bulletin 1403 powermonitor ii tutorial - rockwell automation...standard defines three classes of...

Bulletin 1403Powermonitor II

Tutorial

Steven Lombardi, P.E.Rockwell Automation

OVERVIEW ........................................................................... 1

ACCURACY .......................................................................... 1External Elements Affecting Accuracy............................ 2

Errors Due to User Supplied Transformers.................. 2Ratio Error ............................................................. 2Phase Error ............................................................ 2Bandwidth Error..................................................... 3

Internal Elements Affecting Accuracy............................. 3Errors in the Analog (Continuous Time) Domain........ 3Errors in the Digital (Discrete Sampled) Domain ........ 3

Accuracy vs. Resolution......................................... 4Sampling Effects.................................................... 4

Sample Time Skew............................................ 4Sampling Frequency.......................................... 5Anti-aliasing Filters ........................................... 5

Computational Attributes ............................................ 6

POWERMONITOR II SPECIAL FEATURES .............................. 6Error vs. Frequency......................................................... 6Spectral Analysis............................................................. 8

Sampling Error (Windowing)...................................... 8

Oscillography...................................................................8Selected Computations.....................................................8

Power Factor................................................................8True ........................................................................8Displacement..........................................................8Distortion................................................................9

Demand.......................................................................9Instantaneous ..........................................................9First Order Projection............................................10Second Order Projection........................................10

Symmetrical Component Analysis.............................10Harmonic Distortion..................................................10

IEEE Definition ....................................................10IEC Definition ......................................................11

Crest Factor...............................................................11K-Factor....................................................................11Telephone Influence Factor (TIF)..............................11IEEE-519...................................................................11

WHOSE HARMONICS ARE THESE?......................................12

OVERVIEW

This tutorial is intended to supplement the productliterature for the Powermonitor II. The contents of thedocument are divided into three main sections, Accuracy,Special Powermonitor II Features, and Harmonics. Thesection on accuracy describes various criteria and designissues that affect the ability of a product to accuratelymeasure input waveforms. The Special Powermonitor IIFeatures section provides additional information on someof the unique or specialized functions of the PowermonitorII. It does not cover all of the Powermonitor IIsfunctionality and is not intended to be used in lieu of theproduct literature. The final section on Harmonicsprovides a brief discussion pertaining to who isresponsible for the harmonics present at a specificlocation. This document may be useful to anyoneunfamiliar with electronic product design or desiring abetter understanding of the more specializedPowermonitor II functions.

ACCURACY

For purposes of this discussion accuracy of an idealsystem is defined as:

“The ability to acquire, measure, analyze, andpresent power system data precisely as it exists in thereal world with no error introduced by the measure-ment system.”

The measurement system is composed of the meteringdevice and external support devices such as user suppliedtransformers. Various elements both internal and external tothe measuring system can have detrimental effects on thefinal accuracy of the desired measurements. The followingitems detail how these elements affect the final systemaccuracy.

2 BULLETIN 1403 POWERMONITOR II TUTORIAL

External Elements Affecting AccuracyIn typical installations the user will supply potential

and/or current transformers as level shifting devices betweenthe power system and the power monitoring equipment. Eachof these devices along with the wire run between the devicesand the power monitor can contribute to total system error.In some instances the error, if the magnitude is known, canbe corrected by modifying the scaling in the powermonitoring equipment. In other circumstances the error cannot be removed so easliy.

ERRORS DUE TO USER SUPPLIED TRANSFORMERS

Any errors introduced by the user supplied transformersimpacts the ability of the power monitoring equipment toaccurately measure the system parameters. A brief discussionof the main factors affecting transformer accuracy is below.For a more detailed treatment of issues regarding the use ofinstrument transformers, the user should refer to ANSI/IEEEC57.13, Requirements for Instrument Transformers. Thisstandard defines three classes of transformer accuracy; class1.2, class 0.6, and class 0.3. For optimum performance of themetering equipment, instrument transformer accuracy class0.3 should be used.

The quality of the metering data can be no better thanthe quality of the signals presented to the input terminalsof the metering device. It is the responsibility of the userto ensure that the transformers in the system are adequatefor the desired metering performance.

Ratio Error

The ratio of the voltage on the primary of a potentialtransformer to the voltage on the secondary of the transformeris a function of the ratio of the number of turns of wire on theprimary and the secondary. The formula for this relationshipis expressed as follows:VV

NN

Secondary

imary

Secondary

imaryPr Pr=

Where N represents the respective number of turns of wireon the primary or secondary. When the manufacturerconstructs the transformer, it is possible that the number ofturns on the primary or secondary is slightly more or less thanthe design specifies. If this is the case, an error occurs thataffects the voltage input to the power monitoring equipment.Additional error also occurs due to magnetic core losses andwinding impedance. The combination of these errors isknown as “Ratio Error.” Assuming the magnitude of theRatio Error is known, this can be compensated for byadjusting the configuration entries for PT primary and/orsecondary voltages to correct the ratio error.

In a similar fashion, the ratio of the current in the primaryof a current transformer to the current in the secondary of thetransformer is also a function of the ratio of the number ofturns of wire on the primary and secondary. The formula forthis relationship is expressed as follows:II

NN

imary

Secondary

Secondary

imary

Pr

Pr=

Once again, any error in the number of turns wound onthe transformer in combination with error due to excitationcurrent results in a Ratio Error. In fact, some error in thisratio is quite common in commercial grade PTs and CTs.This error can be corrected by adjusting the configurationentries for CT primary and/or secondary rating.

