Problem 82-1    Source:   Problem elaborated by the author of the site.

In the circuit of the Figure 82-01.1, assuming a direct sequence, we have:

V_CA = 220 ∠30°  , and   Z_A = Z_B = Z_C = 10∠45° . Calculate:

a) The line and phase voltages.

b) The phase and line currents.

c) The total apparent power in the load.

d) The total actual power at the load.

e) The total reactive power in the load.

Solution of the Problem 82-1

Item a

To help you understand how to find line and phase voltages, the Figure 82-01.2 all the voltages are represented. Notice that the information given is V_CA = 220 ∠30°.

Well, the line voltages are calculated. To find the phase voltages, divide the line voltage modulus by √3. And, in addition, it is known that the phase is 30° delayed the corresponding line voltage angle. Mathematically, we must use the eq. 82-01 reproduced below:

As it is known the phase voltages are out of phase with 120°. Soon:

Compare these results with the graph above and notice how all the calculated values are in perfect agreement.

Item b

Since the load is in a Y configuration and the phase voltages have already been calculated, then to find the phase currents simply divide the voltage by the impedance. Do not forget that for this configuration the line current is absolutely equal to the phase current. Soon:

Item c

To calculate the total apparent power in the load, we use the equation eq. 82-03, already studied, or:

In this equation the effective values are of the line voltage and current. And φ is the angle of the impedance, whose value is φ = 45°. Thus:

Efetuando-se o cálculo:

It should be noted that the 45° angle can be used to calculate the power factor of the circuit, or:

Item d

To find the real power P, simply use the eq. 82-04, or:

Item e

To find the reactive power Q, we use the eq. 82-05, or:

Having the value of P and Q, we can write the total power total in its Cartesian form.