Title: High step‐up DC/DC converters based on coupled inductor and switched capacitors
Abstract: IET Power ElectronicsVolume 13, Issue 14 p. 3099-3109 Research ArticleFree Access High step-up DC/DC converters based on coupled inductor and switched capacitors Jie Ding, Jie Ding Electric Power College, South China University of Technology, Guangzhou, People's Republic of ChinaSearch for more papers by this authorShiwei Zhao, Corresponding Author Shiwei Zhao [email protected] Electric Power College, South China University of Technology, Guangzhou, People's Republic of ChinaSearch for more papers by this authorHuajie Yin, Huajie Yin Electric Power College, South China University of Technology, Guangzhou, People's Republic of ChinaSearch for more papers by this authorPing Qin, Ping Qin School of English for International Business, Guangdong University of Foreign Studies, Guangzhou, People's Republic of ChinaSearch for more papers by this authorGuanbao Zeng, Guanbao Zeng orcid.org/0000-0002-5096-9894 Electric Power College, South China University of Technology, Guangzhou, People's Republic of ChinaSearch for more papers by this author Jie Ding, Jie Ding Electric Power College, South China University of Technology, Guangzhou, People's Republic of ChinaSearch for more papers by this authorShiwei Zhao, Corresponding Author Shiwei Zhao [email protected] Electric Power College, South China University of Technology, Guangzhou, People's Republic of ChinaSearch for more papers by this authorHuajie Yin, Huajie Yin Electric Power College, South China University of Technology, Guangzhou, People's Republic of ChinaSearch for more papers by this authorPing Qin, Ping Qin School of English for International Business, Guangdong University of Foreign Studies, Guangzhou, People's Republic of ChinaSearch for more papers by this authorGuanbao Zeng, Guanbao Zeng orcid.org/0000-0002-5096-9894 Electric Power College, South China University of Technology, Guangzhou, People's Republic of ChinaSearch for more papers by this author First published: 01 November 2020 https://doi.org/10.1049/iet-pel.2019.1264Citations: 2AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract This study proposes a series of high step-up DC/DC converters based on the coupled inductor and switched capacitors. The priority of the switched capacitor connection methods of the converters is presented by their contributions to the voltage gain so that the converters can be selected optimally according to the gain demand. For this converter, the voltage gain can be adjusted by both of the turns ratios of the coupled inductor and the number of switched-capacitor units. It makes it possible to further improve the voltage gain when the turns ratio of the coupled inductor is rather high. Moreover, the introduction of switched capacitors can reduce the voltage stress of power components. In this study, the operating principle and performance characteristics of the converters are analysed. Also, a 300 W prototype is set up for verification. The experimental results demonstrate the effect of the theoretical analysis and superiority of the converter. 1 Introduction In recent years, high step-up DC/DC converters have been applied increasingly in many fields such as uninterrupted power supply, vehicle inverter, communication power supply etc. [1-4]. Especially with the increase of energy and environmental problems, engineers and researchers recently pay more attention to clean energy such as photovoltaic, fuel cells, wind energy etc. To convert the low-voltage DC power (20–40 V) of photovoltaic panels and fuel cells into high-voltage DC power (380–400 V), high step-up DC/DC converter plays a more and more important role in new energy field [5-8]. The traditional boost converter can achieve high step-up transformation theoretically, but its voltage gain is limited by its parasitic parameters. Moreover, high-voltage stress of the switch and diode results in high-switching loss and low-converter efficiency. Although the cascade of the converter can improve the voltage gain [9, 10], the efficiency, power density, and reliability of this kind of converter are relatively low, because its power components and magnetic elements are multiplied, and the voltage stress of the rear power tube is still relatively high. The voltage gain can be effectively improved by using a switched capacitor and switched inductor network [11-13]. However, with the increase of voltage gain, the number of power components increases exponentially, which increases the cost and reduces the reliability of the converter. Also, the output voltage can be increased to some extent by integrating the switched capacitor network into the interleaved