A Charge-Equalization Circuit with an Intermediate Resonant Energy Tank

In this paper, we propose a novel charge-equalization circuit for series-connected batteries. The circuit is composed of an intermediate resonant energy tank, formed by an inductor and a capacitor, and selection switches connected to each battery. The advantage of the proposed circuit is that an exchange of unbalanced charge can be established between any two batteries via the intermediate energy tank. In addition, more than one adjacent battery can serve as a charge-transference party on either side through proper activation of the associated selection switches. Moreover, by virtue of the resonant tank, zero-current switching can be achieved to significantly reduce the transference losses. A laboratory circuit with a control unit was designed for eight 2.5 Ah lithium iron phosphate (LiFePO4) batteries in series. The test results demonstrate that the proposed charge-equalization circuit can realize charge balance effectively. With the charge-equalization circuit, the worst open-circuit voltage difference can be reduced to less than 20 mV under offline conditions, which helps the batteries operate more efficiently.


Introduction
Rechargeable batteries have become an integral part of our daily life with the advance of information technology [1]. Lithium-ion batteries, in particular, have been applied in various appliances, ranging from portable devices to high-power electric vehicles and even energy storage systems [2]. However, a single battery cell alone cannot comprehensively cover the range of voltage and power levels [3,4]. Accordingly, numerous battery cells are interconnected in series or parallel to comprise a battery pack, which leads to the charge imbalance issue that frequently accompanies interconnected battery packs. Moreover, a charge imbalance can result in the overcharge/discharge of individual cells, causing the battery's service capacity and service life to decay and the charge and discharge efficiencies to decline [5,6]. The imbalance problem can deteriorate the battery pack severely since the batteries age differently. Moreover, imbalance precipitates aging. In order to avoid irreversible damage to the battery pack in early stages, many studies on the charge balance circuit have been done [7][8][9].
Charge balance schemes can be divided into 'passive balance' and 'active balance' schemes. Passive balance consumes energy from the resistors to provide balance, while active balance delivers energy from the capacitors, inductors, or converters [10]. However, the classification of 'passive' and 'active' is somewhat ambiguous and lacks explanatory power. Here, from the perspective of energy flow between batteries, energy dissipation, energy control, and energy transfer are used to define the charge balance categories shown in Figure 1 [11,12]. In the figure, the "balanced point" denotes the target state-of-charge (SOC) that is shared among the batteries. The red/blue shaped areas indicate the charge that has to be released/accepted from the individual battery to achieve the target. The "energy

Energy Dissipation Circuit
The high-SOC battery dissipates excess charge, denoted as the shaded area, onto some auxiliary components, such as resistors or zener diodes, until its charge is aligned with that of the low-SOC battery. The red arrow indicates that the energy is simply dissipated as heat into the ambience. The energy dissipation scheme is simple and has a low cost [13]. However, the balance efficiency is low and heat management should be considered discretely [14].

Energy Control Circuit
The energy control circuit adjusts the charging and discharging currents for each battery. To achieve the charge balance, the low-SOC battery is charged with a larger current and vice versa. Conversely, the low-SOC battery is discharged with a smaller current while the high-SOC battery is discharged with a larger current [15]. The black arrows of different thicknesses indicate different current (energy) levels. The energy control circuit's advantages include high conversion efficiency, less energy circulating into/out of the batteries, and little heat generation; however, the complexity of the control scheme and difficulties with the integrated circuit are shortcomings [16].

Energy Transfer Circuit
The energy transfer circuit transfers the charge from a high-SOC battery to a low-SOC battery through power converters or storage components. Commonly applied energy transfer circuits include capacitor-type and inductor-type balanced circuits, which have the advantages of a simple structure and a low cost [17]. Through the power switches, the high-SOC battery releases the charge to the capacitor or the inductor, while the low-SOC battery receives the charge in order to balance the charge. However, a surge in the current or voltage will occur at the switches, reducing their reliability. Moreover, if the energy is not transferred to adjacent batteries, sequences of transfers are required, resulting in an accumulation of energy losses [18,19]. In light of the employment of inductor and capacitor (LC) components, with the inductor's ability to suppress the current ripple as well as the capacitor's ability to retard the voltage spike, the authors in [20,21] propose an efficient approach to charge equalization with the energy transfer circuit for series-connected batteries.
As there is a positive correlation between battery voltage and SOC, we use the terms 'battery voltage' and 'battery SOC' interchangeably in this paper. An intermediate energy storage tank is composed of an inductor and a capacitor. By activating the appropriate selection switches, the battery

