal to R (Z = R). In this case, the current amplitude is maximum, and the applied voltage and current change in the same phase (φ = 0). Given
the phenomenon is called stress resonance. Substituting expressions (15.11), (15.14) into the resonance condition (X C = X L), we obtain a formula for calculating the resonant frequency:
RCR chain
Let's consider the flow of current through a parallel RCR chain, which models the conductive properties of biological tissue (Fig. 15.5). If it is connected to an alternating voltage network (15.2), then currents will flow through the lower and upper sections:
Its amplitude vector I equal to the sum of the amplitudes I 1 And I 2, and the advance angle φ is shown in Fig. 15.5, b.
Here is a formula for finding the impedance of an RCR circuit without derivation:
Rice. 15.5. RCR chain and its vector diagram
15.6. Impedance of body tissues. Equivalent electrical circuit of tissues. Impedance dispersion. Rheography
Impedance of body tissues
The electrical properties of body tissues are different. Organic substances (proteins, fats, carbohydrates) are dielectrics. Tissue fluids contain electrolytes.
Tissues are made up of cells, an important part of which are membranes. The phospholipid bilayer likens the membrane to a capacitor.
There are no systems in the body that would be similar to inductance coils, so its inductance is close to zero.
Thus, tissue impedance is determined only by active and capacitive resistance. The presence of capacitive elements in biological systems is confirmed by the fact that the current strength ahead in phase the applied voltage. The value of the advance angle for different biological objects at a frequency of 1 kHz is given in the table.
Equivalent electrical circuit of tissues
In general, organic tissue can be considered as cells located in a conducting medium (R 1), the role of which is played, for example, by intercellular fluid (Fig. 15.6). Cell membranes have capacitive properties, and electrolytes inside the cell have active resistance (R 2).
This idea corresponds to the electrical circuit discussed in section 15.5 (see Fig. 15.5). Figure 15.7 shows the dependence of the impedance on the angular frequency of the current, which is obtained from formula (15.19) after substituting the expression for
Rice. 15.6. Electrical properties of biological tissues
Rice. 15.7. Dependence of impedance on frequency for an RCR chain
Impedance dispersion
Curve in Fig. 15.7 qualitatively correctly describes the change in impedance of biological tissue: a smooth decrease in impedance with increasing frequency. However, for real biological tissues this dependence is more complex. Figure 15.8 shows a graph of the frequency dependence of the impedance of muscle tissue, obtained experimentally (the scale on the vertical axis is logarithmic).
The graph clearly shows three frequency intervals in which the Z value changes more slowly with frequency compared to the overall course of the curve. They are named areas α-, β- and γ-dispersion, respectively. They correspond to three frequency ranges: low frequencies ν< 10 кГц, радиочастоты ν = 0,1-10 МГц, микроволновые частоты ν >0.1 GHz.
The presence of regions of α-, β- and γ-dispersion is associated with the frequency dispersion of the dielectric constant (ε = f(v)), on which the capacitance value depends (see formula 10.20). Figure 15.9 shows the structural elements that make the main contribution to tissue polarization at various frequencies:
- α-dispersion is caused by the polarization of whole cells (1, 2) as a result of ion diffusion, which requires a relatively long time, so this mechanism manifests itself under the action of a low-frequency electric field (0.1-10 kHz). In this region, the capacitance of the membranes is high and the currents flowing through the electrolyte solutions surrounding the membrane fragments predominate.
Rice. 15.8. Frequency dependence of biological tissue impedance
Rice. 15.9. Structural elements that make the main contribution to tissue polarization
Cell polarization is the slowest process among all polarization mechanisms. As the frequency increases, cell polarization almost completely stops.
- β-variance is caused by the structural polarization of cell membranes (3), in which protein macromolecules (4) participate, and at its upper boundary - globular water-soluble proteins (5), phospholipids (6, 7) and the smallest subcellular structures (8). In this case, significantly lower values of dielectric constant are obtained than when polarizing whole cells. This polarization mechanism dominates at frequencies of 1-10 MHz. With a further increase in frequency, this mechanism also stops working.
