1. DEFINITION AND ANALYSIS OF HARMONICS

Electricity generation is normally produced at constant frequencies

of 50 Hz or 60 Hz and the generator’s e.m.f. can be considered

practically sinusoidal. However, when a source of sinusoidal voltage

is applied to a nonlinear device or load, the resulting current is not

perfectly sinusoidal. In the presence of system impedance this

current causes a non-sinusoidal voltage drop and, therefore,

produces voltage distortion at the load terminals, i.e. the latter

contains harmonics.

Power system harmonics are defined as sinusoidal voltage and

currents at frequencies that are integer multiples of the main

generated (or fundamental) frequency. They constitute the major

distorting components of the mains voltage and load current

waveforms.

In the early 1800s, French mathematician, Jean Baptiste Fourier

formulated that a periodic nonsinusoidal function of a fundamental

frequency f may be expressed as the sum of sinusoidal functions of

frequencies which are multiples of the fundamental frequency.

For the periodic nonsinusoidal voltage wave form, the simplified

Fourier expression states:

v(t) = V0 + V1 sin(ωt) + V2 sin(2ωt) + … + Vn sin(nωt) +……. (1)

V0 represents the constant or the DC component of the waveform.

V1, V2, V3, …, Vn are the peak values of the successive terms of the

expression. The terms are known as the harmonics of the periodic

waveform.

FIGURE 1 Nonsinusoidal voltage waveform

FIGURE 2 Creation of Nonsinusoidal waveform by adding the fundamental and third harmonic frequency waveforms.

The fundamental (or first harmonic) frequency has a frequency of f,

the second harmonic has a frequency of 2f, the third harmonic has a

frequency of 3f, and the nth harmonic has a frequency of nf. If the

fundamental frequency is 50 Hz (as in the India), the second

harmonic frequency is 100 Hz, and the third harmonic frequency is

150 Hz.

The ability to express a nonsinusoidal waveform as a sum of

sinusoidal waves allows us to use the more common mathematical

expressions and formulas to solve power system problems. In order

to find the effect of a nonsinusoidal voltage or current on a piece of

equipment, we only need to determine the effect of the individual

harmonics and then vectorially sum the results to derive the net

effect. Figure 2 illustrates how individual harmonics that are

sinusoidal can be added to form a nonsinusoidal wave form.

The Fourier expression in Eq. (1) has been simplified to clarify the

concept behind harmonic frequency components in a nonlinear

periodic function.

v (t )=Vo+∑n=1

∞

❑(an cosnωt+bn sinnωt ) (for n=1¿∞ )(Eq .4 )

w here ,ω=2 πf=2πT

V o=1T∫o

T

❑v (t )dt

an=2T∫0

T

❑v ( t ) cosnωt dt

bn=2T∫0

T

❑v ( t ) sinnωtdt

where an and bn are the coefficients of the individual harmonic terms

or components. Under certain conditions, the cosine or sine terms

can vanish, giving us a simpler expression.

If the function is an even function, meaning f(–t) = f (t), then the

sine terms vanish from the expression. If the function is odd, with

f(–t) = –f(t), then the cosine terms disappear.

Half Wave Symmetry:

From figure 3, we can say that ( t ) , system (main) voltage, possesses half-wave symmetry. i.e.,

v (t )=−v (t−12T ) Or v (t )=−v (t+ 1

2T )

i.e. the shape of the waveform over a period t + T /2 to t + T is the negative of the shape of the waveform over the period t to t + T /2. Consequently ,the voltage waveform of figure below has half wave symmetry

FIGURE 3 Nonlinear waveform with halfwave symmetry by adding the fundamental and third harmonic frequency waveforms.

From the following derivation, it is shown that the Fourier series of any function which has half-wave symmetry contains only odd harmonics.

