Harmonics and PF

7/30/2019 Harmonics and PF

1/20

Technical Document

Copyright 1987. All rights reserved byROBICON CORPORATION, 500 Hunt ValleyDrive, New Kensington, Pennsylvania.

Reproduction or use without the expresspermission of ROBICON CORPORATION isprohibited.

Part 1 Power Factor and Harmonics

Power Factor and Harmonics are topics which generate asmuch confusion and misunderstanding as any two. Its commonknowledge that high power factor is good and that harmonicsare bad, but how good is good and how bad is bad and what arethey really all about?

The widespread use of high power semiconductor equipmenthas tended to bring both power factor and harmonic consider-

ations under closer scrutiny by the electric utility compa-nies. Poor power factors may result in increased demandcharges, and harmonics may have to be controlled by the userof the equipment generating them. Harmonics in power systemsare nothing new, but the widespread use of sophisticated dataprocessing and instrumentation equipment has raised the levelof interference problems.

Power factor and harmonics can be considered separately,but they are very much interdependent when considering powerfactor improvement in the presence of harmonics or filteringof harmonics. This paper will begin by examining the basics ofpower factor.

Consider the most elementary single phase circuit a

sinusoidal voltage feeding a purely resistive load. Add anammeter, voltmeter and wattmeter for quantitative measure-ments.

When an oscilloscope is added for examining the wave-forms, current and voltage are shown to be in phase. The powerflowing in this circuit is the instantaneous product of thevoltage and current. In a resistive circuit, the voltage/current product is always positive. Positive voltage timespositive current is positive and negative voltage times nega-tive current is positive.

The instantaneous power has an offset sinusoidal waveformat twice the line frequency. Our wattmeter reads the averagevalue of this waveform which is just equal to the offsetlevel, and the wattmeter reading agrees with the product ofErms and Irms as measured by the voltmeter and ammeter. OhmsLaw holds. Volts times amperes equals watts. Everything workedas expected. Replace the resistor with an inductor and takethe same measurements.

Assume this inductor draws the same amount of current asthe resistor. According to the voltmeter and ammeter, nothinghas changed. The oscilloscope will show, however, that thecurrent now lags 90 behind the voltage. The instantaneousproduct of voltage and current is no longer positive at alltimes. Now there are regions of positive voltage with negativecurrent and equal periods of negative voltage with positivecurrent. In both regions, the power is negative.

Power Factor and Harmonics

FORWARD


2/20

Technical Document

The power wave form is a double frequency sine wave asbefore, but the offset is gone. There is no net flow of powerto the inductor. Power flows into the inductor momentarily,but it is returned to the line in the next quarter cycle of

supply voltage. The wattmeter reads zero and the power is nolonger equal to the product of voltage and current. E/I nolonger equals R. It seems that Ohms Law has been violated.

Obviously, Ohms Law still holds. This experiment demon-strates the wattless power of AC circuits. The inductordraws current but it consumes no power. Even though there isboth voltage and current in the circuit, there is no power.The wattmeter recognizes this and reads zero.

How does the wattmeter work? Take a dynamometer typewattmeter with a fixed current coil and a moving potentialcoil with an attached pointer. The interacting magnetic fieldsand currents in the two coils produce a torque which deflects

the pointer against a spring force. The torque produced willbe equal at any moment to the instantaneous product of voltageand current. Positive power flow will produce an up scaledeflection and negative power flow will produce a downscaledeflection. The pointer indication will be proportional to theaverage torque since the mechanical inertia damps out thevariations. So the wattmeter will read the average of theinstantaneous products of voltage and current which is justwhat we want. Newer wattmeter types such as Hall effect de-vices and integrated circuit multipliers can perform the samefunctions.

So far, two circuits have been considered: one with aresistor and one with an inductor. Most real circuits, ofcourse, will be a combination of impedances rather than a

single characteristic. If the resistor and inductor loads arecombined into a composite circuit of R and L in parallel, wecan see the effect of adding their currents. These are thewaveforms of voltage and currents in our composite circuit.The resistor current, I

R, is in phase with the voltage, the

inductor current, IL, is lagging the voltage by 90, and the

composite line current lags the voltage by 45 and is 1.414times the resistor or inductor current.

The power waveform is the sum of the resistor and induc-tor power waveforms we had previously. Note that the powerflow is negative for part of each half-cycle of supply volt-age. We can still see the inductor doing its thing.

Vector or phasor diagrams can conveniently represent thecircuit parameters we have been discussing. If voltage ischosen as a reference at zero degrees on the X axis, theresistor current can be represented as a vector inphase withthe voltage and the inductor current as one which lags by 90.Lagging, by convention, means clockwise. The sum of theinductor current and the resistor current must equal the linecurrent, but they cannot simply be added together algebra-ically since they run in different directions. The line cur-rent, being the vector sum of the two currents, has a magni-tude of 1.414 times either and lags the voltage by 45. TheGreek letter, theta, denotes this phase angle. Note that thecircuit must now be supplied with a line current of 1.414times as much as when it was just the resistor alone. Thecurrent draw is 41% higher but there is no more power output.

FORWARD


3/20

Technical Document

Power is the line voltage times the resistor current.Furthermore, power is the product of line voltage and linecurrent multiplied by the cosine of the phase angle, theta.This can also be expressed as the volt-amperes times the

cosine of the phase angle.

Here is the familiar power triangle which describes ACcircuits. Power is volt-amperes times cosine of the phaseangle theta, or power factor. Then, power (in watts) isvolt-amperes times power factor. Theta can then be referred toas the power factor angle and since current is laggingvoltage, it can be defined as a lagging power factor. Thedashed line on the right is equal to line voltage times induc-tor current. It is also equal to volt-amperes times the sineof the phase angle, theta. By extension, then, the sine oftheta is the reactive factor and the dashed line quantity isvolt-amperes reactive, abbreviated as VARs. This is what isoften called wattless power. It is not a real power since it

represents simply the charging and discharging of the inductoron alternate quarter cycles of supply frequency.

Here are the basic relationships between the more famil-iar terms of KVA, KW and KVAR. KVA is the square root of KW2 +KVAR2, KW is the square root of KVA2-KVAR2, KVAR is the squareroot of KVA2 - KW2, KW = KVA * cos (Q), KVAR = KVA * sin (Q)where Q is the power factor angle as before. The power factoris KW/KVA or cos (Q) and the reactive factor is KVAR/KVA orsin (Q).

