Harmonic 1112 Sem2 A1 PERDANA

MEP 1563

HARMONICS

DR AHMAD SAFAWI MOKHTAR

INTRODUCTION

Harmonics have always been present in power systems. Due to the widespread use of power electronic systems resulting in an increase in their magnitude, they have become a key issue in installations.

Harmonic disturbances come generally from equipment with a non‐linear voltage/current characteristic.

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INTRODUCTIONNowadays a large part of industrial, commercial and domestic loads is non‐linear, making the distortion level on the low‐voltage (but not only) supply network a serious concern.

Linear loads are comparatively rare today: the only examples which can be considered as common are undimmed filament bulbs and unregulated heaters.

Non‐linear loads represent a large percentage of the total loads.

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INTRODUCTIONUnder these conditions, total harmonic distortion (THD) may become very high and therefore dangerous for the system.

Harmonic distortion can be considered as a sort of pollution of the electric system which can cause problems if the sum of the harmonic currents exceeds certain limits. Knowledge of electromagnetic disturbances associated with this phenomenon is still developing; for this reason harmonics are currently an issue of great interest.

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DESCRIPTION OF THE PHENOMENA

A harmonic is defined as a component with a frequency that is an integer multiple (the so called order of harmonic h) of the fundamental frequency (Figure 1).

The harmonic number indicates the harmonic frequency: the first harmonic is the fundamental frequency (50 or 60 Hz), the second harmonic is the component with frequency two times the fundamental (100 or 120 Hz), and so on.

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Figure 1: A sinusoidal waveform with fundamental frequency 50 Hz and its harmonics: (a) second (100 Hz); (b) third (150 Hz); (c) fourth (200 Hz); (d) fifth (250 Hz)

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A distorted waveform (i.e. different from the sinusoidal) of a chosen, arbitrary shape can be obtained by superposition of sinusoidal waveforms of various frequencies and amplitudes.

For example, the rectangular waveform in Figure 2 is the result of the summation of an indefinite number of harmonics whose amplitudes are decreasing in inverse proportion to their order (h), and their frequencies f(n) = (2k+1)50 Hz, k = 0 1 2 are odd multiples of the fundamental harmonic f(1) = 50 Hz.

7 Composition of Distorted Waveform


Figure 2: Rectangular waveform as the superposition of the fundamental and odd harmonics



Neglecting harmonics of small amplitude and taking into account a finite number of components softens the sharp top of the waveform and reduces the steepness of its edges.

The waveform shape depends not only on the frequencies and amplitudes of harmonic components but also on their mutual phase shift. Figure 3 illustrates the influence of harmonic amplitude and phase on the resultant waveform shape.



Figure 3 : Waveforms differing in the amplitude and phase of a distorting harmonic: (a) 1‐100 %, 5‐15 %, 15 = 0; (b) 1–100 %, 5–30 %, 15 = 0; (c) 1–100 %, 5–15 %, 15 = 180



The waveforms in Figure 3(a) and (b) contain, apart from the fundamental component whose pu value is 1, also contains the fifth harmonic whose value is 0.15 (Figure 3a) and 0.30 (Figure 3b).

The waveform in Figure 3(c), apart from the fundamental component, also contains the fifth harmonic with amplitude 0.15 shifted in phase by 180 with respect to it.


DESCRIPTION OF THE PHENOMENA12 Decomposition of Distorted WaveformAs each distorted waveform can be composed from harmonic components, so also can any periodic waveform be decomposed into harmonic components (Figure 4).

This analysis technique allows one to consider separately each component of the distorted waveform and, employing standard methods for circuit analysis, obtain a final result being the sum of partial results.


Figure 4: Decomposition of the distorted waveform into harmonic components: the fundamental (1) and the fifth and seventh

13 Decomposition of Distorted Waveform


The French mathematician Jean Baptiste Joseph Fourier (1768–1830), who is considered the author of this technique, demonstrated that any periodic waveform can be deconstructed into a sinusoid at the fundamental frequency with a number of sinusoids at harmonic frequencies.

Depending on the kind of waveform, these coefficients may or may not exist.

