The effective value of the alternating current formula. Effective values ​​of current and voltage

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The current value of the current I is called the force direct current, releasing in the conductor for the same time the same amount of heat as the alternating current.

As can be seen from the figure, at each moment of time, the values ​​​​of voltage and current take on different values. Therefore, in order to judge the magnitude of the current and voltage alternating current, use the current value of the current and voltage. To determine the effective value of the alternating current, it is equated to the direct current, which would release the same amount of heat in the conductor as the alternating current.

A transformer containing 300 turns in the primary winding is connected to an alternating current network with an effective voltage of 220 V. The secondary circuit of the transformer feeds a load with an active resistance of 50 ohms. Find the effective value of the current in the secondary circuit if the voltage drop in secondary winding a transformer containing 165 turns is 50 V.

Thus, when replacing the operation of extracting the root by comparison, the time for which the integrated signal from the CLAY becomes equal to the integral of the square of the measured current strength is proportional to the effective value of the current strength. Prior to this, K2 was open for time t and passed pulses from the GTI clock generator to the SI counter. The number of pulses TV / GIT recorded in the midrange is proportional to the current value of the current. This number is stored in / 77, and at the end of the measurement cycle is displayed on the DIG.

As with mechanical vibrations, in the case of electrical vibrations, we are usually not interested in the values ​​​​of current, voltage, and other quantities at any given time. Important General characteristics fluctuations, such as amplitude, period, frequency, effective values ​​of current and voltage, and average power. It is the effective values ​​​​of current and voltage that are recorded by ammeters and voltmeters of alternating current.

Рх o jjFr In the case of t - n - DRY lamps, the method of a thermometer suspended near the generator lamp is used, and its reading is noted. Then, breaking the circuit of the oscillatory circuit of the generator, they give a positive potential to the grid of the generator lamp until the thermometer is on. In the latter case, taking the values ​​1a and Ea as the initial ones, we determine Px from the relation Px1a Ea. The power in the antenna is determined by the f-le Px - / /, where PY is the power eW, ra is the active resistance of the antenna in Q and 1a is the effective value of the current in the antenna in A. Since, according to modern international standards, transmitter power is usually understood as power in the antenna, then the f-la mentioned above simultaneously determines the transmitter power.

Thermal meters have the widest practical. The action of thermal meters is to elongate a thin wire when it is heated by a high-frequency alternating current passing through it. The effect itself limits the limits of applicability of such meters to currents from a few mA to 1 - 3 A, depending on the material of the thin wire used in the meter. Alloys of silver with platinum, platinum with iridium, etc. are used. If the alloy is used in the form of a wire, then it has a diameter of hundredths of a mm. With a tape, the thickness is 0 01 mm, the width is 3 mm and the length is 25 - 30 mm. The lengthening of the thread by the heated current is proportional to the square of the effective value of the current. The movement on the scale of the meter of the needle, connected to the same wire using a special moving system, is usually proportional to the square root of the current strength. Because of this, the scales of thermal meters have uneven intervals between divisions.

In this case, the current fluctuations are harmonic (oscillation graph is a sinusoid) and forced, since the oscillation parameters (frequency, amplitude) are determined by an external source - a generator. Some electrical devices (for example, an oscillatory circuit) are capable of generating free harmonic oscillations of an electric current. On the left branch of the frame - from us and, since in this case current flows through terminal a in the direction opposite to that shown in fig. 12.1, its polarity is minus. Since at a given position of the loop, the current strength has the greatest value, the phase of the oscillations can be r / 2 or 3 / 2ir, depending on which direction of the current in the loop we take as positive. Comparing formula (12.1) and the given dependence, it is easy to see that 1m 10 A and w 4tgrad / s. Further, using formula (12.2), we determine the oscillation frequency (res. Using the Joule-Lenz law (Q I2Rt), we determine the effective value of the current strength (res.

The strength of an alternating current (voltage) can be characterized using amplitude. However, the amplitude value of the current is not easy to measure experimentally. It is convenient to associate the strength of an alternating current with any action produced by a current that does not depend on its direction. Such is, for example, the thermal effect of the current. The rotation of the needle of an ammeter measuring alternating current is caused by the elongation of the filament, which heats up when current passes through it.

current or effective the value of alternating current (voltage) is such a value of direct current at which the same amount of heat is released on the active resistance over a period as with alternating current.

Let us relate the effective value of the current to its amplitude value. To do this, we calculate the amount of heat released on the active resistance by alternating current for a time equal to the oscillation period. Recall that, according to the Joule-Lenz law, the amount of heat released in the section of the circuit with resistance at permanent current during , is determined by the formula

. Alternating current can be considered constant only for very short periods of time

. Divide the period of oscillation for a very large number of small time intervals

. Quantity of heat

, released on the resistance during

:

. The total amount of heat released over a period is found by summing the heats released over separate small periods of time, or, in other words, by integrating:

.

The current in the circuit changes according to a sinusoidal law

,

.

Omitting calculations related to integration, we write the final result

.

If some direct current were flowing through the circuit , then in a time equal to , it would be warm

. By definition, direct current , which has the same thermal effect as the variable, will be equal to the effective value of the alternating current

. We find the effective value of the current strength, equating the heat released over the period, in the cases of direct and alternating currents

(4.28)

Obviously, exactly the same relationship relates the effective and amplitude values ​​of the voltage in the circuit with a sinusoidal alternating current:

(4.29)

For example, the standard mains voltage of 220 V is the effective voltage. According to formula (4.29), it is easy to calculate that the amplitude value of the voltage in this case will be equal to 311 V.

4.4.5. AC power

Let, in some section of the circuit with alternating current, the phase shift between current and voltage is equal to , i.e. current and voltage change according to the laws:

,

.

Then the instantaneous value of the power released in the circuit section,

Power changes over time. Therefore, we can only talk about its average value. Let us determine the average power released over a sufficiently long period of time (many times greater than the oscillation period):

Using the well-known trigonometric formula

.

the value

no need to average, since it does not depend on time, therefore:

.

Over a long time, the value of the cosine has time to change many times, taking both negative and positive values ​​ranging from (1) to 1. It is clear that the time-averaged value of the cosine is zero

, that's why

(4.30)

Expressing the current and voltage amplitudes in terms of their effective values ​​using formulas (4.28) and (4.29), we obtain

. (4.31)

The power released in the section of the circuit with alternating current depends on the effective values ​​​​of current and voltage and phase shift between current and voltage. For example, if a circuit section consists of only active resistance, then

and

. If the circuit section contains only inductance or only capacitance, then

and

.

The average zero value of the power allocated to the inductance and capacitance can be explained as follows. Inductance and capacitance only borrow energy from the generator, and then return it back. The capacitor is charged and then discharged. The current in the coil increases, then drops back to zero, etc. It is for the reason that the average energy consumed by the generator on the inductive and capacitive resistances is zero, they were called reactive. On the active resistance, the average power is different from zero. In other words, a wire with resistance when current flows through it, it heats up. And the energy released in the form of heat is no longer returned to the generator.

If the circuit section contains several elements, then the phase shift may be different. For example, in the case of the circuit section shown in Fig. 4.5, the phase shift between current and voltage is determined by the formula (4.27).

Example 4.7. A resistor with a resistance is connected to the alternator sinusoidal current . How many times will the average power consumed by the generator change if a coil with an inductive resistance is connected to the resistor

a) in series, b) in parallel (Fig. 4.10)? Ignore the active resistance of the coil.

Solution. When only one active resistance is connected to the generator , power consumption

(see formula (4.30)).

Consider the circuit in fig. 4.10, a. In example 4.6, the amplitude value of the generator current was determined:

. From the vector diagram in fig. 4.11, but we determine the phase shift between the current and voltage of the generator

.

As a result, the average power consumed by the generator

.

Answer: when connected in series to the inductance circuit, the average power consumed by the generator will decrease by 2 times.

Consider the circuit in fig. 4.10b. In example 4.6, the amplitude value of the generator current was determined

. From the vector diagram in fig. 4.11, b we determine the phase shift between the current and voltage of the generator

.

Then the average power consumed by the generator

Answer: when the inductance is connected in parallel, the average power consumed by the generator does not change.

As is known, the variable emf. induction produces an alternating current in the circuit. At highest value emf current will be at its maximum and vice versa. This phenomenon is called phase matching. Although current values ​​can vary from zero to a certain maximum value, there are instruments with which you can measure the strength of alternating current.

