Extension of Zero-Voltage-Switching Range inDual Active Bridge Converter by Switched Auxiliary
Inductance
Hayato Higa and Jun-ichi Itoh
Dept. of Energy and Environment Science Engineering
Nagaoka University of Technology
Nagaoka, Niigata, Japanhhiga@stn.nagaokaut.ac.jp, Itoh@vos.nagaokaut.ac.jp
Abstract— This paper proposes a novel Dual Active
Bridge (DAB) converter, which has an auxiliary inductor
and a bi-directional switch connected in parallel to atransformer in order to achieve zero voltage switching(ZVS) over a wide load range and a wide battery voltage
variation. In the proposed converter, the equivalentmagnetizing inductance is changed by switching theauxiliary inductance depending on the output power. In
order to achieve ZVS at light load, the equivalentmagnetizing current is increased by connecting theauxiliary inductance in parallel. At heavy load, theequivalent magnetizing current is reduced without
connecting the auxiliary inductance because ZVS isachieved without any increment of the equivalentmagnetizing current. In addition, the auxiliary inductanceis switched without the occurrence of DC-offset of theauxiliary-inductor current. In the experimental results, theZVS range is extended by 50% at most. In addition, thehigh efficiency in the wide load is achieved by the switchedauxiliary inductance.
Keywords—Dual Active Bridge Converter; Zero Voltage
Swtichng (ZVS); Magnetizing inductanceI.

INTRODUCTION

Recently, energy storage systems have been actively
installed to electric vehicles (EV) [1-3] and DC micro-gridsystems [4-5]. A bi-directional isolated DC-DC converter isrequired in low voltage battery systems. Due to its flexibility, adual active bridge (DAB) converter is generally employed as
the bi-directional isolated DC-DC converter [6-8]. The DAB
converter can obtain a high efficiency because zero voltage
switching (ZVS) is achieved. ZVS not only reduces switchingloss but also eliminates the voltage surge occurring at the hardswitching operation due to the reserve recovery. Consequently,
a low voltage rating Si-MOSFET can be applied at the low
voltage side (under 100V). However, the ZVS range is limited
due to the operation of the DAB converter over a wide load
range and a wide battery voltage variation [9], which areusually the requirements in the energy storage system.
Furthermore, the transformer current is increased due to theincrement of negative instantaneous power with the voltage
fluctuation [10].

In order to extend the ZVS range, the some modulation
schemes have been presented [11-14]. Pulse Width Modulation(PWM) is employed by using the phase-shift control [11-12].
Nevertheless, it is not always possible to achieve ZVS for all
switches over the wide voltage variation because the directionof the inductor current which discharges the junction capacitor
cannot be controlled at the light load or the medium load. In
addition, the pulse frequency modulation (PFM) is applied
depending on operation conditions [13-14]. In [13], theswitching frequency is adjusted in order to achieve theminimum loss from the loss calculation. In [14], the operation
point at ZVS condition is kept by using PWM and PFM.
Furthermore, the volume of the transformer is decided by theminimum switching frequency.
As a difference approach for extension of the ZVS range
and reduction in the inductor current, the variable leakage
inductance of the transformer has been proposed [15]. In thisliterature, the leakage inductance becomes small at heavy load.
On the other hand, the leakage inductance becomes large atlight load. References [16-17] show the DAB converter withthe tap changing transformer using bi-directional switch. In
these literatures, the turn ratio of transformer is changed by thebi-directional switch in order to accord the input voltage andthe output voltage considering the turn ratio. However, thesequence at the changing leakage inductance is not considered.
In addition, the switching devices used for the variable leakage
inductance must withstand the half value of the main current.
Besides, the coupling coefficient of the transformer kc isdesigned to be low in order to achieve the ZVS range in wide
load range [18]. However, the converter efficiency at the heavy
load decreases compared to that of the high coupling
coefficient (kc≥0.99) due to large magnetizing current.
In this paper, with the aiming for extending the ZVS range
over the wide battery-voltage variation and maintaining thehigh converter efficiency over the wide load range, the DAB
converter with a switched auxiliary inductance is proposed. In
the proposed converter, the bi-directional switch and theauxiliary inductance are connected in parallel to thetransformer. The original contribution in this paper is that theequivalent magnetizing inductance is changed by switching theauxiliary inductance depending on the output power and thevoltage condition. In order to decide the switching sequence ofthe bi-directional switch, the current detection is not requiredbecause the switching timing of bi-directional switch issynchronized to the peak and bottom of the carrier. In addition,
the reactive of the inductor current at low voltage (LV) side isalso reduced by the auxiliary-inductor current. The heavy load
efficiency does not decrease because the auxiliary inductance is
not connected by the turn off of the bi-directional switch when
ZVS can be achieved without the auxiliary-inductor current.
This paper is organized as follows; firstly, the circuit
configuration and the operation mode of the switched auxiliary
inductance are explained. Second, the switching sequence of
the additional switch in order to switch the auxiliary inductanceis introduced. Finally, the effectiveness of the switchedauxiliary inductance is confirmed by the experiment at two
voltage conditions.
II.

