# Portable and Highly Versatile Impedance Meter for Very Low Frequency Measurements
## 3 Results and Discussion  
### Preliminary System Testing  
Measurements were performed with the system in Figure 2 using the \"opt 1\" configuration option, that is, with \(\
u_{VD}=\
u_{DUT}\). The USB audio interface Behringer UMC202 HD was used as the audio board configured for 24 bit and 48 kHz sampling frequency. All measurements were performed with \(R_{R}=1\) k\(\Omega\) an \(C_{R}=680\) nF. As a first experiment, we performed measurements on the DUT in Figure 6a in the frequency range between 10 mHz and 100 Hz. In this experiment, the value of the DC bias voltage was set to 0 and the AC amplitude was set to 10 mV. It is worth noting that with this setting, at the lowest explored frequency (10 mHz), the amplitude of the AC voltage at the output of the TIA is about 100 uV. Since the input dynamic range of \(M_{3}\) in Figure 2 is \(\pm 1\) V, we could have selected different values of \(R_{R}\) and \(C_{R}\) in order to obtain a larger \(\mid A_{R}\mid\) and, hence, better coverage of the input dynamic range. The measurement condition we have selected, on the other hand, allows us to demonstrate that quite accurate results can be obtained also in such an unfavorable condition. QLSA was set to operate with a record length \(N_{R}=2^{15}\) (32,768) and 11 geometrically decreasing ENBs were selected with:  
\[ENB_{j}=\frac{f_{\text{C}K}}{2/N_{R}};\ 1<j<11 \tag{22}\]  
Figure 5: Signal chain equivalent of the relationship between signals \(\
u_{IL3}\) and \(\
u_{IR3}\) as obtained in Equation (16). The largest _ENB_ is therefore about 732 mHz, while the narrowest ENB is as low at 715 \(\mu\)Hz. Different choices are possible, also on a per-measurement basis, and the cost of adding more _ENB_ values is very small in terms of computational load. In principle, one should employ the narrowest ENB in any situation in order to improve the signal-to-noise ratio and reduce leakage effects. However, reducing the ENB results in increasing the measurement time, and therefore, one needs to find a compromise. Moreover, spectra and cross spectra typically need to be averaged a few times in order to reduce statistical errors, and this also contributes to the overall measurement time. When QLSA operates in NB mode, the wait time \(T_{W}(M)\) after which spectra with \(M\) averages corresponding to a given ENB are available is given by:  
\[T_{W}(M)=\frac{\alpha+(M-1)\ \beta}{EBN}\ ;\ \alpha\approx 1.165\,\ \beta\approx 0.35 \tag{23}\]  
When a deterministic sinusoidal signal is present, as in the situations in Figure 4, only a small number of averages are usually required. To provide a better sense of the measurement time required in each case, the wait times for the first estimate (\(T_{W}(1)\)) and the wait time after four averages (\(T_{W}(4)\)) vs. the ENBs in Equation (22) are reported in Figure 7.  
Figure 6: Impedances used for testing the proposed approach. Capacitor \(C_{Pa}\) in (**a**) is a 10 \(\mu\)F, 100 V polyester capacitor; capacitor \(C_{Pb}\) in (**b**) is obtained as the parallel of 20 10 \(\mu\)F, 100 V polyester capacitors. Figure 7: Measurement time for obtaining the first spectra estimation (\(T_{W}(1)\)) and for reaching four averages (\(T_{W}(4)\)) when using QLSA in NB mode vs. the selected ENB. With reference to Figure 4, it can be estimated that in the presence of a negligible DC component, in order to avoid errors due to spectral leakage, we should employ an _ENB_ that is at least 1/10 of the frequency separation between the peaks to be estimated (at \(f_{\text{C}}-f_{LF}\) and \(f_{\text{C}}+f_{LF}\)), that is we must work with _ENB_\(<\)\(f_{LF}/5\). Let us now assume that we want to perform measurements at 10 mHz. According to the criterion above, we should refer to _ENB_\({}_{10}\) = 1.43 mHz among the selected ENBs in Equation (22) to ensure negligible leakage errors. This choice, from Figure 7, would result in a measurement time, for obtaining four averages, of about 1500 s. While this may appear quite a long time, it is worth noticing that it corresponds to just 15 periods of the input signal.