Let us consider the Coefficients an

an=2T

∫−T /2

T /2

❑ f ( t ) cosnωt dt

¿ 2T [ ∫

−T /2

0

❑ f ( t ) cosnω t dt+∫0

T /2

❑ f ( t ) cosnωt dt ]Which we may represent as

an=2T

(I 1+ I 2)

Now we substitute the new variable τ=t+12T in the integral I 1:

I 1=∫0

T2

❑ f (τ−12T )cosnωo(τ−1

2T )dτ

¿∫0

T2

❑−f (τ )(cosnωo τ cosnωT2+Sinnωo τ

SinnωT2 )dτ

But ωT is 2π , and thus, sinnωT

2=Sinnπ=0

Hence

I 1=−Cosnπ ∫0

T /2

❑ f ( τ ) cosnωτdτ

After noting the form of I 2, we therefore may write

an=2T

(1−Cosnπ )∫0

T2

❑ f (t ) cosnωtdt

The factor (1−Cosnπ )indicates that an is zero if n is even.

Thus an=4T∫0

T2

❑ f (t )Cosnωtdt ,when n is odd

¿0w hen n is even

A similar investigation shows that bn is also zero for all even n, and therefore

bn=4T∫0

T2

❑ f (t )Sinnωtdt ,when n is odd

¿0w hen n is even

Thus, For symmetrical waveforms, with half wave symmetry, all the even numbered harmonics are nullified.

2. ODD AND EVEN ORDER HARMONICS

As their names imply, odd harmonics have odd numbers (e.g., 3, 5,

7, 9, 11), and even harmonics have even numbers (e.g., 2, 4, 6, 8,

10). Harmonic number 1 is assigned to the fundamental frequency

component of the periodic wave.

Harmonic number 0 represents the constant or DC component of

the waveform. The DC component is the net difference between the

positive and negative halves of one complete waveform cycle. The

DC component of a waveform has undesirable effects, particularly

on transformers, due to the phenomenon of core saturation.

The majority of nonlinear loads produce harmonics that are odd

multiples of the fundamental frequency. Certain conditions need to

exist for production of even harmonics. Uneven current draw

between the positive and negative halves of one cycle of operation

can generate even harmonics. The uneven operation may be due to

the nature of the application or could indicate problems with the

load circuitry.

3. HARMONIC PHASE ROTATION AND PHASE ANGLE

RELATIONSHIP

In a balanced three-phase electrical system, the voltages and

currents have a positional relationship as shown in Figure 4. The

three voltages are 120˚ apart and so are the three currents. The

normal phase rotation or sequence is a–c–b, which is counter

clockwise and designated as the positive-phase sequence.

For harmonic analyses, these relationships are still applicable, but

the fundamental components of voltages and currents are used as

reference. All other harmonics use the fundamental frequency as

the reference.

The fundamental frequency current components in a three-phase

power system have a positive-phase sequence (counter clockwise).

ia1 = Ia1 sin wt

ib1 = Ib1 sin (wt-120°)

ic1 = Ic1 sin (wt-240°)

The negative displacement angles indicate that the fundamental

phasors ib1 and ic1 trail the ia1 phasor by the indicated angle. Figure

5a shows the fundamental current phasors.

The expressions for the third harmonic currents are:

ia3 = Ia3 sin 3wt

ib3 = Ib3 sin 3(wt-120°) = Ib3 sin (3wt-360°) = Ib3 sin 3wt

ic3 = Ic3 sin 3(wt-240°) = Ic3 sin (3wt-720°) = Ic3 sin 3wt

The expressions for the third harmonics show that they are in phase

and have zero displacement angle between them. Figure 5b shows

the third harmonic phasors. The third harmonic currents are known

as zero sequence harmonics due to the zero displacement angle

between the three phasors.

The expressions for the fifth harmonic currents are:

ia5 = Ia5 sin 5wt

ib5 = Ib5 sin 5(wt-120°) = Ib5 sin(5wt-600°) = Ib5 sin(5wt-240°)

ic5 = Ic5 sin 5(wt-240°) = Ic5 sin(5wt-1200°) = Ic5 sin(5wt-120°)

Figure 5c shows the fifth harmonic phasors. Note that the phase

sequence of the fifth harmonic currents is clockwise and opposite to

that of the fundamental. The fifth harmonics are negative sequence

harmonics.

