How are the effective values of current and voltage determined? Active resistance

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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? The effects of current are determined neither by amplitude nor instantaneous values. To evaluate the effect produced by alternating current, we compare its effect with the thermal effect of direct current.

Power P direct current I, passing through resistance r, will

P = I 2× r .

Power alternating current will be expressed as the average instantaneous power effect i 2× r for the whole period or the average value from ( I m× sin ω t) 2 × r for the same time.

Let the average i 2 per period will be M. Equating DC power and AC power, we have:

I 2× r = M × r ,

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

Average value i 2 with alternating sinusoidal current will be determined as follows. Let's build a sinusoidal curve of current change (Figure 1).

Picture 1. Effective value sinusoidal current

By squaring each instantaneous current value, we obtain the dependence curve i 2 from 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 build a rectangle with a base T and an area equal to the area bounded by the curve i 2 and horizontal axis. Rectangle height M will correspond to the average value i 2 per period. This value for the period, calculated using higher mathematics, will be equal to .

Hence,

Since the effective value of the alternating current I equals , then the formula will finally take the form

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

Effective values variables, that is, the effective value of voltage, current and electromotive force, are indicated by capital letters without subscripts ( U, I, E).

Based on the above, we can say that the effective value of the alternating current is equal to this DC, which, passing through the same resistance as 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 value of current and 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 a factor. This will not change the location of the vectors on the diagram.

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 of different shapes with each other according to the amount of work, they 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. The effective value of the alternating sinusoidal current is 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 is and effective voltage, 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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Current (effective) value of alternating current is equal to the magnitude of such a direct current, which, in a time equal to one period of the alternating current, will produce the same work (thermal or electrodynamic effect) as the alternating current in question.

In modern literature, the mathematical definition of this quantity is more often used - the root mean square value of alternating current.

In other words, the effective value of alternating current can be determined by the formula:

I = 1 T ∫ 0 T i 2 d t . (\displaystyle I=(\sqrt ((\frac (1)(T))\int _(0)^(T)i^(2)dt)).)

For sinusoidal current:

I = 1 2 ⋅ I m ≈ 0.707 ⋅ I m , (\displaystyle I=(\frac (1)(\sqrt (2)))\cdot I_(m)\approx 0(,)707\cdot I_(m ),)

I m (\displaystyle I_(m)) - amplitude current value.

For triangular and sawtooth current:

I = 1 3 ⋅ I m ≈ 0.577 ⋅ I m . (\displaystyle I=(\frac (1)(\sqrt (3)))\cdot I_(m)\approx 0(,)577\cdot I_(m).)

The effective values of EMF and voltage are determined in a similar way.

additional information

In English-language technical literature, the term is used to denote the effective value effective value- effective value. The abbreviation is also used RMS (rms) - root mean square- root mean square (value).

In electrical engineering, devices of electromagnetic, electrodynamic and thermal systems are calibrated 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

Instantaneous, maximum, effective and average values of electrical quantities of alternating current

Instantaneous and maximum values. The magnitude of variable electromotive force, current, voltage and power at any time is called instantaneous values these quantities are designated accordingly lowercase letters (e, i, u, p).
Maximum value(amplitude) variable e. d.s. (or voltage or current) is called the greatest value that it reaches in one period. The maximum value of electromotive force is indicated E m, voltage - U m, current - I m.

Valid (or effective) The value of alternating current is that amount of direct current that, flowing through equal resistance and in the same time as alternating current, releases the same amount of heat.

For sinusoidal alternating current, the effective value is 1.41 times less than the maximum, i.e., times.

Similarly, the effective values of alternating electromotive force and voltage are also 1.41 times less than their maximum values.

From the measured effective values of alternating current, voltage or electromotive force, their maximum values can be calculated:

E m = E· 1.41; U m = U· 1.41; I m = I· 1.41;

Average value= the ratio of the amount of electrical energy passing through the cross-section of a conductor in half a period to the value of this half-cycle.

The average value is understood as the arithmetic mean of its value for half a period.

/ Average and effective values of sinusoidal currents and voltages

The average value of a sinusoidally varying quantity is understood as its average value over half a period. Average current

i.e., the average value of the sinusoidal current is equal to the amplitude one. Likewise,

The concept of effective value of a sinusoidally varying quantity is widely used (it is also called effective or root mean square). RMS current value

Consequently, the effective value of the sinusoidal current is equal to 0.707 of the amplitude current. Likewise,

It is possible to compare the thermal effect of a sinusoidal current with the thermal effect of a direct current flowing at the same time through the same resistance.

The amount of heat released in one period by a sinusoidal current is

The heat released during the same time by a direct current is equal. Let us equate them:

Thus, the effective value of the sinusoidal current is numerically equal to the value of such a direct current, which, in a time equal to the period of the sinusoidal current, releases the same amount of heat as the sinusoidal current.

To establish the equivalence of alternating current in terms of energy and power, the generality of calculation methods, as well as the reduction of computational work, currents vary continuously over time. EMF and voltage are replaced by equivalent time-invariant quantities. The effective or equivalent value is such a time-invariant current at which it is released in resistive element with active resistance r per period the same amount of energy as with a real sinusoidally varying current.

The energy per period released in a resistive element with a sinusoidal current is


		i 2r dt =		I m 2 sin2 ω t r dt..

With a current constant over time, the energy

W=I 2rT

Equating the right sides

I m

0,707I m .

Thus, the effective value of the current is √2 times less than the amplitude current.

The effective values of EMF and voltage are determined similarly:

E = E m / √2, U = U m / √2.

The effective value of the current is proportional to the force acting on the rotor of the AC motor, the moving part of the measuring device, etc. When talking about the values of voltage, EMF and current in AC circuits, they mean their effective values. The scales of AC measuring instruments are calibrated accordingly in effective values of current and voltage. For example, if the device shows 10 A, then this means that the current amplitude

I m = √2I= 1.41 10 = 14.1 A,

and instantaneous current value

i = I m sin (ω t+ ψ) = 14.1 sin (ω t + ψ).

When analyzing and calculating rectifier devices, the average values of current, EMF and voltage are used, which is understood as the arithmetic average value of the corresponding value for half a period (the average value for a period, as is known, is equal to zero):

T 2

2E T

E Wed =

E T sin ω t dt=

sin ω t dω t =

|cos ω t| π 0 =

0,637E T .

Similarly, you can find the average values of current and voltage:

I av = 2 I T /π; U Wed = 2U T /π .

The ratio of the effective value to the average value of any periodically changing quantity is called the curve shape coefficient. For sinusoidal current

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.

RMS value of alternating current- 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 alternating current, we compare its effect with the thermal effect of direct current.

The power P of direct current I passing through resistance r will be P = P2r.

AC power will be expressed as the average effect of instantaneous power I2r over the entire period or the average value of (Im x sinωt)2 x r over 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: I2r = Mr, whence I = √M,

The quantity 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.

RMS value of alternating current

Both halves of this curve lie above the horizontal axis, since the negative values of the current (-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 i2 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 1/2I2m. Therefore, M = 1/2I2m

Since the effective value of I alternating current is 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 actual values of variables are indicated in capital letters without subscripts (I, U, E).

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

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 location of the vectors on the diagram.

List of voltage and current parameters

Due to the fact that electrical signals are time-varying quantities, in electrical engineering and radio electronics they are used as necessary. different ways representations of voltage and electric current

AC voltage (current) values

Instantaneous value

The instantaneous value is the value of the signal at a certain point in time, the function of which is (u (t) , i (t) (\displaystyle u(t)~,\quad i(t))). Instantaneous values of a slowly changing signal can be determined using a low-inertia DC voltmeter, recorder or loop oscilloscope; for periodic fast processes, a cathode-ray or digital oscilloscope is used.

Amplitude value

Amplitude (peak) value, sometimes simply called “amplitude” - the largest instantaneous value of voltage or current over a period (without taking into account the sign):

U M = max (| u (t) |) , I M = max (| i (t) |) (\displaystyle U_(M)=\max(|u(t)|)~,\qquad I_(M)= \max(|i(t)|))

The peak voltage value is measured using a pulse voltmeter or oscilloscope.

RMS value

RMS value (obsolete current, effective) - the square root of the average value of the square of voltage or current.

U = 1 T ∫ 0 T u 2 (t) d t , I = 1 T ∫ 0 T i 2 (t) d t (\displaystyle U=(\sqrt ((\frac (1)(T))\int \limits _(0)^(T)u^(2)(t)dt))~,\qquad I=(\sqrt ((\frac (1)(T))\int \limits _(0)^(T )i^(2)(t)dt)))

RMS values are the most common, as they are most convenient for practical calculations, since in linear circuits with a purely resistive load, alternating current with effective values of I (\displaystyle I) and U (\displaystyle U) does the same work as direct current with the same current and voltage values. For example, an incandescent lamp or a boiler, connected to an alternating voltage network with an effective value of 220 V, operates (lights, heats) in exactly the same way as when connected to a source DC voltage with the same voltage value.

When not specifically stated, they usually mean the root mean square values of voltage or current.

The indicating devices of most AC voltmeters and ammeters are calibrated in rms values, with the exception of special instruments, however these ordinary instruments give correct readings for RMS values only with sine waveform. Devices with a thermal converter are not critical to the signal shape, in which the measured current or voltage is converted using a heater, which is an active resistance, into a further measured temperature, which characterizes the magnitude of the electrical signal. Also insensitive to waveform special devices, squaring the instantaneous value of the signal with subsequent averaging over time (with a quadratic detector) or an ADC, squaring the input signal, also with averaging over time. The square root of the output signal of such devices is precisely the root mean square value.

The square of the rms voltage, expressed in volts, is numerically equal to the average power dissipation in watts across a 1 ohm resistor.

