# 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
By performing Fourier transform on Equation (22), the frequency domain expression for the absolute of the sine wave can be obtained as:  
\[a_{k}=\frac{2}{T}\int_{0}^{\frac{T}{2}}|x(t)|e^{-j2 Ku_{0}t}dt=\frac{2A}{-4\pi k ^{2}+\pi} \tag{20}\]1  
The frequency domain expression of the average circuit in Figure 6 is:  
\[a_{k}=\frac{2}{T}\int_{0}^{\frac{T}{2}}|x(t)|e^{-j2 Ku_{0}t}dt=\frac{2A}{-4\pi k ^{2}+\pi} \tag{20}\]2  
Thus, the frequency domain expression of the output signal for the sine input is as follows:  
\[a_{k}=\frac{2}{T}\int_{0}^{\frac{T}{2}}|x(t)|e^{-j2 Ku_{0}t}dt=\frac{2A}{-4\pi k ^{2}+\pi} \tag{20}\]3  
The time domain expression of \(y(t)\) can be obtained via inverse Fourier transform as:  
\[a_{k}=\frac{2}{T}\int_{0}^{\frac{T}{2}}|x(t)|e^{-j2 Ku_{0}t}dt=\frac{2A}{-4\pi k ^{2}+\pi} \tag{20}\]4  
Similarly, the time-domain expression of \(y(t)\) for the triangular wave input can be obtained as:  
\[y(t)=\Bigg{[} \begin{array}{c}\frac{A}{2}\Big{(}1-e^{-\frac{t}{RC}}\Big{)}+ \\ \sum\limits_{k=1}^{+\infty}\frac{-4A}{\pi^{2}(2k-1)^{2}}\frac{1}{1+[2 RC(2k-1) \omega_{0}]^{2}}\Big{(}\cos[2(2k-1)\omega_{0}t]+2 RC(2k-1)\omega_{0}\sin[2(2k-1 )\omega_{0}t]-e^{-\frac{t}{RC}}\Big{)}\Bigg{]}u(t)\end{array}  \tag{32}\]  
The time-domain expression of \(y(t)\) for periodic pulse input is:  
\[y(t)=\Bigg{[} \begin{array}{c}AD\Big{(}1-e^{-\frac{t}{RC}}\Big{)}+\sum\limits _{k=1}^{+\infty}\frac{A\sin(kD\pi)}{k\pi\big{[}1+(2 RCk\omega_{0})^{2}\big{]}} \Big{\{}[4 RCk\omega_{0}\sin(kD\pi)-2\cos(kD\pi)]\Big{(}e^{-\frac{t}{RC}}-\cos (2k\omega_{0}t)\Big{)}\Big{\}}\\ +\sum\limits_{k=1}^{+\infty}\frac{A\sin(kD\pi)}{k\pi\big{[}1+(2 RCk\omega_{0})^ {2}\big{]}}[4 RCk\omega_{0}\cos(kD\pi)+2\sin(kD\pi)]\sin(2k\omega_{0}t)\end{array}  \Bigg{]}u(t) \tag{33}\]  
Take a finite series \(N=100\) and set the cutoff frequency of the average circuit as \(f_{C}=1/(2\pi RC)=1\) Hz. The three output waveforms are depicted through Equations (31)-(33), as shown in Figure 9.  
All three Equations (31)-(33) are composed of two parts. One is the first-order step response with a time constant of \(RC\) and with amplitude of the rectified mean value. Another is the ripple composed of multi-toned signals. Figure 9 coincides with the above description. In order to ensure that the comparison circuits in Figures 2 and 4 do not repeatedly switch due to a ripple around the comparison threshold, a hysteresis comparison circuit must be used. When the peak to peak value of the ripple \(V_{RPP}\) is less than the hysteresis voltage, the repeated switching issue caused by the ripple can be eliminated. Let the ripple amplitude be \(V_{RA}=0.5V_{RPP}\). In order to evaluate the ripple amplitude, Equation (31) can be transformed into:  
\[y(t)=\left\{\frac{2A}{\pi}+\sum_{k=-\infty}^{+\infty}\frac{-2 Ae^{-\frac{\pi k}{R} }}{(-4\pi k^{2}+\pi)\Big{[}1+(2 RCk\omega_{0})^{2}\Big{]}}+\sum_{k=1}^{+\infty} \frac{4A}{(-4\pi k^{2}+\pi)\sqrt{1+(2 RCk\omega_{0})^{2}}}\sin(2k\omega_{0}t+ \varphi_{k})\right\}u(t) \tag{34}\]  
\[\text{while }\varphi_{k}=\arctan\Big{(}\frac{1}{2 RCk\omega_{0}}\Big{)}.\]  
The output waveform is divided into three parts in Equation (34). The first one is DC signal \(\frac{2\pi}{A}\), which does not change with time. The second part contains the exponential decay function \(e^{-\frac{t}{R}}\), which approaches 0 as time goes on. The third part is composed of multi-toned sine waves, which is the stable contributor of ripple. So the estimation of ripple amplitude is based on the third part. The cutoff angular frequency of the first-order filter is:  
\[\omega_{c}=\frac{1}{RC} \tag{35}\]  
Figure 9: The output waveforms for input as: (**a**) Sine. (**b**) Triangular. (**c**) Periodic pulse. Define the normalized frequency \(\Omega\) as:  
\[\Omega=\frac{\omega_{0}}{\omega_{c}} \tag{36}\]  
The third part of Equation (34) can be transformed into:  
\[