# EETE MAY 2015

Fig. 4: Stationary frame back EMF estimators. Fig. 5: Simulation output showing observer performance. any closed-loop system utilizing negative feedback, we can drive the error signal to nearly zero if we have high enough loop gain. In this case, an error of zero means the phase current estimate is equal to the measured value of phase current. But this can only be true if the signal at the back-EMF input actually equals the missing back-EMF voltage (assuming Ls and Rs estimates are accurate). In other words, the PI controller’s high gain servos the error signal to zero by forcing the controller’s output to become the back-EMF signal. One can use a Spice simulation of a single-phase back- EMF observer running at 10 kHz. The back-EMF signal inside of the motor block is unknown to the observer. But once the simulation starts, the observer output quickly converges on the back-EMF signal as shown in figure 5. Even when you change the back-EMF waveform to different wave shapes, the observer morphs its output to match it! From figure 4, we can create the back-EMF waveforms for the alpha and beta coils, but just having the back-EMF signals doesn’t mean we are done. The angle information is still embedded in these sinusoidal waveforms, and we have to figure out how to extract it. One way is to use the arc tan function: Besides requiring floating point capability to calculate this value, you will also need to trap for the singularity when EMFalpha(t) = 0. But consider another technique (borrowed from the world of resolvers) shown in figure 6. Not only does it solve the problems associated with the arc tan function, it offers another very nice perk: a filtered speed signal! Now that we have solved for the angle, I need to tell you that we solved for the wrong angle. What we wanted was the angle of the rotor flux vector. What we get is the angle of the back-EMF vector. Before you close your browser in disgust, you should also know that the back-EMF vector always lies on the quadrature axis, so we are very close! All that remains is to determine the polarity of the back-EMF vector on the q-axis in order to know exactly where the rotor flux vector is. This is where the frequency signal from figure 7 can help. If the frequency is positive, we subtract 90 degrees from the back-EMF angle. If it is negative, we add 90 degrees instead. Now that we have the angle of the rotor flux, we can proceed with the four steps of FOC. Wow! All these calculations, just to find the angle of the rotor flux vector! But considering the low cost of embedded MIPS compared to the high cost of a motor shaft sensor, it is still way worth the effort in most cases. Fig. 6: Angle demodulator using the relationship: sin(θ) cos(θest)-cos(θ)sin(θest) = sin(θ-θest). CM7V-T1A Thin ceramic package kHz Crystal Consumer & AEC-Q200 automotive compliant, Size: 3.2 x 1.5 x 0.65 mm, Temperature range: -40°C to + 85°C and -55°C to + 125°C A u t h o r i s e d distributor MICRO CRYSTAL SWITZERLAND Our authorised distributor WDI AG is pleased to assist you.  +49 4103 1800-0  microcrystal@wdi.ag  www.wdi.ag/microcrystal www.electronics-eetimes.com Electronic Engineering Times Europe May 2015 39

EETE MAY 2015