Average value

Average value (offset) - constant component of voltage or current

U = 1 T ∫ 0 T u (t) d t , I = 1 T ∫ 0 T i (t) d t (\displaystyle U=(\frac (1)(T))\int \limits _(0)^( T)u(t)dt~,\qquad I=(\frac (1)(T))\int \limits _(0)^(T)i(t)dt)

Rarely used in electrical engineering, but relatively often used in radio engineering (bias current and bias voltage). Geometrically, this is the difference in areas under and above the time axis, divided by the period. For a sinusoidal signal, the offset is zero.

Average rectified value

Average rectified value - average value of the signal module

U = 1 T ∫ 0 T ∣ u (t) ∣ d t , I = 1 T ∫ 0 T ∣ i (t) ∣ d t (\displaystyle U=(\frac (1)(T))\int \limits _( 0)^(T)\mid u(t)\mid dt~,\qquad I=(\frac (1)(T))\int \limits _(0)^(T)\mid i(t)\ mid dt)

Rarely used in practice, most AC magnetoelectric meters (i.e., in which the current is rectified before measurement) actually measure this quantity, although their scale is calibrated according to the rms values for a sinusoidal waveform. If the signal differs noticeably from a sinusoidal one, the readings of the magnetoelectric system instruments have a systematic error. Unlike devices of the magnetoelectric system, devices of electromagnetic, electrodynamic and thermal measurement systems always respond to the effective value, regardless of the form of the electric current.

Geometrically, it is the sum of the areas bounded by the curve above and below the time axis during the measurement time. With a unipolar measured voltage, the average and average-rectified values are equal to each other.

Value conversion factors

The coefficient of the shape of the alternating voltage (current) curve is a value equal to the ratio of the effective value of the periodic voltage (current) to its average rectified value. For sinusoidal voltage (current) is equal to π / 2 2 ≈ 1.11 (\displaystyle (\frac ((\pi )/2)(\sqrt (2)))\approx 1.11) .

The amplitude coefficient of the alternating voltage (current) curve is a value equal to the ratio of the maximum absolute value of the voltage (current) over the period to the effective value of the periodic voltage (current). For sinusoidal voltage (current) is equal to 2 (\displaystyle (\sqrt (2))) .

DC parameters

Voltage (current) ripple range - a value equal to the difference between the largest and smallest values of the pulsating voltage (current) over a certain time interval

Voltage (current) ripple coefficient - a value equal to the ratio highest value the variable component of the pulsating voltage (current) to its constant component.
- Voltage (current) ripple coefficient based on the effective value - a value equal to the ratio of the effective value of the alternating component of the pulsating voltage (current) to its direct component
- Average voltage (current) ripple coefficient - a value equal to the ratio of the average value of the variable component of the pulsating voltage (current) to its constant component

Ripple parameters are determined using an oscilloscope, or using two voltmeters or ammeters (DC and AC)

Literature and documentation

Literature

Handbook of radio-electronic devices: In 2 volumes; Ed. D. P. Linde - M.: Energy, 1978
Shultz Yu. Electrical measuring equipment: 1000 concepts for practitioners: Handbook: Transl. with him. M.: Energoatomizdat, 1989

Regulatory and technical documentation

GOST 16465-70 Radio engineering measuring signals. Terms and Definitions
GOST 23875-88 Quality electrical energy. Terms and Definitions
GOST 13109-97 Electrical energy. Compatibility technical means. Standards for the quality of electrical energy in general-purpose power supply systems

What physical meaning does the effective value of voltage and current have?

Alexander Titov

The effective value of the AC current is the value of the DC current, the action of which will produce the same work (or thermal effect) as the action of alternating current during one period of its action. Let, for example, current pass through a resistor with resistance R = 1 Ohm. Then the amount of heat released in the resistor during the period is equal to the integral of (i(t)^2 * R * T). The figure shows graphs of the current strength and the square of the current strength, related to maximum value. Since R = 1, then the area under the second graph (yellow area) is the amount of heat. And the value of direct current, when flowing through the resistor, will release the same amount of heat, is the effective value of the current. It is not difficult to determine that the indicated area (determined through the integral) is equal to 1/2, i.e. the amount of heat is equal to Im^2 * R * T / 2. This means that if a constant current I flows through the resistor, then the amount of heat released will be is equal to I^2 * R * T. Equating these expressions and reducing by R*T, we obtain I^2 = Im/2, whence I = Im / root of 2. This is the effective value of the current.

The same is true for the effective value of voltage and emf.

Vitas latish

I can say it rudely
- tension - potential energy.... comb - hair.... tension = glow, sparkles, hair lifting... .
- current is work, action, force... heat, combustion, movement, burst of kinetic energy

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 AC circuits, as well as during electronic measurements, it is awkward to use instantaneous or amplitude values of currents and voltages, and their average values over a period are equal to 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

The actual values of variable quantities 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

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 during 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 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 actual values of variables are indicated in capital letters without subscripts (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.

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 location of the vectors on the diagram.