The Power Factor Triangle:
Real , Reactive and Apparent Power

Power Factor Formulas and the Power Factor Triangle, now that you understand the terminology, are easily within your grasp. The key to the Power Factor Triangle is understanding vectors. I'll make vectors easy and in a few minutes you'll have it.

Vector Analysis
Vectors are simply another way to draw sine waves. You'll see its not difficult and actually makes things easier. Vectors, as used in this discussion, are representations of a sine wave of current relative to a sine wave of voltage. Instead of showing the current as a sine wave, the vector shows it as a straight line that points in a direction. The length of the line represents the RMS value of the current (remember, peak value x 0.707) and the direction of the line represents the phase angle of the current relative to the voltage (this is the "offset" as discussed in the "Power Factor" section).

The direction of the arrow is simple. Its like a compass, 0 to 360 degrees. Remember, this is the offset. The compass starts at 3 o'clock and rotates counter-clockwise in a full circle. We establish that the voltage's sine wave peaks at 0 degrees (this is the reference).

Lets look at cases where the currents leads or lags voltage by 45 degrees. See the graphic below.


Current B lags the voltage by 45 degrees so the vector points down and to the right (see below). Current A leads by 45 degrees (it's happening 45 degrees ahead of voltage) so it points up and to the right. Since the peak currents are 1 Amp, the RMS currents are 0.707 Amps.


Adding Vectors
This is almost as fun as connecting the dots (an EE's childhood pastime). To add two RMS currents, simply put one vector at the end of the other (remember to point in the correct direction, not 180 degrees out). The resultant, a straight line from where you started to where you ended, is the sum.


Hopefully that makes sense to you. Now lets apply what you already know....the load current in motors, which does the actual work, is in-phase with the voltage and therefore the load current vector points at 0 degrees (left to right). The magnetizing current in motors (and transformers), which does no work and lags behind the voltage by 90 degrees, points straight down. Oh yeah, capacitive current, which also does no work and leads voltage by 90 degrees, points straight up. (See below)


Good news! Because the Power Factor Triangle is all about examining the relationship between load currents and reactive currents, we will simply be adding currents that are in-phase with voltage (vectors pointing at 0 degrees) to currents that are 90 degrees out of phase (magnetizing or capacitive). See the graphic below, which only has load and magnetizing currents.


Back in the Power Factor Terminology section we learned that Load Current multiplied by the system voltage yields Real Load, also called Real Power. The units of Real Power are Watts (i.e. kW, MW, etc.)

When we multiply the Reactive Current by the system voltage we get the Reactive Load or Reactive Power (also called Imaginary Load). The units of Reactive Power are VARs, which stands for Volts-Amps-Reactive (i.e. kVAR, MVARS, etc.).

If we multiply the Apparent Current by the system voltage we get the Apparent Load or Apparent Power. The units for Apparent Power are VA, for Volt-Amps (i.e. kVA, MVA, etc.).

Changing the above graphic into terms of Power yields the Power Factor Triangle, also called the Power Triangle.
The Power Factor Triangle (below) shows that

(Real Power squared) + (Reactive Power squared) = (Apparent Power squared)

Yup! Pythagorean's Theorem....and you thought you were done with that in high school geometry!


Remember, the units are all in terms of RMS values and RMS values cannot be simply added together unless the components have no phase difference. We use vector addition to add RMS values because Real Power and Reactive Power are 90 degrees apart.

The Power Factor Triangle yields some useful equations.
This one is very useful and easy to remember. Simply stated, the Power Factor is the percentage of Apparent Power that does real work.

Apparent Power x PF = Real Power

Or in terms of units,
VA x PF = Watts

Now lets learn about Power Factor Correction!