# Power on a three-phase system

Given a balanced three-phase voltage source and unbalanced loads connected in wye whose individual apparent power dissipation and power factor are given, find the the line currents.

What is your attack in this kind of problem?

• I would post a question on ee.se and have others solve it... – PlasmaHH May 18 '15 at 12:55

Ultimately you'll need to perform a wye to delta transform: -

But first you need to calculate Z1, Z2 and Z3 and this is done using the apparent power and power factor of each individual Y load. Once you have those impedances do the transform to delta.

Then you have loads connected directly across line voltages and the current into node (a) is the phasor sum of currents thru ZA and ZB. Repeat for the other three nodes and you have your answer.

1. You have not said if it's three or four wire (neutral). I will assume it is 3-phase, 4 wire. Unbalanced current will flow on neutral wire.

2. You have not said if it was wye or delta source. I will assume wye. $V_{Phase_{ Load}} = V_{Phase_{ Source}}$

3. You have not said if pf's lead or lag. I assume lag, as in current lags voltage.

You have individual apparent powers $S_{Phase}$ and power factors $pf_{Phase}$.

$$S_{Phase} = V_{Phase} I_{Phase}$$ $$I_{Phase} = \frac{S_{Phase}}{V_{Phase}}$$

That's the current magnitudes.

$$pf = \frac{P}{S} = cos\ θ$$ $$θ_{Phase} = arccos\ pf_{Phase}$$

That's the phase angles. To make your answer 3-phase, take each phase angle and subtract (due to lagging) from the phase voltage angles (0°, 120° & -120°).

It is a wye, so $I_{Line} = I_{Phase}$

You're given the power factor which means you know the phase. You'll need to represent each voltage, and each current as a complex number and then solve the problem calculating the currents through and voltages across the loads. Probably easiest if you derive mathematical expressions for the currents you want to solve, then plug the complex numbers in and churn it all through.

You may need to convert back and forth between complex numbers represented in polar form (r, theta), and Argand (cartesian) form: a+bj . So learn how to do that, if you don't already.

There's a bit of work involved in all this, but the maths shouldn't be too difficult if you understand complex number theory and feel comfortable with manipulating complex numbers.