## Enhanced Single-Probe Method and Verification
### Sensitivity Analysis for Justification of the Two-Turn Loop Setup  
The derivative-based local sensitivity analysis is proposed to study how sensitive the obtained impedance \(Z_{x}\) is to the measured \(S_{11}\), keeping other parameters (\(k_{1}\), \(k_{2}\), and \(k_{3}\)) fixed. Under the assumption that \(S_{11}\) is a real number (i.e., resistive load), the sensitivity coefficient \(S_{S_{11}}^{Z_{x}}\) is obtained by differentiating (4):  
\[S_{S_{11}}^{Z_{  
Such a probe model represents a large approximation, valid only in an intermediate frequency range (in-band region) of the whole working band declared by probe manufacturers, after the inductive behavior of the low-frequency region and before the very high-frequency region where magnetic-flux leakage and parasitic effects (e.g., stray capacitances) play a fundamental role. Anyway, this basic model is sufficient to investigate and justify the choice of \(N_{turns}\). The ABCD parameters between ports \(a\)-\(a^{\prime}\) and \(b\)-\(b^{\prime}\) can be obtained according to the characteristics of the ideal transformer:  
\[\mathbf{T}= \begin{bmatrix}A&B\\ C&D\end{bmatrix} = \begin{bmatrix}k&0\\ 0&1/k\end{bmatrix}  \tag{9}\]  
where the \(k=N_{probe}/N_{turns}\) is the turn ratio. Substituting the ABCD parameters into (5)-(7), the three coefficients can be computed as:  
\[ \begin{cases}k_{1}=k_{2}=-\frac{Z_{0}}{k^{2}}\\ k_{3}=-1\end{cases}  \tag{10}\]  
In this general scenario, the sensitivity coefficient is associated with the turn ratio of the ideal transformer (\(k\)), the measured scattering parameters (\(S_{11}\)), and the reference impedance of the VNA (\(Z_{0}\)):  
\[S_{S_{11}}^{Z_{x}}=\frac{k_{1}k_{3}-k_{2}}{\left(S_{11}+k_{3}\right)^{2}}= \frac{2Z_{0}}{k^{2}{\left(S_{11}-1\right)}^{2}} \tag{11}\]  
The sensitivity of the setup needs to be increased for high accuracy. To this end, we investigate two extreme cases when the port \(b\)-\(b^{\prime}\) is an open or short circuit. For the open circuit condition (\(S_{11}=1\)), the sensitivity coefficient is singular and reaches infinity, indicating that the single-probe setup can provide high accuracy in measuring high impedances. On the contrary, the short circuit condition (\(S_{11}=-1\)) is the worst case, exhibiting the minimum sensitivity as:  
\[\underset{S_{11}}{\text{min}}S_{S_{11}}^{Z_{x}}=\frac{Z_{0}}{2k^{2}} \tag{12}\]  
Thus, the accuracy of the single-probe setup degrades when measuring low impedances. The minimum sensitivity coefficient in (12) should be kept as high as possible to improve the measurement accuracy for low impedances, i.e., the ideal transformer ratio \(k\) should stay small. This can be achieved in two ways, increasing the turn number \(N_{turns}\) of the loop wire or choosing probes with a small \(N_{probe}\). For the latter, commercial BCI probes are already designed with a small number \(N_{probe}\) (usually one to three) to maximize coupling efficiency in BCI testing. In this respect, the category of BCI probes is best suited for the proposed impedance-measured method compared to other inductive probes, like current-monitor probes, which usually involve many primary turns. Anyway, after a specific BCI probe is  
Figure 2: Equivalent circuit involving an ideal transformer to model the BCI probe. selected (this choice is based on the working band), this number \(N_{probe}\) is fixed and out of the control of the test operator. In terms of selecting the turn number \(N_{turns}\) of the loop wire, there is a trade-off between many factors. On the one hand, increasing this number is helpful in improving the worst-case sensitivity (12). On the other hand, this choice can introduce two critical issues.