Circuit-Parameter-Independent Nonlinearity
Compensation for Boost Converter
Operated in Discontinuous Current Mode
Hoai Nam Le, Student Member, IEEE, Koji Orikawa, Member, IEEE
and Jun-Ichi Itoh, Senior Member, IEEE
Abstract—This paper proposes a current control method for discontinuous current mode (DCM) in order to achieve thesame control performance as continuous current mode (CCM) in a boost converter. By utilizing the duty ratio at theprevious calculation period to compensate for a DCM nonlinearity, the controller which is designed for CCM can also be
used in DCM. In the frequency analysis, the cutoff frequency of the proposed DCM current control agrees exactly to thedesign value which is 2 kHz, whereas the cutoff frequency of the conventional DCM current control results in high error of
47.5%. In the current step response experiment with a 360-W prototype and the switching frequency of 20 kHz, theexperimental DCM current response almost agrees with the conventional CCM current response, which are 380-μs risetime for both CCM and DCM, 9% and 8% overshoot for CCM and DCM, respectively. Furthermore, the computation time ofthe proposed DCM current control is shorter 35% than the conventional DCM current control.
Index Terms—Current control, DC-DC power converters, Discontinuous current mode, Nonlinear control systems.

I. INTRODUCTIONTYPICAL boost DC-DC converters are widely applied in many power conversion systems, e.g. power conditioning system inphoto-voltaic systems. The problem in the typical boost converters is the use of bulky passive components, i.e. an output
capacitor and a boost inductor. One of the challenges to minimize the output capacitor is the dynamic voltage regulation during thefast load transient. In the applications for power conditioning system, the output voltage within a tight tolerance range must be
maintained during the faults of the grid, i.e. a load current step occurs. The load transient requirement can be met even with small
capacitance by a wide bandwidth voltage control [1]-[2]. On the other hand, many minimization methods for the inductor havebeen proposed such as high frequency switching, coupled-inductor or flying-capacitor topologies [3]-[6]. However, higher
switching frequency leads to the increase in both the switching loss and the emission noise [7], whereas the addition components
results in the complexity of the design in the main circuit and the control method.
There are other approaches to minimize the boost inductor in terms of the operation mode. By increasing the current ripple at agiven switching frequency, the typical boost converters can be operated in two main current modes; continuous current mode
(CCM) and discontinuous current mode (DCM) [8]-[9]. The advantages of CCM are: simple controller design due to the linear
duty-ratio-to-inductor-current transfer function, and the simple average current sampling. On the other hand, the boost converterwith the DCM operation achieves the high efficiency over a wide load range because the current ripple in DCM decreases at lightload. Nonetheless, the nonlinear duty-ratio-to-current transfer occurs in DCM, which worsens the current control performance
when the same controller as in CCM is used. In particular, the effects of the DCM nonlinearity on the duty-ratio-to-current transfer
function changes according to the current load [10]. This results in the overcurrent when the current is regulated by a wide
bandwidth controller [11]. In past few years, many researches focusing on the control of DCM have been reported to solve thisproblem [12]-[17]. However, in those control methods, the DCM nonlinearity compensation method becomescircuit-parameter-dependent. Due to this problem, a wide bandwidth current control is still difficult to be applied into DCM.
Consequently, the bandwidth of the voltage control cannot be increased in order to minimize the output capacitor [18], which
restricts the circuit minimization.
This paper proposes a DCM nonlinearity compensation method which is independent from the boost inductance and the loadcondition. The original idea in this paper is that the DCM nonlinearity compensation is constructed by utilizing the duty ratio at theprevious calculation period instead of using the circuit parameter. The advantages of the proposed method are as follows:
circuit-parameter-independence and short computation time. This paper is organized as follows; in section II, the problems of theconventional DCM control are introduced in detail with past works, then in section III the design of the conventional PI controller
is explained because the proposed DCM control is based on the CCM control with a PI controller, next the DCM nonlinearity

Full Load

ic

Current

D2

Average Current

Load

L

iin

C

vin

Light Load

iload

vout

SW1

0

(a) Continuous current mode at light load.
Fig. 1. Circuit diagram of typical boost converter. This paper proposed
the concept where the boost inductor is minimized by DCM and theoutput capacitor is minimized by the wide bandwidth voltage control in
DCM.

