High discharge energy density and fast release speed of (Pb, La)(Zr, Sn, Ti)O3 antiferroelectric ceramics for pulsed capacitors

Accepted Manuscript High discharge energy density and fast release speed of (Pb, La)(Zr, Sn, Ti)O3 antiferroelectric ceramics for pulsed capacitors Jie Shen, Xiucai Wang, Tongqing Yang, Hongsheng Wang, Jing Wei PII:

S0925-8388(17)31951-5

DOI:

10.1016/j.jallcom.2017.05.325

Reference:

JALCOM 42056

To appear in:

Journal of Alloys and Compounds

Received Date: 21 February 2017 Revised Date:

13 May 2017

Accepted Date: 30 May 2017

Please cite this article as: J. Shen, X. Wang, T. Yang, H. Wang, J. Wei, High discharge energy density and fast release speed of (Pb, La)(Zr, Sn, Ti)O3 antiferroelectric ceramics for pulsed capacitors, Journal of Alloys and Compounds (2017), doi: 10.1016/j.jallcom.2017.05.325. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

High discharge energy density and fast release speed of (Pb, La)(Zr, Sn, Ti)O3 antiferroelectric ceramics for pulsed capacitors

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Jie Shen, Xiucai Wang, Tongqing Yang*, Hongsheng Wang, Jing Wei Key Laboratory of Advanced Civil Engineering Materials (Ministry of Education), Functional Materials Research Laboratory, School of Materials Science and

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Engineering, Tongji University, 4800 Cao'an Road, Shanghai, 201804, China

ABSTRACT

The pulsed capacitors have been widely applied in advanced electronic and electrical systems due to their high discharge energy density and fast release speed. In this

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article, the antiferroelectric Pb0.97La0.02(Zr0.60Sn0.35Ti0.05)O3 ceramics were prepared by a rolling method, and the effects of grain size, porosity, thickness as well as electrode area on breakdown strength (EBDS) have been investigated detailedly. The

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discharge properties were directly evaluated by a resistance-inductance-capacitance

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(R-L-C) circuit rather than hysteresis loops. Both underdamped and overdamped state were achieved to explore the current variations, and results show that at 150 kV/cm, the real discharge energy density (Wdis) is about 1.66 J/cm3, much smaller than that indirectly obtained from the hysteresis loops (~2.85 J/cm3, under 100 Hz). The discrepancy is likely caused by the distinct polarization responses to DC and AC voltage of dielectrics. All these consequences demonstrate the potential of antiferroelectric ceramics for pulse power applications. *Corresponding author. E-mail address: [email protected]

ACCEPTED MANUSCRIPT Keywords: antiferroelectric (AFE), breakdown strength (EBDS), R-L-C circuits, underdamped and overdamped state, discharge energy density (Wdis)

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1. Introduction With rapid development of pulse power technology, more and more researches are focused on the dielectrics used for pulsed capacitors due to the severe energy crisis.

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Among these dielectrics, antiferroelectrics (AFEs) have gained ever-increasing

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popularity in the past decades because of their higher energy density and efficiency, smaller remnant polarization, compared with ferroelectrics, which makes AFEs a promising type of dielectrics for pulsed capacitors [1-4]. Till now, various antiferroelectrics based on lanthanum-modified lead zirconate stannate titanate

instance,

at

1120

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(PLZST) have been successfully synthesized and comprehensively studied. For kV/cm,

the

theoretical

energy

storage

density

of

Pb0.97La0.02(Zr0.95Ti0.05)O3 antiferroelectric thin films (1.7 µm, 0.5 mm diameter

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electrode) can reach 12.4 J/cm3 whereas that of (Pb0.88La0.08)(Zr0.91Ti0.09)O3

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antiferroelectric ceramics (250 µm, 7.5 mm diameter electrode) is only 3.04 J/cm3 at 170 kV/cm [5, 6]. Both results were indirectly calculated from hysteresis loops by the following formula:

Wre = ∫

Pmax

Pr

EdP

(1)

Where Wre, Pr, Pmax, E stand for releasable energy density, remnant polarization, maximum polarization and electric field, respectively. However, there are still no accordant test rules on thickness, electrode area that leads to uneven results.

