Visualizing Different Alternatives for ZSM

So that finally brings us to our ZSM viewer (next page of this tutorial).


When you look at the viewer page, you'll see three graphs, two in the top left and one on the right. The two on the top left are time waveforms: the top graph shows the voltage of the three phases as they change over a commutation cycle (with any PWM frequency filtered out), and the bottom graph shows the PWM waveforms corresponding to one commutation angle. The top graph also shows the zero-sequence component in light blue (= the average of the three phase voltage waveforms).


The graph on the right shows the average voltage output of the three phases in a cube:


Still don't believe it's a cube? On the ZSM viewer page, click on the cube with your mouse and drag it around to rotate the viewpoint:


There's a lot going on here, so let's go over what's being displayed:

  • The edges of the cube, with its axes labeled A, B, C
  • The long diagonal of the cube, with that axis labeled 0 (for zero-sequence component)
  • The perpendicular axes X and Y (equivalent to α and β)
  • A black point within the cube showing the instantaneous voltage output of the three-phase bridge
  • A gray hexagon, showing the projection of the cube on the α-β (X-Y) plane, shifted up along the cube's diagonal so that it contains the output voltage point
  • D and Q axes within the plane of the gray hexagon, with the output voltage point on the D-axis
  • A blue curve, showing the trajectory of this voltage point within the cube
  • A blue dashed circle, which is showing that trajectory projected onto the α-β plane through the midpoint of the cube diagonal (0.5, 0.5, 0.5) — this is the resulting trajectory of sine-triangle PWM.

You can also choose one of several types of zero-sequence modulation.


No Shift

This is plain sine-triangle modulation, and the phase voltages are all sinusoidal. Line-to-line amplitudes above $\frac{\sqrt{3}}{2} \approx 0.866$ cannot be achieved. Beyond this point, the trajectory will extend outside the cube.



Midpoint Clamp

This is conventional space-vector PWM (CSVPWM). There's a reason for calling it "midpoint clamp" which we'll discuss in a little bit.



Third Harmonic

The zero-sequence component is a 3rd-harmonic waveform equal to $-\frac{A}{6}sin3\theta$, where A is the amplitude of the output voltage, and θ is the commutation angle. This doesn't get used much, and we'll discuss why a little bit.



Top Clamp (also called "flat-top")

The zero-sequence component is chosen to move the highest output voltage to the positive rail of the DC link. This is sometimes used to reduce switching losses, since that phase with maximum output voltage is at 100% duty cycle and does not need to be switched for 120° in the commutation cycle.



Bottom Clamp (also called "flat-bottom")

The zero-sequence component is chosen to move the lowest output voltage to the negative rail of the DC link. This is also used to reduce switching losses. In low-voltage drives that utilize bootstrapped gate drives, it is more practical than top-clamp since 100% duty cycle on the high-side switches is not maintainable for many PWM cycles.



Top and Bottom Clamp

The zero-sequence component is chosen to use top-clamp or bottom-clamp, whichever requires the lowest shift in zero-sequence voltage. This distributes the switching losses evenly among the six switches. Like top-clamp, however, it is not practical in bootstrapped gate drives, because it requires 100% duty cycle on the high-side switches for extended periods of time.



Minimum Shift

In theory, it is possible to choose the smallest possible adjustment in zero-sequence voltage to keep the output voltages within realizable limits (within the cube). For line-to-line amplitudes below $\frac{\sqrt{3}}{2} \approx 0.866$, the zero-sequence voltage is fixed and the output waveforms are sinusoidal. There's not any real reason to use this method, however, and the choice of zero sequence voltage becomes ambiguous in the overmodulation region (when distortion is permitted in order to achieve larger output voltage amplitudes).



Nip & Tuck

The zero-sequence component is chosen to keep the component with the largest amplitude from the mean at a constant value, so that the waveforms are always flattened at a constant minimum and maximum duty cycle for 60° of the commutation cycle; this follows the waveforms in the King patent. It is very close to the 3rd-harmonic method, and like the third-harmonic method, there are practical reasons not to use it. I am including it here merely because it looks nice, and call it "nip & tuck" for the same reason. A side-image of the cube is not shown; it looks very similar to that of the 3rd-harmonic method. The trajectory consists of six 90° circular arcs from different planes parallel to the cube faces.

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