# Is the Sqrt(3) Factor in 3-phase Power Dependent on Everything Being a Sine Wave?

1. Aug 9, 2012

### Strill

The line current in a 3-phase delta loaded circuit is supposedly equal to sqrt(3) times the phase current. That's dependent on all the signals being sine waves though right? Because the math that gets you to that point is dependent on sine and cosine identities that don't necessarily hold with other waveforms correct?

2. Aug 9, 2012

### milesyoung

Yes, under balanced conditions.

You usually see it derived using phasors, so it naturally assumes sinusoidal signals.

3. Aug 9, 2012

### I_am_learning

4. Aug 10, 2012

### uart

I'm going to go against popular opinion and say that the answer is no, it doesn't depend upon the supply being sinusoidal, merely that it's balanced and without a neutral connection (so no triplen harmonics). This would therefore apply to a star (wye) load provided there was no neutral connection.

Consider the Fourier decomposition of the supply waveform, the absence of neutral forces there to be no triplen harmonics in the current waveform so we don't need to consider these in our power calculations. Now all the non-triplen harmonics individually form a balanced three phase system (either +ive or -ive phase sequence), and their orthogonality means we can separately add their contribution to the total power.

5. Aug 10, 2012

### milesyoung

The title was maybe misleading, but the question was about the relationship between line and phase currents in a delta connected load, not power.

6. Aug 10, 2012

### uart

You're correct miles, I read the words "sqrt(3)" and "power" in the title and assumed the OP was referring to the power equation, $P = \sqrt{3} V_{L-L} \, I_{LINE}$.

Now I see that the OP was merely referring to the relation between line and phase current. Interestingly much of what I said for the case of power does still apply here.

Provided that the line-line voltage contains no triplen (multiple of 3) harmonics, then neither do the phase (load) currents. This means that each of the harmonics individually forms a balanced 3 phase system, eg the 5th harmonic is a balanced -ive phase sequence system, the 7th harmonic is a balanced +ive phase sequence system etc.

So each of the harmonics separately adds vectorially to give the RMS line current equal to sqrt3 times the RMS phase current at any particular harmonic frequency. Since the mean squared harmonic currents add algebraically, then the total MS line current is 3 times the MS phase current.

So to summarize, all that's required is a balanced three phase system with a supply that has no triplen harmonics. Hope that helps. :)

Last edited: Aug 10, 2012