# Failure Mechanism of Multilayer Ceramic Capacitors under Transient High Impact
## 3 Modeling of MLCC Parameter Fluctuation Failure
### Modeling of Capacitance Fluctuation Based on Electrode Deformation  
The basic unit of a multilayer ceramic capacitor is composed of two metal electrodes, which are separated by dielectric layer ceramics and stacked up in layers. The positive and negative charges are stored in the metal electrodes. Both ends of the electrode plate are coated in a ceramic dielectric. Therefore, the equivalent two-dimensional elastic mechanics problem of a ceramic capacitor electrode is shown in Figure 7. The coordinate system is length \(l\) along the X direction, width \(b\) along the Y direction, and thickness \(h\) along the Z direction. When the plate is subjected to the inertial force F, the upper surface is compressed and the lower surface is stretched. The bending moment M produced by any single element under the inertial force is m, and the number of electrode layers is N.  
Figure 5: Diagram of the facing area increase during impact. Figure 6: Schematic diagram of the leakage current increase during impact. Under the impact acceleration \(a\), the uniformly distributed load \(q\) is as follows:  
\[q=\frac{F}{Nl}=\frac{ma}{Nl}=\frac{bh\rho a}{N} \tag{1}\]  
The deflection function of the electrode under the uniform load q is as follows [21]:  
\[w(x)=\frac{ma}{24 NEll}x^{2}(l-x)^{2}=\frac{\rho a}{2 NEh^{2}}x^{2}(l-x)^{2} \tag{2}\]  
where the cross-sectional area of the inner electrode is given by \(A=b\cdot h\), the material density is \(\rho\), and the elastic modulus of the material is \(E\). When \(x=l/2\), the maximum deflection of the plate is as follows:  
\[w_{\max}=\frac{\rho l^{4}a}{32 NEh^{2}} \tag{3}\]  
The rotation function \(\theta(x)\) of the inner electrode is as follows:  
\[\theta(x)=w(x)^{\prime} \tag{4}\]  
By deriving x, we obtain the following results:  
\[\theta(x)=\frac{\rho a}{NEh^{2}}x(l-2x)(l-x) \tag{5}\]  
Then, the length \(l\) of the electrode after impact is bent to the arc length \(l^{\prime}\):  
\[l^{\prime}=\int_{0}^{l}\sqrt{1+\left[\frac{\rho a}{NEh^{2}}x(l-2x)(l-x)\right] ^{2}}dx \tag{6}\]  
Then, the arc length change \(\Delta l\) is as follows:  
\[\Delta l=l^{\prime}-l=\int_{0}^{l}\sqrt{1+\left[\frac{\rho a}{NEh^{2}}x(l-2x) (l-x)\right]^{2}}dx-l \tag{7}\]  
Figure 5:0 Equivalent diagram of the inner electrode: (**a**) schematic diagram of the electrode; (**b**) schematic diagram of electrode equivalent mechanical model. The surface area change \(\Delta S\) of the inner electrode is obtained as follows:  
Figure 5:1  
The capacity of the inner electrode unit of the ceramic capacitor is as follows:  
Figure 5:2  
Therefore, the capacitance change \(\triangle C\) is as follows:  
Figure 5:3  
According to the numerical calculation of the above mechanical model, the results are shown in Figure 5:4 The relationship between the capacity change and the impact acceleration presents two stages: first, it changes gently, then rises rapidly. Further, as shown in Figure 9, the first stage has a slow capacity change because the electrode and ceramic dielectric are closely connected when the impact acceleration is lower than the deformation threshold of the ceramic dielectric and the deflection deformation is small, and the facing area of the inner electrode changes only a little; therefore, the capacity change is not obvious. In the second stage, the impact acceleration exceeds the threshold, the deflection of the electrode is obvious, the facing area of the inner electrode increases, and the capacity and impact acceleration almost increase with a linear relation. The comparison between Tables 2 and 3 shows that the main conclusions of both the simulation and experiment are the same, indicating that the mechanical model proposed above is credible. Specifically, the quantitative value of the absolute capacity change of the simulation is slightly lower than that of the experiment. The reason for this is that the simulation does not consider the mechanical two-dimensional equivalent model and simplifies the influence of the transverse strain.