I searched about this, but couldn't find any helpful explanations. All it says is that magnetic field is generated by either permanent magnets or electromagnets. In both cases, how does the exciter generator start, so that it creates magnetic field according to physics, (e.g. Lorentz law, or another) ?
There are two ways that exciter generators start, so they can then spin up the excitation current to start a larger generator.
The easy way is to use permanent magnets in the exciter generator. As soon as it begins to turn, it will generate the current to get the main generator going.
The tricky way is to rely on the fact that the iron in the armature and rotor of a generator usually retains a small amount of magnetic field even when the generator is shut down. When you start spinning this kind of generator, that residual field allows the generator to make its own excitation current and the generator will then start itself.
The self-starting principle using residual magnetism is commonly used in small gasoline engine powered AC generators.