Transcranial magnetic stimulation (TMS) is the non-invasive stimulus method to the brain by inducing the eddy current within the brain from the TMS coil placed outside the scalp. The induced eddy current stimulates the nerve circuit causes to suppress the brain activity partially in time and space, and it is applied to detect the brain functions etc. When TMS is applied to the brain while measuring the electroencephalography (EEG), the induced artifact caused by TMS covers on the EEG, it is called the TMS artifact. The amplitude of the TMS artifact is generally too large to omit diagnosing the EEG. Therefore some methods have been proposed to negate the TMS artifact from the EEG including the TMS artifact using the EEG only. We proposed a new method for describing the shape of the induced artifact in the EEG applied TMS using two equivalent circuit models, the TMS equipment model and the equivalent circuit model of bioelectric measurement system, under some simplified approximations. This paper shows that some attempts are applied to the proposed method to improve the performance of fitting and eliminating the TMS artifact. One is to derive the strict solution of the TMS artifact by omitting some simplified approximations. Other is the countermeasure against errors calculating inverse matrix while estimating parameters of fitting shape of the TMS artifact. One attempt is found that the strict solution improves the shape fitting but not so far from the approximated solution. Other attempt improves stability of the simulation. These attempts are found that some time parameters which determine the time-transition such as "Tc" should separately treat against other time constant parameters which are derived from circuit parameters. Moreover, considering the residual component this is not described on the circuit model.