# Estimating Hardness of Cosmetic Cream Using Electro-Mechanical Impedance Sensing Technique
## 2 Electro-Mechanical Impedance Sensing Technique  
EMI sensing technology operates on a straightforward principle. When a piezoelectric sensor is connected to a host structure and subjected to a sinusoidal voltage, vibrations occur in the bonded area of the structure due to the direct influence of the piezoelectric materials. The mechanical impedance of the host structure changes with variations in its mechanical properties, leading to a corresponding change in the electrical impedance of the piezoelectric sensor connected to the host structure. The EMI can be calculated by analyzing the input and output signals of the piezoelectric sensor [24; 25; 26]. [PERSON] et al. proposed a one-dimensional electro-mechanical interaction model for the piezoelectric sensor surface bonded to the host structure, as depicted in Figure 1 and Equation (1) [26].  
\[\mathrm{Y}(\omega)=j\omega\frac{w_{A}\,l_{A}}{h_{A}}\left[\frac{d_{31}^{2} \overline{\mathrm{Y}}_{11}^{E}Z_{A}}{Z_{A}\!+\!Z}\frac{tan(kl_{A})}{kl_{A}}+ \,\bar{\mathrm{z}}_{33}^{T}-\overset{2}{31}\overline{\mathrm{Y}}_{11}^{E}\right] \tag{1}\]where Y(\(\omega\)) represents the electrical admittance of a piezoelectric sensor bonded to a host structure (inverse of impedance), \(j\) is the imaginary unit (\(\surd(-1)\)), \(\omega\) is the excitation frequency, \(w_{A}\) is the width of the piezoelectric sensor, \(l_{A}\) is the length of the piezoelectric sensor, \(h_{A}\) is the thickness of the piezoelectric sensor, \(Z\) is the host structural impedance, \(Z_{A}\) is the impedance of the piezoelectric sensor, \(k\) is the wave number, \(\vec{z}_{33}^{T}\) is the dielectric constant of the piezoelectric sensor in a 3-3 direction, \(d_{31}\) is the piezoelectric constant, and \(\vec{Y}_{11}^{E}\) is the complex Young's modulus of the piezoelectric sensor. The equation reveals that, as the mechanical impedance of the piezoelectric sensor remains constant over the monitoring period, any change in the mechanical impedance of the host structure directly impacts the electrical impedance measured by the transducer. Consequently, this method facilitates the monitoring of changes in the mechanical impedance of the host material by measuring the electrical impedance of the piezoelectric sensor combined with the host material.  
When piezoelectric sensors are embedded in cosmetic creams with different hardness levels, the mechanical impedance of the cosmetic cream undergoes changes influenced by the varying hardness of the cream. The relationship between the mechanical properties of the cosmetic cream, such as hardness, and the corresponding changes in the electro-mechanical impedance measured by the embedded piezoelectric sensors forms the basis for estimating the hardness of the cosmetic creams using the EMI sensing technique.  
The objective of this study is to estimate the hardness of cosmetic creams using electro-mechanical impedance (EMI) sensing technology. To achieve this, cosmetic creams with varying hardness levels determined by the wax content are manufactured. A comparative analysis is then conducted, contrasting hardness values obtained through conventional hardness test methods with electro-mechanical impedance values acquired through EMI sensing techniques. The goal is to establish correlations between the two datasets, providing insights into the relationship between wax-content-dependent hardness and EMI signals.