Tutorial 13 - Extension - Unbalanced 3-Phase systems

What happens if the star system is unbalanced?

So far we have kept the star system balanced, and those operating transmission systems attempt to keep the system balanced as far as possible.  However it is possible to have different loads on each phase, which means that the system is unbalanced.  The diagram below shows how it is possible to have an unbalanced star system.

Worked Example

A 415 V, 3-phase star-connected system supplies three resistive loads as shown above.  Determine:

a.       The current in each line;

b.      The current in the neutral line.

 

Answer (a)

All the line voltages are 415 V.  Calculate the phase voltage:

VP = VL 3 = 415 3 = 240 V

 

Calculate the currents using I = P/V:

 

Red phase:

IR = 24000 240 = 100 A

Yellow phase:

IY = 18000 240 = 75 A

Blue phase:

IB = 12000 240 = 50 A

 

Answer (b)

The answer is NOT 100 A + 75 A + 50 A = 225 A

 

By using Kirchhoff I, we can say that the neutral current is the vector sum of all the currents.

 

IN = IR + IY + IB

 

This is because all the vectors are 120o apart.  Our phasor diagram would look like this:

 

 

So we can now put the vectors nose to tail:

 

 

The vector OC is the resultant, IN.  We can solve this by accurate drawing, which gives a resultant current of about 43 A.

 

We can also solve this by adding the components of the vectors.  This is shown below.  Take upwards as positive and downwards as negative.  Left to right is positive, right to left negative.  The vectors are shown below:



 

The vector components are:

           

Vector    Vertical Horizontal
I +100 A   0
IY   -75 sin 30 +75 cos 30
 IB -50 cos 60 - 50 sin 60

                                                   


 

The horizontal component of IN is given by:

 

75 cos 30 + -50 sin 60 = (75 0.866) (50 0.866) = 21.65 A

 

The vertical component of IN is given by:

 

100 + -75 sin 30 + -50 cos 60 = 100 (75 0.500) (50 0.500) = 37.5 A

 

IN is the vector sum of these two:

 

21.652 + 37.52 = 1875

 

I= 43.3 A

 

The neutral current in this case is 43.3 A.

 

Remember that the phase relationship is 120o, so the vertical and horizontal components can be found by simple geometry.  Watch out for the signs on the vertical and horizontal components.  If in doubt, draw a diagram like the one above.

 

Question 15

A three phase industrial motor operates at 1500 V, 3-phase.  Normally the loads of its coils are balanced, each coil taking a current of 200 A.

a.       What is the line voltage?

b.      What is the phase voltage?

c.       What is the neutral line current while in normal operation?

The coils on the red phase develop a fault so that the line current of the red phase is now 250 A.  The yellow and blue phases are not affected.  Calculate the neutral current.  The answer is not 650 A.

Answer

 

Click HERE to go back to Tutorial 13 A

Click HERE to go back to Tutorial 13 B