# Current-Sensing Topology with Multi Resistors in Parallel and Its Protection Circuit
## 2 Materials and Methods
### Rectified Mean Magnitude Detection
#### 2.2.2 Dynamic Characteristics of Rectified Mean Detection  
The rectifier mean-detection circuit in Figure 6 is composed of an absolute value circuit and an average circuit. Its transfer function is simple. The absolutes of the periodic waveforms described in Equations (10)-(12) meet the Dirichlet condition and can be expressed in a time domain via Fourier series [26]. After passing through the absolute value circuit, the sine wave becomes:  
\[|x(t)|=|A\sin(\omega_{0}t)| \tag{19}\]  
The period of \(|x(t)|\) is \(T/2\) and angular frequency is \(2\omega_{0}\). Its Fourier series coefficient is:  
\[a_{k}=\frac{2}{T}\int_{0}^{\frac{T}{2}}|x(t)|e^{-j2 Ku_{0}t}dt=\frac{2A}{-4\pi k ^{2}+\pi} \tag{20}\]  
Thus, its Fourier series is as follows:  
\[|x(t)|=\sum_{k=-\infty}^{+\infty}a_{k}e^{j2 Ku_{0}t}=\sum_{k=-\infty}^{+\infty} \frac{2A}{-4\pi k^{2}+\pi}e^{j2 ku_{0}t} \tag{21}\]  
To analyze its dynamic characteristics, assume that the input signal is applied from \(t=0\). That is \(x(t)=0\) while \(t<0\). The complex exponential expression for the absolute of the sine wave is:  
\[|x(t)|u(t)=\sum_{k=-\infty}^{+\infty}\frac{2A}{-4\pi k^{2}+\pi}e^{j2 ku_{0}t}u(t) \tag{22}\]  
Through [PERSON]'s formula, the trigonometric function expression can be obtained as:  
\[|x(t)|u(t)=\Bigg{[}\frac{2A}{\pi}+\sum_{k=1}^{+\infty}\frac{4A}{-4\pi k^{2}+ \pi}\cos(2k\omega_{0}t)\Bigg{]}u(t) \tag{23}\]  
The complex exponential expression for the absolute of the triangular and periodic pulse wave can also be obtained via by the same transformations as in (20)-(22). For the triangular wave, this is:  
\[|x(t)|u(t)=\Bigg{[}\frac{A}{2}+\sum_{k=-\infty}^{+\infty}\frac{A\Big{(}e^{-j( 2k-1)\pi}-1\Big{)}}{\pi^{2}(2k-1)^{2}}e^{j2(2k-1)\omega_{0}t}\Bigg{]}u(t) \tag{24}\]For periodic pulse wave, it is:  
\[|x(t)|u(t)=\left(AD+\sum\limits_{k=-\infty}^{-1}\frac{A\sin(kD\pi)}{k\pi}e^{-jkD \pi}e^{2 jk\omega_{0}t}+\sum\limits_{k=1}^{+\infty}\frac{A\sin(kD\pi)}{k\pi}e^{- jkD\pi}e^{2 jk\omega_{0}t}\right)u(t) \tag{25}\]  
The trigonometric function expression for the absolute of the triangular wave is further obtained as follows:  
\[|x(t)|u(t)=\Bigg{\{}\frac{A}{2}+\sum\limits_{k=1}^{+\infty}\frac{-4A}{\pi^{2} \left(2k-1\right)^{2}}\cos[2(2k-1)\omega_{0}t]\Bigg{\}}u(t) \tag{26}\]  
For the periodic pulse wave, it is:  
\[|x(t)|u(t)=\left[AD+\sum\limits_{k=1}^{+\infty}\frac{2A\sin(kD\pi)}{k\pi}\cos (2k\omega_{0}t-kD\pi)\right]u(t) \tag{27}\]  
A finite order of expression is taken; this is Equation (21) changing to \(|x(t)|=\sum\limits_{k=-N}^{+N}a_{k}e^{2 jk\omega_{0}t}\). The three waveforms are depicted in Equations (23), (26) and (27), as shown in Figure 8.  
\[a_{k}=\frac{2}{T}\int_{0}^{\frac{T}{2}}|x(t)|e^{-j2 Ku_{0}t}dt=\frac{2A}{-4\pi k ^{2}+\pi} \tag{20}\]0 Three waveforms depicted via Fourier series expressions: (**a**) Sine. (**b**) Triangular. (**c**) Periodic pulse. According to Figure 8, the higher the order of series, the closer its waveform is to the ideal one. While \(N=100\), the Fourier series expressions are usually acceptable for representing the original waveforms. For periodic pulse wave, the oscillations occurring at the rising and falling edges are inevitable, which is called the Gibbs phenomenon.