In the practical application of capacitors, especially in pulsed application, recoverable energy is a key parameter, which represents the ability to store energy. However, many existing evaluation methods for energy storage calculation have not been systematically implemented and comprehensively understood. In this work, four methods were applied to calculate the energy storage in linear, ferroelectric, and antiferroelectric capacitors. All methods were valid when the linear capacitor was examined. In terms of the ferroelectric capacitor, the method of equivalent parameter using DC-bias capacitance was infeasible under the high voltage owing to a massive decrease in field-dependent capacitance. As for the antiferroelectric capacitor with noticeable hysteresis, the maximum of energy storage was obtained by the method of integration of hysteresis loop, while the lower one was obtained in the fast discharge condition by the method of integration of UI (product of voltage and current). In summary, for different materials, both test conditions and calculation methods should be considered to get accurate energy storage, which best fits the working conditions.

Over the past 260 years, capacitors have undergone tremendous development, especially after the time when the vacuum tube was invented.1 As pulsed power technology has been widely applied in electric armor, electric guns, particle beam accelerators, high power microwave sources, nuclear technique, health care, and other electric power systems,2,3 there is a great demand for capacitors with higher energy storage, higher breakdown strength, and longer lifetime.4–6 Just in the aspect of energy storage calculation, many methods have been considered, which may yield quite different results. In this paper, these classic methods are summarized.

This is a widely used method based on Eq. (1) or (2),

$Wrec=−∫DmaxD0E⋅dD,$
(1)
$Energy=−∫QmaxQ0UdQ,$
(2)

where Wrec is the recoverable energy density, which means

$Wrec=Energy/Vol.$
(3)

E, D, U, Q, and Vol are the electric field, electric displacement, voltage across the capacitor, charges on the electrode, and volume of the capacitor, respectively.

Moreover, the test method is usually based on the Sawyer–Tower circuit,10 in which the input voltage signal is a triangle or sine wave with a fixed frequency (0.01 Hz–1000 Hz).

As is known to us all, the output power of a capacitor can be integrated over time to obtain the energy, which is expressed as

$Energy=∫0t′UIdt=−∫QmaxQ0UdQ,$
(4)

where I is the current in the circuit and t′ is the time when U reaches 0.

Moreover Eq. (4) shares the same expression with Eq. (2), that is, they are the same theoretically. However, this method is actually more flexible than method A because it has no limit on the test method, as long as the voltage and current signals can be detected.

In the discharge process of the RLC circuit, most of the energy stored in a capacitor is consumed by the resistance, while the rest is attributed to the dielectric loss, if it exists. The released energy, the former one, can be calculated by

$Energy=∫0∞I2Rdt,$
(5)

where R is the resistance in the circuit.

It is a classic formula that is usually used in a linear capacitor; see Eq. (6). As for nonlinear dielectric capacitors, DC-bias capacitance and the amount of charge released can be used as parameters in

$Energy=12C′U02=12Q′U0,$
(6)

where C′, U0, and Q′ are the DC-bias capacitance, charging voltage, charges on the electrode under U0, respectively.

Ferroelectric (400 nF, Torch Electron) and antiferroelectric (150 nF, TDK) capacitors were chosen as typical nonlinear capacitors. The hysteresis loops of each capacitor were measured by a commercial device (TF analyzer 2000, aixACCT) at room temperature. The discharge signals, i.e., current in circuit and voltage across the capacitor, were captured by a Rogowski coil (CWT MiniHF, PEM) and high-voltage differential probes (THDP0100, Tektronix) and recorded by an oscilloscope (MDO3024, Tektronix).17 The DC-bias capacitance at different voltages was measured by an LCR meter (E4980A, Agilent), which outputted a fixed test signal at 1 V and 100 Hz. As a comparison, energy storage in a linear polypropylene film capacitor (100 nF, KEMET) was investigated by using the same method.

