Problem 81-4    Source:   Problem 11.1 - page 272 -    NILSSON, James W. & RIEDEL, Susan -    Book: Circuitos Elétricos - 5ª edição - Editora LTC - 1999.

Verify that the following voltage assemblies are balanced three-phase assemblies. If the answer is yes, check whether the phase sequence is positive or negative. If the answer is negative, explain why.

a)   v_a = 139 cos 377t,   v_b = 139 cos (377t + 120°)   and   v_c = 139 cos (377t + 120°).

b)   v_a = 381 cos 377t,   v_b = 381 cos (377t + 240°)   and   v_c = 381 cos (377t + 120°).

c)   v_a = 2771 cos (377t - 30°),   v_b = 2771 cos 377t   and   v_c = 2771 cos (377t + 210°).

d)   v_a = 170 cos (ωt + 30°),   v_b = - 170 cos ωt   and   v_c = 170 cos (ωt + 60°).

e)   v_a = 339 cos ωt,   v_b = 339 cos (ωt + 120°)   and   v_c = 393 cos (ωt - 120°).

f)   v_a = 3 983 cos (ωt + 50°),   v_b = 3 983 cos (ωt - 160°)   and   v_c = 3 983 cos (ωt + 70°).

Solution of the Problem 81-4

To solve this problem see the reproduction of the graph with the three-phase voltages in the Figure 81-04.1.

This figure shows the graph of phase and line voltages for the case of positive sequence. Note that in this case the voltage V_CN is advanced of 120° in relation to the voltage V_AN. For this problem, V_CN = v_c and similarly for the other phases. On the other hand, the voltage V_BN is delayed by 120° in relation to the voltage V_AN. This is the case of the positive sequence. For the case of the negative sequence, simply reverse the V_CN with V_BN.

Item a

Note that v_b is advanced of 120° in relation to the voltage v_a. If the sequence was positive v_b should be late. Therefore, this characterizes a negative sequence. And because the voltages are lagged from 120° to each other and their modulus are the same, then it is a balanced three-phase set.

Item b

In this case, note that v_b is advanced by 240° in relation to the voltage v_a. This is the same to say that this voltage is delayed by 120° in relation to v_a. Therefore, the sequence is positive. And because the voltages are lagged from 120° to each other and their modules are the same, then it is a balanced three-phase set.

Item c

Although the voltages modulus is the same, there is no gap of 120° between them, consisting of an unbalanced three-phase set.

Item d

Same as above. The lag between the voltages is different from 120° making a set three-phase unbalanced.

Item e

There is a lag between the voltages equal to 120°, but the modulus of v_c differs from the others by making an unbalanced three-phase array.

Item f

Although the modulus of the voltages are the same, there is no lag of 120° between the voltages, making a three-phase array unbalanced.

Observation

As in all items the voltages had the same frequency, which is necessary for the circuit to be considered balanced, we omit this information to avoid being repetitive.