**2. Principle of the pulsed electroacoustic (PEA) method**

The PEA method is one of the widely used techniques for measuring the charge distribution in dielectrics. The principle of the PEA method is explained at the followings [1].

### *Polyimide for Electronic and Electrical Engineering Applications*

**Figure 1** shows the schematic diagram of the PEA method. A charged sample with thickness a is used, and its both sides are sandwiched by electrodes. The pulse voltage generator is connected to both electrodes. When the pulse electric field *e*p (t) is applied with the electrode induced charges *σ* (0) and *σ* (a) accumulated on both electrode surfaces due to the accumulated charge ρ in the sample, a pulsed pressure wave *p*(t) is generated from *σ*(0), *σ*(a) and *ρ*(z) in the sample. Those pulsed pressure wave *p*(t) is shown by the convolution equation between each accumulated charges (*σ*(0), *σ* (a), *ρ*(d)) and *e*p(t). The convolution equation is shown in the following formula.

$$\begin{split} p(t) &= p\_1(t) + p\_2(t) + p\_3(t) \\ &= \frac{Z\_{Al}}{Z\_{sa} + Z\_{Al}} \cdot \sigma(\mathbf{0}) \cdot \boldsymbol{e}\_p(t) + \frac{2Z\_{Al}}{Z\_{sa} + Z\_{Al}} \cdot \frac{1}{2} \cdot u\_{sa} \cdot \int\_{-a}^{a} \rho(\tau) \cdot \boldsymbol{e}\_p(t-\tau) d\tau + \\ &\quad \frac{2Z\_{Al}}{Z\_{sa} + Z\_{Al}} \cdot \frac{Z\_{sa}}{Z\_{sa} + Z\_{RS}} \cdot \sigma(a) \cdot \boldsymbol{e}\_p\left(t - \frac{a}{v\_{sa}}\right) \end{split} \tag{1}$$

where, ZAl, Zsa, ZBS is the acoustic impedance of Al, sample and backing material, respectively. usa is the sound velocity of sample.

**Figure 1.** *The schematic diagram of the principle of PEA method.*

*Space Charge Accumulation Phenomena in PI under Various Practicable Environment DOI: http://dx.doi.org/10.5772/intechopen.96786*

### **Figure 2.**

*The schematic diagram of convolution model on the time function between the pulsed pressure wave and the piezoelectric device.*

Those pulsed pressure *p*(t), described the above on the Eq. (1), the intensity of the pressure wave is proportional to the accumulated charges in the sample. The propagation delay of the pressure wave is related to each charge accumulated position.

A piezoelectric transducer is used to detect the pressure wave by transforming into an electric charge signal. The electric charge signal q(t) on the piezo electric device is expressed by the convolution model between pressure wave function p(τ) and piezoelectric device function h(τ) as shown in formula (2)

$$q(t) = \frac{\upsilon\_p}{b} \int\_{-\upsilon}^{\upsilon} h(\tau) \, p(t - \tau) d\tau \tag{2}$$

where, vp and b is sound velocity and thickness of piezoelectric device, respectively. The convolution model is also shown in **Figure 2**. Normally, we observed charge distribution signals obtained by the above convolution model. However, as the real measurement system has a system function, which is the low pass filter between the amplification circuit and and the capacitance of the piezoelectric device, the observed wave form signal is distorted like as the differentiated waveform by the system function.

The details of the principle of PEA are described elsewhere [1].
