Most people don't realize that harmonics have been around a
long time. Since the first AC generator went online more than 100 years ago,
electrical systems have experienced harmonics. The harmonics at that time were
minor and had no detrimental effects.
Basic Concept
A pure sinusoidal voltage is a conceptual quantity produced
by an ideal AC generator built with finely distributed stator and field
windings that operate in a uniform magnetic field. Since neither the winding
distribution nor the magnetic field are uniform in a working AC machine,
voltage waveform distortions are created, and the voltage-time relationship
deviates from the pure sine function. The distortion at the point of generation
is very small (about 1% to 2%), but nonetheless it exists. Because this is a
deviation from a pure sine wave, the deviation is in the form of a periodic
function, and by definition, the voltage distortion contains harmonics.
When a sinusoidal voltage is applied to a certain type of
load, the current drawn by the load is proportional to the voltage and
impedance and follows the envelope of the voltage waveform. These loads are
referred to as linearloads (loads where the voltage and current follow one
another without any distortion to their pure sine waves). Examples of linear
loads are resistive heaters, incandescent lamps, and constant speed induction
and synchronous motors.
In contrast, some loads cause the current to vary
disproportionately with the voltage during each half cycle. These loads are
classified as nonlinear loads, and the current and voltage have waveforms that
are nonsinusoidal, containing distortions, whereby the 60-Hz waveform has
numerous additional waveforms superimposed upon it, creating multiple
frequencies within the normal 60-Hz sine wave. The multiple frequencies are
harmonics of the fundamental frequency.
Normally, current distortions produce voltage distortions.
However, when there is a stiff sinusoidal voltage source (when there is a low
impedance path from the power source, which has sufficient capacity so that
loads placed upon it will not effect the voltage), one need not be concerned
about current distortions producing voltage distortions.
Examples of nonlinear loads are battery chargers, electronic
ballasts, variable frequency drives, and switching mode power supplies. As
nonlinear currents flow through a facility's electrical system and the
distribution-transmission lines, additional voltage distortions are produced
due to the impedance associated with the electrical network. Thus, as
electrical power is generated, distributed, and utilized, voltage and current
waveform distortions are produced.
Power systems designed to function at the fundamental
frequency, which is 60-Hz in the United States, are prone to unsatisfactory
operation and, at times, failure when subjected to voltages and currents that
contain substantial harmonic frequency elements. Very often, the operation of
electrical equipment may seem normal, but under a certain combination of
conditions, the impact of harmonics is enhanced, with damaging results.
Motors
There is an increasing use of variable frequency drives
(VFDs) that power electric motors. The voltages and currents emanating from a
VFD that go to a motor are rich in harmonic frequency components. Voltage
supplied to a motor sets up magnetic fields in the core, which create iron
losses in the magnetic frame of the motor. Hysteresis and eddy current losses
are part of iron losses that are produced in the core due to the alternating
magnetic field. Hysteresis losses are proportional to frequency, and eddy
current losses vary as the square of the frequency. Therefore, higher frequency
voltage components produce additional losses in the core of AC motors, which in
turn, increase the operating temperature of the core and the windings
surrounding in the core. Application of non-sinusoidal voltages to motors
results in harmonic current circulation in the windings of motors. The net rms
current is
Irms = √[(I1)2 + (I2)2 + (I3)2 + …], where the subscripts 1,
2, 3, etc. represent the different harmonic currents. The I2R losses in the
motor windings vary as the square of the rms current. Due to skin effect,
actual losses would be slightly higher than calculated values. Stray motor
losses, which include winding eddy current losses, high frequency rotor and
stator surface losses, and tooth pulsation losses, also increase due to harmonic
voltages and currents.
The phenomenon of torsional oscillation of the motor shaft
due to harmonics is not clearly understood, and this condition is often
disregarded by plant personnel. Torque in AC motors is produced by the
interaction between the air gap magnetic field and the rotor-induced currents.
When a motor is supplied non-sinusoidal voltages and currents, the air gap
magnetic fields and the rotor currents contain harmonic frequency components.
The harmonics are grouped into positive (+), negative (-)
and zero (0) sequence components. Positive sequence harmonics (harmonic numbers
1, 4, 7, 10, 13, etc.) produce magnetic fields and currents rotating in the
same direction as the fundamental frequency harmonic. Negative sequence
harmonics (harmonic numbers 2, 5, 8, 11, 14, etc.) develop magnetic fields and
currents that rotate in a direction opposite to the positive frequency set.
