# Calculating the True Power Factor in non-linear loads

In my circuit I am taking in 60 Hz 120 VAC from the power company and hooking it up to a AC/DC non-linear switching buck converter I am designing.

I know that: true power factor = (displacement power factor) x (distortion power factor) But how can I calculate the displacement power factor for a non-sinusoidal or transient current?

Example waveform:

PF := RealP / S -- by definition of power factor PF
S  := Vrms * Irms -- by definition of apparent power S


If you have those two waveforms (the instantaneous voltage and instantaneous current waveform), you can

• find Irms, the RMS flow of electrons through the load over a full cycle. Electric power supply companies prefer to make Irms as small as possible, so their power lines don't melt from the I2R heating caused by this current. (They can successfully reduce this number while keeping the real power the same by using very high voltages, mandating power factors closer to 1, etc.)
• find Vrms, the RMS voltage across the load over a full cycle. (In your case, this is pretty close to 120 VAC).
• find the apparent power S := Vrms * Irms
• find a third waveform: the instantaneous real power waveform. At any one instant, the product -- of the instantaneous voltage and the instantaneous current -- gives the power going into the load at that instant. With some loads -- such as inductive motor loads -- the power waveform alternates between positive (power going into the load) and negative (power coming out of the load).
• find the average real power RealP over a full cycle (the plain average, not the RMS).
• find the power factor PF := RealP / S

As far as I know, that's the only way to calculate the power factor for non-sinusoidal loads. (All other formulas I've seen for power factor end up doing the same or more calculations, perhaps in some other order with other names for the intermediate terms, or else are wrong).

unnecessary rant

Some textbooks imply that calculating the power factor involves first taking the cosine of something, perhaps something related to zero-crossings. That's doing it wrong. It may happen to give the currect result for perfectly linear loads with ideal sine-wave power, but not for nonlinear loads, or linear loads with our less-than-ideal not-exactly-sine-wave power. Instead, good power monitoring devices use the above equation, which gives the correct power factor even with nonlinear loads or non-sine-wave power or both.

Some textbooks imply that one can calculate the real power by multiplying the power factor times something. That is an unnecessarily round-about method of getting the real power, since the device must have already calculated the real power in order to get the power factor.

None of the formulas for calculating power factor in terms of phase shift will work for you. Those "phase angle" formulas only apply when both the voltage and the current are sinusoidal.

• Now i get it! :) Thank you! Thant makes alot of sense. Basically, pf=avg(v)*avg(i)/(rms(v)*rms(i)), over a cycle – hassan789 Nov 29 '12 at 18:27
• Almost. You really want pf=avg(v*i)/(rms(v)*rms(i)). – davidcary Nov 30 '12 at 19:27
• true! my mistake... good catch! :D – hassan789 Dec 5 '12 at 18:28

p.f. = kd * kφ

• kφ = displacement factor = cos φ1
• kφ = distortion factor = I1rms / Irms
• where, I1rms = fundamental current & Irms = total current
• φ1 = phase shift of fundamental current

You would have to filter both the voltage and current so match phase shift @ fundamental freq and measure phase shift, cos φ1 of the current.

An approximation might be to detect & measure the peak of voltage and current & measure phase shift, cos φ1, but this is subject to impulse noise but satisfactory for intuititive quick check.

• A simple FFT for the fundamental, equivalent of DQ transform, will do: [v(i)*sin,v(i)*cos] > both filtered for DC > DC1*sin+DC2*cos which is the fundamental with displacement. You can also use this (the sin component) to further calculate THD. – Vlad Nov 27 '12 at 20:30

If the peak of the current is at the same phase with the peak of the Voltage u do not need to calculate the Displacement Power Factor, only need to calculate the Distortion Power Factor. You need to calculate the THD of that waveforms, Use Fast Fourier Transform to know the exact value of the THD. Then from the THD you can count the Distortion Power Factor.

So the value of the Displacement PF = 1 and the value of the Distortion PF < 1.

Multiply it then you get True PF.