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The Harmonic Spectra of InterharmonicsGary MalhoitSRPAugust 26, 2010Two house keeping items before we start:

If you would like a copy of this presentation, you can get a copy on Slide share. I uploaded this presentation a couple of days ago.

Feel free to interrupt this discussion with your comments and questions at any time. OverviewBrief overview on harmonics and periodicityFouriers Method and the DFTInterharmonic Defined Picket-Fence Effect & Spectral LeakageGenuine & Non-Genuine InterharmonicsInterharmonic standards and allowed limitsSources of InterharmonicsInterharmonic ProblemsMeasuring InterharmonicsIEC Grouping Standard and Understanding Spectra Measurements

2Heres an overview of this discussion.

A good grasp of harmonics is a prerequisite to understanding interharmonic and provides a good background for todays 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 slide2SinusoidsSinusoids 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

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.3Fourier1822, a French mathematician named Joseph Fourier, claimed that continuous periodic signals can be represented by the sum of properly chosen sinusoids. 4

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 Fouriers method for discrete signals (predominately used today) is exact.

4Limitations to Fourier Methods55Source: Wikipedia

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.5Fourier Tool Kit6

The Scientist and Engineer's Guide to Digital Signal ProcessingBy Steven W. Smith, Ph.D.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 todays discussion I will refer to the DFT in a broad context to include the other methods.

6Assumptions in Applying DFTfor PQ MeasurementsThe 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.

7When these conditions are satisfied the measurements are accurate.7Non-Stationary Signal82Pi

DFT WindowSource: A Notebook Compiled While Reading Understanding Digital Signal Processing by Lyons 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. 8Non-periodic Signal9

DFT WindowHalf a sinusoidIn time domainSpectral Leakagein the frequencydomainSource: A Notebook Compiled While Reading Understanding Digital Signal Processing by Lyons The vertical axis is normalized voltage or current.

The horizontal axis is time.9What is Harmonic Spectra?Harmonic spectra includes sub-harmonics, harmonics and interharmonics.

10HarmonicInterharmonicSubharmonicf = n f1 where n is an integer > 0.f = nf1 where n is an integer > 0.0 < f < f1f1= fundamental frequencySource: Power Quality Application Guide: European Copper Institute, AGH University of Science and Technology and Copper Development Association.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.

10Harmonic SpectraCharacteristic HarmonicsThose 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.

11Source: IEEE 519

Each piece of equipment has its own harmonic signature.11Harmonic Spectra (cont.)Non-Characteristic HarmonicsHarmonics 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.

12Source: IEEE 519Cycloconverter 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 519 infringes 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.

12InterharmonicsInterharmonics- 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

13This 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. 13Interharmonics RedefinedInterharmonics- Any frequency which is not an integer multiple of the fundamental frequency

Source: IEC-61000-2-2

14One-Cycle Window151560 Hz

The 60 Hz component competes 1 cycle within the DFT window.DFT Window16.67 msThe vertical axis is normalized voltage and the horizontal axis is time.

Discrete calculation are made across the DFT window

As 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.

15Frequency Resolution16

CBATimeThe 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)

16Picket Fence Effect17

Source: Azima DLIThe 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 l