Here’s an overview of this discussion.A good grasp of harmonics is a prerequisite to understanding interharmonic and provides a good background for today’s discussion.Since most of you all have been exposed to harmonics, I will briefly cover harmonics and quickly transition into the concept of the DFT.This will lay the ground work for our discussion of interharmonics.Next, follow slide
1. A sinusoid is the lowest common denominator of all periodic signals.2. Also, in engineering sinusoids are a preferred tool because of Sinusoidal Fidelity. If you apply a sinusoid to a linear system the output is always a sinusoid no other waveform has that characteristic. 3. The atom is depicted as the basic unit in chemistry. For this discussion, the sinusoid is analogous and the basic unit for periodic waveforms.4. Utilities such as SRP are know as voltage providers. It should be noted that the voltage at least from the generator is near perfect.
Fourier came up with his claim from his attempt to solve a heat transfer problem of a metal plate.This claim did not go without a challenge. Joseph LaGrange another famous French mathematician of the day vehemently objected. He said such an approach cannot be used to represent signals with corners such as square waves.The mathematical question of determining when a Fourier series converges has been fundamental for centuries.It turns out that using Fourier’s method for discrete signals (predominately used today) is exact.
1. The top figure depicts the construction in the time domain of a square wave from a plurality of sinusoids.2. The resultant square wave in the time domain of the bottom figure shows the ringing (Gibbs Phenomenum) caused by applying the Fourier method to make a square-wave function at a discontinuity. Even if the number of harmonics approaches infinity. The fact is the Fourier expansion fails to converge uniformly at discontinuities.4. In the next few slides, we will discuss related issues with the Fourier method as it relates to interharmonics.5. The point of this slide is that the Fourier method is not perfect. For example, the unit-step function and undamped sinusoids do not have Fourier transforms.
1. Fourier methods allow the user to look at time domain signal in the frequency domain. The frequency domain that is provided via the transform is a discrete spectra, not continuous.2. Talk to the four methods.3.The DFT can be substantially slow. Hence, methods such as the Fast Fourier Transform (1965) and Damn Fast Fourier Transform can be used in lieu of the Discrete Fourier Transform. This is due to the time constraints associated with the DFT.4. For today’s discussion I will refer to the DFT in a broad context to include the other methods.
When these conditions are satisfied the measurements are accurate.
The vertical axis is normalized voltage or current.The horizontal axis is time.The scope is depicting the whole DFT window.Only the 2 Pi portion of the DFT is stationary.DFT does a poor job of resolving the frequencies for non-stationary signals.
The vertical axis is normalized voltage or current.The horizontal axis is time.
Harmonics must be a multiple of the fundamental frequency.Interharmonic are not a multiple of the fundamental frequency.Subharmonics are a special type of interharmonics and not discussed in this presentation.
Each piece of equipment has its own harmonic signature.
Cycloconverter or a cycloinverter converts an AC waveform, such as the mains supply, to another AC waveform of a lower frequency, synthesizing the output waveform from segments of the AC supply without an intermediate direct-current link.This definition of non-characteristic harmonics in IEEE 519infringes upon the definition of interharmonics, because of the words “beat frequencies”, “demodulation” and “cycloconverter”. Hence, non-characteristic harmonics can include interharmonics based on the IEEE 519 definition. The next update of IEEE 519 is expected to include a more comprehensive definition of interharmonics.
This definition puts words to the table on a previous slide.Signals across the electromagnetic spectrum is an example of wide band spectrum.One of the things SRP does for its customers is answer complaints concerning RF interference caused by arcing within the distribution system.
The vertical axis is normalized voltage and the horizontal axis is time.Discrete calculation are made across the DFT windowAs illustrated a DFT is taken of a 60 Hz fundamental power frequency over one cycle or about 16.67 ms.We probably cannot do justice to this subject without going through the math. However, we will instead plug through this subject using graphics in the charts to follow.