- γ-dispersion is caused by the processes of orientational polarization of molecules (9, 10) of free and bound water, as well as low-molecular substances such as sugars and amino acids. In this case, the dielectric constant decreases even more. This polarization mechanism dominates at frequencies above 1 GHz.
IN frequency ranges, corresponding to the main areas of dispersion, the greatest energy losses of alternating electric current (field) occur. The release of energy occurs at the structural level that is responsible for this area dispersive
these. This is what the action is based on. various methods physiotherapy using alternating currents and fields.
Tissue impedance depends not only on frequency, but also on the condition of the tissue. The frequency dependence of impedance allows us to assess the viability of body tissues. This is used in tissue and organ transplantation. For example, determining the viability of a graft is one of the primary tasks of ophthalmic surgery. Such an assessment is also needed when determining treatment tactics for corneal burns, during keratoplasty and keratoprosthetics in eyes with a cataract (clouding of the cornea of the eye), when monitoring the course of keratitis (inflammation of the cornea), to determine the suitability of conservative donor material.
Rheography
The impedance of tissues and organs depends on their physiological state and on the degree of filling of the blood vessels passing through these tissues. When the tissue is filled with blood during systole, the total resistance of the tissue decreases, and during diastole it increases. Impedance changes in time with the work of the heart. This is used for diagnostic purposes.
Rheography - a diagnostic method based on recording changes in tissue impedance during cardiac activity.
These changes are presented in the form of a rheogram. An example of a rheogram of the lower leg of a healthy person is shown in Fig. 15.10.
Rice. 15.10. Rheogram of the lower leg of a healthy person
When the vessels are filled with blood, the electrical conductivity of the tissues changes, and along with it the impedance changes.
By the rate of change in impedance, one can judge the speed of blood inflow during systole and blood outflow during diastole.
Using this method, rheograms of the brain (rheoencephalogram), heart (rheocardiogram), great vessels, lungs, liver, and limbs are obtained. The study of rheograms is used in the diagnosis of diseases of peripheral blood vessels, accompanied by changes in their elasticity, narrowing of arteries, etc.
15.7. Basic concepts and formulas
End of the table
15.8. Tasks
1. The voltage and current in the circuit vary according to the law U = 60sin(314t + 0.25) mV, i = 15sin(314t) mA. Determine the circuit impedance Z and the phase angle between the current and voltage.
2. Is it permissible to include a capacitor in a 220 V AC circuit whose breakdown voltage is 250 V?
5.
The AC frequency is 50 Hz. How many times per second is the voltage zero?
Answer: 100 times.
6. Find the total resistance to alternating current if the following are connected in series:
a) a resistor with resistance R 1 = 3 Ohms and a coil with inductive reactance X L = 4 Ohms;
b) a resistor with a resistance of R 2 = 6 Ohms and a capacitor with a capacitance of X C = 8 Ohms;
c) a resistor with resistance R 3 = 12 Ohms, a capacitor with capacitive resistance X C = 8 Ohms and a coil with inductive reactance X L = 24 Ohms.
Answer: a) 5 Ohm; b) 10 Ohm; c) 20 Ohm.
7. How long will a neon lamp burn if it is connected to an alternating current network with an operating voltage of 120 V and a frequency of 50 Hz for 1 minute? The light comes on and goes off at a voltage of 84.5 V.
The graph of U(t) is shown in Fig. 15.11.
Rice. 15.11.
The graph shows the lamp ignition voltage U з and the two times corresponding to it: t 1 - ignition time
lamps, when instantaneous voltage values become greater than U z; t 2 is the time when the light goes out, since the instantaneous voltage values become less than the voltage U. It is obvious that the duration of one flash
During one voltage fluctuation, the light bulb lights up 2 times, since the operation of a neon lamp does not depend on the polarity of the applied voltage (see Fig. 15.11). Therefore, the number of voltage oscillations during time t 0 is equal to (t 0 - ν), and the number of flashes during this time h = 2t 0? v.