compensation method is proposed as the main part of this paper.
After that, the validity of the proposed DCM nonlinearity
compensation method is verified in section IV. In addition, inorder to achieve higher efficiency in the boost converter, thesimple DCM synchronous switching method is realized basedon the proposed DCM current control. Finally, the comparisonsof the efficiencies and the current-control computation time aredemonstrated in order to confirm the effectiveness of the
proposed DCM current control.
II. PROBLEMS OF CONVENTIONALDISCONTINUOUS-CURRENT-MODE CONTROLMETHODS

Full Load

Light Load
Current
Average Current0(b) Discontinuous current mode at light load.
Fig. 2. Inductor current waveform of CCM and DCM when the
inductance is minimized at given switching frequency. At light load, theconstant current ripple of CCM increases the ratio between the
average current and the current ripple, whereas the current ripple of
DCM decreases. Therefore, the high efficiency at wide load range with
DCM is achievable.

Fig. 1 shows the typical boost converter configuration. In
order to minimize the boost converter at the given switchingfrequency, this paper proposed the concept where the boost inductor is minimized by the DCM operation and the output capacitor
is minimized by the wide bandwidth voltage control in DCM. In other words, at the given switching frequency, the boost converter
can achieve higher power density by the application of DCM. Note that the high current ripple in DCM can be overcome simply
with the interleaved boost converter topologies as shown in [8]-[9].
Fig. 2 illustrates the current waveforms of CCM and DCM when the inductance is reduced at the given switching frequency.
Note that the boost converter is designed to be operated at critical current mode (CRM) at full load and the constant switchingfrequency is applied over all load range, i.e. the requirement to apply the pulse-width modulation. At light load, the boost inductor
current becomes discontinuous. Consequently, the transfer function becomes nonlinear and this leads to the instability of thecurrent control [10]. In order to avoid the discontinuous current, the switch is used instead of the upper diode. The CCM
synchronous switching by applying this switch improves the efficiency at full load and maintains the continuity of the current.
However, the current ripple becomes much higher than the average current at light load. This results in the poor power conversion
efficiency at light load. On the other hand, the current ripple in DCM decreases at light load as shown in Fig. 2(b). Hence, when theconverter is operated in DCM, the high efficiency is expected to be maintained over wide load range.
The current control for DCM is generally divided into two main methods: feedforward control and feedback control. The
principle of the DCM current feedforward control is to design the controller based on the reduced-order model or the full-ordermodel [9], [12]. An advantage of the feedforward control is the unrequired current sensor. However, the mismatch between thenominal values and the actual values of the circuit parameters results in the instability of the control system when a wide bandwidth
control is applied [11], [18]. On the other hand, the feedback control in DCM has to deal with the nonlinear duty-to-current transfer
function [10]. In [13] and [17], the PI controller with constant coefficients is used in order to control DCM. The DCM nonlinearity,
however, implies that the duty-ratio-to-inductor-current transfer function depends on the load. With a constant-coefficient PI
controller, the cutoff frequency of current controller varies dependently on load, which results in the instability of the control
system when the load varies. In order to deal with the DCM nonlinearity and also avoid the PI controller design with online-tuning
coefficients, the hybrid control between the feedforward control and feedback control has been proposed in [15]-[16]. The principleof these control methods is to estimate the duty ratio from the current command, then feed forward this estimated duty ratio tocompensate for the DCM nonlinearity. In [15], the estimated duty ratio is calculated by a complex function based on the
reduced-order model as same as [9], [12]. In [16], the calculation time is reduced by estimating the duty ratio through a referencetable.

Inductor current iin
iavg*

Averagecurrentiavg

PI Controller
Kp(sTi+1) D1
sTi

iavg_sampled

0

D1Tsw

D2Tsw

D3Tsw

Tsw
Fig. 3. Inductor current waveform in DCM. The zero current interval
occurring in DCM introduces the nonlinearity into the duty-to-current
transfer function [10].

