# Current-Sensing Topology with Multi Resistors in Parallel and Its Protection Circuit
## 2 Materials and Methods
### Rectified Mean Magnitude Detection
#### 2.2.1 Protection Validity of Rectified Mean Detection  
Both the input and output of the rectified mean-detection circuit are voltages. Let the input voltage \(V_{l}\) be \(x(t)\) and the output voltage \(V_{O}\) be \(y(t)\). Common waveforms under test include sine, triangular and periodic pulse waves. The expressions are (10)-(12), respectively:  
\[x(t)=A\sin\omega_{0}t \tag{10}\]  
\[x(t)=\left\{ \begin{array}{rl}&4A(t-NT)/T,\quad NT\leq t<NT+T/4\\ &2A-4A(t-NT)/T,\quad NT+T/4\leq t<NT+3T/4\\ &-4A+4(At-NT)/T,\quad NT+3T/4\leq t<NT+T\end{array} \right. \tag{11}\]  
\[x(t)=\left\{ \begin{array}{rl}&A,\quad NT\leq t<NT+DT/2\\ &-A,\quad NT+T/2\leq t<NT+T/2+DT/2\\ &0,\;\;others\end{array} \right. \tag{12}\]  
where \(N\) is an integer, \(T\) is the waveform period, \(\omega_{0}=2\pi/T\), and \(D\) is the duty cycle of periodic pulse wave. Take \(A=1\), \(T=0.1\) s, \(D=0.5\) (the same values in Figures 8 and 9). The three waveforms are shown as Figure 7.  
Figure 6: Rectified mean circuit. The effective and rectified mean values of waveforms can be calculated with Equations (13) and (14), respectively. \[X_{RMS}=\sqrt{\frac{\int_{0}^{T}x^{2}(t)}{T}} \tag{13}\]  
\[X_{RM}=\frac{\int_{0}^{T}|x(t)|}{T} \tag{14}\]  
Form factor \(W_{F}\) is defined as the ratio of effective value to rectified mean value of waveforms:  
\[W_{F}=X_{RMS}/X_{RM} \tag{15}\]  
The three waveforms are calculated through Equations (13)-(15) and the results are shown in Table 1.  
Figure 7: Three common waveform: (**a**) Sine. (**b**) Triangular. (**c**) Periodic pulse. If the rectified mean circuit in Figure 6 is applied for rough measurement of the effective value and the form factor of the sine wave is used as the conversion coefficient from the rectified mean value to the effective value, the relative errors \(e_{R}\) of other waveforms are as in Table 1.  
\[e_{R}=\frac{1.111X_{RM}-X_{RMS}}{X_{RMS}} \tag{16}\]  
If the rectified mean circuit is applied for measurement of the effective value and all waveforms with the same effective value are set as the input, the DC or periodic pulse waveform with \(D=1\) (that is the square wave) would result in the highest output. Therefore, To avoid protection lag, the protection thresholds in Figures 2 and 4 should be calculated based on DC or square wave. The actual protection threshold of various waveforms with the same effective value \(V_{0}\) is shown in Table 2:  
In Table 2, \(e_{RM}\) represents the relative error between the actual protection point and the threshold. It was calculated as follows:  
Figure 6:0  
The crest factor of the periodic pulse at \(D=1/9\) is just 3. \(e_{PK}\) represents the relative error between the actual protection point and the threshold, when the peak-detection circuit is applied for magnitude detection and the protection threshold is set based on periodic pulse at \(D=1/9\). It was calculated as follows:  
Figure 6:1  
\(e_{RM}\) and \(e_{PK}\) can be used to indicate the protection validity of these two magnitude-detection methods. The higher the value, the worse the validity. As concluded from the data in Table 2, when the rectified mean circuit is used for magnitude detection, it has