Problem + Hard 54-2    Source:  Problem 6.7 - page 95 -   EDMINISTER, Joseph A. -   Book: Electric Circuits - 2nd Edition - Ed. McGraw Hill - 1971.

The effective values ​​​​of the currents I₁, I₂ and I_T are respectively 18 , 15 and 30 A. Determine the unknown impedances R and X_L.

Solution of the Problem + Hard 54-2   

Let's take I₂ = 15 ∠0°. In this way we can calculate the value of V_ab. Using Ohm's law, we have:

Applying Kirchhoff's law for currents, one can write:

Note that the parallel formed by R and X_L, impedance which we'll call it Z_L, form a circuit with inductive predominance. With this we can say that I₁ will be lagging behind I₂. And since I_T is the phasor sum between I₁ and I₂, we conclude that the values ​​of these three phasors form the sides of a triangle, as shown in the figure below.

As we know the three sides of the triangle we can calculate the φ angle, using the law of cosines. Then:

Substituting numerical values ​​and carrying out the calculation, we find:

From this data we can calculate the value of the angle θ, which is given by:

Note that we are using the convention where phasors below the horizontal axis must have negative angles. So, we have I₁ = 18 ∠-49.46°. Now we can calculate the impedance of the parallel circuit, Z_L.

On the other hand we can write the complex admittance for Z_L, or:

Therefore, calculating the inverse of the values ​​we obtain:

e,