Values ​​of effective voltage and current. Definition. Amplitude ratio for different shapes. (10+)

The concept of effective (effective) values ​​of voltage and current

When we talk about alternating voltages or currents, especially of a complex shape, the question arises of how to measure them. After all, the tension is constantly changing. You can measure the signal amplitude, that is, the maximum modulus of the voltage value. This measurement method is fine for relatively smooth signals, but the presence of short bursts spoils the picture. Another criterion for choosing a measurement method is for what purpose the measurement is made. Since in most cases the power that a particular signal can give is of interest, the effective (effective) value is used.

Consider the following chain.

It consists of an AC voltage source, connecting wires and some load. Moreover, the load inductance is very small, and the resistance R is very high. We used to call this load resistance. Now we will call it active resistance.

Active resistance

Resistance R called active, since if there is a load in the circuit with such a resistance, the circuit will absorb the energy coming from the generator. We will assume that the voltage at the terminals of the circuit obeys the harmonic law:

U = Um * cos (ω * t).

The instantaneous value of the current strength can be calculated according to Ohm's law, it will be proportional to the instantaneous value of the voltage.

I = u / R = Um * cos (ω * t) / R = Im * cos (ω * t).

Let's conclude: in a conductor with active resistance, there is no phase difference between voltage and current fluctuations.

RMS value of current

The amplitude of the current is determined by the following formula:

The average value of the square of the current strength over the period is calculated using the following formula:

Here Im there is the amplitude of the current fluctuation. If we now calculate the square root of the average value of the square of the current strength, we get a value that is called the effective value of the alternating current.

To designate the effective value of the current strength, the letter I is used. That is, in the form of a formula, it will look like this:

I = √ (i ^ 2) = Im / √2.

The effective value of the alternating current will be equal to the strength of such a direct current, at which the same amount of heat will be released in the conductor under consideration over the same period of time as with alternating current. The following formula is used to determine the effective voltage value.

U = √ (u ^ 2) = Um / √2.

Now let's substitute the effective values ​​of the current and voltage in the expression Im = Um / R. We get:

This expression is Ohm's law for a section of a circuit with a resistor through which an alternating current flows. As in the case of mechanical vibrations, in alternating current we will be of little interest in the values ​​of the current strength, voltage at some particular moment in time. It will be much more important to know the general characteristics of oscillations - such as amplitude, frequency, period, effective values ​​of current and voltage.

By the way, it is worth noting that voltmeters and ammeters designed for alternating current record the actual values ​​of voltage and current.

Another advantage of rms values ​​over instantaneous ones is that they can be immediately used to calculate the average power value P of the alternating current.

Definition 1

Effective (effective) is called the value of an alternating current equal to the value of an equivalent direct current, which, when passing through the same resistance as the alternating current, emits the same amount of heat on it for equal periods of time.

The quantitative relationship of the amplitudes of the strength and voltage of the alternating current and the effective values

The amount of heat that is generated by an alternating current on the resistance $ R $ for a short period of time $ dt $ is equal to:

Then, in one period, the alternating current releases heat ($ W $):

Let us denote by $ I_ (ef) $ the strength of the direct current, which on the resistance $ R $ emits the same amount of heat ($ W $) as the alternating current $ I $ for a time equal to the period of alternating current oscillations ($ T $). Then we express $ W $ in terms of direct current and equate the expression to the right-hand side of equation (2), we have:

We express from equation (3) the strength of the equivalent direct current, we get:

If the current strength changes sinusoidally:

substitute expression (5) for alternating current into formula (4), then the value of direct current will be expressed as:

Therefore, expression (6) can be transformed to the form:

where $ I_ (ef) $ is called the effective value of the current strength. The expressions for the effective (rms) voltage values ​​are written in a similar way:

Application of effective values ​​of current and voltage

When in electrical engineering they talk about AC strength and voltage, they mean their effective values. In particular, voltmeters and ammeters are usually calibrated to effective values. Consequently, the maximum value of the voltage in the AC circuit is about 1.5 times that of the voltmeter. This fact should be taken into account when calculating insulators, researching safety problems.

RMS values ​​are used to characterize the AC waveform (voltage). So, the crest factor ($ k_a $) is introduced. equal:

and form factor ($ k_f $):

where $ I_ (sr \ v) = \ frac (2) (\ pi) \ cdot I_m $ is the rectified average current value.

For sinusoidal current $ k_a = \ sqrt (2), \ k_f = \ frac (\ pi) (2 \ sqrt (2)) = 1.11. $

Example 1

Exercise: The voltage shown by the voltmeter is $ U = 220 V $. What is the amplitude of the voltage?

Solution:

As it was said, voltmeters and ammeters are usually calibrated for effective voltage values ​​(current), therefore, the device shows in our notation $ U_ (ef) = 220 \ V. $ In accordance with the well-known relation:

find the amplitude value of the voltage as:

Let's calculate:

Answer:$ U_m \ approx 310.2 \ V. $

Example 2

Exercise: How is the AC power on the resistance $ R $ and the effective values ​​of current and voltage related?

