Physical Parameters Affecting Stall Torque of a Brushless DC Motor
Torque, moment, or moment of force is a rotational force. Just as a force is a push or a pull, a torque can be thought of as a twist or turning force on an object. Stall torque is the torque available from a mechanical device, such as a motor, whose output rotational speed is zero. Alternately, it may also denote the torque load required to cause stalling. Motors in a stalled condition may be prone to overheating and possible damage if the current flowing under these conditions becomes higher than the maximum continuous rating. The purpose of this document is to explain which parameters contribute to the tolerance and variation of the stall torque value of a brushless DC motor.
1. How to calculate the stall torque:
Stall torque of a brushless DC motor is denoted by:
T_{s}=k_{t} . I_{s }T_{static friction}
Neglecting static friction: T_{s}=k_{t} . U/R (1)
Where,
T_{static friction}: Static friction torque (Nm)
T_{S}: Stall torque (Nm)
k_{t}: Torque constant (Nm/A)
l_{s}: Stall current (A)
U: Voltage (V)
R: Phase resistance (Ohms)
Part of the losses in a brushless DC motor is due to the friction of the bearings ( ) – typically ball bearings. The friction is linked to the bearing size, the preload force, the actual loading of the bearing, sealing if any, and to the type / quantity of the lubricant. Ball bearing suppliers provide literature to estimate this starting moment with sufficient accuracy. In this technical note, we will neglect the static friction assuming that the stall torque is much higher, but it might not necessarily be true for all cases, when considering the stall condition.
2. What is contributing to our standard tolerance?
Let us now understand the factors contributing to the tolerance of stall torque, based on equation (1).
2.1 Torque constant value: ±10%
The torque constant has a typical tolerance of ±10%. This is caused by variation of the magnetic strength of the permanent magnet, i.e. variation of the magnet remanence. Remanence is a physical property of magnetic material, which is the amount of magnetization remaining in zero field after a large magnetic field is applied. The tolerance of this value depends on the manufacturing processes capability of the supplier,generally around ±2.5%.
The last part of the tolerance (±7.5%) is mainly due to phase setting, with mechanical tolerances having a lower impact. The phase setting tolerance could be reduced through more automated processes. Another solution could be to demagnetize the motor precisely which would ensure tighter tolerance of the torque constant value in the range of few %. However, this would lower the torque constant value, which reduces the torque capability of the motor.
 Resistance of the coil: ±8%
Phase resistance of the motor (ohms) also affects the standard tolerance, based on the same equation (1), and is usually given by ± 8%. This is due to the variation in the length of the wire while winding it, as well as diameter variation when forming the coil. This is inherent to the manufacturing process.
 Theoretical calculation of the uncertainty on the stall torque value
Now let us calculate the uncertainty of the stall torque value. Applying a logarithmic function to the above formula and differentiating it, we can directly obtain the absolute uncertainty on the stall torque value as
In (T_{s})_{}= (k_{t} . U/R) = In (k_{t})_{} In (U)  In (R)
As input voltage U is a constant, on differentiating the above equation, we get:
dT_{s}/T_{s }= dk_{t}/k_{t}  dR/R
(∆T_{s})/T_{s} =(∆k_{t}/k_{t}t +∆R/R
Putting into numbers:
(∆T_{s})/T_{s} =10%+8%=18%
The absolute uncertainty on stall torque value is thus 18%.
Let us now study the behavior of Portescap’s 16ECP528B112 brushless dc motor. Considering the torque constant and resistance value at the limit of their tolerances (table 1), we get an output speed as shown in Figure 1 on the right.
DC electrical parameters 
1 
2 

Torque constant 
[mNm/A] 
16.56 
20.24 
Resistance 
[Ohm] 
6.7 
5.7 
Supply voltage 
[V] 
24.0 
24.0 
No load current 
[mA] 
Negligible 
negligible 
Stall Torque 
[mNm] 
59.3 
85.2 
The motor parameters are measured at 24V, considering the extreme tolerances of the resistance and torque constant values. The tolerance range is given by (85.2 – 59.3), i.e. 25.9 mNm, and absolute tolerance is given by (25.9 / (59.3 + 85.2)/2), i.e. ±18 %.
3. What is influencing the stall torque value?
Now, let’s focus on the factors influencing stall torque.
One key environmental factor is temperature, which can be an extrinsic factor (environmental temperature) or an intrinsic factor (joule losses of the coil), or both. That is the reason why phase resistance is always given for a certain temperature, which is typically room temperature of 22°C.
Additionally, you will need a controller to drive the motor. Depending on the type of controller, it will directly impact the way you energize the motor phases and hence it will impact the performances of the motor at stall condition.
3.1 Thermal impact:
Let us consider the thermal impact first, namely the influence of the temperature.
Resistance of the winding:
Phase to phase resistance of the motor might have a significant impact on actual performances, as the phase resistance R depends to a great extent on the temperature:
R=R_{0}.(1+γ.∆T)
∆T=TT_{0}
where,
T_{0}: temperature at which has been measured
R_{0}: phase resistance at temperature
y: resistivity temperature coefficient of copper (y = 0.0039 °C ^{1})
The resistance temperature is thus directly proportional to coil temperature with a linear increase of 0.39 %/ °C.