Then the time during which the lamp glows is
8.
A neon lamp is connected to an alternating current network with an effective value of 71 V and a period of 0.02 s. The lamp ignition voltage, equal to 86.7 V, is considered equal to the extinguishing voltage. Find: a) the value of the period of time during which the lamp flash lasts; b) flash frequency.
Answer: a) 3.3 ms; b) 100 Hz.
9. The current voltage in the electrical network is 220 V. What voltage should the wire insulation be designed for?
Solution
Lecture 13-14.
1 . The principle of obtaining variable EMF
Alternating current has a number of advantages over direct current: an alternating current generator is much simpler and cheaper than a direct current generator; alternating current can be transformed; alternating current is easily converted into direct current; AC motors are much simpler and cheaper than DC motors.
In principle, alternating current can be called any current that changes its magnitude over time, but in engineering, alternating current is such a current that periodically changes both magnitude and direction. Moreover, the average value of the strength of such a current over the period T is equal to zero. The alternating current is called periodic because after intervals T, the physical quantities characterizing it take on the same values.
In electrical engineering, sinusoidal alternating current is most widespread, i.e. a current whose value varies according to the law of sine (or cosine), which has a number of advantages compared to other periodic currents.
Alternating current industrial frequency obtained at power plants using alternating current generators (three-phase synchronous generators). These are quite complex electrical machines, let’s just consider physical basis their actions, i.e. the idea of producing alternating current.
Let a permanent magnet rotate uniformly with angular velocity in a uniform magnetic field? frame with area S (Fig. 1).
The magnetic flux through the frame will be equal to:
Ф=BS cos? (1.1)
Where? - the angle between the normal to the frame n and the magnetic induction vector B. Since with uniform rotation of the frame? = ?/t, then the angle? will be changed by law?= ? t and formula (1.1) will take the form:
Ф=BScos?t (1.2)
Since when the frame rotates, the magnetic flux crossing it changes all the time, then according to the law of electromagnetic induction, an induced emf E will be induced in it:
E = -d F /dt =BS?sin?t =E0sin?t (1.3)
where E0 = BS? - amplitude of sinusoidal EMF. Thus, a sinusoidal EMF will arise in the frame, and if the frame is closed to a load, then a sinusoidal current will flow in the circuit.
The quantity?t = 2?t/T = 2?ft, standing under the sine or cosine sign, is called the phase of the oscillations described by these functions. The phase determines the value of the EMF at any time t. Phase is measured in degrees or radians.
The time T of one complete change in the emf (this is the time of one revolution of the frame) is called the period of the emf. The change in EMF over time can be depicted on a time diagram (Fig. 2).
The reciprocal of the period is called frequency f = 1/T. If the period is measured in seconds, then the frequency of alternating current is measured in Hertz. In most countries, including Russia, the industrial frequency of alternating current is 50 Hz (in the USA and Japan - 60 Hz).
The magnitude of the industrial frequency of alternating current is determined by technical and economic considerations. If it is too low, then the dimensions of electrical machines increase and, consequently, the consumption of materials for their manufacture; The flickering of light in light bulbs becomes noticeable. At too high frequencies, energy losses in the cores of electrical machines and transformers increase. Therefore, the most optimal frequencies turned out to be 50 - 60 Hz. However, in some cases alternating currents of both higher and lower frequency are used. For example, airplanes use a frequency of 400 Hz. At this frequency, it is possible to significantly reduce the size and weight of transformers and electric motors, which is more significant for aviation than an increase in losses in the cores. On railways use alternating current with a frequency of 25 Hz and even 16.66 Hz.
