# Power factor and cos φ

I am confused about the difference between "power factor" and "cos φ".

Some textbooks state that those two quantities are equal. Other textbooks state that those two quantities are not equal. And only in one internet resource, I have read that power factor equals cos φ plus non-linear distortion factor.

As far as I know, those two quantities are equal when we have ideal sinusoidal current.

Also I will be very happy if you can give me a link to some textbook, in which these topics are covered in more detail.

• The Wikipedia article is a good place to start. It also has links to more resources. en.wikipedia.org/wiki/Power_factor – Mattman944 Apr 29 '19 at 8:16
• I have red this article already. But it do not answer my question. if I would be able to understand this article, I probably would not have asked such a question – Salekh Apr 29 '19 at 11:52
• Would be mind linking the internet resource that mentions distortion? – Sean Dever Feb 11 at 20:29

The power factor is the ratio between real power and apparent power. It is a generalization of the concept of cos φ. In case of a sinusoidal current, the power factor is just plain cos φ, but in case of non-linear current consumption (which is typical for phase-angle control and rectifiers, so a whole lot of electronic devices today), the power factor is affected by the current waveform as well.

Power grid operators prefer power factors close to one, because power is (by residential households) paid for real power, whereas the energy losses in distribution depend mostly on apparent power, so power factor compensation, the act of getting the power factor close to unity, is a great deal.

In the case of phase shift, the power factor can be brought to nearly one by just adding a parallel inductor or capacitor to the load, so that their reactive powers cancel out and just the true power remains as apparent power.

In case of non-sinusoidal current consumption, adding inductors or capacitors still is able to change the apparent power (and thus the power factor), but no amount of parallel inductors or capacitors can bring the power factor to one. So you can split the power factor into two parts: The displacement power factor is introduced by phase shift (called φ) and can be compensated using suitable reactance, while the distortion power factor is introduced by distortion and can not be compensated that way. The total power factor is the product of the displacement power factor and the distortion power factor.

• Thanks for your answer! So while there is not any distortion we may say that power factor equels cos fi. Yes? If so then I grasp the idea. – Salekh Apr 29 '19 at 11:20
• You got it. And if you are lazy or ignorant, you might call the power factor cos φ event when distortions are present. Some people on the net are lazy and/or ignorant. – Michael Karcher Apr 29 '19 at 11:34
• It may not be a case of lazyness. Nearly all the texts give the classic sine phase-shift description only. That covered most industrial stuff 30 or 40 years ago but with the advent of VFDs, etc., current waveform distortion is much more of a factor. Invertek have a series of articles on VFDs that may help. – Transistor Apr 29 '19 at 11:45
• "Some people on the net are lazy and/or ignorant." - There speaks a master of the art of understatement. – Martin Bonner supports Monica Apr 29 '19 at 16:23
• Good answer. Worth adding that some relevant terminology is "displacement power factor" for p.f. due to phase shift, and "distortion power factor" for p.f. due to non-linear load / current waveform distortion. – Li-aung Yip Apr 29 '19 at 23:52

Power factor (PF) is defined as: $$\\dfrac{\text{RealPower}}{\text{ApparentPower}}\$$ where ApparentPower is simply the RMS voltage multiplied by the RMS current.

RealPower can be more complicated to calculate if the voltage and current are not perfect sine waves of the same frequency

(Image source: Envirotec Magazine - Monitoring power factor for effective energy management)

In the case where they are however from the diagram above we can see $$\\text{RealPower} = \text{ApparentPower}\cdot \cos(\varphi)\$$ and $$\\text{PF} = cos(\varphi)\$$

Where $$\\varphi\$$ is the phase difference between the voltage and current.

For the more general case $$\\text{RealPower} = \dfrac{1}{T} \cdot \int_0^T v(t) \cdot i(t) \text{ d}t \$$ where $$\v(t)\$$ and $$\i(t)\$$ are the instantaneous voltage and current values, with respect to time, $$\T\$$ is the time for any whole number of cycles.

• Thanks for your answer! I think I begin to understand. So when there are distortions (voltage and current are not perfect sine waves of the same frequency) the power factor and cos fi are not equal. And when we say power factor equels cos fi we assume that there are not any distortions. – Salekh Apr 29 '19 at 11:27
• ###RealPower can be more complicated to calculate if the voltage and current are not perfect sine waves of the same frequency### It would be nice to read about it in more detail. – Salekh Apr 29 '19 at 11:43

Best way to understand this is to think of Power Triangle.

(Image source: Envirotec Magazine - Monitoring power factor for effective energy management)

Because true power and apparent power form the adjacent and hypotenuse sides of a right triangle, respectively, the power factor ratio is also equal to the cosine of that phase angle.

Power Factor of 1 implies that all power is utilized and there is no reactive power. This can be seen by realizing simple values:

• PF = True Power/Apparent Power
• PF = 150W/165VA
• PF = 0.909, which can also be realised as cos(24.6) = 0.909

Looking from the graph above, you can see that as the reactive power is decreased in magnitude - the phase angle becomes smaller. With no reactive power component the angle becomes 0. PF of cos(0) = 1.

• That's all correct for sinusoidal voltage and current. What happens power factor when, for example, the current is non-sinusoidal as happens when there is an inverter, VFD or SMPS? – Transistor Apr 29 '19 at 9:42
• Thanks for your answer! I understand that. It is written in the same manner in textbooks. If these two quantities are not equel in general case then I am interested what the difference is that – Salekh Apr 29 '19 at 11:41