The Ratio Error for both PTs and CTs is also affected bythe burden, or load, on the transformer secondary. For a PTthe Ratio Error increases as the current supplied by thetransformer increases. Therefore, it is desirable to keep thetotal load impedance seen by the potential transformer ashigh as possible. For a CT the Ratio Error increases as thevoltage supported by the transformer secondary increases.Therefore, it is desirable to keep the total load impedanceseen by the current transformer, including the impedance ofthe wire connecting the CTs to the metering device, as low aspossible. This is one of the main reasons for the commonspecification of #10 wire when five amp CTs are used.

Phase Error

Phase shift of the transformed signal from primary tosecondary is another source of error introduced by the usersupplied transformers. This error is generally not of anyconcern for simple voltage or current measurements, such asline to neutral voltage or line current. However, when thesesignals are combined in order to determine other quantities,such as line to line voltage or phase power, they can becomeanother error source in addition to the ratio error. Theabsolute value of the phase error is not a factor. It is thedifference in phase error between different transformers thatcauses measurement errors. If all the PTs and CTs in thesystem introduced a five degree phase shift, there would be norelative phase error and therefore no error in the measuredquantities. However, if the PTs had a phase error of onedegree and the CTs had a phase error of six degrees, therewould be a five degree phase error introduced into the powercalculation. This would manifest itself as power factor/varerrors. Phase errors can not be corrected by adjusting theconfiguration entries to the metering device since the errorswill vary based on the state of the power system at any givenmoment.

When operated within the rated voltage and frequencyrange of the PT, the phase error varies from approximately±1° to ±0.25° depending on the accuracy class of thetransformer. Operation at higher voltages will increase thephase error. Continuing to increase the applied voltage willsaturate the transformer and cause larger errors. The phase

BULLETIN 1403 POWERMONITOR II TUTORIAL 3

error in a current transformer is sensitive to the magnitude ofthe current and increases as the current decreases. For bestaccuracy measurements, the current should be greater than80% of the transformer rating. Significant error can beintroduced when measurements are made at currents less than20% of rating. Therefore, CTs sized for protection do notperform as well when used in a metering application.

The phase error of both PTs and CTs are also affected bythe power factor of the load on the secondary. For bestaccuracy, the impedance on a PT should be large andresistive. For best CT accuracy, the impedance on thesecondary should be as small as possible and also resistive.

Bandwidth Error

For normal 50 Hz or 60 Hz measurements, there is nosignificant bandwidth error. However, for waveforms withsignificant harmonic content, the user supplied transformerscan significantly attenuate the higher harmonics. Most usersupplied instrument quality potential transformers have areasonably flat frequency response out to about 3 KHz, or thefiftieth harmonic on a 60 Hz system. Current transformerstend to be more frequency sensitive, especially older,previously installed units. These transformers may only havea flat response out to about 300 Hz, or the fifth harmonic on a60Hz system. Wideband instrument CTs are available forimproved frequency response. 1Bandwidth error can not becorrected by adjusting the configuration entries fortransformer primary and/or secondary rating.

In addition to the limited high frequency response of thetransformers, operation of either the PTs or CTs at lowfrequencies, less than 90% of rating, may also causesignificant magnitude and phase errors due to saturation ofthe transformer core.

Internal Elements Affecting AccuracyModern, high performance power metering devices

combine analog, digital, and microprocessor/DSP technologyto achieve their comprehensive range of functionality. Thecharacteristics of these internal technologies, in addition tothe external elements discussed above, also have an effect onthe measurement system accuracy. Some of the major itemsare discussed below.

ERRORS IN THE ANALOG

(CONTINUOUS TIME ) DOMAIN

Analog components can be divided into two basic classes,passive components (such as resistors and capacitors) andactive components (such as transistors and op amps). Analogerrors due to passive components are typically due tocomponent tolerances and variations due to temperature

1 G.T. Heydt, Electric Power Quality, Indiana: Stars in aCircle Publications, 1991.

changes. In an analog only system, errors due to componenttolerances can be reduced by including adjustable componentsin the circuit, such as potentiometers. However, adjustablecomponents are more expensive and they require additionalmanufacturing steps, and cost, in order to properly adjustthem. Furthermore, adjustable components need to beperiodically readjusted in order to maintain proper settings.Errors that result from component variations due totemperature changes can be compensated for by selectingcomponents whose temperature variations tend to offset eachother. Unfortunately, this approach is usually only partiallysuccessful at eliminating the problem while it adds additionalcomponents and cost to the design.

Analog errors due to active components include DCvoltage offset, gain, and linearity errors. In an analog onlysystem, offset and gain errors can be corrected through theuse of adjustable components. Once again, as statedpreviously, the disadvantages of using adjustable componentsis a factor. Depending on the design, it may also be possibleto add extra components, along with the associated cost, inorder to minimize the effects of temperature variation. In ananalog only system, component nonlinearity is difficult orimpossible to correct, and errors due to these factors musteither be tolerated or reduced by selecting more expensivecomponents.

ERRORS IN THE DIGITAL

(DISCRETE SAMPLED ) DOMAIN

In a system that combines analog components withmicroprocessor/DSP technology, many of the analog errorsources can be economically minimized on an individualdevice basis at the time of manufacture. Embedded productfirmware, in combination with an interactive calibrationprocess that doesn’t require any adjustable components, canbe used to significantly reduce or eliminate errors due tocomponent tolerances, DC voltage offset, gain, and linearityerrors. Errors due to temperature variation can be minimizedby selecting components that are relatively stable overtemperature. When this is not sufficient, an inexpensivetemperature sensor can be added to allow embedded firmwareto correct for these variations. Therefore, this combination oftechnologies allows nearly all of the analog error sources tobe significantly reduced.

