AC for a long time did not find practical application. This was due to the fact that the first generators of electrical energy produced direct current, which completely satisfied the technological processes of electrochemistry, and the DC motors had good control characteristics. However, as production developed, direct current increasingly less began to meet the increasing demands of an economical power supply. Alternating current made it possible to efficiently crush electrical energy and change voltage with the help of transformers. It has become possible to produce electricity at large power plants with its subsequent economical distribution to consumers, and the radius of power supply has increased.

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

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

The range of frequencies used in technology: from ultralow frequencies (0.01¸10 Hz - in automatic control systems, in analog computing) - to ultrahigh (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 value of power.

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

Current amplitude;

Voltage amplitude;

The amplitude of the EMF.

Ac current value

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

Similarly, the effective values ​​of emf and voltage are determined.

Sinusoidal current

Of all the possible forms of periodic currents, the most common is sinusoidal current. Compared to other types of current, sinusoidal current has the advantage of allowing, in the general case, the most economical production, transmission, distribution and use of electrical energy. Only with the use of a sinusoidal current, it is possible to maintain unchanged the shape of the voltage and current curves in all parts of a complex linear circuit. The theory of sinusoidal current is the key to understanding the theory of other circuits.

The image of sinusoidal emf, voltage and current on the plane of Cartesian coordinates

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

Given in Fig. 1, 2 graphs of two sinusoidal EMF e 1   and e 2    match the equations:

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

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

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

For sinusoidal emf e 1   and e 2   phase angle:

Vector image of sinusoidally varying values

On the Cartesian plane from the origin, vectors are taken that are equal in magnitude to the amplitude values ​​of sinusoidal values, and rotate these vectors counterclockwise ( in TOE this direction is taken as positive) with an angular frequency equal to w. The phase angle during rotation is counted from the positive semi-axis of the abscissa. The projections of the rotating vectors on the y-axis are equal to the instantaneous values ​​of the emf e 1   and e 2    (Fig. 3). The set of vectors representing sinusoidally varying emf, voltages and currents is called vector diagrams.When constructing vector diagrams, it is convenient to arrange vectors for 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 system of Cartesian coordinates itself rotates counterclockwise with speed w. Thus, in this coordinate system, the vectors are fixed (Fig. 4). Vector diagrams are widely used in the analysis of sinusoidal current circuits. Their use makes the calculation of the circuit more intuitive and simple. This simplification lies in the fact that the addition and subtraction of the instantaneous values ​​of the quantities can be replaced by the addition and subtraction of the corresponding vectors.

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

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

The resulting current will also be sinusoidal:

Determining the amplitude and initial phase of this current by means of the corresponding trigonometric transformations is quite cumbersome and not very visual, especially if a large number of sinusoidal quantities are summed up. Much easier this is done using a vector diagram. In fig. 6 shows the initial positions of the current vectors, the projections of which on the y-axis give instantaneous currents for t=0. By rotating these vectors with the same angular velocity wtheir relative position does not change, and the phase angle between them remains equal.

Since the algebraic sum of the projections of the vectors on the y-axis is equal to the instantaneous value of the total current, the total current vector is equal to the geometric sum of the current vectors:

.

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

The physical meaning of these concepts is approximately the same as the physical meaning of the average velocity or other quantities averaged over time. At different points in time, the AC power and its voltage take on different values, so it’s generally possible to talk about the AC power only conditionally.

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

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

Effective values ​​of periodic and sinusoidal currents

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

The alternating sinusoidal current during the period has various instantaneous values. 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 in 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 electric effect of a periodically varying current (the amount of heat released, the perfect work, etc.) cannot be judged by the amplitude of this current.

The introduction of the concepts of the so-called current values ​​of current and voltage. These concepts are based on the thermal (or mechanical) action of the current, independent of its direction.

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

To evaluate the action of an alternating current, we compare its actions with the thermal effect of a direct current.

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

The AC power is expressed as the average instantaneous power effect I 2 r over the whole period or the average value from (Im x sinωt) 2 x r over the same time.

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

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

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

Construct a sinusoidal curve of the current. Having squared each instantaneous value of the current, we obtain the curve of dependence of P on time.

Ac current value

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

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 with the help of higher mathematics, will be equal to 1 / 2I 2 m. Therefore, M = 1 / 2I 2 m

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

Similarly, the relationship between the current and amplitude values ​​for the voltage U and E is:

U = Um / √2, E = Em / √2

Valid values ​​of variables are indicated by capital letters without indices (I, U, E).

Based on the above, it can be said that the effective value of the alternating current is equal to such a direct current, which, passing through the same resistance as the alternating current, releases the same amount of energy during the same time.

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

When constructing vector diagrams, it is more convenient to postpone not the amplitude, but the effective values ​​of the vectors. For this length of the vectors is reduced by √2 times. From this the location of the vectors in the diagram does not change.