FIGURE 5 (a) Fundamental phasors. (b) Third harmonic phasors. (c) Fifth harmonic phasors. (d) Seventh harmonic phasors.

Similarly the expressions for the seventh harmonic currents are:

ia7 = Ia7 sin 7wt

ib7 = Ib7 sin 7(wt-120°) = Ib7 sin(7wt-840°) = Ib7 sin(7wt-120°)

ic7 = Ic7 sin 7(wt-240°) = Ic7 sin(7wt-1680°) = Ic7 sin(7wt-240°)

Figure 5d shows the seventh harmonic current phasors. The seventh

harmonics have the same phase sequence as the fundamental and

are positive sequence harmonics. Similar way phase sequence of

even harmonics current phasors can be calculated.

Table 1 categorizes the harmonics in terms of their respective

sequence orders.

4. CAUSES OF VOLTAGE AND CURRENT HARMONICS

A pure sinusoidal waveform with zero harmonic distortion is a

hypothetical quantity and not a practical one. The voltage

waveform, even at the point of generation, contains a small amount

of distortion due to nonuniformity in the excitation magnetic field

and discrete spatial distribution of coils around the generator stator

slots. The distortion at the point of generation is usually very low,

typically less than 1.0%.

The generated voltage is transmitted many hundreds of miles,

transformed to several levels, and ultimately distributed to the

power user. The user equipment generates currents that are rich in

harmonic frequency components, especially in large commercial or

industrial installations.

As harmonic currents travel to the power source, the current

distortion results in additional voltage distortion due to impedance

voltages associated with the various power distribution equipment,

such as transmission and distribution lines, transformers, cables,

buses, and so on.

Figure 6 illustrates how current distortion is transformed into

voltage distortion. Not all voltage distortion, however, is due to the

flow of distorted current through the power system impedance.

Static uninterruptible power source (UPS) systems can generate

appreciable voltage distortion due to the nature of their operation.

Normal AC voltage is converted to DC and then reconverted to AC in

the inverter section of the UPS. Unless waveform shaping circuitry is

provided, the voltage waveforms generated in UPS units tend to be

distorted.

As nonlinear loads are propagated into the power system, voltage

distortions are introduced which become greater moving from the

source to the load because of the circuit impedances. Current

distortions for the most part are caused by loads. Even loads that

are linear will generate nonlinear currents if the supply voltage

waveform is significantly distorted.

When several power users share a common power line, the voltage

distortion produced by harmonic current injection of one user can

affect the other users. This is why standards are being issued that

will limit the amount of harmonic currents that individual power

users can feed into the source.

The major causes of current distortion are nonlinear loads due to

adjustable speed drives, fluorescent lighting, rectifier banks,

computer and data-processing loads, arc furnaces, and so on. One

can easily visualize an environment where a wide spectrum of

harmonic frequencies are generated and transmitted to other loads

or other power users, thereby producing undesirable results

throughout the system.

FIGURE 6 Voltage distortion due to current distortion. The gradient graph indicates how distortion changes from source to load.

5. HARMONIC SIGNATURES

Many of the loads installed in present-day power systems are

harmonic current generators. Combined with the impedance of the

electrical system, the loads also produce harmonic voltages. The

nonlinear loads may therefore be viewed as both harmonic current

generators and harmonic voltage generators. Prior to the 1970s,

speed control of AC motors was primarily achieved using belts and

pulleys. Now, adjustable speed drives (ASDs) perform speed control

functions very efficiently. ASDs are generators of large harmonic

currents. Fluorescent lights use less electrical energy for the same

light output as incandescent lighting but produce substantial

harmonic currents in the process. The explosion of personal

computer use has resulted in harmonic current proliferation in

commercial buildings.A. FLUORESCENT LIGHTING

Figure 8 shows a current waveform at a distribution panel supplying

exclusively fluorescent lights. The waveform is primarily comprised

of the third and the fifth harmonic frequencies. The individual

current harmonic distortion makeup is provided in Table 2. The

waveform also contains slight traces of even harmonics, especially

of the higher frequency order. The current waveform is flat topped

due to initiation of arc within the gas tube, which causes the voltage

across the tube and the current to become essentially unchanged

for a portion of each half of a cycle.