ACCEPTED MANUSCRIPT Quite a great number of studies would often attach significant importance to the theoretical value of Wre simply calculated by P-E hysteresis loops (viewed as a static way), but fail to take the real discharge energy density (Wdis) of direct-current circuits

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into account [5-9]. In fact, the latter is usually much smaller than the former [10, 11]. According to Ran Xu, the difference between Wre and Wdis lies in that the depolarization speed of hysteresis loops is far slower than that of impulse discharge

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process [12]. Namely, materials with high Wre in theory do not necessarily have high

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Wdis in essence. So it is not suitable to evaluate energy release properties just by the estimation on the hysteresis loops, and more factors should be taken into account, such as Wdis, discharge time, circuit parameters (resistance, capacitance and inductance). Therefore, a dynamic test (a sort of R-L-C circuit) is introduced to

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calculate the total released energy density of antiferroelectric capacitors since the pulse circuit itself belongs to the category of R-L-C circuits, which is more persuasive and practical than a static one [13].

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In this study, the antiferroelectric Pb0.97La0.02(Zr0.60Sn0.35Ti0.05)O3 ceramics were

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prepared by rolling method and sintered at different temperatures. The investigations of microstructure, relative density, XRD patterns, P-E hysteresis loops and charge-discharge tests with fitting results are presented, which aims at their energy release characteristics and impacts on the breakdown strength (EBDS). Considering the energy density estimated by hysteresis loops may have its limitations, the real released energy density of antiferroelectric capacitors were measured by the R-L-C circuit with DC voltage instead of hysteresis measurements with AC voltage. Beyond

ACCEPTED MANUSCRIPT that, a typical R-L-C series circuit model and current simulations are discussed to understand the trend of current variations in the discharge process. This dynamic way

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might be closer to the practical working state of pulsed capacitors.

2. Material and methods

Firstly, the powders of PbO (99%), La2O3 (99.99%), ZrO2 (99%), SnO2 (99.5%)

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and TiO2 (99%) were mixed in ethanol via ball-milling for 24 h, according to the

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stoichiometric ratio of Pb0.97La0.02(Zr0.60Sn0.35Ti0.05)O3. To make up for the lead loss during the later sintering, 2 wt% PbO was added. Secondly, the dried mixture was presintered at 900 °C for 2 h with a heating rate of 2 °C/min to take shape the perovskite structure. Thirdly, the presintered compound was ball-milled again for 24 h.

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After drying off, the compound was uniformly blended with polyvinyl alcohol binder (PVA, mass concentration: ~18%) under a mass ratio of compound/PVA= 4:1, and then the slurry was formed. Next, the slurry was rolled into 200 µm films through a

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two-roller machine at a speed of 25 r/min. The desired thickness of the films could be

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achieved by adjusting the gap between the two rollers, as Fig. 1(a) shows. The films were cut into square samples (10 mm×10 mm). Then these samples were calcined at 600 °C for 2 h (heating rate was controlled at 1 °C /min) to burn up the polyvinyl alcohol binder. After that, the samples were sintered at 1050 °C, 1100 °C, 1150 °C, 1200 °C and 1250 °C for 2 h, respectively. Finally, all the samples were polished to 100-150 µm and both sides were coated with rounded silver electrodes of 2-6 mm in diameter for electrical measurement.

ACCEPTED MANUSCRIPT The Archimedes method was employed to calculate the relative density. The surface topography of the samples was performed by a scanning electron microscopy (SEM, JSM EMP-800; JEOL, Tokyo, Japan) and the phase structure was examined

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via the X-ray diffractometer (D8 Advance, Bruker, Karlsruhe, Germany). Hysteresis loops (P-E curves) were characterized under 100 Hz at room temperature by a precision ferroelectric analyzer (Premier II, Radiant Technologies Inc.), equipped with

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a high voltage power supply (Trek Model 663A). Fig. 2 displays the circuit for

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charge-discharge test set up by ourselves. The current is captured and converted by an oscilloscope connected with a Pearson current monitor (Pearson Electronics, Inc., Palo Alto, CA, U.S.A. Model: 6595), seen in Fig. 1(b). The overdamped state is obtained by applying an external resistor (300 Ω) whereas the underdamped one is

circuit is about 3.5 Ω.

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achieved when removing the external resistor. The total internal impedance of the

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3. Discussion

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The surface topography of the samples can be seen clearly from Figs. 3(a)-(e). We can jump to conclusions that the sintering temperature has a significant influence on sample’s porosity and grain size. In Figs. 3(a)-(c), porous characteristics are obvious owe to the incomplete growth of grains as well as the volatilization of organic residues. These samples usually are pale in color and fragile in mechanical strength. With sintering temperature increases, fewer pores and larger grains can be observed. Furthermore, the grain sizes in Fig. 3(e) are more uniform than those in Fig. 3(d).