The experimental results of the polypropylene film capacitor are shown in Fig. 1. It exhibits absolutely linear dielectric response in Figs. 1(a) and 1(b). The characteristics of the discharge signals at different voltages are similar, so only 600 V-discharge signals are shown in Figs. 1(c) and 1(d). The loads of circuits 1, 2, and 3 were slightly different; only the length and winding of the wire were increased. However, discharge signals were totally controlled by circuit parameters, i.e., resistance and inductance, which led to strikingly different discharge rates and signal waveforms.18 Following method B, the authors integrated the current signals over time and then plotted the results vs voltage signals, which are shown in Fig. 2. Compared with the hysteresis loop, there is no obvious quantitative difference in recoverable energy, which is detailed in Table I. The values of energy storage calculated by other methods are also shown in Table I. In addition, in different circuits, methods B, C, and D could be reused. Here, the authors took the value calculated by method A as a reference. All four methods got similar results under the same voltage (±6.9%), no matter which circuit was used. This is because the polypropylene film has nearly no dielectric loss and a fixed dielectric constant within a certain range.19 Here, ±10% was set as the conservative limit that distinguished the difference of the calculated energy.

FIG. 1.

Typical experimental results of the linear capacitor: (a) hysteresis loops at 600 V, 800 V, and 1000 V. The inset table shows the recoverable energy that is calculated by method A, (b) the bias field capacitance vs bias voltage curve during the rise and fall of voltage, and discharge signals (c) current and (d) voltage with standard deviation in three circuits with different loads at 600 V.

FIG. 1.

Typical experimental results of the linear capacitor: (a) hysteresis loops at 600 V, 800 V, and 1000 V. The inset table shows the recoverable energy that is calculated by method A, (b) the bias field capacitance vs bias voltage curve during the rise and fall of voltage, and discharge signals (c) current and (d) voltage with standard deviation in three circuits with different loads at 600 V.

Close modal
FIG. 2.

Comparison of recoverable energy among circuits 1–3 and hysteresis loops at a voltage of (a) 600 V, (b) 800 V, and (c) 1000 V.

FIG. 2.

Comparison of recoverable energy among circuits 1–3 and hysteresis loops at a voltage of (a) 600 V, (b) 800 V, and (c) 1000 V.

Close modal
TABLE I.

Energy storage in the linear capacitor.a

600 V800 V1000 V
0.017 91 (100.0%) 0.031 50 (100.0%) 0.049 09 (100.0%)
Circuit1 0.017 01 (95.0%) 0.031 44 (99.8%) 0.048 92 (99.7%)
Circuit2 0.017 34 (96.8%) 0.031 82 (101.0%) 0.048 80 (99.4%)
Circuit3 0.017 69 (98.8%) 0.031 64 (100.4%) 0.049 48 (100.8%)
Circuit1 0.016 67 (93.1%) 0.029 77 (94.5%) 0.046 88 (95.5%)
Circuit2 0.018 36 (102.5%) 0.032 43 (103.0%) 0.050 55 (103.0%)
Circuit3 0.017 98 (99.9%) 0.031 98 (101.5%) 0.050 14 (102.1%)
C′ 0.017 91 (100.0%) 0.031 92 (101.3%) 0.049 78 (101.4%)
Q′-Circuit1 0.017 66 (98.6%) 0.031 43 (99.8%) 0.048 69 (99.0%)
Q′-Circuit2 0.017 68 (98.7%) 0.031 79 (100.9%) 0.049 11 (100.0%)
Q′-Circuit3 0.017 84 (99.6%) 0.031 80 (101.0%) 0.049 63 (101.1%)
600 V800 V1000 V
0.017 91 (100.0%) 0.031 50 (100.0%) 0.049 09 (100.0%)
Circuit1 0.017 01 (95.0%) 0.031 44 (99.8%) 0.048 92 (99.7%)
Circuit2 0.017 34 (96.8%) 0.031 82 (101.0%) 0.048 80 (99.4%)
Circuit3 0.017 69 (98.8%) 0.031 64 (100.4%) 0.049 48 (100.8%)
Circuit1 0.016 67 (93.1%) 0.029 77 (94.5%) 0.046 88 (95.5%)
Circuit2 0.018 36 (102.5%) 0.032 43 (103.0%) 0.050 55 (103.0%)
Circuit3 0.017 98 (99.9%) 0.031 98 (101.5%) 0.050 14 (102.1%)
C′ 0.017 91 (100.0%) 0.031 92 (101.3%) 0.049 78 (101.4%)
Q′-Circuit1 0.017 66 (98.6%) 0.031 43 (99.8%) 0.048 69 (99.0%)
Q′-Circuit2 0.017 68 (98.7%) 0.031 79 (100.9%) 0.049 11 (100.0%)
Q′-Circuit3 0.017 84 (99.6%) 0.031 80 (101.0%) 0.049 63 (101.1%)
a

