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

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

$$Z' = \frac{R\_1}{1 + \left(\alpha C\_1 R\_1\right)^2} + R\_2 \qquad , \qquad Z' = \frac{\alpha C\_1 R\_1^2}{1 + \left(\alpha C\_1 R\_1\right)^2} \tag{6}$$

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 at high-frequency side.

Ceramic Materials ‐ Synthesis, Characterization, Applications and Recycling

#### Figure 4.

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

## 3.3 Model 3: parallel combinations R1C1 connected in series with C2

Parallel combinations R1C1 connected in series with C2 is shown in Figure 5(a). The values of Z<sup>0</sup> and Z<sup>00</sup> are given as [17]

$$Z' = \frac{R\_1}{1 + \left(\alpha C\_1 R\_1\right)^2} \qquad , \qquad Z' = \frac{\alpha C\_1 R\_1^2}{1 + \left(\alpha C\_1 R\_1\right)^2} + \frac{1}{\alpha C\_2} \tag{7}$$

3.4 Model 4: parallel combinations R1C1, R2C2 connected in series with CPE

(a) Equivalent circuit model containing series combination of parallel R1C1 and C2. Normalized (b) Z″/R1 vs

element (CPE) are shown in Figure 6(a). The impedance of CPE is given as

ZCPE <sup>¼</sup> ð Þ YCPE �<sup>1</sup> <sup>¼</sup> A0ð Þ <sup>j</sup><sup>ω</sup> <sup>ψ</sup> ½ ��<sup>1</sup> <sup>¼</sup> <sup>1</sup>

] and (e) <sup>ɛ</sup>″/[1/(C0/C1 + C0/C2)] vs <sup>ɛ</sup><sup>0</sup>

Impedance Spectroscopy: A Powerful Technique for Study of Electronic Ceramics

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

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

Figure 5.

(C2/(C1 + C2))<sup>2</sup>

Z0

87

Parallel combinations R1C1, R2C2 connected in series with a constant phase angle

cos

ψπ 2

/[C0(1/C1 + 1/C2)], (d) Y″/[(1/R1)(C2/(C1 + C2))<sup>2</sup>

� <sup>j</sup> <sup>1</sup>

A0ωψ

/[1/(C0/C1 + C0/C2)] for C2/C1 = 0.5, 1, 5 and 100.

sin ψπ 2 (8)

] vs Y<sup>0</sup>

/[(1/R1)

A0ωψ

Plot of imaginary part vs. real part of ZCPE is a straight line with slope tan(ψπ/2), which remains constant as ω varies (hence the name CPE). For ψ = 1, the real part becomes zero and ZCPE = 1/(jωA0) and the CPE behaves like an ideal capacitor having capacitance A0. For ψ = 0, the imaginary part becomes zero and ZCPE = 1/A0

The immittance plots for this model are shown in Figure 5(b-e). It is found that Z<sup>00</sup> vs. Z<sup>0</sup> has a low-frequency rising branch where as a shift is seen in M<sup>00</sup> vs. M<sup>0</sup> plot.

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

#### Figure 5.

3.3 Model 3: parallel combinations R1C1 connected in series with C2

Ceramic Materials ‐ Synthesis, Characterization, Applications and Recycling

(a) Equivalent circuit model containing series combination of parallel R1C1 and R2. Normalized (b) Z″/

] for R2/R1.

The values of Z<sup>0</sup> and Z<sup>00</sup> are given as [17]

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

/[(C1/C0)(R1/(R1 + R2))<sup>2</sup>

1 þ ð Þ ωC1R<sup>1</sup>

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

M<sup>0</sup> plot.

86

Figure 4.

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

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

] vs ɛ<sup>0</sup>

Parallel combinations R1C1 connected in series with C2 is shown in Figure 5(a).

The immittance plots for this model are shown in Figure 5(b-e). It is found that Z<sup>00</sup> vs. Z<sup>0</sup> has a low-frequency rising branch where as a shift is seen in M<sup>00</sup> vs.

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

/(C0/C1), (d) Y″/(1/R2) vs Y<sup>0</sup>

1 1 þ ð Þ ωC1R<sup>1</sup>

<sup>2</sup> þ

1 ωC<sup>2</sup>

/(1/R2) and (e) <sup>ɛ</sup>″/[(C1/C0)

(7)

(a) Equivalent circuit model containing series combination of parallel R1C1 and C2. Normalized (b) Z″/R1 vs Z0 /R1, (c) M″/[C0(1/C1 + 1/C2)] vs M<sup>0</sup> /[C0(1/C1 + 1/C2)], (d) Y″/[(1/R1)(C2/(C1 + C2))<sup>2</sup> ] vs Y<sup>0</sup> /[(1/R1) (C2/(C1 + C2))<sup>2</sup> ] and (e) <sup>ɛ</sup>″/[1/(C0/C1 + C0/C2)] vs <sup>ɛ</sup><sup>0</sup> /[1/(C0/C1 + C0/C2)] for C2/C1 = 0.5, 1, 5 and 100.