ACCEPTED MANUSCRIPT The optimum sintering temperature was decided by monitoring the average relative density achieved for two-hour dwell time with heating/cooling rates of 2 °C/min from 1050-1250 °C, seen in Fig. 3(f). The samples sintered at 1250 °C have

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the highest average relative density of 94.8%, while the value is only 81.2% at 1050 °C. Normally, dense samples have high mechanical strength and good abrasion resistance with yellow transparent appearance.

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As is manifested in Fig. 4, all the peaks are marked in the XRD patterns, which

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ascertains that the majority is tetragonal phase of the perovskite structure [14]. In the enlarged region near 2θ=44⁰, the split peaks (200) and (002) that depicts the tetragonal phase can be easily captured as the sintering temperature increases (from 1100-1250 ⁰C). It is interesting that the peak (002) cannot be observed apparently in

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the first sample as a result of the lower sintering temperature (1050 °C). Specifically, nonuniform element distribution may arise since low sintering temperature can lead to a poor ability of phase diffusion.

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Figs. 5(a)-(f) illustrate the samples’ double hysteresis loops under 100 Hz with

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different sintering temperatures, and the variations of remnant polarization (Pr). Ideally, antiferroelectrics have no remnant polarization but nearly all the samples have small values of Pr. It is likely that there is still a very small amount of electric-field induced ferroelectric phase when removing the electric field [15, 16]. Meanwhile, for electric domains, the existence of defects in a crystal as well as the mechanical constraints among grains and boundaries, make them impossible to reverse thoroughly. When the sintering temperature is 1050 °C, though, the hysteresis loop is evidently

ACCEPTED MANUSCRIPT not closed, for the structure is not dense [17]. With higher sintering temperature, phase diffusion tends to be easier and more uniform, which causes the decrease of porosity and Pr. On the contrary, Pmax increases

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continuously from ~18 µC/cm2 to ~35 µC/cm2 in Fig. 6(a), attributing to larger grain size and density. Meanwhile, the Wre keeps increasing with the higher sintering temperature and reaches maximum ~4.1 J/cm3 at 1250 °C, as showed in Fig 6(b).

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As for EBDS, the value is largely affected by the grain size, porosity and external

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test conditions (such as samples’ thickness, electrode area) [18-20]. Generally, ceramics of fine grains and dense structure tend to have high values of EBDS [21]. Consequently, the value of EBDS increases firstly with the shrinkage of porosity, and maximizes when sintered at 1150 °C. We can see from the Figs. 3(a)-(c) that the grain

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size changes little and thus the EBDS is dominated by porosity in such circumstances. After that, the value of EBDS will reduce notably from ~230 kV/cm to ~180 kV/cm in Fig. 6(a), indicating that the growth of grain size may have adverse effects on the

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breakdown strength. Although samples sintered at 1250 °C have the highest relative

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density, their mean values of EBDS are still smaller than those of sintered at 1150 °C. The most likely explanation is that grain size effect accounts for more proportion in breakdown strength if no great density disparity exists among samples. Besides the influence of grain size and porosity on EBDS, some other factors

(electrode area and samples’ thickness) are also investigated. Approximately, the breakdown strength reduces linearly, which can be clearly seen in Fig. 6(c) and (d). The values of Pmax, EBDS, theoretical Wre and other parameters are summarized in

ACCEPTED MANUSCRIPT Table 1. There is no doubt that defects are inevitable in ceramics, and the thinner size, smaller electrode area tend to lessen the number or occurrence probability of defects. Thus the value of EBDS and theoretical Wre are always overestimated to some extent.

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To evaluate the energy release performance of PLZST antiferroelectrics, the R-L-C circuit is introduced and the AFE samples sintered at 1250 °C are chosen to study. Fig. 7(a) is the P-E curve under 100 Hz, and the maximum polarization is about

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35 µC/cm2. Figs. 7(b)-(c) show the discharge current as a function of time in the

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underdamped (with no external resistor) and overdamped (with an external resistor: 300 Ω) state with an initial DC voltage of ~2250 V. The whole underdamped discharge process is faster than the overdamped one (~1 µs vs ~3 µs). It was reported that the discharge time is mainly controlled by the external resistor and the capacitor

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[22]. Meanwhile, the peak current of the overdamped state (~6.5 A) is much smaller than that of the underdamped one (~195 A). Consequently, we can adjust the

degree.

