Equivalent transformations in electric circuits

Two sites of the electric circuit are called equivalent if the replacement of one by a new one the currents and voltages in the rest of the circuit will not change.

When calculating the electric circuit it is necessary to formulate and to solve a of linearly independent equations system of electrical balance. Equivalent transformations of electrical circuits are based on equivalent transformations of the corresponding equations system of electrical balance.

Series connection of elements

Let us consider an electrical circuit with a serial connection active resistances r , r ,..., r , inductances L , L , ..., L , capacitances C , C ,..., C and voltage sources with the EMF e , e , ... e (Fig.4.1).

Fig. 4.1

According to the law of Kirchhoff’s low for the voltage we get for the instantaneous values

(4.18)

or

(4.19)

and finally

(4.20)

where:

(4.21)

(4.22)

(4.23)

(4.24)

I.e. by a series connection of resistances or inductances the equivalent resistance or inductance equals to the sum of series-connected resistances or inductances.

By a series connection capacitances the value of inverse equivalent capacitance equals to the sum of the inverse values of each series connected capacitances.

By a series connection of the voltage sources value of equivalent voltage source equals to the algebraical sum of the values of each of series-connected voltage source.

Serial connection ideal current sources is impossible.

Similar relations can get for complex resistances and EMFs

(4.25)

On the rules of a serial connection elements is based device voltage divider (Fig. 4.2)

Fig. 4.2

Here

(4.25.а)

Parallel connection of elements

Let us consider an electrical circuit with a parallel connection of a resistances r , r , ..., r , inductances L , L , ..., L ,capacitances C , C , ... , C and current sources j , j , ... , j (Fig. 4.3).

Fig. 4.3

According to the of Kirchhoff law for the currents we get to the instantaneous values

(4.26)

or

(4.27)

and finally

(4.28)

where:

(4.29)

(4.30)

(4.31)

(4.32)

I.e. by parallel connection оf resistances or inductances the value of inverse equivalent resistance or inductance equals to the sum of the inverse values of each parallel connected resistances or inductances.

By parallel connection of capacitances the equivalent capacitance equals to the sum of parallel-connected capacitances.

By parallel connection of the current sources value equivalent current source equals to the algebraic sum of the values of each parallel-connected current sources.

Parallel connection ideal voltage sources is impossible.

Similar relations can get ratios for complex conductances and current sources.

(4.33)

On the rules of parallel connection elements is based device current divider (Fig. 4.4).

Fig. 4.4

Here

(4.34)

or through the resistance (Fig.4.5)

Fig. 4.5

Here

(4.35)

The ratio of (4.35) expresses a rule of "the alien resistance": current in one of two parallel-connected resistances equals to the total current, divided by the sum of these resistances and multiplied by the other ("alien") resistance.

Date: 2015-12-18; view: 786