# Portable and Highly Versatile Impedance Meter for Very Low Frequency Measurements
## 2 Materials and Methods
### Digital Signal Elaboration Software
phenomenon is normally referred to as \"spectral leakage\". If a sufficiently narrow NEB is used (for example 11 mHz or 2.9 mHz), we can assume that the maximum of the spectrum at \(f_{\text{C}}-f_{LF}\) is not significantly modified by the presence of the sinusoid at \(f_{\text{C}}+f_{LF}\) with respect to the case in which only a sinusoid at \(f_{\text{C}}-f_{LF}\) was present, and the same is true for the peak value at \(f_{\text{C}}+f_{FL}\). On the other hand, if the NEB becomes larger (for instance 46 mHz as in Figure 4a), the effect of the spectral leakage is significant and the extraction of the amplitude of the sinusoids contributing to the overall spectrum becomes challenging because of the reciprocal influence among them. The situation becomes much more severe in the case in which a third sinusoid is present with a much larger amplitude. This is indeed quite a common occurrence in our application. Indeed, we typically would like to extract the impedance of the DUT using a small AC excitation in the presence of non-zero DC voltage across the DUT. In our system, a large DC component across the DUT translates into a large sinusoidal component at the carrier frequency. The resulting spectra as estimated by QLSA in the very same conditions as before, but with a 100 mV DC component across the DUT, are reported in Figure 4b. As it is apparent, the large leakage from the large sinusoidal component at \(f_{\text{C}}\) completely masks the presence of the frequency components at \(f_{\text{C}}-f_{LF}\) and \(f_{\text{C}}+f_{LF}\) for an NEB of 46 mHz and it is clear that we need to operate with NEBs much smaller than in the case of Figure 4a if we have to make the influence from the leakage of the sinusoid at \(f_{\text{C}}\) negligible as far as the estimation of the components at \(f_{\text{C}}-f_{LF}\) and \(f_{\text{C}}+f_{LF}\) are concerned. In principle, using QLSA and setting a proper value of \(\text{p}_{\text{i}}\) in Equation (19) we can obtain an NEB as narrow as necessary, but this might result in extremely long measurement times. The problem is especially relevant for the lowest frequencies of interest, where measurement time is already intrinsically long. As an alternative, one could numerically subtract the components at \(f_{\text{C}}\) in both \(v_{IL}\) and \(v_{IR}\) in Figure 3 before sending the data stream to QLSA. This clearly requires exact knowledge of the amplitude and phase of the sinusoidal signal to be subtracted, which can be obtained as part of the measurement configuration sequence: any time a new DC value is set across the DUT, the system is operated with no AC component so that proper coefficients can be adjusted in the carrier subtraction block in Figure 3.  
When spectra (and cross spectra) with well-defined peaks at \(f_{\text{C}}-f_{LF}\) and \(f_{\text{C}}+f_{LF}\) are obtained, we can extract the value of the impedance of the DUT according to the following procedure. The relationship between signals and DUT impedance summarized by Equation (16) can be represented by the signal chain reported in Figure 5. As long as  
Figure 4: Spectra estimated by QLSA in NB mode when synthetic data are used for creating the signal \(V_{OL}\) in Equation (2) with different settings for ENB. In (**a**) the DC level of the test signal across the DUT is set to 0, while in (**b**) we set a DC level of 100 mV across the DUT. the frequency response H is such that at frequency \(f_{\text{C}}+f_{\text{LF}}\) we have:  
\[H(f_{\text{C}}+f_{\text{LF}})=A_{RLF}Y_{DLF}e^{i(\varphi_{RLF}+\varphi_{DLF})} \tag{20}\]  
the relationship between \(\
u_{\text{IL3}}\) and \(\
u_{\text{IR3}}\) obtained from Figure 5 is the same as the one obtained in Equations (13) and (14).