Contents
7 The Display Obtained
8 The Classic 6000 FFT
8.1 FFT Length
8.2 Averaging FFTs
8.3 High Frequency FFTs
8.4 The Classic FFT Nomogram
8.5 Speed of Execution
9 Getting the Best Results
9.1 Filling the time record with appropriate data
9.2 Selecting the appropriate window
10 Summary
7 The Display Obtained
Having set up the FFT menu what happens? Depending upon several factors - e.g., the number of samples, the window function, log or linear scaling there will be a display similar to figure 7.1 below.
Typical Display
Figure 7.1: Illustration of the Typical Display Obtained
We will look at several different elements of the result. In the Gould FFT, the display obtained the left hand side of the display is not 0Hz, but is equal to the line spacing.
As discussed previously, the line spacing is the reciprocal of the time record length. The FFT is always a series of discrete lines - there is no envelope of the lines formed. The FFT is a power spectrum, and the unit used is the volt. This is not incorrect as we assume that because the load to the input signal is unknown the power cannot be calculated and the only presentation that can be made is in terms of volts - where this corresponds to the power for one amp RMS flowing.
The height of each line represents an RMS power. The total RMS power in the FFT display is the sum of all the line heights. Because the FFT display covers a broad range of frequencies, the power in the FFT display is close to the RMS power in the selected trace. This is the case when the rectangular window is used. However, when the Hanning window is selected the total power is close to twice that of the RMS power of the input trace. This is because of the scaling of this window.
8 The Classic 6000 FFT
8.1. FFT Length
FFTs, as implemented on Classic, are always displayed with the same store length as that of the main time domain traces. This means that a combination of interpolation and expansion is generally required during computation because the FFT algorithm ordinarily operates upon and produces data files of binary length - 256, 512, 1024, etc. However, Classic trace memory lengths are related in a 1, 2, 5 sequence - 1004, 2008, 5020, etc. So, for example, a 256 point FFT would ordinarily produce a 128 point output. Classic interpolates such that 251 output points are produced instead. But, because the shortest Classic acquisition is 502 points, each of the 251 outputs is repeated. If the store length chosen was longer than 502 points, say 10 k, then each of the 251 original points is repeated 40 times. This makes for a very coarse display.
On longer FFTs, the situation is different. For an 8 k FFT and 10 k store length, the usual 4 k output points are interpolated to 5020 and then each point is repeated to reach the full 10,040 store length. In this case, there must be an amount of display compression applied, depending upon the display zoom factor. For example, if the display is zoomed by x2, this is equivalent to the display of 5020 points within the actual FFT output data file. So, compression must be performed which involves generating max-min 'frames' of 10 points each (502 max-min pairs to be developed from the 5020 data points. Classic displays 501 pairs of max-min points, so 1002 compressed points developed in this process are displayed as 501 vertical line segments.
Figure 8.1 shows how the max-min points developed during compression of the interpolated and expanded FFT output data file 'map' to the display trace. Note in particular that if the cursor continues to be moved in the direction of increasing frequency, the readout will continue to increase but the value increments will not be uniform. This is because the readings are the actual max-min values, which themselves can have occured anywhere within each max-min frame. However, because the display always presents max's and min's in the order in which they occur within a frame, there will be no reversals - no frequency decreases as the cursor is moved toward the higher frequency end of the FFT trace.
Display Max - Min
Figure 8.1 Display Max-Min
8.2 Averaging
When a time domain signal is averaged, the effect is to reinforce those parts of the acquired waveform which remain substantially the same from acquisition to acquisition, while minimizing those parts which change Ð such as random noise. The reason that averaging works this way is that each acquisition is accurately related to a reliable trigger event. The results can be quite different if, for example, auto trigger is applied, or the trigger source is a different channel with an unrelated signal.
In the frequency domain, FFT components which remain substantially the same from acquisition to acquisition will be reinforced, while those frequency components which change (noise) will be minimized. Time domain averaging actually produces higher resolution trace data which can be viewed by applying vertical zoom to the trace of interest. Frequency domain averaging has a similar benefit, but because a log (dB) scale is usually applied to the vertical axis, improved resolution appears as a lower 'noise floor', or expressed another way, as a greater 'dynamic range'. For short FFTs (256 to 1 k input points) without averaging, a 50 dB dynamic range is typical. For an 8 k FFT with 128 averages, over 70 dB can be achieved. This means that very low level signal components can be identified. Figure 8.2.1 shows a very noisy trace in which the RMS noise voltage is actually 1000 x the rms signal voltage. However, with averaging of the FFT, the resulting trace clearly shows the 252 kHz signal about 8 dB above the noise floor. Both the input trace and its FFT have been expanded to show more detail.