Remanence of the magnet:
Temperature will have an impact on the remanence of the magnet, which will then impact the torque constant of the motor itself. If we know the working point at which the magnet is operating, then we can derive from the B/H curves of the magnet (physical characteristic provided by magnet suppliers) the impact of temperature on remanence. With typical magnets used in brushless DC motors, assuming the temperature remains below the temperature of demagnetization, the temperature increase will result in a drop of the torque constant of roughly 0.11% / °C
3.2 Driving method:
There are basically three different ways to drive brushless DC motors:
 Sensorless
 Conventional control method using hall effect sensors (6step drivers)
 Field oriented control
As well as impacting the stall torque, the driving method will also impact the motor performances and behavior, like vibration level, smoothness of operation or noise. Let us see how the motor behaves for each of these three driving methods.
Sensorless
This method does not require any sensors to know the rotor position. Though different techniques exist to know the rotor position of a brushless DC motor without using sensors, the most common technique is to sense the back EMF (electro motive force) of the nonenergized phase of star connected windings. This requires a minimum speed, as the back EMF is proportional to the motor speed. The back EMF of a motor at stand still is equal to zero. Generally, an openloop driving sequence is used to start the motor. Therefore, this type of drive method is not appropriate for an application reaching stall conditions. It is not recommended to use such driving method when you need your motor to operate at stall condition.
Conventional control method using hall sensors (6step drivers)
Let’s consider a typical brushless DC motor with 3 phases and one pole pair, which is the easiest case to understand. Such brushless DC motors can be equipped with 3 hall sensors shifted by 120 electrical degrees, in order to sense the rotor position.
The 3 digital hall sensors will generate high output or low output signals depending on the direction of the rotor flux, therefore it will generate a 6step logical commutation table over 360° electrical degrees, which is equal to one motor rotation for a 1 pole pair motor. Then, the driver will energize the right phases to keep the angle between the stator flux and the rotor flux close to 90°, so that it maximizes the torque generated. See Figure 2 on the right the typical shape of phase current vs. each logical state of the 3 hall sensors.
Since it is a discreet 6step commutation, the torque generated will not be constant and it will be subject to ripple. See Figure 3 on the right which shows the shape of the BEMF vs. 360° and the derived shape of the BEMF. The output torque of the motor would have the exact same shape.
Let us now understand the influence of torque ripple. The available torque will be lower, right at the moment of the theoretical commutation and at its peak (exactly between 2 commutations). This depends on the angle between the rotor flux and the stator flux (over one sixth of entire revolution). This difference between to top and the bottom of this ripple is equal to 13.4% of the maximum torque available (top of the ripple).
Taking the example of an electric gripper application, where the jaws will grip and hold an object. In this case, the motor will run the mechanical system and will reach stall conditions when the object is gripped. At this moment, the resistance torque will be equal to the motor torque. When gripping the object, the load torque reflected on the motor shaft can be assimilated to a spring having a high rigidity. It is represented by the slope curve in red, whereas the motor torque has a typical ripple shape and is represented in blue in figure 4. The intersection between these two curves, highlighted by the red dot in figure 4 on the right, represents the equilibrium position at which the motor will stall. It can be seen that the actual stall torque of the motor will be within the 13.4% tolerance range of the torque ripple.
Field Oriented Control
This driving method will apply sinusoidal current in each of the three phases, as shown in Figure 5 on the right.
Similar to the 6step driver which requires hall sensors in order to know the rotor position, field orientated control method requires a high resolution in order to adjust the current more often depending on the rotor position. As a result, the output torque of the motor will be constant no matter the rotor position. This means smooth operation of the motor, especially at low speed (close to stall condition).
Also, the motor will feature slightly higher torque, up to 5% more vs. RMS torque compared with 6step driver operation. Taking the same example of an electric gripper application, in this case the gripping torque will be the same no matter the rotor position, thus providing consistent gripping force.
It is represented graphically in the figure 6 on the right by the intersection between the motor torque (horizontal line in blue) with the load torque (slope curve in red).
Conclusion
The tolerance on the electrical parameters of a BLDC motor has a significant impact on the calculation of the stall torque value.
It can go up to ±18% combining the tolerances on the torque constant value (magnet, phase setting) and the resistance value (manufacturing).
Other factors, like temperature variations, which can be either linked to the application itself or the heating of the motor itself, will also impact significantly the electrical parameters of the motor.
Variations due to tolerance and thermal perturbations are inherent to the real world, and design engineers need to consider these when selecting the right brushless DC motor for their application.
Selection of the driver is also a key factor to take into consideration, as the drive technology itself will impact the stall performances. For examples, when looking for repeatable stall torque value nomatter the rotor position, engineers need to consider field orientated drive instead of the widespread 6step drive.
Figure 4 – Motor torque ripple and load torque intersecting