Energy Dissipation Circuit
The high-SOC battery dissipates excess charge, denoted as the shaded area, onto some auxiliary components, such as resistors or zener diodes, until its charge is aligned with that of the low-SOC battery. The red arrow indicates that the energy is simply dissipated as heat into the ambience. The energy dissipation scheme is simple and has a low cost [13]. However, the balance efficiency is low and heat management should be considered discretely [14].

Energy Control Circuit
The energy control circuit adjusts the charging and discharging currents for each battery. To achieve the charge balance, the low-SOC battery is charged with a larger current and vice versa. Conversely, the low-SOC battery is discharged with a smaller current while the high-SOC battery is discharged with a larger current [15]. The black arrows of different thicknesses indicate different current (energy) levels. The energy control circuit's advantages include high conversion efficiency, less energy circulating into/out of the batteries, and little heat generation; however, the complexity of the control scheme and difficulties with the integrated circuit are shortcomings [16].

Energy Transfer Circuit
The energy transfer circuit transfers the charge from a high-SOC battery to a low-SOC battery through power converters or storage components. Commonly applied energy transfer circuits include capacitor-type and inductor-type balanced circuits, which have the advantages of a simple structure and a low cost [17]. Through the power switches, the high-SOC battery releases the charge to the capacitor or the inductor, while the low-SOC battery receives the charge in order to balance the charge. However, a surge in the current or voltage will occur at the switches, reducing their reliability. Moreover, if the energy is not transferred to adjacent batteries, sequences of transfers are required, resulting in an accumulation of energy losses [18,19]. In light of the employment of inductor and capacitor (LC) components, with the inductor's ability to suppress the current ripple as well as the capacitor's ability to retard the voltage spike, the authors in [20,21] propose an efficient approach to charge equalization with the energy transfer circuit for series-connected batteries.
As there is a positive correlation between battery voltage and SOC, we use the terms 'battery voltage' and 'battery SOC' interchangeably in this paper. An intermediate energy storage tank is composed of an inductor and a capacitor. By activating the appropriate selection switches, the battery Energies 2020, 13, 6566 3 of 14 with the highest SOC in the pack can transfer the excess charge to the LC energy tank. Subsequently, as the battery with the lowest SOC is connected to the energy tank, the voltage gap between the capacitor and the battery will complete the transfer of unbalanced charge. As a result, the process for balancing the highest-and lowest-SOC batteries can be established using only two charge transference procedures. In addition, zero-current switching (ZCS) can be achieved to effectively reduce the switching losses at the selection switches. Our experimental results demonstrate that the voltage difference can be reduced to less than 20 mV by the proposed circuit and the balance strategy. Figure 2 shows the proposed charge-transfer balance circuit with an LC intermediate resonant energy tank. The proposed circuit is inspired by the switched-capacitor method, whose main problems are high current spikes and an elongated balancing time. As a result, the resonant energy tank, which is composed of an inductor L r and a capacitor C r , is capable of suppressing the surges in the current and the voltage, respectively [22][23][24]. A total of n batteries are connected in series to form the battery pack. Each battery, e.g., the kth, can be attached to the resonant tank by its associated AC switches, i.e., S ka , S kb and S kc , and S kd . The power switches are referred to as 'selection switches' here because they are responsible for sifting out a specific battery or even adjacent batteries to mount on the intermediate energy tank and process the charge transfer. Moreover, as the resonant current rewinds back to zero, the selection switches secure the breach to turn ZCS off, reducing the switching loss. energy tank. The proposed circuit is inspired by the switched-capacitor method, whose main problems are high current spikes and an elongated balancing time. As a result, the resonant energy tank, which is composed of an inductor Lr and a capacitor Cr, is capable of suppressing the surges in the current and the voltage, respectively [22][23][24]. A total of n batteries are connected in series to form the battery pack. Each battery, e.g., the k th , can be attached to the resonant tank by its associated AC switches, i.e., Ska, Skb and Skc, and Skd. The power switches are referred to as 'selection switches' here because they are responsible for sifting out a specific battery or even adjacent batteries to mount on the intermediate energy tank and process the charge transfer. Moreover, as the resonant current rewinds back to zero, the selection switches secure the breach to turn ZCS off, reducing the switching loss.