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discharge time and peak current by altering the value of external resistor in some

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Based on the circuit theory, the discharge energy density Wdis could be given by the following formula:

Wdis =

2 R i (t ) dt ∫ V

(2)

Where R, i(t) are the current, resistance of the circuit and V represents the volume of the sample. Thus, the discharge energy density Wdis could be easily calculated via Formula (2). When the overdamped circuit reaches its steady state, Wdis is ~1.66 J/cm3 [Fig. 7(d)], much smaller than that indirectly obtained from the hysteresis loops under

ACCEPTED MANUSCRIPT 100 Hz (~2.85 J/cm3). This phenomenon is similar to other dielectric materials [23, 24], and the discrepancy is likely caused by the distinct polarization responses to DC and AC voltage of dielectrics. So evaluating the energy performance just by hysteresis

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loops with a constant frequency (usually between 1 Hz and 1000 Hz) is inappropriate and deficient. Because both the polarization value and transition field of AFEs will change as the frequency [2, 12, 13], which might lie in the fact that the speed of some

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dielectrics’ polarization relaxation process cannot keep up with that of the electric

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field variations in a transient circuit. More specifically, the value of Wre will decline as the frequency increases.

In the electrostatic field, dielectrics have enough time to reach their steady polarization state, while in a transient circuit, the polarization procedure is incomplete

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and unsaturated. What’s more, the actual energy dissipation in a transient circuit can be quite different from the value predicted by the steady-state model (0.5CU2) [25]. And then the calculation by hysteresis loops with a fixed frequency is invalid to some

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extent. Instead, the charge-discharge test might be closer to the practical working state

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of pulsed capacitors.

Just take a simple R-L-C series circuit model for example (Fig. 8), when t≥0, we

can list the equation according to the circuit theory: L

di 1 + Ri + ∫ idt = U 0 dt C

(3)

Where L, R, C, i represent the inductance, resistance, capacitance and the current of the circuit, respectively. By taking a derivative, the equation can be written as:

L

d 2i di i +R + =0 2 dt dt C

(4)

ACCEPTED MANUSCRIPT This second-order differential equation can be solved via advanced mathematics. To conveniently discuss, we introduce the concept of damping coefficient k ( k =

R C ⋅ ). 2 L

the above differential equation, as described in the following: For the underdamped state (k <1), the current is:

U0

i (t ) =

1− k 2

⋅

C kt 1− k 2 ⋅ exp(− ) ⋅ sin( t) L LC LC

(5)

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①

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Suppose the initial voltage in a capacitor is U0, then we may get the final solutions of

expressed as:

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② For the overdamped state (k >1), the relevant parameters can similarly be

C kt k 2 −1 i (t ) = ) ⋅ sinh( t) ⋅ ⋅ exp(− LC LC k 2 −1 L U0

(6)

From Formulas (5)-(6) and Fig. 9, we can know that if k <1, the current curve

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behaves like a kind of damped shockwave, and then the damped frequency f is

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determined by the following relationship:

f =

1 1− k 2 ⋅ 2π LC

(7)

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So the damped frequency is irrelevant to the charging voltage, simply related to the circuit parameters (R, L and C). If k >1, the current curve shows no oscillating and keeps attenuating after reaching its maximum. Compare Fig. 9(a) and (b), with the total resistance increases, both the current amplitude (from ~195 A to ~6.5 A) and its corresponding time (from 70 ns to 10 ns) reduces. What has been discussed above is the ideal capacitor with constant value. However, for antiferroelectric capacitors, their capacitance will change as the applied

ACCEPTED MANUSCRIPT voltage [26], making the situation even more complicated. During the whole discharge process, if we regard the varied capacitance as a fixed value (average capacitance), then the analysis could be simplified greatly. Therefore, we can still use

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the Formula (6) and the modified Formula (8) to approximately predict the current variations in the R-L-C series circuit. The final fitting results can be seen in Fig. 9 and the specific expressions are:

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(1) Underdamped state: i (t ) = 240.83746 exp(−0.00348t ) ⋅ sin(0.01498t 1.14057 ) ;

i (t ) =

U0 1− k 2

⋅

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Modified formula:

1− k 2 δ C kt ⋅ exp(− ) ⋅ sin( t ), δ > 1 L LC LC

(8)

(2) Overdamped state: i (t ) = 13.24 exp(−0.21154t ) ⋅ sinh(0.21066t ) . These results manifest that we can still approximately predict the trend of current

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Formulas (6) and (8).