In order to accomplish this economical reduction ofanalog error components, the measured signals aretransformed from a continuous time signal to a series ofdiscrete samples separated in time by a constant increment.The value of each discrete sample is assigned a numericalvalue that is proportional to the value of the continuous timesignal at the instant of each time increment. Thistransformation is accomplished through the use of an Analog-to-Digital (A/D) converter. Once the signals have beentransformed, a microprocessor or Digital Signal Processor canbe used to minimize the analog errors, as previously stated,


and to perform other mathematical calculations on the data.While the transformation into a digital representation is verybeneficial, it also creates its own set of error sources that mustbe considered. The major error sources are discussed below.

Accuracy vs. Resolution

Accuracy and resolution are two terms that are frequentlymisused in digitally sampled systems. A concise definitionfor accuracy has already been given. The definition ofresolution relates to the granularity to which an A/Dconverter is able to transform an analog signal into a digitalrepresentation. The resolution is determined by the numberof binary “bits” the A/D converter uses to quantify the analogvalue. A “bit” is simply a binary, or base 2 (instead of thetypical base 10 used in every day math) digit. It can onlyhave one of two values, 0 or 1. Each binary digit is related toits adjacent digits by a power of 2. This is similar to a base10 system where each place is related to its adjacent place bya power of 10. For example, in a base 10 system the digitplaces are “ones” (100), “tens” (101), “hundreds” (102), and soon. In a base 2 system the digit places are “ones” (20), “twos”(21), “fours” (22), “eights” (23), and so on. Once the numberof “bits” is known the resolution is determined as follows:

Resolution = One part in 2B where B is the number of“bits” utilized by the A/D

For instance, a 10-bit A/D converter has a resolution of 1part in 1,024. One count of the A/D converter represents0.0977% of the full scale input of the device. A 16-bit A/Dconverter has a resolution of 1 part in 65,536, or 0.0015% offull scale. The smallest quantity than can be resolved isknown as the quantization error of the A/D converter.Quantization error is a randomly distributed, bipolar errorthat can usually be minimized by averaging over time.

Clearly a measurement system with a 16-bit A/Dconverter has much finer resolution than one with a 10-bitconverter, but is it more accurate? The answer is maybe!First of all, a system with a 16-bit A/D converter that ispreceded by a poorly designed analog front end, with nofirmware compensation for the analog domain errors, mayprovide very fine resolution of an input signal that is onlyaccurate to within a range of ±2% to ±5% of the ideal signal.Conversely, a measurement system with a 10-bit A/Dconverter and a well designed analog front end withcompensation for analog domain errors may easily beaccurate to within ±0.2 % or better of the ideal signal.

In addition to resolution, there are other characteristics ofA/D converters that contribute to the final output accuracy.These include harmonic distortion, nonlinearity, and gainerror. A detailed discussion of these items is beyond thescope of this paper. However, it is worth noting that thesefactors can vary considerably between differentmanufacturers’ components even if the number of “bits” isthe same for all the devices.

Therefore, it is obvious that resolution is an importantcontributor to total system accuracy, but it is not, by itself, anindicator of true system accuracy.

Sampling Effects

Up to this point all of the error sources discussed were dueto physical deficiencies in electronic components. However,this is not the only source of error in a digitally sampledsystem. Additional errors can result from the rate at which asignal is sampled and the relative timing of samples whenmore than one signal must be sampled. These issues arediscussed below.

Sample Time Skew

In a system that measures signals on multiple inputchannels, it is common for a single A/D converter to be usedwith a multiplexer in front of it to route each of the desiredsignals to it for conversion. This is done to reduce cost,required circuit board area, and ultimately the size of the finalproduct. Unfortunately, this is also the cause of time skew ina sampled system. When a measurement system must samplemultiple channels of data, the time relationship of thesamples may or may not be important. For example, considera measurement system in which one channel of “current” datais being acquired and one channel of “voltage” data is beingacquired for the purpose of computing line current, linevoltage, and power. The time relationship between thecurrent input samples and the voltage input samples is of noimportance for calculating the respective magnitude of thosesignals. However, the time relationship is very important forcalculating power. If the two signals are sampled at differentinstants in time, this creates time skew between the two setsof sampled data. The time skew error manifests itself as aphase error in the power calculation and results in degradedaccuracy for power, var, power factor, and harmoniccalculations.

The effects of time skew error can be reduced oreliminated by appropriate firmware/hardware design. For aninput that contains a fundamental frequency component only,the error is a linear function of frequency and can becorrected by the system firmware. However, for an input thatis composed of a fundamental component and significantharmonic content, the spectral content of the input signalsmust be computed and the error determined individually foreach of the harmonics present. The corrected power at thefundamental and each of the harmonics present then has to bemathematically combined to obtain the final result. This isnot a practical approach for a measurement system thatprovides real-time data.


The need for a firmware based solution to this problemcan be eliminated by proper circuit design. In order toaccomplish this, the time skew must be reduced to asufficiently small value or eliminated all together. The timeskew can be minimized by selecting a multiplexer and A/Dconverter that are capable of acquiring and converting theinput signals in as small a time increment as possible. Thetradeoff for smaller time increments and the resultingreduction in error is increased cost. This must be evaluatedbased on the requirements of the design.

In systems that demand the highest possible accuracy, allof the input signals should be sampled simultaneously. Thiseliminates time skew and the need to correct for it.Therefore, high accuracy real-time measurements arepossible. This solution also results in the most complexcircuitry and the highest cost.

Sampling Frequency

The sampling frequency specifies how many equallyspaced samples per second are taken on each channel of themeasuring device. This number can be used to determinehow many samples the measuring device will take in onecycle of the fundamental frequency of an input waveform.The number of samples per cycle of the fundamentalfrequency is computed as follows:

Samples per cycle Sampling FrequencyFundamental Frequency_ _ _

_=

For example, the number of “Samples per Cycle” with a960 Hz Sampling Frequency and a 60 Hz fundamental is 16.If the Sampling Frequency is increased to 10,800 Hz, then theresult is 180 samples per cycle.