The volume of the capacitor is 8.544 cm3.

b

The percentages in brackets represent the ratio of energy to that calculated by method A.

Figure 3(a) shows the hysteresis loops for the ferroelectric capacitor, measured at different voltages. The shape of the loops is slim. However, the slope decreases significantly as voltage rises. Figure 3(b) also shows decreased DC-bias capacitance as voltage rises and a slight mismatch in capacitance during voltage rises and falls, which indicates that the ferroelectric capacitor is partly in the paraelectric state at room temperature20,21 or is in the relaxor ferroelectric state.22Figures 3(c) and 3(d) show the current and voltage signals in two circuits with totally different resistances. In circuit 2, the discharge of the capacitor lasted about 30 ms, while it only took about 1 microsecond to discharge all charges in circuit 1. In addition, the rest of the process in the circuit is called oscillation. In Fig. 4, the shaded area, the recoverable energy calculated by method B, is almost equal to that in the hysteresis loop, which is detailed in Table II. Owing to its small dielectric loss, method C also got similar results (±3.3%). It is worth mentioning that, in relaxor ferroelectric materials with obvious hysteresis,23,24 the hysteresis loops vary under different test frequencies. In other words, methods A, B, and C may get different recoverable energies in such material, similar to the antiferroelectric capacitor mentioned below. As for method D, especially the one with the equivalent parameter C′, the percentage tumbled to 81.4% when the voltage rose to 2500 V, which means the method is a failure. This is due to the decreased DC-bias capacitance. In other words, this method is not suitable for nonlinear materials or needs further correction. As for the ones with the equivalent parameter Q′, they are highly dependent on the linearity of the capacitor during the process of discharge. Among these materials whose capacitance decreases with the increasing external field, mathematically like a concave function, the calculated results are higher than method A’s theoretically.

FIG. 3.

Typical experimental results of the ferroelectric capacitor: (a) hysteresis loops at 1500 V, 2000 V, and 2500 V. The inset table shows the recoverable energy that is calculated by method A, (b) the bias field capacitance vs bias voltage curve during the rise and fall of voltage, and discharge signals (c) current and (d) voltage in three circuits at 1500 V.

FIG. 3.

Typical experimental results of the ferroelectric capacitor: (a) hysteresis loops at 1500 V, 2000 V, and 2500 V. The inset table shows the recoverable energy that is calculated by method A, (b) the bias field capacitance vs bias voltage curve during the rise and fall of voltage, and discharge signals (c) current and (d) voltage in three circuits at 1500 V.

Close modal
FIG. 4.

Comparison of recoverable energy among circuits 1 and 2 and hysteresis loops at a voltage of (a) 1500 V, (b) 2000 V, and (c) 2500 V.

FIG. 4.

Comparison of recoverable energy among circuits 1 and 2 and hysteresis loops at a voltage of (a) 1500 V, (b) 2000 V, and (c) 2500 V.

Close modal
TABLE II.

