Amplitude value of current and voltage. Active resistance

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An alternating sinusoidal current has different second values during a period. It is natural to ask the question: what current value will be measured by an ammeter connected to the circuit?

When calculating circuits alternating current, also with electronic measurements It is awkward to use instantaneous or amplitude values of currents and voltages, and their average values over the period are zero. In addition, the electronic effect of a periodically changing current (the amount of heat released, the work done, etc.) cannot be judged by the amplitude of this current.

It turned out to be more comfortable to introduce the so-called concepts effective values of current and voltage. These concepts are based on the thermal (or mechanical) effect of current, independent of its direction.

- this is the value of constant current at which during the period of alternating current the same amount of heat is released in the conductor as with alternating current.

To evaluate the effect produced by alternating current, we compare its effect with the thermal effect of constant current.

The power P of a constant current I passing through resistance r will be P = P 2 r.

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

Let the average value of t2 for the period be M. Equating the power of a constant current and the power of an alternating current, we have: I 2 r = Mr, whence I = √ M,

Magnitude I is called the effective value of the alternating current.

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

Let's construct a sinusoidal curve of the current configuration. By squaring each second value of the current, we obtain a curve of P versus time.

Both halves of this curve lie above the horizontal axis, because negative current values (-i) in the 2nd half of the period, when squared, give positive values.

Let's construct a rectangle with a base T and an 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 for the period. This value for the period, calculated using higher arithmetic, will be equal to 1/2I 2 m. As follows, M = 1/2I 2 m

Because the effective value of I alternating current is equal to I = √ M, then absolutely I = Im / √ 2

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

U = Um / √ 2 , E= Em / √ 2

Effective values variables are indicated by lowercase characters without subscripts (I, U, E).

Based on the above, we can say that The effective value of an alternating current is equal to such a constant current, which, passing through the same resistance as the alternating current, releases the same amount of energy in the same time.

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

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

Electrician school

RMS values of current and voltage

As is known, variable emf. Induction causes alternating current in a circuit. At highest value e.m.f. the current will have a maximum value and vice versa. This phenomenon is called phase matching. Despite the fact that current values can fluctuate from zero to a certain maximum value, there are instruments with which you can measure the strength of alternating current.

The characteristic of alternating current can be actions that do not depend on the direction of the current and can be the same as with direct current. These actions include thermal action. For example, 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 the following force value direct current, so that on the same conductor during the same time the same amount of heat is generated by this current as with alternating current. This value of direct current is called the effective value of alternating current.

IN given time widespread in global industrial practice three phase alternating current, which has many advantages over single-phase current. A three-phase system is a system that has three electrical circuits with its variable e.m.f. with the same amplitudes and frequency, but shifted in 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 alternating current generators and 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 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 compared to the emf. the first generator, i.e. it will be delayed by 120°. E.m.f. the third generator will also be delayed in relation to the second by 120°.

However, this method of producing 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 begins to rotate, a

a) b)

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

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

changing e.m.f. induction. Due to the fact that the windings shift in space, the oscillation phases 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 windings of the generator and receivers need to be connected to each other, forming three-phase system. There are two types of connections: star and triangle. When using both methods, you can save electrical 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 beginnings of the windings are usually designated 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 “star”. In this case, wires called linear 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., right). In this case, the wire that connects the zero point of the generator and receivers is called zero. This system three-phase current has two different voltages: between the linear and neutral wires or, what is the same, between the beginning and end of any stator winding. This value is called phase voltage ( Ul). Since the circuit is three-phase, the line voltage will be v3 times more than phase, i.e.: Ul = v3Uф.

An alternating sinusoidal current has different instantaneous values during a period. It is natural to ask the question: what current value will be measured by an ammeter connected to the circuit?

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

It turned out to be most convenient to introduce the concepts of the so-called effective values of current and voltage. These concepts are based on the thermal (or mechanical) effect of current, independent of its direction.

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

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

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

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

Let the average value of t2 for the period be M. Equating the power of direct current and power with alternating current, we have: I 2 r = Mr, whence I = √ M,

Magnitude I is called the effective value of the alternating current.

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

Let's construct a sinusoidal curve of current change. By squaring each instantaneous current value, we obtain 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 a base T and an 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 for the period. This value for the period, calculated using higher mathematics, will be equal to 1/2I 2 m. Therefore, M = 1/2I 2 m

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

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

U = Um / √ 2 E= Em / √ 2

The effective values of the variables are indicated in capital letters without indices (I, U, E).