So far, only circuits containing inductance have beendiscussed. If the circuit contains resistance and capacitance,all the relationships discussed to this point are still true,with one exception: the current now leads the voltage in phase

angle. The capacitor acts as an energy storage element andalternately receives power from the line and returns it aquarter cycle later. The result: the same wattless power aswith an inductor except for one very important difference.The power factor diagram is now angled in the opposite direc-tion, and the KVARs are in the opposite direction. If aninductive load consumes VARs, then a capacitive load must beone which generates VARs. With this distinction in mind, powerfactor correction can now be explained.

POWER FACTOR CORRECTION

Here is a deceptively simple circuit one which is reallyfar from simple when the intricacies of the circuit are con-

sidered. Lets assume that the resistor, inductor and capaci-tor all draw equal currents. The line current, of course, isthe sum of the three.

The resistor current is in phase with the line voltage,the inductor current lags by 90 and the capacitor currentleads by 90. Note that the inductor and capacitor currentsare directly opposite to each other 180 out of phase. Theline current shows that the inductor and capacitor currentscancel each other and the line sees only the resistor current.As far as the line is concerned, the inductor and capacitor donot exist. Because they have equal but opposite reactancevalues, they form a parallel resonant circuit of very highimpedance and do not draw any current from the line.

FORWARD


4/20

Technical Document

The power factor triangle is affected in the same fash-ion. The lagging KVARs consumed by the inductor are exactlybalanced by the leading KVARs consumed by the capacitor. Theline does not see any load reactance only a resistance.

Theta is zero. Cos (Q), the power factor, is unity and sin(Q), the reactive factor, is zero. With only the resistor andinductor, there would be an inductive load with current lag-ging voltage by 45, 0.707 lagging power factor and a KVA of1.414 times the KW. Conversely, with only the resistor andcapacitor, there would be a capacitive load with the currentleading voltage by 45, 0.707 leading power factor and a KVAof 1.414 times the kW. If we had started with just the induc-tor and capacitor, there would have been no current at all inthe line!

From all this, it would seem that the power factor of aload or a plant can be brought to any desired level by the useof capacitors and such is the case. Whatever lagging KVARs

the plant consumes can be offset by capacitors which consumean appropriate level of leading KVARs, or in other words,generate an appropriate level of lagging KVARs. The calcula-tions to bring a plant from 1000 KW at 0.8 power factorlagging to 1000 KW at 0.95 power factor lagging are givenbelow.

A load of 1000 KW at 0.8 power factor will have a powerfactor angle of arccos (0.8) = 37. The KVA is 1000/0.8 = 1250KVA and the reactive factor is sin (37) or 0.6. The KVARdemand is 1250 * 0.6 = 750 KVAR.

To achieve a 0.95 power factor, the required angle isarccos (0.95) = 18. The KVA is 1000/0.95 = 1053 KVA and thereactive factor is sin (18) = 0.31. The allowable KVAR demand

is then 1053 * 0.31 = 329 KVAR.

Because the plant now consumes 750 KVAR, there must be anoffset of 750 - 329 = 421 KVAR to reach the 329 KVAR level for0.95 power factor. So it appears a capacitor bank of 421 KVARrating would be desirable. In reality, a 450 or 500 KVAR bankwould be used. The calculations and relationships are the samewhether applied to a single phase or a three phase circuit.This is an important milestone in the analysis of power fac-tor.

In summary, the example began with 1000 KW at 0.8 powerfactor which means a reactive consumption of 750 KVAR. Addedto that were 421 KVAR of capacitors which netted the reactive

demand down to 329 KVAR for a 0.95 power factor with the 1000KW load. Note that the line current has come down in the ratio1053/1250 or a 16% reduction. This reduction in current mayprovide some useful relief on cables and switchgear, but thatis not usually the big picture on power factor correction. Ananalysis of the electric power bill will reveal more.

ENERGY AND DEMAND

Most, but not all, electric utility tariffs are arrangedfor industrial customers to be billed for energy used and forthe maximum power required, the power bill actually consistingof two parts, energy and demand. The energy part representsthe kilowatt hours consumed in the billing period. It repre-sents the cost of the coal, oil, gas, water, atoms or Diesel

ORIGINAL

1000KW 0.8 PF750 KVAR 1250 KVA

ADDED

421 KVAR (LEADING)

RESULT1000KW 0.95 PF

329 KVAR 1053 KVA

FORWARD


5/20

Technical Document

fuel required to generate and distribute the energy consumed.This charge may be embellished with fuel adjustment clauses,taxes and the like, but basically it represents the cost ofsupplying the energy. It has nothing to do with power factor

and is based solely on kilowatt hours.

The second part of the bill, the demand charge, is some-what more complicated. The demand charge represents the costof providing the generation, transmission and distributionfacilities necessary to furnish the peak power requirement ofthe customer. Peak power requirement is usually measured by awatt-hour meter with a demand register. This device integratesthe power used over a certain time period, usually 15 or 30minutes, to determine the average kilowatt requirement duringthat time period. The demand pointer is coupled to a peakretaining pointer which registers the maximum value of 15 or30 minute demand since it was last reset usually at the timethe meters are read. From this pointer, the meter reader

determines the maximum 15 or 30 minute kilowatt demand for thebilling period.

But this is not the end of the demand matter. Most utili-ties also adjust this demand charge based, in some manner, onthe load power factor. Although practices may vary, one methodused is to pair the kilowatt-hour meter with a kilovar-hourmeter. This latter device is simply a kilowatt-hour meterconnected to a transformer set which rotates the line voltagesby 90 so that the meter is driven by VARs rather than watts.Now there are two meters. With kilowatt-hours and kilovar-hours in hand, a sort of average power factor for the monthcan be computed. The power factor angle, Q, is given as arctan(KVARH/KWH). If the utility wanted to charge for the currentor KVA demand rather than Kw, it could, in principle, divide

the kilowatt demand by this average power factor. The moreusual method, however, is to use a formula which does notassign any penalty for power factors above some base level,usually 0.90 or 0.95, and approximates the inverse powerfactor for power factors below this value.

One system is based on a multiplier factor given as F =0.8 + 0.6 * (KVARH/KWH) where F is not less than 1.0 norgreater than 2.0. When this multiplier is plotted along withreciprocal power factor, the correlation is quite good overthe usual range of power factors.