converter [14, 15]. Unfortunately, it requires two magnetic elements and two switches, with low-power density and high cost. As we all know, conventional isolated topology can improve transformer ratio to improve voltage gain [16], but this always leads to a large leakage inductor of the transformer and high-voltage stress of the output rectifier diodes, thus reducing the reliability and efficiency of the whole converter. The voltage gain can be increased by adding a second winding to the boost converter to form a coupled inductor converter [17]. Unfortunately, when the switch is off, leakage energy will be released at both ends of the switch, resulting in a high-voltage spike. Active or passive clamping circuits are proposed in [18, 19] to recycle the leakage inductor energy, thus alleviating the voltage spike of the switch. However, the voltage stress of the output diode is still greater than the output voltage, and the resonant voltage spike with the leakage inductor will further increase the voltage stress of the output diode. To solve the resonant voltage spike between the output diode and the leakage inductor, the voltage doubling circuit can be increased in the secondary winding of the coupled inductor [20, 21]. Unfortunately, the voltage gain of the converter is still limited. When the voltage conversion ratio is relatively high, the coupled inductor turns ratio is too high, which will lead to an increase of leakage inductor and the voltage stress of the power device. It is a very effective method to increase voltage gain and reduce voltage stress of power devices by integrating switched capacitors on the coupled inductor converter [22-35]. In some converters, clamping capacitors and switched capacitors are in series as the output voltage [22-24]. However, on the one hand, the capacitor series will lead to the increase of equivalent series resistance (ESR), on the other hand, the total capacitance will be reduced, thus increasing the output voltage ripple. The authors of [25, 26] integrated the coupled inductor and switched capacitors based on the SEPIC converter. Although the low-input current ripple can be realised, the increased input inductor will bring additional loss, and the power density of the converter will also be reduced. Some converters integrate coupled inductors based on the z-source converter [29, 30]. However, the converters have either more coupled inductors [29] or more windings of the coupled inductor [30], resulting in a large volume of the converter and increased loss. At the same time, the low voltage and high current in the high-voltage gain application scene make the diode conduction loss at the input terminal large, which reduces the efficiency of the converter. Moreover, the converter proposed in [29-31] is not conducive to the design of the driving circuit because the main switch is not grounded. Although many switched capacitors are integrated into a coupled inductor converter [24, 32-34], the voltage gain is still not high, and the voltage stress of components is still relatively high. The voltage gain can also be increased by integrating coupled inductors and switched capacitors into interleaved converter [35]. However, the number of coupled inductors and switches is large and is mainly used in high-power applications. To avoid the coupled inductor turns ratio is too high, which can lead to increased leakage inductor and excessive voltage stress of components, this study proposes a series of high step-up DC/DC converters based on the coupled inductor and switched capacitors. The introduction of the switched capacitor unit (SCU) makes it possible for the converter to improve the voltage gain when the turns ratio of the coupled inductor is very high. In addition, the voltage stress of power devices will be further reduced. Therefore, the efficiency of the converter can be improved by using the switch with low-voltage level and low-conduction resistance. The paper is organised as follows. Firstly, the construction of circuit topologies and operation principles are introduced in Section 2; secondly, the performance of the proposed converter is analysed in Section 3; thirdly, the key parameters of the converter are designed in Section 4; then, the efficiency of the converter is estimated in Section 5; finally, a prototype of a 300 W output power and 380 V output voltage in Section 6 is implemented to verify the excellent performance of the proposed converter in Section 7. These characteristics show that the proposed converters can be used in distributed power generation systems, such as a photovoltaic inverter. 