The vertical axis is normalized voltage and the horizontal axis is time.As depicted, a DFT samples at 16 points over a DFT window of one cycle or DFT Window A. The sampling at discrete points is not shown in the remaining DFT window or slides.The angular frequency resolution can be used to determine the frequency buckets for collecting spectra of the DFT. The 0 to A, B and C are the boundaries of the DFT windows of 1 cycle, 2 cycles and 5 cycles, respectively.For example, a DFT window A of one cycle only collects spectra or in this case harmonics at multiples of 60 Hz (i.e., 120, 180, 240 and the like.Further, DFT window B up over two cycles collects spectra every 30 Hz including interharmonics of 30 Hz (90Hz, 150Hz…)Further yet, DFT window C over three cycles collects spectra every 12 Hz including interharmonics or subharmonics of (12Hz, 24Hz…)
The DFT is analogous to looking at the world through a picket fence.The frequencies determined by the DFT have nothing to do with the signal being analyzed.Sometimes signals of interest can be interharmonics outside the DFT window.As an example, the signal at the frequency as circled in red is not seen by the DFT.As the frequency resolution increases, the picket fence can be said more like looking through a screen porch.
As we will see by opening up the DFT window measurements can be deceiving.
The vertical axis is normalized voltage and the horizontal axis is time.In the real world we typically do not see the harmonics broken up with the fundamental as depicted above. Instead, we see a distorted waveform. But for this discussion various harmonics are shown with the 60 Hz fundamental (red).The figures on the left and right illustrate the fundamental and harmonic in the time-domain and frequency-domain, respectively.Fundamental frequency (in red) is depicted with some of its harmonics.Note that all the harmonics close out as a complete period for their respective frequency within the DFT window.For example, count the number of cycles of the third harmonic (black) within the DFT window.The fundamental and all the harmonics can be measured within one 16.67 ms DFT window.Further, the frequency domain from the DFT of the fundamental and harmonics are shown on the right.The main takeaway here is that if one period of the fundamental is selected for the DFT only the fundamental and integral multiples of the fundamental frequency (harmonics) can be seen in the DFT.
The vertical axis is normalized voltage and the horizontal axis is time for the plot on the left.The important point to note is that both sinusoids close on complete cycles at the end of the DFT. For the 60 Hz waveform, the two cycles are completed at the end of the DFT.For the 90 Hz waveform, three cycles are completed at the end of the DFT.The frequency domain plot on the right (i.e., the result of the two cycle DFT) depicts the frequency buckets with the appropriate frequencies.
The vertical axis is normalized voltage and the horizontal axis is time for the plot on the left.The figures on the left and right illustrate the 60 Hz and 100 Hz waveforms in the time and frequency domains, respectively.As depicted on the left, the 100 Hz waveform (black) does not complete the DFT. Instead, the 100 Hz waveform completes 3.33 cycles within the DFT window.On the right, the frequency components are in buckets of 60, 90, 120 and 150 Hz. The 60 Hz component is correctly bucketized. In contrast, the 100 Hz signal is distributed into the 90, 120 and 150 Hz buckets. It can be said the components of the 100 Hz waveform are smeared, thereby generating false components.This phenomenon is referred to as spectral leakage. Hence, we can conclude that not all interharmonic measured are necessarily real. Instead, they can be a manifestation of the DFT.
Harmonics between 3 and 6 percent of Nominal Fundamental per IEC 61000-2-2 & Table 11.1 of IEEE 519-1992Interharmonics per IEC 61000-2-2White Noise per 1000-2-1The power system voltage contains a background Gaussian noise with a continuous spectrum. Typicallevels of this disturbance are in the range (IEC 1000-2-1).The takeaway here is the relative differences in the limits of interharmonics compared to harmonics (i.e., harmonic limits are 25 time greater).And, the interharmonic limit is about 10 times (or 20 db) the noise occurring on the power system.
I understand the next generation of IEEE 519 will include interharmonics limits.
The low limit will guarantee compliance of interharmonic distortion with light systems. However, due to measurement difficulties other limits are being evaluated.Difficulties in measurement are due to at least:Low values or magnitudesVariability of their frequency and magnitudesVariability of their waveform periodicityHigh sensitivity to spectral leakage*Interharmonics per IEC 61000-2-2
Interharmonics caused by asynchronous switching tend to be at discrete frequencies.In contrast, rapid changes can be generated by loads operating in a transient state, either continuously or temporarily. This can cause amplitude modulation of currents and voltages and are largely random in nature. The interharmonics are typically random in nature and highly a function of the changes in load current.Interharmonics caused by rapid changes in load current tend to be wide band.In referring to harmonics and interharmonics, it is important to distinguish between voltage and current.The sources of interharmonics are due to a change in current. These current changes can be either a rapid irregular/random current changes quasi-regular current changes caused by asynchronous switching (i.e., switching not synchronized with the power frequency.
