# Current-Sensing Topology with Multi Resistors in Parallel and Its Protection Circuit
## 3 Results
### Time Delay Property of Protection Circuit  
In order to verify the performance of the protection circuit, an experimental platform was built as shown in Figure 15.  
The acquisition module was installed in the self-made power analyzer, as shown in Figure 15b. The current waveforms can be displayed on the screen of the power analyzer. A multifunctional calibrator model 5502A provided current input for the current-measurement channel. The output waveforms of rectified mean-detection circuits were captured through the oscilloscope model MDO3104. The delays of rectified mean-detection circuits in the flow chart of Figure 5 were not considered. When the input current has a step change from a value below the protection threshold \(I_{T2}\) to a value above, overcurrent protection is generated by closing \(S_{1}\). Due to the time delay property of rectified mean-detection circuits, the overcurrent state could not be latched if the time constants or hysteresis ratios were not properly designed. Figure 14: Hardware circuit. In order to prevent repetitive switching caused by AC ripple, hysteresis comparison circuits were used in Figure 13. The single supply voltage of the comparators was 4.8 V and the thresholds were set to half the supply voltage \(V_{T1}=V_{T2}=2.4\) V. Hence, the hysteresis ratios in both directions of each comparison circuit were the same:  
\[\delta_{H1}=\frac{V_{T1+}-V_{T1}}{V_{T1}}=\frac{V_{T1}-V_{T1-}}{V_{T1}}=\frac{R_ {I1}}{R_{F1}} \tag{47}\]  
where \(\delta_{H1}\) is the hysteresis ratio of the comparison circuit #1; \(V_{T1+}\) is the positive threshold voltage; \(V_{T1-}\) is the negative threshold voltage. Similarly, the hysteresis ratio of comparison circuit #2 is:  
\[\delta_{H2}=\frac{R_{I2}}{R_{F2}} \tag{48}\]  
Since all circuits from current sensing to comparator input are linear ones, the hysteresis ratios of current thresholds are the same as the voltage hysteresis ratios mentioned above. One can obtain the following:  
\[I_{T1+}=(1+\delta_{H1})I_{T1} \tag{49}\]  
\[I_{T2+}=(1+\delta_{H2})I_{T2} \tag{50}\]  
\[I_{T1-}=(1-\delta_{H1})I_{T1} \tag{51}\]  
\[I_{T2-}=(1-\delta_{H2})I_{T2} \tag{52}\]  
\(I_{T1+}\) and \(I_{T2+}\) are the current thresholds flowing through two sensing resistors, while protections are triggered and the outputs of comparators switch from low level to high level. Figure 15: Experimental platform: (**a**) Front view. (**b**) Rear view. According to Equation (7), the input current threshold while protection circuit 1 is triggered is:  
Figure 14:0  
When hysteresis was considered, the hiccup-free condition of the protection process in Figure 5 becomes \({I_{T1+}}^{\prime}<{I_{T2+}}\). Even if this condition is met, delay of the rectified mean circuit may still cause hiccups. Let time constants of the first-order filters for two rectified mean circuits be \(\tau_{1}\) and \(\tau_{2}\), respectively. If both comparator hysteresis and filter delay are considered and the input current \(I_{IN}\) is DC, the workflow of the two protection circuits in Figure 13 is as follows:  
1. Before the switch \(S_{1}\) is closed, the currents flowing through the sensing resistor \(R_{1}\) and \(R_{2}\) are 0 and \(I_{IN}\), respectively. 2. After the switch is closed, the currents rapidly change to \(I_{R1}\) and \(I_{R2}\), determined by Equations (4) and (5), respectively. Due to a delay of the first-order filter, the changes of output voltages of rectified mean circuits take times. Let the back-calculated currents corresponding to the output voltages be \({I_{R1}}^{\prime}\) and \({I_{R2}}^{\prime}\), respectively. They are determined by:  
Figure 14:1  
Figure 14:2
3. \({I_{R1}}^{\prime}\) gradually increases from 0 to \({I_{R1}}\) and \({I_{R2}}^{\prime}\) gradually decreases from \({I_{IN}}\) to \({I_{R2}}\). Make \({I_{R1}}^{\prime}(t_{1})={I_{T1+}}\) and \({I_{R2}}^{\prime}(t_{2})={I_{T2-}}\): At time \(t_{1}\), protection circuit 1 enters protection state; At time \(t_{2}\), protection circuit 2 exits protection state.