# Dielectric Spectroscopy Studies and Modelling of Piezoelectric Properties of Multiferroic Ceramics
## 3 Results and Discussion
### Modelling of Impedance--Frequency Characteristics of the Piezoelectric Equivalent Circuit  
Let us first explain the notation used. In the adopted notation, square or box brackets [ ] denote that elements are in series-connected, whereas round brackets or parentheses ( ) denote the parallel connection of electric elements. According to the adopted notation, (RC) is a parallel circuit, while [RC] is a series connection of the elements \(R\) and \(C\). To accurately approximate the behaviour of the piezoelectric specimen close to its fundamental resonance, it can be represented by the electric equivalent circuit ([L\({}_{1}\)R\({}_{1}\)C\({}_{1}\)]C\({}_{0}\)) consisting of a \"mechanical arm\" (L, C, and \(R\) connected in series) and \(C_{0}\) (which corresponds to the electrical capacitance of the specimen) connected in parallel [24]. In this connection, it is worth remembering that the impedance of the parallel circuit can be represented by the equivalent series circuit consisting of equivalent in series-connected resistance and reactance values. In the case of our simulation and fitting, the modified electric equivalent circuit [RsCPE\({}_{1}\)([L\({}_{1}\)R\({}_{1}\)C\({}_{1}\)]C\({}_{0}\))] including resistance Rs and constant phase element CPE\({}_{1}\) connected in series with the \"piezoelectric\" equivalent circuit was used. Figure 9 shows the electric equivalent circuit used for the simulation and fitting of the impedance response of the ceramic specimen vibrating close to its fundamental resonance. It should be noted that the resonance frequency \(\
u_{\rm r}\) and antiresonance frequency \(\
u_{\rm a}\) correspond to the \"zero\" value of reactance for the electric equivalent circuit (Figure 9). The reactance of the \"mechanical arm\" is zero at the series resonant frequency \(\
u_{\rm s}\) when:  
\[\omega=1/\sqrt{L_{1}C_{1}} \tag{2}\]  
where \(\omega\) is the angular frequency (\(\omega\) = 2 \(\tau\
u\)). The reactance of the parallel circuit is zero at the parallel resonant frequency (\(\
u_{\rm p}\)). The parallel resonance \(\
u_{\rm p}\) occurs when the currents flowing in the two arms are in antiphase, which is when:  
\[\omega=\left(C_{0}+C_{1}/L_{1}C_{0}C_{1}\right)^{1/2} \tag{3}\]  
In this connection, it must be pointed out that the relation between the above-mentioned characteristic frequencies of the equivalent circuit is as follows: \(\
u_{\min}<\
u_{\rm s}<\
u_{\rm r}\). However, the difference between them is very small (\(\
u_{\min}\)-\(\
u_{\rm s}\)-\(\
u_{\rm r}\)) [24]. Similar relation exists between antiresonance, parallel resonance, and \(|\,Z\,|\,_{\max}\) frequencies: \(\
u_{\rm a}<\
u_{\rm p}<\
u_{\max}\) (\(\
u_{\rm a}\)-\(\
u_{\rm p}\)-\(\
u_{\rm max}\)). What is important is that values of \(\
u_{\min}\) an \(\
u_{\max}\) can be readily measured using an impedance analyser (Alpha-AN High Performance Frequency Analyzer). An example of the fitting results obtained for impedance characteristics measured under the influence of a weak circumferential magnetic field is shown in Figure 10. Figure 10a shows the spectra measured at \(H_{\rm ac}\) = 0.05 A/m, (\(E\) = 166 V/m; \(U\) = 0.1 V\({}_{\rm rms}\)) at 20 \({}^{\circ}\)C, whereas Figure 10b shows the results obtained for spectrum measured under the following conditions: \(H_{\rm ac}\) = 0.49 A/m, (\(E\) = 1333 V/m; \(U\) = 2 V\({}_{\rm rms}\)) at 20 \({}^{\circ}\)C.  
Figure 9: Electric equivalent circuit of a piezoelectric specimen used for simulation and fitting of the dielectric response. Experimental data were fitted to the electric equivalent circuit using the ZView programme (Scribner Associates, Inc. Southern Pines, NC, USA). Complex non-linear least squares method (CNLS) was employed for the analysis of the impedance/frequency data of the electroceramics [17]. The fitting procedure was limited to the frequency range of the radial resonances \(\Delta\
u\) = (2-6) \(\times\) 10\({}^{5}\) Hz.