Proposed Charge-Equalization Circuit
The circuit has no impact on the charging/discharging functions; that is, charge equalization can be performed concurrently as the AC breaker on the main charge/discharge path is closed. The control of the activation of selection switches and the breaker can be integrated into and performed by a control unit.
The elongated balancing time issue arises from the diminished battery voltage difference as balance processing, which degrades the charge transference current. When the battery voltage difference is relatively large, the capacitor's voltage, which is a replica of the high-SOC battery's voltage, is sufficiently higher to transfer charge to the low-SOC battery in a one-to-one battery charge transfer. However, as the battery voltage difference gradually diminishes, the balance current decreases accordingly. To solve this problem, in this paper we propose an enhanced balance mode. Two adjacent batteries are connected in series to replenish the capacitor, which in turn increases the balance capacity.
To illustrate the circuit's operation, we take the highest voltage battery Bi and the lowest voltage battery Bj as examples to perform an operation mode analysis. The on-state resistance of the selection switches and the stray resistances through the current's path are aggregated into a lumped resistance Rs.   The circuit has no impact on the charging/discharging functions; that is, charge equalization can be performed concurrently as the AC breaker on the main charge/discharge path is closed. The control of the activation of selection switches and the breaker can be integrated into and performed by a control unit.
The elongated balancing time issue arises from the diminished battery voltage difference as balance processing, which degrades the charge transference current. When the battery voltage difference is relatively large, the capacitor's voltage, which is a replica of the high-SOC battery's voltage, is sufficiently higher to transfer charge to the low-SOC battery in a one-to-one battery charge transfer. However, as the battery voltage difference gradually diminishes, the balance current decreases accordingly. To solve this problem, in this paper we propose an enhanced balance mode. Two adjacent batteries are connected in series to replenish the capacitor, which in turn increases the balance capacity.
To illustrate the circuit's operation, we take the highest voltage battery B i and the lowest voltage battery B j as examples to perform an operation mode analysis. The on-state resistance of the selection switches and the stray resistances through the current's path are aggregated into a lumped resistance R s .

Normal Balance Mode
Normal balance mode features a one-to-one energy transfer between batteries. Figure 3 shows the theoretical waveforms. v Bi and v Bj are the instantaneous voltages of B i and B j . S ix and S jx are the switches connected to B i and B j , respectively, where x denotes a, b, c, and d. v ix and v jx are the gating signals controlling the power switches S ix and S jx , respectively. We use the exaggeratedly diminished voltage difference to illustrate that the transferred charges Q 1 and Q 2 , which represent the accumulated charging/discharging current, are reduced greatly when the voltage difference nearly vanishes, i.e., the ability to transfer charge is becoming limited. The small voltage difference is also the reason for the decreases in i L and v c . According to the switching actions of the selection switches, the charge transfer is decomposed into four intervals. Figure 4 depicts the current conduction path of each interval.

Normal Balance Mode
Normal balance mode features a one-to-one energy transfer between batteries. Figure 3 shows the theoretical waveforms. vBi and vBj are the instantaneous voltages of Bi and Bj. Six and Sjx are the switches connected to Bi and Bj, respectively, where x denotes a, b, c, and d. vix and vjx are the gating signals controlling the power switches Six and Sjx, respectively. We use the exaggeratedly diminished voltage difference to illustrate that the transferred charges Q1 and Q2, which represent the accumulated charging/discharging current, are reduced greatly when the voltage difference nearly vanishes, i.e., the ability to transfer charge is becoming limited. The small voltage difference is also the reason for the decreases in iL and vc. According to the switching actions of the selection switches, the charge transfer is decomposed into four intervals. Figure 4 depicts the current conduction path of each interval.