Control System Circuit Model
D1vout iavgsL
D1_s

vout_sampled a/bbavin_sampled
vout_sampled vin
Duty-ratio feedforward
a/b
a

vout
b
vout

Fig. 4. Current control system for CCM. By applying the duty-ratio
feedforward in CCM, the PI controller compensates only the small
difference between the duty ratio at steady state and the instantaneous
duty ratio [15].

However, the penalties of these control methods are as follows;
the control system becomes circuit-parameter-dependent and along computation time is required. In case of the power conditioning system application, the boost converter is usually required todeal with the severe change of the ambient environment, where the circuit condition such as the operation temperature varies
frequently. When the actual values of the circuit parameters such as, e.g. inductance and capacitance, are different from thenominal values due to the variation of the circuit condition, the stability of the circuit-parameter-dependent control system can nolonger be guaranteed. As the motivation for the achievement of the wide bandwidth DCM current control, it is necessary to realize
the DCM nonlinearity compensation with the features as circuit-parameter-independence and short computation time.
III. DISCONTINUOUS-CURRENT-MODE NONLINEARITY COMPENSATION BY DUTY RATIO AT PREVIOUS CALCULATION PERIOD
A. Conventional Current Loop Design in Continuous Current Mode
Fig. 3 depicts the inductor current waveform in DCM, where D1, D2 and D3 denote the duty ratios of the first, the second and thezero-current interval. Average small signal modeling technique is used to model the boost converter for the inner current control
loop design [10], [16]. The average inductor voltage during a switching period Tsw is given by (1),

L

diavg
dt

 D1vin  D2 vin  vout  ............................. (1)where vin and vout are the instantaneous input voltage and the instantaneous output voltage, iavg is the average inductor current during
a switching period Tsw, and L is the inductance. In case of CCM, the relationship between D1 and D2 becomes,
D1  D2  1 .......................................................... (2)By substituting (2) into (1) in order to represent (1) as a function of only the duty ratio D1, (3) is obtained,
L diavg vin  vout
 D1 .................................... (3)vout dtvout
When the boost converter operates in steady state, the output voltage, the input voltage, the average input current and the duty ratio
in each switching period Tsw can be expressed as,
vout  Vout_ s  vout ............................................... (4)vin  Vin _ s  vin .................................................... (5)iavg  I avg _ s  iavg ................................................ (6)D1  D1 _ s  D1 .................................................... (7)where Vout_s, Vin_s, Iavg_s and D1_s are the output voltage, the input voltage, the average input current and the duty ratio at steady state,and vout, vin, iavg andD1 are the small signals of the output voltage, the input voltage, the average input current and the duty
ratio, respectively. In the current design step, the input and output voltages are considered to be at the steady state. Consequently,
the small signals of the input voltage and the output voltage, i.e. the input and output voltage dynamics are considered to be
negligible. By substituting (4)-(7) into (3), (8) is obtained,Ld iavg

Vout_ sdt

Vin _ s Vout_ s

Vout_ s D1 _ s  D1 .......... (8)The values of the input voltage, the output voltage and the duty-ratio at steady state are determined by letting the left-hand side ofdifferential equation (8) and the small signal of the duty ratio D1 equal to zero,

D1_ s Vo u t_ s  Vin _ s
Vo u t_ s............................................. (9)Substituting (9) into (8) and applying the Laplace transform, then the duty-ratio-to-inductor-current transfer function in CCM isgiven by (10),GCCM ( s) vin

iavg ( s)

Vout_ s............................... (10)
D1 ( s)sL
Fig. 4 depicts the current control system in CCM designed
based on (3) and (10). The integral period Ti and the proportional
gain Kp of PI controller are designed based on the second-orderstandard form expressed by (11),

D1

1
sL

vout

D3

Ti 

2
........................................................... (14)c

The digital PI controller has the same coefficients as theanalogous controller. To sum up, the design of the controller isperformed in the frequency domain, whereas for the digital
implementation, the obtained parameters are linked to theparameters of a digital PI controller [15].