Gain
Fig 8.2.1
As the averaging operates, the 'gain' of the displayed FFT increases to show small components better because the noise floor is being reduced. Assuming a log vertical scale, this is accomplished by changing the top grat line reference - for the same (dB) dynamic range, a smaller value for this reference means that small components appear larger.
The important distinction made above about time domain triggering, as compared with frequency domain averaging, means that many new applications can now be addressed by Classic. These include any signals which are continuous in nature - not transients - but which have no obvious point in time which can be used as a trigger. One type is signals obscured by noise, such as the 252 kHz sinewave in figure 8.2.1. Noise from rotating bearings and generally from machinery are two practical examples.
The combination of greater amplitude sensitivity afforded by averaging, and better frequency resolution resulting from longer FFTs, is exactly what is needed for applications such as non-invasive induction motor fault analysis. Especially with very large motors, itÕs very attractive to be able to analyse suspected faults without removing the motor from its associated driven machinery, dismantling it, and then isolating the suspect winding Ð all very costly and perhaps destructive if the motor is damaged by this work. See Figures 8.2.2 and 8.2.3.
Healthy Motor
Figure 8.2.2 FFT of Current from a Healthy Motor
Damaged Motor
Figure 8.2.3 FFT of Current from a Damaged Motor
Note: Both Figure 8.2.2 and Figure 8.2.3 were reproduced from an article by David Rankin in Power Engineering Journal (IEE), April 1995, p. 80, with permission of the author.
The technique of examining the FFT of the current drawn by a motor, while the motor is run normally, is very much cheaper, and clearly much easier to do. This technique was developed by D. R. Rankin and others during the mid-1980's. He is now technical director of Electro Electrical Projects Ltd., in Scotland. Lots of work has been done with the School of Electronic and Electrical Engineering at Robert Gordon University, Aberdeen. So, the technique of using the FFT is well supported by theory, and in fact, Electro Electrical Ltd., has developed an 'expert system' using a PC database, so that FFTs from a suspect motor can be compared with those from similar motors having known faults. However, the basic technique is used widely, and usually provides sufficient clues to motor problems that the FFT alone is really the key part of the process - the 'expert system' being offered by Electro Electric Ltd is a means of being almost 100% sure of the cause of a motor fault before it has been dismantled.
One key requirement of the FFT technique very fine frequency resolution, so that the 'sidebands' caused by faulty rotor bars, for example, can be clearly separated from the nearby fundamental (50 Hz) signal. This is why the longer FFT capability in Classic is so important, because longer FFTs produce fine frequency resolution FFTs. The idea is to include many AC supply cycles within the FFT, so that frequency resolution is increased. At least one motor researcher we've contacted uses 32 k FFTs and sometimes 64 k.
8.3 FFTs on ETS (long stores)
Figures 8.3.1 and 8.3.2 show a comparison between short and long FFTs performed on ETS data. Classic originally allowed only 502 point acquisitions in ETS mode, so the longest FFT which could be used at high frequencies was only 256 points. Now that ETS allows up to 50 k long acquisitions, a 32 k FFT could be performed (with 200 k memory option 136 fitted) which would have 200 x better frequency resolution.
256 point 10 k
Figure 8.3.1 A 256 point FFT with 10 k memory
8192 point 10 k
Figure 8.3.2 An 8192 point FFT with 10 k memory
8.4 Nomogram
Many factors are involved in the effective use of the FFT on Classic, and several examples have been given above to highlight these. The Nomogram presented in Figure 8.4.1 is a comprehensive way of relating all these factors, providing a quick and easy to use solution to the problem.
Instructions for use of the Classic FFT Nomogram
The Classic FFT Nomogram provides a very quick solution to the problem of determining the maximum frequency and resolution of an FFT, for a given timebase range, memory length and FFT length.
To use:
Find the column labeled with the memory length which you have selected.
Find the row in that column having the timebase value that you are using.
Follow that row to the right, reading the value of the maximum FFT frequency. The example shows a 50Ês/div timebase with 50Êk word memories having a maximum frequency of 50ÊMHz. This value depends only upon the sample rate.
