RMS values ​​of current and voltage

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

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 technical literature the term is used to denote the actual meaning 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

Links

• RMS values ​​of current and voltage
• RMS value

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

Instantaneous and maximum values. The value of the variable electromotive force, current, voltage and power at any time are 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 a 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			2E T		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 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 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 a network with an alternating voltage with an effective value of 220 V, operates (lights, heats) in exactly the same way as when connected to a direct voltage source 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 lowest values 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

Links

• DC electrical circuits
• Alternating current. Picture of sinusoidal variables
• Amplitude, average, effective
• Periodic non-sinusoidal EMF, currents and voltages in electrical circuits
• Current systems and rated voltages of electrical installations
• Electricity
• Problems of higher harmonics in modern systems power supply

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 the 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

Alternating current has not found practical use for a long time. This was due to the fact that the first electrical energy generators produced direct current, which fully satisfied the technological processes of electrochemistry, and direct current motors have good control characteristics. However, as production developed, direct current became less and less suitable for the increasing requirements for economical power supply. Alternating current made it possible to effectively split electrical energy and change the voltage using transformers. It became possible to produce electricity at large power plants with its subsequent economical distribution to consumers, and the radius of power supply increased.

Currently, the central production and distribution of electrical energy is carried out mainly on alternating current. Circuits with changing - alternating - currents have a number of features compared to direct current circuits. Alternating currents and voltages cause alternating electrical and magnetic fields. As a result of changes in these fields in circuits, the phenomena of self-induction and mutual induction arise, which have the most significant impact on the processes occurring in the circuits, complicating their analysis.

Alternating current (voltage, emf, etc.) is a current (voltage, emf, etc.) that varies over time. Currents whose values ​​are repeated at regular intervals in the same sequence are called periodic, and the shortest period of time through which these repetitions are observed is period T. For periodic current we have

Frequency range used in technology: from ultra-low frequencies (0.01¸10 Hz – in automatic control systems, in analog computer technology) – up to ultra-high (3000 ¸ 300000 MHz – millimeter waves: radar, radio astronomy). In the Russian Federation, industrial frequency f= 50Hz.

The instantaneous value of a variable is a function of time. It is usually denoted by a lowercase letter:

i- instantaneous current value;

u – instantaneous voltage value;

e - instantaneous value of EMF;

R- instantaneous power value.

The largest instantaneous value of a variable over a period is called amplitude (it is usually denoted capital letter with index m).

Current amplitude;

Voltage amplitude;

EMF amplitude.

RMS value of alternating current

The value of a periodic current equal to the value of direct current, which during one period will produce the same thermal or electrodynamic effect as the periodic current, is called effective value periodic current:

The effective values ​​of EMF and voltage are determined similarly.

Sinusoidally varying current

Of all the possible forms of periodic currents, the sinusoidal current is most widespread. Compared to other types of current, sinusoidal current has the advantage that it allows, in general, the most economical production, transmission, distribution and use of electrical energy. Only when using sinusoidal current is it possible to keep the shapes of voltage and current curves unchanged in all sections of a complex linear circuit. The theory of sinusoidal current is the key to understanding the theory of other circuits.

Image of sinusoidal emfs, voltages and currents on the Cartesian coordinate plane

Sinusoidal currents and voltages can be represented graphically and written using equations with trigonometric functions, represent them as vectors on the Cartesian plane or as complex numbers.

Shown in Fig. 1, 2 graphs of two sinusoidal EMFs e 1 And e 2 correspond to the equations:

The values ​​of the arguments of sinusoidal functions are called phases sinusoid, and the phase value at the initial time (t=0): And - initial phase( ).

The quantity characterizing the rate of change of the phase angle is called angular frequency. Since the phase angle of a sinusoid during one period T changes by rad., then the angular frequency is , Where f– frequency.

When considering two sinusoidal quantities of the same frequency together, the difference in their phase angles, equal to the difference in the initial phases, is called phase angle.

