18
14 16
12 10 8 6 4 2
00 2 4 6 8 10 12 14 16
force sine
step size (step register value + 1)
max. diff. between consecutive data
Figure 59: Maximal difference between consecutive waveform data points plotted for theforce andsinewaveforms against the waveform counter increment value (step sizes).
Fortunately, this is not a dead-end situation. If one was able to choose variablesand dvalues in (135) in a way to arrive at a closest frequency match to the desired frequency within given bounds for both independent parameters, the frequency errors would decrease without violating the tolerance safety margin.
6.3 continued fractions approach in frequency
The number 1 is a divider of bothaandb. However,aandbare not necessarily integer numbers. Also, it is not known if they have another common divider greater than 1. Still, we are not looking for an exact solution to (139) but are rather interested in keeping the frequency error in (135) as low as possible. Thus, we could either round the coefficients aandbto the nearest integer values, check if these values are coprime, and then look for a nonnegative integer solution to s and dor develop an algorithm which can cope with real coefficients as well. The latter approach is pursued in appendixB, in which it is shown that the problem of finding two integer numbers ξ and η satisfying
aξ+bη=1
is closely related to finding the continued fraction expansion ofa/b.
If a/bis real (irrational), this expansion is infinite and approximates a/bbetter and better with every further expansion term.
At this point, two questions still need to be answered. First, how good is the solution computed by the algorithm from appendixBin terms of frequency error, especially if upper bounds onsanddare given due to a fixed width of the registers in the waveform generator. Second, how to cope with the fact thats anddin (135) have to be nonnegative integers, ifξ andηcan be negative in general.
In order to answer the latter question, note that for frequencies f below 12 kHz,ain (139) is negative andbis positive. From this, one can conclude that the sign of sandd has to be the same. Since we are interested in approximative solutions, we may take the absolute value ofs anddas our solution. This is motivated by the fact thats andd grow with every iteration. For positive sanddwe have
(−C)|s|+ f|d|=C− f ⇒ f(s,d) =C|s|+1
|d|+1 and for negative sandd(139) becomes
C|s|+ (−f)|d|=C− f ⇒ f(s,d) =C|s| −1
|d| −1.
Already after a few iterations the difference between ||ds||++11 and ||ds|−|−11 will become relatively small.
As far as the quality of the solution is concerned, there are fixed bounds on the maximal values ofs andd(available register widths) and the iterative approach has to be terminated before these bounds are reached. Fig.60illustrates frequency errors for a varying step regis-ter width between5and12bits and two different numbers of maximal iterations. Recall, that the waveform generator is implemented with 6 and 16bits for the s and d registers, respectively. For this reason, only the bounds on s were varied ass saturates much faster than d.
In Fig.60(a) and (b), box-and-whisker plots are used to illustrate the distribution of frequency errors in dependency of step register width.
5 6 7 8 9 10 11 12 0
10 20 30 40 50 60 70 80 90 100
step register width [bits]
N = 5
freq. error [%]
5 6 7 8 9 10 11 12
0 10 20 30 40 50 60 70 80 90 100
step register width [bits]
N = 10
freq. error [%]
(a)
(b)
(c) 0 500 1000 1500 2000 2500 3000
100 101 102
frequency [Hz]
log of freq. error [%]
Figure 60: Frequency error distribution for integer frequencies between 1 and 3000 Hz when the iterative B´ezout’s identity solving algo-rithm is used (see sect.B.3). The distributions are illustrated with help of box-and-whisker plots in (a) and (b) for an increasing step register width and for different numbers of iterations (N=5 and N=10). Median values are indicated with horizontal lines;
thick vertical bars correspond to the interquartile range (IQR);
the whiskers span the range±1.5IQR and the outliers are shown with help of dots having a random horizontal distribution for the sake of clearer illustration. In (c) the frequency errors are shown directly for the step register width of6and N=10.
More specifically, for each step register width value wand the integer frequencies f ∈ [0, . . . , 3000]the iterative algorithm from sect.B.3was run and terminated before the computed s exceeded 2w−1 or the computed d exceeded 216−1 or the maximal number of iterations Nwas reached. The so computed frequencies were subtracted from the corresponding desired values and the differences expressed as absolute percentage errors. Clearly,5iterations are not sufficient since the median of frequency error is close to 10% and do not improve considerably with increasingw. For10iterations and wof6, it drops
below3.2% and forw=8already below1%. However, even for large register widths, there are still outliers close to100%. Even for20bits long registers, the calculations for several frequencies still produce large errors. The situation improves only when no bounds ons andd are set but the obtained solutions are orders of magnitude too large for a practical implementation. Note that if s0 andd0 are a solution to
as+bd=1
then s0+kb and d0−ka with k ∈ Zare solutions as well [184]. Ac-cordingly, by defining
w1|s0+kb|+w2|d0−ka|=s(k) +h(k) = f(k) (140) with some weighting factors w1 and w2 we could try to minimize f(k)and find an optimal but possibly smaller/minimized solution.
However, the involved computations quickly overpower the capacities of the integrated circuits employed in the motor electronics. Another problem concerns the fact that even with a moderate register width, as with the 6 bits used for the storage of s, we should be reluctant to accept high s values even within the given bounds, because they degrade waveform resolution and because of the tolerance issues mentioned before.
In practical terms, whensand dare constrained according to the register widths in the actual implementation of the waveform gener-ator, the frequency errors take the form of Fig.60(c). This solution is not satisfactory since even for a low-median error, there are too many high-error outliers to treat them as special cases with dedicated solutions. The most practical way out seems to be the pre-computation of optimal solutions and their storage in a look-up table for a later access. This was done for the actual register widths in the waveform generator but the maximal value ofswas additionally constrained to 8 because of the tolerance window considerations at the end of the previous section. The results are illustrated in Fig.61. Even with the narrow interval of the allowed s values, the computed frequencies differ from the desired ones by at most1.2%. The minimal amount of storage required for the look-up table equals 3000(4+16)≈7.32 kB for integer frequencies between1and3000Hz. For frequencies below 100Hz a finer sampling is advisable but for these frequencies a con-stantsof0and the computation ofdaccording to (136) is sufficient in terms of the error magnitude below1%.
0 500 1000 1500 2000 2500 3000 0
0.2 0.4 0.6 0.8 1 1.2 1.4
frequency [Hz]
freq. error [%]
0 500 1000 1500 2000 2500 3000
frequency [Hz]
optimal step value
0 2 4 6 8
(a)
(b)
Figure 61: Optimal step register values (a) resulting in minimal frequency errors (b) precomputed and stored in a look-up table. The step register values have an upper bound of8. Note the large spread of these values within the allowed range. Also note that the frequency error remains below 0.2% for most frequencies and never exceeds 1.2%.