L
Vin

ipr
vpr

Low voltage (LV) side

S1

iaux

S2
iLm

ise

Vout

vse

N:1
Lm
Laux
Switched auxiliary Transformer
inductanceFig. 1. Configuration of DAB converter with switched auxiliary
inductance. When the additional switch is turned on at light load, ZVS isachieved by the auxiliary current. At the heavy load, the bi-directionalswitch is turned off in order to reduce the equivalent magnetizng current.

PROPOSED DUAL ACTIVE BRIDGE CONVERTER

A. Circuit configulation
Fig. 1 shows the configuration of a DAB converter with theswitched auxiliary inductance. Two H-bridge converters
output square waveforms to a high frequency transformer.
Transferred power is controlled by a phase difference between
a high voltage (HV) side converter and a LV side converter. In
the DAB converter at light load, the inductor current ipr
becomes too low to satisfy the condition of ZVS in the HVside switches. The transferred power PZVS that limits the ZVS
range is calculated by (1) [19].
PZVS =High voltage (HV) side δ 
Vin NVoutδ ZVS 1 − ZVS 

ωL
π 


(a) Without switched Laux.
(b) With switched Laux
Fig. 2. Operation waveforms of DAB converter with or without theswitched auxiliary inductance. In order to achieve ZVS at light load, theauxiliary-inductor current is increased with the auxiliary inductor current.
In the inductor current of LV side, the negative instantaneous power isreduced.

(1)

The phase-shift angle δZVS is calculated at the condition of
NVout≤Vin and the condition of NVout>Vin, in (2) and (3)respectively [19].

δ ZVS =δ ZVS =π
2Lm + L NVout+ 4βf swTd 
1 −
2
2Lm
Vin


π


2 Lm
Vin
+ 4αf swTd 1 −2  2 Lm + L NVout


(2)

(3)

where Lm is the equivalent magnetizing inductance, α and β are
the coefficients which are decided by the power flow, and Td isdead time. The coefficients are decided to be α=1 β=-1 in the
charge mode, whereas the coefficients are decided to be α=-1
β=1 in the discharge mode.
Note that the parasitic capacitance is excluded in thecalculation. The ZVS range is extended by the smaller
auxiliary inductance Lm as shown in (2), whereas the ZVS
range is not extended by the smaller auxiliary inductance asshown in (3). Therefore, the turn ratio of the transformer N is
designed in order to obtain the condition of Vin>NVout.
Fig. 2 shows operation waveforms at light load with
/without switching the auxiliary inductance. At light load
where ZVS is not achieved by only the inductor current of theFig. 3. Phaser diagram among voltages and the current in fundamental
wave model of DAB converter. In Fig. 3, the reactive current is reducedby the equivalent magnetizng current ILm.

HV side ipr and the LV side ise, ZVS is achieved due to theauxiliary-inductor current when the auxiliary inductance isswitched by turning on the bi-directional switch. At the heavyload, the bi-directional switch is turned off because ZVS can beachieved by only the inductor current at HV side and LV side.
In Fig. 2 at the inductor current of the LV side, the negative
instantaneous power is reduced. As a result, the reactive currentof the inductor current at LV side is reduced by the auxiliaryinductor current. This reason is explained by using thesinusoidal wave model of the DAB converter with only thefundamental component of output voltage of each inverter Vpr
and Vse [20].
Fig. 3 shows phaser diagram between the voltage andcurrent of transformer [21]. In the sinusoidal model of the
DAB converter, the output voltage of each inverter Vpr, Vse, arecalculated by (4), (5) [20].