Devices using PWM and cycloconverters can result in asynchronous switching. In contrast, arc furnaces can cause rapid changes in the load current.
A typical application of a cycloconverter is for use in controlling the speed of an AC traction motor and starting of synchronous motor. Most of these cycloconverters have a high power output –on the order a few megawatts – and silicon-controlled rectifiers (SCRs) are used in these circuits. Common applicationssuch as for rolling mills in ore processing, cement kilns, and also for azimuth thrusters in large ships.The cycloconverter is the ideal source of interharmonics.The cycloconverter directly connects the input and output frequencies, because there is no DC node.The formula shown can be used to calculate the interharmonic generated from the cycloconverter.
Though not as a reliable source of interharmonic as the cycloconverter, variable speed drives and also generate interharmonics.
The characteristic harmonics of an ideal six-pulse rectifier is F ,which is the calculated per the depicted formula.
A little background on the electric arc furnace.The use of EAFs allows steel to be made from a 100% scrap metal feedstock. The primary benefit of this is the large reduction in specific energy (energy per unit weight) required to produce the steel. Another benefit is flexibility: while blast furnaces cannot vary their production by much and are never stopped, EAFs can be rapidly started and stopped, allowing the steel mill to vary production according to demand.
Another source of varying loads includes an arc furnace. The frequency of the load ωm determines the interharmonic generated. As the load varies the interharmonics vary. Typically, an the load of an arc furnace varies as the state of the molten metal changesThe solution I(t) is a series of sum and difference frequencies. Effectively, there is intermodulation between the power frequency and the operating frequency of the load.
60 Hz waveform is modulated by interharmonic sidebands.
The variation of the RMS value is a function of the magnitude and frequency of the interharmonics.Here, the magnitude is held constant at 0.2 and the frequency is allowed to vary.Interharmonics cause a variation in the RMS value.The closer the interharmonic is to the fundamental frequency the greater the fluctuation or deviation of the RMS value.At interharmonic frequencies higher than twice the power frequency (here 50 Hz), RMS deviationis small compared to the RMS deviation below the second harmonic.Interharmonics in Power Systems byErich W. Gunther
The graph shown plots Pst on the vertical axis and frequency on the horizontal axis.Pst - Short term flicker “perceptibility”. The average person borders on irritation for a level of 1.0Since the eye is sensitive to frequencies around 8 Hz. The interharmonics that cause the biggest lamp flicker issues are near the fundamental or harmonic frequencies. Interharmonics further away for the fundamental or harmonic frequencies are less of a lamp flicker problem.When the absolute value of the difference between the fundamental or harmonic frequency and the interharmonic frequency is about 8 Hz, the Pst approaches a maximum, as depicted above.
The minimum interharmonic amplitude (%) generating perceptible flicker.Since these plots are the minimum interharmonic amplitude to cause perceptible flicker, these plot are the inverse of the plot shown on the previous slide.On moving toward the power frequency or harmonic frequency, the interharmonic amplitude needed to generate perceptible flicker diminishes.Flicker can be detected with an IEC flicker meter for interharmonics less than 102 Hz for 60 Hz systems.Above 102 Hz the IEC flicker meter cannot detect flicker.3. Incandescent, LED and Fluorescent lamps are all exhibit a flicker response to frequencies less than 120 Hz.4. Only the LED and Fluorescent lamps have a flicker response above 120 Hz (due to electronics).The deficiency of the IEC flicker meter is due to the squaring and filtering processes in Blocks 2 and 3, respectively.
Lights in the office area of a steel manufacturing plant were flickering.A rolling milling using a cycloconverter is a source of interharmonics and is connected to a 30 KV bus.Two power factor correction capacitor banks of 4800 KVA are connected to the bus. Further, single-tuned notch type passive filter capacitors for the 5th, 7th and 11th harmonics are also connected to the bus.