Lecture 13-14
Alternating current
As is known, the current strength at any time is proportional to the emf of the current source (Ohm’s law for complete chain). If the emf of the source does not change over time and the parameters of the circuit remain unchanged, then some time after the circuit is closed, the changes in current strength stop, and direct current flows in the circuit.
However, in modern technology, not only direct current sources are widely used, but also various electric current generators, in which the EMF changes periodically. When an alternating EMF generator is connected to an electrical circuit, forced electromagnetic oscillations or alternating current occur in the circuit.
Alternating current – these are periodic changes in current and voltage in an electrical circuit that occur under the influence of alternating EMF from an external source
or
Alternating current is an electric current that changes over time according to a harmonic law.
In the future, we will study forced electrical oscillations that occur in circuits under the influence of voltage that varies harmoniously with frequencyω according to the sinusoidal or cosine law:
u = Um ⋅ sinωt
oru = Um ⋅ cosωt
Whereu – instantaneous voltage value,U m – voltage amplitude,ω – cyclic frequency of oscillations. If the voltage changes with frequencyω , then the current strength in the circuit will change with the same frequency, but the current fluctuations do not necessarily have to be in phase with the voltage fluctuations. Therefore, in the general case
i = Im ⋅ sin(ωt + φc )
Whereφ c – phase difference (shift) between current and voltage fluctuations.
Alternating current ensures the operation of electric motors in machines in plants and factories, powers lighting fixtures in our apartments and outdoors, refrigerators and vacuum cleaners, heating appliances, etc. The frequency of voltage fluctuations in the network is 50 Hz. The alternating current has the same oscillation frequency. This means that within 1 s the current will change its direction 50 times. A frequency of 50 Hz is accepted for industrial current in many countries around the world. In the USA, the frequency of industrial current is 60 Hz.
Resistor in AC circuit
Let the circuit consist of conductors with low inductance and high resistanceR (from resistors). For example, such a circuit could be the filament of an electric lamp and the supply wires. SizeR , which we have hitherto called electrical resistance or simply resistance, will now be calledactive resistance . There may be other resistances in an AC circuit, depending on the inductance of the circuit and its capacitance. ResistanceR It is called active because only it releases energy, i.e.
The resistance of an electrical circuit element (resistor) in which electrical energy is converted into internal energy is calledactive resistance .
So, there is a resistor in the circuit, the active resistance of which isR , and the inductor and capacitor are missing (Fig. 1).
Rice. 1
u = Um ⋅ sinωt
As with direct current, the instantaneous value of the current is directly proportional to the instantaneous value of the voltage. Therefore, we can assume that the instantaneous value of the current is determined by Ohm’s law:
i = UR = Um ⋅ sinω tR = Im ⋅ sinω t
Consequently, in a conductor with active resistance, current fluctuations in phase coincide with voltage fluctuations (Fig. 2), and the current amplitude is equal to the voltage amplitude divided by the resistance:
Rice. 2
At low frequencies of alternating current, the active resistance of the conductor does not depend on frequency and practically coincides with its electrical resistance in a direct current circuit.
Coil in AC circuit
Inductance affects the strength of alternating current in a circuit. This can be discovered by simple experiment. Let's make a circuit of a high-inductance coil and an incandescent lamp (Fig. 3). Using a switch, you can connect this circuit to either a DC voltage source or an AC voltage source. In this case, the direct voltage and the effective value of the alternating voltage must be the same. Experience shows that the lamp glows brighter at constant voltage. Consequently, the effective value of the current in the circuit under consideration is less than the direct current.
Rice. 3
This is explained by self-induction. When the coil is connected to a constant voltage source, the current in the circuit increases gradually. The vortex electric field that appears as the current increases slows down the movement of electrons. Only after some time does the current reach its highest (steady) value corresponding to a given constant voltage. If the voltage changes quickly, then the current strength will not have time to reach those steady-state values that it would acquire over time at a constant voltage equal to the maximum value of the alternating voltage. Consequently, the maximum value of the alternating current (its amplitude) is limited by the inductanceL circuit and will be smaller, the greater the inductance and the greater the frequency of the applied voltage.