The sampling frequency is important because itdetermines the maximum harmonic frequency that can bedetected by the system. This maximum frequency is knownas the “Nyquist” frequency and is equal to one-half thesampling frequency. In addition the sampling frequency isalso a significant factor in defining the frequency resolutionof any spectral analysis performed on the sampled data. Thesampling frequency also affects the ability of the measuringdevice to accurately measure the input signal, particularly ifthe input frequency varies.

Higher sampling frequencies can theoretically providemore information about the signal being measured. However,the additional data requires more processing time and slowsdown the response time of the measurement system. Inaddition, if the sampling frequency is much greater thantwice the highest frequency component in the signals beingmeasured, very little improvement in accuracy is obtainedeven though the computation requirements significantlyincrease. When designing or selecting a measurementsystem, these tradeoffs must be carefully considered andimplemented.

Anti-aliasing Filters

In the previous section it was stated that the sigNal beingmeasured should not contain any frequency componentsgreater than one-half the sampling frequency. This is arequirement of Shannon’s Sampling Theorem which statesthat in order for a discrete sampled system to accuratelyreproduce a continuous analog signal, the analog signal mustbe sampled at a frequency at least twice the bandwidth of thesampled signal. If this requirement is not met, the sampledsystem transposes the “out of band” frequencies into lowerfrequencies within the band being measured. The transposedfrequency components are indistinguishable from thosefrequency components that are actually present. This effect iscalled “aliasing.” Aliasing results in erroneous spectralinformation and the inability to correctly reproduce thesampled waveform.

In order to prevent problems due to aliasing, analog low-pass filters are typically used at the front end of themeasurement system. Ideally these filters pass all thefrequencies within the desired bandwidth without affectingtheir relative magnitude and phase while completelyeliminating all frequencies outside the target bandwidth.Unfortunately, realizable analog filters can not provide thatkind of performance and therefore represent another errorsource in the measurement system. A very complex,expensive filter would be required to even approximate theideal performance. Therefore, less than ideal performance isusually accepted. The typical filters used will increasinglyaffect both the magnitude and phase of the analog signal asthe frequency increases. In addition, the filter will not be ableto completely eliminate the frequencies outside the desiredrange. The unwanted, frequency dependent effects of theinput filters can be reduced by appropriate softwarealgorithms if the filter characteristics are known andsufficient processing time is available.

Some measurement systems attempt to eliminate theanti-aliasing filters by assuming that the samplingfrequency of the system, and therefore the Nyquistfrequency, is sufficiently higher than the anticipatedbandwidth of the analog input to prevent problems due toaliasing. If this type of approach is implemented byarbitrarily increasing the sample rate, unnecessary costs,increased processing time, and increased memoryrequirements may result. If the sample rate is pickedbased on the highest anticipated frequency content of theanalog input and no anti-aliasing filters are used, thereare still two possible error sources. The first is thepossibility that the input signal actually has frequencycomponents higher than anticipated and therefore thesampling frequency is too low. The second, less obvious,error source is the presence of high-frequency electricalnoise at the inputs to the measurement system (e.g.,interference from radio transmitters). With no anti-aliasing filters in the system, high frequency noise


coupled to the inputs can easliy cause aliased frequencycomponents to be present in the sampled data. Both ofthese possible error sources result in spectral errors. It istherefore prudent design practice to include analog anti-aliasing filters in the signal input circuitry.

COMPUTATIONAL ATTRIBUTES

Once the input signals have been properly conditionedand sampled by the analog front end circuitry, anti-aliasingfilters, and A/D converter the resulting data stream isprocessed by the microprocessor/DSP to provide usefulinformation about the input signal. The word size (8, 16, or32 bit) of the processor, the word type (fixed point or floatingpoint), and the algorithms used to compute the metering dataaffect the accuracy of the final result. Word size by itself isnot a definite indication of computational accuracy sincemultiple words can be strung together to maintain accuracy.For example, two 8-bit words can be used to provide the sameinformation that one 16-bit word would contain. However,the complexity of the software and the execution timeincrease. A processor with an 8-bit word size is typically lessexpensive than a processor with a 16-bit word size. Thisforces a cost versus complexity and execution time tradeoff tobe made.

A “fixed point” processor expresses the value of a numberas a standard binary number with the adjacent bits related bythe power of 2 in the same manner as previously discussed forthe A/D converter. A “floating point” number consists of twoparts, a mantissa and an exponent. The value of the numberis the value of the mantissa times the number two raised tothe power specified by the value of the exponent. Thefollowing examples, which for simplicity assume positivenumbers only, illustrate the difference in the two differentformats. In a “fixed point” system the largest number thatcan be defined by a 16-bit machine is (216 -1), or 65,535. Fora “floating point” system, again assuming the same 16-bitlimitation, with 12 bits for the mantissa and 4 bits for theexponent, the largest number that can be represented is ((212 -1) * 215), or 134,184,960. Obviously, the “floating point”system has a much larger dynamic range than the “fixedpoint” system. However, in this example the resolution of the“floating point” system is 1/16 the resolution of the “fixedpoint” system because the number of bits in the mantissa isfour less than the number of bits in the “fixed point” system.A “floating point” processor is usually more expensive than a“fixed point” processor. Therefore, tradeoffs betweendynamic range, resolution, and cost must be evaluated. Dueto its more complex structure, the “floating point” processormay process data slower than its fixed point counterpart eventhough the clock rates are the same. This becomes significantin the choice of processor word type.