V pr = V1α + jV1β

(4)

(5)

V se = V 2α

The inductor current of LV side Ise with equivalentmagnetizing inductance Lm and the additional inductor L is
calculated by (6) [20].
Ι se = I pr − I Lm =

V1β

ωL

V − V1α  V2α + j  2α−  −
 ωL ωLm (6)

From (6), when the real part of primary voltage V1α is larger
than the real part of secondary voltage V2α, the reactive currentof the inductor current at LV side is reduced by the auxiliary
inductance Laux compared with only the magnetizing
inductance Lm. In addition, the active current is not varied by
the auxiliary-inductor current because only the reactive current
is reduced by the auxiliary-inductor current.

III.

SWITCHING SEQUENCE OF ADDITIONAL SWITCH

Fig. 4 shows the relationship between the transient
waveforms and the switching sequence of the additional switch
to achieve the switched auxiliary inductance. Fig. 4(a) shows
operation waveforms at the turn on of the additional switch,whereas Fig. 4(b) shows operation waveforms at the turn off ofthe additional switch. In the Fig. 4(a), the DC offset mightoccurs depending on the turn-on timing of bi-directional switch.
In order to avoid this problem, the turn-on timing of S1 and S2
has to be set at the zero-crossing point of the magnetizing
current iLm. Otherwise, the large inductor and the high current
rating switch are required. In Fig. 4(b), the surge voltage occurs
at the turn-off timing. The high voltage rating device is also
required. In order to avoid the hard switching occurring in thebi-directional switch, the following switching sequence is
proposed.
Fig. 5 shows the proposed switching sequence at switching(a) Turn on of bi-directional switch(b) Turn off of bi-directional switch
Fig. 4. Operation waveforms of DAB converter with switched auxiliary inductance. At the turn-on of the bi-directional switch, the DC-offset might occur
depending on the turn-on timing. At the turn-off of the bi-directional switch, the hard switching also occurs. As a result, the surge voltage occurs.

(a) Operation waveform at turn on of bi-directional switch.

(b) Operation waveform at turn off of bi-directional switch.

(c) Turn-on of bi-directional switch(d) Turn-off of bi-directional switch
Fig. 5 Proposed switching sequence in order to switch auxiliary inductance. In the proposed switching sequence, the current detection is not required because
the switching timing is synchronized to the carrier peak and bottom. The turn on of S1 or the turn off of S2 are ZVS because the body diode in S1 or S2 isturned on before the turn on of S1 or S2. Moreover, the DC-offset current does not occur. ZCS is achieved at the turn on of S2 and the turn off of S1, whereas
ZVS is achieved at the turn off of S1. Hence, no surge voltage occurs at the turn off of S1 and S2.

the auxiliary inductance. Fig. 5(a) shows waveforms at the
turn-on timing of the bi-directional switch, whereas Fig. 5(b)shows the waveforms at the turn-off timing of the bidirectional switch. In Fig. 5(a), there are four switching steps inorder to turn on the bi-directional switch at the soft switchingcondition. First in step 1 of Fig. 5(a) and (c), S1 and S2 is off.
Then in step 2, S2 is turned on at the point when themagnetizing current iLm is zero cross, i.e. the point of the carrier
peak. At the step 2, the turn on of S2 is the zero current
switching (ZCS) operation because the auxiliary-inductor
current iaux has not been flown yet. Then in step 3, the body
diode of MOSFET in S1 is naturally turned on at iLm<0. Finally,
S1 is turned on at the point when iLm is zero cross point, i.e. theperiod of the carrier bottom. In addition, the DC offset does notoccur because the turn-on timing of the body diode in
MOSFET is zero cross point of the auxiliary current. In Fig.
5(b), there are also four switching steps in order to turn off thebi-directional switch at the soft switching condition. First in
step 1 of Fig. 5(b) and (d), S1 and S2 are on. Then in step 2, S2
is turned off during the period when iLm is positive, i.e. theperiod between the carrier bottom and half of the carrier peak.
Note that at step 2, the auxiliary-inductor current iaux is notcorrupted because of the commutation to the body diode in S2.
Therefore, the turn off of S2 is ZVS. When iLm becomes
negative at step 3, the body diode in S2 is turned off naturally.
Finally at step 4, S1 is turned off during the period when iLm isnegative, i.e. the period between the half of carrier peak and thecarrier peak. Turn off of S2 and turn on of S1 are ZVS because
the terminal voltage of S1 and S2 at the switching timing iszero. If the auxiliary inductance has to be inactive after the S1off range, the switching timing order of S1 and S2 are switched.
In the proposed switching sequence, the current detection is not
required because switching timings of S1 and S2 aresynchronized to the peak and bottom of the LV side carrier.
IV.