Let's prove this mathematically. Let an ideal coil with an electrical resistance of the wire equal to zero be included in an alternating current circuit (Fig. 4). When the current changes according to the harmonic law
i = Im ⋅ cosωt
self-induced emf occurs in the coil
e =− L ⋅ i ′= Im ⋅ L ⋅ ω ⋅ sinωt
WhereL – coil inductance,ω – cyclic frequency of alternating current.
Rice. 4
Since the electrical resistance of the coil is zero, the self-induction EMF in it at any moment of time is equal in magnitude and opposite in sign to the voltage at the ends of the coil created by an external generator:
u =− e =− Im ⋅ L ⋅ ω ⋅ sinωt
Consequently, when the current in the coil changes according to a harmonic law, the voltage at its ends also changes according to a harmonic law, but with a phase shift:
u = Im ⋅ L ⋅ ω ⋅ cos(ωt + π 2)
Consequently, the voltage fluctuations across the inductor lead the current fluctuations by π/2, or, which is the same, the current fluctuations lag behind the voltage fluctuations by π/2.
At the moment when the voltage on the coil reaches its maximum, the current strength is zero (Fig. 5). At the moment when the voltage becomes zero, the current strength is maximum in magnitude.
Rice. 5
WorkIm ⋅ L ⋅ ω
is the amplitude of voltage oscillations on the coil:
Um = Im ⋅ L ⋅ ω
The ratio of the amplitude of voltage fluctuations on the coil to the amplitude of current fluctuations in it is calledinductive reactance (denotedX L ):
XL = UmIm = L ⋅ ω
The relationship between the amplitude of voltage fluctuations at the ends of the coil and the amplitude of current fluctuations in it coincides in form with the expression of Ohm’s law for a section of a direct current circuit:
Im = UmXL
Unlike the electrical resistance of a conductor in a DC circuit, inductive reactance is not a constant value that characterizes a given coil. It is directly proportional to the frequency of the alternating current. Therefore, the amplitude of current oscillations in the coil at a constant value of the amplitude of voltage oscillations should decrease in inverse proportion to the frequency. Direct current does not “notice” the inductance of the coil at all. Atω = 0 inductive reactance is zero (X L = 0).
The dependence of the amplitude of current oscillations in the coil on the frequency of the applied voltage can be observed in an experiment with an alternating voltage generator, the frequency of which can be changed. Experience shows that doubling the frequency of alternating voltage leads to a halving of the amplitude of current fluctuations through the coil.
Capacitor in AC circuit
Let's consider the processes occurring in an alternating current electrical circuit with a capacitor. If you connect a capacitor to a direct current source, a short-term current pulse will appear in the circuit, which will charge the capacitor to the source voltage, and then the current will stop. If a charged capacitor is disconnected from a direct current source and its plates are connected to the terminals of an incandescent lamp, the capacitor will be discharged, and a short-term flash of the lamp will be observed.
When a capacitor is connected to an alternating current circuit, its charging process lasts a quarter of a period. After reaching the amplitude value, the voltage between the plates of the capacitor decreases and the capacitor is discharged within a quarter of the period. In the next quarter of the period, the capacitor is charged again, but the polarity of the voltage on its plates is reversed, etc. The processes of charging and discharging the capacitor alternate with a period equal to the oscillation period of the applied alternating voltage.
As in a DC circuit, through the dielectric separating the capacitor plates, electric charges don't pass. But as a result of periodically repeating processes of charging and discharging the capacitor, alternating current flows through the wires connected to its terminals. An incandescent lamp connected in series with a capacitor in an alternating current circuit (Fig. 6) appears to burn continuously, since the human eye high frequency fluctuations in current strength does not notice the periodic weakening of the glow of the lamp filament.