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

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

We talked about power and AC operation. Let me remind you that then we calculated it through some integral, and at the very end of the article I casually said that there are ways to make an already difficult life easier and often you can do without taking the integral at all, if you know about effective current value. Today we’ll talk about him!

Gentlemen, it will probably not be a secret to you that in nature there is big number types of alternating current: sinusoidal, rectangular, triangular and so on. And how can they even be compared to each other? In form? Hmm...I guess so. They are visually different, you can’t argue with that. By frequency? Yes, too, but sometimes it raises questions. Some people believe that the definition of frequency itself is only applicable to a sinusoidal signal and cannot be used, for example, for a sequence of pulses. Perhaps formally they are right, but I do not share their point of view. How else is it possible? And, for example, in terms of money! Suddenly? In vain. Current costs money. Or rather, it costs money to operate the current. In the end, those same kilowatt hours for which you all pay every month on the meter are nothing more than the work of current. And since money is a serious thing, it’s worth introducing a separate term for this. And to compare currents with each other various shapes by the amount of work introduced the concept effective current.

So, the effective (or root mean square) value of the alternating current is the amount of some direct current that, in a time equal to the period of the alternating current, will generate the same amount of heat on the resistor as our alternating current. It sounds very tricky and, most likely, if you are reading this definition for the first time, you are unlikely to understand it. This is fine. When I heard it for the first time at school, it took me a long time to figure out what it meant. Therefore, now I will try to analyze this definition in more detail so that you understand what is hidden behind this tricky phrase faster than I did in my time.

So we have alternating current. Let's say sinusoidal. It has its own amplitude A m and period T period(or frequency f). Per phase in in this case don’t care, we consider it equal to zero. This alternating current flows through some resistor R and this resistor releases energy. For one period T period Our sinusoidal current will release a very certain amount of joules of energy. We can accurately calculate this number of joules using the integral formulas that I cited last time. Let's say we calculated that in one period T the period of the sinusoidal current will be highlighted Q joules of heat. And now, attention, gentlemen, important point! Let's replace alternating current with direct current, and choose it of such a value (well, that is, so many amperes) that on the same resistor R for the same timeT period exactly the same number of joules was releasedQ. Obviously, we must somehow determine the magnitude of this direct current, which is equivalent to alternating current from an energy point of view. And when we find this value, it will be exactly the same effective value of alternating current. And now, gentlemen, return once again to that sophisticated formal definition that I gave at the beginning. It's better understood now, isn't it?

So, the essence of the question, I hope, has become clear, so let’s translate everything said above into the language of mathematics. As we already wrote in the previous article, the law of change in alternating current power is equal to

The amount of energy released during current operation over time T period- accordingly, equal to the integral over the period T period:

Gentlemen, now we need to take this integral. If, due to your dislike of mathematics, this seems too complicated to you, you can skip the calculations and see the result right away. And today I’m in the mood to remember my youth and carefully deal with all these integrals.

So how should we take it? Well, the quantities I m 2 and R are constants and can be immediately taken out of the integral sign. And for the square of sine we need to apply the formula reduction in degree from a trigonometry course. I hope you remember her. And if not, then let me remind you again:

Now let's split the integral into two integrals. You can use the fact that the integral of a sum or difference is equal to the sum or difference of integrals. In principle, this is very logical if you remember that the integral is an area.

So we have

Gentlemen, I have simply excellent news for you. The second integral is equal to zero!

Why is this so? Yes, simply because the integral of any sine/cosine at a value that is a multiple of its period is equal to zero. A most useful property, by the way! I recommend you remember it. Geometrically, this is also clear: the first half-wave of the sine goes above the x-axis and the integral from it Above zero, and the second half-wave goes below the x-axis, so its value is less than zero. And in modulus they are equal to each other, so their addition (in fact, the integral over the entire period) will result in a zero.