Solution:

The average value of the AC power in the circuit is

\ [\ left \ langle P \ right \ rangle = \ frac (A_T) (T) = \ frac (U_mI_mcos \ varphi) (2) \ left (2.1 \ right), \]

where $ cos \ varphi $ is the power factor, which shows the efficiency of power transfer from the current source to the consumer. On the other hand, the average current powers on individual circuit elements are $ \ left \ langle P_ (tC) \ right \ rangle = 0, \ left \ langle P_ (tL) \ right \ rangle = 0, \ left \ langle P_ (tR) \ right \ rangle = \ frac (1) (2) (I ^ 2) _mR, $ and the resulting cardinality can be found as the sum of the cardinalities:

\ [\ left \ langle P \ right \ rangle = \ left \ langle P_ (tC) \ right \ rangle + \ left \ langle P_ (tL) \ right \ rangle + \ left \ langle P_ (tR) \ right \ rangle \ left (2.2 \ right). \]

Therefore, we can write that:

\ [\ left \ langle P \ right \ rangle = P_ (tR) = \ frac (1) (2) (I ^ 2) _mR = \ frac (U_mI_mcos \ varphi) (2) \ left (2.3 \ right), \]

where $ I_m \ $ is the amplitude of the current, $ U_m $ is the amplitude of the external voltage, $ \ varphi $ is the phase difference between the current and the voltage.

For direct current, the instantaneous power coincides with the average. For $ I_ (ef) $ = const, you can put $ cos \ varphi = 1, \ $ means formula (2.3) can be written as:

if instead of amplitude values ​​($ U_m \ and \ I_m $) we use their effective (effective) values:

Therefore, the current power can be written as:

where $ cos \ varphi $ is the power factor. In technology, this coefficient is made as large as possible. With a small $ cos \ varphi $, in order for the required power to be released in the circuit, a large current must be passed, which leads to an increase in losses in the supply wires.

The same power (as in expression (2.3)) is developed by a direct current, the strength of which is presented in formula (2.5).

Answer:$ P_ (tR) = U_ (ef) I_ (ef) cos \ varphi. $

The alternating sinusoidal current has different instantaneous values ​​during the period. It is natural to ask the question, what value of the current will be measured by the ammeter included in the circuit?

When calculating alternating current circuits, as well as electrical measurements, it is inconvenient to use instantaneous or amplitude values ​​of currents and voltages, and their average values ​​over a period are zero. In addition, the electrical effect of a periodically varying current (the amount of heat released, the perfect work, etc.) cannot be judged by the amplitude of this current.

The most convenient was the introduction of the concepts of the so-called effective values ​​of current and voltage... These concepts are based on the thermal (or mechanical) action of the current, which does not depend on its direction.

This is the value of direct current at which the same amount of heat is generated in the conductor during the period of alternating current as during alternating current.

To evaluate the action produced, we compare its action with the thermal effect of direct current.

The power P of direct current I passing through the resistance r will be P = P 2 r.

AC power will be expressed as the average effect of the instantaneous power I 2 r over the whole period or the average value of (Im x sinω t) 2 x r in the same time.

Let the average value of t2 over the period be M. Equating the DC power and AC power, we have: I 2 r = Mr, whence I = √ M,

The magnitude I is called the rms value of the alternating current.

The average value of i2 at alternating current is determined as follows.

Let's build a sinusoidal current curve. Squaring each instantaneous current value, we get a curve of P versus time.

Both halves of this curve lie above the horizontal axis, since negative current values ​​(-i) in the second half of the period, when squared, give positive values.

Let's construct a rectangle with base T and area equal to the area bounded by the curve i 2 and the horizontal axis. The height of the rectangle M will correspond to the average value of P over the period. This value for the period, calculated using higher mathematics, will be equal to 1 / 2I 2 m. Therefore, М = 1 / 2I 2 m

Since the rms value of the alternating current is equal to I = √ M, then finally I = Im / √ 2

Similarly, the relationship between the effective and amplitude values ​​for the voltage U and E has the form:

U = Um / √ 2 E = Em / √ 2

The effective values ​​of variables are indicated by capital letters without subscripts (I, U, E).

Based on the above, we can say that the effective value of the alternating current is equal to that direct current, which, passing through the same resistance as the alternating current, releases the same amount of energy at the same time.

Electrical measuring instruments (ammeters, voltmeters) connected to the alternating current circuit show the effective values ​​of current or voltage.

When constructing vector diagrams, it is more convenient to postpone not the amplitude, but the effective values ​​of the vectors. For this, the lengths of the vectors are reduced by √ 2 times. This does not change the location of the vectors on the diagram.

RMS values ​​of current and voltage

As you know, the variable emf induction induces an alternating current in the circuit. At the highest value of the emf the current will have a maximum value and vice versa. This phenomenon is called phase coincidence. Despite the fact that the values ​​of the current strength can fluctuate from zero to a certain maximum value, there are devices with which you can measure the strength of the 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. These actions include heat. 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. You can choose such a value of the DC current so that the same amount of heat is released by this current on the same conductor for the same time as with an alternating current. This DC value is called the RMS AC current.

At this time, in world industrial practice, it is widespread 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 its own variable emf. with the same amplitudes and frequencies, but out-of-phase relative to each other by 120 ° or 1/3 of the period. Each such chain is called phase.

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

According to the above conditions, it turns out that the emf arising in the second generator will not have time to change in comparison with the emf. the first generator, that is, it will be late by 120 °. E.m.s. the third generator will also be 120 ° late in relation to the second.

However, this method of producing alternating three-phase current is very cumbersome and economically unprofitable. To simplify the task, it is necessary 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 there is a shift of the windings in space, the phases of the oscillations in them are also shifted relative to each other by 120 °.

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

Star connection

Usually, a three-phase current generator is depicted in the form of 3 stator windings, which are located at an angle of 120 ° to each other. The beginnings of the windings are usually denoted by letters A, B, C and the ends are 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 "star". In this case, wires called linear wires are connected to the beginning of the windings (Figure 3.5 on the left).

The 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 line and neutral wires or, equivalently, between the beginning and the end of any stator winding. This value is called phase voltage ( Uл). Since the circuit is three-phase, the line voltage will be in v3 times more than phase, i.e.: Ul = v3Uph.