The metering accuracy of a measurement system is alsoaffected by the algorithms used to compute the value of themeasured quantities. For example, the RMS value of awaveform is mathematically defined as follows:

( )VT

v t d tr m sT

= ∫1 2

0

In a digitally sampled system, there are several ways tocompute the value of the integral in this equation. The mostcommon methods are the Rectangular Approximationmethod, the Trapezoidal Rule, and Simpson’s Rule. All threeof these methods provide a numerical approximation to theexact value of the integral. The Rectangular Approximationis the least complex method to implement and also the leastaccurate of the three methods. The Simpson’s Rule methodfor approximating the value of the integral is the mostcomplex to implement and provides the best accuracy.Obviously, the Trapezoidal Rule is in the middle for bothcomplexity and accuracy of results. These differences shouldbe kept in mind when comparing metering systems since twodifferent products may both provide the same function but oneof them may use an inferior method to accomplish the endresult.

POWERMONITOR II SPECIAL FEATURES

The first portion of this document discussed generalinformation that pertains to any measurement system. Theinformation to follow specifically relates to the PowermonitorII measurement system.

Error vs. FrequencyEven after selecting the most accurate algorithms, (The

Powermonitor II utilizes Simpson’s Rule in conjunction witha patent pending algorithm for calculating RMS values.)errors can be introduced as the frequency of the measuredsignals change. Most power monitoring devices have asampling frequency that is optimized for 50 and 60 Hzsystems. However, it is unrealistic to expect the inputfrequency to remain exactly at those frequencies. Slightvariations from those frequencies can result in significanterror, especially for fast, single cycle measurements. Thegraph shown in Figure 1 shows the percent error introducedinto the RMS calculation (Typical result for competitivedevices when the classical Simpson’s Rule method is usedalone.) for a single cycle measurement as the frequency isvaried from 40 to 75 Hz. It does not include any errors due toother factors (such as hardware induced errors). Obviously,there is a significant amount of error introduced as thefrequency changes. The Powermonitor II uses a unique,patent-pending algorithm to virtually eliminate thesevariations due to frequency. An example of the results of thisalgorithm is shown in Figure 2. The accuracy improvementis easily seen by noting the difference in the “Percent Errort”scales.


40 45 50 55 60 65 70 750.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fundamental Frequency in Hz

Per

cent

Err

or

Figure 1 - Percent RMS Error vs. Frequency (Basic Simpson’s Rule Method)

40 45 50 55 60 65 70 750.02

0.015

0.01

0.005

0

0.005

Fundamental Frequency in Hz

Per

cent

Err

or

Figure 2 - Percent RMS Error vs. Frequency (Improved Powermonitor II Method)


Spectral Analysis

SAMPLING ERROR (WINDOWING )

In a typical power system monitored by the PowermonitorII, the input signals are a combination of the fundamentalfrequency, 50 or 60 Hz, and other additional frequencies thatare integer multiples of the fundamental. The additionalfrequencies, called harmonics of the fundamental, are theresult of non-linear loads that are present on the powersystem. In an ideal power system only the fundamental ispresent. Harmonics can cause overheating of the wires,transformers, and other devices connected to the system. Inaddition, they can cause other electronic devices connected tothe power system to fail or perform unreliably.

The Powermonitor II uses a “Fast Fourier Transform” toperform a spectral analysis function on the voltage andcurrent waveforms to determine the harmonic content presentin the system being monitored. The magnitude and phase ofeach of the harmonics, through the 41st, is provided by thedevice. The spectral data can then be utilized to implementappropriate actions based on the measured harmonic content.Typical actions that may be implemented by a user include:removal of specific loads from the power system,reconfiguration of the power distribution system, time shiftingof loads known to be harmonic sources, or automatedswitching of filter banks to remove the unwanted harmonics.

In review, a metering device’s sample rate can beoptimized for 50 or 60 hertz but significant errors areintroduced as the power system’s frequency drifts. The errorsmanifest themselves by indicating incorrect magnitude andphase information for the harmonics actually present in thesystem and by indicating frequency components that don’treally exist. In most instruments, these errors are reduced bypre-processing the waveform data with a method known as“windowing.” Windowing can provide some improvement inthe accuracy of the spectral analysis, but the improvement isnot uniform across the desired bandwidth of operation and itdistorts the original data in the process. The Powermonitor IIimplements a patent pending algorithm that maintainsaccuracy as the fundamental frequency varies withoutdistorting the original data. As a result, the Powermonitor IIsability to determine the harmonic content of the waveform issignificantly improved when compared to other devices in itsclass.

OscillographyThe Powermonitor II is capable of providing multi-cycle

oscillographs of any of the voltages or currents that itmeasures (not including the auxiliary input). Theoscillographs can be triggered manually or configured toautomatically occur when an user entered setpoint isexceeded. All input channels are continuously sampled at a10.8 KHz rate. This results in 180 samples per cycle at 60

hertz. This sample rate provides a new sample point every92.5 micro-seconds and yields better resolution than the 4,16, 64, or 128 samples per cycle utilized by most otheravailable devices.

There are two different groups of oscillographs available.The first group consists of all seven input channels (Fourcurrent and three voltage) and is two cycles in length. All ofthe signals in this group are simultaneously saved. Thisallows the user the ability to see the waveform status on allchannels when a trigger occurs. The oscillograph data in thisgroup is all post-trigger data.

The second group of oscillograph data consists of twochannels and is 12 cycles in length. This group of data istypically used to obtain more in depth information on specificsignals of interest. Any of the seven input channels may beconfigured to either of the channels in this group. Theoscillograph data in this group is configurable for 0 to 40percent pre-trigger data with the remainder of the data beingpost-trigger.

