# A Demagnetisation Risk Index for a sensorless brushless DC drive

The article Demagnetisation in a sensorless brushless DC drive gave a broad overview of demagnetisation in a sensorless brushless DC drives that depend on Zero Crossing (ZC) detection to synchronise the next commutation phase.

There is no widely recognised method of prediction whether a drive is at risk of excessive demagnetisation time, not even in the steady-state wide open throttle (WOT) scenario.

This article proposes a statistic that might be used as a risk indicator, it is dimensionless and approximately proportional to the ratio of energy stored in the self inductance of a coil to the energy consumed by the motor in the time available for demagnetisation.

The concept is that most of the energy stored in the self inductance of a coil is returned to the motor producing output power and internal losses in much the same proportions as the primary input power.

The calculation of each of these quantities is quite complex, but broadly, making some approximations and ratioing the quantities gives a dimensionless statistic that might well be a good indicator of potential problems.

The method relates to energy, a statistic is proportional to the ratio of energy stored in the winding field due to current to the energy delivered in the available interval for demagnetisation.

Energy stored in the magnetic field due to coil current is equal to 0.5LI^2, and though L is the phase inductance, we can use the line to line inductance and say that energy stored in the magnetic field due to coil current is proportional to LI^2.

This energy is returned to the motor (shaft torque and losses), and this must be achieved in less than MotorPeriod/12. The average energy consumed by the motor in time t =Pin*MotorPeriod/12 =Pin*60/rpm*2/P/12 =Vbat*I*60/rpm*2/Poles/12, we can say that the energy consumed by the motor in the available time is approximately proportional to V*I/rpm/P where P is the number of poles, V is the loaded battery voltage, I is the current under load and rpm is the loaded rpm.

Ratioing these quantities gives the statistic LIrpmP/V, and large values indicate high risk of demagnetisation overrunning the available time, failure of ZC detection and sync loss.

A number of drives were tested on different voltages and with different propellers, and the LIrpmP/V statistic calculated. Experience was that small drives that appeared to have reliable sync even under rapid acceleration had LIrpmP/V well less than 5, and drives that would not make WOT before sync loss had LIrpmP/V well over 5.

A series of SPICE models were run to model the behaviour of some drives and results were consistent with tests on those drives.

In drives that could not make WOT, examination of the statistic gives a hint about possible measured to configure a workable drive. For a given motor, L and P are fixed. Reducing I  by decreasing the propeller load helps but typically reduced output power results in higher rpm for the same battery voltage somewhat offseting the benefit of reduced I. However if I is reduced sufficiently, even at the higher rpm, reliable sync may be obtained.

For example, a Hobbyking 4220-650Kv 16 pole motor with 1245SF propeller on 14V drew 15A at less than WOT and lost sync. The calculated index was 9.0, and would have been higher had sync not been lost.

Changing to 1145SF and 1045SF at 14V did not solve the problem, but a 0945SF allowed 8190rpm at WOT on 24V, I=6.1A and the calculated index was 3.2. Motor power at 146W is way less than you might expect from the specifications.

All of these tests were steady state, and drives that just make WOT with reliable sync may come undone under rapid acceleration, though the rpm term might be lower, the I term might be much greater.

## Issues with usage

It seems that as time passes, fewer and fewer motors are advertised with inductance specified, so it must be measured by the user / community.

Whilst inexpensive inductance meters abound online, I have not verified they are up to the task. All measurements of L for this study were done with a quality LCR meter at 10kHz.