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Home Electric Networks Equivalent Networks  


Equivalent NetworksIn the preceding section the value of each impedance of Fig. 9 (a) was known, and the reduction to the simple T network was made by transformation equations. Sometimes a circuit or device is in a "black box" (Fig. 10), the values of the impedances are unknown, and it is desired to determine the simple equivalent T or π network. For limited equivalence between the two sets of input terminals 12 and the two sets of output terminals 34, the input currents I_{1} to the two networks must be identical and the output currents I_{2} to the loads Z_{L} must be identical (in both magnitude and phase). Of course, this also implies that the voltage across terminals 12 must be identical, and the output voltages across terminals 34 must be identical when the networks are connected between the same generator and the same load impedance Z_{L}. An investigation will show that if the input current I_{1} does not equal the output current I_{2}, there must be a shunt element such as Z_{3} in an equivalent network (Fig. 9 or 10). Similarly, if the input voltage E_{12 }does not equal the output voltage E_{34}, there must be series elements such as Z_{1} and Z_{2}. Thus, three impedances are required in an equivalent network. These three impedances may be in a T configuration as in Fig. 10, or may be transformed into a π configuration.
The individual elements within the black box of Fig. 10 in general cannot be determined. It is known, however, that, at a given frequency, some T network will be equivalent. The values of Z_{1}, Z_{2}, and Z_{3} of this equivalent T network must be determined from measurements on the unknown network. Thus, if the input impedance between terminals 12 is measured with the output terminals 34 open circuited and if this is called Z_{12oc}, then
If the input impedance Z_{12sc} is measured between input terminals 12 with the output terminals 34 short circuited, then
If the output impedance Z_{34oc} is measured between output terminals 34 with the input terminals 12 open circuited, then
From these three equations,
and
Equation 21 is found by substituting equations 19 and 20 in equation 17 and solving for Z_{3}.


Home Electric Networks Equivalent Networks 