2 Circuit topology and its operating principle 2.1 Circuit topology Fig. 1 shows the evolution of the basic four high step-up converters. The voltage gain can be further increased by adding the SCU based on Fig. 1d, as shown in Figs. 2a and b. The internal structure of the SCU is shown in Fig. 3. These circuit topologies contain five switched capacitor connection methods that can be combined into a variety of high step-up converters. Figs. 1 and 2 are only plotted from high to low according to priority (the degree to which each diode, capacitor (DC) increases the voltage gain) to increase the voltage gain. Table 1 shows the performance parameters of the converters and the expressions for the voltage gain will be derived in the next section. Table 2 shows the performance parameters of the switched capacitor connection methods ①–⑤ in Figs. 1 and 2. Fig. 1Open in figure viewerPowerPoint Evolution of high step-up circuit topology (I) (a) Boost, (b) Coupled inductor converter I, (c) Coupled inductor converter II, (d) Coupled inductor converter III Fig. 2Open in figure viewerPowerPoint Evolution of high step-up circuit topology (II) (a) Coupled inductor converter with nSCU, (b) Coupled inductor converter with nSCU and pump capacitor Fig. 3Open in figure viewerPowerPoint Switched capacitor unit (SCU) Table 1. Performance parameters of various topologies High step-up converter Number of switches Number of diodes and capacitors Output voltage gain M (1a) –M (2b) Fig. 1a 1 1 Fig. 1b 1 3 Fig. 1c 1 4 Fig. 1d 1 5 Fig. 2a 1 5 + 2m Fig. 2b 1 6 + 2m Table 2. Performance parameters of various switched capacitor connection methods Switched capacitors Increased voltage gain Number of diodes and capacitors Average voltage gain per DC priority ① 2 I ② 1 II ③ 1 III ④ 2m IV ⑤ 1 V 2.2 Operating principle of the converter For the similar analysis method, one of the topologies as shown in Fig. 2b is discussed in this section. Without loss of generality, only SCU1 is considered as shown in Fig. 4a and its equivalent circuit is shown in Fig. 4b, where n 1 /n 2 is the ideal transformer ratio, L m is the magnetising inductor of the transformer, and L k is the sum of leakage inductor from the secondary side to the primary side. For simplifying, some assumptions are considered as follows: (i) All devices are ideal devices without considering the influence of parasitic parameters. Fig. 4Open in figure viewerPowerPoint Topology and its equivalent circuit in Fig. 2 b with SCU1 (a) Topology of the converter mentioned, (b) Equivalent circuit (ii) The capacitors C C, C m1, C m2, C m3, C 0, C 1, C 2, and C 3 are large enough that their voltage ripple is negligible. (iii) The magnetising inductor L m is large enough and the magnetising current iL m is continuous. The converter has seven modes in a switching cycle T S. The main operating waveforms are shown in Fig. 5 and the equivalent circuits of each mode are shown in Figs. 6 and 7. The main operating process is described as follows: 1 Mode I [t 0 –t 1]: as shown in Fig. 6a, switch S, diodes D 1, D 4, and D 5 are on, diodes D C, D 0, D 2, D 3, and D 6 are off. In this mode, the magnetising inductor current iL m and leakage inductor current iL k of coupled inductor linearly increases under the action of the input voltage U in. The clamping capacitor C C charges C 2 and C m3 through D 5. Input voltage U in charges capacitor C m3 through D 4. At the same time, capacitors C C, C 3, and C 1 are in series and charge C m1 and C m3 through secondary side winding, providing conditions for realising high step-up of the output voltage. At time t 1, the sum of capacitors C m1 and C m3 voltages is equal to the sum of capacitors C C, C 3, C 1, and C m2 voltages. Diode D 3 is turned on and enters mode II. The currents of magnetising inductor and leakage inductor can be expressed as follows: (1) (2) where UC 1, UC 2, UC 3, and UC m1 are, respectively, the voltage of capacitors C 1, C 2, C 3, and C m1 ; N = n 2 /n 1. Fig. 5Open in figure viewerPowerPoint Main operating waveforms of the converter Fig. 6Open in figure viewerPowerPoint Equivalent circuits for modes I–IV (a) Mode I [t 0 –t 1], (b) Mode II [t 1 –t 2], (c) Mode III [t 2 –t 3], (d) Mode IV [t 3 –t 4] Fig. 7Open in figure viewerPowerPoint Equivalent circuits for modes V–VII (a) Mode V [t 4 –t 5], (b) Mode VI [t 5 –t 6], (c) Mode VII [t 6 –t 7] 2 Mode II [t 1 –t 2]: as shown in Fig. 6b, switch S, diodes D 1, D 3, D 4, and D 5 are on, diodes D C, D 0, D 2, and D 6 are off. With the same operation of mode I, the secondary side winding starts to charge C m2 in this mode. At time t 2, switch S is turned off to enter mode III. The currents of the magnetising inductor and leakage inductor can be written as follows: (3) (4) where UC m2 is the voltage of capacitor C m2. 