vinTsw
2L 0.5ipeak

Do not exist
in CCM1iavg

ωc
2 ................ (11)2s  2ωc s  ωcwhere ωc is the cutoff angular frequency and ζ is the damping
factor, which are designed in order to achieve the desired current
response. The closed loop transfer function of the current control
loop in Fig. 4 is derived by (12),K pVout_ s1  sTi iavg ( s)LTi
H CCM ( s) 
(12)
K pVout_ s
K pVout_ siavg * ( s)
s2 
s
L
LTi
In order to make the design of PI parameters simple, a low passfilter whose role is to filter the command current iavg* is necessary
for matching (11) and (12). However, a low pass filter is notessentially required. Matching (11) and (12), the PI parameters
are obtained by (13) and (14),2 c LKp ..................................................... (13)Vo ut_ siavgvout - vin

2

G2 n do rd erd ela y( s ) vout

0.5ipeakiavg

Nonlinear factor
Fig. 5. Circuit model of boost converter operated in DCM. The
zero-current interval D3Tsw introduces the nonlinearity into the DCM
transfer function, which worsens the current control performance.

ΔD1

1
sL

Vout_siavg

Vout_s - Vin_s
Vin_sTsw
2L

2Ipeak_s4Iavg_s
Ipeak_s2
Nonlinear factor

Fig. 6. Linearized circuit model. Controlling DCM by PI controller of
CCM is achieved by compensating the DCM nonlinearity at the PI
output.

ΔD1

Vout_s1
sL

Vout_s2L(Vout_s – Vin_s)
D1_sTswVin_siavg

B. Proposed
Nonlinearity
Compensation
for
Discontinuous Current Mode
Nonlinear factor
In order to design the nonlinearity compensation part for DCM, Fig. 7. Simplified and linearized circuit model. The duty ratio at steadythe circuit model in DCM is required. First, the average current state in the circuit model is approximated as the duty ratio at previous
iavg and the current peak ipeak, which are shown in Fig. 3 can be sampling period in order to eliminate the DCM nonlinearity.
expressed as,iavg 

i peak( D1  D2 ) ......................................... (15)2v
i peak  in D1Tsw ................................................. (16)L
Substituting (16) into (15) and solving the equation for the duty ratio D2, then the duty ratio D2 is expressed by (17),2 Lia vg
 D1 ........................................... (17)D1Tswvin
Substituting (17) into (1) in order to remove the duty ratio D2 and representing (1) as a function of only the duty ratio D1, (18) is
obtained [10].
di2 Lia vg 
 (18)
L a vg  vin  vo u t  D1vo u t  (vo u t  vin )1 dt
 vin D1Tsw 
Then, the circuit model in DCM is established based on (18).
Fig. 5 illustrates the circuit model of the boost converter in the DCM operation which is based on (18). In CCM, the dash line partdoes not exist, because the average current iavg equals to the half current peak ipeak/2. On the other words, this makes theD2 zero-current interval D3Tsw in Fig. 3 become zero. However, in
DCM, the zero-current interval introduces the nonlinearity into
the DCM transfer function. This worsens the current response in
Nonlinear factor
DCM when same PI controller is applied for both CCM and DCM
Fig. 8. Discretized circuit model. Utilizing the duty ratio at previous [10], [15]. Therefore, the output of the PI controller is necessary
sampling period in order to eliminate the DCM nonlinearity makes the to be compensated when the circuit is operated in DCM. The
current control independent from inductance.
design of the compensation part for the DCM nonlinearity is
explained as follows. First, the circuit model in Fig. 6 islinearized at steady state.
Nonlinearity Compensation
vout_sampled
vout_sampled
Fig. 6 depicts the linearized circuit model. In order to simplify
the coefficients in the linearized circuit model, the relationships
vin_sampled
vin_sampled
between such coefficients at steady state are derived byb
substituting the differential of the inductor current diavg/dt in (18)2Laa/b
as zero and calculating the current peak,
Tsw
D
ΔD1

D1_sVin_s
Vout_s – Vin_sz-1z

Vout_sΔiavg

Tsw
L(z-1)