Selected Computations

POWER FACTOR

Power Factor is a term used to describe what portion ofthe voltage and current supplied to a power system is actuallyperforming work. If this number is less than 100 percent,then a portion of the voltage and current supplied is beingutilized by inductive or capacitive reactance and provides nowork. The Powermonitor II calculates three different PowerFactor numbers. Each of these methods are defined below.

True

True Power Factor is the value obtained by computing theclassical definition of Power Factor and is expressed inpercent:

PFWatts

VA% = ∗100

This value of Power Factor takes into account the effect ofany phase shift between the voltage and current as well as theeffect of any harmonics present.

Displacement

Displacement Power Factor is the value obtained bycomputing the cosine of the phase angle between thefundamental component of the voltage and the fundamentalcomponent of the current (see Figure 3 below) andmultiplying by 100. This is the value that a typical analogPower Factor meter would measure. The value of the TruePower Factor and the Displacement Power Factor are equal ifthere are no harmonics present in either the voltage or thecurrent. The formula for Displacement Power Factor is:Displacement_Power_Factor% = 100∗cos(θvoltage−θcurrent)


Figure 3 - Phase Angle Between Voltage and Current

Distortion

Distortion Power Factor is a value that is computed toindicate what percentage of the current is attributable to thefundamental component. The formula for Distortion PowerFactor is:

Distortion Power FactorFundamental Current

Total Current_ _ %

_

_= ∗100

When no harmonics are present the value of theDistortion Power Factor is 100%.

DEMAND

A typical industrial utility bill consists of at least twoitems, the Kilowatt-Hour charge and the Demand charge.Demand is equal to the average power (current, VA, or Varsmay be used instead) level during a predetermined timeinterval. This interval continuously repeats and is typicallybetween five and 30 minutes in length. The formula for KWdemand is shown below:

DemandT

P t dtt

t T

= ∗+

∫1( )

T = Demand interval duration,t = Time at beginning of intervalP(t) = Power as a function of time

The peak Demand that occurs during a period specified bythe utility, or agreed to by contract, is used to determine theDemand charge. This period may be one month, one year,the length of the contract, or some other agreed uponduration. As a result, only one occurrence of a high Demandlevel can have a long range effect on the user’s utility bill.The peak Demand value provides an indication of the reservecapacity the utility must be capable of supplying to satisfy theshort term requirements of the user even though the user isnot continuously using that level of power. The instantaneousavailability of this capacity is what the user is paying forwhen the Demand charge is levied.

The utility may provide a pulse to allow the user to knowwhen the beginning of each Demand interval begins. Thedemand value is then updated at the end of each interval andthe highest value obtained during any interval is maintained.This method is known as the thermal demand. If the utilitydoes not provide a pulse at the beginning of each interval, theuser is not able to synchronize with the utility to control hisDemand level. In this instance the sliding window method isused. This method breaks the Demand interval into manysub-intervals and updates the Demand value at the end ofeach sub-interval. For example a five minute interval mightbe divided into five one-minute sub-intervals. The Demandfor each one minute interval is calculated and at the end offive minutes the average value of the sub-intervals iscomputed to obtain a Demand value. At the end of the sixthminute, the value for sub-interval one is discarded and a newDemand value is computed based on sub-intervals twothrough six. In this manner a new five minute demand valueis obtained every minute. The maximum value obtained inthis manner is then maintained as the peak Demand. Thisallows a reasonable approximation to the actual Demand thatthe utility measures.

The user typically would like to minimize his peakDemand in order to reduce his utility demand penaltycharges. One method to accomplish this is to measure thepower being used and to project what the Demand level willbe at the end of the interval. In this manner the user canreduce power consumption when the projected Demandreaches a predetermined threshold and prevent the finalDemand from exceeding the desired level.

The Powermonitor II computes Demand levels for watts,VA, amps, and Vars, and provides three different methods forprojecting the value of the Demand at the end of the interval.In addition the Powermonitor II is capable of accepting asynchronizing pulse from the utility in order to implement thethermal Demand method or it can be configured to implementthe sliding window method. The user should select the bestprojection method for his system by comparing the projectedvalues from each method with the actual demand at the end ofthe interval. The three methods of projecting Demand aredescribed below:

Instantaneous

This Demand calculation computes instantanteousdemand by substituting the elapsed interval duration for thetotal interval duration (T) in the above equation. It istherefore identical to the standard computation except itintegrates the power only over the elapsed interval durationand calculates the average value over the elapsed duration.The modified equation thus becomes:

Demandt t

P t dtt

t

=−

∗ ∫1

2 11

2

( )

(t2-t1) = Elapsed interval duration and is less than T

0 30 60 90 120 150 180

10

0

10

voltagecurrent

Phase Angle in Degrees

Mag

nitu

de


First Order Projection

The first order Demand projection utilizes theInstantaneous Demand as a starting point, computes the trendof the Instantaneous Demand, computes the time remainingin the interval, and performs a first order projection of whatthe final Demand will be at the end of the interval. Thisprojection method is most useful for power utilizationscenarios that have a significant base load with additionalloads that are switched in and out during the interval.

Second Order Projection

The second order Demand projection utilizes theInstantaneous Demand as a starting point, computes the trendof the Instantaneous Demand, computes the rate of change ofthe trend, computes the time remaining in the interval, andperforms a second order projection of what the final Demandwill be at the end of the interval. This projection method ismost useful for power utilization scenarios that have a little orno base load and a load profile that increases over theduration of the interval. Due to the characteristics of asecond order projection, the projected Demand value for thismethod is more sensitive to rapid load changes than the othermethods.

