## Enhanced Single-Probe Method and Verification  
### Enhanced Single-Probe Method  
In this section, an enhanced single-probe setup to measure the unknown impedance (\(Z_{x}\)), as shown in Figure 1a, is presented. In the proposed setup, the bulk current injection (BCI) probe is connected to one port of the vector network analyzer (VNA) through a coaxial cable. A short wire connecting the unknown impedance is wound on the core of the BCI probe forming multiple turns. In each of these turns, the wire enters the probe inner window from a side, crosses the probe leaving the opposite side, and runs back to the entering side. The turn number of this looped wire is called \(N_{turns}\). Inside the BCI probe, there is a feeding winding whose number of turns is \(N_{probe}\) (this number is out of the operator's control since it is related to the probe manufacturer and model). It is worth noting that it is possible to use not only BCI probes but also current monitor probes, as they are basically the same instrument, and many models are declared for both purposes by manufacturers. However, their design is usually optimized for one of the two scopes. Monitor probes have a high number of turns \(N_{probe}\) in order to reduce the loading impedance (\(50/N_{probe}^{2}\)\(\Omega\)) on the clamped cable, which would perturb the circuit under test. Conversely, BCI probes usually have a very low \(N_{probe}\) to be effective in inducing a longitudinal voltage source onto the clamped cable. The main improvements of this setup over previous solutions come in two folds. First, the signal amplification and protection module in [25], which comprises extra radio-frequency components (including a directional coupler, a signal amplifier, and surge protectors) to allow a gain/phase meter to operate as VNA, is eliminated, and substituted by a real VNA with low-frequency extension capabilities (Keysight, E5061B, 5 Hz--1.5 GHz). As a result, the lowest frequency can be extended to 10 kHz instead of 150 kHz (directional couplers and other radio-frequency devices for use below hundreds of kHz are often commercially unavailable or exhibit limited performance parameters). Second, in order to improve the sensitivity of this setup in a simplified approach, i.e., to enhance coupling effects and signal-to-noise ratio (SNR), we propose the use of multiple turns \(N_{turns}\) instead of only one turn [21], and to specifically make the number of wire turns \(N_{turns}\) exactly equal to the turns \(N_{probe}\) of the feeding winding inside the BCI probe. The sensitivity analysis in Section 2.2 will support this choice. Figure 1: (a) Enhanced single-probe setup and (b) its corresponding equivalent circuit. The instrument calibration is carried out at the probe port (_a-a'_). Thus, the input impedance (\(Z_{m}\)) seen at _a-a'_ can be extracted from the measured scattering parameter (\(S_{11}\)) by [28]:  
\[Z_{m}=Z_{0}\frac{1+S_{11}}{1-S_{11}} \tag{1}\]  
where \(Z_{0}\) is the internal reference impedance of the VNA. The equivalent circuit of the setup is shown in Figure 1b, where the transmission (ABCD) matrix \(\mathbf{T}\) of the two-port network (port _a-a'_ and _b-b'_) accounts for the BCI probe and the two-turn loop wire and its parasitic components:  
\[\mathbf{T}= \begin{bmatrix}A&B\\ C&D\end{bmatrix}  \tag{2}\]  
Thus, the unknown impedance \(Z_{x}\) can be computed from [25]:  
\[Z_{x}=-\frac{Z_{m}D-B}{Z_{m}C-A} \tag{3}\]  
Substituting (1) into (3) and further simplifying, \(Z_{x}\) can be expressed in terms of \(S_{11}\) and three coefficients (\(k_{1}\), \(k_{2}\), and \(k_{3}\)) [25]:  
\[Z_{x}=\frac{k_{1}S_{11}+k_{2}}{S_{11}+k_{3}} \tag{4}\]  
where  
\[k_{1}=-\frac{Z_{0}D+B}{Z_{0}C+A} \tag{5}\]  
\[k_{2}=-\frac{Z_{0}D-B}{Z_{0}C+A} \tag{6}\]  
\[k_{3}=\frac{Z_{0}C-A}{Z_{0}C+A} \tag{7}\]  
The three coefficients depend on the internal impedance of the VNA (\(Z_{0}\)) and the ABCD parameters of the two-port network, which is determined by the calibration setup (the BCI probe, the multiple-turn wire, and the parasitic components).