Problem 10.10    Source:    Problem 3.52 - page 61 -    NILSSON & RIEDEL -    Book: Circuitos Elétricos - 8ª edição - Ed. Pearson Education do Brasil - 2010.

Use Delta-Star transformation and calculate the voltages V_b and V_c in the circuit show in Figure 10-10.1.

Solution of the problem 10.10

Note that the highlighted circuit in green represents a Delta or Triangle circuit. There is a need to transform it in a Star circuit. Using the equations studied in Basic Theory the equivalent circuit is calculated, as can be seen in the Figure 10-10.2.

Now there is how to replace the Star circuit in the original circuit. See the Figure 10-10.3 for the new configuration. Note, in the right circuit, that resulted in two resistors of equal value (34 ohms) in parallel. Therefore, for each one will pass the same current, that is, i_total / 2.

To calculate i_total we must calculate the resistance that the circuit offers to the voltage source. However, the parallel of the two 34 ohms resistors results in 17 ohms, which together with the 3 ohms resistor results in an equivalent resistance of 20 ohms. So:

Returning to the original circuit we can calculate the currents through the whole circuit. The current passing through the voltage source of 80 volts is 4 A, as calculated. Then 2 A will go through the resistors of 28 and 6 ohms (path a-c-d, as shown above). And by the resistors of 16 and 18 ohms (path a-b-d, as shown above) also circulates 2 A. Thus, one can calculate the tension in the point b.

To calculate the voltage at point c, one can use the mesh by the resistor of 28 ohms, or:

To conclude the calculation of all currents, simply use the values of V_b and V_c. The calculated values are shown in the Figure 10-10.4.

As an exercise take balance of potencies and verify that the values found are correct. Notice that the power supply to the circuit is the voltage source through which a current of 4 A flows. Therefore, the power supplied by the voltage source is calculated by 80   (-4) = - 320 watts. It is obvious that the algebraic sum of all dissipated powers in the resistors should be equal to 320 watts.     Check !!