Finally, if the demand multiplier is plotted as a func-tion of the power factor, this curve results. Now the high

cost of low power factor has become abundantly clear. A powerfactor of 0.8 increases demand charges by 25%, a 0.7 powerfactor by 41% and a 0.6 power factor by 60%. Energy and demandrates vary widely among the various electric utility compa-nies, but we can give order of magnitude costs as perhaps 3-5cents per kilowatt hour of energy use and 10-15 dollars permonth per kilowatt of power factor adjusted demand. Someutilities base an entire year of demand charges on the maximumvalue for the past year, so it will pay one to analyze thepower bill carefully and to enlist the assistance of the powercompany representative in order to get a full understanding.

Note that the maximum kilowatt demand and the averagepower factor are not necessarily related. For example, it ispossible for a major piece of equipment with high kilowatt

ENERGYKILOWATT HOURS

TONS COAL

ACRE FEET WATER

BARRELS OILGALLONS DIESEL

CUBIC FEET GAS

OUNCES URANIUM

TO TURN GENERATORS

DEMANDKW CAPACITY OF

GENERATORS

TRANSMISSION

TRANSFORMERS

SWITCHGEAR

DISTRIBUTION TO FURNISH PEAK POWER

FORWARD


6/20

Technical Document

demand at unity power factor to be used only once a month.This could set the demand level. However, this kilowatt demandlevel would be adjusted for power factor based on the averagepower factor for the month, and the average power factor could

be based on totally unrelated equipment. Even though theequipment which created the high demand operated at unitypower factor, there is still a demand power factor penalty topay. Is this fair? What if the reverse is the case?

If the seldom used piece of equipment which created thispeak demand operated at a very poor power factor and the restof the plant equipment operated at a good power factor, youwould probably not pay any power factor penalty even thoughthe KVA demand of the major load may have been much greaterthan the KW demand. Average power factor for the month isgenerally all the utility has to go on, so the power factor ofa given piece of equipment is of little interest except as itaffects the ratio of KVARH to KWH for the month.

There is one final item to examine before ending thissection on power factor, and that is feeder regulation. Theinstallation of power factor correction capacitors will im-prove feeder regulation by reducing quadrature line currents,those components of current which are 90 out of phase withthe line voltage, but over correction can cause problems. Manyutilities are glad to get excess capacitors on line, espe-cially if they are switched on only during working hours.Some, however, have problems with excessive voltage rise atlight load and may have tariffs penalizing leading powerfactors as well as lagging power factors. And leading powerfactors do, of course, require just as much additional feedercapacity as lagging loads of the same power factor. So theremay be penalties for over-correction. Except for this proviso,

capacitors may often be added in large banks for overall plantcompensation rather than going after every piece of equipmentindividually. This offers important economies when harmonicprotected capacitor banks or filters are considered.

INTRODUCTION TO HARMONICS

There are a number of ways to approach the subject ofharmonics, but it is best to begin by defining harmonics forthese purposes as those frequencies which bear an integerrelationship to the power line frequency of 60 Hz. Thus, 180Hz and 600 Hz are harmonics by this definition, 93 Hz and 275Hz are not. Also, exclude those frequencies which fall below60 Hz. These distinctions are important because variable speed

AC drives can generate non-integer harmonics and sub-harmonicsof the power line frequency. Treatment of these effects isbeyond the scope of this presentation, but they are not usu-ally of great consequence in the first place. Begin by examin-ing what happens when two harmonically related frequencies areadded.

Here we have a sine wave of a given frequency, the funda-mental, and one which is three times that frequency, a thirdharmonic, with one-third the amplitude. This third harmonic isphased so that it starts out in-phase with the fundamental.

When these two sine waves are added together, the resultis a distorted waveshape with a squashed top. The harmonicadds to the beginning and end of the half cycle and subtracts

FORWARD


7/20

Technical Document

in the middle.

If the phase of the third harmonic wave relative to thefundamental is then reversed, there is a totally different

result. Note that the fundamental and harmonic start out inopposite directions.

The composite waveform changes from the relatively flatform to a peaked form. The difference between the squashedwaveform and the peaked waveform is solely due to the phase ofthe third harmonic. Both contain the same frequencies in thesame amplitudes and have the same RMS values.

From what was just demonstrated, harmonics in a voltageor current imply non-sinusoidal, distorted waveforms. And,conversely, a distorted waveform means harmonics are present.The logical question now is, Where do harmonics originate?

The peaked waveform is characteristic of the excitingcurrent of transformers and reactors. Induction motors alsotend to draw non-sinusoidal components of current due toexciting current and slot ripple. Arc furnaces are prodigioussources of harmonic currents. Thyristor drives of all typeswill generate harmonic line currents as will all rectifiers.In fact, nearly any load on an AC power system will generateharmonics to some degree, however small they may be.

In 1822, the great French mathematician, Fourier, showedin a paper on heat conduction, that any periodic function, orany repetitive waveform, can be resolved into a fundamentalcomponent at the repetition frequency and a series of integralharmonics of that frequency, each with a particular amplitudeand phase relationship relative to the fundamental. It is best

to begin with the basics before exploring the formulas forFourier series.

Examine a 120 square wave, a current waveform typical ofthree-phase thyristor DC motor drives. Technically, this is aquasi-square wave, but for simplification here it will bereferred to simply as a square wave. Just eyeballing thissquare wave, it seems that the fundamental frequency sine wavecomponent would be symmetrical with respect to the square wave that is, the peak of the sine wave would occur at the midpoint of the square wave. Remembering, also, the waveformsexamined earlier, it would seem that the square wave containsout-of-phase harmonics which knock down the front and back ofthe fundamental sine wave.

In the most general case of a repetitive function, f(x),the various frequency components are given by the equationsshown here. These are known as the discrete forms of a Fourierseries in which the harmonics are referenced to a fundamentalwave of sin(x). A

orepresents a DC component while A

nand B

n

represent, respectively, the amplitudes of sine waves ofharmonic order n which are in-phase with the fundamental and90 out of phase. The complex waveform is then simply the sumof all these components.

If the waveform is symmetrical about the vertical axis atp/2, and the negative half cycle is the inverse of the posi-tive half cycle, the integrals reduce to a much more conve-nient form for calculation. There is no DC component, all even

HARMONIC SOURCESTRANSFORMERS

MOTORS

ARC FURNACES

THYRISTOR DRIVES

RECTIFIERS

(NEARLY ALL LOADS)

FORWARD


8/20

Technical Document

order harmonics (2, 4, 6, etc.) vanish, all Bn(cosine) terms

vanish and the limits of integration need only be from 0 to p/2.

