Real and Imaginary Impedance

To summarize the impedance discussion so far an electrochemical cell is constructed, and its impedance Z determined as a function of frequency. From these impedance values, the real and imaginary impedances, Z and Z", respectively, are computed and hence a Nyquist plot is drawn. 

Equations (19.24) to (19.29) can be applied for complex noiUinear least-squares regression by concatenating the real and imaginary impedance data Z,- to form a data vector with length equal to twice the number of measured frequencies. A similar concatenation applies for the model values Z(Ti P). Press et al. provide a very approachable discussion of the least-squares methods and their implementation. 

In the absence of bias errors, the errors in the real and imaginary impedance are uncorrelated and the variances of the real and imaginary parts of the complex impedance are equal. Some specific identities are given in Table 21.1. 

Figure 13. Real and imaginary impedance as a function of the concentration of BSA in aqueous phase at indicated frequencies. Water 0.01 mol/L LiCl nitrobenzene 0.01 mol/L TBATPB. Applied interfacial potential U = 0.35 V vs. TBA + ion selective electrode t — 25 °C. (Reproduced with permission from reference 32. Copyright 1990 Elsevier.)...

The real and imaginary impedances ZR and Zl are directly accessible from the impedance measurement. Rs, the solution resistance, has to be obtained from examination of the complex plane impedance plot (cf., Figure 9). In the impedance plane plot, the imaginary value decreases with increasing frequency until the curve approaches the real impedance axis. The real impedance is equal at this point to the solution resistance Rs. The value is independent of applied interfacial potential, but it depends on the position of the reference electrodes (different uncompensated resistance). Because the calculated capacitance is very sensitive to the calculated Rs, the placement of the reference electrodes must be carefully controlled. 

The impedance of the equivalent circuit in Figure 4.10a is obtained by the reciprocal combination of Zg and Z. . The result is linear (serial) combined with Z, = This leads to the following real and imaginary impedance terms, Z and respectively... 

The relationship between the real and imaginary impedance of eireuit that eon-sists of elements sueh as resistors and eapaeitors can be represented by a Nyquist plot. If the impedanee of a circuit exhibits a Nyquist plot that is similar to that exhibited by a polymer electrolyte, then this circuit model can be used to represent the electrolyte and is known as the equivalent circuit for the electrolyte. The impedance data is commonly analyzed by fitting them to the impedance equation of the equivalent circuit. Now, we will diseuss the selected eireuit elements together with their corresponding Nyquist plots. 

If impedance measurements are carried out at the same sensitivity scale for the real and imaginary components, the stochastic errors of the real and imaginary impedances will be similar, and one can use modulus weighting. Modulus weighting assumes the same statistical weights for real and imaginary parts, and they are proportional to the impedance modulus. This means that small and large impedances contribute in a similar way to the sum of squares and are equally important. 

After determining the above values, from (D.26), (D.27) and (D.28), we can calculate both R and C using the real and imaginary impedance values measured in the empty and filled cells, as+... 

The circuit analysis results in the expressions for total, real, and imaginary impedance of the circuit as ... 

Proton conductivity of composites, obtained by this method, was measured by impedance spectrometry over frequency range of 1010 Hz. In Figure 6 one can see the dependence of real and imaginary impedance constituents on current frequency as well as Nyquist plot for the sample PC SGS = 50 50 (% v.) at catalyst concentration in SGS of 20% V. 

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