Zero sequence harmonics (harmonic numbers 3, 9, 15, 21, etc.) do not develop
usable torque, but produce additional losses in the machine. The interaction
between the positive and negative sequence magnetic fields and currents
produces torsional oscillations of the motor shaft. These oscillations result
in shaft vibrations. If the frequency of oscillations coincides with the
natural mechanical frequency of the shaft, the vibrations are amplified and
severe damage to the motor shaft may occur. It is important that for large VFD
motor installations, harmonic analyses be performed to determine the levels of
harmonic distortions and assess their impact on the motor.
Transformers
The harmful effects of harmonic voltages and currents on
transformer performance often go unnoticed until an actual failure occurs. In
some instances, transformers that have operated satisfactorily for long periods
have failed in a relatively short time when plant loads were changed or a
facility's electrical system was reconfigured. Changes could include
installation of variable frequency drives, electronic ballasts, power factor
improvement capacitors, arc furnaces, and the addition or removal of large
motors.
Application of nonsinusoidal excitation voltages to
transformers increase the iron lesses in the magnetic core of the transformer
in much the same way as in a motor. A more serious effect of harmonic loads
served by transformers is due to an increase in winding eddy current losses.
Eddy currents are circulating currents in the conductors induced by the
sweeping action of the leakage magnetic field on the conductors. Eddy current
concentrations are higher at the ends of the transformer windings due to the
crowding effect of the leakage magnetic fields at the coil extremities. The
eddy current losses increase as the square of the current in the conductor and
the square of its frequency. The increase in transformer eddy current loss due
to harmonics has a significant effect on the operating temperature of the
transformer. Transformers that are required to supply power to nonlinear loads
must be derated based on the percentages of harmonic components in the load
current and the rated winding eddy current loss.
One method of determining the capability of transformers to
handle harmonic loads is by k factor ratings. The k factor is equal to the sum
of the square of the harmonic currents multiplied by the square of the frequencies.
k = [([I.sub.1]).sup.2]([1.sup.2]) +
[([I.sub.2]).sup.2]([2.sup.2]) + [([I.sub.3]).sup.2]([3.sup.2]) + . . . +
[([I.sub.n]).sup.2]([n.sup.2]).
where [I.sub.1] = ratio of fundamental current to total rms
current, [I.sub.2] = ratio of second harmonic current to total rms current,
[I.sub.3] = ratio of third harmonic current to total rms current, etc., and
1,2,3, ... n are harmonic frequency numbers. The total rms current is the
square root of the sum of square of the individual currents.
By providing additional capacity (larger-size or multiple
winding conductors), k factor rated transformers are capable of safely
withstanding additional winding eddy current losses equal to k times the rated
eddy current loss. Also, due to the additive nature of triplen harmonic (3, 9,
15, etc.) currents flowing in the neutral conductor, k rated transformers are
provided with a neutral terminal that is sized at least twice as large as the
phase terminals.
Example: A transformer is required to supply a nonlinear
load comprised of 200A of fundamental (60 Hz), 30A of 3rd harmonic, 48A of 5th
harmonic and 79A of 7th harmonic. Find the required k factor rating of the
transformer:
Total rms current, I = [square root of [([I.sub.1]).sup.2] +
[([I.sub.3]).sup.2] + [([I.sub.5]).sup.2] + [([I.sub.7]).sup.2]]
Total rms current, I = [square root of [(200).sup.2] +
[(30).sup.2] + [(48).sup.2] + [(79).sup.2]] = 222.4A
[I.sub.1] = 200 / 222.4 = 0.899
[I.sub.3] = 30 / 222.4 = 0.135
[I.sub.5] = 48 / 222.4 = 0.216
[I.sub.7] = 79 / 222.4 = 0.355
k = [(0.899).sup.2][(1).sup.2] + [(0.135).sup.2] [(3).sup.2]
+ [(0.216).sup.2]([5).sup.2] + [(0.355).sup.2][(7).sup.2] = 8.31
To address the harmonic loading in this example, you should
specify a transformer capable of supplying a minimum of 222.4A with a k rating
of 9. Of course, it would be best to consider possible load growth and adjust
the minimum capacity accordingly.
The photo (on page 33) shows one of the things that can
happen when large nonlinear loads are present in a transformer. In this case,
the nonlinear loads caused a substantial temperature rise. The unit had been
installed to serve an online UPS source that produced high harmonic currents in
the lines coming from the transformer. The darkened areas of the coils are due
to the effect of heat caused by excess eddy current losses in the transformer's
windings. Very often, the damage to the coils in a transformer is not known
until a failure occurs.