3 Mode III [t 2 –t 3]: as shown in Fig. 6c, switch S, diodes D 0, D 1, D 2, D 4, and D 5 are off, diodes D C, D 3, and D 6 are on. In this mode, diode D 3 is still on due to the leakage inductor L k, and the current iD 3 through D 3 gradually decreases under the action of the voltage UC m2. Meanwhile, the leakage inductor current iL k linearly decreases under the action of UC c, which are the voltages of capacitor C C, while magnetising current iL m continues to linearly rise. At time t 3, diode D 3 current iD 3 drops to zero, D 3 is turned off with zero current, D 0 is turned on with zero current, and mode IV starts. The current leakage inductor can be expressed as follows: (5) where UC m3 is the voltage of capacitor C m3. 4 Mode IV [t 3 –t 4]: as shown in Fig. 6d, switch S, diodes D 1, D 2, D 3, D 4, and D 5 are off, diodes D C, D 0, and D 6 are on. At this time, the energy starts to transfer from magnetising inductor L m and capacitors C m1, C m2, and C m3 to the load, magnetising current iL m starts to linearly drop, and leakage inductor current iL k continues to linearly drop. At time t 4, diode D 2 is turned on and enters mode V. The currents of the magnetising inductor and leakage inductor can be written as follows: (6) (7) where U o is the output voltage. 5 Mode V [t 4 –t 5]: as shown in Fig. 7a, switch S, diodes D 1, D 3, D 4, and D 5 are off, diodes D C, D 0, D 2, and D 6 are on. With the same operation of mode IV, the secondary side winding starts to charge capacitor C 1 through diode D 2 in this mode. At time t 5, switch S is turned on and enters mode VI. 6 Mode VI [t 5 –t 6]: as shown in Fig. 7b, switch S, diodes D 0, D 2, D 4, and D 5 are on, diodes D C, D 1, D 3, and D 6 are off. In this mode, the current iD 0 through diode D 0 linearly drops. At time t 6, the current of D 0 drops to zero, D 0 is turned off and enters mode VII. The currents of magnetising inductor and leakage inductor can be expressed as follows: (8) (9) 7 Mode VII [t 6 –t 7]: as shown in Fig. 7c, switch S, diodes D 2, D 4, and D 5 are on, diodes D C, D 0, D 1, D 3, and D 6 are off. In this mode, diode D 2 continues to conduct due to leakage inductor L k and the current iD 2 flowing through D 2 decreases gradually under the action of UC 1. At time t 7 (t 0), when iD 2 falls to zero, D 2 is turned off and enters mode I. 3 Steady-state analysis of the converter 3.1 Voltage gain M For simplicity, the transient modes III, VI, and VII are ignored. As a result, there are only four modes in a working cycle. The main operating waveforms are shown in Fig. 8. Fig. 8Open in figure viewerPowerPoint Simplified operating waveform of the converter When switch S is on, the voltage of the magnetising inductor UL m-charge and the secondary winding voltage U r-charge can be expressed as follows: (10) (11) When switch S is off, the voltage of the magnetising inductor UL m−discharge and the secondary winding voltage U r-discharge can be written as follows: (12) (13) Since the capacitors are very large, the ripple is ignored. The voltages of capacitors C C, C 1, C 2, and C m3 in the circuit meet the following relationship: (14) According to the volt-second balance of magnetising inductor, it can be obtained as follows: (15) where D is the duty cycle of switch S drive signal. From (10)–(15), it can be obtained as follows: (16) (17) (18) (19) (20) (21) (22) Formula (22) is the voltage gain expression of the proposed converter in Fig. 2b with one SCU. When the converter contains m SCU, its voltage gain expression is shown in (23) with similar derivation (23) Fig. 9 shows the variation of voltage gain with turn ratio N of the coupled inductor and the number of SCU m when duty cycle D is 0.6. As can be seen from Fig. 9, the voltage gain increases with the increase of turn ratio N, and the number of SCU m when duty cycle D is constant. The voltage gain can be increased by increasing the number of SCU when the turns ratio is very high. Fig. 9Open in figure viewerPowerPoint Variation of voltage gain with N and m (D = 0.6) 3.2 Effect of leakage inductor on voltage gain of the converter The calculated voltage gain of the converter does not take into account the leakage inductor of the coupled inductor. The calculated voltage gain mentioned above is slightly larger than its actual value since leakage inductor would cause the loss of the duty cycle. When leakage inductor is considered, the deduction process of the converter's voltage gain could be revealed as follows. According to the charge conservation of the capacitor, the average current of diodes D 0, D 1, D 2, and D 3 is equal. When switch S is off, the current i r-off and peak current I r-peak-off of the secondary side winding can be expressed