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)
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(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)
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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)
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(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)
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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.
Mutual equivalent transformations of the parallel and series connection of elements
The task of equivalent transformation of a series connection of elements of the same type in parallel connection and inverse is ambiguous. Let us consider the transformation of the series and parallel connection of elements of various types in the harmonic current circuit. On Fig. 4.6 the equivalent series and parallel connection of elements are shown.
Fig. 4.6
Obviously, you can write down
(4.35,а)
Hence
(4.35,b)
I.e.
(4.35,в)
Similarly may be obtained ratio for other elements. In the Table 4.1 presents the ratio of the equivalent transformation of the series connection of R , L , C - elements to parallel connection of elements of R , L , C , and back.
Table 4.1
The task of equivalent transformation series connection of L , C to parallel connection of L , C - elements is also ambiguous.
It should be noted that the expression of the Table 4.1. for the transformed network is valid for the same frequency. When you change the frequency values of the transformed network parameters are changed. It follows that the mutual equivalent transformations of the series and parallel connection of various types elements for nonlinear circuits, or for linear of nonharmonic current, generally speaking, is impossible.
The transformation of delta – to star – connection and back
The connection of the elements in Fig. 4.7.a and Fig. 4.7.b is called the delta - and the star connection, respectively.
Fig. 4.7
Let us consider the connection of complex resistance Z , Z , Z in the triangle (delta connection). For the circuit of Fig. 4.7, a in according to the Kirchhoff's laws for the currents and voltages we can be recorded for nodes 1,2 and the loop Z - Z - Z .
(4.36)
Expressing the current I from the first equation in (4.36) and substituting to the rest equations , we get
(4.37)
Expressing the current I from the first equation in (4.37) and substituting to second, we get
(4.38)
Hence
(4.39)
Voltage U
(4.40)
Consider the connection of complex resistance Z , Z , Z to a star. For the circuit Fig. 4.7 a and Fig. 4.7.b according to the laws of Kirchhoff for currents and voltages can be recorded for node "0" and the loop Z - Z - U
(4.41)
Expressing the current I of the first equation and substituting the result into the second equation, we get
(4.42)
From here
(4.43)
As for equivalent transformation the currents I , I , I and voltages U , U , U in both networks of Fig. 4.7 a and Fig. 4.7.b are the same, then (4.40) and (4.43) , we obtain from (4.40), (4.43)
(4.44)
(4.45)
Substituting (4.45) into (4.44), get
(4.46)
Similar transformations can get
(4.47)
Defining of (4.45) - (4.47) Z , Z , Z from (4.45) – (4.47,) we obtain
(4.48)
(4.49)
(4.50)
Replacing in (4.45) - (4.47) resistance to the conductances, we get
(4.51)
(4.52)
(4.53)
It is visible, the relation (4.51) - (4.53) for the conductances of the star-connection are identical on structure with the relations (4.48) - (4.50) for the resistance of the delta-connection. Obviously, it should be expect that the ratio of (4.45) - (4.47) for the resistance of the star- connection have the same structure ratio for the conductances of the delta-connection. Indeed, replacing in (4.45) - (4.47) resistances by the conductances, we get
(4.54)
(4.55)
(4.56)
Fig. 4.8
The obtained relations are to a delta and to three beams star-connection. In the general case, by transformation of the N - beam star connection in N –angular network (Fig. 4.8,a, b) , we obtain the same (4.54) - (4.56)
(4.57)
The reverse conversion of N – angular network into the N –beam star connection is impossible in the general case.
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