Analysis Methods Used for Flicker Calibration Signals
Time Domain Analysis of the Flicker Calibration Signal
In the case of slow modulations of a few changes per minute, time domain measurements are used. This involves calculating regular RMS values from the samples obtained by the ADC. For example, a RMS value can be found once per mains frequency cycle, hence building-up sets of measurements at the modulation levels. Further processing on these measurements can be carried-out and the results used to find the modulation depth.
For faster modulation rates, a time domain approach is not practical as there are fewer full mains cycles at each level available for analysis. In this case a frequency domain approach is used which makes use of AM theory to find the modulation depth.
Frequency Domain Analysis of the Flicker Calibration Signal
AM theory predicts sideband frequency components at the sum and difference of the modulating and carrier frequencies. Consider a sinewave amplitude modulation signal nm at a repeat rate frequency wm . The mains-level signal is given by nm at frequency wm.
nm=Vmsin(wm) nc=Vcsin(wc)
The amplitude modulation of the mains-level signal is given by nam.
nam= nc.(1+nm) = Vcsin(wc) + VmVcsin(wm)sin(wc)
(Note that when the modulation Vm is zero the equation gives the un-modulated signal)
using the trigonometric function for the product of two sine functions,
nam= Vcsin(wc) + ½ VmVc[cos(wc - wm) + cos(wc + wm)]
This equation shows that the frequency spectrum of the AM wave consists of a component at wc and two so-called sideband components either side of it at the sum and difference frequencies. This is shown in Figure 6.
Figure 6: Sidebands produced by sinewave amplitude modulation.
Using the samples obtained on the ADC, a Fourier Transform can be used to find the amplitudes at wc and the two sideband components. This gives enough information to find Vm and Vc and hence the depth of modulation dV.
The above analysis applies only to a sinewave modulation signal but it can be readily adapted to squarewave modulation. A squarewave can be represented by a sum of frequency components at all the odd harmonic frequencies, for example a 15 Hz squarewave is made up of the addition of components at 15, 45, 75, 105 Hz… This summation is predicted using Fourier series which gives the amplitude of each component. In theory these components run to infinity, with their amplitudes getting progressively smaller at each higher order.
For squarewave AM, each component in the series forms a sum and difference sideband with the mains carrier frequency. For a 15 Hz squarewave used to amplitude modulate 50 Hz mains, the sum sidebands will be at (50+15), (50+45), (50+75) etc. The difference sidebands are at (50-15), (50-45), (50-75) etc. Note that the difference sideband series has given rise to apparently negative frequencies. The anomaly is handled by folding the frequencies around the 0 Hz origin, for example -25 Hz becomes +25 Hz. Figure 7 shows this effect, it can be imagined folding the paper around the y-axis at 0 Hz so that lower sidebands LS4 and LS5 appear on the positive side.
Figure 7: Square Wave Amplitude Modulation Spectrum.
(x-axis is frequency; y-axis is amplitude; LS is Lower Sideband; US is Upper Sideband; Fmains is the carrier frequency)
Squarewave AM gives rise to many sideband components, but it is only necessary to pick one of the sidebands, together with the carrier frequency component. Using these two values, the depth of modulation can be found in a similar way to the sinewave case. It is a useful check to perform the calculation on some of the other available sidebands and compare the results.
The concept of negative frequencies does not cause a problem to the measurement provided the wrapped frequency components do not have the same frequency as any other components. The reader may wish to consider the example of the 25 Hz squarewave modulation of a 50 Hz mains signal.