SIZE RANGE
Having praised the simplicity and elegance of the induction motor, and noting that they can be provided up to multi-MW powers, we might wonder why there are no very small ones. Industrial (3-phase) induction motors are rarely found below about 200 W, and even single-phase versions, which might be expected to dominate the domestic scene, rarely extend below about 50 W.
We will see that when we scale down a successful design, the excitation or Xux-producing function of the windings becomes more and more demanding until eventually the heat produced in the windings by the excitation current causes the permissible temperature to be reached. There is then no spare capacity for the vital function of supplying the mechanical output power, so the machine is of no use.
Scaling down – the excitation problem
We can get to the essence of the matter by imagining that we take a successful design and scale all the linear dimensions by half. We know that to fully utilise the iron of the magnetic circuit we would want the air-gap Xux density to be the same as in the original design, so because the air-gap length has been halved the stator MMF needs to be half of what it was. The number of coils and the turns in each coil remains as before, so if the original magnetising current was Im, the magnetising current of the half-scale motor will be Im=2.
Turning now to what happens to the resistance of the winding, we will assume that the resistance of the original winding was R. In the half- scale motor, the total length of wire is half of what it was, but the cross- sectional area of the wire is only a quarter of the original. As a result, the new resistance is twice as great, i.e. 2R.
The power dissipated in providing the air-gap Xux in the original motor is given by I 2 R, while the corresponding excitation power in the
half-scale motor is given by (Im=2)2 x 2R ¼ 1=2I 2 R.
When we consider what determines the steady temperature rise of a body in which heat is dissipated, we Wnd that the equilibrium condition is reached when the rate of loss of heat to the surroundings is equal to the rate of production of heat inside the body. And, not surprisingly, the rate of loss of heat to the surroundings depends on the temperature diVerence between the body and its surroundings, and on the surface area through which the heat escapes. In the case of copper windings in a motor, the permissible temperature rise depends on the quality of insu- lation, so we will make a reasonable assumption that the same insulation is used for the scaled motor as for the original.
We have worked out that the power dissipation in the new motor is half of that in the original. However, the surface area of the new winding is only one quarter, so clearly the temperature rise will be higher, and if all other things were equal, it will double. We might aim to ease matters by providing bigger slots so that the current density in the copper could be reduced, but as explained in Chapter 1 this means that there is less iron in the teeth to carry the working Xux. A further problem arises because it is simply not practicable to go on making the air-gap smaller because the need to maintain clearances between the moving parts would require unacceptably tight manufacturing tolerances.
Obviously, there are other factors that need to be considered, not least that a motor is designed to reach its working temperature when the full current (not just the magnetising current) is Xowing. But the fact is that the magnetisation problem we have highlighted is the main obstacle in small sizes, not only in induction motors but also in any motor that derives its excitation from the stator windings. Permanent magnets therefore become attractive for small motors, because they provide the working Xux without producing unwelcome heat. Several examples of small permanent magnet d.c. motors were included in Chapter 3, not- ably the toy motor discussed in Section 3.8.