Normal Balance Mode
Normal balance mode features a one-to-one energy transfer between batteries. Figure 3 shows the theoretical waveforms. vBi and vBj are the instantaneous voltages of Bi and Bj. Six and Sjx are the switches connected to Bi and Bj, respectively, where x denotes a, b, c, and d. vix and vjx are the gating signals controlling the power switches Six and Sjx, respectively. We use the exaggeratedly diminished voltage difference to illustrate that the transferred charges Q1 and Q2, which represent the accumulated charging/discharging current, are reduced greatly when the voltage difference nearly vanishes, i.e., the ability to transfer charge is becoming limited. The small voltage difference is also the reason for the decreases in iL and vc. According to the switching actions of the selection switches, the charge transfer is decomposed into four intervals. Figure 4 depicts the current conduction path of each interval. Figure 3. The theoretical waveforms of the normal balance mode.  Interval I begins when the selection switches attached to battery B i , i.e., S ia , S ib , S ic , and S id , are turned on. The voltage equation can be expressed as: Energies 2020, 13, 6566 5 of 14 Before the selection switches are turned on, i L is zero. Therefore, the current can be expressed as follows: where V c0 is the capacitor's voltage at t 0 , α is the damping factor and is expressed as α = R s /2L r , and ω d is the damped natural frequency, which can be expressed as: Here, the resonance is assumed to be underdamped due to the low stray resistance. The resonant current, i L , is transferred from B i to the capacitor, and the capacitor's voltage can be expressed as: To prevent i L flowing back to the battery B i , switch S ib is turned off at t 1 to block the reverse current path.
In this interval, S ib is turned off and i L continuously charges capacitor C r via the antiparallel diode of S ib . As i L decreases to zero, the other switches S ia , S ic , and S id can turn ZCS off, ending this interval.
Interval III starts when the selection switches, S ja , S jb , S jc , and S jd , are turned on. Battery B j closes the voltage loop and is characterized as: The resonant current reverses and charges the battery B j , which is expressed as: where V c2 is the capacitor's voltage at t 2 .
The voltage across the capacitor can be expressed as: Similarly, switch S ja is turned off earlier to prevent the current flow from being reversed.
This interval is followed by the switch S ja being turned off. The resonant current charges B j through the antiparallel diode of S ja . When the resonance current reaches zero, the entire charge transfer cycle is completed. Figure 5 shows the theoretical waveforms of the enhanced balance mode, which describe the energy transfer between two series-connected batteries and another battery. v kx are the gating signals to the power switches s kx . Given that B i has the highest voltage, and j = I ± 1, then B j is adjacent to B i . The two batteries collaboratively discharge to the lowest-voltage battery B k . Due to the nearly doubled charging voltage of C r , i L and v c are only slightly affected and Q 1 and Q 2 are degraded only a little, even when the difference between V Bi and V Bk is small.

