Problem 83-4    Source:   Problem 78 - page 226 - FOERSTER, Gerd & TREGNAGO, Rodrigo - Book: Circuitos Elétricos - 1ª edição - Ed.da Universidade - Ufrgs - 1987.

For the three-phase symmetric network shown in Figure 83-4.1, sequence ABC, it is known that the line voltage V_AB = 100√3∠0° and that Z = 10√3∠83.13°. Determine:

a) The currents I₁, I₂ and I_A.

b) The voltage V₁₂.

Solution of the Problem 83-4

Item a

Since we have series impedances with a three-phase circuit balanced in delta, the best solution is to transform the delta circuit into a star circuit, so we can sum the impedances in series. Therefore, to find the impedances (Z'_S) of the circuit star, it is enough to divide the impedances of the circuit delta by 3, because the circuit is balanced. Like this:

Now you can add the impedances in such a way that:

On the other hand, because the circuit is balanced, we can calculate the currents using the phase voltage, as shown in the equation below.

Thus, the line voltage must be divided by √3 and the phase must be delayed by 30°. Then:

Now we can calculate the current I₂.

Once again, because it is a balanced circuit we can calculate I₁ from I₂ using the equation below:

Thus, the line current must be divided by √3 and the angle of the phase should be advanced by 30°. So:

To calculate I_A, we must add I₂ with the current that pass through the impedance Z. So:

Substituting for the numerical values, we have:

Performing the calculation:

Item b

In order to calculate the voltage V₁₂ it is necessary to know the current that flows through impedance Z and the current that flows through the impedance 2-j6 , which interconnects the points B and 2. The current that flows through Z, called the I_Z, is given by:

Performing the calculation:

Therefore the voltage (V_Z) on the impedance Z of under is:

To calculate the current ( I_B2) which circulates through the impedance 2-j6 that interconnects the points B and 2, we can relate the currents in node 2 :

Note that I₂₄, that circulates in the delta circuit, is the current I₁ delay of 120°. Soon,I₂₄ = 11.547∠-173.13°. Then:

Performing the calculation:

To calculate V₁₂ add up the vpltages V_Z and V_B2 = I_B2 (2+j6), or:

Performing the calculation: