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.

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

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

Let's connect the effective value of the current with its amplitude value. To do this, we calculate the amount of heat released on the active resistance by an 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 only be considered constant for very short periods of time.
... Divide the oscillation period for a very large number of small periods of time
... Quantity of heat
released on 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 down the final result

.

If there was some constant current running 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 alternating one, 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 cases of direct and alternating currents

(4.28)

Obviously, exactly the same ratio connects the effective and peak values ​​of the voltage in the circuit with a sinusoidal alternating current:

(4.29)

For example, the standard voltage of 220 V is the effective voltage. According to the 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 strength and voltage change according to the laws:

,
.

Then the instantaneous value of the power allocated in the section of the circuit is

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 exceeding the oscillation period):

Using the famous trigonometric formula

.

The value
there is no need to average, since it does not depend on time, therefore:

.

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

, That's why
(4.30)

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

. (4.31)

The power released in the section of the AC circuit depends on the effective values ​​of current and voltage and phase shift between current and voltage... For example, if a section of a circuit consists of only one 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 strength of the current in the coil increases, then drops again to zero, etc. Precisely 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 nonzero. In other words, a wire with resistance when a current flows through it, it heats up. And the energy released in the form of heat does not return back to the generator.

If the chain section contains several elements, then the phase shift may be different. For example, in the case of the chain 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 resistance is connected to the alternating sinusoidal current generator ... How many times will the average power consumed by the generator change if a coil with inductive resistance is connected to the resistor
a) in series, b) in parallel (Figure 4.10)? Disregard the active resistance of the coil.

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

(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, and we determine the phase shift between the current and voltage of the generator

.

As a result, the average power consumed by the generator is

.

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

Consider the circuit in fig. 4.10, b. 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.

The physical meaning of these concepts is approximately the same as the physical meaning of the average speed or other quantities averaged over time. At different points in time, the strength of the alternating current and its voltage take on different values, therefore, talking about the strength of the alternating current in general can only be conditional.

At the same time, it is quite obvious that different currents have different energy characteristics - they produce different work in the same period of time. The work performed by the current is taken as the basis for determining the effective value of the current strength. They are set for a certain period of time and calculate the work done by alternating current during this period of time. Then, knowing this work, the reverse calculation is performed: they find out the strength of the direct current, which would produce similar work in the same period of time. That is, power averaging is performed. The calculated force of a hypothetically flowing direct current through the same conductor, producing the same work, is the effective value of the original alternating current. Do the same with tension. This calculation is reduced to determining the value of such an integral:

Where does this formula come from? From the well-known formula for the power of the current, expressed in terms of the square of its strength.

RMS values ​​of periodic and sinusoidal currents

Calculating the effective value for arbitrary currents is an unproductive exercise. But for a periodic signal, this parameter can be very useful. It is known that any periodic signal can be decomposed into a spectrum. That is, it is represented as a finite or infinite sum of sinusoidal signals. Therefore, to determine the magnitude of the effective value of such a periodic current, we need to know how to calculate the effective value of a simple sinusoidal current. As a result, by adding the RMS values ​​of the first few harmonics with the maximum amplitude, we obtain an approximate value of the RMS current value for an arbitrary periodic signal. Substituting the expression for the harmonic vibration into the above formula, we obtain the following approximate formula.

When calculating alternating current circuits, they usually use the concept of effective (effective) values ​​of alternating current, voltage and e. etc. with.

RMS values ​​of current, voltage and e. etc. with. indicated by capital letters.

The effective values ​​of the quantities are also indicated on the scales of measuring instruments and technical documentation.

The effective value of the alternating current is equal to the value of such an equivalent direct current, which, passing through the same resistance as the alternating current, releases the same amount of heat in it during the period.

The amount of heat generated by an alternating current in resistance in an infinitely small period of time

and for the period of alternating current T

Equating the resulting expression to the amount of heat released in the same resistance by direct current for the same time T, we get:

Reducing the common factor, we get the effective value of the current

Rice. 5-8. AC and square current graph.

In fig. 5-8, a curve of instantaneous values ​​of current i and a curve of squares of instantaneous values ​​are plotted.The area bounded by the last curve and the abscissa is, on a certain scale, a value determined by the expression square r.m.s. current

If the current changes according to the sine law, i.e.

Similarly for the rms values ​​of sinusoidal voltages and e. etc. with. you can write:

In addition to the effective value of current and voltage, sometimes they also use the concept of the average value of tbka and voltage.

The average value of the sinusoidal current over the period is zero, since during the first half of the period a certain amount of electricity Q passes through the cross-section of the conductor in the forward direction. During the second half of the period, the same amount of electricity passes through the cross-section of the conductor in the opposite direction. Consequently, the amount of electricity that has passed through the cross-section of the conductor during the period is zero, equal to zero, and the average value of the sinusoidal current over the period.

Therefore, the average value of the sinusoidal current is calculated over a half period during which the current remains positive. The average value of the current is equal to the ratio of the amount of electricity passed through the conductor cross-section in half a period to the duration of this half-period.

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.