Energy storage in the ferroelectric capacitor.a

1500 V2000 V2500 V
0.3818 (100.0%) 0.6305 (100.0%) 0.9186 (100.0%)
Circuit1 0.4053 (106.2%) 0.6077 (96.4%) 0.8529 (92.8%)
Circuit2 0.3894 (102.0%) 0.6367 (101.0%) 0.9138 (99.5%)
Circuit1 0.3802 (99.6%) 0.6394 (101.4%) 0.9487 (103.3%)
Circuit2 0.3832 (100.4%) 0.6314 (100.1%) 0.9343 (101.7%)
C′ 0.3493 (91.5%) 0.5421 (85.8%) 0.7477 (81.4%)
Q′-Circuit1 0.4166 (109.1%) 0.6613 (104.9%) 0.9649 (105.0%)
Q′-Circuit2 0.3786 (99.2%) 0.6401 (101.5%) 0.9614 (104.7%)
1500 V2000 V2500 V
0.3818 (100.0%) 0.6305 (100.0%) 0.9186 (100.0%)
Circuit1 0.4053 (106.2%) 0.6077 (96.4%) 0.8529 (92.8%)
Circuit2 0.3894 (102.0%) 0.6367 (101.0%) 0.9138 (99.5%)
Circuit1 0.3802 (99.6%) 0.6394 (101.4%) 0.9487 (103.3%)
Circuit2 0.3832 (100.4%) 0.6314 (100.1%) 0.9343 (101.7%)
C′ 0.3493 (91.5%) 0.5421 (85.8%) 0.7477 (81.4%)
Q′-Circuit1 0.4166 (109.1%) 0.6613 (104.9%) 0.9649 (105.0%)
Q′-Circuit2 0.3786 (99.2%) 0.6401 (101.5%) 0.9614 (104.7%)
a

The volume of the capacitor is 1.766 cm3.

b

The percentages in brackets represent the ratio of energy to that calculated by method A.

The unipolar hysteresis loops of the antiferroelectric capacitor at different voltages are displayed in Fig. 5(a). The recoverable energy sharply increased as the field enhanced. The corresponding energy is recorded in the inset of Fig. 5(a). The DC-bias capacitance is shown in Fig. 5(b). Around 650 V, the DC-bias capacitance reached a maximum. Above 650 V, the curves are almost coincident; otherwise, the curves are separated. Similar results were obtained on a modified lead-zirconate (PbZrO3) capacitor.25Figures 5(c) and 5(d) show the discharge signals of the three circuits. In circuits 1 and 2, the discharge time is similar to each other, about 1.1 ms (the time taken for the voltage to drop to zero for the first time). In circuit 3, it took about 100 ms to discharge. However, these processes were too fast to release all stored energy compared with that shown in Fig. 5(a) (250 000 ms), see Fig. 6. This phenomenon is explained by the motion of the domain wall.14,26 In the process of quick discharge, dipole reversion cannot keep up with the external field, which causes dielectric loss. In brief, the shorter the time, the more the dielectric loss. The results of method C given in Table III support the point because the released energy was consumed by the resistor, while the rest was wasted as heat. That is to say, the energy calculated by method C was the recoverable energy. Besides, in method D, C′ showed the same problem as the ferroelectric capacitor. When the voltage reached 900 V, the percentage was only 48.4%. This phenomenon is also a consequence of the reduced DC-bias capacitance. In contrast, in method D, Q′ yielded a slightly smaller result than that in the ferroelectric capacitor owing to the convex function property of the hysteresis loop.

FIG. 5.

Typical experimental results of the antiferroelectric capacitor: (a) hysteresis loops at 400 V, 600 V, and 900 V. The inset table shows the recoverable energy that is calculated by method A, (b) the bias field capacitance vs bias voltage curve during the rise and fall of voltage, discharge signals (c) current and (d) voltage in three circuits at 600 V.

FIG. 5.

Typical experimental results of the antiferroelectric capacitor: (a) hysteresis loops at 400 V, 600 V, and 900 V. The inset table shows the recoverable energy that is calculated by method A, (b) the bias field capacitance vs bias voltage curve during the rise and fall of voltage, discharge signals (c) current and (d) voltage in three circuits at 600 V.

