Power factor and cos φ

Question

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.

Please help me, as I cannot continue my education without understanding.

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.

Answer 1

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.

Answer 2

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.

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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.

There is more information on calculation here on Wikipedia

Answer 3

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.