Which factors affect this ratio. I am aware of the general statement that Induction machines are bulky and have seen DC machines that appear to be compact and light weight, but I am stumped at comparing the torque produced by them.
2 Answers
There is a wide torque range for which induction machines, separately excited or permanent-magnet commutator DC machines and brushless permanent-magnet AC or DC machines are all available. Those machine types are also available with similar power and speed ratings. Considering just the machines without any required control equipment, a comparison of machines offered for sale will probably show that the permanent-magnet machine have the highest ratios of torque to mass and torque to volume for machines with equal torque and power ratings.
In considering the required control equipment, it is necessary to consider if there is any need for variable speed. If there is no need for speed control, relatively simple on/off control is all that is required for induction machines with AC power available or for commutator DC machines with DC power available. Brushless permanent-magnet machines require electronic control regardless of the need for variable speed or whether AC or DC power is available. Considering the machine and control units together may alter both the torque/power density and the efficiency comparison.
Electric machines that are offered for sale are designed to meet performance, purchase price, efficiency, reliability, expected lifetime, mass and volume objectives that are estimates of anticipated customer requirements and competitive offerings. Separate products are often offered for various market segments such as general industry, specific industry, land vehicles, aircraft, watercraft, military, consumer products.
There seems to be some interest in electric machines for aircraft for which the top priority is torque to mass ratio with purchase price and volume being lower priorities. Rare-earth, air-core permanent magnet machines seem to be of great interest for that market.
In a motor (and in fact in many other situations, power supplies for example), some things are controlled by the physics, but for other things there is more of a free choice dictated by economics, or even the market fashion.
In a motor, power density is very closely controlled by physics. The power to weight is limited by the amount of copper, iron and magnetic material in the motor. Here the permanent magnet DC motor wins, especially strong-field brushless types now available.
Power is the product of speed and torque, and for any given motor, there is more of a choice between them. Not an unconstrained choice, but there is some freedom within the power limit. Altering the number of copper turns alters the ratio of torque and speed the motor produces. Some motors, for instance vacuum cleaners, are optimised for very high speed, and have very mechanically strong rotors and few turns. Those intended for large-prop quad-copters have high torque and relatively low speed, certainly not challenging the rotor strength.
So your highest torque motor is likely to be found in the PMBLDC catalogues, and it's likely to one of the bigger one your dollar budget can afford. Which of those motors though requires you to read all the specs and compare.