This document is part of the hoverboard firmware documentation.
The modulation method used in the firmware is described in this document.
In this chapter the original BLDC trapezoidal PWM scheme is described. The operation is analyzed from the NiklasFauth's source code.
The modulation is a trapezoidal modulation, in which one phase is not switching and the two others have positive (> 50 %) or negative (< 50 %) PWM. First, the hall sensors are used to determine the motor position and select which channels to use, according to following table.
| Sector | Hall A | Hall B | Hall C | Output A | Output B | Output C |
|---|---|---|---|---|---|---|
| 0 | 1 | 1 | 0 | 0 | + | - |
| 1 | 0 | 1 | 0 | - | + | 0 |
| 2 | 0 | 1 | 1 | - | 0 | + |
| 3 | 0 | 0 | 1 | 0 | - | + |
| 4 | 1 | 0 | 1 | + | - | 0 |
| 5 | 1 | 0 | 0 | + | 0 | - |
The PWM duty cycle is set directly by the given (and low-pass filtered) speed reference.
This basically means that the motor applies a torque which is dependent on the requested speed reference, and as a result, the speed settles to a point in which losses (air drag, friction) equal the torque generated.
SVM (Space Vector Modulation) is the modulation scheme used in this firmware.
The space vector hexagon is seen in image below. The bold numbers inside the sectors correspond to the sector numbers used in the earlier BLDC modulation scheme. The oblique numbers at the corners is the output switch states, given in ABC order. It can be seen that for example at corners of sectors 0 and 1 the B phase is high (1), which is similar to the BLDC modulation where the B output was positive.
The given voltage vector Uref can be realized using the neighboring corners and one or both zero
vectors. First, it is necessary to select the correct sector to know which vectors to use. The sector
and vectors are selected according to following table.
| Sector | Angle | Zero 1 | Active 1 | Active 2 | Zero 2 |
|---|---|---|---|---|---|
| 0 | 60...120 | 000 | 010 | 110 | 111 |
| 1 | 120...180 | 000 | 010 | 011 | 111 |
| 2 | 180...240 | 000 | 001 | 011 | 111 |
| 3 | 240...300 | 000 | 001 | 101 | 111 |
| 4 | 300..360/0 | 000 | 100 | 101 | 111 |
| 5 | 0...60 | 000 | 100 | 110 | 111 |
The modulation pattern is Zero 1 -> Active 1 -> Active 2 -> Zero 2 -> Active 2 -> Active 1 -> Zero 1.
Active vectors are selected in such order that only one transistor needs to be switched to transit
from one vector to another. The angle is the desired voltage vector angle, while amplitude only
affects the vector times.
Knowing which vectors to use, vector times are calculated using
Ta1 = Ts * 2/sqrt(3) * M * sin(pi/3 - angle)
Ta2 = Ts * 2/sqrt(3) * M * sin(angle)
T0 = Ts - Ta1 - Ta2
Where Ta1 and Ta2 are active vector times and T0 is the sum of the zero vector times. M is
the modulation index, i.e. the voltage amplitude and angle is the desired voltage angle inside the
sector, i.e. 0...60 degrees.
Ta1 always references to the vector that is clockwise next to the reference, and Ta2 to the one on
the counter-cockwise side. According to the modulation table above, sometimes we wish to use first the
clockwise one and sometimes the other. Thus it is necessary to swap Ta1 and Ta2 depending on the
sector we are in, to achieve correct switching order and correct vector times.
The sectors during which we wish to use the counter-clockwise vector first and need to swap the times are 0, 2, and 4.
Since it is quite time consuming to calculate sinusoidal function, the value of Ts * 2/sqrt(3) * sin(angle) is
pre-calculated into an array for 0...60 degrees angle. The pi/3 - angle is achieved by reading the
array backwards. Since Ts is pre-calculated into the array, it is necessary to re-compute the array
if switching frequency is changed.
Functions above require modulation index to be equal or smaller than sqrt(3)/2=0.866, which is the
maximum radius of a circle that fits inside the SVM hexagon. In practice, the equations are normalized
by dividing with that value, so that modulation index of one gives the maximum circle.
The modulator requires, as an input, the desired voltage vector length and angle, and as a result it gives out the switching instants of the three switches.