The AC characteristic can be actions that are independent of the direction of the current and can be the same as with DC. Such actions include thermal. For example, an alternating current flows through a conductor with a given resistance. After a certain period of time, a certain amount of heat will be released in this conductor. It is possible to choose such a value of the direct current strength so that on the same conductor for the same time the same amount of heat is released by this current as with alternating current. This value of direct current is called the effective value of the alternating current.

At present, in the world industrial practice, it is widely used three-phase alternating current, which has many advantages over single-phase current. A three-phase system is called a system that has three electrical circuits with their own emf variables. with the same amplitudes and frequency, but shifted in phase relative to each other by 120 ° or 1/3 of the period. Each such circuit is called phase.

To obtain a three-phase system, you need to take three identical single-phase alternating current generators, connect their rotors to each other so that they do not change their position during rotation. The stator windings of these generators must be rotated relative to each other by 120 ° in the direction of rotation of the rotor. An example of such a system is shown in Fig. 3.4.b.

According to the above conditions, it turns out that the emf that occurs in the second generator will not have time to change, compared with the emf. the first generator, i.e., it will be late by 120 °. emf the third generator will also be late in relation to the second one by 120°.

However, this method of obtaining an alternating three-phase current is very cumbersome and economically unprofitable. To simplify the task, you need to combine all the stator windings of the generators in one housing. Such a generator is called a three-phase current generator (Fig. 3.4.a). When the rotor starts to rotate, in each winding there is

a) b)

Rice. 3.4. Example of a three-phase AC system

a) three-phase current generator; b) with three generators;

changing emf induction. Due to the fact that the windings are shifted in space, the phases of the oscillations in them also shift relative to each other by 120 °.

In order to connect a three-phase alternator to a circuit, you need to have 6 wires. To reduce the number of wires, the winding of the generator and receivers must be interconnected, forming a three-phase system. There are two connections: star and delta. When using both methods, you can save wiring.

Star connection

Typically, a three-phase current generator is depicted as 3 stator windings, which are located at an angle of 120 ° to each other. The beginning of the windings is usually denoted by letters A, B, C, and the ends X, Y, Z. In the case when the ends of the stator windings are connected to one common point (zero point of the generator), the connection method is called a "star". In this case, wires called linear wires are connected to the beginnings of the windings (Fig. 3.5 on the left).

Receivers can be connected in the same way (fig. 3.5., on the right). In this case, the wire that connects the zero point of the generator and receivers is called zero. This three-phase current system has two different voltages: between the linear and neutral wires or, equivalently, between the beginning and end of any stator winding. This value is called the phase voltage ( Ul). Since the circuit is three-phase, the line voltage will be in v3 times more phase, i.e.: Ul = v3Uf.

Delta connection.

Figure 3.6. Delta connection example

When using this connection method, the end X the first winding of the generator is connected to the beginning AT the second of his windings, the end Y second winding - to the beginning FROM third winding, end Z third winding - to the beginning BUT first winding. An example connection is shown in fig. 3.6. With this method of connecting phase windings and connecting a three-phase generator to a three-wire line, the linear voltage is compared in its value with the phase one: Up = Ul

test questions

1. List the main parameters characterizing alternating current.

2. Give the definition of frequency and its unit of measurement.

3. Define the amplitude and its unit of measurement.

4. Give the definition of the period and its units of measurement.

5. The difference between the simplest three-phase current generator and a single-phase current generator.

6. What is a phase?

7. What is a three-phase current generator rotor?

8. Why are the stator windings of a three-phase current generator shifted in phase?

9. Feature of a symmetrical system of three phases.

10. The principle of connecting the phase windings of three-phase generators and transformers according to the "star" scheme.

11. The principle of connecting the phase windings of three-phase generators and transformers according to the "triangle" scheme.

3.2. Types of resistance in AC circuits

In AC circuits, resistances are divided into active and reactive.

AT active resistance , included in the alternating current circuit, electrical energy is converted into heat. active resistance R have, for example, wires electrical lines, windings of electrical machines, etc.

AT reactances the electrical energy generated by the source is not consumed. When reactance is included in an alternating current circuit, only an exchange of energy occurs between it and a source of electrical energy. Reactance is created by inductances and capacitances.

