## 3 Results and Discussion
### Antiferroelectricity of La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) Ceramics
As the magnetic field increases, the complex impedance decreases, causing the real part (\(Z^{\prime}\)) and imaginary part (\(Z^{\prime\prime}\)) of the complex impedance to decrease as the magnetic field increases. However, in the simplified model of the actual capacitor, when the frequency is small, we take the inductor as a path, thus becoming the series connection of the \(ESR\) and the ideal capacitor. At this time, the complex impedance can be expressed as \(z(\omega)=R-j\frac{1}{\omega c}=z^{\prime}-jz^{\prime\prime}\). The impedance complex plane composed of the real part and the imaginary part of the complex impedance should be a straight line with an intercept of \(R\) and being perpendicular to the real part of the complex impedance, but the actual impedance complex plane is a semicircle, as shown in Figure 4:5 This is because the capacitor actually has a dielectric loss resistance \(R_{e}\) in parallel with the ideal capacitor \(C\), so the full equivalent circuit of the capacitor is shown in Figure 4:6 Therefore, the full equivalent circuit can be divided into lead resistances \(R_{s}\) that do not vary with frequency, and a parallel circuit of dielectric loss resistors \(R_{e}\) and ideal capacitor \(C\). The real and imaginary parts of the complex impedance of the parallel circuit of \(R_{e}\) and \(C\) can be expressed as a semicircle of expression \(\left(z^{\prime}-\frac{R_{e}}{2}\right)^{2}+\left(z^{\prime\prime}\right)^{2}= \left(\frac{R_{e}}{2}\right)^{2}\) on the impedance complex plane, but due to  
Figure 4:7 Simplified model of actual capacitance. \(ESR\) is the equivalent series resistance, \(ESL\) is the equivalent series inductance, and C is the ideal capacitance. Figure 4:8 Variation of complex impedance with frequency for La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) ceramics under different M fields. the series connection of \(R_{s}\), the complex impedance of the actual capacitor is a semicircle of \(\left(z^{\prime}-\frac{R_{e}}{2}-R_{s}\right)^{2}+\left(z^{\prime}\right)^{2}= \left(\frac{R_{e}}{2}\right)^{2}\) in the impedance complex plane. It can be seen in Figure 13 that the radius of the Core-Core circle also decreases with the increase in the magnetic field, that is, the lead resistance \(R_{s}\) is basically unchanged, while the dielectric loss resistance \(R_{e}\) of the capacitor itself decreases with the increase in the magnetic field. This is due to the fact that in the full equivalent model of the actual capacitance in this experiment, the lead resistance \(R_{s}\) varies with the magnetic field strength as shown in the inset in Figure 4:9 The inset shows that the value of \(R_{s}\) is small and does not show a clear trend with the magnetic field. While the dielectric loss resistance \(R_{e}\) is caused by the sample itself, the value should be closely related to the magnitude of the magnetic field. \(R_{e}\) is the radius of the core circle along the x-axis. In order to get closer to the actual value of \(R_{e}\), we processed the lowest point of each core circle in Figure 13 and made its variation with the M-field in Figure 5:0 As the magnetic field increased, \(R_{e}\) changed very little until 600 mT, above which an increase in the magnetic field led to a rapid decrease in \(R_{e}\). These behaviours are more indicative of a GMR effect for La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\). As the magnetic field increased to 1.1 T, the real part of the complex impedance decreased from 1901 \(\Omega\) to 1114 \(\Omega\), and the imaginary part of the complex impedance decreased from 623 \(\Omega\) to 394 \(\Omega\). It can be seen in Figure 13 that when the imaginary part was greater than 0, the real and imaginary parts of the complex impedance were no longer part of the Core-Core circle. This is because the parasitic inductance had a more significant effect on the complex impedance at higher frequencies. Figure 5:1 The full equivalent circuit of the actual capacitor.