# 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
\begin{array}{l}V_{RA}=\max\biggl{\{}\biggl{|}\sum\limits_{k=1}^{+\infty}\frac{ 4A}{(-4\pi k^{2}+\pi)\sqrt{1+(2k\Omega)^{2}}}\sin(2k\omega_{0}t+\varphi_{k}) \biggr{|}\biggr{\}}\\ \leq\frac{2A}{\Omega\pi}\biggl{|}\sum\limits_{k=1}^{+\infty}\frac{1}{(-4k^{2}+ 1)k}\biggr{|}=\frac{2A}{\Omega\pi}(\ln(4)-1)\end{array}  \tag{37}\]  
The estimated value of the ripple amplitude is:  
\[{V_{RA}}^{\prime}=\frac{2A}{\Omega\pi}(\ln(4)-1) \tag{38}\]  
The estimated value \({V_{RA}}^{\prime}\) in Equation (38) is simply the sum of amplitudes of all sinusoidal components. It cannot be met while considering the influence of phase values \(\varphi_{k}=\arctan\biggl{(}\frac{1}{2k\Omega}\biggr{)}\). The actual ripple amplitude will be smaller. The ratio of ripple amplitude to its estimated value is defined as:  
\[K_{RA}=\frac{V_{RA}}{{V_{RA}}^{\prime}} \tag{39}\]  
Figure 10 shows the relationship between \(K_{RA}\) and \(\Omega\):  
As shown in Figure 10, when \(\Omega\) continues to increase, \(K_{RA}\) approaches a constant 0.856. The minimum value of \(\Omega\) is set to 10; that is, only the condition when the signal frequency is greater than 10 times the cutoff frequency of the first-order low-pass filter is considered. Other subsequent waveform analyses will also be carried out according to this condition. With the correction constant of 0.856, the ripple amplitude estimation Equation (38) can be rewritten as:  
\[{V_{RA}}^{\prime}=\frac{0.21A}{\Omega} \tag{40}\]  
Figure 10: Ripple estimation accuracy for sine wave. For the triangular wave, transformations similar to Equations (34)-(37) are performed on Equation (32) to obtain:  
\[ \begin{array}{l}V_{RA}=\max\biggl{\{}\biggl{|}\sum\limits_{k=1}^{+\infty}\frac{-4 A}{\pi^{2}(2k-1)^{2}\sqrt{1+[2(2k-1)\Omega]_{k}^{2}}}\sin[2(2k-1)\omega_{0}t+ \varphi_{k}]\biggr{|}\biggr{\}}\\ \leq\frac{2A}{\Omega\pi^{2}}\sum\limits_{k=1}^{+\infty}\frac{(2k-1)}{(2k-1)^{ 3}}\approx\frac{2A}{\Omega\pi^{2}}\times 1.052\end{array}  \tag{41}\]  
while \(\varphi_{k}=\arctan\Bigl{(}\frac{1}{2K\subset(2k-1)\omega_{0}}\Bigr{)}\). For the periodic pulse wave, the following can be obtained:  
\[ \begin{array}{l}V_{RA}=\max\biggl{\{}\biggl{|}\sum\limits_{k=1}^{+\infty} \frac{2A\sin(kD\pi)}{k\pi\sqrt{1+(2k\Omega)^{2}}}\sin(2k\omega_{0}t+\varphi_{ k})\biggr{|}\biggr{\}}\\ \leq\frac{A}{\Omega\pi}\sum\limits_{k=1}^{+\infty}\frac{\sin(kD\pi)}{k^{2}}< \frac{A}{\Omega\pi}\sum\limits_{k=1}^{+\infty}\frac{1}{k^{2}}=\frac{\pi A}{ \delta\Omega}\end{array}  \tag{42}\]  
while \(\tan\varphi_{k}=\frac{2\cos(kD\pi)-4 RC\omega_{0}\sin(kD\pi)}{4 RC\omega_{0}\cos (kD\pi)+2\sin(kD\pi)}\). The ripple amplitudes estimated via Equations (41) and (42) are also based on the assumption that all sinusoidal components are maximized at the same time and the influence of phase is ignored. For periodic pulse input, the final approximation of Equation (42) also ignores the effect of duty cycle \(D\). In actual calculation, different values of \(D\) have a great impact on ripple estimation. According to the simulation, when \(\Omega\) is extra high, \(K_{RA}\) only depends on \(D\). Figure 10(b) shows the relationship between \(K_{RA}\) and \(D\), while \(\Omega=10^{4}\). From Figure 10(a), a correction constant for triangular wave input is obtained as 0.9228. The ripple estimation equation is rewritten as:  
\[{V_{RA}}^{\prime}=\frac{0.197A}{\Omega} \tag{43}\]  
For periodic pulse wave, the correction value of ripple estimation can be fitted as a quadratic function related to duty cycle: \(-2.985D^{2}+2.985D+0.00075\). The estimation equation can be rewritten as:  
\[{V_{RA}}^{\prime}=\frac{0.524A}{\Omega}\Bigl{(}-2.985D^{2}+2.985D+0.00075 \Bigr{)} \tag{44}\]  
Figure 11: Ripple estimation accuracy for: (**a**) Triangular (depends on \(\Omega\)). (**b**) Periodic pulse (depends on \(D\), while \(\Omega=10^{4}\)).