SYMMETRICAL COMPONENT ANALYSIS

In a three phase system it is very desirable for both themagnitude and relative phase displacement of each of thevoltage sources to be the same (balanced). It is also desirablefor the load on each phase to be identical (balanced).Whenever either of these conditions is not met, the threephase system is unbalanced. Rotating loads, such as motorsor generators, operating on an unbalanced system will exhibitsignificantly higher losses (heating) than the same loadsdoing the same amount of work when supplied by a balancedsystem.

Symmetrical Component Analysis is a method ofmathematically transforming a set of unbalanced three phasevectors (voltage or current) into three sets of balanced vectors.One set of vectors, the positive sequence component, rotates inthe same direction as the original set of unbalanced vectors. Thepositive sequence vectors represent that portion of the voltage (orcurrent) supplied to a rotating load that is capable of doing work.The second set of vectors, the negative sequence component,rotates in the opposite direction as the original set of unbalancedvectors. The negative sequence vectors represent that portion ofthe voltage (or current) supplied to a rotating load that result insystem losses due to the unbalance. The third set of vectors, thezero sequence component, is actually a single vector with norotation. The zero sequence vector represents ground or neutralcurrent (voltage). The effects of each of the sequence componentscan now be analyzed separately using simple equations restrictedfor use with balanced systems. The individual results can then becombined by the laws of superposition.

Typically, the negative sequence impedance of rotatingloads is much greater than the positive sequence impedance.Therefore, a small negative sequence component can causesignificantly more heating than one might expect based onthe magnitude of the original unbalanced vectors. Manymotors have been unnecessarily damaged due to thiscondition. In addition, the ratio of the negative sequencecomponent to the positive sequence component is the mostaccurate measurement of system unbalance.

The equations for computing the sequence components fora set of voltage vectors are shown below. To compute forcurrent vectors, simply substitute current for voltage in theequations.

Vpos seqV V Vab bc ca_

( )=

∠ ∗ ∠ + ∠ ∗ ∠ + ∠ ∗ ∠θ θ θ1 2 31 0 1 120 1 240

3

Vneg seqV V Vab bc ca_

( )=

∠ ∗ ∠ + ∠ ∗ ∠ + ∠ ∗ ∠θ θ θ1 2 31 0 1 240 1 120

3

Vzero seqV V Vab bc ca_

( )=

∠ + ∠ + ∠θ θ θ1 2 3

3

The Powermonitor II implements patented algorithms thatare capable of computing the positive and negative sequencecomponents in real time. These algorithms are bothfrequency and phase rotation insensitive. Zero sequencecurrent is typically measured directly with a zero sequencetransformer connected to the I4 input of the monitor. Withthis information, the user can now protect his system fromoverheating due to unbalance and, if desired, adjust theallowable system loading in real time based on the level ofunbalance present.

HARMONIC DISTORTION

Harmonic distortion is typically expressed in one of twoways. Both methods provide a summary indication of theamount of distortion due to harmonics present in a system.However, neither one of them provides any information as towhat harmonics are present or their relative magnitudes. ThePowermonitor II is capable of presenting the information inboth formats.

IEEE Definition

The standard IEEE definition of harmonic distortion iscalled the “Total Harmonic Distortion (THD).” It iscomputed as follows:

THD

H

H

nn= =

∞

∑ ( )2

2

1

Hn = magnitude of the n’th harmonic (n ≤ 41 in thePowermonitor II), H1 = magnitude of fundamental


IEC Definition

The standard IEC or DIN definition of harmonicdistortion is called the “Distortion Index (DIN).” It iscomputed as follows:

DIN

H

H

nn

nn

= =

∞

=

∞

∑

∑

( )

( )

2

2

2

1

Hn = magnitude of the n’th harmonic (n ≤ 41 in thePowermonitor II)

The two different equations for THD and DIN providebasically the same information but in different formats. Theyare related by the following equation:

THDDIN

DIN=

−1 2

CREST FACTOR

Crest Factor is another quantity that is sometimes used todescribe the amount of distortion present in a waveform. Itcan also be used to express the dynamic range of a measurentdevice. This value is simply the ratio of the peak to the RMS.The formula is shown below:

Crest FactorPeak Value

RMS Value_

_

_=

Therefore, for a pure sinusoid the Crest Factor is equal to

the 2 .

K-FACTOR

K-Factor is a value that takes into account the additionalheating in a power transformer that results from the presenceof harmonics in the system. These harmonics causeadditional heating due to increased core losses that occur athigher frequencies. The increased losses are related to thesquare of the harmonic frequency. Therefore, just a fewharmonics can significantly increase the heat rise in a powertransformer. The additional harmonic heating may cause atransformer to exceed designed temperature limits eventhough the RMS current is less than the transformer rating.The K-Factor can be used as justification to oversize a powertransformer to provide extra margin for harmonic losses or toselect an appropriate K-Factor rated transformer. A K-Factorrated transformer is the preferred choice because it providesknown performance in the presence of harmonics. Theformula for K-Factor is as follows:

K Factor

H n

H

n

n

nn

− =∗

=

∞

=

∞

∑

∑

( )

( )

2 2

1

2

1

Hn = magnitude of the n’th harmonic (n ≤ 41 in thePowermonitor II)

TELEPHONE INFLUENCE FACTOR (TIF)

The Telephone Influence Factor (TIF), sometimes calledthe Telephone Interference Factor, is another method ofmeasuring signal distortion and is used to estimate the effectthat power line harmonics will have on nearby analogtelephone conductors. In this method, the magnitude of eachof the harmonics is weighted based on the physiological andaudiological characteristics of the human ear. The harmonicsare additionally weighted to reflect the relationship ofharmonic frequency and degree of coupling to the phonelines. These weights are called the single frequency TIFweights. The Powermonitor II uses the most recent TIFweights (updated in 1960). The single frequency factors areused to compute the total Telephone Interference Factor. Theuser typically multiplies the TIF numbers by the RMSmagnitude of the voltage or current in the power lines toobtain an index for estimating the amount of interferingenergy that is coupled to the telephone system. The formulafor the total TIF is as follows:

TIF

w X

X

i ii

ii

= =

∞

=

∞

∑

∑

( )

( )

1

2

1

2

Where Xi = single frequency RMS current or voltage atharmonic i, and wi = single frequency TIF weighting factor atharmonic i.