For current measurement purposes we wish to guarantee certain vector lengths, during which the current is sampled. As a result, we need to modify the calculated ideal vector times in such a way that we have proper vector lengths and also proper zero vector time, i.e. to keep the resulting vector as close to the reference as possible.
One method to do this limitation is to first calculate the ideal vectors, and then if we assume minimum
pulse lengths Tm0 for zero vector and Tma for active vector minimum length, we calculate
Tma2 = Ts - Tma - Tm0
T0err = max(T0 - Tm0, 0)
Ta1 = min(max(Ta1 - T0err/2), Tma2)
Ta2 = min(max(Ta2 - T0err/2), Tma2)
T0 = Ts - Ta1 - Ta2
This has the downside that the voltage is distorted in sector changes as well as with small or high modulation indices. Following figure shows how the actual voltage looks like compared to the ideal reference, with 3 different modulation indices: 0.05, 0.6 and 0.85 (note that in this case 0.866 is maximum linear region).
At low modulation index the active pulse length limitation, forcing the usage of two short active pulses, distorts the waveform. The same also affects sector changes where only one active vector would be needed.
At high modulation index the forced usage of zero pulse distorts the voltage at the center of sectors where there would be no need for zero vector.
The limitations mentioned above are caused mainly by requiring the vectors for current measurement reasons. If phase current measurement is not needed (or its accuracy can be decreased), the pulse limitations may not be needed and more accurate voltage generated.
One thing that I noticed when running the space vector modulator without any control loop was that the voltage quality was very poor and as a result the motor would not run nicely, even without any pulse width limitations applied. I figured this must be due to the commutation delays distorting the short pulses generated by SVM.
First I measured the commutation delays without any load connected. In the following figures the gate voltages of lower and upper transistors is drawn together with the drain-source voltage of the lower transistor. First transition from lower transistor conducting to upper transistor conducting is shown.
The next figure shows transition from upper transistor conducting to lower transistor conducting.
During the measurements, the dead time of the PWM was set to 500 ns.
As can be seen, the transistor turn off is very fast. Turn on, on the other hand, is very slow and gate voltage rises only gradually. In this case because there was no load, the voltage changes state only when the on-turning transistor's gate voltage reaches the miller plateau at about 4 volts.
For some reason the events are not symmetric at all. It seems like there is a significantly more delay when going from high to low than other way around. I have not yet studied why this is.
An automatic dead time compensation method is started in branch deadtime.
In that branch the phase voltage measurement pins, PA0, PC3, PC4, and PC5, are configured as digital inputs and
are set to generate interrupts (EXTI). The interrupt stores the current counter value of the PWM timer, and knowing when
the output was supposed to change state (from the modulator), the commutation delay can easily be calculated.
The default phase voltage measurement lies somewhere between digital states 0 and 1 when phase voltage is zero, and for this
reason the EXTI does not quite work properly. It seems to always generate the up-going edge interrupt, but rarely
generates the down-going edge. To overcome this, I modified the phase voltage circuit (temporarily) to shift the zero point
down so that at 0 V the phase voltage measurement is definitely digital zero (less than 1 V or so). The modification is
according to following figure, note the added Radd from op-amp negative input to +3.3V rail.
With this modification and SVM, it is possible to control the output voltage angle over the complete circle (electrical angles) and measure the commutation delay for going from low to high and vice versa. Next figure shows the results; for right motor the measurement is in phases V and W.
The blue and gray lines show the low-to-high delay while the orange and yellow and high-to-low. Since the PWM counter counts downwards during high-to-low transition, those commutation delays show as negative; it is really a delay.
Since the rotation speed was very slow, the current is in phase with the voltage. We can see that the low-to-high delay does not vary much but is typically around 2 us. The high-to-low depends highly on the current polarity, when the phase is positive the delay is about 2 us and when it is negative the delay is 4 us. It is expected that when current is out from the phase (phase is positive, high), current immediately commutates to the lower diode when the upper transistor is turned off. For negative current (into the phase, negative phase in this case as current is in phase with voltage), the lower transistor must be turned on for the current to commutate from upper diode to it, and thus additional delay is introduced.
There are some jumps up and down in the measured values, which might be caused by the (other) interrupt priorities delaying the measurement, or other noise in the interrupt system, which is not very precise. I would also expect the low-to-high transition delay to vary, but there is only some variation in the V phase and I'm not sure if it is only some artifact or not.
Copyright (C) 2019 Lauri Peltonen