Rice. 6
Let us establish a connection between the amplitude of voltage fluctuations on the capacitor plates and the amplitude of current fluctuations. When the voltage on the capacitor plates changes according to the harmonic law
u = Um ⋅ cosωt
the charge on its plates changes according to the law:
q = C ⋅ u = Um ⋅ C ⋅ cosωt
Electric current in the circuit arises as a result of a change in the charge of the capacitor:i = q ’. Therefore, current fluctuations in the circuit occur according to the law:
i =− Um ⋅ ω ⋅ C ⋅ sinωt = Um ⋅ ω ⋅ C ⋅ cos(ωt + π 2)
Consequently, voltage fluctuations on the capacitor plates in an alternating current circuit lag in phase behind current fluctuations by π/2, or current fluctuations lead in phase voltage fluctuations by π/2 (Fig. 7). This means that at the moment when the capacitor begins to charge, the current is maximum and the voltage is zero. After the voltage reaches its maximum, the current becomes zero, etc.
Rice. 7
WorkUm ⋅ ω ⋅ C
is the amplitude of current fluctuations:
Im = Um ⋅ ω ⋅ C
The ratio of the amplitude of voltage fluctuations on the capacitor to the amplitude of current fluctuations is called the capacitance of the capacitor (denotedX C ):
XC = UmIm =1 ω ⋅ C
The relationship between the amplitude value of the current and the amplitude value of the voltage coincides in form with the expression of Ohm's law for a section of a direct current circuit, in which instead of electrical resistance the capacitance of the capacitor appears:
Im = UmXC
The capacitive reactance of a capacitor, like the inductive reactance of a coil, is not a constant value. It is inversely proportional to the frequency of the alternating current. Therefore, the amplitude of current fluctuations in the capacitor circuit at a constant amplitude of voltage fluctuations on the capacitor increases in direct proportion to the frequency.
Ohm's law for an alternating current electrical circuit
Let's consider an electrical circuit consisting of a resistor, capacitor and coil connected in series (Fig. 8). If you apply to the terminals of this electrical circuit electrical voltage, varying according to a harmonic law with frequencyω and amplitudeU m , then forced oscillations of current strength will occur in the circuit with the same frequency and a certain amplitudeI m . Let us establish a connection between the amplitudes of current and voltage fluctuations.
Rice. 8
At any moment of time, the sum of the instantaneous voltage values on the series-connected circuit elements is equal to the instantaneous value of the applied voltage:
u = uR + uL + uC
. (1)
In all series-connected circuit elements, changes in current strength occur almost simultaneously, since electromagnetic interactions propagate at the speed of light. Therefore, we can assume that fluctuations in current strength in all elements of a series circuit occur according to the law:
i = Im ⋅ cosωt
. (2)
Voltage fluctuations across the resistor are in phase with current fluctuations, voltage fluctuations across the capacitor are out of phase byπ /2 from current fluctuations, and voltage fluctuations on the coil are ahead in phase of current fluctuations byπ /2. Therefore, equation (1) can be written as follows:
u = URm ⋅ cosωt + ULm ⋅ cos(ωt + π 2)+ UCm ⋅ cos(ωt − π 2)
, (3)
WhereU Rm , U Cm AndU Lm – amplitudes of voltage fluctuations across the resistor, capacitor and coil.
The amplitude of voltage fluctuations in an alternating current circuit can be expressed through the amplitude values of the voltage on its individual elements using the vector diagram method.
When constructing a vector diagram, it is necessary to take into account that voltage fluctuations across the resistor coincide in phase with current fluctuations, therefore the vector depicting the voltage amplitudeU Rm , coincides in direction with the vector depicting the amplitude of the currentI m . Voltage fluctuations across the capacitor are out of phase byπ /2 from current fluctuations, so the vectorU ⃗ Cm
lags behind the vectorI ⃗ m at an angle of 90°. Voltage fluctuations on the coil are ahead of current fluctuations in phase byπ /2, so the vectorU ⃗ Lm ahead of the vectorI ⃗ m
at an angle of 90° (Fig. 9).