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variations in the R-L-C series circuits containing antiferroelectric capacitors based on

4. Conclusion

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In brief, antiferroelectric ceramic capacitors Pb0.97La0.02(Zr0.60Sn0.35Ti0.05)O3 with

high discharge energy density and fast release speed were prepared by a rolling method. The final discharge energy density Wdis calculated from DC charge-discharge tests is smaller than that obtained from hysteresis loops. This discrepancy is likely caused by the distinct polarization responses to DC and AC voltage of dielectrics. Besides, the current variations in R-L-C series circuits containing antiferroelectric capacitors can be approximately predicted by Formulas (6) and (8). So discharge

ACCEPTED MANUSCRIPT current can be adjusted by circuit parameters (R, L, and C) or initial voltage U0. The charge-discharge tests might be closer to the practical working situation of pulsed capacitors than hysteresis loops with fixed frequency. These results indicate the

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potential of antiferroelectric ceramics for pulse power applications.

Acknowledgements

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This work was supported by the National Natural Science Foundation of China (No. 51472181), the Innovation Program of Shanghai Municipal Education Commission (No.

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14ZZ041), and the Open Foundation of National Engineering Research Center of Electromagnetic Radiation Control Materials (ZYGX2014K003-3). We also sincerely acknowledge the Professor Jian Yu, Ke Yang, Zhongbin Pan and Feng Li for enlightening discussions.

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Figure captions and Table 1 Fig. 1. (a) The sketch of the rolling process, (b) the Pearson current monitor. Fig. 2. The circuit for charge-discharge test.

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Fig. 3. (a)-(e) The SEM images of AEF samples sintered at 1050 °C, 1100 °C, 1150 °C, 1200 °C, 1250 °C, respectively, (f) the relative density of samples as a function of sintering temperatures.

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temperature and the enlarged region near 2θ=44⁰.

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Fig. 4. X-ray diffraction patterns of Pb0.97La0.02(Zr0.60Sn0.35Ti0.05)O3 with different sintering

Fig. 5. (a)-(e) Hysteresis loops of Pb0.97La0.02(Zr0.60Sn0.35Ti0.05)O3 at 100 Hz with different sintering temperatures, (f) the variations of remnant polarization (Pr). Fig. 6. (a) Variations of maximum polarization (Pmax) and breakdown strength (EBDS), (b) trend of

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theoretical value Wre at different sintering temperature, (c) electrode diameter dependence of EBDS, (d) thickness dependence of EBDS. Fig. 7. (a) The first quadrant P-E curve of Pb0.97La0.02(Zr0.60Sn0.35Ti0.05)O3 with an AC voltage of

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~2250 V, (b)-(c) the discharge current as a function of time in the underdamped and

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overdamped state with an original DC voltage of ~2250 V, (d) The discharge energy density variations under the overdamped state with an initial DC voltage of ~2250 V.

Fig. 8. The diagram of a simple charge-discharge R-L-C series circuit model. Fig. 9. The real current vs its fitting results in the R-L-C circuit: (a) underdamped, (b) overdamped.

ACCEPTED MANUSCRIPT Table 1 Some parameters of Pb0.97La0.02 (Zr0.60Sn0.35Ti0.05) O3 antiferroelectric ceramics. Thickness(µm)

Average Pmax(µC/cm2)

Average Pr(µC/cm2)

Average EBDS(kV/cm)

Electrode diameter(mm)

Theoretical Wre(J/cm3)

1050°C 1100°C 1150°C 1200°C 1250°C 1250°C

100 100 100 100 100 150

18 22 26 31 35 35

1.85 0.84 0.66 0.62 0.52 0.52

170 190 230 180 200 145

3 3 3 3 3 6

1.41±0.05 2.32±0.07 2.63±0.10 3.22±0.06 4.10±0.10 2.85±0.10

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Sintering temperature

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Figures:

slurry

(b)

Thickness

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Fig. 1. (a) and (b)

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wire

rotating

rotating

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(a)

A

Pearson Current Monitor

Vacuum Switch

B

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DC High Voltage

C

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Sample External resistor

Fig. 2.

Oscilloscope

100

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Average relative density(%)

(f) 95

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90 85 80 75 70

1000

1050

1100

1150

1200

1250

Sintering temperature(°C)

Fig. 3. (a)-(f)

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Fig. 4.