1_s

iavg*

PI

iavg_sampledD1
Designed as in CCM

2

D1

I avg _ s 

Tsw D1_ s VinVout......................................(19)L 2 Vout  Vin

Vin
D1_ sTsw ..................................................(20)L(a) Conventional DCM nonlinearity compensation in [15]-[16].
where Iavg_s, and Ipeak_s are the average current and the peakcurrent at steady state, respectively. Then, (19) and (20) are
Nonlinearity Compensation
substituted into Fig. 6 in order to express all coefficients as
vin_sampled
vout_sampled
functions of D1_s.
Designed as in CCM
Fig. 7 depicts the simplified circuit model. In order to eliminate
D1
a
a/b
iavg*
D1
PI
the dash line part in Fig. 5, in the control system, the value of D1_s
D1_s
is approximated as the duty ratio of SW 1 at the previousbcalculation period D1[n-1]. As a result, the circuit model is
iavg_sampledz-1
necessary to be analyzed in the discrete model.
vin_sampled
Fig. 8 depicts the discretized circuit model. In order tocompensate the DCM nonlinearity at the output of the PI(b) Proposed DCM nonlinearity compensation.
Fig. 9. Conventional and proposed DCM nonlinearity compensation. controller designed in CCM, the dash line part in Fig. 8 is
The main difference in the proposed method is that, the DCM
necessary to be set as 1 when the circuit is operated in DCM.
nonlinearity compensation is constructed by utilizing the duty ratio at
the previous calculation period. This makes the control system Therefore, in the control system, the inverse part of the dash line
circuit-parameter-independent and achieves the fast computation of part in Fig. 8 is multiplied at the output of the PI controller inthe current control.
order to compensate for the DCM nonlinearity.
Fig. 9 illustrates the conventional DCM nonlinearity
compensation in [15]-[16], and the proposed DCM nonlinearity
compensation. Note that the PI controllers in both the conventional and proposed method are designed by (13)-(14). Besides, in the
power conditioning system application, the sampled input voltage which is used for the maximum power point tracking can be also
used for the DCM nonlinearity compensation. The principle of two methods is the estimation of the duty ratio at steady state inorder to compensate for the DCM nonlinearity. In the conventional method, the duty ratio at steady state is estimated by using
current command iavg* and (19). This leads into many disadvantages: circuit-parameter-dependence, and long computation time. On
the other hand, in the proposed method, the duty ratio at steady state is estimated by utilizing the duty ratio at the previous
calculation period. This provides the control system circuit-parameter-independence, and short computation time. Moreover,because the dash-line part in Fig. 9(b) is always the inverse part of the nonlinear factor in Fig. 8 by utilizing the duty ratio at theprevious calculation period, the DCM nonlinearity is eliminated over entire load range.
I peak_ s Asynchronous
Switching

Synchronous
SwitchingDuty-to-SW1

D1

D1

Duty-to-SW2

0

D1vin
vout - vin

C. Realization of Synchronous Switching in Discontinuous
Current Mode
Fig. 10 depicts two switching patterns which occur in DCM
when the switch (MOSFET) is used instead of the upper diode.
This converter can improve the efficiency because the conduction
loss of MOSFET is smaller than that of diode. This converter canrealize three operation modes at light load depending on the turnon and off of SW2; i) Simple CCM synchronous switching inwhich SW2 is switched alternately to SW1, ii) DCM asynchronous
switching in which SW2 is turned off at light load and iii) DCM
synchronous switching in which SW2 is turned on only during theperiod D2Tsw as depicted in Fig. 3.
The boost inductor current of the simple CCM synchronous
switching becomes continuous. Therefore, the boost inductor
current ripple is constant. However, the boost inductor current
increases at light load compared to that of the conventional circuit
which is shown in Fig. 2. In order to improve the efficiency atlight load, the DCM asynchronous switching is applied. Although
the DCM asynchronous switching can reduce the boost inductor
current in light load, the conduction loss of the body diode in the
MOSFET is still high. In the DCM synchronous switching, in casethat SW2 is turned on exactly only in period D2Tsw, the current
flows through MOSFET. As a result, the conduction loss is
reduced.
The main key of the DCM synchronous switching technique isto detect or estimate the moment when the current reaches zero inorder to turn off SW2. The zero-current period can be detected bycomparing the detected current with a threshold value [17] or by
the modified Gate Drive Unit (GDU) in [19]. However, bothtechniques suffers following penalties; the experimental tuning of
the threshold value in the current detection or the complex GDU.
In order to realize the DCM synchronous switching without any
modifications of the main circuit, the duty ratio D2 is calculated by(21) in the proposed DCM feedback control and (22) in theconventional DCM feedforward control [9], respectively,

