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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.

Power triangle from Wikipedia

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:

Screenshot of a small part of the Wikipedia article on power factor

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

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.

Power triangle from Wikipedia

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:

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

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.

Power triangle from Wikipedia

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:

Screenshot of a small part of the Wikipedia article on power factor

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

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source | link

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.

Power triangle from Wikipedia

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:

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