FIGURE 8 Nonlinear current drawn by fluorescent lighting

B. PERSONAL COMPUTER

Figures 10 show the nonlinear current characteristics of a personal

computer. Tables 3 provide the harmonic content of the currents.

The predominance of the third and fifth harmonics is evident. The

current THD for both devices exceeds 100%, as the result of high

levels of individual distortions introduced by the third and fifth

harmonics. The total current drawn by a personal computer and its

monitor is less than 2 A, but a typical high-rise building can contain

several hundred computers and monitors. The net effect of this on

the total current harmonic distortion of a facility is not difficult to

visualize. So far we have examined some of the more common

harmonic current generators. The examples illustrate that a wide

spectrum of harmonic currents is generated. Depending on the size

of the power source and the harmonic current makeup, the

composite harmonic picture will be different from facility to facility.

FIGURE 10 Nonlinear current drawn by single personal computer.

C. Variable Speed Drives / UPS:

Variable speed controllers, UPS units and DC converters in general

are usually based on the three-phase bridge, also known as the six-

pulse bridge because there are six pulses per cycle (one per half

cycle per phase) on the DC output.

The six pulse bridge produces harmonics at (6n 1), i.e. at one

more and one less than each multiple of six. In theory, the

magnitude of each harmonic is the reciprocal of the harmonic

number, so there would be 20% fifth harmonic and 9% eleventh

harmonic, etc.

The magnitude of the harmonics is significantly reduced by the use

of a twelve-pulse bridge. This is effectively two six-pulse bridges, fed

from a star and a delta transformer winding, providing a 30 degrees

phase shift between them.

The 6n harmonics are theoretically removed, but in practice, the

amount of reduction depends on the matching of the converters and

is typically by a factor between 20 and 50. The 12n harmonics

remain unchanged. Not only is the total harmonic current reduced,

but also those that remain are of a higher order making the design

of the filter much easier.

Often the equipment manufacturer will have taken some steps to

reduce the magnitudes of the harmonic currents, perhaps by the

addition of a filter or series inductors. In the past this has led some

manufacturers to claim that their equipment is ‘G5/3’ compliant.

Since G5/3 is a planning standard applicable to a complete

installation, it cannot be said to have been met without knowledge

of every piece of equipment on the site.

A further increase in the number of pulses to 24, achieved by using

two parallel twelve-pulse units with a phase shift of 15 degrees,

reduces the total harmonic current to about 4.5%. The extra

sophistication increases cost, of course, so this type of controller

would be used only when absolutely necessary to comply with the

electricity suppliers’ limits.

6. EFFECT OF HARMONICS ON POWER SYSTEM DEVICES

Transformers

Harmonics can affect transformers in two ways.

Voltage harmonics produce additional losses in the transformer core as the higher frequency harmonic voltages set up hysteresis loops, which super impose on the fundamental loop.

Each loop represents higher magnetization power requirement and higher core losses.

The harmonic currents increase the net RMS current flowing in the transformer winding which results in additional I 2R losses.

Winding eddy current losses are also increased winding eddy current are circulating current s induced in the conductor by the leakage magnetic flux.

Eddy current due to harmonics an significantly increase the transformer winding temperature.

Transformer that are required to supply large nonlinear loads must be derated to handle the harmonics.

This derated factor is based on the percentage of the harmonic current in the load and the rated winding eddy current losses.

Also, transformers that supply large third harmonic generated loads should have neutral oversized.

Alternators:

In alternators, the primary source orcause, of harmonics in the emf wave form is due to the non-sinusoidal field flux wave form.

Due to non-uniform distribution of the field flux and armature reaction in ac machines, the current and voltage waves may get distorted. Such wave forms are referred to as non-sinusoidal or complex wave forms.

The sinusoidal components of a complex wave are called harmonics.