Table I Simulation condition(a) Turn-on of bi-directional switch.

SIMULATION RESULTS

In this section, the simulation results are demonstrated inorder to evaluate the effectiveness of the switched auxiliary
inductance. Table I shows the simulation conditions. The
auxiliary inductance Laux is 700 μH. It is noted that thetransferred power which ZVS is achieved without the auxiliary
inductance is 311 W from Eq. (1) and (2) in the discharge
operation.
Fig. 6 shows the transient response of the switchedauxiliary inductance at 173 W. Fig. 6(a) shows the transient
response when turning on the bi-directional switch, i.e. theactive of the switched auxiliary inductance. Fig. 6(b) shows thetransient response at turning off the bi-directional switch, i.e.
the inactive of the auxiliary inductance. In Fig. 6(a), the turn onof S2 is confirmed to be ZCS. In addition, the turn on of S1 isalso ZVS At the turn on of bi-directional switch, the DC offsetof the auxiliary-inductor current iaux does not occur. In Fig.
6(b), ZVS is achieved at the turn-off of S2. At the turn off of
S1, ZCS is achieved. In the terminal voltages of S1 and S2, nosurge voltage is confirmed. By using the switched auxiliaryinductor, the inductor current of the LV side is reduced by 43%.
Fig. 7 shows waveforms of the gate signal and drain-source
voltage of the LV-side inverter at the condition of Fig. 6. Fig.

(b) Turn-off of bi-directional switch.
Fig. 6. Transient responses of switched auxiliary inductor in discharge
operation at 173 W. At the switching timing of S1 and S2, ZVS or ZCS isachieved. Consequently, no surge voltage occurs. Furthermore, DC-offset
does not occur in the auxiliary current(b) With switching Laux(a) Without switching Laux
Fig. 7. Waveforms of gate signal and drain-source voltage at 173 W. ZVSis achieved by the additional auxiliary current iaux.

Fig. 8 shows characteristics of the inductor current at LV
side. In Fig. 8, the ZVS range with switching the auxiliary
inductance is extended by 32% compared to that withoutswitching the auxiliary inductance. Moreover, the inductor
current of the LV side is reduced by 46.9% at most. However,
the inductor current increases at the heavy load. Therefore, thelight load efficiency can be improved by using the auxiliary
inductance. Therefore, it is confirmed that the auxiliary
inductance is changed according to the output power in order toachieve the high efficiency over the wide load range.
V.

EXPERIMENTAL RESULTS

In this section, experimental results are demonstrated inorder to evaluate the switched auxiliary inductance in Table II.
It is should be noted that the auxiliary inductor Laux and the
additional inductor L are connected to the primary side of thetransformer. In the LV-side inverter of MOSFET,
IRFP4110PbF (Infenion) is selected. IRFP4110PbF has the onstate resistance Ron of 3.7 mΩ, the voltage rating Vrate of 100 V
and the current rating Irate of 120 A. On the other hand,
SCT3030AL-E (Rohm) is selected at the HV-side inverter andthe switched auxiliary inductor. SCT3030AL has the on-state
resistance Ron of 30 mΩ. In this paper, the diode clamp snubber
circuit is used in order to reduce the resonant current between
the auxiliary inductance and the parasitic capacitance of(a)Pout=65.1 W, without switching Laux

MOSFET (170 pF). The switching devices with low parasitic
capacitance will be selected in the future work.
Fig. 9 shows operation waveforms with/without switching
30
Inductor current at LV side [A]7(a) shows waveforms without the auxiliary inductance. Fig.
7(b) shows waveforms with switching the auxiliary inductance.
In Fig. 7(b), ZVS is confirmed with the switched auxiliary
inductance because MOSFET of the LV-side inverter are
turned on at zero voltage.

