**4.3 Parameter identification for the two-decade model**

The following models are build starting from measurement of the body impedance frequency characteristic |*ZRC*(*jω*)| reported in [3, 8]. The simulations presented in Section 2 show that the result of the frequency characteristic measurement is an accurate representation of the human body impedance modulus. Using the above algorithm it follows:

$$\begin{aligned} Y\_{RC}(\mathbf{s}) &= \\ &= \\ \frac{17.81 \cdot 10^{-4} (15.92 \cdot 10^{-5} \mathbf{s} + 1) \left(10.23 \cdot 10^{-6} \mathbf{s} + 1\right) \left(26.82 \cdot 10^{-7} \mathbf{s} + 1\right) \left(28.82 \cdot 10^{-8} \mathbf{s} + 1\right)}{\left(15.91 \cdot 10^{-5} \mathbf{s} + 1\right) \left(86.25 \cdot 10^{-7} \mathbf{s} + 1\right) \left(22.70 \cdot 10^{-7} \mathbf{s} + 1\right)} \end{aligned} \tag{4}$$

Using a Cauer synthesis, the continued fraction expansion in Eq. (5) is obtained, and the circuit is given in **Figure 9**. The parameter values extracted from Eq. (5) are given below.

$$Y\_{RC}(s) = $$

$$\text{F1.97} \cdot 10^{-11}s + \cfrac{1}{410.65 + \cfrac{1}{31.08 \cdot 10^{-9}s + \cfrac{1}{106.17 + \cfrac{1}{18.16 \cdot 10^{-8}s + \cfrac{1}{44.66 + \cfrac{1}{15.69 \cdot 10^{-3}s + 98.64}}}}} \tag{10.17}$$

A comparison between the frequency characteristic of the *RC* ladder model, the Cole model and the measured results is given in **Figure 10**.

#### *Electrochemical Impedance Spectroscopy*

#### **Figure 9.**

*Cauer synthesis of the* RC *ladder circuit model [8].*

**Figure 10.**

*The frequency characteristic for the Cole model, the two-decade* RC *ladder model and the measured results.*

#### **4.4 Parameter identification for the three-decade model**

Using the same algorithm presented above applied for the measurement set of three decades, a more elaborated admittance expression, given in Eq. (6), is obtained [18]. The Cauer synthesis starting from the continued fraction expansion in Eq. (7) gives the circuit presented in **Figure 11**. The parameter values are given above the model. A comparison between the frequency characteristic of the new model and the measured results is given in **Figure 12**.

**5.** *RC* **parallel model**

for a model of this kind.

described in the previous section.

**Figure 13**.

**91**

A behavioral model, as a linear circuit which can be an extension of the Cole model is the best choice, taking into account that the intracellular and the extracellular water volumes are related to the real part of the model impedance computed at minimum and maximum frequencies [3], this impedance being well defined only

*The frequency characteristic of the Cole model, the three-decade* RC *ladder model and the measured results.*

An *RC* parallel model, valid for a frequency range of three decades, which can be reduced to the Cole model for a narrow frequency interval, is presented in this section.

For the parameter identification of the *RC* parallel model, only the measured frequency characteristic |*YRC*(*jω*)| is used [10]. In order to build this model, the approximation method is employed, followed by the circuit synthesis as it is

Using the above algorithm, the frequency characteristic |*YRC*(*jω*)| corresponding to the data in [3] has been approximated by the admittance in Eq. (6) with an error *ε* = 0.95 dB using a sweeping step Δ*ω*<sup>m</sup> *=* 8315 Hz. The synthesis of this admittance can be made by the Foster II method which gives the most interesting circuit in

**5.2 Parameter identification for the** *RC* **parallel model**

**5.1 Introduction**

**Figure 12.**

**Figure 11.**

*The Cauer synthesis of the three-decade* RC *ladder model.*

*Circuit Models of Bioelectric Impedance DOI: http://dx.doi.org/10.5772/intechopen.91004*

$$\begin{aligned} Y\_{RC}(s) &= \\ \hline \frac{17.81 \cdot 10^{-4} \left(15.92 \cdot 10^{-5}s + 1\right) \left(13.04 \cdot 10^{-6}s + 1\right) \left(43.11 \cdot 10^{-7}s + 1\right) \left(18.59 \cdot 10^{-7}s + 1\right) \left(75.82 \cdot 10^{-9}s + 1\right)}{\left(15.91 \cdot 10^{-5}s + 1\right) \left(11.55 \cdot 10^{-6}s + 1\right) \left(38.08 \cdot 10^{-7}s + 1\right) \left(16.47 \cdot 10^{-7}s + 1\right)} \\ \text{s.t.} \\ Y\_{RC}(s) &= \end{aligned} \tag{6}$$

C5 = 16.99 mF, R5 = 106.80 Ω, C4 = 500.71 nF, R4 = 17.39 Ω, C3 = 107.12 nF, R3 = 42.54 Ω, C2 = 23.94 nF, R2 = 109.02 Ω, C1 = 194.77 pF, R1 = 392.53 Ω.

*Circuit Models of Bioelectric Impedance DOI: http://dx.doi.org/10.5772/intechopen.91004*

#### **Figure 11.**

*The Cauer synthesis of the three-decade* RC *ladder model.*

**Figure 12.** *The frequency characteristic of the Cole model, the three-decade* RC *ladder model and the measured results.*
