Impedance-measurement parameters Bandwidth

We now check whether Eq. (1), with S /3 = e2/ and modified as above to account for finite propagation time, can explain our data. The unknown parameters are the resistance Rq and the effective environment noise temperature Tq. We checked that the impedance of the samples was frequency independent up to 1.2 GHz within 5%. Fig. 2 shows the best fits to the theory, Eq. (1), for all our data. The four curves lead to Ro = 42 12, in agreement with the fact that the electromagnetic environment (amplifier, bias tee, coaxial cable, sample holder) was identical for the two samples. We have measured the impedance Zenv seen by the sample. Due to impedance mismatch between the amplifier and the cable, there are standing waves along the cable. This causes Zenv to be complex with a phase that varies with frequency. We measured that the modulus Zenv varies between 30 12 and 70 12 within the detection bandwidth, in reasonable agreement with f o = 42 12 extracted from the fits. 

The motional resistance, Ri (Sect. 6), is also used as a measure of dissipation. i i is an output parameter of some instriunents based on advanced oscillator circuits. However, experiments based on impedance analysis show that Ri usually is not strictly proportional to the bandwidth (although it should be, according to the Butterworth-van Dyke (BvD) circuit. Appendix A). Also, in absolute terms, Ry—being an electrical quantity and not a frequency— is affected by calibration problems much more than the bandwidth. In the author s opinion, P or D are better measures of dissipation than Ri. 

The functional distance measures considered are defined for functions on the same domain, the interval [S, If impedance spectra are measured on different domains, the distance can only be calculated for the overlapping area of the frequency interval. The distribution of the frequencies within the overlapping interval is of minor interest. If the spectra are measured on the same frequency interval but on different frequency grids they can be transformed to functional form, as mentioned above, by using smoothing or interpolation methods. The bandwidth is not a parameter of the distance measure, it is rather a fixed value which has to be stated together with the distance measure applied. In the following it is assumed that all impedance spectra are measured on the same bandwidth. 

Antennas can find use in systems that require narrow or large bandwidths depending on the intended application. Bandwidth is a measure of the frequency range over which a parameter, such as impedance, remains within a given tolerance. Dipoles, for example, by their nature are very narrow band. 

Figure 8.1 Sketch of QCM. The figure on the left is not to scale. The crystal thickness is around 300 jim. The sample, on the other hand, typically has a thickness of well below a micron. Right Conductance spectrum as obtained in impedance analysis. These measurements may be carried out on different harmonics. The ring-down technique (QCM-D] yields equivalent parameters," where the "dissipation" is given as D = 2T/f. Resonance frequency (/) and resonance bandwidth (F) are derived by fitting resonance curves to the experimental conductance spectra. The presence of the sample changes both / and F. in the modeling process one tries to reproduce the experimental values of A/ and AF.

The calculated impedance for the improved model is compared to the measured impedance in Fig. 1 lb. The DE used is the same as compared to the SLS model before. The improved model using the fiuctional element fits quite well in the complete analyzable frequency range. The analysis of the applicability of the firactional model for a larger bandwidth will be a focus of future work. The calculated parameters for the improved model are summarized in Table 3. 

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