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Space vectors
John Dunn - April 03, 2017
Consider the task of driving a three-phase motor along the lines of the following sketch:
Figure 1 Three phase motor drive using pulse width modulation
Six power MOSFETs are driven with pulse width modulation (PWM) to provide effectively sinusoidal
motor winding excitation. A closer look at the sinusoidal waves versus the PWM is as follows:
Figure 2 Expanded view of PWM waveform
This works very nicely, but there is a further refinement that can be employed to reduce the peak
drive amplitudes demanded of the MOSFETs for delivery of sinusoidal drive to the windings while
using some given rail voltage.
Looking at the three drives from a vector standpoint, we can examine the issue of dynamic range
versus rail voltage. If there is a vector in collision with the rail voltage limit as in middle level of the
following sketch, that is bad news, but we can mitigate that if we dynamically move the vectors up
and down as shown in the bottom level of the sketch.
Figure 3 Rail voltage limit issue and space vector compensation:
1.
2.
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4.
Rotating vectors of motor drive signals (typ.)
With adequate Vcc, motor drive vectors are contained within Vcc rail limits.
If the Vcc is too low, the vectors can collide with the rail limits.
Using space vectors, motor drive can be revised to preserve containment.
Instead of using pure sinusoidal drives, we write (with normalization, of course) our drive waveform
equations as follows:
V=3*SIN(THETA*PI/180)
IF THETA>30 THEN V=SQR(3)*SIN((THETA+30)*PI/180)
IF THETA>90 THEN V=SQR(3)*SIN((THETA-30)*PI/180)
IF THETA>150 THEN V=3*SIN(THETA*PI/180)
IF THETA>210 THEN V=SQR(3)*SIN((THETA+30)*PI/180)
IF THETA>270 THEN V=SQR(3)*SIN((THETA-30)*PI/180)
IF THETA>330 THEN V=3*SIN(THETA*PI/180)
We do this for Theta from zero to 360 degrees and repeat it for each cycle for all three drives while
including 120 degree phase shifts between drives to account for the drives being three phase.
The following illustrates the benefit we achieve from the above equations:
Figure 4 Chip outputs (top) versus winding drives (bottom), without space vectors (left) and with
space vectors (right)
Here, for winding-to-winding voltages of 10V peak, pure drive consisting of pure sinusoids would
have to reach 5.7735V peak but with the revised drive, that 10V peak of output to the windings is
achieved with only 5V peak drive coming from the MOSFETs. Rail voltage collision becomes less of
an issue.
It may be qualitatively instructive to look at an expanded view of the chip output waveforms:
Figure 5 Diminished peak of modified waveform
This simple example is by no means the end of the story. Much has been written on the use of space
vectors, which is what this is called, and some of that can be found in this Google search result.
Also see:
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FPGA high efficiency, low noise pulse frequency space vector modulation
Field-oriented control by the numbers