# Failure Mechanism of Multilayer Ceramic Capacitors under Transient High Impact
## 3 Modeling of MLCC Parameter Fluctuation Failure
### Modeling of Leakage Current Fluctuation Based on the Piezoresistive Effect
#### 3.2.2 Theoretical Model of Equivalent Circuit for Multilayer Ceramic Capacitor  
The characteristics of an MLCC are not ideal due to the parasitic effects. Many scholars use the following equivalent circuit model to describe the capacitance [23], as shown in Figure 13a. This paper  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(R_{0}\) & \(A_{1}\) & \(T_{1}\) & \(M_{1}\) & \(A_{2}\) & \(T_{2}\) & \(M_{2}\) \\ \hline
457 M\(\Omega\) & \(-\)248 & \(-\)18.9 & 11.9 & \(-\)204 & 2.34 & 2.33 \\ \hline \hline \end{tabular}
\end{table}
Table 4: P–R model parameters.  
Figure 11: Relationship between pressure and ceramic conductivity: (**a**) the change of ceramic current in static pressure experiment; (**b**) the relationship between ceramic dielectric pressure and resistance.  
Figure 12: Relationship between pressure and ceramic resistance.  
focuses on the influence of the mechanical impact on the MLCC (mainly determined by the ceramic dielectric resistance, Rp), which is not significantly related to the parasitic effects. So, the equivalent circuit model is simplified as Figure 13b:  
Where \(C\) is MLCC capacity, \(C_{k}\) is the storage capacity of one pair of positive and negative electrodes, ESR(\(R_{s}\)) is the equivalent series resistance, and \(R_{p}\) is the equivalent resistance of the ceramic dielectric:  
According to the equivalent circuit, the mathematical model can be established as follows:  
\[I_{I,0}R_{s}+V_{p,0}=U_{0} \tag{13}\]  
\[\bigg{(}C_{k}\frac{V_{p,k}}{\Delta t}+\frac{V_{p,k}}{R_{p,k}}\bigg{)}R_{s}=U_{ 0}-V_{p,k} \tag{14}\]  
\[V_{p,k}=\frac{U_{0}}{C_{k}R_{s}}e^{-\frac{1}{C_{k}}\frac{\pi}{k-1}(\frac{1}{R_ {p,k}}+\frac{1}{R_{s}})\Delta t}\big{[}\int e^{-\frac{1}{C_{k}}\frac{\pi}{k-1} (\frac{1}{R_{p,k}}+\frac{1}{R_{s}})\Delta t}dt\big{]} \tag{15}\]  
\[I_{I,k}=\frac{U_{0}-V_{p,k}}{R_{s}} \tag{16}\]  
where \(U_{0}\) is the constant voltage applied to the MLCC, \(V_{p,k}\) are the voltages of the ceramic dielectric between the \(k\)th pair of positive and negative electrodes, and the conductivity of \(R_{p}\) varies with the impact.  
After combining the equivalent circuit model (Equations (13)-(16)) and the P-R model (Equations (11) and (12)), perform the process calculation in Figure 14 to obtain the relationship between acceleration and leakage current:  
Figure 14: Flow chart of the leakage current calculation.  
Figure 13: MLCC equivalent circuit model: (**a**) general equivalent circuit model of the MLCC; (**b**) simplified equivalent circuit model of the MLCC.  
The piezoresistive effect of the ceramic dielectric is the main factor that causes the change in leakage current. The impact acceleration has the characteristics of a pulse signal. Therefore, the corresponding impact stress waves at different moments of the impact have different pressures on the ceramic dielectric, which in turn causes pulsed changes in the leakage current.  
According to the static pressure data model of the ceramic dielectric and the mathematical model of an equivalent circuit, the relationship between the conductivity of the ceramic dielectric and the leakage current is presented, as shown in Figure 15. As the conductivity increases, the leakage current increases.  
In addition, the leakage current variation of a multilayer ceramic capacitor under different impacts is simulated, as shown in Figure 16.  
The results show that the leakage current increases approximately exponentially with the increasing impact acceleration, which is in good agreement with the experimental results (Figure 4). The sudden increase in leakage current of the ceramic capacitors causes the loss of energy storage, which makes the detonator energy insufficient. Therefore, the design of the fuze's energy-storage capacitor must consider the leakage current during the impact and increase the design margin to ensure the reliability of the fuze effect.  
Figure 16: Relationship between leakage current and impact acceleration of the MLCC.  
Figure 15: Relationship between leakage current and ceramic conductivity.