Problem 55-14    Source:   Problem elaborated by the author of the site.

From the phasor diagram shown in the Figure 55-14.1 determine the equivalent circuit and the angle θ, knowing that |V| = 200 V, |I₁| = 10 A e |I₂| = 8 A.

Solution of the Problem 54-14

Item a

From the figure, note that the voltage and currents are referenced to the axis x. Thus, we can write the phasor values of these variables by taking the axis x as a reference, that is, V = 200 ∠-30°  ,  I₁ = 10 ∠30°   and   I₂ = 8 ∠-60°. In addition I₁ is advanced by 60° in relation to Vand this characterizes a circuit with capacitive predominance. As we know V and I₁, then it is possible to calculate this impedance, denoting it by Z_C and substituting the numerical values, in the equation below, we find:

On the other hand, I₂ is delayed at 60° - 30° = 30° with respect to V, characterizing an impedance with inductive predominance. Denoting this impedance by Z_L and after the substitution by the numerical values, in the equation below, we obtain:

We must calculate the value of the modulus of I_T. As I₁ and I₂ make an angle of 90° between them, we can use Pythagoras. Like this:

In order to calculate the angle θ, which is the angle between I_T and the axis x, first calculate the angle between I_T and I₁ calling it φ. Thus, having the value of this angle and subtracting from 60° will find the value of θ.

With the value of I_T , I₁ and I₂, we can use the law of cosines to find the value of φ, using the eq. 51-01. After the calculation, we find:

And the value of θ is:

With the values ​​of Z_C and Z_L calculated above, one can draw the electrical diagram of the circuit. There are several circuits that meet this specification, however, a very simplified circuit is shown in Figure 55-14.2.

Item b

We have already calculated the module of I_T and θ. Writing in phasorial form:

Knowing the value of V and I_T, we can calculate the total equivalent impedance that the circuit offers to the voltage source. Like this:

Performing the calculation:

Note that the equivalent impedance has capacitive predominance. This conclusion can also be reached by looking at the phasor diagram and verifying that I_T is advanced of 30° - 8.66° = 21.34° in relation to V. Of course you can get the value of Z_eq calculating the parallel between Z_C and Z_L. The result has to be the same. Check!!!!.