For sinusoidal EMF e 1 And e 2 phase angle:

Vector image of sinusoidally varying quantities

On the Cartesian plane, from the origin of coordinates, draw vectors equal in magnitude to the amplitude values ​​of sinusoidal quantities, and rotate these vectors counterclockwise ( in TOE this direction is taken as positive) with angular frequency equal to w. The phase angle during rotation is measured from the positive semi-axis of abscissa. Projections of rotating vectors onto the ordinate axis are equal to the instantaneous values ​​of the emf e 1 And e 2 (Fig. 3). A set of vectors representing sinusoidally varying emfs, voltages and currents is called vector diagrams. When constructing vector diagrams, it is convenient to place the vectors at the initial moment of time (t=0), which follows from the equality of the angular frequencies of sinusoidal quantities and is equivalent to the fact that the Cartesian coordinate system itself rotates counterclockwise at a speed w. Thus, in this coordinate system the vectors are stationary (Fig. 4). Vector diagrams have found wide application in the analysis of sinusoidal current circuits. Their use makes circuit calculations more clear and simple. This simplification lies in the fact that addition and subtraction of instantaneous values ​​of quantities can be replaced by addition and subtraction of the corresponding vectors.

Let, for example, at the branch point of the circuit (Fig. 5) the total current is equal to the sum of the currents of the two branches:

,

After substituting the current value i and subsequent transformations we find that the effective value of the alternating current is equal to:

Similar relationships can also be obtained for voltage and emf:

Most electrical measuring instruments measure not instantaneous, but effective values ​​of currents and voltages.

Considering, for example, that the effective voltage value in our network is 220V, we can determine the amplitude value of the voltage in the network: U m =UÖ2=311V. The relationship between the effective and amplitude values ​​of voltages and currents is important to take into account, for example, when designing devices using semiconductor elements.

RMS value of alternating current

Theory/ TOE/ Lecture No. 3. Representation of sinusoidal quantities using vectors and complex numbers.

I haven't found alternating current for a long time practical application. This was due to the fact that the first electrical energy generators produced direct current, which completely satisfied technological processes electrochemistry, and DC motors have good control characteristics. However, as production developed, direct current became less and less suitable for the increasing requirements for economical power supply. Alternating current made it possible to effectively split electrical energy and change the voltage using transformers. It became possible to produce electricity at large power plants with its subsequent economical distribution to consumers, and the radius of power supply increased.

Currently, the central production and distribution of electrical energy is carried out mainly on alternating current. Circuits with changing - alternating - currents have a number of features compared to direct current circuits. Alternating currents and voltages cause alternating electric and magnetic fields. As a result of changes in these fields in circuits, the phenomena of self-induction and mutual induction arise, which have the most significant impact on the processes occurring in the circuits, complicating their analysis.

Alternating current (voltage, emf, etc.) is a current (voltage, emf, etc.) that varies over time. Currents whose values ​​are repeated at regular intervals in the same sequence are called periodic, and the shortest period of time through which these repetitions are observed is period T. For periodic current we have

The range of frequencies used in technology: from ultra-low frequencies (0.01-10 Hz - in automatic control systems, in analog computer technology) - to ultra-high frequencies (3000 ¸ 300000 MHz - millimeter waves: radar, radio astronomy). In the Russian Federation, industrial frequency f= 50Hz.

The instantaneous value of a variable is a function of time. It is usually denoted by a lowercase letter:

i- instantaneous current value;

u– instantaneous voltage value;

e- instantaneous value of EMF;

R- instantaneous power value.

The largest instantaneous value of a variable over a period is called amplitude (it is usually denoted by a capital letter with a subscript m).

Current amplitude;

Voltage amplitude;

EMF amplitude.

The value of a periodic current equal to the value of direct current, which during one period will produce the same thermal or electrodynamic effect as the periodic current, is called effective value periodic current:

,

The effective values ​​of EMF and voltage are determined similarly.