A quick review of the response or harmonic impedance parallel (left) and series (right) configurations. At resonance, the current is minimum and maximum in the parallel and series resonant configurations, respectively.
A simulation of the 30 KV bus was done to examine the harmonic impedance.The chart plots the per unit impedance on the vertical axis and frequency in Hz on the horizontal axis.As depicted, the harmonic impedance (blue response curve) includes substantial peaks and valleys.As is typical for notch filter, there are parallel resonant frequencies just below each notch at 180 Hz, 330 Hz and 485 Hz.If any of the interharmonics produced by the rolling mill should coincide, or almost coincide, with one or more of these peak of harmonic impedance, this can generate substantial voltage components, thereby causing interharmonic flicker.The Q of the filter can be reduced by adding damping resistors. This will reduce the peaks and tend to flatten out the frequency response.Unfortunately, this is expensive and results in real power losses.Another difficulty with interharmonic filter design is that the interharmonics tend to move around as the load varies.In some cases, this can lead to multi-stage filters adding to the cost. Dynamic filters using electronic controls can be used to real-time monitor and control the harmonic and interharmonic levels.advanced power electronic techniques to continuously control harmonic and interharmonic levels in real-time.
IEC 61000-4-7 applies a DFT of 12 cycles to provide a frequency resolution of 5 Hz. The sampling is synchronized with the power frequency using a phase-locked loop.Depicted here are Harmonic Groups and Harmonic Subgroups.The RMS values of bands of frequencies are calculated. Go over the rules per above.
Depicted here are Interharmonic Groups and Interharmonic Subgroups.Go over rules above.
Here only the harmonics are used.
Here all components at 5 Hz increments are depicted.
Transcript
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The Harmonic Spectra of Interharmonics Gary Malhoit SRP August 26, 2010
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Overview Brief overview on harmonics and periodicity Fourier’s Method and the DFT Interharmonic Defined Picket-Fence Effect & Spectral Leakage Genuine & Non-Genuine Interharmonics Interharmonic standards and allowed limits Sources of Interharmonics Interharmonic Problems Measuring Interharmonics IEC Grouping Standard and Understanding Spectra Measurements 2
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Sinusoids Sinusoids are the basic building block of all periodic signals. Periodic waveforms are comprised of component sinusoids having distinct frequencies. This includes distorted periodic waveforms. 3
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Fourier 1822, a French mathematician named Joseph Fourier, claimed that continuous periodic signals can be represented by the sum of properly chosen sinusoids. 4
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Limitations to Fourier Methods 5 Source: Wikipedia 5
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Fourier Tool Kit 6 The Scientist and Engineer's Guide to Digital Signal ProcessingBy Steven W. Smith, Ph.D.
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Assumptions in Applying DFTfor PQ Measurements The signal is strictly periodic and stationary. The sampling frequency is an integer multiple of the fundamental. The sample frequency is at least twice the highest frequency being measured. 7
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Non-Stationary Signal 8 DFT Window Source: A Notebook Compiled While Reading Understanding Digital Signal Processing by Lyons 2Pi
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Non-periodic Signal DFT Window Half a sinusoid In time domain Source: A Notebook Compiled While Reading Understanding Digital Signal Processing by Lyons Spectral Leakage in the frequency domain 9
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What is Harmonic Spectra? Harmonic spectra includes sub-harmonics, harmonics and interharmonics. 10 f = n f1 where n is an integer > 0. Harmonic f = nf1 where n is an integer > 0. Interharmonic 0 < f < f1 Subharmonic f1= fundamental frequency Source: Power Quality Application Guide: European Copper Institute, AGH University of Science and Technology and Copper Development Association.