14 Representation of Harmonics


A d.c. component may complete these purely sinusoidal terms. This concept can be explained by the following equation:

(1)



f(t) is a periodic function of frequency fo, angular frequency ωo = 2πfo and period T = 1/fo = 2π/ωo

represents the fundamentalcomponent

represents the hth harmonic ofamplitude Ch, frequency hfo andphase ψh



The Fourier series coefficients are given by

(2)

A0 is the d.c. component;



Example 1Express the waveform in time function



Computing the Fourier series coefficients gives



The waveform’s time function is expressed as


DESCRIPTION OF THE PHENOMENA21 Representation of Harmonics


Figure 5 shows the current waveform of an example non‐linear load and its harmonic spectrum. It is possible to observe that current at fundamental frequency (F) is only a percentage of total current.



Figure 5 The current waveform of a compact fluorescent lamp (CFL) and its spectrum


CHARACTERISTICS OF HARMONICS IN POWER SYSTEMS

• Odd symmetry is characterised by f(‐t) = ‐ f(t) and results in no cosine terms in the waveform Fourier series expansion

• Even symmetry is characterised by f( ‐t) = f(t) and results in the waveform Fourier series expansion having no sine terms, that is only cosine terms

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Symmetry


• Waveforms with half‐wave symmetry, f(t ± T/2) = ‐f(t), have zero dc components and result in the cancellation of even‐order (2, 4, 6, ... ) harmonics. This feature leads us to ignore even harmonics in power systems since they consist of bilateral components that produce voltages and currents which are half‐wave symmetric

25

Symmetry


Example 2Synthesise the waveform of Figure below, which is typical of the phase current of a 6‐pulse converter with a Y‐ Y transformer.

26

Symmetry


Applying the rules of symmetry, one would automatically say that the Fourier series expansion of this waveform exhibits:

• no cosine terms since the waveform is odd, f( ‐t) = ‐ f(t), thus Ah = O.

• zero dc‐component and no even harmonics since the waveform is half‐wave symmetric,

f(t ± T/2) = ‐ f(t).

27

Symmetry


Computing the Fourier series coefficients gives

28

Symmetry

CHARACTERISTICS OF HARMONICS IN POWER SYSTEMS29

Symmetry


Now, the waveform’s time function can be expressed as

30

Symmetry

Harmonics in a three‐phase system, similar to the fundamental component, can be considered by applying the notion of symmetrical components.

In a balanced three‐phase system for the fundamental component and where each harmonic exists, there is a simple relation between the harmonic order and the corresponding phase sequence (Table 1).

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Phase sequence


Table 1 Relations between symmetrical components and harmonic orders

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Phase sequence

Figure 6 shows the third‐harmonic waveform as related to phase voltages or currents.

It can be seen that in given phases these components are cophasal. Similarly, it can be demonstrated that the fifth‐order harmonics in given phases are shifted with respect to each other by 120, and their sequence is: L1 → L3 → L2.

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Phase sequence


A

B

C

Figure 6: A three‐phase balanced voltages or currents system with the third harmonic

34 Harmonics and Symmetrical Components

MEASURES OF HARMONIC DISTORTION

The expressions for the RMS voltage and current are

(3)

and

(4)

Here, it is assumed that Vh and Ih are also given in RMS

35 RMS Voltage and Current


The most widely used indices is the Total harmonic distortion (THD). The THDv also known as the voltage distortion factor (VDF) is defined as

(5)

or(6)

36 Total Harmonic Distortion


where Vh is the harmonic voltage at harmonic frequency h in RMS, V1 is the rated fundamental voltage in RMS, and h is the harmonic order (h = 1 corresponds to the fundamental).

Similarly, the total harmonic distortion THDI , also known as the current distortion factor (CDF) is defined as



(7)

or(8)

where Ih is the harmonic current at harmonic frequency h in RMS and I1 is the rated fundamental current in RMS



The RMS voltage and current can now be expressed in terms of THD as

(9)and

(10)



Example 3Calculate the RMS value and THD for the waveform of Example 1

Basic definition for RMS


MEASURES OF HARMONIC DISTORTION41 Total Harmonic Distortion


The fundamental component has an RMS value of

Total harmonic distortion is



Example 4Calculate the RMS value and THD for the waveform of Example 2



The fundamental component has an RMS value of

Total harmonic distortion is



Harmonic distortion is most meaningful when monitored at the point of common coupling (PCC) — usually the customer’s metering point — over a period that can reflect maximum customer demand, typically 15 to 30 minutes as suggested in Standard IEEE‐519.

Weak sources with a large demand current relative to their rated current will tend to show greater waveform distortion.