Sinusoidally varying current

Of all the possible forms of periodic currents, the sinusoidal current is most widespread. Compared to other types of current, sinusoidal current has the advantage that it allows, in general, the most economical production, transmission, distribution and use of electrical energy. Only when using sinusoidal current is it possible to keep the shapes of voltage and current curves unchanged in all sections of a complex linear circuit. The theory of sinusoidal current is the key to understanding the theory of other circuits.

Image of sinusoidal emfs, voltages and currents on the Cartesian coordinate plane

Sinusoidal currents and voltages can be represented graphically, written using equations with trigonometric functions, represented as vectors on a Cartesian plane or complex numbers.

Shown in Fig. 1, 2 graphs of two sinusoidal EMFs e 1 And e 2 correspond to the equations:

The values ​​of the arguments of sinusoidal functions are called phases sinusoid, and the phase value at the initial time (t=0): And - initial phase ( ).

The quantity characterizing the rate of change of the phase angle is called angular frequency. Since the phase angle of a sinusoid during one period T changes by rad., then the angular frequency is , Where f– frequency.

When considering two sinusoidal quantities of the same frequency together, the difference in their phase angles, equal to the difference in the initial phases, is called phase angle.

For sinusoidal EMF e 1 And e 2 phase angle:

Vector image of sinusoidally varying quantities

On the Cartesian plane, from the origin of coordinates, draw vectors equal in magnitude to the amplitude values ​​of sinusoidal quantities, and rotate these vectors counterclockwise ( in TOE this direction is taken as positive) with angular frequency equal to w. The phase angle during rotation is measured from the positive semi-axis of abscissa. Projections of rotating vectors onto the ordinate axis are equal to the instantaneous values ​​of the emf e 1 And e 2 (Fig. 3). A set of vectors representing sinusoidally varying emfs, voltages and currents is called vector diagrams. When constructing vector diagrams, it is convenient to place the vectors at the initial moment of time (t=0), which follows from the equality of the angular frequencies of sinusoidal quantities and is equivalent to the fact that the Cartesian coordinate system itself rotates counterclockwise at a speed w. Thus, in this coordinate system the vectors are stationary (Fig. 4). Vector diagrams have found wide application in the analysis of sinusoidal current circuits. Their use makes circuit calculations more clear and simple. This simplification lies in the fact that addition and subtraction of instantaneous values ​​of quantities can be replaced by addition and subtraction of the corresponding vectors.

Let, for example, at the branch point of the circuit (Fig. 5) the total current is equal to the sum of the currents and two branches:

Each of these currents is sinusoidal and can be represented by the equation

The resulting current will also be sinusoidal:

Determining the amplitude and initial phase of this current by means of appropriate trigonometric transformations turns out to be quite cumbersome and not very visual, especially if it is summed big number sinusoidal quantities. This is much easier to do using a vector diagram. In Fig. Figure 6 shows the initial positions of the current vectors, the projections of which onto the ordinate axis give instantaneous current values ​​for t=0. When rotating these vectors with the same angular velocity w their mutual arrangement does not change, and the phase shift angle between them remains equal.

Since the algebraic sum of projections of vectors onto the ordinate axis is equal to the instantaneous value total current, the total current vector is equal to the geometric sum of the current vectors:

.

Plotting a vector diagram to scale allows you to determine the values ​​of and from the diagram, after which a solution for the instantaneous value can be written by formally taking into account the angular frequency: .

RMS and average values ​​of alternating current and voltage.

Mean or arithmetic mean Fcp arbitrary function of time f(t)for a time interval T determined by the formula:

Numerically average value Fav equal to the height of a rectangle equal in area to the figure bounded by the curve f(t), axis t and limits of integration 0 – T(Fig. 35).