D2 _ prop. _ fb  D1
D2 _ conv. _ ff 

TABLE IISW
PARAMETERS
OF Cturned
IRCUIT off
ANDin
CONTROLLERS IN EXPERIMENTS.
2 is completely
D2Tsw and D3Tsw.
SymbolQuantity
Value
Vin
Input Voltage SW2
180 V
iin
L
Vout
Output Voltage
280 V
vout
P
Rated Output Power
360 W
fsw
Switching Frequency
20 kHzFrequency Analysis&
vin
SWSimulation
1
Current Step Response – Load Transient Response Experiment
ΔiHCR
Current Ripple at Rated Load
100% (High)ΔiLCR
Current(a)Ripple
at
Rated
Load
10% (Low)
Asynchronous switching.
LHCR
High Current Ripple Inductance 655 μH
LLCR
Low Current Ripple Inductance 6680 μH
iacSW is turned
Injected
ACduring
current (Magnitude) 0.5 A (0.25 p.u.)
on only2
fD
InjectedACcurrent (Frequency) 0.1-5 kHz
iac
2Tsw and turned off in D3Tsw.
vac
Injected AC voltage (Magnitude) 0.1 p.u.
SW(Frequency)2
fvac
Injected
AC voltage
0.1-100 kHz
iin
L
idc
Offset of AC current command
1.25 A
vout
fc
Cutoff frequency (Current Con.) 2 kHz
cDamping factor (Current Con.)
0.707
C vin
Output Capacitance
33 mF
SW1
fv
Cutoff frequency (Voltage Con.) 0.2 kHzv
Damping factor (Voltage Con.)
0.707
Efficiency Comparison(b) Synchronous switching.
Switching device
TPH3006PS (Transphorm)

Fig. 10. Asynchronous switching and synchronous switching in DCM.
The loss reduction is achieved by applying synchronous switching to
DCM.1

0
Designed control gain (CCM)-1-2
Magnitude [dB]TABLE I
DUTY GENERATION OF DCM ASYNCHRONOUS SWITCHING AND DCM
SYNCHRONOUS SWITCHING.

Proposed DCM control
gain in Fig. 9(b)

-3
-4
-5
Conventional DCM
control gain in Fig. 9(a)

-6
-7
-8

Increase 47.5%

-9
-10

0

500

1000

1500 2000 2500
Frequency [Hz]

3000

3500

Fig. 11. Frequency characteristics of conventional and proposed DCM
current feedback control. The agreement between the proposed DCM
control gain and the designed gain confirms the complete elimination of
the DCM nonlinearity, whereas the conventional DCM control gain
results in high error of 47.5%.

vin.................................. (21)vout  vin*
2Liavg(vout  vin )
2
sw in

T v

vin
. (22)
vout  vin

Table I illustrates the duty generation of the DCM asynchronous switching and the DCM synchronous switching. Because of themismatch between the nominal values and the actual values, the duty ratio calculated from the feed forward control method in (22)makes SW2 turn off too early or too late when the inductor current reaches zero. This leads to the increase of the switching losses
and the conduction losses. On the other hand, the duty ratio D1 calculated from the feedback control method in (21) represents
precisely for the value of the inductor current. This enables SW2 to be turned off exactly at the moment when the inductor current