Find the FFT length (the number of points to be included in the FFT data) which you have selected, and draw a straight line through the maximum frequency found in step 3, passing through the selected FFT length, intersecting the FFT resolution line.
Read the FFT resolution figure from the final column - 20 kHz in the example. This is the frequency spacing of the 2510 (interpolated) 'lines' produced by the 4 k FFT. Each point is repeated 20 times to make up the total 50 k store length in this example
Expanded Cell
Figure 8.4.1 Expanded View of a Typical Chart Cell
Note: The vertical scales on the actual Nomogram are logarithmic, and in particular both the time/div and the FFT Frequency Resolution values increase moving towards the bottom of the chart, while both the Samples/s and FFT Max. Freq values increase moving towards the top of the chart. The note at the bottom of the chart about FFT resolution values being only 20% accurate results because a given point on the resolution axis can be reached via a large number of different routes, involving different interpolation factors depending upon store length and FFT length. However, in most practical situations, it should be quite acceptable to know, for example, that an FFT resolution could be 20 Hz or 24 Hz, but that it certainly is not 10 Hz or 50 Hz.
FFT Nomogram
Figure 8.4.2 FFT Nomogram
8.5 Speed of execution.
Because the Classic hardware includes a closely coupled DSP, the FFT and other analysis functions can be executed very quickly. Table 8.5.1 shows actual measured execution rates vs. store length and FFT length. It's obvious that even an 8 k FFT runs at a fast enough rate to remain 'live' for purposes of interactive adjustments, for example. Note: All measurements were made using the fastest transient timebase available for the chosen memory length, to minimise the effects of acquisition time upon the FFT rate.
FFT execution rates
Figure 8.5.1 Classic FFT Execution Rates
9 Getting the best results
To obtain the best results involves two parts:
a. Filling the time record with appropriate data.
b. Selecting the appropriate window.
9.1 Filling the time record with appropriate data.
Because of the relationship between the time record length and line spacing, it is important to fill the trace with many cycles of a periodic waveform. This will give the best frequency resolution. To a scope user this seems wrong because he normally expands a trace to obtain the best time resolution. Let us look at two examples. In figure 9.1.1 there is a single cycle of a 1 kHz sinewave captured at 0.1 ms/div.
1 cycle
Figure 9.1.1: How the length of the Time Record affects the FFT - Capture of 1 cycle
This is a total time record length of 1 ms and a line spacing of 1/1 ms = 1 kHz. So in the lower drawing the signal corresponds to a line at 1 kHz. And because the frequency resolution is quite coarse the line is not very easy to see.
In figure 8.1.2, there is the same signal but now with 20 cycles at 2 ms/div captured and a total time record of 20 ms. Here the line spacing is 50 Hz and the frequency resolution is finer and the line at 1 kHz is more visible.
20 cycles
Figure 9.1.2: How the length of the Time Record affects the FFT - Capture of 20 cycles
It is also useful to try to avoid discontinuities between the left and right hand sides of the display. Obviously, windowing can be used to reduce this problem.
9.2 Selecting the appropriate window.
As a rule, use the Hanning window for periodic signals and the rectangular window for transient signals. However, as with all rules there can be exceptions. Leakage can be an unpredictable effect and sometimes the rectangular window can give better results. However, the best one to try first is Hanning.
10 Summary
This is a collection of points extracted from this material. Time and Frequency Domain
The Time Domain has time on the X axis.
The Frequency Domain has frequency on the X axis.
The FT, DFT AND FFT
A complex waveform can be thought of as the sum of many harmonically related sinusoids of different amplitudes.
Periodic waveform gives discrete line spectra.
Line spacing increases as the period increases.
Non-periodic waveforms give continuous spectra.
The DFT approximates to the FT.
The FFT is a special case of the DFT. The FFT produces line spectra only. Lines are spaced at equal frequency. The line spacing is given by 1/(duration of the time record) When the lines in the Fourier Transform donÕt match those available in the FFT, leakage occurs. Windows control the degree of leakage.
For the Gould FFT:
The line heights represent RMS power in units of volts.
Use the Hanning window for periodic waveforms - not a strict rule.
Use the rectangular window for non-periodic waveforms.
Try to get many cycles on the display - this will improve frequency resolution .
Try to make the two ends of the Time Record match up.
Averaging and filtering can be used to reduce unwanted high frequency components prior to the FFT function. Remember that this can be automated using the programmable sequence function.
Glitch detect will cause distortion, most noticeably at the higher frequency end of the FFT.