## 3 Simulation Results and Analysis
### Simulation Results and Error Analysis of the Frequency Response of C-MHD ARS  
As mentioned earlier, it is difficult to set the exact value of the energized current for compensating according to the analytic expression. To investigate the appropriate frequency response through the whole bandwidth, we repeatedly tried to study the combined effect under the energized current. It is basic to give the frequency response of the sensing bottom channel without the energized current. The amplitude ratio and the phase differences between the voltage and angular velocity are calculated, when the reference rotating with \(\Omega=\sin(2\pi f\cdot t)\) at different frequencies. 500 intervals are set in every period for acquiring the phase accurately. Based on the modified simplified model analysis in our paper [12], the results plotted in Figure 12 are utilized to make fit the frequency domain to the method based on [PERSON]'s work using the Gauss-Newton method for an iterative search. The pre-assigned orders of the numerator and denominator are respectively 1 and 1. The fitting results can be written as (18). \[H_{\mathrm{I}}(s)=\frac{0.0000132s}{s+2.512} \tag{18}\]  
Based on the modified analytic theory, the simplified expression (1) added modified factor \(k\) (\(0<k<1\)) can approximately be given. After substituting the geometric and physical parameters as listed in Table 1, the results can be written as (19) when the magnetic field \(B\) is set as the average value 0.12 T. As expected, the factor \(k\) is about 0.18 and the amplitude in simulation is smaller for the inhomogeneity of the magnetic field. \[\frac{\varphi(s)}{\Omega(s)}=\frac{0.000016s}{s+k\cdot 13.847+0.093} \tag{19}\]  
Figure 11: The radial velocity \(u_{r}\) and relative circumferential velocity \(u_{\theta}\) of the test point \(P(0.010,0,-0.001)\) in the sensing channel under an energized current \(I=0\) A and \(I=1.5\) A at the frequency \(f=10\) Hz. Then energized current along the positive direction of the axis \(y\) is applied in the simulation. The current is increased gradually to improve the amplitude at \(f=0.01\) Hz until the ration is closed to 0.0000132. As presented in Figure 12, it is obvious that a significant improvement in the frequency response was obtained in the majority of cases. It should, however, be noted that the errors exist from 0.1 Hz to 2 Hz. When the energized current I is set at 2.9 A, the amplitude \(f=0.01\) Hz is a little smaller than the one at \(f=10\) Hz while the phase about \(0^{\circ}\). As the rotating frequency accelerates, the amplitude increases until 0.2 Hz while the phase adds until 0.1 Hz. The amplitude peak at 0.2 Hz is 0.4 dB higher than that at 10 Hz, while the phase is about 3.5\({}^{\circ}\). In the next, the amplitude and phase both decrease while the increase of frequency ranges from 0.2 Hz to 1 Hz, then the frequency response curve is close to a line above 1 Hz. Apparently, if the energized current continues to increase, the amplitude and phase at 0.01 Hz would be equal to the one at 10 Hz while the fluctuation in the transition zone would be larger. This indicates that the corner frequencies of the two effects of the Coriolis effect and MHD effect are a little different. Thus, we tried to illustrate the compensated frequency response by the approximate analytic expression according to the fitting method based on [PERSON]'s principle. The fitting result is written as (20) and its goodness of fit is up to 0.9. According to the record in simulation, the generated radial velocity under the energized current \(I=2.9\) A is about 0.015 m/s. Compared with the simplified model shown in Equation (2), we suppose that the corner frequency of the Coriolis effect in C-MHD ARS is higher than one of MHD effect. \[H_{2}(s)=\frac{0.0000132s}{s+2.512}+\frac{0.00001298*3.2}{s+3.2} \tag{20}\]  
A specific analysis of the model errors will be given in the following section.