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

bioimpedance spectroscopy is presented in this section. This approach employs the approximation of the measured body admittance modulus |*YRC*(*jω*)| with a physically realizable function followed by the circuit synthesis [16]. This model is a linear *RC* circuit with frequency independent values of resistances and capacitances. As the frequency dependence of the phase angle *arg*(*YRC*(*jω*)) can be computed from |*YRC*(*jω*)| using the Bayard-Bode relationships [17], the measured values of *arg* (*YRC*(*jω*)) are not needed for the parameter identification of this model. Two equivalent circuits of the human body, built using this approach, have been proposed [10–11]. These are ladder circuits which cannot be considered as extensions

The synthesis method for an *RC* admittance (*YRC*) developed in [15] can be used also for an *RC* impedance (*ZRC* = 1/*YRC*) with some minor modifications, and it is

A function *F*(*s*) of a complex variable *s* is an *RC* admittance if and only if the

• the poles and zeros of *F*(*s*) are simple and alternate on the negative real axis,

• the number of zeros of *F*(*s*) is equal or greater with one with respect to the

Replacing *s* with *jω*, (where *ω=2πf* and *f* is the frequency) the shape of the *RC* admittance modulus curve |*YRC*(*jω*)| versus *ω* is defined by the poles and zeros location. Sweeping the *ω* axis starting from the origin, it can be observed that the location of a zero is associated with a slope change of 20 dB/decade and the location of a pole is associated with a slope change of 20 dB/decade. This is because the characteristic |*YRC*(*jω*)| has asymptotes whose slopes are 20 dB/decade, 0, 20 dB/ decade, 0, and so on. The |*YRC*(*jω*)| characteristic approximation by asymptotes has

A natural way to approximate the |*YRC*(*jω*)| characteristic is to consider a smaller

the maximum error of 3 dB at the asymptote intersection (**Figure 8**) [16].

asymptote number than that corresponding to the measured characteristic. A greater error *ε* between measured and simulated values leads to a simpler circuit

of the Cole model.

**4.2** *RC* **admittance synthesis**

*Electrochemical Impedance Spectroscopy*

presented in the following.

following conditions are fulfilled:

number of poles *F*(*s*).

(**Figure 8**).

**Figure 8.**

**88**

*The* |YRC*(*jω*)*| *characteristic approximation by asymptotes.*

• *F*(*s*) is a rational fraction of *s* with real coefficients;

the closest to the origin being a zero; and

The algorithm for the synthesis of a *RC* one-port in the angular frequency band [*ωm*, *ωM*], where *ω<sup>m</sup>* is the minimum value and *ω<sup>M</sup>* is the maximum value, has the following steps [16]:


Sweeping the frequency axis with a step Δ*ωm*, the algorithm checks the error between the 20 dB/decade asymptote and the given characteristic. This error cannot be greater than an imposed value *ε*. The first pole *p1* is assigned to the last value before that corresponding to an error of *2ε* or greater. If this error occurs after the first angular frequency step Δ*ωm*, then *p1* is placed in the vicinity of *z1*. Afterwards, the first asymptote is translated so that a maximum error of *ε* is obtained. The other asymptotes are determined similarly, in order to fulfill the condition *error* ≤ *ε* for each asymptote.