So, discarding the cosine integral, we get

Well, you don’t have to be a big math guru to say that this integral is equal to

And thus we get the answer

This is how we got the number of joules that will be released on the resistorRwhen a sinusoidal current with amplitude flows through itI mduring the periodT period. Now, to find what in this case is equal to effective current we need to proceed from the fact that on the same resistorR for the same timeT period the same amount of energy will be releasedQ. Therefore we can write

If it’s not entirely clear where it came from left side, I recommend that you repeat the article about the Joule-Lenz law. Meanwhile, we will express the effective value of the currentI action. from this expression, having previously reduced everything that is possible

This is the result, gentlemen. Effective value alternating sinusoidal current to the root of two times less than its amplitude value. Remember this result well, it is an important conclusion.

Generally speaking, no one bothers, by analogy with current, to introduce effective voltage value. In this case, our dependence of power on time will take the following form:

It is this that we will substitute for the integral and perform all the transformations. Gentlemen, each of you can do this at your leisure if you wish, but I’ll just give final result, since it is completely similar to the case with current. So, the effective value of the sinusoidal current voltage is equal to

As you can see, the analogy is complete. The effective voltage value is also exactly two times less than the amplitude.

In a similar way, you can calculate the effective value of current and voltage for a signal of absolutely any shape: you just need to write down the law of power change for this signal and perform all the above-described transformations step by step.

All of you have probably heard that our sockets have a voltage of 220 V. What volts? After all, we now have two terms - amplitude and effective value. So it turns out that 220 V in sockets is the current value! Voltmeters and ammeters connected to alternating current circuits show exactly the actual values. And the shape of the signal in general and its amplitude in particular can be viewed using an oscilloscope. Well, we have already said that everyone is interested in money, that is, the work of current, and not some incomprehensible amplitude. Nevertheless, let’s still determine what the voltage amplitude in our networks is equal to. Using the formula we just wrote, we can write

From here we get

That's it, gentlemen. In our sockets, it turns out, we have a sine wave with an amplitude of as much as 311 V, and not 220, as one might think at first. To remove all doubts, I will present you with a picture of what the law of voltage changes in our sockets looks like (remember that the network frequency is 50 Hz or, which is the same, the period is 20 ms). This law is presented in Figure 1.

Figure 1 - Law of voltage changes in sockets

And especially for you, gentlemen, I looked voltage in the outlet using an oscilloscope. I watched it through voltage divider 1:5. That is, the signal shape will be completely preserved, and the signal amplitude on the oscilloscope screen will be five times less than what is actually in the socket. Why did I do this? Yes, simply because, due to the large input voltage swing, the entire image does not fit on the oscilloscope screen.

ATTENTION! If you do not have sufficient experience working with high voltage, if you do not have an absolutely clear idea of how currents can flow during measurements in circuits that are not galvanically isolated from the network, I strongly do not recommend carrying out such an experiment yourself, it is dangerous! The fact is that with such measurements using oscilloscope connected to a grounded outlet there is a very high chance that this will happen short circuit through the internal grounds of the oscilloscope and the device will burn out without the possibility of recovery! And if you make these measurements using oscilloscope connected to an ungrounded outlet, its housing, cables and connectors may contain lethal potential! This is not a joke, gentlemen, if you don’t understand why this is so, it’s better not to do it, especially since the oscillograms have already been taken and you can see them in Figure 2.

Figure 2 - Voltage oscillogram in the socket (divider 1:5)

In Figure 2 we see that the amplitude of the sine wave is about 62 volts and the frequency is exactly 50 Hz. Remembering that we are looking through a voltage divider, which divides the input voltage by 5, we can calculate the actual voltage value in the outlet, it is equal to

As we can see, the measurement result is very close to the theoretical one, despite the measurement error of the oscilloscope and the imperfection of the voltage divider resistors. This indicates that all our calculations are correct.

That's all for today, gentlemen. Today we learned what effective current and effective voltage are, learned how to calculate them and checked the calculation results in practice. Thanks for reading this and see you for more articles!

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additional information

In English technical literature to denote the actual meaning the term “ effective value" - literally translated " effective value»

In electrical engineering, devices of electromagnetic, electrodynamic and thermal systems respond to the effective value.

Sources

“Handbook of Physics”, Yavorsky B. M., Detlaf A. A., ed. "Science", 19791
Physics course. A. A. Detlaf, B. M. Yavorsky M.: Higher. school, 1989. § 28.3, paragraph 5
“Theoretical foundations of electrical engineering”, L. A. Bessonov: Higher. school, 1996. § 7.8 - § 7.10

Amplitude value of current and voltage. Active resistance

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