4.2.3. 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.

4.2.4. 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.