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Harmonic Spectra Characteristic Harmonics “Those harmonics produced by semiconductor converter equipment in the course of normal operation. In a six-pulse converter, the characteristic harmonics are the non-triple odd harmonics, for example, the 5th, 7th, llth, 13th, etc.” 11 Source: IEEE 519
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Harmonic Spectra (cont.) Non-Characteristic Harmonics “Harmonics that are not produced by semiconductor converter equipment in the course of normal operation. These may be a result of beat frequencies; a demodulation of characteristic harmonics and the fundamental; or an imbalance in the ac power system, asymmetrical delay angle, or cycloconverter operation.” 12 Source: IEEE 519
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Interharmonics Interharmonics- “Between the harmonics of the power frequency voltage and current, further frequencies can be observed which are not an integer of the fundamental. They can appear as discrete frequencies or as a wide-band spectrum.” Source: IEC 61000-2-1 13
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Interharmonics Redefined Interharmonics- “Any frequency which is not an integer multiple of the fundamental frequency” Source: IEC-61000-2-2 14
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One-Cycle Window 15 DFT Window 16.67 ms 60 Hz 15 The 60 Hz component competes 1 cycle within the DFT window.
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Expanded DFT Window In order to see interharmonics the DFT window must be larger than one cycle of the fundamental frequency. 18 Source: Interharmonics: basic concepts & techniques for their detection & measurement, Chun Li, et al.
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The Fundamental with Harmonics 19 DFT Window 60 180 300 Time 60 Hz-Fundamental Frequency 180 Hz-Third Harmonic 16.67 ms 300 Hz-Fifth Harmonic 19
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A Genuine Interharmonic DFT Window Genuine Interharmonic Magnitude 60 90 Frequency (Hz) Source: Interharmonics: Theory and Modeling, IEEE Task Force on Harmonics Modeling and Simulation 33.34 ms The 60 Hz component competes 2 cycles within the DFT window. 60 Hz 90Hz The 90 Hz component completes 3 cycles within the DFT window. 20
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Non-Genuine Interharmonics 21 DFT Window Non-Genuine Interharmonics (Black) Magnitude Source: Interharmonics: Theory and Modeling, IEEE Task Force on Harmonics Modeling and Simulation 33.34 ms 60 90 120 150 180 Frequency (Hz) 60 Hz The 60 Hz component completes two cycles within the DFT window. 100 Hz The 100 Hz component completes 3.33 cycles within DFT window.
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DFT Assumes Signal Repetitive Discontinuities Magnitude Time DFT Window Source: Oppenheim, et al. “Discrete-Time Signal Processing” Frequency Domain 22 Time Domain
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Determining if Interharmonics Are Real The voltage and current spectral components should show correlation. If the magnitude of the signal appears modulated, it is highly likely that the signal contains interharmonics. Interharmonics usually coexist with harmonics. If the signal is substantially non-varying or stationary, a longer DFT window can improve the frequency resolution. 23 Source: Interharmonics: basic concepts & techniques for their detection & measurement, Chun Li, et al.
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Interharmonics The main reason for lack of interharmonic concerns is that interharmonics are produced by relatively few types of loads, unlike harmonics. 24 Survey of Interharmonics in Indian Power System Network, B.E. Kushare, et al.
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Spectra & Noise Magnitudes 25 5% Harmonics 10.0 Interharmonics 1.0 % of Nominal Voltage of Fundamental .2% .1 White Noise .02% .01 Sources: IEEE 519 & IEC 61000-2-2 & IEC 1000-2-1
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Interharmonic Limits 26 Limits Standard IEEE 519-1992 Not covered IEC 1000-2-2 0.2% at ripple control frequencies IEC 1000-2-4 0.2% for classes 1 & 2, up to 2.5% for class 3 EN 50160 Under consideration All %’s are of nominal fundamental frequency Survey of Interharmonics in Indian Power System Network, B.E. Kushare, et al.
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Proposed Interharmonic Limits Current Standards* use 0.2% Other Proposed Limits Less than 1%, 3% or 5% depending on the voltage level. Adopt limits correlated with Pst Develop appropriate limits for particular equipment and systems. 27 All %’s are of nominal fundamental frequency *IEC 61000-2-2 Source: Survey of Interharmonics in Indian Power System. Network, B.E. Kushare, et al.
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Causes of Interharmonics Asynchronous switching (i.e., not synchronized with the power system frequency); and Rapid changes of the load current causing the generation of sideband components adjacent to the fundamental supply frequency and its harmonics; and A combination of the above can occur at the same time in many kinds of equipment. 28 Source: Power Quality Application Guide: European Copper Institute, AGH University of Science and Technology and Copper Development Association.
