Complex plane plots interpretation

An impedance response can be interpreted graphically as a vector on the complex plane. The imaginary axis is the out-of-phase response (Z"), and the real axis is the in-phase response (Z ). The magnitude of the impedance response Z is the length of the vector, and the phase angle (]) describes its direction (Fig. 3). Each point on the plane defines an impedance response at a particular frequency. Such representations are commonly referred to as complex plane plots, Nyquist diagrams, or Cole-Cole plots. However, the Cole-Cole plot is actually the complex plane representation of the dielectric response of a material. 

De Levie [1963,1964] was the first, it seems, to examine the frequency response of a wire-bmsh electrode of the above kind and provide the background mathematical treatment required to interpret the experimental behavior. The important aspect of the observed impedance spectroscopy over a wide range of frequencies was that transmission-line (Figure 4.5.22e) behavior resulted, as illustrated in Figure 4.5.30. The characteristic behavior is a linear complex-plane plot having a 45° phase-angle... 

Figure 12.27 shows a schematic IMPS respouse for a mouocrystalhue semiconductor electrode. It is easy to confuse IMPS plots with EIS plots since both contain semicircles. However the quantity displayed in an IMPS plot is not an impedance or an admittance it is the dimensionless transfer function corresponding to the ratio of the elecuon flux to the photon flux. The interpretation of the phase relationship between photocurrent and illumination requires some care. A response in the first upper quadrant indicates that the photocurrent leads the illumination. A response in the lower quadrant indicates that the photocurrent lags behind the illumination. In the schematic response shown in Pig. 12.27, the low-frequency response in the upper quadrant of the complex plane arises from surface... 

In general, a driving force, which may be optical or electrical is applied and the phase and amplitude of the response, which may be a current, voltage, or optical density, is measured as a function of modulation frequency. For compactness, the results are often presented in the complex plane, as the locus of the complex response as frequency is varied, although phase and amplitude data plotted against frequency may be easier to interpret. For a simple system with a single time constant, r, the response is a semicircle in the complex plane, and it is easy to show that the frequency at the point where the imaginary component of the response is a minimum, time constants in the time domain. 

The following problem arises in the interpretation of such semicircles in the complex plane impedance plots every parallel combination of a constant resistance and constant capacity leads to a semicircle in the Nyquist plot of the impedance. To verify a charge transfer, for instance, the potential dependence of the charge-transfer resistance should be investigated to demonstrate the Butler-Volmer potential dependence of the exchange current. 

The first system called LiSSA has been developed for interpretation of data from eddy-current inspection of heat exchangers. The data that has to be interpreted consists of a complex impedance signal which can be absolute and/or differential and may be acquired in several frequencies. The interpretation of data is done on the basis of the plot of the signal in the impedance plane the type of defect and/or construction is inferred from the signal shape, the depth from the phase, and the volume is roughly proportional to the signal amplitude. 

Thus, in the complex (e, e")-plane, the locus of e((tx) is a semicircle which intersects the real axis (e ) at Cg and its centre being on the real axis at (Cg+e, ). This type of dielectric plot, introduced by Cole Cole (1941), is, as we shall see, of great help in interpreting experimental measurements. It also has a simple variant, for the case where there is a spread of relaxation times in the specimen, the centre of the circular plot being displaced slightly below the real axis. 

Bozler and Cole (1935) measured the electrical impedance of frog sartorius muscle from 1.1 kHz to 1.1 MHz. Measurements were first done approximately 2 h after dissection. The tissue was then stimulated to induce contraction, and the tissue was measured again, approximately 3 h after dissection. They found a minor arc of a circle when the data were plotted in the complex impedance plane. Between the relaxed and contracted state, Rq was found to increase by 75% whereas Roo increased only 2%. As usual, Rq and Ra> denote the resistances measured at very low and very high frequency, respectively. The significant increase in Rq was interpreted as a reduced ionic conduction through the cell membranes. 

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