Measuring thermal parameters empirically

In the last post, I gave an overview of what thermal metrics are relevant to motor drive applications and how they drive the important performance metrics of controllers and motors. In this post, I’m going to look at how to measure those thermal parameters empirically in at least a crude way, but with enough accuracy to be useful for practical design applications.

What do we want to measure?

There are a set of parameters that we would like to be able to measure that have some overlap between the controller and motor case. For both, we want to be able to measure:

• R_lab: The thermal resistance between the controller or motor and ambient under lab conditions with no cooling.
• R_cooling: The thermal resistance under various cooling conditions.
• C_bare: The heat capacity with no cooling present.
• C_cooling: The heat capacity with various additional features present, like heat spreaders.

Then, some desired mappings are unique to one of controllers or motors:

• Pc(V, PWM, I): For controllers, a function that determines the waste thermal energy from supply voltage, PWM frequency, and output phase current.
• Pm(I, ω): For motors, a function that determines the waste thermal energy from phase current and motor angular velocity.

How to measure the controller parameters

With the newly improved dynamometer, it should be possible to measure all of these parameters for the controller in a relatively straightforward manner, although the exact order of which parameters to measure first is critical. Here’s my plan

1. Measure R_lab: The first parameter to measure is the thermal resistance of the controller. To do this, I’m going to take advantage of two facts. First, that the thermal loss varies based on supply voltage and PWM frequency even if no motor is connected or zero current is being driven by the motor. Second, when using the Chroma source for the DUT in the dynamometer, I can programmatically measure the power input very accurately. If no power is going to the motor, then every watt of power coming out of the supply must by necessity ending up as thermal loss.

Thus the idea is to step through a few different supply voltages and PWM frequencies. At each step, wait for the temperature of the controller to completely stabilize. Then I have a mapping between power input and temperature which can be used to calculate R_lab.

2. Sample Pc(V, PWM, I): Now that R_lab is known, we can use it to measure how much thermal loss is present in scenarios where some power is going to the motor. Just wait for the temperature to stabilize, and solve for P in the thermal resistance formula:

$$P = {{T_{infinity} - T_{ambient}} \over R}$$

To get a better sense of the mapping, the sampling process I used was to roughly equally sample across the supply voltage and PWM space. For each point in that space, ramp up current linearly until the maximum temperature I was willing to test with was hit. This generates a 3 dimensional sparse mapping of the Pc(V, PWM, I) function.

3. Measure C_bare: The heat capacity is measured by using a technique similar to that for R_lab. First, the controller is soaked at the supply voltage and PWM which results in the lowest thermal loss. The temperature and power required is recorded. Then it is soaked at the supply voltage and PWM which results in the highest thermal loss, at which point the temperature and power is recorded. Finally, the controller is switched back to the original supply voltage and PWM frequency, and the time required to reach 63.2% of the way back to the original temperature is measured. Those parameters give us enough information to calculate the heat capacity.

$$C\_bare = {\Delta t \over R}$$

4. Measuring R_cooling: This can be done in the same way as R_lab, but separately for each different cooling method. For all moteus controllers I tested, I used at least the largest fan that would fit board mounted, the heat spreader for the board, and the heat spreader for the board with a heat sink attached.

5. C_cooling Can be measured in the same was a C_bare, but with each of the different cooling methods attached. This will only really be different when the cooling method involves the addition of significant thermal mass, like the heat spreader or a heat sink.

How to measure the motor parameters

Measuring the motor parameters doesn’t necessarily require the dyanamometer, as we don’t need to apply non-zero torque. Instead we can generate copper losses by applying current purely in the D axis, which makes the setup a bit simpler. In contrast to the controller parameter measurement though, a thermistor does need to be installed in the windings of each motor so that we can sample the motor temperature directly.

The procedure I used for the motors was to perform an integrated measurement of thermal resistance and heat capacity in one go. This was similar to the procedure for the heat capacity of the controllers, but does require knowing the thermal resistance of the controller as well in order to accurately know the actual electrical energy making it into the windings of the motor. During the initial soak phase, no power is applied to the motor and its temperature is sampled. When applying power, a D axis voltage of slowly varying angle is applied so as to ensure all coils are evenly heated. The input electrical power to the motor is estimated by first estimating the thermal loss in the controller from its temperature, then subtracting that power from the power supply output.

$$P_{motor} = P_{psu} - {{T_{controller} - T_{ambient}} \over {R_{controller}}}$$

With the power to the motor and its temperature at the two operating points, and the time required to fall 63.2% of the way back to the lower point, it is possible to estimate both the thermal resistance and heat capacity of the motor.

$$R_{motor} = {{T_{peak} - T_{ambient}} \over P_{motor}}$$

$$C_{motor} = {\Delta t \over R_{motor}}$$

Once again, the values of these for different cooling options can be measured as well. In order to just capture the spectrum, I measured motors at ambient, and with the biggest fan I had on hand blowing directly through the windings in order to get an estimate of the “maximimal” cooling scenario.

For now, I’m going to ignore measuring the iron loss component of Pm(I, ω), not because it isn’t important, but just because I ran out of time and it isn’t that large of a factor in many robotics applications.

Next steps

In the next post, I’ll cover how I turned all this data into the first version of something of a useful tool that can be used to help design systems with moteus controllers.