IEEE-519

IEEE-519 is the IEEE standard for “RecommendedPractices and Requirements for Harmonic Control inElectrical Power Systems.” The Powermonitor II refers tothe 1992 version of this standard. The intent of IEEE -519 is to provide recommended limits for the level ofharmonic current injection at the Point of CommonCoupling (PCC) between the utility and the user. ThePCC is typically defined as that location in the powerdistribution system where the utility meters are connected.The standard provides recommended limits for individualharmonic components as well as a limit for Total DemandDistortion (TDD). Total Demand Distortion is defined asthe root sum square of the current distortion expressed asa percent of the maximum fundamental demand loadcurrent (based on the maximum demand over the last 12months over the applicable demand interval). Table 10.3of the standard specifies the limits. The appropriate limitsare selected by computing the ratio of the available shortcircuit current to the maximum fundamental demand loadcurrent. The row of the table that corresponds to the ratiois then used to determine the proper limits for each of theindividual harmonics and the TDD specified in the tablecolumns. IEEE-519 also recommends maximum voltagedistortion levels that the utility should remain below.Table 11.1


Rockwell Automation Headquarters, 1201 South Second Street, Milwaukee, WI 53204 USA, Tel: (1) 414 382-2000 Fax: (1) 414 382-4444Rockwell Automation European Headquarters, Avenue Hermann Debroux, 46, 1160 Brussels, Belgium, Tel: 32-(0) 663 06 00, Fax: 32-(0) 2 663 06 40Rockwell Automation Asia Pacific Headquarters, 27/F Citicorp Centre, 18 Whitfield Road, Causeway Bay, Hong Kong, Tel: (852) 2887 4788, Fax: (852) 2508 1846World Wide Web: http://www.ab.com

Publication 1403-1.0.2 November 1996Copyright 1996 Rockwell International Corporation Printed in USA

specifies these limits based on the magnitude of the line toline voltage at the PCC. For more detailed informationrefer to IEEE Standard 519-1992.

The Powermonitor II allows the user to configure thedevice to automatically monitor the system voltage andcurrent for IEEE-519 compliance. The user must configurethe Powermonitor II with the appropriate available shortcircuit current and the maximum demand current. ThePowermonitor II then uses this information along with theconfigured PT primary (or direct connected) voltage and thereal time harmonic analysis values to compute compliancewith the distortion limit tables. In the event that one of theindividual harmonics or the TDD exceeds the recommendedlimits, the Powermonitor II records and time stamps theevent. The event is communicated to the user and can also beused to affect the status of one of the output relays.

WHOSE HARMONICS ARE THESE?With the ease of use and affordability of the Powermonitor

II, it is now a relatively simple task to determine theharmonic content of the voltage and current at almost anypoint in the power distribution system. While this greatlyincreases the available knowledge about the system, it alsocreates new questions. Is this level of harmonics a problem?Why are there harmonics present in the current and not in thevoltage? Where do these harmonics come from? Who isresponsible for these harmonics? These are important andsometimes perplexing questions. While a full treatment ofthese questions is beyond the scope of this document, thefollowing notes may be helpful.

• The exact level of harmonics that may create a problemrequires a detailed harmonic flow analysis that is specificto each individual system. However, estimates of theseverity of any harmonic presence can be made byutilizing various data items from the Powermonitor II.These items include Crest Factor, K-Factor, THD,Distortion Power Factor, Positive and Negative Sequencecomponents, and comparison of harmonic content toIEEE-519. Values for these quantities can be comparedto reasonably expected values or examined for significantchanges from nominal readings.

• The presence of significant harmonics in the current andnot in the voltage is not unusual. Most harmonics in apower system are created by loads that draw non-sinusoidal current. These loads can be visualized asharmonic current sources and the resultant harmoniccurrents are measured by the Powermonitor II. Thesystem impedance at the frequency of each of theharmonics present determines the magnitude of anyresultant harmonic distortion in the voltage. Therefore, asystem with a large available current rating andcorresponding low source impedance will have lessharmonic voltage distortion than a system with a smalleravailable current and a corresponding higher sourceimpedance when both are subjected to the same harmoniccurrent flow.

It is generally accepted that the provider of the power isresponsible for harmonics present in the voltage delivered tothe service entrance connection and the user of the power isresponsible for harmonic content in the current. This conceptmakes sense when there is little or no voltage distortion andthere is significant current distortion. However, when voltagedistortion is present on the incoming supply, this approach isan oversimplification. This is due to the fact that theexcitation due to each of the voltage harmonics will cause acorresponding harmonic current to flow in the system. Themagnitude of these currents is dependent on the systemimpedance at each of the harmonic frequencies. In order todetermine what harmonic currents are the responsibility ofthe user, the currents created as a result of the voltagedistortion should be subtracted from the total current in orderto determine the harmonic current that is the responsibility ofthe user. The procedure to accomplish this is described byK. Srinivasan of Hydro Quebec in the July / August 1995issue of Power Quality Assurance magazine.2 The issue ofharmonics is rarely a simple one. However, thePowermonitor II provides a significant amount of data toallow intelligent analysis and resolution of harmonicproblems.

2 Krishnaswamy Srinivasan, “How Much Harmonics Is YourResponsibility?”, Power Quality Assurance, July/August1995, p. 62 - 65.

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