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30 2 P(µ µC/cm )

2 P(µ µC/cm )

10

10 0

0

-10

-10 -20

-20

-30

-30

-40 -250 -200 -150 -100 -50

0

50

E(kV/cm)

30

(c)

40

1150°C

30

P(µ µC/cm2)

0

0

-10

-10

-20

-30

-30 50

100 150 200 250

E(kV/cm)

(e)

1250°C

2 P(µ µC/cm )

20 10 0

-30

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-10

-40 -250 -200 -150 -100 -50

0

50

100 150 200 250

-40 -250-200-150-100 -50 0 50 100 150 200 250 E(kV/cm)

Remanent polarization(µC/cm2)

0

2.0

(f)

1.0

0.5

0.0 1050

Fig. 5. (a)-(f)

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Pr

1.5

E(kV/cm)

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50 100 150 200 250

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-20

-20

1200°C

10

10

30

(d)

20

-40 -250 -200 -150 -100 -50

0

E(kV/cm)

2

P(µ µC/cm )

-40 -250 -200 -150 -100 -50

100 150 200 250

20

40

1100°C

20

20

40

(b)

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30

40

(a)

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40

1100

1150

T(°C)

1200

1250

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EBDS

270

25

240

20 210

15 10

180

5 0 1100

1150

1200

3.5 3.0 2.5 2.0 1.5 1.0

150 1050

theoretical Wre

1050

1250

1100

240

200 180

140 120

220

1250

(d) Samples sintered at 1250°C (3mm electrode in diameter)

200 180 160

M AN U

160

1200

SC

Breakdown Strength(kV/cm)

Breakdown Strength(kV/cm)

240 220

1150

T(°C)

T(°C)

(c) Samples sintered at 1250°C (thickness: 100µm)

RI PT

30

(b)

4.0 3

35

Wre(J/cm )

Pmax(µ µC/cm2)

4.5

300

Pmax

(a)

Breakdown field(kV/cm)

40

140 120

2mm

3mm

4mm

5mm

Electrode diameter

6mm

100

AC C

EP

TE D

Fig. 6. (a)-(d)

110

120

130

Thickness(µ µm)

140

150

40

200

(a) Thickness: 0.15mm 35 Frequency: 100Hz πmm2 30 Electrode area: 9π

150

Underdamped discharge DC charge voltage: 2250V

(b) 100

Current(A)

25 20 15

50 0 -50

10

-100

5

-150

0 0

20

40

60

80

0

100 120 140 160 180

300

(c)

Overdamped discharge DC charge voltage: 2250V

6

4 3 2 1 0 0

500

1000

1500

2000

(d)

1.5

2500

Time(ns)

900

1.2 0.9

Released energy density

0.6 0.3 0.0

0

3000

500

1000

1500

2000

Time(ns)

EP

TE D

Fig. 7. (a)-(d)

R

2

AC C

1

L

i

3

U

L

R

3

1200

1.66J/cm3

M AN U

Current(A)

5

1.8

600

Time(ns)

SC

7

Released energy density(J/cm 3)

E(kV/cm)

RI PT

P(µ µC/cm2)

ACCEPTED MANUSCRIPT

2

i

C -

t=0

t≥0

Fig. 8.

C U0

2500

3000

ACCEPTED MANUSCRIPT 200

(a) 70ns

150

194.70162

Adj. R-Square

0.9535

100

b c

Value 240.83746

Standard Error 0

-0.00348 0.01498 1.14057

4.81766E-5 2.11509E-5 0

d e

50 0 -50

Only internal resistance: ~3.5Ω Ω

-100

real current curve fitted current curve

-150 0

200

400

600

(b)

10ns

0.23774

Adj. R-Square

0.91425

M AN U

Reduced Chi-Sqr

Value

4 3 2

Standard Error

a

13.24

0

b

-0.21154

4.46135E-5

c

0.21066

4.50937E-5

Additional external resistance: 300Ω Ω

TE D

Current(A)

5

1

y = a*exp(b*t)*sinh(c*t)

Equation

6

800

SC

Time(ns) 7

RI PT

Current(A)

y =b*exp(c*t)*sin(d*t^e)

Equation Reduced Chi-Sqr

real current curve fitted current curve

0 -1

500

AC C

EP

0

1000

1500

Time(ns)

2000

Fig. 9. (a) and (b)

2500

3000

ACCEPTED MANUSCRIPT

Highlights: The Wdis should be calculated by R-L-C circuit instead of hysteresis loops.

●

The underdamped state lasts longer than the overdamped one.

●

Current expression is deduced from the R-L-C series circuit model.

●

Discharge current can be adjusted by circuit parameters or initial voltage.

●

Approximate prediction of the current is given.

AC C

EP

TE D

M AN U

SC

RI PT

●