Sinusoidal current


Features of the sinusoidal current

The most desirable form of the curve for instantaneous values of alternating current and tension is the sinusoidal form.

Схема синусоидального тока

Scheme of the sinusoidal current.

In mathematics sinusoidal changes are considered as the simplest harmonious form of the batch process therefore calculation of chains of the sinusoidal current is rather simple and in such chains there are no undesirable by-effects.

Построение синусоидальной кривой при помощи вращающего вектора

Figure 1. Creation of the sinusoidal curve by means of the rotating vector.

For creation of the sinusoidal curve we will take some piece of OA (fig. 1) which length on the scale of creation is equal to the maximum value of sinusoidal size, is the vector of sinusoidal size.

For example, Im = OA x n = 10 and, scale n = 0,1 and/mm; OA = 10: 0,1 = 100 mm. In the rectangular coordinate system we will direct this vector at first on the horizontal axis — it will be the starting position of the vector at the time of time reference mark, i.e. at t = 0.

The vector rotates with constant angular speed counterclockwise. During the T period the vector turns on 2? the radian (is glad). Therefore, its angular speed

As in expression? alternating-current frequency enters, that the angular speed of the vector is usually called angular frequency.

When from the moment of the reference mark will some time of t1 pass  then the vector of OA will turn on the corner? t1. From the vector head of OA which is in new situation we will lower the perpendicular on the horizontal axis. Length of this perpendicular will be OA x sin ? t1. At some next moment of t2 the vector forms the corner with the horizontal axis? t2, and length of the perpendicular lowered from its end will be, respectively, OA xsin t2. Later the quarter of the period from the moment of time reference mark, i.e. at the time of t3 = T/4 the vector of OA will become perpendicular to the horizontal axis, and perpendicular length

Синусоидальная величина с положительной начальной фазой

Figure 2. Sinusoidal size with positive initial phase.

Now near the circle described by the end of the rotating vector we will construct the curve of dependence of the size OA x sin in rectangular system ? t from? t or from t is and the sinusoidal curve for the period from t = About to t = t3 will be.

At the time of t3 = T/4 sinusoidal size reaches the maximum value. In process of further rotation of the vector  the size OA x sin ? t decreases (the moments of t4 and t5). At last, at the time of t6 = T/2, having described the arch equal? to radians, the vector will accept horizontal position. At the moment, when OA x sin ? t6 = OA x sin? =0", sinusoidal size passes through zero value.

At further rotation of the vector perpendicular of OA x sin ? we will consider t negative (the moments of t7, ts, ts). Respectively, we will build from the horizontal axis this site of the sinusoidal curve down.

If at the initial moment of t = 0 vector forms some corner with the horizontal axis and, then at the time of the reference mark sinusoidal size is not equal to zero, and matters OA x sin 0 (fig. 2). The corner and is called the initial phase angle, or initial phase. In this case length of the perpendicular lowered from OA vector head on the horizontal axis at the time of t will be:

OA x sin (? t + a),

according to it the sinusoidal curve at the initial moment will not pass through zero. Thus, generally it is desirable that alternating current changed in time according to expression

I = Im x sin (? t + a).

In this expression of i - the instantaneous value of current intensity, Im — the maximum value (amplitude). It is necessary for receiving the sinusoidal current that the EMF of alternating current generators was sinusoidal too,

here? - any initial phase of this EMF. If EMFs е and i current, belonging to the same chain, not at the same time pass through zero or maximum value, then they are dephased relatively each other (fig. 3). In the presence of phase shift of EMF in the chain it can be equal to zero, and current will pass in it still, or current can be equal to zero in the presence of considerable EMF,

Сдвиг фаз между э. д. с. и током

Figure 3. Phase shift between э. of page and current.

Phase shift ф is measured by the difference of initial phases of sinusoidal sizes. In the case considered by us ф = ? - and, and EMF advances current on the phase. Respectively, vectors of Em and Im form the corner ф which remains invariable at their rotation.

Sinusoidal sizes, for example tension and current, match on the phase if their initial phases are identical; they are opposite on the phase if their phase shift ф = ±?. If one of sinusoidal sizes changes on the sinusoid, for example i = Im x sin    ? t, and the second — on the cosinusoid, for example u = Um cos ? t, phase shift between them ф ?/2 (to what there corresponds the quarter of the period), as

It must be kept in mind that the rotating vectors of alternating-current sizes significantly differ from vectors of the physical quantities (force, speed, magnetic induction, electric field intensity, etc.) having determinate direction in space.

The alternating-current vectors called also radius vectors represent only the convenient mathematical form of the image of the sizes changing in time sinusoidalno. Radius vectors, as well as space vectors, often briefly call equally vectors. Alternating-current vectors distinguish the point over the letter designating this or that sinusoidal size, for example ? m or? m

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