Automated Frequency Response Analysis

In general, direct methods can be used to acquire impedance data significantly more rapidly than bridge methods. This is particularly true for digitally demodulated, phase-sensitive detectors, for which only a single cycle is required. Nevertheless, in unstable systems, such as rapidly corroding specimens, acquisition rate is an important consideration, and a major criticism of PSD methods is that these must be performed frequency by frequency. Fortunately, this often is not a serious hindrance when such equipment is automated. In the past decade, a number of experimenters have used automated frequency response analyzers as digitally demodulated, stepped-frequency impedance meters. Typical of this class are the Solartron 1170 and 1250 series frequency response analyzers (FRAs). 

The real and imaginary components of the impedance are given by the integrals 

If the noise is completely random (i.e. asynchronous), then the last integrals in Eqs (28) and (29) are equal to zero provided that they are carried out over infinite time. If the integration is carried out over Nf periods of the sinusoidal perturbation, the equivalent filter selectively is given by (Gabrielli [1981]) 

Then the integrals involving the harmonics in Eqs (31) and (32) are identically equal to zero provided that the integrals are carried out over multiples of 2n. Accordingly, FRAs effectively reject the harmonics. Application of the above identities to the fundamental components in Eqs (28) and (29) therefore yields the real and imaginary outputs from the integrators as 

FRAs are also readily used to determine the harmonics contained within the output from the cell. This is done by multiplying the reference signal to the multipliers (but not to the cell) by the harmonic coefficient (2 for the second harmonic, 3 for the third, and so forth). The ability of FRAs to characterize the harmonics provides a powerful tool for investigating nonlinear systems a topic that is now being actively developed (McKubre [1983]). 

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Analysis of Lissajous figures on oscilloscope screens was the accepted method of impedance measurement prior to the availability of lock-in amplifiers and frequency response analyzers. Modern equipment allows automation in applying the voltage input with variable frequencies and collecting the output impedance (and current) responses as the frequency is scanned from very high (MHz-GHz) values where timescale of the signal is in micro- and nanoseconds to very low frequencies (pHz) with timescales of the order of hours. 

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