25
20
15
10
5
0
-800

-600

-400

-200
0
200
Output power[W]

400

600

800

Fig. 8. Characteristic of inductor current with/without switchedauxiliary inductance. The inductor current of LV side at the light load is
reduced by 46.9% at most compared with the no switched auxiliary
inductance.
Table II Experimental condition(b) Pout=70.4 W, with switching Laux(c) Pout=168 W, without switching Laux(d) Pout=164 W, with switching Laux
Fig. 9. Operation waveforms with/without switching the auxiliary inductance when input voltage is 190 V, output voltage is 15 V. At light load as shown in(a) and (b), the inductor current in (b) is reduced by the increment of the equivalent magnetizing current. At heavy load as shown in (c) and (d), the inductor
current is almost same because the real part of the HV side voltage is close to that of the LV side voltage.

Output voltage of HV-side inverter 250 V/div0

Output voltage of HV-side inverter 250 V/div0
Output voltage of LV-side inverter 50 V/div0

Output voltage of LV-side inverter 50 V/div

0
Auxially inductor current 1 A/div

Auxially inductor current 1 A/div

0

0

0

0
Inductor current of LV side 10 A/div(b)

NIse=4.6A

4 s/div

Pout=58.1 W, without switching Laux

Inductor current of LV side 10 A/div

NIse=3.7A

4 s/div(b) Pout=58.4 W, with switching Laux(c) Pout=200 W, without switching Laux(d) Pout=194 W, with switching Laux
Fig. 10. Operation waveforms with/without switching auxiliary inductance when input voltage is 190 V, output voltage are 18 V. At light load as shown (a)and (b), the inductor current in (b) can still be reduced by the increment of the equivalent magnetizing current even at the different output voltage of 18V
compared to Fig. 9.

the auxiliary inductor at the input voltage of 190 V, the output
voltage of 15 V. The voltage ratio NVout/Vin is 0.63. Fig. 9(a)and (b) show operation waveforms at the light load
with/without switching the auxiliary inductor. When theswitched auxiliary inductance is active, the inductor current isreduced by the increment of the equivalent magnetizing
inductance. In the inactive of the switched auxiliary inductor,
the resonance between the auxiliary inductance Laux and theparasitic capacitance of the MOSFET Cds occurs. Besides, theoutput power is slightly different with/without switching theauxiliary inductance because the dead time error [22-23] occurs
when the switching timing of the LV side inverter is close tozero cross point of the inductor current at the LV side in thecondition of Fig. 9(b). Fig. 9(c) and (d) show operation
waveforms at heavy load with/without switching the auxiliary
inductor. Although the reactive current is reduced by theswitching the auxiliary inductance, the inductor current isalmost same as shown in Fig 9(c) and (d). This is because thereal part of the LV side voltage V2α is close to that of the HVside voltage V1α. Therefore, only the light load efficiency canbe improved by the increment of equivalent magnetizing
current.

Fig. 12 shows operation waveforms of the gate signal andthe drain-source voltage with/without switching the auxiliary
inductance. In Fig. 12(a), the occurrence of the hard switchingis confirmed, i.e. the turn on of MOSFET at the output voltage.
The surge voltage occurs due to the reverse recovery current ofbody diode in MOSFET. In Fig. 10(b), ZVS is confirmed
because MOSFET is turned on at zero voltage. In addition, therecover surge does not occur compared without the switching
the auxiliary inductance.

Fig. 10 shows operation waveforms with/without switching
the auxiliary inductor at the input voltage of 190 V, the output
voltage of 18 V. The voltage ratio NVout/Vin is 0.76. Fig. 10(a)and (b) show operation waveforms at the light load with andwithout the auxiliary inductor. The inductor current of the LVside is reduced by 19%. When the auxiliary inductance isswitched, ZVS can be achieved because the direction of theinductor current is adjusted in order to discharge the drainsource capacitance due to the switching of the auxiliaryFig. 13 shows characteristics of the inductor current at theLV side with or without the auxiliary inductance. Fig. 13 (a)shows the inductor current of the LV side at the input voltage
of 190 V and the output voltage of 15 V. In Fig. 13(a), theinductor current of the LV side with the auxiliary-inductor
current is reduced by 25% at most compared with that of no
auxiliary-inductor current. Fig. 13(b) shows the inductor
current of the LV side at the input voltage of 190 V and theoutput voltage of 18 V. In Fig. 13(b), the inductor current of

inductor. However, the inductor current at the heavy load isincreased by the auxiliary inductor current as shown in Fig
10(c) and (d). This is because the real part of the LV side
voltage V2α is larger than the real part of the HV side voltage
V1α from (6). As a conclusion, the converter efficiency isimproved by switching the auxiliary inductance depending on
the voltage ratio and the output power.
Fig. 11 shows the transient response of the switching
auxiliary inductance. At the turn on of bi-directional switch, it
is confirmed that the DC offset of the auxiliary-inductor current
iaux does not occur.

