  We derive the approximate, ``analytic-kludge'' (AK) waveforms for the
inspiral of a charged stellar-mass compact object (CO) into a charged massive
Kerr-Newman (KN) black hole (BH). The modifications of the inspiral orbit due
to the charges in this system can be attributed to three sources: the electric
force between the CO and the MBH, the energy flow of the dipole electromagnetic
radiation, and the deformation of the metric caused by the charge of the MBH.
All these are encoded explicitly in the fundamental frequencies of the orbits,
which are calculated analytically in the weak-field regime. By calculating the
mismatch between the waveforms for charged and neutral EMRI systems with
respect to space-borne detectors TianQin and LISA, we show that tiny charges in
the system can produce distinct imprints on the waveforms. Finally, we perform
parameter estimation for the charges using the Fisher information matrix method
and find that the precision can reach the level of $10^{-5}$ in suitable
scenarios. We also study the effects of charges on the parameter estimation of
charge, where the effects from the charge of the MBH can be well explained by
its effects on the cutoff of the inspiral.
