3.1 Model 1: parallel combinations R1C1 and R2C2 connected in series

Parallel combinations R1C1 and R2C2 connected in series [8] are shown in Figure 2(a) and the values for Z<sup>0</sup> and Z<sup>00</sup> are given in Eq.(5):

low-frequency side will give the value of R. The value of C can be obtained by noting the frequency where the arc peaks and using the relation ωCR =1.

, M<sup>00</sup>

<sup>þ</sup> <sup>M</sup>″

Eq. (4) indicates that M<sup>00</sup> vs. M<sup>0</sup> plot would be a semicircular arc passing through the origin and having intercept at M<sup>0</sup> = C0/C and centre at (C0/2C, 0) (Figure 1(c)). On this plot, M<sup>00</sup> would peak at frequency where ωCR = 1. Value of M<sup>0</sup> becomes equal to C0/C as ω ! ∞. Thus, by using M<sup>00</sup> vs. M<sup>0</sup> plot of the same experimental data, C can be obtained by noting the high-frequency intercept in the M<sup>00</sup> vs. M<sup>0</sup> plot. The value of R can be obtained by using this value of C and noting the

frequency where M<sup>00</sup> peaks. In the M<sup>00</sup> vs. M<sup>0</sup> plot, the arc traverses from left to right whereas in Z<sup>00</sup> vs. Z<sup>0</sup> plot, it traverses from right to left as the frequency is increased. Choice of a model to represent a system is a difficult process, becoming more so

In order to facilitate a prompt development of an equivalent circuit model for electronic ceramics, some very important clues have been given in [9] and are

1. The experimental data can be represented in any of the formalisms (Z, M, Y and ε) and interpreted. However, analysis by using only one formalism might lead to erroneous conclusions and two or more functions such as Z and M

since same behaviour can be simulated by different models [6]. The choice is greatly facilitated by comparing the experimental plots with those simulated for various models and considering the processes that might be present in the system with a preference to simple models to start with. When the complex plane plots display more than one arcs, presence of more polarisation/charge transfer processes is inferred. A general practice is to consider one RC circuit to represent one process and connect more RCs in series to develop an equivalent circuit model. Thus for representing grain-grain boundary-electrode system in a ceramic, a model comprising three parallel RCs connected in series might be taken [1]. It has been found that usually combinations of resistances (R) and capacitances (C) suffice for dielectrics, combinations of R and inductance L suffice for magnetic systems, and combinations of R, L and C suffice for ferro-/piezoelectrics [6–8, 10–20]. Sometimes, it is found that the lumped-component type of models do not yield good fits and their simulated patterns do not show even qualitative resemblance with the experimental plots. In those cases, attempt is made to represent the data by models involving constant phase angle elements (CPE) [6, 9] that are considered to correspond to some sort of distribution in the material properties. The values of the components used in the model are estimated by comparing the experimental plots with the simulated ones. These values are obtained more accurately by fitting the experimental data with complex non-linear least-square (CNLS) procedure using these values as initial guesses. For cases, where it is not possible to decide upon an appropriate model, equivalent circuits models that seem to be most probable to represent the processes thought to be possibly present, or to be dormant but becoming dominant as some variables change, in the system may be considered and the model that yields the lowest value of sum of squares of errors in the CNLS runs

¼ ωC0

<sup>2</sup> <sup>¼</sup> C0 2C <sup>2</sup>

R <sup>1</sup> <sup>þ</sup> <sup>ω</sup>2C<sup>2</sup>

R2

(3)

(4)

R2

ωCR<sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>ω</sup>2C<sup>2</sup>

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<sup>M</sup><sup>0</sup> � C0 2 C <sup>2</sup>

For this model, M<sup>0</sup> and M<sup>00</sup> are given as

M<sup>0</sup> ¼ ωC0

which satisfy the relation

may be accepted [6].