In an alternator field flux wave is symmetrical, i.e. field flux wave has equal positive and negative half cycles and as a consequence field flux wave can’t contain even field space harmonics.

Thus no even harmonics can be generated-hence the output emf is free from even harmonics.

Since the negative of the wave is a reproductionof the positive half, the even harmonics are absent.

AC Motors

Application of distorted voltage to a motor results additional losses in the magnetic core of the motor.

Hysteresis and eddy current losses in core increase as higher frequency harmonic voltages are impressed on the motor windings.

Hysteresis losses increase with frequency and eddy current losses increase as the square of the frequency.

Also harmonic currents produce additional I 2R losses in the motor windings which must be accounted for.

Large motors supplied from adjustable speed drives (ASD,s) are usually provided with harmonic filters to prevent motor damage due to harmonics.

Capacitor Banks

Capacitor banks are designed to operate at a maximum voltage of 110 % of their rated voltages and at 135% of their rated kVARS.

When large levels of voltage and current harmonics are present, the ratings are quite often exceeded, resulting in failures.

Because the reactance of a capacitor bank is inversely proportion to frequency, harmonic currents can find their way into a capacitor bank.

Capacitor bank acts as sink, absorbing stray harmonic currents and causing over loads and subsequent failure of the bank.

A more serious condition with potential for substantial damage occurs due to a phenomenon called harmonic resonance.

Resonant conditions are created when the inductive and capacitive reactance’s become equal at one of the harmonic frequencies.

If a high level of harmonic voltage or current corresponding to the resonance frequency exists in a power system, considerable damage to the capacitor bank as well as other power system devices can result.

Cables

Current flowing in a cable produces I 2R losses. When the load current contains harmonic current, additional losses are introduced.

The effective resistance of the cable increases with frequency because of the phenomenon known as skin effect.

Skin effect is due to unequal flux linkage across the cross section of the conductor which causes AC currents to flow only on the outer periphery of the conductor.

Protective Devices

Harmonic currents influence the operation of protective devices.

Fuses and motor thermal overload devices are prone to nuisance operation when subjected to nonlinear currents.

This factor should be given due consideration when sizing protective devices for use in a harmonic environment.

Electromechanical relays are also affected by harmonics.

Depending on the design an electromechanical relay may operate faster oy slower than the expected times for operation at the fundamental frequency alone.

Neutral conductor over-heating in distridution system

In a three-phase system, when each phase is equally loaded,

current in the neutral is zero. When the loads are not balanced only

the net out of balance current flows in the neutral. In the past,

installers have taken advantage of this fact by installing half-sized

neutral conductors.

However, the positive sequence and negative sequence harmonics

as categorized in above table 1 cancel out, the harmonic currents

that are an odd multiple of three times the fundamental, the ‘triple-

N’ harmonics, add in the neutral as these are zero sequence

currents. 70% third harmonic current in each phase results in 210%

current in the neutral.

Case studies in commercial buildings generally show neutral

currents between 150% and 210% of the phase currents, often in a

half-sized conductor.

7. REMEDIES TO REDUCE HARMONICS PROBLEMS

(1) Over sizing Neutral Conductors

In three phase circuits with shared neutrals, it is common to

oversize the neutral conductor up to 200% when the load served

consists of non-linear loads. For example, most manufacturers of

system furniture provide a 10 AWG conductor with 35 amp

terminations for a neutral shared with the three 12 AWG phase

conductors.

In feeders that have a large amount of non-linear load, the feeder

neutral conductor and panel board bus bar should also be oversized.

(2) Using Separate Neutral Conductors

On three phase branch circuits, another philosophy is to not

combine neutrals, but to run separate neutral conductors for each

phase conductor. This increases the copper use by 33%. While this

successfully eliminates the addition of the harmonic currents on the

branch circuit neutrals, the panel board neutral bus and feeder

neutral conductor still must be oversized.

Oversizing Transformers and Generators: The oversizing of

equipment for increased thermal capacity should also be used for

transformers and generators which serve harmonics-producing

loads. The larger equipment contains more copper.