45 Total Demand Distortion


Conversely, stiff sources characterized for operating at low demand currents will show decreased waveform distortion. The total demand distortion is based on the demand current, IL , over the monitoring period:

(11)

46 Total Demand Distortion


Every harmonic provides a contribution to the average power that can be positive or negative. However, the resultant harmonic power is very small relative to the fundamental frequency active power.

(12)

(13)

47 Active (Real) and Reactive Power


(14)or

(15)or

(16)or

(17)

where S1 is the apparent power at the fundamental frequency

48 Apparent Power


Power factor (18)(displacement power factor)

Noting that when harmonics are present, apparent power S is not only comprised of active power P and reactive power Q, distortion power (volt‐amperes) D is defined to account for the difference

Distortion power (19)

49 Power Factor


True power factor (20)

or (21)

50 Power Factor


TPF can be expressed as

(22)

or (23)

where (24)

or (25)

DPF = distortion power factor

51 Power Factor


One should not be misled by a nameplate power factor of unity; unity power factor being attainable only with pure sinusoids.

What is actually provided is the displacement power factor. With reference to Equation (22), the actual power factor, when harmonics are involved, is

(25a)

52 Power Factor


(26)

(27)

Neglecting phase angles, the total peak current or voltage would be

(28)or

(29)

53 Current and Voltage Crest Factors


The corresponding pu increase in total peak current or voltage is then

(30)or

(31)

Note that Ipeak/IRMS = is only true for the case of a pure sinusoid and the same applies for voltage

54 Current and Voltage Crest Factors


Harmonic generate telephone interference through inductive coupling. The I.T product, used to measure telephone interference is defined as

(32)

where Th is the telephone interference weighting factor at the hth harmonic. (It includes the audio effect as well as inductive effects)

55 Telephone Interference and The I.T Product


TIF is a dimensionless value used to describe the interference of a power transmission line on a telephone line. The telephone interference factor (TIF) is defined as

(33)

Table 2 gives the telephone interference weighting factors for various harmonics based on Table 12.2 of IEEE Std 519‐1992



Table 2


Example 5:

Based on the output of a harmonic analyser, it has been determined that a nonlinear load has a total RMS current of 75A. It also has 38, 21, 4.6, and 3.5 for the third, fifth, seventh and ninth harmonic currents respectively. The instrument used has been programmed to present the resulting data in amps rather than in percentages. Based on the given information, determine the following:(a) The fundamental current in amps(b) The amounts of the third, fifth, seventh and ninth harmonic currents in percentages.(c) The amount of the THD

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Solution

(a)

(b) Hence

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(c) Since

Thus

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or

or

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Example 6:A 4.16 kV three‐phase feeder is supplying a purely resistive load of 5400 kVA. It has been determined that there are 175 V of zero sequence third harmonic and 75 V of negative‐sequence fifth harmonic. Determine the following:(a) The total voltage distortion(b) Is the THD below the IEEE Std 519‐1992 for the 4.16 kV distribution system?Solution(a)

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(b) From Table 11.1 of IEEE Std 519‐1992, THDV limit for 4.16 kV is 5%. Since the THD calculated is 4.58%, it is less than the limit of 5% recommended for 4.16 kV distribution system.

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Example 7:According to ANSI 368 Std., telephone interference from 4.16 kV distribution system is unlikely to occur when the I.T index is below 10000. Consider the load given in example 6 and assume that TIF weighting for the fundamental, the third and fifth harmonics are 0.5, 30, and 225, respectively. Determine the following:(a) The I1, I3, and I5 current in amps(b) The I.T1, I.T2, and I.T5 indices(c) The total I.T index(d) Is the total I.T index less than ANSI 368 Std limit (e) The total TIF index

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Solution(a)

And the resistance is

The harmonic currents are

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(b) The I.T indices are

(c)

(d) Since 3202.02 < 10,000 limit, it is well below the ANSI Std limit.

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(e) The total TIF index for this case is67

POWER IN PASSIVE ELEMENTS

Real (or active) power dissipated in a resistor is given by

where Rh is the resistance at the hth harmonic.

If the resistance is assumed to be constant, that is, ignoring the skin effect, then

68 Power in a Pure Resistance


(34)

Alternatively, expressed in terms of current,



(35)

Note that these equations can be re‐expressed in pu as

(36)



where P is the total power loss in the resistance, P1is the power loss in the resistance at the fundamental frequency,

Vh(pu) =Vh/V1, and Ih(pu) =Ih/I1.