An analysis of the original 120 square wave will showthat the frequency spectrum contains an infinite number ofharmonics of order 6n1, each of which has an amplitude rela-tive to the fundamental of 1/n, where n is the order of theharmonic. Thus, there is, relative to the fundamental, 20% ofthe 5th harmonic, about 14% of the 7th harmonic, 9% of the11th, 7% of the 13th and so on. Note that these magnitudes arerelative to the fundamental and that they are peak values. Thefundamental itself has a peak amplitude of 110% of the squarewave amplitude. The RMS value of the fundamental or any har-monic is 0.707 times its peak value since all are sine waves.Note that the 120 square wave has no third harmonic. Bycontrast, a 180 square wave, and this is a true square wave,

has 33% of third harmonic. The difference is due simply to thedifferent conduction angles.

Lets take a more general look at this business of RMS.Its an important concept, especially when we consider non-sinusoidal waveforms. The acronym RMS stands for root meansquared and says simply that to find the RMS value of acomplex wave, integrate the square of the values over thewaveform, divide by the base period to get the mean value andthen extract the square root.

Here is the 120 square wave again. If it has a peakamplitude of 1.00 unit, the square at each point is also 1.00and the mean value is 120/180 or 2/3. The RMS, then, is justthe square root of 2/3 or 0.816. It can easily be shown that

any pulsed waveform with constant unit amplitude during thepulse has an RMS value equal to the square root of the dutycycle.

When the RMS value during the pulse period has some valueother than unity, the RMS of the pulsed waveform has a valueof the square root of the duty cycle multiplied by the RMSvalue during the pulse. This is a handy relationship to remem-ber for quick calculations.

If the development of the harmonic structure is valid,the fundamental plus all the harmonics must yield the same RMSvalue as a direct analysis of the complex wave itself. For the120 square wave of unit height, the RMS value has been deter-mined to be 0.816 by direct methods.

When two or more sine waves of different frequencies areadded, the RMS value of the composite wave is the square rootof the sum of the squares of the individual RMS values. Eachfrequency is another dimension in an orthogonal vector space,and this is stated without proof. Returning to the 120 squarewave, if the squares of the harmonics are added and the squareroot is extracted, we should wind up with 0.816.

This is the harmonic structure developed earlier with thepeak values converted to RMS. For the harmonics through the13th, the RMS value progressively increases, and its reason-able to expect the rest of the harmonics to bring the finalvalue up to 0.816. It seems to work.

FORWARD


9/20

Technical Document

There is another very useful way to consider these har-monics. The terms under the square root sign can be groupedand represented as shown in the second equation. Now there isan expression for the non-sinusoidal current in terms of the

fundamental current and all the harmonic currents grouped. Thetotal current can be considered as a sinusoidal fundamentalcurrent plus a distortion current which represents the effectof all harmonics. The utility of this representation canperhaps best be demonstrated by recognizing that if the com-posite RMS value of a non-sinusoidal wave is known, and if theRMS value of the fundamental is known, the RMS value of allharmonics grouped together can be easily calculated. This issimply the square root of the composite RMS squared minus thefundamental RMS squared. Similarly, the fundamental is thesquare root of the composite RMS squared minus the square ofthe distortion terms.

DISPLACEMENT AND APPARENT POWER FACTOR

So far, its been sort of glossed over that it may notalways be the ideal to have the Fourier harmonic series phasereferenced to the complex wave itself. In particular, itsusually important to know the phase relationship between thefundamental component and some other wave.

Here is a voltage wave and the 120 square wave current.The fundamental of the current, I

f, is in phase with, that is,

centered on the current wave. Therefore, the phase relation-ship between the voltage and the fundamental component ofcurrent is apparent, and this will be defined to be the phaseangle of the circuit, and theta will still be used to desig-nate it. The vector relationships are the same as before.

With a sinusoidal line voltage, the power is determinedsolely by the fundamental component of line current and itsphase relationship to the voltage. Using RMS values through-out, the expression for power is simply E * I

f* cos(Q) where

Ifis the fundamental frequency component of line current.

Similarly, the VARs are given as E * If* sin(Q). Neither the

watts nor the VARs are affected in any way by the harmonics inthe current, as long as the voltage is sinusoidal. But now aproblem arises. The KVA is E * I where I is the RMS currentincluding harmonics, but the KVA implied by the watts and VARsis E * I

f. The two are obviously not equal. If E * I is the

true KVA, then the power is E * I * cos(f) where f is a newpower factor angle chosen to make the power come out right.But there already was a perfectly good power factor angle, Q,

based on watts and VARs. Are two power factor angles and twopower factors really necessary?

The answer is yes! The power factor, cos(Q), determinedfrom Q=arctan(KVAR/KW) in the kilowatts and kilovars triangle,is defined as the displacement power factor, so-calledbecause it measures the displacement angle between the linevoltage and the fundamental component of line current. Theother power factor, cos(f), is called the apparent powerfactor because that is the factor which is used to obtainkilowatts from the measured line KVA. It is given byf=arccos(KW/KVA). The displacement power factor is alwayshigher than the apparent power factor when harmonics arepresent.

FORWARD


10/20

Technical Document

The wattmeter knows the difference between these twopower factors and does its job correctly; it simply ignoresthe harmonics in the current. Similarly, a VARmeter, connectedas described earlier, will ignore the harmonics. The case for

watt-hour meters and VARhour meters is a little less clear,but they can be trusted to give good approximations to thecorrect values. In most cases, the differences would not beworth arguing about.

These relationships can be further developed by consider-ing the characteristics of a three-phase thyristor DC motordrive. The most common type of thyristor drive has a 120square wave line current as discussed earlier. The magnitudeof this current is directly proportional to the armaturecurrent of the motor and is independent of speed. Speed,however, determines the phase displacement of the line currentfrom the line voltage. At full DC output voltage, the currentis in phase with the line-to-neutral voltage. At 0 DC voltage,

the thyristors are phased fully back and line current lagsline-to-neutral voltage by 90. In between, the current phaseangle moves forward as the DC voltage is increased. If IR dropand demagnetization effects are ignored and constant fieldcurrent is assumed, the motor speed is directly proportionalto the DC armature voltage. The power developed by the motoris also directly proportional to speed for any given armaturecurrent. These relationships can be expanded to include kilo-watts, kilovars, KVA and power factors.