82

summarised below for a ready reference.

should be used [6, 8].

$$Z' = \frac{\mathbf{R}\_1}{\mathbf{1} + \left(\alpha \mathbf{C}\_1 \mathbf{R}\_1\right)^2} + \frac{\mathbf{R}\_2}{\mathbf{1} + \left(\alpha \mathbf{C}\_2 \mathbf{R}\_2\right)^2},\\ Z'' = \frac{\alpha \mathbf{C}\_1 \mathbf{R}\_1^2}{\mathbf{1} + \left(\alpha \mathbf{C}\_1 \mathbf{R}\_1\right)^2} + \frac{\alpha \mathbf{C}\_2 \mathbf{R}\_2^2}{\mathbf{1} + \left(\alpha \mathbf{C}\_2 \mathbf{R}\_2\right)^2} \tag{5}$$

Complex plane plots for various immittance functions are shown in Figure 2(b-e) for R2/R1 = 1. It is seen that when the values of time constants are widely separated (R2C2/R1C1 = 100), two clear arcs appear indicating the presence of two processes. For R2C2/R1C1 > 1 but below 5, a depressed looking arc appears and when R2C2/ R1C1 = 1, a single arc appears (Figure 2(b)).

Now let us look at the plots of Figure 3(b-e) for R2/R1 = 100. Here, Z<sup>00</sup> vs. Z<sup>0</sup> plot obtained for different ratios of time constants looks like a single semicircular arc indicating presence of one process. However, the corresponding M<sup>00</sup> vs. M<sup>0</sup> (Figure 3(c)) has two clear arcs indicating the presence of two processes. This indicates that inference derived from only one formalism might lead to wrong conclusions, and plots using more formalisms should be looked at together.

Figure 2.

(a) Equivalent circuit model containing series combination parallel R1C1 and R2C2. Normalized (b) Z″ (R1 + R2) vs Z<sup>0</sup> (R1 + R2), (c) M″/[C0(1/C1 + 1/C2)] vs M<sup>0</sup> /[C0(1/C1 + 1/C2)], (d) Y″/[1/(R1 + R2)] vs Y<sup>0</sup> [1/ (R1 + R2)] and (e) <sup>ɛ</sup>″/[(C1/C0)(R1/(R1 + R2))<sup>2</sup> + (C2/C0)(R2/(R1 + R2))<sup>2</sup> ] vs ɛ<sup>0</sup> /[(C1/C0)(R1/ (R1 + R2))<sup>2</sup> + (C2/C0)(R2/(R1 + R2))<sup>2</sup> ] for R2/R1 = 1 and R2C2/R1C1.

3.2 Model 2: parallel combinations R1C1 connected in series with R2

Impedance Spectroscopy: A Powerful Technique for Study of Electronic Ceramics

DOI: http://dx.doi.org/10.5772/intechopen.81398

<sup>Z</sup><sup>0</sup> <sup>¼</sup> <sup>R</sup><sup>1</sup>

1 þ ð Þ ωC1R<sup>1</sup>

(R1 + R2), (c) M″/[C0(1/C1 + 1/C2)] vs M<sup>0</sup>

[1/(R1 + R2)] and (e) <sup>ɛ</sup>″/[(C1/C0)(R1/(R1 + R2))<sup>2</sup> + (C2/C0)(R2/(R1 + R2))<sup>2</sup>

given as

85

Figure 3.

(R1 + R2) vs Z<sup>0</sup>

(R1 + R2))<sup>2</sup> + (C2/C0)(R2/(R1 + R2))<sup>2</sup>

at high-frequency side.

The model is schematically shown in Figure 4(a). The values of Z<sup>0</sup> and Z<sup>00</sup> are

(a) Equivalent circuit model containing series combination of parallel R1C1 and R2C2. Normalized (b) Z″

The immittance plots for this model are shown in Figure 4(b-e) for various values of R2/R1. It is seen that Z<sup>00</sup> vs. Z<sup>0</sup> plot (Figure 4(b)) shows a shift towards right, and the corresponding M<sup>00</sup> vs. M<sup>0</sup> plot (Figure 4(c)) has steeply rising branch

<sup>2</sup> <sup>þ</sup> <sup>R</sup><sup>2</sup> , Z<sup>0</sup> <sup>¼</sup> <sup>ω</sup>C1R<sup>2</sup>

] for R2/R1 = 100 and different values of R2C2/R1C1.

