HARMONICS AND HOW THEY RELATE TO POWER FACTOR
W. Mack Grady
The University of Texas at Austin
Austin, Texas  78712
Robert J. Gilleskie
San Diego Gas & Electric
San Diego, California  92123
Abstract
We are all familiar with power factor, but are we using it to its true potential?  In this paper we investigate the effect of harmonics on power factor and show through examples why it is important to use true power factor, rather than the conventional 50/60 Hz displacement power factor, when describing nonlinear loads.
Introduction
Voltage and current harmonics produced by nonlinear loads increase power losses and, therefore, have a negative impact on electric utility distribution systems and components.  While the exact relationship between harmonics and losses is very complex and difficult to generalize, the well-established concept of power factor does provide some measure of the relationship, and it is useful when comparing the relative impacts of nonlinear loads–providing that harmonics are incorporated into the power factor definition.
Power Factor in Sinusoidal Situations
The concept of power factor originated from the need to quantify how efficiently a load utilizes the current that it draws from an AC power system.  Consider, for example, the ideal sinusoidal situation shown in Figure 1.
Vsin(wt)
Motor Load
(Linear) Figure 1:  Power System with Linear Load
The voltage and current at the load are
v(t)=V1sin (ωo t+δ1), (1) i(t)=I1sin (ωo t+θ1), (2)
where V1 and I1 are peak values of the 50/60 Hz voltage and current, and δ1 and θ1 are the relative phase angles.  The true power factor at the load is defined as the ratio of average power to apparent power, or
pf
P
S
P
V I
true
avg avg
rms rms
== . (3)
For the purely sinusoidal case, (3) becomes
pf true=pf disp=
P avg
P2+Q2
=
V1
2
I1
2
cosδ1−θ1
()
V1
2
I1
2
=cosδ1−θ1
() , (4)
where pf disp is commonly known as the displacement power factor, and where δ1−θ1
() is known as the power factor angle.  Therefore, in sinusoidal situations, there is only one power factor because true power factor and displacement power factor are equal.
For sinusoidal situations, unity power factor corresponds to zero reactive power Q, and low power factors correspond to high Q.  Since most loads consume reactive power, low power factors in sinusoidal systems can be corrected by simply adding shunt capacitors.
Power Factor in Nonsinusoidal Situations
Now, consider nonsinusoidal situations, where network voltages and currents contain harmonics.  While some harmonics are caused by system nonlinearities such as transformer saturation, most harmonics are produced by power electronic loads such as adjustable-speed drives and diode-bridge rectifiers.  The significant harmonics (above the fundamental, i.e., the first harmonic) are usually the 3rd, 5th, and 7th multiples of 50/60 Hz, so that the frequencies of interest in harmonics studies are in the low-audible range.
When steady-state harmonics are present, voltages and currents may be represented by Fourier series of the form
v(t)=V k sin (k ωo t +δk )k =1∞
∑, (5)
i(t)=I k sin (k ωo t +θk )k =1∞
∑, (6)  whose rms values can be shown to be
V rms = V k
2
2
k =1
∞∑=
V krms
2
k =1
∞
∑, (7)
I rms = I k
2
2
k =1
∞∑
=
I krms
2
reactivepowerk =1
∞
∑. (8)  The average power is given by
P avg =V krms I krms cos δk –θk ()k =1∞
∑=P 1avg +P 2avg +P 3avg + L , (9)
where we see that each harmonic makes a contribution, plus or minus, to the average power.
A frequently-used measure of harmonic levels is total harmonic distortion (or distortion factor),
which is the ratio of the rms value of the harmonics (above fundamental) to the rms value of the fundamental, times 100%, or
THD V =V krms 2k =2
∞
∑
V 1rms
• 100%=
V k 2
k =2
∞
∑V 1
• 100% ,
(10)
THD I I I I I krms k rms
k k =
•=
•=∞
=∞
∑
∑22
12
2
1
100%100%    . (11)
Obviously, if no harmonics are present, then the THD s are zero.  If we substitute (10) into (7), and (11) into (8), we find that
()V V THD rms rms V =+12
1100/, (12)
()I I THD rms rms I =+12
1100/. (13)
Now, substituting (12) and (13) into (3) yields the following exact form of true power factor, valid for both sinusoidal and nonsinusoidal situations:
pf true =P avg
V 1rms I 1rms 1+THD V /100()2
1+THD I /100()
2
. (14)
A useful simplification can be made by expressing (14) as a product of two components,
pf true =P avg V 1rms I 1rms
•
1
1+THD V /100()2
1+THD I /100()
2
,
(15)
and by making the following two assumptions:
1.
In most cases, the contributions of harmonics above the fundamental to average power in (9) are small, so that
P avg ≈P 1avg .
2. Since
THD V  is usually less than 10%, then from (12) we see that  V rms ≈V 1rms .
Incorporating these two assumptions into (15) yields the following approximate form for true power factor:
pf true ≈P avg1V 1rms I 1rms
•
1
1+THD I /100()
2
=pf disp  • pf dist . (16)
Because displacement power factor
pf disp  can never be greater than unity, (16) shows that the
true power factor in nonsinusoidal situations has the upper bound
pf true ≤pf dist =1
1+THD I /100()
2
. (17)
displacement power factors.
It is important to point out that one cannot, in general, compensate for poor distortion power factor by adding shunt capacitors.  Only the displacement power factor can be improved with capacitors.  This fact is especially important in load areas that are dominated by single-phase power electronic loads, which tend to have high displacement power factors but low distortion power factors.  In these instances, the addition of shunt capacitors will likely worsen the power factor by inducing resonances and higher harmonic levels.  A better solution is to add passive or active filters to remove the harmonics produced by the nonlinear loads, or to utilize low-distortion power electronic loads.
Power factor measurements for some common single-phase residential loads are given in Table 1, where it is seen that their current distortion levels tend to fall into the following three categories:  low (THD I≤ 20%), medium (20% <THD I≤ 50%), high (THD I> 50%).

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