116
Functional Description
fundamental, absolute phase. This is because the subtraction is actually one of time; the purpose
being to re-align the reference point to correspond to zero degrees on the fundamental. The phase
angle corresponding to this time shift is proportional to the frequency.
11.14
Harmonics
Harmonics are measured using overlapping Hanning window FFT’s of 1024 samples and 100 ms
window length. Based on the instantaneous frequency, the location of the bins containing significant
energy for each harmonic are determined. This is a total of three bins, one approximately centered
on the harmonic and those two adjacent to it. Then, the energy in those three bins is totaled,
resulting in the energy for that harmonic. This can then easily be expressed as a percent of the rms
signal level, or in whatever form is required. While there is a closed-form correction, which can be
employed to find the harmonic magnitude in the presence of frequency errors, this approximation
was chosen because it is much faster and gives adequate performance.
There is an error in this approximation, due to the fact that there will be a small amount of
energy leakage into nearby bins, which will not be included in the three measured bins. This is
generally of little consequence when the frequency is close to being accurate (which is most of the
time), since the amount of energy outside of the three bins summed is so low. Finally, with the
Hanning window, there is no signal at all outside of these three bins if the frequency is exact.
In the real world, however, the frequency will be off somewhat, and it is reasonable to ask
how large the error can be. For small frequency errors, say 0.01 Hz, the 50th harmonic will be
0.5 Hz from the center of the nominal bin. This results in an error of about 0.005%, which is
insignificant. The worst-case error will occur when a harmonic is very nearly centered between two
bins. In this case, the algorithm described above will miss a bin containing a signal with amplitude
of about 17% of the actual harmonic amplitude. The energy contained in this bin is then (0.17)
2
or
about 2.9% of the total energy, resulting in a measured energy 0.971–times what it should be. The
measured harmonic amplitude will then be
√
0
.
971 or about -1.5% in error. This is well within the
specification limits (5%) of IEC 61000-4-7. This worst-case error would occur for the 50th harmonic
with a fundamental frequency error of 0.1 Hz. At lower harmonics, the frequency error must be
progressively greater; for example, to result in a -1.5% error in measuring the 9th harmonic would
require a fundamental frequency offset of 0.556 Hz.
The phase angle of the harmonics is determined by taking the arctangent of the real and
imaginary components of the bin closest to each harmonic. This information cannot be used,
however, in the averaging process described in IEC 61000-4-7, because this specification requires
the rms average of a series of measurements. This, by definition, requires magnitude data only.
Therefore, there are two different harmonic tables available from the unit. The averaged harmonic
data, in accordance with IEC 61000-4-7, is provided in a 300-element array (6 channels x 50
harmonics), averaged over the interval specified by the user (typically 10 or 15 minutes). The
harmonic magnitude and phase data are provided in a 600-element array, once per second. The
harmonic magnitude is the rms value over the preceding second, and the phase angle is the
instantaneous, absolute (based on a 100 ms window centered at top of second) phase angle, as
described in the preceding paragraph.
