Problem 5-1    Source: Problem elaborated by the autror of the site.

Prove that the resistance seen by the terminals a - b is equal to R in the circuit shown in Figure 05-01.1.

Solution of the Problem 5-1

To solve this problem we have to use the transformations Star-Triangle. Initially we will calculate the equivalent circuit for points 1-2-b, as shown below. Note that in the figure, for facilitating understanding of the transformation, we the three resistors of value R, which belong to the circuit triangle and will be replaced by three resistors of value R_eq, which will be part of the star circuit.

In the Figure 05-01.2 we replace the triangle circuit with its corresponding circuit star connecting the 1-2-b points. As the three resistors of the Triangle circuit are equal then the equivalent resistance ( R_eq ) is given by R /3.

In the Figure 05-01.3 we can see how the circuit was after the transformations. Note that we replace the delta circuit with the star circuit with their respective values.

In the Figure 05-01.4, the resistors of values equal to 4 R / 3 appear, which is the result of the sum of R with R / 3 since they are in series. Like this, we have two associated 4 R / 3 resistors in parallel. As a result of this parallel we find a resistor of value equal to 2 R/3.

And finally, in the Figure 05-01.5 the resulting circuit appears. Note that the two resistors are in series (left side of figure). Therefore, adding their values we find a single resistor of value equal to R (right side of the figure). And so, we conclude that we can replace the entire circuit by a single value resistor R.