(3) Passive filters

Passive filters are used to provide a low impedance path for

harmonic currents so that they flow in the filter and not the supply.

The filter may be designed for a single harmonic or for a broad band

depending on requirements.

Simple series band stop filters are sometimes proposed, either in

the phase or in the neutral. A series filter is intended to block

harmonic currents rather than provide a controlled path for them so

there is a large harmonic voltage drop across it.

This harmonic voltage appears across the supply on the load side.

Since the supply voltage is heavily distorted it is no longer within

the standards for which equipment was designed and warranted.

Some equipment is relatively insensitive to this distortion, but some

is very sensitive. Series filters can be useful in certain

circumstances, but should be carefully applied; they cannot be

recommended as a general purpose solution.

(4) Isolation transformers

As mentioned previously, triple-N currents circulate in the delta

windings of transformers. Although this is a problem for transformer

manufacturers and specifiers – the extra load has to be taken into

account it is beneficial to systems designers because it isolates

triple-N harmonics from the supply.

The same effect can be obtained by using a ‘zig-zag’ wound

transformer. Zig-zag transformers are star configuration auto

transformers with a particular phase relationship between the

windings that are connected in shunt with the supply.

(5) Active Filters

The solutions mentioned so far have been suited only to particular

harmonics, the isolating transformer being useful only for triple-N

harmonics and passive filters only for their designed harmonic

frequency. In some installations the harmonic content is less

predictable.

In many IT installations for example, the equipment mix and

location is constantly changing so that the harmonic culture is also

constantly changing. A convenient solution is the active filter or

active conditioner.

The active filter is a shunt device. A current transformer measures

the harmonic content of the load current, and controls a current

generator to produce an exact replica that is fed back onto the

supply on the next cycle. Since the harmonic current is sourced

from the active conditioner, only fundamental current is drawn from

the supply. In practice, harmonic current magnitudes are reduced

by 90%, and, because the source impedance at harmonic

frequencies is reduced, voltage distortion is reduced.

(6) K-Rated Transformers

Special transformers have been developed to accommodate the

additional heating caused by these harmonic currents. These types

of transformers are now commonly specified for new computer

rooms and computer lab facilities.

(7) Special Transformers

There are several special types of transformer connections which

can cancel harmonics. For example, the traditional delta-wye

transformer connection will trap all the triplen harmonics (third,

ninth, fifteenth, twenty-first, etc.) in the delta.

Additional special winding connections can be used to cancel other

harmonics on balanced loads. These systems also use more copper.

These special transformers are often specified in computer rooms

with well balanced harmonic producing loads such as multiple input

mainframes or matched DASD peripherals.

(8) Filtering

While many filters do not work particularly well at this frequency

range, special electronic tracking filters can work very well to

eliminate harmonics.

These filters are presently relatively expensive but should be

considered for thorough harmonic elimination.

(9) Special Metering

Standard clamp-on ammeters are only sensitive to 50 Hertz current,

so they only tell part of the story. New “true RMS” meters will sense

current up to the kilohertz range. These meters should be used to

detect harmonic currents. The difference between a reading on an

old style clamp-on ammeter and a true RMS ammeter will give you

an indication of the amount of harmonic current present.

The measures described above only solve the symptoms of the

problem. To solve the problem we must specify low harmonic

equipment. This is most easily done when specifying electronic

ballasts. Several manufacturers make electronic ballasts which

produce less than 15 % harmonics. These ballasts should be

considered for any ballast retrofit or any new project. Until low

harmonics computers are available, segregating these harmonic

loads on different circuits, different panel boards or the use of

transformers should be considered. This segregation of “dirty” and

“clean” loads is fundamental to electrical design today. This equates

to more branch circuits and more panel boards, thus more copper

usage.

Reference Books:

1. Power Quality by C Sankaran

2. Power System Harmonics, Second Edition J. Arrillaga, N.R. Watson

3. Engineering Circuit Analysis, Seventh Edition, William H.Hayt,Jr, Jack

E.Kemmerly, Steven M. Durbin