Correct power factor formua

Question

My instructor asked me why have I written the power factor formula as

If \$\varphi\$ is the phase angle between the current and voltage, then the power factor is equal to the cosine of the angle, \$cos\,\varphi\$: $$|P| = |S|\,\cdot\,cos\,\varphi$$

with his reasoning that \$P\$ must always be positive with the exception when the resistance is negative, so I should not have the absolute value there.

Now I cannot remember the reason why, but I am pretty sure I have found it somewhere, but since I did a poor job with references, now I cannot find the origin of this formula.

I have found it on wikipedia.

Is this formula wrong? If yes, how can it be written correctly in this form. If no, can you forward me to some more reading on this? Many thanks.

Answer 1

I would say that the most general formula is:

$$P=Re\{\underline{S}\}=Re\{\underline{V}·\underline{I}^{\star}\}=S\cos{\theta}$$

where I'm using underline to identify phasors.

S is the module of \$\underline{S}\$ and it is always positive (it is simply the voltage amplitude times the current amplitude).

\$\theta\$ is the angle between voltage and current. If this angle is bigger than 90º it means that the load is in reality providing active power to the source and not the other way around.

Therefore P can be positive or negative.

Answer 2

The real power (P) should not be an absolute value. Let's look at the math. Power factor is defined as the ratio of real power (P) to apparent power (S). But S is a complex number, so a simple ratio is meaningless. We need to use the magnitude of S:

$$pf = \frac {P} {|S|}$$

If we only have S, we can use it to get a formula for P. P is the real part of S:

$$P = Re\{S\} = |S| \cos \varphi$$

where \$\varphi\$ is the argument of S -- the angle between the complex vector S and the positive real axis. If S has a negative real component, then P should be negative. And indeed, \$\cos \varphi\$ is negative when \$-90^\circ < \varphi < 90^\circ\$.

This is a bit confusing because power triangles are almost always drawn with positive real power, which makes it look like the angle is between P and S. But it's not -- the angle is a property of the complex number S alone.

So a negative power factor tells you that the load is supplying power to the generator. If you don't care about that, you can take the absolute value of the cosine to keep the power factor positive:

$$|pf| = \frac {|P|} {|S|} = |\cos \varphi|$$

UPDATE: The Wikipedia article is incorrect. It contradicts itself:

The formula contains the absolute value of P, but then immediately says the power factor can be negative.