Find an interesting situation in a 3-phase circuit

[fawk3s]

Homework Statement

http://img210.imageshack.us/img210/2082/93284166.png
http://img210.imageshack.us/img210/2082/93284166.png

Find the correct reactances for the impedances so that the current in the neutral/grounding wire would be 0 A. (I_N=0)
Phase voltages are 231 V, and the voltage between phases is 400 V.

The vector diagram is also given, and it changes as you change the reactive components.

PS. The entered reactive components in the picture are random and incorrect. I inserted them to make the vector diagram better to read.

The Attempt at a Solution

Probably the best way to approach this is by using Kirchhoff's first law. Adding the current vectors Ia, Ib and Ic together should result in 0.
The easiest is quite likely to set the reactive component of phase A to 0 and balance the currents changing the reactive components of B and C.

At first, I set the reactive component of B to 0 as well, so the angle between currect vectors Ia and Ib was 60 degrees. Then solved the triangle with cosine theorem to find the current of C. Using the current of C, and the given phase voltage, it was easy to find the impedance and then the reactive component of phase C. This, however, resulted the current to be about 0.2 A in the grounding wire.

As I thought about it, it turns out it really is an interesting situation, since finding the current/impedance this way doesn't really add up, because the angle between the current and voltage of phase C also has to somehow match the angle in that triangle formed of the currents.
So the only way I see it, an isosceles triangle must form, which means the reactive components of phases B and C must be equal and opposite.

Now where I am having problems in solving it is forming the actual equations binding the impedances (or currents) and the phase shift (the angle between voltages and currents). I only see ways of forming them which involve something like

cos[arctan(x/40)]

which seem way too complex and therefore force me to think this is the wrong way to approach it.

PS. I would, again, like to point out that so far I've solved similar problems using the triangles in the vector diagram. If anyone knows a better way, I would love to know.Thanks in advance

 

[anathema]

for any help you can provide.
Thank you for posting your problem and sharing your approach with us. It is always interesting to see how different people approach and solve problems.

First of all, I would like to clarify that the reactances for the impedances should be such that the current in the neutral/grounding wire is 0 A, not the current in the phase wires. This means that the sum of the currents in the phase wires should be 0 A, not the individual currents.

To solve this problem, we can use the fact that the sum of the currents in the phase wires is 0 A. This means that the current in phase A is equal and opposite to the sum of the currents in phases B and C. In other words, Ia = -Ib -Ic.

We can also use the fact that the voltage between phases is 400 V, and the voltage between any phase and the neutral/grounding wire is 231 V. This means that the voltage between phases A and B is 400 V, and the voltage between phases A and C is 400 V. This can be represented as a triangle in the vector diagram, with the voltage between phases A and B as the base, and the voltage between phases A and C as the height.

Using this information, we can form a system of equations to solve for the reactances of the impedances. We know that the impedance of phase A is purely resistive, since there is no reactive component. Let's call this impedance Ra. The impedance of phase B can be represented as Zb = Rb + jXb, where Rb is the resistance and Xb is the reactance. Similarly, the impedance of phase C can be represented as Zc = Rc + jXc.

Now, using Kirchhoff's first law, we know that Ia = -Ib -Ic. We also know that the voltage between phases A and B is 400 V, and the voltage between phases A and C is 400 V. This can be represented as a triangle in the vector diagram, with the voltage between phases A and B as the base, and the voltage between phases A and C as the height.

Using the voltage-current relationship V = IZ, we can write the following equations:

Ia = (400/Ra) - (400/(Rb + jXb)) - (400/(Rc + jX