Enhanced Balance Mode
Energies 2020, 13, x FOR PEER REVIEW 6 of 14 The two batteries collaboratively discharge to the lowest-voltage battery Bk. Due to the nearly doubled charging voltage of Cr, iL and vc are only slightly affected and Q1 and Q2 are degraded only a little, even when the difference between VBi and VBk is small. Figure 6 shows the circuit configuration and the selection switches. The positive and negative arms of the series-connected batteries are turned on to charge Cr through Lr and Rs. After the charging phase of Cr is complete, the selection switches change states to transfer the charge to Bk and complete the transfer.   Figure 6. The circuit's operation in enhanced mode. Figure 7 shows the control flowchart for the balance strategy. At the beginning, the voltages of each cell in the battery pack are collected and sorted. When the voltage difference between the highest-voltage battery and the lowest-voltage battery is greater than 20 mV, the battery pack is tagged as unbalanced and the balance procedure is initiated.  Figure 6 shows the circuit configuration and the selection switches. The positive and negative arms of the series-connected batteries are turned on to charge C r through L r and R s . After the charging phase of C r is complete, the selection switches change states to transfer the charge to B k and complete the transfer. The two batteries collaboratively discharge to the lowest-voltage battery Bk. Due to the nearly doubled charging voltage of Cr, iL and vc are only slightly affected and Q1 and Q2 are degraded only a little, even when the difference between VBi and VBk is small. Figure 6 shows the circuit configuration and the selection switches. The positive and negative arms of the series-connected batteries are turned on to charge Cr through Lr and Rs. After the charging phase of Cr is complete, the selection switches change states to transfer the charge to Bk and complete the transfer.   Figure 6. The circuit's operation in enhanced mode. Figure 7 shows the control flowchart for the balance strategy. At the beginning, the voltages of each cell in the battery pack are collected and sorted. When the voltage difference between the highest-voltage battery and the lowest-voltage battery is greater than 20 mV, the battery pack is tagged as unbalanced and the balance procedure is initiated.  highest-voltage battery and the lowest-voltage battery is greater than 20 mV, the battery pack is tagged as unbalanced and the balance procedure is initiated.

Special Case
There is a special case that should be considered when using the enhanced mode. This case is when B1 or Bn, one of the two geographical extremes of the battery string, is the highest-voltage battery while its adjacent neighbor, B2 or Bn−1, happens to be the lowest-voltage battery. In this case, the operation is different during charging and discharging. During charging, the highest-and lowest-voltage batteries are connected together to transfer the charge to the battery with the second-lowest voltage, which extends the charging time. In contrast, during discharging, in order to prolong the usage time, the battery with the second-highest voltage works together with the adjacent battery, depending on which neighbor possesses the higher voltage, to transfer the charge to the lowest-voltage battery.  At the beginning, the question of whether normal mode or enhanced mode should be adopted is determined by the voltage difference between the highest-voltage battery and the lowest-voltage battery. To ensure a sufficiently high balance current, a boundary threshold voltage, V en , is preset. If the voltage difference is greater than V en , normal mode is adopted; otherwise, enhanced mode is adopted. Figure 8 shows the loaded voltage and temperature waveforms of the experimental lithium iron phosphate batteries (LiFePO 4 ) over the accumulated charge during constant current-constant voltage (CC-CV) charging and CC discharging, which are standard charging and discharging methods [25]. Lithium iron phosphate batteries are relatively safe and thermally stable and have a long lifecycle and high power density [26]. Obviously, there is a flat-voltage region (approximately between 3.3 and 3.4 V in the figure). Outside of this flat-voltage region, the loaded voltage varies swiftly over the accumulated charge.

Experimental Results
We performed offline and online balancing experiments with eight 2.5 Ah LiFePO4 batteries in series, whose total rated voltage is 25.6 V, to verify the proposed balance circuit and strategy.  As shown in Figure 7, after the decision is made on the appropriate operation mode, the charge-equalization circuit operates for a few seconds, which is hereafter referred to as the 'balance duration,' to balance the batteries. However, due to the sharp deviation in voltage when the battery's voltage is outside the flat-voltage region, the balance duration should be limited. As a result, when the enhanced mode is adopted, i.e., there is a sufficiently high transference current, as well as a less than 200 mV voltage difference under the condition that the battery's voltage is outside the flat-voltage region, the balance duration needs to be shortened to 5 s to prevent the dramatic variation in the battery's voltage. Finally, the balance circuit idles for 5 s to wait for the battery's voltage to rebound in order to ensure accurate detection.

Special Case
There is a special case that should be considered when using the enhanced mode. This case is when B 1 or B n , one of the two geographical extremes of the battery string, is the highest-voltage battery while its adjacent neighbor, B 2 or B n−1 , happens to be the lowest-voltage battery. In this case, the operation is different during charging and discharging. During charging, the highest-and lowest-voltage batteries are connected together to transfer the charge to the battery with the second-lowest voltage, which extends the charging time. In contrast, during discharging, in order to prolong the usage time, the battery with the second-highest voltage works together with the adjacent battery, depending on which neighbor possesses the higher voltage, to transfer the charge to the lowest-voltage battery.

