## Enhanced Single-Probe Method and Verification
### Calibration Procedure and Validation of the Proposed Method  
Purpose of the calibration procedure is to directly measure the three coefficients (\(k_{1}\), \(k_{2}\), and \(k_{3}\)) needed in (4), in order to be able to correctly measure the impedance of the circuit under test with no need for an explicit probe model [25]. This allows users to avoid the determination of an accurate model of the BCI probe as a two-port network to be used for computing A, B, C, and D. Additionally, the calibration procedure can be repeated with any BCI probe with no need for a specific setup. Calibration is performed from 10 kHz to 30 MHz. The calibration setup shown in Figure 3a consists of a short loop wire with two turns. The calibration procedure requires measuring the \(S_{11}\) parameters of three known resistors (nominal values 1.1 \(\Omega\), 50 \(\Omega\) and 1 k\(\Omega\)), whose actual impedances (see Figure 3b) are determined with an impedance analyzer by reflectometric measure. The VNA is configured to measure 1601 points from 10 kHz up to 30 MHz, with 100 Hz resolution bandwidth and 8 dBm transmitted power. The same configurations have been used both for the calibration discussed in this section and for the tests reported in the following sections. The three coefficients can be calculated by this equation system derived from (4):  
\[ \begin{cases}Z_{1.1\Omega}=\frac{k_{1}S_{110}+k_{2}}{S_{110}+k_{3}}\\ Z_{50\Omega}=\frac{k_{1}S_{500}+k_{2}}{S_{500}+k_{3}}\\ Z_{1\Omega}=\frac{k_{1}S_{510}+k_{2}}{S_{110}+k_{3}}\end{cases}  \tag{13}\]  
where \(Z_{1.1\Omega}\), \(Z_{50\Omega}\), and \(Z_{1\Omega}\) are actual impedances of three resistors measured by an impedance analyzer, and \(S_{1.1\Omega}\), \(S_{50\Omega}\), and \(S_{1\Omega\Omega}\) are the measured \(S_{11}\) parameters in the calibration setup (see Figure 3a). The amplitude and phase of \(k_{1}\), \(k_{2}\), and \(k_{3}\) obtained from the calibration procedure are plotted in Figure 4a, and are consistent with the ideal theoretical expectations (10) in the in-band region (1-5 MHz). Specifically, since the turn ratio \(k\) is equal to 1 in our setup, the coefficients become \(k_{1}=k_{2}=-Z_{0}=-50\)\(\Omega\) and \(k_{3}=-1\). Outside that narrow in-band region, the coefficients exhibit a non-trivial frequency response, which is determined by all the possible complex phenomena (e.g., lossy and dispersive ferrite core, inductive and capacitive parasitic effects) associated with the specific BCI probe model. Several discrete components have been used in the calibrated setup, not only by resistors (like the three calibration loads) but also by capacitors and inductors. For instance, some discrete RLC components are reported in Figure 4b. Their impedances have been  
Figure 4: (**a**) Calibrated three coefficients and (**b**) verification with discrete RLC components. Figure 3: (**a**) Calibration setup and (**b**) measurements of the magnitude and phase of the three resistors used for calibration (obtained by reflectometric measure with impedance analyzer). measured both through the proposed methodology using the calibrated coefficients \(k_{1}\), \(k_{2}\), and \(k_{3}\), and by an impedance analyzer. The minor discrepancies of the 3.3 \(\mu\)F capacitor around its self-resonance frequency are expected due to the low sensitivity at low impedance, confirming the conclusions of sensitivity analysis. The deviations of the 100 \(\mu\)H inductor are likely due to the different shape of the impedance analyzer fixture (involving a small ground plane below the component under test) and the proposed method (without ground plane), which has an impact on capacitive parasitic effects. The magnitude relative error calculated from the data in Figure 4b is reported in Figure 5a, while the phase error calculated from the data in Figure 4b is reported in Figure 5b.