## 3 Electrical Tomography Principles and Applications
### Fundamentals of Electrical Tomography
#### 3.2.2 Resistivity/Impedance Tomography  
The impedance variation distribution of a mixture is the focus of EIT. In EIT, the stimulus used is a sinusoidal electrical signal (current or voltage) with a certain frequency and amplitude. Then, the real and imaginary parts of the impedances are calculated using phase sensitivity measurements [66; 67]. The resistive component is obtained by an in-phase measurement with the stimulus signal, while the capacitive component is found by a quadrature of phase measurement with the excitation signal [53]. The invention credit for EIT is given to [PERSON], who reported it in a 1978 publication [68]. Still, the first known experimental application of EIT took place in 1984, for which the authors were [PERSON] and [PERSON] [69]. For investigating processes where the continuous phase is electrically conductive, either ERT or EIT can be used. Specifically, when the inspected medium is purely resistive, then ERT should be used. Similarly to ECT, in ERT/EIT an electrical current is injected through a pair of electrodes that are placed along the boundary of the domain under study. Consequently, the resulting electrical field distribution in the domain will be conditioned by the material distribution within it [70]. The potential voltage differences between all pairs of electrodes placed around the domain perimeter, apart from the pair used for exciting the domain, are measured, and used to nourish a non-linear algorithm. This algorithm solves the inverse problem described above and permits the calculation of the unknown conductivity/resistivity distribution along the domain. However, the procedure only ends when all electrodes are used for stimulating the domain, and thus a full cycle has as many projections as the number of electrodes comprising the sensing system (see Figure 2) [71]. Therefore, in summary, for estimating the material distribution in the domain, an electrical field distribution characterisation is used. In ERT, which is easier to implement [67; 72], the aim is to obtain a conductivity/resistivity distribution along the domain. The working principle used in ERT to obtain the data related to a full set of projections, is identical to that described for EIT (requiring the use of all electrodes to introduce the electrical excitation signal in the domain, and to measure all the potential voltage differences between the remaining ones). The single difference is that in ERT, the electrical signal used as stimulus is a continuous current. Therefore, for EIT, the data acquisition cycle only ends when a full rotation of the applied electric field excitation is complete. Consequently, for both EIT and ERT, if the sensing system is composed of \(N\) electrodes, the number of independent differential voltage measurements is \(N(N-3)/2\). Typical values for the frequencies used in EIT systems lie in the range 20-150 kHz, so quasi-static conditions can be assumed when a full rotation data is  
Figure 1: Schematic representation of the measurement principle of an ECT system (reproduced with permission from [PERSON] et al. [62]). acquired. In recent years, efforts have been directed towards the development of acquisition systems and to the conception of sensors (materials used in their fabrication, dimensions and location of the sensors, etc.). The use of a discretised electrode system is restricted to based flows with constant conductance [73]. When large bubbles are present in the fluid, those bubbles can cause electrodes to lose contact with the fluid, and consequently, the obtained images of the conductance distribution will be inconsistent. When faced with this type of suspension, one possible solution would be the use of continuous electrode rings instead of a discrete electrode sensing system [73].