Assume a constant armature current at all speeds. Sincewatts are proportional to speed and watts are also propor-tional to the displacement power factor, cos(Q), displacementpower factor is proportional to speed. On the other hand, VARsbehave in the opposite fashion being highest at zero speed

and zero at full speed. The reactive factor is given as thesquare root of one minus the square of the power factor. KVAis independent of speed since the simplified drive model isassumed, but they are still very useful for analyzing powerfactor correction requirements of DC motor drives under vari-ous operating conditions.

LINE NOTCHING

Digressing from the theoretical 120 square wave justdiscussed, consider the phenomenon of notching. The actionof rectification in a thyristor or diode converter involvesthe sequential transfer of load current from one line phase toanother. AC lines inevitably contain some inductance which

prevents this current from being transferred instantaneously,so the square wave we looked at is actually a trapezoid. Thetime to complete current transfer is directly proportional toboth the line inductance and the load current. It is alsoaffected by the angle of phase delay in a converter, and is amaximum at full output.

During the current transfer between phases, the line isshort circuited and the line-to-line voltage on the commutat-ing phase is zero at the converter. This voltage notch maylast for a few tens of microseconds or may continue for amillisecond or more. It depends on the line characteristics,the converter output voltage and the line current, all thevariables mentioned earlier.

FORWARD


11/20

Technical Document

Whereas the commutation notches bring the voltage down tozero at the converter terminals, the disturbances becomeprogressively less as one proceeds toward a stiff powersource. The notch at any point is characterized by the product

of notch depth in volts and notch width in microseconds.

It may seem paradoxical, but this notch area in volt-microseconds remains constant regardless of how much induc-tance might be added on the load side of the point in ques-tion. As load side inductance is added, the notch depth isreduced but the width is increased, and the only practical wayto reduce the notch area is to provide some of the commutationenergy from a capacitor bank. The subject is further consid-ered later in this discussion. Note that the commutationnotches generate their own set of harmonic voltages, ofcourse, but it is much easier to treat them directly in thetime domain rather than to use their Fourier expansion.

The commutation process can be looked at as problematicbecause of the line voltage notching it produces. However, theharmonic spectrum of the current is actually softened, andhigh frequency harmonic current amplitudes are reduced. Thehigh frequencies drop off as 1/n2 rather than as 1/n, where nis the harmonic number. The frequency at which the break pointis reached depends on the commutation angle which, in turn, isa function of source inductance, phaseback angle and loadcurrent. The break point for the high frequency rolloff willincrease as the load voltage is reduced by thyristor phasecontrol. At any given load current, thyristor converters willgenerally have their worst high frequency harmonic noiseoutput at low output voltages where commutation times areshortest.

DISTORTION FACTOR

A useful measure of the severity of harmonics in voltagesor currents is the distortion factor. This is simply 100 timesthe RMS equivalent of all the harmonics divided by the RMSvalue of the fundamental. The term THD for Total HarmonicDistortion is often used, usually as a percentage. It is thesame as the distortion factor.

The waveforms and harmonic spectra of SCR controlledsystems have been used as examples. PWM motor drives andchopper DC supplies generally have rectifier inputs withcapacitor filters. This type of circuit is characterized by ahigh level of low frequency harmonics but relatively low high

frequency harmonics. The particular spectrum depends on thesource inductance. A low source reactance will greatly in-crease the fifth and seventh harmonic currents.

Harmonics in power systems are addressed in IEEE 519 1992Recommended Practices and Requirements for Harmonic Controlin Electrical Power Systems. Limits on individual currentharmonics and THD are shown in Table 10.3 at the end of thisbooklet. Limits on voltage distortion are 3% for any oneharmonic and 5% THD. It should be noted that current distor-tion levels are based on the total plant demand current ofboth harmonic producing and sinusoidal loads.

FORWARD


12/20

Technical Document

ISOLATION TRANSFORMERS

Now for a few words about isolation transformers. Isola-tion transformers are often regarded as cure-alls for harmonic

production and line notching. Unfortunately, they do little ornothing to cure either of these problems. Isolation transform-ers do slow the commutation rate and will decrease the fre-quency at which the higher harmonics fall off as 1/n2. Theywill also decrease the depth of line notches but will notreduce the volt-microseconds of notch area. If such transform-ers are equipped with well-designed Faraday electrostaticshields, they can also reduce common-mode high frequencyharmonic transmission to the power line, but there are lessexpensive ways to accomplish all of these improvements byusing line reactors and simple filters. Isolation transformersserve mainly to control ground fault currents in equipmentsupplied from solidly grounded power systems, and even here,they are not necessary for current source inverters withreactors in both lines of the DC link.

To summarize quickly, its been shown that harmonicsarise from non-sinusoidal currents or voltages, that they canbe defined by Fourier series expansions and that they giverise to two types of power factor, displacement and appar-ent. Also considered was the line voltage notching caused bycommutation and the definition of distortion factor. In thenext and final part, harmonic interaction with the electricutility supply and harmonic control measures will be examined.

POWER FACTOR AND HARMONICS INTERACTIONS

Having covered the fundamentals of power factor and theproduction of harmonics, the next step is to see how harmonics

interact with the electric utility and connected loads andhow to control them.

When a source of harmonic currents is connected to afeeder, two related but distinctly different effects arise.First, the currents flow into the electric utility systemthrough the point of common connection. Their potential ef-fects within the utility system will be examined later. Sec-ond, the utility supply has some source impedance and, inflowing through this source impedance, the harmonic currentswill produce distortion in the feeder voltage. Commutationnotching is one aspect of this phenomenon. Connected loads maybe affected by these harmonics in the line voltage, particu-larly those which are sensitive to zero crossings in thevoltage waveform or which have potential internal resonances.Computer disk drives, fluorescent lamp ballasts, telephonecircuits, PA and paging systems, lamp dimmers and radios arejust a few of the many connected devices which may experienceinterference from harmonics. These problems may range fromminor glitches in operation to catastrophic failure.

POWER FACTOR CAPACITOR RESONANCE

If the feeder has power factor correction capacitors, theresult of harmonic injection may be spectacular. The utilitysource impedance at the point of common connection is gener-ally inductive, especially so if the feeder is supplied from alocal transformer. This source inductance and the power factorcorrection capacitor form a parallel L/C circuit, and if this

FORWARD


13/20

Technical Document

circuit resonates on a frequency at which harmonic currentsare being sourced, there can be trouble. This review of thecharacteristics of resonant circuits can explain why.

