# 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
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Waveform** & \(D\) & \(X_{RMS}\) & \(X_{PK}\) & \(X_{RM}\) & \(e_{PK}\)* & \(e_{RM}\)* \\ \hline \hline \begin{tabular}{c} Sine \\ Triangular \\ DC \\ \end{tabular} & \(/\) & \(V_{0}\) & \(\sqrt{2}V_{0}\) & \(0.9V_{0}\) & \(112\%\) & \(11.1\%\) \\ \hline \
\begin{tabular}{c} Sine \\ Triangular \\ DC \\ \end{tabular} & \(/\) & \(V_{0}\) & \(\sqrt{3}V_{0}\) & \(0.866V_{0}\) & \(73.2\%\) & \(15.5\%\) \\ \hline \hline \multirow{4}{*}{
\begin{tabular}{c} Pulse \\ \end{tabular} } & \(D\) & \(V_{0}\) & \(V_{0}/\sqrt{D}\) & \(\sqrt{D}V_{0}\) & \(3\sqrt{D}-1\) & \(1/\sqrt{D}-1\) \\ \cline{1-1} \cline{2-6}  & \(1\) & \(V_{0}\) & \(V_{0}\) & \(V_{0}\) & \(200\%\) & \(0\%\) \\ \cline{1-1} \cline{2-6}  & \(0.5\) & \(V_{0}\) & \(\sqrt{2}V_{0}\) & \(V_{0}/\sqrt{2}\) & \(112\%\) & \(41.4\%\) \\ \cline{1-1} \cline{2-6}  & \(1/3\) & \(V_{0}\) & \(\sqrt{3}V_{0}\) & \(V_{0}/\sqrt{3}\) & \(73.2\%\) & \(73.2\%\) \\ \cline{1-1} \cline{2-6}  & \(1/9\) & \(V_{0}\) & \(3V_{0}\) & \(V_{0}/3\) & \(0\%\) & \(200\%\) \\ \cline{1-1} \cline{2-6}  & \(0.1\) & \(V_{0}\) & \(\sqrt{10}V_{0}\) & \(V_{0}/\sqrt{10}\) & \(-5\%\) & \(216\%\) \\ \hline \hline \end{tabular}  
* * Defined in Equations (17) and (18).  
\end{table} 
Table 2: The actual protection threshold of various waveforms. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Waveform** & \(D\) & \(X_{RMS}\) & \(X_{PK}\) & \(X_{RM}\) & \(e_{PK}\)* & \(e_{RM}\)* \\ \hline \ \begin{tabular}{c} Sine \\ Triangular \\ DC \\ \end{tabular}  & \(/\) & \(V_{0}\) & \(\sqrt{2}V_{0}\) & \(0.9V_{0}\) & \(112\%\) & \(11.1\%\) \\ \cline{1-1} \cline{2-6}  & \(/\) & \(V_{0}\) & \(\sqrt{3}V_{0}\) & \(0.866V_{0}\) & \(73.2\%\) & \(15.5\%\) \\ \cline{1-1} \cline{2-6}  & \(/\) & \(V_{0}\) & \(V_{0}\) & \(V_{0}\) & \(200\%\) & \(0\%\) \\ \hline \hline \multirow{4}{*}{ \begin{tabular}{c} Pulse \\ \end{tabular}  } & \(D\) & \(V_{0}\) & \(V_{0}/\sqrt{D}\) & \(\sqrt{D}V_{0}\) & \(3\sqrt{D}-1\) & \(1/\sqrt{D}-1\) \\ \cline{1-1} \cline{2-6}  & \(1\) & \(V_{0}\) & \(V_{0}\) & \(V_{0}\) & \(200\%\) & \(0\%\) \\ \cline{1-1} \cline{2-6}  & \(0.5\) & \(V_{0}\) & \(\sqrt{2}V_{0}\) & \(V_{0}/\sqrt{2}\) & \(112\%\) & \(41.4\%\) \\ \cline{1-1} \cline{2-6}  & \(1/3\) & \(V_{0}\) & \(\sqrt{3}V_{0}\) & \(V_{0}/\sqrt{3}\) & \(73.2\%\) & \(73.2\%\) \\ \cline{1-1} \cline{2-6}  
better protection validity for most waveforms, such as DC, sine, triangular and periodic pulse with \(D>1/3\). For the periodic pulse with \(D<1/3\), peak detection results in better protection validity. When \(D=1/3\), these two magnitude-detection methods have the same validity and the actual protection point is \(\sqrt{3}\) times the threshold. Therefore, the overcurrent protection method combing rectified mean and peak detection should be used. Peak detection does not need to be performed through a hardware circuit as in Figures 2 and 4. It can be executed by monitoring the real-time sample value of ADC. While the full scale sampling value is met, the protection can be executed by changing the signal level of \(CTR_{N}\) (\(N\) is 1 or 2). It has similar timeliness to the hardware circuit because neither storage nor a computing module is required to process the sample value. In summary, with the combination of rectified mean-detection circuit and peak detection via ADC sample value, overcurrent protection for common current waveforms can be realized. The thermal design of sensing resistors and switches shall ensure that they can conduct a current at \(\sqrt{3}\) times the rated value continuously without damage, in response to periodic pulse at a specific duty cycle.