For a purely resistive element, it can be observed from Equation (36) that

THDV = THDI



Power in a pure inductance can be expressed as

(37)where

V1 = j2πf1LI1 , Vh = j2πhf1LIh

andf1 is the fundamental frequency

Thus

(38)

72 Power in a Pure Inductance


so that

(39)

or(40)



whereQL is the total reactive power absorbed by the

inductor

QL1 is the reactive power absorbed by the inductorat the fundamental frequency

QL(pu) is the total pu reactive power absorbed by theinductor, QL(pu) = QL / QL1



Vh(pu) is the hth harmonic inductor voltage in pu ofrated inductor voltage, Vh(pu) = Vh / V1

Ih(pu) is the hth harmonic inductor current in pu of rated inductor current, Ih(pu) = Ih/I1

h is the harmonic order, h = 1 corresponds to thefundamental.



Power in a pure capacitance can be expressed as

(41)

The negative sign indicates that the reactive power is delivered to the load

and

76 Power in a Pure Capacitance


Thus,(42)

hence

(43)

or(44)



whereQC is the total reactive power absorbed by the

capacitor

QC1 is the reactive power delivered by the capacitorat the fundamental frequency

QC(pu) is the total pu reactive power delivered by thecapacitor, QC(pu) = QC / QC1



Vh(pu) is the hth harmonic capacitor voltage in pu ofrated capacitor voltage, Vh(pu) = Vh / V1

Ih(pu) is the hth harmonic capacitor current in pu of rated capacitor current, Ih(pu) = Ih/I1

h is the harmonic order, h = 1 corresponds to thefundamental.


Harmonics are produced by nonlinear loads or devices that draw no sinusoidal currents.

An example of a nonlinear load is a diode, which permits only one‐half of the otherwise sinusoidal current to flow.

Another example is a saturated transformer, whose magnetizing current is no sinusoidal.

But, by far the most common problem‐causing nonlinear loads are large rectifiers and ASDs.

80SOURCES OF HARMONICS

SOURCES OF HARMONICS

Some harmonic sources are not related to power electronics and have been in existence for many years. Examples are

Transformers One source of harmonics using transformers with iron cores is the nonlinear B/H curve associated with these devices.

Figure shows a representative B/H curve for electrical steel that may be used in a transformer core.

81 Classical Nonlinear Loads


A representative nonlinear B/H curve



Note that the core material saturates at a magnetic flux density B ~ 2 Tesla or so.

The slope of the B/H curve at any point is the magnetic permeability of the core material, or

From this equation, note that as the core material saturates, the magnetic permeability decreases.



For economic reasons, power transformers are often operated at a relatively high magnetic flux density, above the B/H curve “knee.”

The resulting magnetizing current is slightly peaked and rich in harmonics. The third harmonic component dominates.



An inductor that draws nonlinear current. (a) The circuit. The inductor has N turns; a mean path length, lc; core permeability, μc; and a core crosssectionalarea, Ac. (b) The waveforms of inductor voltage, v(t), and the inductor current, i(t).



Fortunately, magnetizing current is only a few percent of full‐load current. The magnetizing current for a 25 kVA, 12.5kV/240V transformer is shown in Figure 7. (Spectrum in Table 3)

The fundamental current component lags the fundamental voltage component by 66°.

Even though the 1.54 Amps magnetizing current is highly distorted (76.1%), it is relatively small compared to the rated full‐load current of 140 Amps



Figure 7: Magnetizing Current for Single‐Phase 25 kVA. 12.5kV/240V Transformer. THD = 76.1%


Table 3: Spectraldata for Figure 7

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Machines As with transformers, machines operate with peak flux densities beyond the saturation knee.

Unless blocked by a delta transformation, a three‐phase synchronous generator will produce a 30% third harmonic current.

There is considerable variation among single‐phase motors in the amount of current harmonics they inject.



Most of them have THDI in the 10% range, dominated by the 3rd harmonic.

The current waveforms for a refrigerator and residential air conditioner are shown in Figures 8 and 9, respectively. The corresponding spectra are given in the Table 4 and 5.



Figure 8: 120V Refrigerator Current. THDI = 6.3%.



Figure 9:240V Residential Air Conditioner Current. THDI = 10.5%.