In a series resonant circuit, the inductive and capaci-tive reactances are equal and the impedance is just the resis-tance, R. Circuit Q is defined as X

L/R or X

C/R. Note that the

circuit impedance can never be less than the resistor value.In the parallel resonant circuit, X

Lis approximately equal to

XCand Q is defined in the same fashion as with the series

circuit as XL/R or X

C/R. The impedance of the parallel resonant

circuit is then Q*XLor Q*X

C, and the current circulating

within the resonant circuit is Q times the current in theexternal circuit. If the Q is high, the impedance is muchhigher than either the inductive or capacitive reactance andthe current multiplication is correspondingly high. Now toreexamine the feeder problems.

This is the equivalent circuit of the feeder, the utilitysource being represented by an inductor with a loss resis-tance. For harmonic currents, the capacitor appears in paral-lel with this impedance. To a first approximation, the har-monic currents produced by the harmonic source are just that harmonic currents, and they will flow regardless of the cir-cuit characteristics. When they encounter this parallel L/Ccircuit, however, any harmonic current at the resonant fre-quency encounters a very high circuit impedance, and even asmall current flowing into a high impedance has the capabilityof producing high harmonic voltage. This voltage can be allout of proportion to the voltage which would be present with-out the capacitor. In fact, the resonance can produce a severeovervoltage on the feeder with obvious potential for damage toconnected equipment. As if this were not enough, there are

still more potential problems.

Because the harmonic is a current source flowing into aparallel resonant circuit, the currents in the utility lineand in the capacitor will be approximately Q times the actualharmonic current sourced by the load and the circuit Q may beten or more under some conditions. Thus, the harmonic currentat resonance is amplified many times by this effect and mayblow capacitor fuses or cause serious interference effectswithin the electric utility system. The larger the capacitorbank is relative to the utility short circuit MVA, the morelikely trouble becomes. In fact, the potential problems atpower levels of several thousand KVA are so severe as tosuggest a system study before any capacitors are added to a

plant which has large harmonic producing loads. And even smallmotor drives may get into trouble if capacitors are added on afeeder. The commutation rate of current change, di/dt, mayincrease to a point where low cost motor drives with no lineinductions may fail.

The harmonic order (multiple of supply frequency) of theresonance can be found as the square root of the source shortcircuit kVA divided by the capacitor bank kVAR.

There is no safe rule to avoid trouble, but resonancesabove 1000 Hz will probably not cause problems except, possi-bly, to telephone circuits. This means the capacitor KVARshould not exceed roughly 0.3% of the system short circuit KVAat the point of connection unless control measures are taken

FORWARD


14/20

Technical Document

to deal with the harmonics.

RESONANCE SOLUTIONS

One way of dealing with these potential resonances is toforce them to a frequency below that at which any harmonicenergy is normally produced. The addition of a series inductorwith the capacitor means the resonance can be kept below anydesired frequency independent of the utility source impedance.For example, if an inductor with a reactance of one-tenth thecapacitor reactance at the supply frequency is added, theresonance can never lie above 3.16 times the supply frequencysince the utility source reactance, nearly always inductive,will simply add to the inductor reactance and lower the reso-nant frequency from that produced by the capacitor and induc-tor alone. Since normal three-phase thyristor controllers donot source appreciable harmonic energy below the fifth har-

monic, this approach will generally yield satisfactory re-sults. With large single-phase controllers, however, an induc-tive reactance as large as 20% of the capacitor reactance maybe required to prevent third harmonic problems. The use of aninductor to protect the capacitor from harmonic currents andresonances generally allows power factor correction capacitorsto be applied successfully in the presence of harmonics. Butthere is yet another problem to recognize before rushing outto buy reactors.

The presence of the reactor means the voltage on thecapacitor at the fundamental frequency will rise. The inductorreactance subtracts from the capacitor reactance, lowers thecircuit reactance and increases the fundamental current in thecapacitor. The increased current raises the capacitor voltage,

usually to unacceptable levels. For example, an inductor ofone-tenth the capacitor reactance will raise the fundamentalfrequency voltage on the capacitor by 11%. Since ANSI capaci-tor ratings allow only a 10% overvoltage, this is a problem even in the absence of line voltage variations. The capaci-tors, therefore, must be rated for a voltage higher than thenominal line voltage of the feeder. If the operating voltageis then less than the rated capacitor voltage, because capaci-tor voltage ratings come in steps, the bank rating will usu-ally be reduced from the nameplate value. The actual KVAR willbe equal to the nameplate KVAR multiplied by the square of theratio of actual operating voltage with the inductor to thenameplate voltage rating of the capacitors.

TELEPHONE INFLUENCE FACTOR (TIF)

So far, we have just considered the local effects ofharmonic currents. After examining the local effects of har-monic currents, it makes sense to study those which flow intothe utility lines. There is the possibility that harmoniccurrents will be trapped in remote power factor correctioncapacitors. There is even the possibility that resonances willarise on a feeder or within a customer plant on the system.When transmission lines or long distribution circuits areinvolved with harmonic current, coupling to adjacent telephonecircuits may also be a problem. Whereas the higher harmonicsare not usually a problem with power circuits, the voltageinduced into a telephone line is proportional to the frequencyof the current, so even a small amount of high frequency

FORWARD


15/20

Technical Document

harmonic current can affect voice or data communication cir-cuits. For at least the telephone voice circuits, we canquantify the exposure.

The potential interference to telephone circuits is gov-erned by the Telephone Influence Factor, abbreviated TIF. Thisfactor weighs the significance of various harmonic frequenciesaccording to their coupling coefficients and their potentialfor interference with voice communications. The relativesensitivity of telephone circuits to interference is definedby the C Message Weighting characteristic which combinessubjective analysis of interference with the frequency re-sponse of the telephone set and human ear. It is a weightingfunction which defines the interference potential of a givenfrequency relative to the interference potential of a 1000 Hzsignal. When combined with an empirical constant and a cou-pling coefficient proportional to frequency, a dimensionlessweighting curve known as the TIF weighting function, W

f, re-

sults. This function defines an interference level for anygiven single frequency. Numerically, Wf= 5 * P

f* f where P

fis

the C Message Weighting for Frequency f.

