Problem 83-9    Source:   Adapted of Problem 12.25 - page 486 -    SADIKU, Mathew & ALEXANDER, Charles -    Book: Fundamentos de Circuitos Elétricos - Ed. McGraw Hill - 5ª edição - 2013.

Considering the problem 83-8, with the circuit shown in Figure 83-09.1 and repeating that

V_AB = 440 ∠ 10°, V_BC = 440 ∠ -110° and V_CA = 440 ∠ 130°.

We also know that Z_L = 3 + j2 Ω  and  Z = 10 - j8 Ω.

a) Calculate the currents at each source, ie I_AB , I_BC and I_CA.

b)Check that the energy conservation law is met.

Solution of the Problem 83-9

Item a

As was studied in item 2.3 , the currents that circulate through the voltage sources are considered phase currents. Therefore, to find your values we must use eq. 83-04 , repeated below.

This means that we must divide the line current by √3 and add 30° to its angle. As calculated in the problem 83-8, I_a = 17.74 ∠ 4.78°   A and we know that I_AB is related to the line current I_a. So, we can write:

The others currents, I_BC and I_CA, are 120° lagged of I_AB. Then:

Item b

Taking the A phase as a reference, the total active power that the circuit consumes is given by:

where φ is the angle of the impedance between the points A and n or, alternatively, represents the phase difference between the voltage V_An and the current I_a, given by φ = tg^-1 (-6/13) = -24.78°. Using this data in the previous equation, we have:

Likewise, we can find the total reactive power of the circuit, or:

Therefore, the total apparent power is:

The calculated powers refer to the consumption of the circuit. Now let's calculate the power delivered by the three-phase system, represented by the three sources of tension. Let's calculate the power provided by each source and multiply the value by three. Thus, we find the total power supplied to the circuit. So:

We use the negative sign to indicate that the system is supplying power to the circuit, according to the adopted convention. So, the energy conservation law is proven.