The characteristic of inductance in the alternating current circuit

In the chain containing inductance of L (fig. 1), electric current is caused by the combined effect of tension and the power source and EMF of the self-induction е, arising in the chain owing to changes of current:

Схема цепи переменного тока

Scheme of the alternating current circuit.

I = (u + e): r


u = (— e) + ir

Let's address the simplest conditions when r = 0. In this case:

u = - e = L (? i:? t)

where ? i:? tskorost of change of current in time.

Кривые мгновенных значений напряжения и тока в цепи, содержащей только индуктивность L

The drawing 1 Curves of instantaneous values of tension and the circuit current, containing only inductance of L.

Let's consider how tension on inductance clips has to change in time in order that through it there passed sinusoidal alternating current:

I = Imsin? t

For the sinusoidal current size? i:? t has the certain nature of change in time.

It too sinusoidal, but on the phase advances current by the quarter of the period.

It can be proved as follows.

At the time of t current intensity:

I = Imsin? t

and later very small period? tsila of current will be:

I +? i = i = Im sin? (t + ? t)

zmeneny current intensities:

i = Im [sin? (t + ? t) - sin? t]

Графики напряжения u и тока i в цепи переменного тока при сдвиге фазы

Schedules of tension of u and current of i in the alternating current circuit at phase displacement.

Sum sine:

sin (? t +?? t) = sin? t cos?? t + cos? t sin?? t

And the cosine of very small corner what is?? t, is equal to unit: cos?? t = 1, and the sine of very small corner is equal to the corresponding arch, therefore:

sin?? t =?? t

on the basis of it:

i = Im  (sin? t +?? t cos? t-sin? t) = Im?? t xcos? t

Thus, speed of change of the sinusoidal current:

? i:? t = Im? cos? t

and tension proportional to it on inductance:

u = L (? i:? t) = Im? cos? t

Therefore, the sinusoidal current in inductance is created by sine wave voltage too, only this tension advances current on the phase by the quarter of the period what there corresponds the arch to: (?: 2) or corner 90o

Thus, tension on clips of inductance advances current on the phase, or, otherwise, inductive current is the current which is lagging behind on the phase tension.

In the right member of equation only cos depends on time? t, which greatest value cos? t = 1. Therefore, the maximum value of tension on inductance:

Um = Im? L

Let's substitute in these formulas instead of the maximum values their operating values:

Let's receive:

U = I? L or I = (U:? L)

It will also be the Ohm's law for the chain (or the site of the chain) with one inductance.

Size? The dimension of resistance as dimension has L? = (1: sec.), and inductance unit гн = ohm x sec. Size? Is called as L as induced resistance and often it is in abbreviated form designated x or xL =? L

In essence, this size is the conditional resistance by means of which we consider counteraction of EMF of the self-induction to alternating-current changes, otherwise, reaction (counteraction) of inductance to alternations of the sinusoidal current. Induced resistance is proportional to the alternating-current frequency therefore at the direct current it is equal to zero.

Many devices and alternating-current machines cannot be turned on under the constant voltage as at alternating current they have big induced resistance, and for the direct current their resistance is not enough and force of the direct current can be for them destructive (for example, primary winding of the transformer in the radio receiver).

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