as follows: (24) (25) Under this mode, the voltage on the L k can be written as follows: (26) where f s is the working frequency. The output voltage of this process is expressed as (27) When switch S is on, the current i r-on and peak current I r-peak-on of the secondary side winding can be represented as follows: (28) (29) The voltages of L k and C m2 under this mode can be obtained as follows: (30) (31) By solving (24)–(31), the voltage gain can be obtained as follows: (32) where , R 0 is the output resistance load. Similarly, when the leakage inductor is considered, the voltage gain expression of the converters mentioned in Figs. 1b –d and Figs. 2a and b can be written as follows: (33) (34) (35) The voltage gain of the converters in Figs. 1 and 2 is shown in Fig. 10 along with the change of duty ratio D under different leakage inductors when R 0 = 1000 Ω, f s = 100 kHz, N = 2.3, and m = 1. As shown in Fig. 10a, when the leakage inductor is not considered, the voltage gain of the proposed converter increases with the number of switched capacitors, and the gain curve is monotonous. However, the actual gain curves are shown in Figs. 10b and c due to their leakage inductor. It can be seen from Figs. 10b and c that the greater the leakage inductor is the more obvious the voltage gain decreases but the voltage gain still increases with the number of switched capacitors increasing. In addition, as the duty cycle approaches 1, the voltage gain decreases with the increase of duty cycle, but this does not affect their performances, because the duty cycle will not exceed 0.8 usually. Therefore, the converters could be selected according to the demand for voltage gain. To make the voltage gain as high as possible, the leakage inductor of the coupled inductor should be designed with as small as possible. Fig. 10Open in figure viewerPowerPoint Voltage gain curve of the converters (a) L k = 0, (b) L k = 5 uH, (c) L k = 10 uH 3.3 Voltage stress of the converter The voltage stress of switch S is (36) The voltage stress of diodes D 0 and D 1 is (37) The voltage stress of diodes D2 and D3 is (38) The voltage stress of diodes DC, D4, D5, and D6 is (39) Fig. 11 shows the curve of components voltage stress along with duty cycle D when N = 2.3 and m = 1. It can be seen from Fig. 11 and (36)–(39) that the voltage stress of all components is much less than the output voltage U o, and the voltage stress of the switch and diodes increases with duty cycle D increasing and decreases with the number of SCUs m increasing. Fig. 11Open in figure viewerPowerPoint Components’ voltage stress curve 3.4 Current stress of the converter According to the previous section, the component current stress is as follows. The rms current of switch S is (40) The rms current of diodes D 1 and D 3 is (41) The rms current of diodes D 0, D 2, and D C is (42) The rms current of diodes D 4 and D 5 is (43) where ESR is the parasitic resistance of the capacitor, x = m3, 2. The rms current of diode D 6 is (44) 3.5 Input current ripple analysis According to the charge conservation of the capacitor, the average value of the secondary winding current I r2 = 0, so the average value of the primary winding current I r1 = 0. It can be obtained as follows: (45) where IL k, IL m, I in, and ID 4 are the average currents of magnetising inductor, leakage inductor, input end, and diode D 4, respectively. According to Fig. 8, the maximum of the leakage inductor current can be expressed as follows: (46) where is the peak value of the sum of diodes D 1 and D 3, which can be expressed as (47) Equation (48) could be derivated by combining (45)–(47) (48) Also, the minimum of leakage inductor current can be written as follows: (49) The input current ripple can be obtained as (50) (50) According to (50), the larger the turns ratio N of the coupled inductor is, the larger the input current ripple will be. If the duty cycle D is too large or too small, the input current ripple will be large. 3.6 Converter performances comparison The high turns ratio of the coupled inductor not only causes high-voltage stress of some diodes but also increases the leakage inductor. The increased leakage inductor causes a loss of duty cycle. For the proposed converters, the high-voltage gain can be achieved by using the SCU without a high turns ratio. Table 3 shows the performance comparison between the high step-up DC/DC converters proposed in this study and those of references [23, 24, 27, 28, 31-34]. As can be seen from Table 3, the voltage gain of the proposed converters is gradually increased. When the turns ratio is very high, the switched capacitor circuits can be added regularly. Table 3. Performance comparison of the