# Electrically Equivalent Head Tissue Materials for Electroencephalogram Study on Head Surrogates
## 3 Experimental Procedure
### Conductivity Measurement Process  
In a regular conductor, the valence band and conduction band overlap, creating a continuous band of energy states that allows charge carriers to move freely. On the other hand, PDMS (Polydimethylsiloxane) is nonconductive, displaying a disordered and amorphous structure, along with a substantial energy band gap [37] which causes the charge carriers to become confined within localized states, giving rise to a phenomenon known as hopping conductance. As a result, charge carriers can only migrate to neighboring empty states by jumping over a potential barrier and using quantum mechanical tunneling [38]. In  
Figure 3: (**a**) A 4-layer head tissue sample model. (**b**) This shows the EEG testing setup of a 7-layer sample. A 7-layer head tissue sample model is highlighted with the red line. The dimensions of these are mentioned in Table 1. For testing in the conductivity measuring machine, the signal generator was skipped from the layer. The signal generator was inserted in the grey matter area while processing the grey matter and CSF layer only for performing the EEG test. The signal generator tip was made of silver chloride and a signal was supplied from DAQ to the signal generator using copper wire. The dimensions of the sample shown here are for the EEG testing sample which was prepared according to the EEG electrode placing requirement. this study, the conductivity of the PDMS sample was measured using Broadband Dielectric Spectroscopy (BbDS) (Novocontrol Technologies GmbH & Co. KG (Montabaur, Germany)). This technique has been widely employed in the polymer industry to analyze the electrical properties of polymeric materials. The experimental setup, illustrated in Figure 4, involves placing the sample material between two conductive electrodes. The diameter of the electrodes was maintained at 20 mm. A sinusoidal voltage (1 V\({}_{\text{rms}}\)) is applied to one electrode, while the output current is measured through the other electrode. Depending on the material's characteristics, the output current can either be in phase or out of phase with the input voltage signal. The conductivity of the material was measured over a frequency range of 1 Hz to 1 kHz at room temperature. To determine the complex conductivity, the first step is to measure the complex dielectric permittivity, \(\varepsilon^{\star}\), as a function of frequency, \(\omega\). Once the complex permittivity data are obtained, mathematical relations between permittivity and conductivity are used to calculate the complex conductivity of the material. When a voltage \(U_{0}(V)\) of fixed frequency (\(\omega 2\pi\)) is applied to the sample, it results in a current \(I(A)\) at the same frequency, but with a phase shift (\(\Phi\)). These relations can be expressed in complex notation as shown below [39; 40]. \[U(t)=U_{0}cos(\omega t)=\text{Re}\Big{(}\text{U}^{*}\text{e}^{\text{i}\omega t }\Big{)} \tag{1}\]  
\[I(t)=I_{0}\text{cos}\left(\omega t+\Phi\right)=\text{Re}\Big{(}\text{I}^{*} \text{e}^{\text{i}\omega t}\Big{)} \tag{2}\]  
Here, U* = U\({}_{0}\), I* = \(I^{\prime}+iI^{\prime\prime}\), and Re indicates the only real part of the complex notation. The magnitude of the current can be measured by the following equation:  
\[I_{0}=\sqrt{\big{(}I^{\prime 2}+I^{\prime\prime}2\big{)}} \tag{3}\]  
Applying an electric field to a material leads to a phase shift between the current and voltage responses, and this phase angle can be measured by the following equation. \[\tan(\Phi)=\frac{I^{\prime}}{I^{\prime\prime}} \tag{4}\]  
The impedance and capacitance of the sample can be determined using the equations provided below. \[Z^{*}=Z^{\prime}+iZ^{\prime\prime}=\frac{U^{*}}{I^{*}}=\frac{U_{0}}{I^{\prime} +iI^{\prime\prime}} \tag{5}\]  
\[C^{*}(\omega)=C^{\prime}+iC^{\prime\prime}=\frac{1}{j\omega z^{*}} \tag{6}\]  
Figure 4: The experimental setup for dielectric measurement of the sample.