Rice. 9
In a vector diagram, the instantaneous voltage values on the resistor, capacitor and coil are determined by projections onto the horizontal axis of the vectorsU ⃗ Rm
, U ⃗ Cm AndU ⃗ Lm , rotating at the same angular speedω counterclock-wise. The instantaneous voltage value in the entire circuit is equal to the sum of the instantaneous voltagesu R , u C Andu L on individual elements chains, i.e. the sum of projections of vectorsU ⃗ Rm , U ⃗ Cm AndU ⃗ Lm
to the horizontal axis. Since the sum of the projections of vectors onto an arbitrary axis is equal to the projection of the sum of these vectors onto the same axis, the amplitude of the total voltage can be found as the modulus of the sum of vectors:
U ⃗ m = U ⃗ Rm + U ⃗ Cm + U ⃗ Lm
From Figure 9 it can be seen that the voltage amplitude throughout the entire circuit is equal to
Um = U 2 Rm +(ULm − UCm )2−−−−−−−−−−−−−−−−− √
, (4)
or
Um =(ImR )2+(ImXL − ImXC )2−−−−−−−−−−−−−−−−−−−−−− √ = Im ⋅ R 2+(XL − XC )2−−−−−−−−−−−−−− √ = Im ⋅ R 2+(Lω −1 Cω )2−−−−−−−−−−−−−− √
.
From here
Im = UmR 2+(Lω −1 Cω )2 √
. (5)
By introducing the designation forimpedance AC circuits
Z = R 2+(Lω −1 Cω )2−−−−−−−−−−−−−− √
, (6)
Let us express the relationship between the amplitude values of current and voltage in an alternating current circuit as follows:
Im = UmZ
. (7)
This expression is calledOhm's law for an alternating current circuit .
From the vector diagram shown in Figure 9, it is clear that the phase of the total voltage oscillations is equal toω∙t + φ . Therefore, the instantaneous value of the total voltage is determined by the formula:
u = Um ⋅ cos(ωt + φ )
. (8)
Initial phaseφ can be found from the vector diagram:
cosφ = URmUm = Im ⋅ RIm ⋅ R 2+(Lω −1 Cω )2 √ = RZ
. (9)
Cos valueφ plays an important role in calculating power in an AC electrical circuit.
AC power
Power in a DC circuit is determined by the product of voltage and current:
P = U ⋅ I
.
The physical meaning of this formula is simple: since the voltageU is numerically equal to the work of the electric field to move a unit charge, then the productU∙I characterizes the work of moving a charge per unit of time flowing through the cross section of a conductor, i.e. is power. The power of the electric current in a given section of the circuit is positive if energy comes to this section from the rest of the network, and negative if energy from this section returns to the network. Over a very short period of time, alternating current can be considered constant. Therefore, instantaneous power in an alternating current circuit is determined by the same formula:
p = u ⋅ i
.
Let the voltage at the ends of the circuit vary according to the harmonic law
u = Um ⋅ cosωt
(with the same success, of course, instead u = Um ⋅ cosωt
could be written down u = Um ⋅ sinωt
), then the current strength will change over time harmoniously with the same frequency, but in the general case it will be phase shifted relative to the voltage:
i = Im ⋅ cos(ω t + φ c )
,
Whereφ c – phase shift between current and voltage. Therefore, for instantaneous power we can write:
p = u ⋅ i = Um ⋅ Im ⋅ cosωt ⋅ cos(ωt + φc )
.
In this case, the power changes over time both in magnitude and sign. During one part of the period, energy flows to a given section of the circuit (R > 0), but during the other part of the period some energy returns to the network again (R < 0). Как правило, во всех случаях нам надо знать среднюю мощность на участке цепи за достаточно большой промежуток времени, включающий много периодов. Для этого достаточно определить среднюю мощность за один период.
