## 5 Experiment and Analysis
### Problem and Countermeasures  
During the optimization of the sensor's data output rate, researchers identified that the limiting factor is not the closed-loop control algorithms, but rather the open-loop measuring circuit of the sensor.  
In the closed-loop system, the MCU starts capturing the open-loop output \(V_{open}\) and then executes the PID-based closed-loop algorithms to output the control voltage \(V_{DAC}\), which takes a time interval referred to as \(t_{p}\). Then, the voltage \(V_{DAC}\) controls the feedback current drive circuit to generate an appropriate feedback current, which acts on an open-loop measuring circuit until the corresponding open-loop output occurs. The time required for this process can be called the control response time \(t_{c}\). A crucial precondition for the proper operation of the PID controller is that the outcome of the previous control cycle has been incorporated into the PID controller's calculations before initiating the next control cycle. Therefore, the prerequisite for ideal closed-loop control is as follows:  
\[\mathbf{t_{p}}\geq\mathbf{t_{c}}. \tag{22}\]  
Since data output occurs at each program loop, the response time of the closed-loop sensor is nearly equal to the program's single run time, i.e., \(\mathbf{t_{p}}\). Equation (22) shows that the response time of the closed-loop sensor is limited by the control response time \(\mathbf{t_{c}}\). In the developed closed-loop fluxgate current sensor, \(\mathbf{t_{c}}\) is determined by the response times of both the open-loop sensing circuit and the feedback current drive circuit. The open-loop sensing circuit used in this paper is based on the circuit of the open-loop fluxgate  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Input (A)** & \(\mathbf{V_{DAC}}\) **(V)** & **Theor. Value (V)** & **Diff (V)** & **Non-Linearity** \\ \hline
0.012 & 0.00033776 & 0.002367913 & 0.002030153 & 0.06\% \\ \hline
1.98 & 0.6365235 & 0.634203051 & \(-\)0.002320449 & \(-\)0.07\% \\ \hline
3.966 & 1.273239 & 1.271817169 & \(-\)0.001421831 & \(-\)0.04\% \\ \hline
5.961 & 1.91007 & 1.912320777 & 0.002250777 & 0.07\% \\ \hline
7.938 & 2.54812 & 2.547045405 & \(-\)0.001074595 & \(-\)0.03\% \\ \hline
9.931 & 3.186371 & 3.186906904 & 0.000535904 & 0.02\% \\ \hline \end{tabular}
\end{table}
Table 4: Measuring results under the input DC current range of 0-10 A.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Input (A)** & \(\mathbf{V_{DAC}}\) **(V)** & **Theor. Value (V)** & **Diff (V)** & **Non-Linearity** \\ \hline
0.001 & 0.0008649 & 1.8965 \(\times\) 10\({}^{-5}\) & \(-\)0.000845935 & \(-\)0.03\% \\ \hline
0.197 & 0.6369142 & 0.637992105 & 0.001077905 & 0.03\% \\ \hline
0.394 & 1.280448 & 1.27922021 & \(-\)0.00122779 & \(-\)0.04\% \\ \hline
0.592 & 1.921707 & 1.92370328 & 0.00199628 & 0.06\% \\ \hline
0.789 & 2.56533 & 2.564931385 & \(-\)0.000398615 & \(-\)0.01\% \\ \hline
0.987 & 3.210018 & 3.209414455 & \(-\)0.000603545 & \(-\)0.02\% \\ \hline \end{tabular}
\end{table}
Table 3: Measuring results under the input DC current range of 0-1 A.  
current sensor CYCT03 series, which is developed by ChenYang Technologies GmbH & Co. KG and has a response time of about 120 ms [40]. As a result, the actual effective response time of the developed closed-loop sensor is limited to at least 120 ms, although the minimal response time from the experiments can be 10 ms. To respond to this issue, increasing the response speed of the open-loop measuring circuit is necessary, e.g., by increasing the excitation voltage frequency and increasing the integration speed over the excitation current.