What are called effective values ​​of current and voltage. Effective voltage, current

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 awkward to use instant or amplitude values currents and voltages, and their average values ​​over the 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

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

Current (effective) value of alternating current equal to the value of this direct current, which, in a time equal to one period of 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 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 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 current or voltage is measured using a heater, which is active resistance, is converted 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

• Curve shape factor AC voltage(current) - 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

Definition 1

Effective (effective) is the value of alternating current equal to the value of the equivalent direct current, which, when passing through the same resistance as alternating current, releases the same amount of heat over equal periods of time.

Quantitative relationship between the amplitudes of AC force and voltage and effective values

The amount of heat released by alternating current at resistance $R$ over a short period of time $dt$ is equal to:

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

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

Let us express from equation (3) the strength of the equivalent direct current, we obtain:

If the current varies according to a sinusoidal law:

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

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

where $I_(ef)$ is called the effective current value. Expressions for effective (effective) stress values ​​are written similarly:

Application of effective values ​​of current and voltage

When we talk about alternating current and voltage in electrical engineering, we mean their effective values. In particular, voltmeters and ammeters are usually calibrated to effective values. Hence, maximum value the voltage in the AC circuit is approximately 1.5 times greater than what the voltmeter shows. This fact should be taken into account when calculating insulators and studying safety problems.

Effective values used to characterize the shape of an alternating current (voltage) signal. Thus, the amplitude coefficient ($k_a$) is introduced. equal:

and shape factor ($k_f$):

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

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

Example 1

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

Solution:

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

Let's find the amplitude value of the voltage as:

Let's calculate:

Answer:$U_m\approx 310.2\ V.$

Example 2

Exercise: How is the alternating current power across resistance $R$ related to the effective values ​​of current and voltage?

Solution:

The average value of alternating current power in the circuit is

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

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

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

Therefore, we can write that:

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

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

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

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

Therefore, the current power can be written as:

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

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

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

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

Effective values ​​of current, voltage and e. d.s. are indicated by capital letters.

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

The effective value of the alternating current is equal to the value of the equivalent direct current, which, passing through the same resistance as the alternating current, releases the same amount of heat over a period.

The amount of heat released by alternating current in resistance in an infinitesimal period of time

and for the period of alternating current T

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

By reducing common multiplier, we obtain the effective current value

Rice. 5-8. Graph of alternating current and current squared.

In Fig. 5-8, a curve of instantaneous values ​​of current i and a curve of squared instantaneous values ​​are plotted. The area bounded by the last curve and the abscissa axis is, on a certain scale, a value determined by the expression The height of a rectangle equal to the area bounded by the curve and the abscissa axis, equal to the average value of the ordinates of the curve, is square of effective current value

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

Similarly for the effective values ​​of sinusoidal voltages and e. d.s. you can write:

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

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

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

In a mechanical system, forced vibrations occur when an external periodic force acts on it. Similarly, forced electromagnetic oscillations in an electrical circuit occur under the influence of an external periodically varying EMF or an externally varying voltage.

Forced electromagnetic oscillations in an electrical circuit are variable electricity .

• Alternating electric current is a current whose strength and direction change periodically.

In the future, we will study forced electrical oscillations that occur in circuits under the influence of voltage that varies harmoniously with frequency ω according to the sinusoidal or cosine law:

\(~u = U_m \cdot \sin \omega t\) or \(~u = U_m \cdot \cos \omega t\) ,

Where u– instantaneous voltage value, U m is the voltage amplitude, ω is the cyclic frequency of oscillations. If the voltage changes with a frequency ω, then the current in the circuit will change with the same frequency, but the current fluctuations do not necessarily have to be in phase with the voltage fluctuations. Therefore, in the general case

\(~i = I_m \cdot \sin (\omega t + \varphi_c)\) ,

where φ c is the phase difference (shift) between current and voltage fluctuations.

Based on this, we can give the following definition:

• Alternating current is an electric current that changes over time according to a harmonic law.

Alternating current ensures the operation of electric motors in machines in plants and factories, powers lighting fixtures in our apartments and outdoors, refrigerators and vacuum cleaners, heating appliances, etc. The frequency of voltage fluctuations in the network is 50 Hz. The alternating current has the same oscillation frequency. This means that within 1 s the current will change its direction 50 times. A frequency of 50 Hz is accepted for industrial current in many countries around the world. In the USA, the frequency of industrial current is 60 Hz.

Alternator

The bulk of the world's electricity is currently generated by alternating current generators, which create harmonic oscillations.

• Alternator called electrical device, designed to convert mechanical energy into alternating current energy.

The induction emf of the generator changes according to a sinusoidal law

\(e=(\rm E)_(m) \cdot \sin \omega \cdot t,\)

where \((\rm E)_(m) =B\cdot S\cdot \omega\) is the amplitude (maximum) value of the EMF. When connected to the terminals of the load frame with resistance R, alternating current will pass through it. According to Ohm's law for a section of a circuit, the current in the load

\(i=\dfrac(e)(R) =\dfrac(B \cdot S \cdot \omega )(R) \cdot \sin \omega \cdot t = I_(m) \cdot \sin \omega \cdot t,\)

where \(I_(m) = \dfrac(B\cdot S\cdot \omega )(R)\) is the amplitude value of the current.

The main parts of the generator are (Fig. 1):

• inductor- an electromagnet or permanent magnet that creates a magnetic field;
• anchor- winding in which an alternating EMF is induced;
• commutator with brushes- a device by means of which current is removed from rotating parts or supplied through them.

The stationary part of the generator is called stator, and movable - rotor. Depending on the design of the generator, its armature can be either a rotor or a stator. When receiving high-power alternating currents, the armature is usually made motionless in order to simplify the current transmission circuit to the industrial network.

In modern hydroelectric power plants, water rotates the shaft of an electric generator at a frequency of 1-2 revolutions per second. Thus, if the generator armature had only one frame (winding), then an alternating current with a frequency of 1-2 Hz would be obtained. Therefore, to obtain alternating current with an industrial frequency of 50 Hz, the armature must contain several windings that allow increasing the frequency of the generated current. For steam turbines, the rotor of which rotates very quickly, an armature with one winding is used. In this case, the rotor rotation frequency coincides with the alternating current frequency, i.e. the rotor should make 50 rps.

Powerful generators produce a voltage of 15-20 kV and have an efficiency of 97-98%.

From the history. Initially, Faraday detected only a barely noticeable current in the coil when a magnet moved near it. "What's the use of this?" - they asked him. Faraday replied: “What use can a newborn baby have?” A little more than half a century has passed and, as the American physicist R. Feynman said, “the useless newborn turned into a miracle hero and changed the face of the Earth in a way that his proud father could not even imagine.”

*Operating principle

The operating principle of an alternating current generator is based on the phenomenon of electromagnetic induction.

Let the conducting frame have an area S rotates with angular velocity ω around an axis located in its plane perpendicular to a uniform magnetic field with induction \(\vec(B)\) (see Fig. 1).

With uniform rotation of the frame, the angle α between the directions of the induction vector magnetic field\(\vec(B)\) and the normal to the plane of the frame \(\vec(n)\) changes with time according to a linear law. If at the moment t= 0 angle α 0 = 0 (see Fig. 1), then

\(\alpha = \omega \cdot t = 2\pi \cdot \nu \cdot t,\)

where ω - angular velocity rotation of the frame, ν is the frequency of its rotation.

In this case, the magnetic flux passing through the frame will change as follows

\(\Phi \left(t\right)=B\cdot S\cdot \cos \alpha =B\cdot S\cdot \cos \omega \cdot t.\)

Then, according to Faraday's law, an induced emf is induced

\(e=-\Phi "(t)=B\cdot S\cdot \omega \cdot \sin \omega \cdot t = (\rm E)_(m) \cdot \sin \omega \cdot t.\ )

We emphasize that the current in the circuit flows in one direction during half a turn of the frame, and then changes direction to the opposite, which also remains unchanged during the next half turn.

RMS values ​​of current and voltage

Let the current source create an alternating harmonic voltage

\(u=U_(m) \cdot \sin \omega \cdot t.\;\;\;(1)\)

According to Ohm's law, the current in a section of a circuit containing only a resistor with a resistance R, connected to this source, also changes with time according to a sinusoidal law:

\(i = \dfrac(u)(R) =\dfrac(U_(m) )(R) \cdot \sin \omega \cdot t = I_(m) \cdot \sin \omega \cdot t,\; \;\; (2)\)

where \(I_m = \dfrac(U_(m))(R).\) As we see, the current strength in such a circuit also changes over time according to a sinusoidal law. Quantities Um, I m are called amplitude values ​​of voltage and current. Time-dependent voltage values u and current strength i called instant.

In addition to these quantities, one more characteristic of alternating current is used: current (effective) values ​​of current and voltage.

• Current (effective) force value alternating current is the strength of such a direct current, which, passing through a circuit, releases the same amount of heat per unit time as a given alternating current.

Denoted by the letter I.

• Current (effective) voltage value alternating current is the voltage of such direct current, which, passing through the circuit, releases the same amount of heat per unit time as the given alternating current.

Denoted by the letter U.

Active ( I, U) and amplitude ( I m, U m) values ​​are related to each other by the following relationships:

\(I = \dfrac(I_(m) )(\sqrt(2)), \; \; \; U =\dfrac(U_(m) )(\sqrt(2)).\)

Thus, the expressions for calculating the power consumed in direct current circuits remain valid for alternating current if we use the effective values ​​of current and voltage in them:

\(P = U\cdot I = I^(2) \cdot R = \dfrac(U^(2))(R).\)

It should be noted that Ohm's law for an alternating current circuit containing only a resistor with a resistance R, is carried out both for amplitude and effective, and for instantaneous values ​​of voltage and current, due to the fact that their oscillations coincide in phase.