The potential of a single harmonic for telephone inter-ference is given by I

f* W

fwhere I

fis the RMS current at

frequency f and Wfis the TIF weighting factor for that fre-

quency. The effect of a series of harmonics can then be devel-oped by summing the squares of each I

f* W

fvalue, extracting

the square root and dividing by the RMS current. This formsthe TIF value T, for that waveform and harmonic series. Theinterference level is given by I * T, where I is the RMScurrent. Even though harmonic currents generally decrease withincreasing frequency, the coupling factor makes harmonics upto about 5000 Hz potentially troublesome. A sample calculation

is shown for a 120 square wave. With no commutation sloping,this waveform yields a TIF value, T, of 800.

A better idea of the significance of the various harmon-ics can be obtained by plotting the individual I

f* W

fvalues

and the cumulative square root of the sum of the squares. Thisis shown for the 120 square wave with square edges and withthe trapezoidal shape typical of thyristor controllers. Al-though the difference appears dramatic, it represents only a 3dB change in total T level barely enough to hear in a tele-phone circuit. The harmonics from the 11th at 660 Hz to the25th as 1500 Hz account for the vast majority of the T valueand, hence, the potential for interference. This, of course,assumes no resonance effects in either the power or communica-

tions circuits to aggravate the situation.

IEEE 519

The question naturally arises of what limits may be ap-propriate for harmonic currents, line notching, distortionproducts and I * T levels in various classes of apparatus. Thematter has been extensively studied and is summarized in IEEEStandard 519. The recommended practices involve a number ofconsiderations of feeder voltage and capacity, sensitivity ofconnected equipment, in-plant versus system interferencepotential and other related matters. It is not feasible tosummarize the recommendations in a discussion of this sort.Let it suffice to say that the allowable limits on harmonicsare already difficult to meet in some cases, and that they

FORWARD


16/20

Technical Document

will probably be lower in the future. The current issue ofIEEE 519 should be consulted for details.

MULTIPURPOSE SYSTEMS HARMONIC DISTORTION SOLUTIONS

What can be done to control line notching and voltagedistortion from harmonic currents? One possibility is tospecify the type of controller or drive to minimize harmonicproduction. The 6-pulse thyristor DC motor drive and itsharmonic spectrum have already been discussed. If two suchdrives are connected in series or parallel with a 30 phaseshift between them, a 12-pulse circuit results, and the har-monic production is greatly reduced. Whereas the 6-pulsecircuit generated harmonics at 6f 1, the 12-pulse generatedharmonics mostly at 12f 1. Fifth and seventh harmonics aregreatly reduced, as are the 17th and 19th.

Pulse multiplication and PWM circuitry have been used tocreate the Robicon Perfect Harmony line of medium voltageVFDs. These units meet all applicable requirements of IEEE519-1992 with no filtering required. At lower voltages, pulsemultiplication and minimal filtering are used to meet currentdistortion requirements.

While discussing load current waveforms, it should bementioned that some small motor drives with rectifier inputsto capacitor filters are advertised as being free from lineharmonics. They are free from commutation notches, but theyare most definitely not free from harmonics. In fact, they maywell have a higher level of low frequency harmonics than the120 square wave has.

LOWER IMPEDANCE BUS

Another approach is to get a stiffer feeder. In fact, ifthe feeder short circuit ratio, the ratio of short circuitcurrent to rated drive or controller current, is greater than100, the installation will probably have an acceptable levelof harmonics in the first place. If there is a choice to bemade in laying out the distribution system, one should isolateharmonic producing equipment as much as possible. Two differ-ent arrangements, taken from IEEE 519, illustrate this point.The preferred arrangement can be expected to make a worthwhilereduction in higher order harmonic voltage distortion at thesensitive loads and to reduce coupling to plant telephone anddata communication circuits. Another possibility is to filterthe input of critical loads. For example, it is much lessexpensive to filter the input power to a computer than toharmonic suppress a 1000 HP motor drive. If the harmonicproblems lie within the utility system, however, these ap-proaches will not help, and the only recourse then is a diver-sionary tactic.

SHUNT NETWORKS

Because the harmonics are current sources, something todivert these currents from the electric utility at the pointof common connection is necessary. If a shunt network, whichhas an impedance at a harmonic frequency that is much lowerthan the utility source impedance at that frequency, can becreated, we can divert most of the harmonic current into our

FORWARD


17/20

Technical Document

network. Now, how is that done? What forms can these networkstake?

In principle, a capacitor bank could be installed on the

feeder and make it resonate with the source inductance off anyharmonic frequency. Its reactance at harmonic frequenciescould be made as low as necessary by choosing an appropriatesize. However, if the resonance falls between two harmonics,the system will probably amplify those harmonic currentsbecause of their proximity to the resonant frequency. Iftuning below the lowest harmonic is attempted, the capacitorbank usually becomes very large and expensive, and wouldprobably create an unacceptable rise in the feeder voltage.Despite these disadvantages, it is sometimes necessary to usea brute force approach like this. A shunt capacitor is usuallyimplemented with a series resistor to control the high fre-quency resonances and a shunting inductor around the resistorto reduce the losses at supply frequency. The circuit forms a

high pass filter which may allow low frequency harmonics topass with little attenuation, but which provides increasingattenuation up to some fixed level at high frequencies.

By itself, the shunt capacitor is used primarily fortelephone interference control and reduction of line notching,but it often forms part of an overall filter system. For anysingle frequency, a low impedance path can be created by usinga series resonant circuit in shunt with the feeder. The con-stants can be chosen for any desired degree of attenuation atresonance. The trap must be capable of being tuned accuratelyand it must stay acceptably close to resonance with tempera-ture variations or small frequency excursions. Further, thetrap will sink harmonic currents which may be present on theutility lines, so a survey of line distortion may be required

prior to filter design.

FILTER NETWORKS

As far as any one harmonic frequency is concerned, thisis the end of the matter. Unfortunately, a profusion of har-monic frequencies typically needs to be dealt with, and amultiplicity of traps ultimately are involved. Intuition saysthat the traps will not ignore each other, so consider allpossible system resonances and interactions. It is worthwhileto examine first the effect of adding just a single seriesresonant filter to a system.

The filter will have a very low impedance at its resonant

frequency, but the capacitor, filter inductor and sourceinductance also form a parallel resonant circuit. Since thesource inductance adds to the filter inductance, the parallelresonance is at a lower frequency than the series resonance.The curve shows the system absolute impedance characteristicsover a range of frequencies. This filter has a series trapresonance at the 7th harmonic but it also has a potentiallyfatal parallel resonance with the power system at the 5thharmonic. Of course, this example was chosen specifically tocreate this situation, but it can arise in the real world aswell.

