## 2 Materials and Method
### Method
#### 2.2.3 Rotation Speed Calculation Method  
According to the method introduced in Section 2, a frequency analysis of the conditioning signal in Section 3 is required to obtain the main frequency of the AC signal, which is used to calculate the generator speed. Fast Fourier Transform (FFT) is commonly used in spectrum analysis, but its accuracy is affected by the sampling frequency and the number of sampling points. For short time signals, the frequency is calculated using the phase difference with higher accuracy [38]. FFT Accuracy Analysis  
The frequency of the FFT is calculated as  
\[f_{FFT}=i\times\frac{f_{\mathrm{s}}}{N} \tag{1}\]  
where \(i\) is defined as the horizontal coordinate sequence corresponding to the maximum amplitude point in the frequency spectrum. \(f_{\mathrm{s}}\) represents the sampling frequency of the original signal. \(N\) denotes the total number of sampling points used for FFT analysis. This fundamental FFT frequency resolution formula follows the standard spectral analysis theory [39]. Figure 3: Flowchart of the incremental iteration method. In practical engine speed measurement applications, the sampling duration for a single analysis should not be excessively prolonged. When \(f_{\text{s}}\) = 5000 Hz is adopted and 4096 points are selected as the total sampling points for FFT analysis, a frequency resolution of 1 Hz is achieved. The engine rotational speed is subsequently calculated using the derived frequency component through the following relationship:  
\[r=60\times f\times j \tag{2}\]  
where \(r\) is defined as the engine speed in revolutions per minute (RPM). \(f\) is identified as the fundamental frequency of the acquired signal. \(j\) is specified as the transmission ratio between the engine and alternator (typically 0.5). [PERSON] (2018) investigated the modeling of belt drive systems in vehicles ([PERSON], 2018). When conventional FFT analysis is employed to determine engine speed, a speedy resolution of 30 RPM is obtained, which is considered insufficient for precise measurement applications. This limitation is primarily attributed to the inherent frequency resolution constraints of standard FFT algorithms when applied using practical sampling parameters. Phase-Difference Method Used to Analyze the Main Frequency  
A higher measurement precision can be achieved through phase-difference frequency analysis, but this requires that the phase information at each sampling point is accurately determined. The phase information at individual sampling points is obtained using the sinusoidal least squares fitting method. Subsequently, the fundamental frequency is calculated based on the phase-difference method. Sinusoidal Least-Squares Fitting Method  
In a single period, the original sampled signal can be regarded as a single period sinusoidal signal, which can be expressed by the following formula:  
\[x(k)=A\sin\varphi_{k}+c+\varepsilon_{k} \tag{3}\]  
where \(A\) is the amplitude of the signal. \(k\) is the sampling sequence, \(k\in[0,\,n-1]\). \(\varphi_{\text{k}}\) is the phase; \(\varphi_{\text{k}}=\varphi_{0}+k\delta\). \(\varphi_{0}\) is the initial phase. \(\delta\) is the interval phase; \(c\) is the direct current bias. \(\varepsilon_{\text{k}}\) is the sampling error. In order to obtain the values of parameters \(A\), \(\varphi_{\text{k}}\), and \(c\), let \(a=A\cdot\sin\varphi_{0}\); \(b=A\cdot\cos\varphi_{0}\). In a single period, the original sampled signal can be regarded as a single-period sine signal, expressed as follows:  
\[x(k)=a\cdot\cos(k\delta)+b\cdot\sin(k\delta)+c+\varepsilon_{k} \tag{4}\]  
Taking \(\sum\varepsilon_{\text{k}}{}^{2}\) = min as the target for least squares fitting, if the total fitted data are \(n=2\pi/\delta\) and \(n\) is an integer, the simplified fitting parameter calculation formula can be obtained as follows:  
\[ \begin{bmatrix}a\\ b\\ c\end{bmatrix} =