  The conversion of external energy into electricity is the foundation of power
station and energy harvesting operation. The external source supplies an
impressed force per unit charge to free or bound charge to produces AC
electricity. We analyze the electrodynamics of ideal electricity generators
through a time dependent permanent polarization without any applied electric
field, which modifies the constitutive relations and is essential to oscillate
charge in a lossless way. For both cases, we show that Maxwell's equations, and
in particular Faraday's law are modified, along with the required boundary
conditions through the addition of an effective impressed magnetic current
boundary source. For the free charge case, we highlight the example of an
electromagnetic generator based on Lorentz force, where the impressed force per
unit charge that polarizes the conductor comes from mechanical motion of free
charge with an impressed velocity of a conductor relative to a stationary DC
magnetic field. The bound charge generator is an idealized permanently
polarized bar electret with a time dependent permanent polarization, the
underlying principle behind piezoelectric nano-generators. In the open circuit
state, both electricity generators are equivalent to idealized Hertzian
dipoles, with the open circuit voltage equal to the induced emf. Analyzing the
short circuit responses, we show that the bound charge electricity generator
has a capacitive source impedance. In contrast, we show for the ideal free
charge AC electricity generator, the back emf from the inductance of the loop
that defines the short circuit, directly cancels the source emf, so the voltage
across the inductor is solely determined by the magnetic current boundary
source. Thus, we determine the magnetic current boundary source is the
topological invariant that best describes the output voltage of an AC
generator.