Inductor current at LV side [A](a) Vin=190 V, Vout= 15 V

Inductor current at LV side [A]Fig. 11. Transient response of switched auxiliary inductance in discharge
operation. By using the proposed switching sequence, DC-offset does notoccur in the auxiliary current(a) Without switching Laux(b) With switching Laux
Fig. 12. Waveforms of gate signal and drain-source voltage at light load.
ZVS is not achieved without the auxiliary inductance, whereas ZVS isachieved by the switched auxiliary inductance. Therefore, the recovery
surge is reduced by ZVS.

the LV side with the auxiliary-inductor current is also reducedby 28% at most compared with that of no auxiliary inductor. At
heavy load with the auxiliary inductance, the inductor current
is increased compared to that of the auxiliary inductance.
Fig. 14 shows the efficiency characteristics with/without
the auxiliary inductance. Fig. 14(a) shows the efficiency
characteristics at the input voltage of 190 V, the output voltageof 15 V, whereas Fig. 14(b) shows the efficiency characteristics
at the input voltage of 190 V, the output voltage of 18 V. The
maximum efficiencies in two voltage conditions are 95.9% and95.4% when switching the auxliary inductance. By applying
the switched auxilary inductance, the converter loss is reduced
at light load region. At heavy load, the converter efficiency
with Laux is decreased compared to that without Laux because
the inductor current is increased by the incremnet of theauxiliary-inductor current. In Fig 14(a), the converter loss is
reduced by 18% at most, whereas the ZVS range is extended
by 50%. In Fig. 14(b), the converter loss is reduced by 21% at
most compared without swithing the auxiliary inductance.
Moreover, the ZVS range is extended by 29.2%. In Fig. 14(a)and (b), There are the changing points between two operation(b) Vin=190 V, Vout= 18V
Fig. 13. Characteristic of inductor current with/without switchedauxiliary inductance. With the auxiliary inductance Laux, the inductor
current of LV side at the light load is reduced by 40% at most comparedto the no switched auxiliary inductance. In Fig. (a) and (b), Thechanging point between two operation mode are 144 W and 137 W.

modes of 129 W and 110 W. In the proposed method, the highefficiency in wide load is achieved when the auxiliary-inductor
is switched at the changing point. From above results that, theconverter efficiency is impoved by the switching auxiliary
inductance depending on the output power and the voltage
condition.
VI.

CONCLUSION

This paper proposed the switching auxiliary inductor for the
DAB converter in order to reduce the reactive current of theinductor current at the LV side and extend the ZVS range. In
the proposed method, the equivalent magnetizing current ischanged by switching the auxiliary-inductor depending on the
output power. The switching sequence of the switchedauxiliary inductor was also introduced in order to reduce theDC-offset current.
In the experiment, the operation with/without the switchedauxiliary inductor was confirmed. Two maximum efficienciesof 95.4% and 95.9% at different output voltages of 15V and
18V were achieved by the switched auxiliary inductor. In
addition, the converter loss was reduced by 21% at most bychanging the equivalent magnetizing current. Moreover, theinductor current of the LV side was also reduced by 40%,

whereas the ZVS range was extended by 50%. In the proposed
switching sequence, the transient response without DC-offset
current the auxiliary inductor was also confirmed.
In future work, the converter efficiency will be improvedby the optimizing design of the auxiliary inductance and the
additional inductor.
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Vin=190 V, Vout= 18 V
Fig. 14 Efficiency characteristics of prototype with switched auxiliary
inductor. At the light load, the converter loss is reduced by 38.3% at
most when the switched auxiliary inductor is active. In addition, theZVS range is extended by 50% at most. In Fig. (a) and (b), Thechanging points between two operation modes are 129 W and 110 W.
Therefore, the high efficiency in wide load is achieved when theauxiliary-inductor is switched at the changing point.

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