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Sources of Interharmonics Includes at least: PWM power electronic systems (Asynchronous Switching) Arc Furnaces (Rapid Current Changes) Cycloconverters (Asynchronous Switching) 29 Source: Power Quality Application Guide: European Copper Institute, AGH University of Science and Technology and Copper Development Association.
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Variable Speed Drives 31 DC-Link Ld Id AC/DC f1 f2 DC/AC Converter 2 Converter 1
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Variable Speed Drives (cont.) 32 If the reactor and/or capacitor at the DC Link is infinite there will not be any DC ripple at the DC Link. Source: Interharmonics: basic concepts & techniques for their detection & measurement, Chun Li, et al.
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Varying Loads 34 Source: Interharmonics: basic concepts & techniques for their detection & measurement, Chun Li, et al.
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Modulated Power 35 Source: Interharmonics: basic concepts & techniques for their detection & measurement, Chun Li, et al.
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RMS Deviation from Interharmonics 36 36 Source: Power Quality Application Guide: European Copper Institute, AGH University of Science and Technology and Copper Development Association. 0.2 0.15 % RMS Deviation Due to Interharmonics 0.1 0.05 0.0 0 50 100 150 200 Interharmonic Frequency
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Problems Caused by Interharmonics Lamp Flicker; Heating; and Interharmonics vary with the operating conditions of the interharmonic producing load. This makes interharmonics more difficult to mitigate than harmonics. 37
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Lamp Flicker 38 58 Hz 42 Hz Source : EPRI Source: Interharmonics: basic concepts & techniques for their detection & measurement, Chun Li, et al. Human eye is sensitive to frequencies between about 8 Hz and 12 Hz
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Minimum Interharmonic Amplitude Causing Perceptible Flicker 39 Flickermeter Source: Detection of Flicker Caused by Interharmonics Taekhyun Kim, Student Member, IEEE, Edward J. Powers, Fellow, IEEE, W. Mack Grady, Fellow, IEEE, and Ari Arapostathis, Fellow, IEEE
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Rolling Mill Case 40 Bus Source: Leonardo Energy by Michele De Witte 40
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Harmonic Impedance at Resonance 41 Source: Harmonic Impedance Study for Southwest Connecticut Phase II Alternatives by KEMA, Inc.
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Rolling Mill Case (cont.) 42 180 485 330 notch notch notch Source: Leonardo Energy by Michele De Witte
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Interharmonic Conclusions Interharmonics have always been around, they are just becoming more important and visible. Power electronic advances are resulting in increasing levels of interharmonic distortion. Traditional filter designs can result in resonances that make interharmonic problems worse. Light flicker is the most common impact. Measurement is difficult, but standards make them possible and the results comparable. 43
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IEC Groupings Number of cycles to sample chosen to provide 5 Hz frequency bins 10 Cycles for 50 Hz Systems 12 Cycles for 60 Hz Systems Grouping concept Harmonic factors calculated as the square root of the sum of the squares of the harmonic bin and two adjacent bins. Interharmonic factors calculated as the square root of the sum of the squares of the bins in between the harmonic bins (not including the bins directly adjacent to the harmonic bin). 44 Source: Power Quality Application Guide: European Copper Institute, AGH University of Science and Technology and Copper Development Association.
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IEC 61000-4-7 (Groupings) 45 Harmonic subgroup Harmonic group Harmonic subgroup The RMS value of the two harmonic components immediately adjacent to the fundamental . The RMS value of the fundamental and adjacent Harmonic components n n+1 n+2 The time-window is 12 cycles at 60 Hz and has 5 Hz resolution. Source: Power Quality Application Guide: European Copper Institute, AGH University of Science and Technology and Copper Development Association.
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IEC 61000-4-7 (Groupings) 46 The RMS value of all interharmonics components in the interval between two consecutive harmonics. The RMS value of all interharmonic components in the interval between two consecutive harmonic frequencies, excluding components adjacent to the harmonic frequencies Interharmonic subgroup Interharmonic group n n+1 n+2 Source: Power Quality Application Guide: European Copper Institute, AGH University of Science and Technology and Copper Development Association.
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