However, while back EMF depends on the inductances, flux linkage, currents, and speed, torque is independent of speed - the same \((I_d, I_q)\) will always produce the same torque, no matter what speed the motor is at. Furthermore, we already know the relationship between torque and the axis currents from stall testing, and we can used this data as a black box to look up torque outputs from \(I_d\) and \(I_q\) inputs.
We are going to make an additional huge assumption: at high speeds, the current is low. This is not necessarily true, but for motors designed to be aggressively field weakened, the achievable current is likely low due to the high inductances. This assumptions means we can use the voltage equations to compute the back EMF for most of the field weakened operating regime. Of course, there will be a transition around base speed where this assumption doesn't hold, but we can "fix that in post".
Armed with this, we can write a simple C++ program (source, executable, sample input) to search the entire space of \(I_d\) and \(I_q\) values. The program is not particularly good or fast, but the brute-force approach makes it very robust and trivially extensible to a saturated motor (just override the Vs2() function in the MotorModel class with a lookup table based one). In contrast, Newton's-method based approaches seem to fail if the voltage surface is too complex.
The program generates some very reasonable output; for example, the following plot of power and torque versus speed for the HSG at 160V:
The flat part of the torque-speed curve extends up to what would traditionally be called "base speed" [1]. A surface PM machine spends most of its time operating in this regime, as operating over base speed results in reduced power output and efficiency. In contrast, an IPM is a constant-power device past base speed; this has several implications for system design:
Hybrid vehicles: Field weakening is very important for hybrid vehicles. Consumer hybrids have electric subsystems optimized for city driving. In order to optimize efficiency in this scenario, it is beneficial to have a high reduction between the motor and the wheels, to reduce the motor current required to accelerate the car. This typically means putting base speed somewhere around 40 mph, which means at highway speeds, the motor is operating well beyond base speed. Being able to produce power at these speeds is important for consistent performance.
There is also a class of emerging high-performance hybrids. Typically, these use a combination of one or two motors, a medium sized (around 5KWh) battery pack, and a very high power forced-induction internal combustion engine. The electric subsystem is used to compensate for the narrow power band of the ICE by adding additional low-speed torque. It also usually provides power to all four wheels, improving handling and launch performance. Finally, it improves the regulatory status of such cars by at least nominally increasing the fuel economy. Once again, we find it beneficial to place base speed at a relatively low speed in order to maximize the launch torque delivered to the wheels (and reduce the weight of motor required to deliver that torque to the wheels); consequently, field weakening is needed to prevent the top speed of the car from being voltage-limited.
Pure electric vehicles: It is widely known that most EV's have a single-speed gearbox. This is entirely due to the power-speed profile of an IPM [2]; as the motor can reach peak power at very low speeds, a variable-speed transmission is not necessary to maximize power output across the entire operating range of the vehicle.
In fact, we can simulate the broad power band of an IPM with a surface PM machine and a continuously-variable transmission. It is usually not desirable to do so [3]; multi-speed transmissions incur additional complexity, weight, cost, and losses, usually negating the improved torque density of the surface PM motor. The only cars that use surface PM motors (Honda, Hyundai) are hybrids which are strongly derived from existing gas-only cars and already have manual transmissions.
Combat robots [4]: Spinner weapons are very similar to cars - both are inertial loads that have highly variable speed profiles. Interior PM machines have obvious mechanical benefits, as the rotors are much more robust. In addition, having a virtually unlimited top speed makes match-ups more consistent. Having moderate weapon speeds is usually beneficial, as it improves energy transfer and tooth engagement. However, in the vertical-on-vertical matchup (which is becoming much more common), the robot with the higher blade speed hits first. In this case, being able achieve very high speeds can greatly improve chances of victory.
And of course, higher-speed weapons hit harder if they do engage, so having the option to spin up to very high energies can be beneficial in certain situations.
Notes
[1] Technically, base speed also depends on stator current, so the correct terminology would be 'the base speed of the motor is 2000 rpm at 180A'.
[2] Induction machines (Tesla) and synchronous reluctance motors (no one yet) have similar characteristics, and trade off torque density for reduced cost.
[3] There are some designs which use a 2-speed transmission to further improve efficiency below base speed.
[4] No one has done this yet, but someone should!
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