converters Converter Number of components Voltage gain M Voltage stress on the switch Voltage stress on output diode Current stress on the switch Switch Capacitor Diode Fig. 1c 1 4 4 Fig. 1d 1 5 5 Fig. 2a 1 5 + 2m 5 + 2m Fig. 2b 1 6 + 2m 6 + 2m [23] 1 4 4 [27] 1 4 4 [28] 2 4 3 [31] 1 5 5 [24, 32] 1 6 6 [33] 1 6 6 [34] 1 8 8 Fig. 12 shows the performance comparison curves between the proposed converters and other converters when N = 2. It can be seen from Fig. 12 that without considering the number of components, the proposed converters can undoubtedly achieve the highest voltage gain, the lowest voltage stress of the switch and output diode. Of course, the converters presented in this study also have certain advantages under the condition of the same number of components. The converter proposed in Fig. 1c uses the same number of components as those proposed in [23, 27, 28], but the converter proposed in Fig. 1c can achieve higher voltage gain and lower current stress of switch S. When duty cycle D is >0.6, the voltage stress of the switch and output diode is lower. Compared with reference [31], the proposed converter in Fig. 1d can achieve higher voltage gain and lower current stress of the switch. When duty cycle D is >0.5, the voltage stress of switch S and the output diode is lower. The voltage gain of the converter proposed in Fig. 2b at m = 0 is higher than that of the converter proposed in [33] under the full duty cycle, and the voltage stress of the switch and output diode is lower than the converter proposed in [33]. When duty cycle D is <0.67, the voltage gain of this converter is higher and the voltage stress of the switch is lower than those of the converter proposed in [24, 32]. When m = 1, the number of components used in the proposed converter in Fig. 2b is the same as that in [34], but the voltage gain of the proposed converter in Fig. 2b is higher and the voltage and current stress of switch S is lower. In practical high step-up converter design, duty cycle D is generally in the range of 0.5–0.7 to make the efficiency as high as possible and current ripple as low as possible. Therefore, compared with references [23, 24, 27, 28, 31-34], the converters proposed in this study have certain advantages in the same number of components. Fig. 12Open in figure viewerPowerPoint Converter performance comparison (a) Voltage gain comparison, (b) Comparison of the switch current stress, (c) Comparison of the switch voltage stress, (d) Comparison of the output diode voltage stress 4 Key parameter design 4.1 Turns ratio of coupled inductor According to (23), it can be obtained (51) The turns ratio N of the coupled inductor could be calculated with voltage gain M, duty cycle D, and SCU number m. For this converter, duty cycle D should be 0.5–0.7 in general with consideration of input current ripples and operating efficiency. The SCU number m could be adjusted by the voltage stress of related components. 4.2 Magnetising inductor of coupled inductor To make the magnetising inductor operate in a continuous state, the magnetising inductor L m should satisfy (52) 4.3 Main switch and diodes selection According to (36)–(39), the voltage stress of the active devices can be known. In practical applications, due to parasitic parameters such as a parasitic capacitor and parasitic inductor of active devices and printed circuit boards, voltage spikes may be generated on components when switching on or off. Therefore, considering the above factors, the voltage and current ratings of the selected active devices could be generally >50% of their calculated values. 4.4 Capacitor selection The voltage ripple of the capacitor depends on its capacity and the operating frequency of the converter. To limit the voltage ripple to an acceptable range, the capacitor should be met by (53) where is the voltage ripple of the capacitor. 5 Efficiency estimation 5.1 Loss of diodes According to the charge conservation of capacitors, the average current of all diodes is equal to the output current I o, so the loss P D-loss of the eight diodes can be expressed as follows: (54) where V f is the diode threshold voltage. 5.2 Loss of the switch The conduction loss of switch S is (55) where R on is the on-resistance of switch S. Since the leakage inductor current iL k is very small and not mutational, switch S approximately realises zero current opening. Also, opening loss of the switch is about zero, hence the switching loss of switch S can be shownas follows: (56) where t off is the turning off time of switch S. The loss of switch S can be written as follows: (57) 5.3 Loss of coupled inductor The rms current of the leakage inductor can be obtained as follows: (58) The rms current