If filters tuned to higher frequencies continue to beadded, each added filter will yield a low shunt impedance atits self-resonant frequency, but will always add a parallel

FORWARD


18/20

Technical Document

resonance at a lower frequency. This is the impedance curvefor a five element filter with a high pass element. It hasbeen configured for a 12-pulse converter, which has onlyresidual fifth and seventh harmonic output, so these elements

can be relatively high impedance. It is only necessary tocontrol these harmonics so they do not create overvoltages,and because of inductor costs, the lower frequency traps aremade as small as possible. The 11th and 13th harmonics havelarge filter elements so as to obtain a low shunt impedance,and the high pass filter provides attenuation of the remainingharmonics. The overall network reduces voltage distortion tolevels recommended in IEEE 519.

Lets see how these shunt filters function. A computerwas used to show what happens as harmonics are filtered from adistorted wave. The example chosen here is a 120 square wavecurrent with 10 commutation time, a typical line currentwaveform for a DC motor drive and for many AC drives. This is

the square wave before any filtering. The distortion factor is26% not too pretty a waveform. Now take out the fifth har-monic.

This may not look much better, but the distortion factoris down from 26% to 18%, so things are improving. Now removethe seventh as well.

Things are actually looking better now. The sine wave isstarting to emerge. The distortion factor is down to 11%. Nexttake out the 11th.

This still isnt a great waveform, but the distortionfactor is now only 8%. Add in the final element and remove the13th harmonic.

This is the final current waveform. The distortion factoris 6%, so a reasonable current is being put into the utility.Of course, the significance of this current waveform to thevoltage distortion would depend on the source impedance andthe current level.

The design and analysis of these multiple section filtersis usually done with a relatively complex computer program.The assumption of a purely inductive utility source is seldomcorrect, since the source impedance is affected by transformer

FORWARD


19/20

Technical Document

and transmission line characteristics, neither of which have alinear reactance with frequency. In fact, it may be necessaryto model some transmission lines as long lines with hyperbolicfunctions, since electrical line lengths may exceed a wave-

length at frequencies of interest for I * T analysis. Trans-former characteristics are determined by leakage reactancesand stray losses in the iron, copper and frame. Their imped-ance may increase with frequency up to some limit and thenremain more or less constant for further increases in fre-quency. Some guidelines for system source impedances arecontained in IEEE 519.

FILTER GUIDELINES

Despite the difficulties of filter design, some generalground rules can be set down. First, start with the lowestharmonic frequency present and design a shunt filter to pro-vide sufficient attenuation at that frequency. Then, work up

in frequency, taking one harmonic at a time, and design theindividual filters. The L/C ratio of each can be determinedfrom cost considerations and the total capacitive KVARs allow-able or desirable on the system from power factor consider-ations. After low frequency harmonics have been handled, itmay be necessary to install a high pass filter as describedearlier to handle line notching and I * T limits. When theentire filter has been defined, the system impedance charac-teristic must be determined to see if the parallel resonanceswill cause problems with ringing. If so, the filter sectionsmust be damped, the usual technique being to install resistorsin parallel with the inductors so as to limit the power fre-quency losses.

Experimental evidence indicated that parallel resonances

near the fourth harmonic of power frequency may cause insta-bility in some high performance drive systems. If the totalcapacitance on the system is not critical, it may be feasibleto tune the parallel resonance off this frequency band. Tuningparameters should be chosen with an eye to the possibility offuture changes in the utility source impedance. The resonancemay also be controlled to some extent by adding damping resis-tors to the traps. Otherwise, it may be necessary to add aseries resonant fourth harmonic trap. These considerationsbecome especially important in filters for arc furnaces sincethe arc generates a wide band of both even and odd orderharmonics.

HARMONIC PREDICTION

So far, the assumption has been made that a harmonicproblem exists. If you have one, you are likely to have heardabout it one way or another. But if a plant expansion isplanned or new equipment is to be installed, how does oneestimate the new harmonic output of a number of sources?

The net harmonic effect of a number of rectifiers and/ormotor drives depends on the circuitry used in each. Rectifierunits and the diodes which supply PWM type drives have harmon-ics which tend to be in phase, so they add directly. SCRcontrolled equipments, however, tend to have harmonics inrandom phase relationships unless several items operate syn-chronously. In general, the net harmonic effect of SCR equip-ments will be given by square root sum of squares of the

FILTER GUIDELINES

START WITH LOWEST HARMONI

ADD SECTIONS AS REQUIRED

ADD HIGH PASS IF NEEDED

COORDINATE WITH SYSTEM

VARS

CHECK RESONANCES AND

DAMPING

FORWARD


20/20

Technical Document

individual currents.

Since rectifiers, rectifiers with capaci-tor filters and SCR equipments have different

harmonic spectra, the only way to combinethem is to develop the spectrum for eachtype. Then, for each harmonic frequency, addthe harmonic currents of the same type eitheralgebraically (rectifiers and rectifiers withfilters) or as square root sum of squares(SCR units).

SUMMARY

This report reviewed how to reduce har-monic currents so that the limits of feedervoltage distortion recommended in IEEE 519are not exceeded. But this by no means as-

sures that equipment connected to that feederwill perform properly. It is an unfortunatetruth that a great deal of electronic equip-ment and even some solid state power equip-ment is designed by engineers who are misin-formed or unaware of power line distortion.

Make IEEE 519 a company standard as re-gards procurement specifications foranything that will attach to an AC line.Get a copy of 519 and use it. Tell company

vendors that their equipment must operateproperly with the distortion inherent in afeeder which is in conformance with IEEE519 recommendations as regards harmonicdistortion and line notching.

Only the highlights of harmonic con-trol were discussed in this booklet. Thereis a wealth of information available onthese subjects, including IEEE 519-1992Recommended Practices and Requirementsfor Harmonic Control in Electric PowerSystems, and the UK Engineering Recommen-dation G.5/3, Limits for Harmonics in the

United Kingdom Electricity Supply System.Both contain a wealth of background infor-mation, including bibliographies andpractical suggestions as well as recommen-dations for limits on harmonic levels.

Harmonics and PF

Documents

Transcript of Harmonics and PF