Close modal
FIG. 6.

Comparison of recoverable energy among circuits 1–3 and hysteresis loops at the voltage of (a) 400 V, (b) 600 V, and (c) 900 V.

FIG. 6.

Comparison of recoverable energy among circuits 1–3 and hysteresis loops at the voltage of (a) 400 V, (b) 600 V, and (c) 900 V.

Close modal
TABLE III.

Energy storage in the antiferroelectric capacitor.a

400 V600 V900 V
0.015 09 (100.0%) 0.052 14 (100.0%) 0.132 9 (100.0%)
Circuit1 0.013 84 (91.7%) 0.046 64 (89.5%) 0.116 0 (87.3%)
Circuit2 0.013 83 (91.7%) 0.046 47 (89.1%) 0.114 3 (86.0%)
Circuit3 0.014 48 (96.0%) 0.049 83 (95.6%) 0.122 2 (91.9%)
Circuit1 0.014 62 (96.9%) 0.044 61 (85.6%) 0.117 2 (88.2%)
Circuit2 0.014 43 (95.6%) 0.044 90 (86.1%) 0.114 7 (86.3%)
Circuit3 0.014 41 (95.5%) 0.050 76 (97.4%) 0.125 2 (94.2%)
C′ 0.014 46 (95.8%) 0.049 12 (94.2%) 0.064 30 (48.4%)
Q′-Circuit1 0.013 94 (92.4%) 0.045 43 (87.1%) 0.119 0 (89.5%)
Q′-Circuit2 0.013 93 (92.3%) 0.045 35 (87.0%) 0.118 3 (89.0%)
Q′-Circuit3 0.014 37 (95.2%) 0.046 06 (88.3%) 0.118 7 (89.3%)
400 V600 V900 V
0.015 09 (100.0%) 0.052 14 (100.0%) 0.132 9 (100.0%)
Circuit1 0.013 84 (91.7%) 0.046 64 (89.5%) 0.116 0 (87.3%)
Circuit2 0.013 83 (91.7%) 0.046 47 (89.1%) 0.114 3 (86.0%)
Circuit3 0.014 48 (96.0%) 0.049 83 (95.6%) 0.122 2 (91.9%)
Circuit1 0.014 62 (96.9%) 0.044 61 (85.6%) 0.117 2 (88.2%)
Circuit2 0.014 43 (95.6%) 0.044 90 (86.1%) 0.114 7 (86.3%)
Circuit3 0.014 41 (95.5%) 0.050 76 (97.4%) 0.125 2 (94.2%)
C′ 0.014 46 (95.8%) 0.049 12 (94.2%) 0.064 30 (48.4%)
Q′-Circuit1 0.013 94 (92.4%) 0.045 43 (87.1%) 0.119 0 (89.5%)
Q′-Circuit2 0.013 93 (92.3%) 0.045 35 (87.0%) 0.118 3 (89.0%)
Q′-Circuit3 0.014 37 (95.2%) 0.046 06 (88.3%) 0.118 7 (89.3%)
a

The volume of the capacitor is 0.1567 cm3.

b

The percentages in brackets represent the ratio of energy to that calculated by method A.

The energy storage of three types of dielectrics, polypropylene, ferroelectric, and antiferroelectric, was investigated and compared systematically by four methods. Method A is the most widely used one that can get accurate results when the discharge rate is slow. However, in pulsed power applications with noticeable hysteresis materials, the dipole reversion cannot keep up with the external field, leading to energy dissipation. In such situations, methods B and C could make up for this shortcoming, neglecting the dielectric loss of materials. Method D may not be appropriate for nonlinear dielectrics, but it may be adopted as a reference for qualitative estimation to some extent.

All authors contributed equally to this work.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

This work was supported by the National Natural Science Foundation of China (Grant No. 11774366).

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