For a sinusoidal function, the average value over a full period T(or for an integer number of full periods) is equal to zero, since the areas of the positive and negative half-waves of this function are equal. For alternating sinusoidal voltage, the average absolute value for the full period is determined T or the average value for half the period ( T/2) between two zero values ​​(Fig. 36):

Ucp = Um∙ sin wt dt = 2R. Thus, the quantitative parameters of electrical energy on alternating current (amount of energy, power) are determined by the effective voltage values U and current I. For this reason, in the electric power industry, all theoretical calculations and experimental measurements are usually performed for effective values ​​of currents and voltages. In radio engineering and communications technology, on the contrary, they operate with the maximum values ​​of these functions.

The above formulas for energy and power of alternating current completely coincide with similar formulas for direct current. On this basis, it can be argued that the effective value of alternating current is energetically equivalent to direct current.

What is taken as the effective value of alternating current and alternating voltage

what is taken as the effective value of alternating current and alternating voltage?

Battle Egg

Alternating current, in a broad sense electricity, changing over time. Typically in technology, current flow is understood as a periodic current in which the average value over a period of current and voltage is zero.

Alternating currents and alternating voltages constantly change in magnitude. At every other moment they have a different magnitude. The question arises, how to measure them? To measure them, the concept of effective value has been introduced.

The effective or effective value of an alternating current is the value of a direct current that is equivalent in its thermal effect to a given alternating current.

The effective or effective value of an alternating voltage is the value of such a direct voltage, which in its thermal effect is equivalent to a given alternating voltage.

All alternating currents and voltages in technology are measured in effective values. Measuring devices variables show their actual meaning.

Question: the mains voltage is 220 V, what does this mean?

This means that a 220 V DC source has the same thermal effect as the mains.

The effective value of a sinusoidal current or voltage is 1.41 times less than the amplitude of this current or voltage.

Example: Determine the voltage amplitude of an electrical network with a voltage of 220 V.

The amplitude is 220 * 1.41 = 310.2 V.

Values effective voltage and current strength. Definition. Relationship with amplitude for different shapes. (10+)

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

When we talk about variable voltage or current strength, especially of complex shape, then the question arises of how to measure them. After all, the tension is constantly changing. You can measure the amplitude of the signal, 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 the purpose for which the measurement is being made. Since in most cases the interest is in the power that a particular signal can produce, the effective (effective) value is used.

Here is a selection of materials: RMS value for standard waveforms Sine wave (sine, sinusoid) [Effective value] = [Amplitude value ] / [Square root of 2] Square wave (square wave) [Effective value] = [Amplitude value] Triangle signal [Effective value] = [Amplitude value] / [Square root of 3] Ohm's law and power for effective values ​​of voltage and current The effective value of voltage is measured in Volts, and current in Amperes. For effective values Ohm's law is true: = / [ Load resistance, Ohm] [Power dissipated by ohmic load, W] = [Effective current value, A] * [Effective voltage value, V] Unfortunately, errors are periodically found in articles; they are corrected, articles are supplemented, developed, and new ones are prepared. Subscribe to the news to stay informed. If something is unclear, be sure to ask! Ask a Question. Discussion of the article. More articles Microcontrollers - an example of the simplest circuit, a sample application. Fuzy (... Your very first circuit on a microcontroller. A simple example. What is fuzy?... Practice of electronic circuit design. Electronics tutorial.... The art of device development. Element base radio electronics. Typical schemes.... Power powerful pulse transformer, choke. Winding. Make... Techniques for winding a pulse inductor / transformer.... Power resonant filter for obtaining a sine wave from an inverter... To obtain a sine wave from the inverter, we used a homemade power resonator... Do-it-yourself uninterruptible power supply. Do it yourself UPS, UPS. Sine, sinusoid... How to make an uninterruptible power supply yourself? Pure sinusoidal output voltage, with... Operating principle, independent production and adjustment of pulsed power transducer... Single-phase to three-phase voltage converter. Operating principle,... Operating principle, assembly and adjustment of the converter single phase voltage in three... Electrical voltage. Signal amplitude. Amplitude. Volt. Volt.... The concept of voltage and electrical potential difference. Amplitude. Units of measure...