Problem 55-21    Source:   Adapted from Problem 8.63 - page 497 - Ulaby, Maharbiz & Furse - Book: Circuit Analysis and Design - 2018.

Determine the equivalent impedance between points a - b in the circuit shown in Figure 55-21.1.

Solution of the Problem 55-21

To solve this problem we will calculate the impedances between the points a-m, b-m and d-m. Note that the impedances between points b-m and d-m will be equal, as the values of the components are the same. Therefore, we have:

See Figure 55-21.2 for the new circuit with the impedances calculated above.

In the figure above, it is noticeable that the reactive elements of the circuit are connected in a star configuration. Now we can transform the star circuit into a delta circuit with what was learned in chapter 5   see here!

Then, using the equations mentioned in chapter 5, and naming the impedance between the points a-d of the star configuration as Z'_ad, we obtain:

Similarly, the other impedances are:

Substituting these values into the circuit in Figure 55-21.2, we obtain the circuit shown in Figure 55-21.3.

Now it is necessary to calculate the parallel between the components that appear in the circuit shown in Figure 55-21.3. Let us call the parallel of the components between the points a-b of Z_ab. This value will be equal to Z_ad, as the components have the same values. Calculating, we obtain:

We must pay attention to the fact that the impedances Z_ad and Z_db are connected in series configuration. And we will call this association Z_S. Thus, adding the two impedances, we obtain:

And obviously, Z_S is connected in parallel with Z_ab. Calculating this parallel we will find the value of the equivalent impedance between the points a-b, which we will call Z_eq. Then:

Carrying out the calculation, we obtain: