The length of air gap is inversely proportional to power factor and hence result in poor operating power factor of an electrical machine. Larger air gap length means higher resistance to magnetic flux which causes undesirable increases in magnetizing current and the associated electrical loss. Why would that not mean that air gap length would be minimized in all types of electric machines?
length of air gap is inversely proportional to power factor
The statement is true only for induction (asynchronous) machines. Reactive current is only required in the armature to magnetize the field. In permanent-magnet synchronous machines, no external magnetizing current is required. In wound-field synchronous machines, as much DC field excitation as is desired is supplied by a separate excitation system. For a synchronous generator, lagging reactive current requirement of the load can be supplied by increasing the excitation. For a synchronous motor, the DC excitation can be adjusted to keep the AC supplied to the motor at 1.0 pf. The motor can also be "over excited" to supply reactive current to other loads.
Induction machines are designed with the smallest air gap that mechanical construction will permit. Mechanical considerations require the air gap to be directly proportional to rotor diameter. Synchronous machines can have a larger air gap than induction machines and empirical data indicates that they do. That might be desirable for cooling or to make the machine more "rugged".
Source: John. H. Kuhlmann, Design of Electrical Apparatus, 1940
In machines in which the magnetizing current is taken from the supply network, the length of the air gap is dimensioned to produce a minimum magnetizing current and, on the other hand, an optimal efficiency. In principle, a small air gap gives a low magnetizing current, while the eddy current losses of the rotor and stator surface increase because of permeance harmonics created by the open or semi-closed slots. A small air gap also increases the surface losses in the rotor caused by the current linkage harmonics of the stator. Although the air gap is of great significance, no theoretical optimum has been solved for its length, but usually empirical equations are employed instead in the definition of the length of the air gap…. In DC and synchronous machines, the air gap is basically defined by the permitted armature reaction. We have to ensure that the armature reaction (flux caused by the current linkage of the armature) does not reduce the flux density excessively on one side of a magnetic pole.
Source: Juha Pyrhonen, Tapani Jokinen , Valeria Hrabovcova, Design of Rotating Electrical Machines 1st Edition 2008