1 1 þ ð Þ ωC1R<sup>1</sup>

/[C0(1/C1 + 1/C2)], (d) Y″/[1/(R1 + R2)] vs Y<sup>0</sup>

/[(C1/C0)(R1/

] vs ɛ<sup>0</sup>

<sup>2</sup> (6)

Impedance Spectroscopy: A Powerful Technique for Study of Electronic Ceramics DOI: http://dx.doi.org/10.5772/intechopen.81398

Figure 3.

<sup>Z</sup><sup>0</sup> <sup>¼</sup> R1

Figure 2.

84

(R1 + R2) vs Z<sup>0</sup>

(R1 + R2))<sup>2</sup> + (C2/C0)(R2/(R1 + R2))<sup>2</sup>

1 þ ð Þ ωC1R1

2 þ

R1C1 = 1, a single arc appears (Figure 2(b)).

R2 1 þ ð Þ ωC2R2

Ceramic Materials ‐ Synthesis, Characterization, Applications and Recycling

<sup>2</sup> , Z 00

Complex plane plots for various immittance functions are shown in Figure 2(b-e) for R2/R1 = 1. It is seen that when the values of time constants are widely separated (R2C2/R1C1 = 100), two clear arcs appear indicating the presence of two processes. For R2C2/R1C1 > 1 but below 5, a depressed looking arc appears and when R2C2/

Now let us look at the plots of Figure 3(b-e) for R2/R1 = 100. Here, Z<sup>00</sup> vs. Z<sup>0</sup> plot obtained for different ratios of time constants looks like a single semicircular arc indicating presence of one process. However, the corresponding M<sup>00</sup> vs. M<sup>0</sup> (Figure 3(c)) has two clear arcs indicating the presence of two processes. This indicates that inference derived from only one formalism might lead to wrong conclusions, and plots using more formalisms should be looked at together.

(a) Equivalent circuit model containing series combination parallel R1C1 and R2C2. Normalized (b) Z″

] for R2/R1 = 1 and R2C2/R1C1.

/[C0(1/C1 + 1/C2)], (d) Y″/[1/(R1 + R2)] vs Y<sup>0</sup>

/[(C1/C0)(R1/

] vs ɛ<sup>0</sup>

[1/

(R1 + R2), (c) M″/[C0(1/C1 + 1/C2)] vs M<sup>0</sup>

(R1 + R2)] and (e) <sup>ɛ</sup>″/[(C1/C0)(R1/(R1 + R2))<sup>2</sup> + (C2/C0)(R2/(R1 + R2))<sup>2</sup>

<sup>¼</sup> <sup>ω</sup>C1R2

1 1 þ ð Þ ωC1R1

2 þ

ωC2R2 2 1 þ ð Þ ωC2R2

<sup>2</sup> (5)

(a) Equivalent circuit model containing series combination of parallel R1C1 and R2C2. Normalized (b) Z″ (R1 + R2) vs Z<sup>0</sup> (R1 + R2), (c) M″/[C0(1/C1 + 1/C2)] vs M<sup>0</sup> /[C0(1/C1 + 1/C2)], (d) Y″/[1/(R1 + R2)] vs Y<sup>0</sup> [1/(R1 + R2)] and (e) <sup>ɛ</sup>″/[(C1/C0)(R1/(R1 + R2))<sup>2</sup> + (C2/C0)(R2/(R1 + R2))<sup>2</sup> ] vs ɛ<sup>0</sup> /[(C1/C0)(R1/ (R1 + R2))<sup>2</sup> + (C2/C0)(R2/(R1 + R2))<sup>2</sup> ] for R2/R1 = 100 and different values of R2C2/R1C1.