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94



Fluorescent Lamps (with Magnetic Ballasts) Fluorescent lamps extinguish and ignite each half‐cycle, but the flicker is hardly perceptible at 50 or 60Hz.

Ignition occurs sometime after the zero crossing of voltage. Once ignited, fluorescent lamps exhibit negative resistive characteristics.

Their current waveforms are slightly skewed, peaked, and have a characteristic second peak.



The dominant harmonics is the 3rd, on the order of 15% ‐ 20% of fundamental.

A typical waveform is shown in Figure 10 and the spectrum is given in Table 6


SOURCES OF HARMONICS97 Classical Nonlinear Loads

Figure 10: 277V Fluorescent Lamp Current (with Magnetic Ballast). THDI = 18.5%.

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Arc Furnaces These are not strictly periodic and, therefore, cannot be analyzed accurately by using Fourier series and harmonics.

Actually, these are transient loads for which flicker is a greater problem than harmonics.



Examples of power electronic loads are

Line Commutated Converters. These are the workhorse circuits of AC/DC converters above 500HP. The circuit is shown in Figure 11.

These are sometimes described as six pulse converters because they produce six ripple peaks on Vdc per AC cycle.

100 Power Electronic Loads


ia with delta‐wye or wye‐delta transformer ia with delta‐delta or wye‐wye transformerTHDI = 30.0% in both cases


Figure 11: Three‐Phase, Six‐Pulse Line Commutated Converter


In most applications, power flows to the DC load. However, if the DC circuit has a source of power, such as a battery, power can flow from DC to AC in the inverter mode.

The DC choke smooths Idc, and since Idc has low ripple, the converter is often described as a “current source.”



Two or more six‐pulse converters can be operated in parallel through phase‐shifting transformers to reduce the harmonic content of the net supply‐side current. This principle is known as phase cancellation.

A twelve‐pulse converter has two six‐pulse converters connected in parallel on the AC side and in series on the DC side (Figure 12).



Figure 12: Reduction of the converter input current harmonics: schematic diagram of a 12‐pulse converter



One load‐side transformer winding is delta and the other is wye.

As a result, half of the harmonic currents cancel (notably, the 5th and 7th), producing an AC current waveform that is much more sinusoidal than that of each individual converter alone. Higher pulse orders (i.e., eighteen pulse, twenty‐four pulse, etc.) can also be achieved.



The AC current harmonic multiples produced by a P‐pulse converter are

h = PN ± 1, N = 1,2,3, ... ,P = an integer multiple of 6.



Voltage‐Source ConvertersFor applications below 500HP, voltage source converters employing pulse‐width modulators with turn‐on/turn‐off switches on the motor side are often the choice for ASDs.

Since both power and voltage control is accomplished on the load side, the SCRs in Figure 11 can be replaced with simple diodes.



The circuit is shown in Figure 13.

The diode bridge and capacitor provide a relatively stiff Vdc source for the PWM drive, hence the term “voltage source.”

Since voltage‐source converters do not employ phase control, their displacement power factors are approximately 1.0.



ia with high power. THDI = 32.6%. ia with low power. THDI = 67.4%.(delta‐wye or wye‐delta) (delta‐wye or wye‐delta)


Figure 13: Three‐Phase, Six‐Pulse Voltage‐Source Converter


Unfortunately, current distortion on the power system side is higher for voltage‐source converters than for line commutated converters, and the current waveshape varies considerably with load level.

Typical waveforms are shown in Figure 13. Even though lower load levels have higher THDI , the harmonic amperes do not vary greatly with load level because fundamental current is proportional to load level.



The higher current distortion created by these drives is one of the main reasons that voltage‐source inverters are generally not used above 500HP.

Switched‐Mode Power SuppliesThese power supplies are the "front‐end" of singlephase 120V loads such as PCs and home entertainment equipment.



Typically, they have a full‐wave diode rectifier connected between the AC supply system and a capacitor, and the capacitor serves as a low‐ripple “battery” for the DC load.

Unfortunately, low ripple means that the AC system charges the capacitor for only a fraction of each half‐cycle, yielding an AC waveform that is highly peaked, as shown in Figure 14.



AC Current for Above Circuit. THDI = 134%.


Figure 14 : Single‐Phase Switched‐Mode Power Supply and Current Waveforms

Harmonic 1112 Sem2 A1 PERDANA

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