Definition of motor constant (Km)

For certain applications, a high motor constant (K_m) can be one of the primary performance indicators. K_m shows how much torque the motor can deliver for the amount of heat generated inside the motor due to losses.

K_m is defined and can be calculated as follows:

K_m = T/sqrt(P_l), (1)

where T is torque in Nm, P_l is power dissipation in W.

Under certain assumptions, described below, K_m can also be expressed through another constant – torque constant K_t:

K_m = K_t/sqrt(R), 2)

where K_t is in Nm/A, R is resistance in Ohms.

K_t shows how much torque the motor can deliver when supplied by a given amount of current.

When using the expressions (1) and (2) it is important to have the parameters and the units correct, for example if K_t is in Nm/A_rms, then R (in expression 2 above) needs to be the three-phase equivalent of resistance, not the measured line-line value R_L-L:

R = 3/2*R_L-L, (3)

Using Km as the Key Performance Indicator

The key assumption

It is important to realize that the use of K_m, especially defined as in expression (2), is based on the assumption that all the generated heat is coming from copper losses due to current flowing in the motor winding (P_l =I²R). All other sources of losses are ignored. So, the K_m only accurately depicts losses when the motor is delivering torque without moving or moving at low speeds.

Thus, motor designs where the other losses are not negligible are less suitable for the evaluations based on the value of K_m. Usually other losses, such as core losses, losses in magnets, windage losses can be more pronounced at higher operational speeds.

Use of motor constant in benchmarking

Keeping in mind the assumptions above, one can still say that K_m is very useful in comparing multiple motors of similar size, voltage, speed and power. The motor with higher K_m can be expected to provide higher torque, have lower surface temperature and consume less power (i.e. have higher efficiency).

When benchmarking, it is important to make sure that the parameters and units used are correct and the same for both cases being compared, e.g., that R is the three-phase equivalent of resistance, not the measured line-to-line value (as already mentioned above).

Motor constant as the factor in motor sizing

Checking the value of motor constant is especially important when sizing motors for applications where heat dissipation possibilities are limited and the motor surface cannot be very hot, like in medical applications or in humanoid robots.

The designers need to evaluate how much heat can be comfortably absorbed into the system. Typically, thermal resistances can be identified and then used to set a limit on heat generation. If it is known, the robot joint can dissipate not more than 100 W of heat before the surface temperature exceeds the design limit, this information can be used to find allowable K_m. For example, if there is needed 10 Nm of torque, and the mechanical structure can only dissipate 100 W, then one needs a motor with minimum K_m of 1 Nm/W^1/2. This motor will also have to fit into the available space and operate from the given power source at the specified speed.

When one should be careful using motor constant as the primary performance indicator

As mentioned previously, K_m is not as useful in power conversion applications with motors constantly running at speeds of a few thousands of RPM compared to low-speed high-precision applications. Again, the reason is non-ohmic losses starting to become considerable at higher speeds.

In slotless motors the speed-dependent losses are lower than in iron-cored slotted motors. In slotless motors even at speeds of 3000-5000 RPM the fundamental conduction losses can be around 90% of the total losses. Therefore, benchmarking of slotless vs slotless on K_m is possible even for relatively high speeds. However, when benchmarking slotless vs slotted for high-speed operation, K_m is a less reliable parameter.

In any case, K_m can be a useful performance indicator, when taken together with other application related criteria.

Derivative figures of merit

Depending on the application, various derivatives of K_m can be useful as the figures of merit, such as K_m/weight and K_m/volume.

For example, low weight is important for exoskeletons and aerospace applications; therefore, K_m/weight can be used as the Key Performance Indicator. However, size is also important so K_m/volume can be good to use as well.

The importance of having a high motor constant in certain applications

As it was mentioned above, K_m is critical to understanding heat generation at different operation points. The applications where K_m is actively applied today include robotic joints, gimbals, fine-positioning systems.

For example, in the domain of medical devices it will often be reasonable to assume the temperature in the operating room of 25 ^oC. If a robotic assisted surgery system is in question, it most probably will not have any large heatsinks for heat dissipation so thermal issues should be considered very seriously. If say, the patient surface temperature limit is set to 45 ^oC, so the available temperature rise over ambient, which dictates the power that can be dissipated, is only 20 ^oC. Therefore, reducing the heat generated, through using high-K_m motor technology is key.

Gimbal systems are often battery powered. High K_m will mean lower power consumption and as the result longer operation time.

To put it simple, in practice high K_m means either colder motor running at the same torque or more torque produced for the same motor surface temperature.

Application of FiberPrinting™ results in the motors with high motor constant

Thanks to the intrinsic advantages of FiberPrinting^TM, such as high copper fill factor and optimal winding geometry, Alva’s motors usually have higher motor constant than other motors of comparable size and weight available on the market (Fig. 1).

In Alva’s motors the motor constant isn't affected by core saturation at high loads and is valid even at very high peak currents and torques.