Experimental Results
We performed offline and online balancing experiments with eight 2.5 Ah LiFePO 4 batteries in series, whose total rated voltage is 25.6 V, to verify the proposed balance circuit and strategy.

Parameters
In order to employ a medium capacitor, a switching frequency, f sw , of 4.15 kHz was chosen to transfer electric charge between batteries and the intermediate energy tank. Since the resonant frequency should be slightly higher than the switching frequency, it was set to 5 kHz. Furthermore, the balance circuit requires both the maximum and minimum charge-transfer currents to be limited. Therefore, under these three specifications, the resonant tank can be accordingly designed. As a result, the following equation can be formulated: where f r is the resonant frequency. The maximum average current can be calculated within half of the resonant cycle. Simplifying (2) and (6) by approximating e −αt as (1 − αt) and averaging the discharging and charging currents, the average currents I disch and I ch can be expressed as: where V c0 is assumed to be 0 V to obtain the maximum value of I ch , V c2 is approximated as double the discharged battery voltage, and R s is aggregated to be 0.5 Ω.
When substituting the possible maximum V Bi with two fully charged series of cells, that is, 3.6 × 2 V, and taking the minimum V Bj to be (3.6 − V en ) V and V c2 to be 14.4 V in the above equations, I ch and I disch should be less than 3.75 A (equivalent to 1.5 C). On the other hand, to prevent the transfer Energies 2020, 13, 6566 9 of 14 current from being too low, under the condition of V Bi and V c2 being 3.6 V and 7.2 V, respectively, I ch and I disch should be greater than 1.25 A, i.e., 0.5 C.
Obviously, I ch is the critical threshold in (8) and (9). V en was calculated to be 800 mV, which was used as the watershed between normal balance mode and enhanced balance mode in the balance strategy. Accordingly, C r was calculated to be 16 µF. However, since 16 µF is not an off-the-shelf specification, 20 µF was applied. Consequently, the inductor was set to 50 µH. The circuit parameters for the experiments are listed in Table 1.  Figure 9a,b shows the inductor current and capacitor voltage waveforms in the normal mode and the enhanced mode, respectively. According to the experimental waveforms, zero-current switching is achieved by the characteristics of the LC resonant tank. Moreover, no current surge was detected. It is worth noting that i L and v c are larger in the enhanced mode than in the normal mode, indicating that more energy can be transferred in a cycle.   Figure 9a,b shows the inductor current and capacitor voltage waveforms in the normal mode and the enhanced mode, respectively. According to the experimental waveforms, zero-current switching is achieved by the characteristics of the LC resonant tank. Moreover, no current surge was detected. It is worth noting that iL and vc are larger in the enhanced mode than in the normal mode, indicating that more energy can be transferred in a cycle.  Figure 10 shows the voltages of the batteries in the pack during offline balancing; that is, with no charging/discharging current flowing through the entire pack. Table 2 summarizes the initial and balanced voltages of each battery in the pack. The initial maximum and minimum voltages are 3.075 V (VB7) and 2.170 V (VB2), respectively, with a voltage difference of 905 mV. At the beginning, the voltage difference is greater than 800 mV and, as expected, the normal mode is entered into to transfer electric charge from battery B7 to battery B2, which results in the observed voltage fluctuation in both batteries. After 30 s, the maximum voltage difference between batteries shrinks to less than 800 mV, triggering the enhanced mode.

Experimental Result: Offline Balancing
At 185 s, VB7 is the highest voltage (2.941 V), while VB1 is the lowest (2.742 V). Since the voltage difference is lower than 200 mV, the balance duration is reduced to 5 s to avoid overcharging and over-discharging. This is the reason why the battery voltage varies more rapidly after this instant.
At around 482 s, the peak and valley voltages are 2.788 V and 2.737 V, respectively, corresponding to a 15 mV difference. Since the voltage difference is less than 20 mV, the balance circuit is inactivated.   Figure 10 shows the voltages of the batteries in the pack during offline balancing; that is, with no charging/discharging current flowing through the entire pack. Table 2 summarizes the initial and balanced voltages of each battery in the pack. The initial maximum and minimum voltages are 3.075 V (V B7 ) and 2.170 V (V B2 ), respectively, with a voltage difference of 905 mV. At the beginning, the voltage difference is greater than 800 mV and, as expected, the normal mode is entered into to transfer electric charge from battery B 7 to battery B 2 , which results in the observed voltage fluctuation in both batteries. After 30 s, the maximum voltage difference between batteries shrinks to less than 800 mV, triggering the enhanced mode.

Experimental Result: Offline Balancing
At 185 s, V B7 is the highest voltage (2.941 V), while V B1 is the lowest (2.742 V). Since the voltage difference is lower than 200 mV, the balance duration is reduced to 5 s to avoid overcharging and over-discharging. This is the reason why the battery voltage varies more rapidly after this instant.
At around 482 s, the peak and valley voltages are 2.788 V and 2.737 V, respectively, corresponding to a 15 mV difference. Since the voltage difference is less than 20 mV, the balance circuit is inactivated.   Figure 11a,b shows the battery voltages when the battery pack is discharged by a 1.5 A current with or without activation of the balance circuit. In Figure 11a, the voltages undulate due to the balance operation. In contrast, Figure 11b shows smooth voltage variations. As demonstrated in Table 3, the initial battery voltages are adjusted to be as similar as possible in both cases. When B8 in Figure 11a and B2 in Figure 11b satisfy the discharging cut-off condition, 2 V, the discharging operation stops.

Experimental Result: Discharging
With the balance operation, the discharging operation lasts for 254 s, whereas the discharging time without the balance operation is only 190 s, which is about 60 s shorter. Moreover, the final pack voltages are 18.803 V with the balance operation and 19.914 V without the balance operation. These results show that the energy of a battery pack can be better utilized by employing the proposed balance operation.     Figure 11a,b shows the battery voltages when the battery pack is discharged by a 1.5 A current with or without activation of the balance circuit. In Figure 11a, the voltages undulate due to the balance operation. In contrast, Figure 11b shows smooth voltage variations. As demonstrated in Table 3, the initial battery voltages are adjusted to be as similar as possible in both cases. When B 8 in Figure 11a and B 2 in Figure 11b satisfy the discharging cut-off condition, 2 V, the discharging operation stops.
With the balance operation, the discharging operation lasts for 254 s, whereas the discharging time without the balance operation is only 190 s, which is about 60 s shorter. Moreover, the final pack voltages are 18.803 V with the balance operation and 19.914 V without the balance operation. These results show that the energy of a battery pack can be better utilized by employing the proposed balance operation.   Figure 11a,b shows the battery voltages when the battery pack is discharged by a 1.5 A current with or without activation of the balance circuit. In Figure 11a, the voltages undulate due to the balance operation. In contrast, Figure 11b shows smooth voltage variations. As demonstrated in Table 3, the initial battery voltages are adjusted to be as similar as possible in both cases. When B8 in Figure 11a and B2 in Figure 11b satisfy the discharging cut-off condition, 2 V, the discharging operation stops.