If we do not take into account the mutual influence of individual elements electrical circuit, then in the general case, the electric circuit of a sinusoidal current can be represented by three passive elements: active resistance R, inductance L and capacitance C.

Active resistance in the AC circuit.

When an active resistance is connected to an alternating current circuit, the current and voltage coincide in phase (Fig. 3.7) and change according to the same sinusoidal law: u=U m sinωt. They simultaneously reach their maximum values ​​and simultaneously pass through zero (Fig. 3.7.b).

For an AC circuit containing only active resistance, Ohm's law has the same form as for a DC circuit: I=U/R.

Electric power R in a circuit with active resistance at any time is equal to the product of instantaneous current values i and stress u: p=ui.

Figure 3.7. Scheme of inclusion in the alternating current circuit of active resistance R (a), current curves i, voltage u and power p(b) and vector diagram.

It can be seen from the graph that the change in power occurs at a double frequency with respect to the change in current and voltage, i.e. one period of change in power corresponds to half the period of change in current and voltage. All power values ​​are positive, which means that energy is transferred from the source to the consumer.

Average power Рcp, consumed by active resistance, P=UI=I 2 R- That's what it is active power.

Under inductance L we will understand an element of an electric circuit (an inductor, the losses of which can be neglected), capable of storing energy in its magnetic field, which does not have active resistance and capacitance FROM ( Fig.3.8).

When an inductance is included in an alternating current circuit, the changing current continuously induces an emf in it. self-induction e L = LΔi/Δt, where ∆i/∆t is the rate of current change.

When the corner ωt equal to 90° and 270° rate of change of current ∆i/∆t=0, so emf. e L=0.

The rate of current change will be greatest when the angle ωt is 0°, 180° and 360°. In these minutes of time, emf. is of the greatest importance.

The power curve is a sinusoid that changes at twice the frequency of current and voltage changes. Power has positive and negative values, i.e. there is a continuous oscillatory process of energy exchange between the source and the inductance.

Figure 3.8. Scheme of inclusion in the alternating current circuit of inductance (a), current curves i, voltage u, emf e L(b) and vector diagram (c)

emf self-induction according to Lenz's rule is directed so as to prevent a change in current. In the first quarter of the period, when the current increases, emf. has a negative value (directed against the current).

In the second quarter of the period, when the current decreases, emf. has a positive value (coincides in direction with the current).

In the third quarter of the period, the current changes its direction and increases, so the emf. is directed against the current and has a positive value.

In the fourth quarter of the period, the current decreases and emf. self-induction tends to maintain the current position and has a negative value. As a result, the current lags the voltage in phase by an angle of 90 O.

The resistance of a coil or conductor to alternating current, caused by the action of emf. self-induction is called inductive reactance X L[Ohm]. Inductive reactance does not depend on the material of the coil and on the cross-sectional area of ​​the conductor.

In AC circuits, inductors are connected in series and in parallel.

When the coils are connected in series, the equivalent inductance Le and equivalent inductive reactance X L e will be equal:

Le \u003d L 1 + L 2 + ... X L e \u003d X L 1 + X L 2 + ...

When coils are connected in parallel:

1 / Le \u003d 1 / L 1 + 1 / L 2 + ... 1 / X L e \u003d 1 / X L 1 + 1 / X L 2 + ...

test questions

1. What types of resistance in AC circuits do you know?

2. What does active resistance mean?

3. What is reactance?

4. What circuit elements create reactance?

5. What is active power?

1. Define inductance.

2. What happens in the first quarter of the period of the oscillatory process of energy exchange between the source and the inductance?

3. What happens in the second quarter of the period of the oscillatory process of energy exchange between the source and the inductance?

4. Define inductive reactance.

3.3. Capacitors. Capacitance in the AC circuit

Capacitor - a device capable of storing electrical charges.

The simplest capacitor consists of two metal plates (electrodes) separated by a dielectric.

Each capacitor is characterized by a nominal capacitance and allowable voltage. The voltage of the capacitor is indicated on the case, and it must not be exceeded. Capacitors differ in the shape of the electrodes (flat), the type of dielectric and capacitance (constant and variable).