To find the average power over a period, we transform the resulting formula in such a way as to highlight a term in it that does not depend on time. For this purpose, we will use the well-known formula for the product of two cosines:
cosα ⋅ cosβ =cos(α − β )+cos(α + β )2
.
In this caseα = ω∙t Andβ = ω∙t + φ c . That's why
p = Um ⋅ Im 2= Um ⋅ Im 2cosφc + Um ⋅ Im 2cos(2ωt + φc )
.
The expression for instantaneous power consists of two terms. The first does not depend on time, and the second changes sign twice for each period of voltage change: during some part of the period, energy enters the circuit from an alternating voltage source, and during the other part it returns back. Therefore, the average value of the second term over the period is zero. Therefore, the average powerR over a period is equal to the first term independent of time:
P = Um ⋅ Im 2cosφc
. (10)
When the phase of current and voltage oscillations coincides (for active resistanceR ) the average power value is:
P = Um ⋅ Im 2= I 2 m ⋅ R 2
.
In order for the formula for calculating the power of alternating current to coincide in form with a similar formula for direct current (R = I∙U = I 2 ∙ R ), the concepts of effective values of current and voltage are introduced. From the equality of powers we obtain
P = I 2 m ⋅ R 2= I 2 ⋅ R
or I 2 m 2= I 2
.
Effective current value name the quantity in 2–√
times less than its amplitude value:
I = Im 2√
.
RMS current value is equal to the strength of such a direct current at which the average power released in a conductor in an alternating current circuit is equal to the power released in the same conductor in a direct current circuit.
Similarly, it can be proven that
effective value of alternating voltage V 2–√
times less than its amplitude value:
U = Um 2√
.
Note that usually electrical equipment in alternating current circuits shows the effective values of the measured quantities. Turning to the effective values of current and voltage, equation (10) can be rewritten:
P = Um 2√ ⋅ Im 2√cosφc = U ⋅ I cosφc
. (10)
Thus, the alternating current power in a section of the circuit is determined precisely by the effective values of current and voltage. It also depends on the phase shiftφ c between voltage and current. cos multiplierφ c in the formula is calledpower factor .
In caseφ c = ± π /2, the energy supplied to a section of the circuit during a period is zero, although there is a current in the circuit. This will be the case, in particular, if the circuit contains only an inductor or only a capacitor. How can the average power be equal to zero in the presence of current in the circuit? This is explained by the graphs of changes over time in the instantaneous values of voltage, current and power atφ c = - π /2 (purely inductive reactance of the circuit section). A graph of instantaneous power versus time can be obtained by multiplying the values of current and voltage at each time. From this graph it can be seen that during one quarter of the period the power is positive and energy flows to this section of the circuit; but during the next quarter of the period the power is negative, and this section transfers the previously received energy back into the network without loss. The energy arriving during a quarter of the period is stored in the magnetic field of the current, and then returned to the network without loss.
Rice. 10
Only in the presence of a conductor with active resistance in a circuit that does not contain moving conductors, electromagnetic energy is converted into internal energy of the conductor, which heats up. The reverse conversion of internal energy into electromagnetic energy in the area with active resistance no longer occurs.
When designing AC circuits, it is necessary to ensure that cosφ c was not small. Otherwise, a significant part of the energy will circulate through the wires from the generator to consumers and back. Since the wires have active resistance, energy is spent on heating the wires.
Unfavorable conditions for energy consumption arise when electric motors are connected to the network, since their windings have low active resistance and high inductance. To increase cosφ c in power supply networks of enterprises with a large number of electric motors, special compensating capacitors are included. It is also necessary to ensure that electric motors do not run idle or underload. This reduces the power factor of the entire circuit. Increase cosφ c is an important national economic task, as it makes it possible to use power plant generators with maximum efficiency and reduce energy losses. This is achieved by proper design of electrical circuits. Do not use devices with cosφ c < 0,85.
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