Experimental Result: Discharging
With the balance operation, the discharging operation lasts for 254 s, whereas the discharging time without the balance operation is only 190 s, which is about 60 s shorter. Moreover, the final pack voltages are 18.803 V with the balance operation and 19.914 V without the balance operation. These results show that the energy of a battery pack can be better utilized by employing the proposed balance operation.      Table 4 shows the voltages in both cases. As B 5 and B 7 satisfy the cut-off condition, 3.6 V, the circuit operation terminates. With the balance operation, the total charging time is 360 s. However, without the balance operation, the charging time is only 100 s, meaning that less charge can be transferred to the battery pack. Moreover, the final pack voltages are 28.328 V with the balance operation and 27.866 V without the balance operation. Again, these results indicate that the balance circuit can help to optimize the charge storage capability of a battery pack.   Table 4 shows the voltages in both cases. As B5 and B7 satisfy the cutoff condition, 3.6 V, the circuit operation terminates. With the balance operation, the total charging time is 360 s. However, without the balance operation, the charging time is only 100 s, meaning that less charge can be transferred to the battery pack. Moreover, the final pack voltages are 28.328 V with the balance operation and 27.866 V without the balance operation. Again, these results indicate that the balance circuit can help to optimize the charge storage capability of a battery pack.    Figure 13a,b presents the experimental results of balancing in the special case. Table 5 lists the voltage of each battery in this case. It can be noticed that the highest-voltage battery is B8 with 3.425 V, and its adjacent battery B7 has the lowest voltage of 3.303 V. As a result, the battery that is to receive the charge shifts from the one with the lowest voltage to the one with the second-lowest voltage, B1. According to the figure, it is obvious that B7 and B8 collaboratively charge B1. On the other hand,   Figure 13a,b presents the experimental results of balancing in the special case. Table 5 lists the voltage of each battery in this case. It can be noticed that the highest-voltage battery is B 8 with 3.425 V, and its adjacent battery B 7 has the lowest voltage of 3.303 V. As a result, the battery that is to receive the charge shifts from the one with the lowest voltage to the one with the second-lowest voltage, B 1 .
According to the figure, it is obvious that B 7 and B 8 collaboratively charge B 1 . On the other hand, similar to the results of the previous experiments, the charging time of the battery pack without the balance operation is less than that of the battery pack with the balance operation.    Table 6 presents a comparison between the proposed method and some existing methods dedicated to balancing charge. It can be observed that a current spike occurred with the capacitortype balance method, which is a severe problem because it might damage the components [27]. As for the component count, the number of required capacitors and inductors increases as the number of batteries increases for the inductor-type balance method [28]. For the proposed circuit, only one capacitor and one inductor are needed no matter how many batteries there are. Another important point is that the charge transfer path of the proposed method can directly transfer charge between any two batteries compared with the other types of balance circuits, which not only reduces unnecessary switching loss but also shortens the balance duration.    Table 6 presents a comparison between the proposed method and some existing methods dedicated to balancing charge. It can be observed that a current spike occurred with the capacitor-type balance method, which is a severe problem because it might damage the components [27]. As for the component count, the number of required capacitors and inductors increases as the number of batteries increases for the inductor-type balance method [28]. For the proposed circuit, only one capacitor and one inductor are needed no matter how many batteries there are. Another important point is that the charge transfer path of the proposed method can directly transfer charge between any two batteries compared with the other types of balance circuits, which not only reduces unnecessary switching loss but also shortens the balance duration.

Conclusions
In this study, we proposed a charge-equalization circuit composed of an intermediate resonant energy tank to transfer unbalanced charge between batteries connected in series. The operation and the circuit parameters were determined according to the preset requirements. When the battery pack is charging or discharging, the balance operation may not finish because the charging or discharging cut-off conditions have been satisfied. As expected, the experimental results show that the proposed balance circuit can help the battery pack to fully utilize its available capacity.
Consequently, the proposed circuit enables batteries connected in series to transfer the charge from any number of adjacent batteries to any other number of adjacent batteries. Moreover, during a transfer of charge, the potential for current surges is eliminated due to the characteristics of LC resonance, and the switching loss is also minimized. Furthermore, we proposed an enhanced balance mode to sustain a sufficient amount of balance current. Our experimental results show that the voltage difference can be shrunk to less than 20 mV during offline balancing. As for online operation, balance during discharging can increase the load usage time, while balance during charging is able to increase the total charge of the battery pack. Finally, a comparison with the capacitor-type and inductor-type balance methods was carried out to demonstrate that the proposed balance circuit has the advantage of no current or voltage spikes. Moreover, the proposed balance method not only possesses the benefits of less components when balancing a large number of batteries, but also the shortest charge transfer path, which can shorten the balance time and decrease transfer loss.