reaches zero, which results in Zero Current Switching (ZCS).
IV. EXPERIMENTAL RESULTS
Table II shows the parameters of circuit and controllers in experiments. While the proposed DCM compensation is applicable forthe boost converter at any power rate and any switching frequency, a boost converter is designed for a specific switching frequency
at 20 kHz due to the computation time limitation of the low cost controller. Besides, the sampling method for the instantaneous
average current in DCM which has been described thoroughly in [15]-[16] is also applied simply in the proposed DCM current
control.
A. Frequency Analysis, Current Step Response and Load Transient Response
Fig. 11 illustrates the frequency characteristics of the conventional and proposed DCM current feedback control as shown in Fig.
9. From Table II, it is noted that the magnitude of the injected AC current command is 0.25 p.u., which can be considered as a large
signal. This makes the elimination of nonlinearity in DCM by the proposed control be confirmed not only in the small-signal model
but also in the large-signal model. Besides, the designed gain in Fig. 11 is calculated from the discretized second-order standard
form which is derived from (11). Note that the frequency analysis is conducted in the simulation in order to confirm the cutoff
frequency between the conventional and proposed DCM current feedback control. As shown in Fig.11, the frequency characteristicof the proposed DCM control agrees exactly with the designed values (CCM), whereas the cutoff frequency of the conventional
DCM control results in high error of 47.5%. Furthermore, the relationship between the cutoff frequency of the outer voltage control
loop and the output capacitance is expressed as (23) [1],CI lo ad........................................... (23)2Vovershoo tf v

where C is the output capacitance, Iload is the load current variation, Vovershoot is the overshoot voltage during the load transientvariation and fv is the cutoff frequency of the outer voltage control loop. It can be understood from (23) that, under the same
condition of the load current variation Iload and the overshoot voltage Vovershoot, the output capacitance can be minimized by a widevoltage control bandwidth. Besides, the cutoff frequency fv of the outer voltage control loop depends linearly on the cutoff
frequency of the inner current control loop [18]. Hence, the bandwidth of the outer voltage control loop can be improved by 47.5%
with the proposed DCM current control comparing to the voltage control bandwidth of the conventional DCM current control. As
a result, the output capacitance C can be minimized by 47.5% with the proposed DCM current control.
In this paper, the implementation of the proposed DCM nonlinearity is carried out by the simplified model as shown in Fig. 6, in
which the complete model in Fig. 5 is linearized and the small signal of the output voltage, i.e. the output voltage dynamic is
neglected. This simplification can be considered to be reasonable in PFC applications where the output capacitor is designed basedon the line frequency. However, for DC-DC conversion applications, the output voltage dynamic has to be considered in the DCM
nonlinearity compensation. Therefore, AC voltages at different frequency are injected into the output voltage in order to investigate
the simplification of the output voltage dynamic in the proposed DCM nonlinearity compensation.

2

Average Current iavg (0.25 p.u./div)

Resonance occurs when fvac equals to fiac.
Output Voltage Simplified
vout (250 V/div)
model

0V

Complete model
AC coupling output voltage vout_ac
V/div)
f (10
: 100
Hz

1

0V

Magnitude |G| [dB]vacfvac : 1 kHz
fvac : 10 kHz
fvacCurrent: 100 kHzInput
iin (2 A/div)

0V0

0

Rise time: 380 ms
Overshoot: 9%
vgs1
vgs2
( 10V/div)( 10V/div)
Command iavg* (0.25 p.u./div)

-1

0A
-2

0A
0A

Time (800 ms/div)

Input Current i (2A/div)
CCM Current iinin (2A/div)

Time (20 ms/div)
Time (400 ms/div)(a) Switching signal at light load (0.3 p.u.).
(a) Conventional CCM current response.

(a) Load step change: 1.0 p.u. to 0.4 p.u.

-3

0 V-4

0

Max. gain difference:
smaller than 0.5 dB
Output Voltage vout (250 V/div)
500
1000
1500
2000
2500
Frequency
fiac [Hz]AC coupling
output voltage
vout_ac (10 V/div)

Fig. 12. Frequency characteristics of closed-loop transfer function
betweenaverage inductor current and its reference using simplified
0V
model and complete model that includes also output voltage dynamic.
The resonance occurs due to the second-order nature of the transfer
function between the duty-ratio and the inductor current. Apart of that,the maximum gain difference between the simplified model and thecomplete model is smaller than 0.5 dB.

0A

Time (800 ms/div)

Input Current iin (2 A/div)(b) Load step change: 0.4 p.u. to 1.0 p.u.
Fig. 14. Load transient response by output voltage control with 0.2-kHz
cutoff frequency. The overshoot voltage and the settling time are 9 V
and 1660 ms, the errors of which compared to the designed values are
4.3% and 1.6%, respectively [18].

Average Current iavg (0.25 p.u./div)
0V

Rise time: 380 ms
8%
vgs1 Overshoot:vgs2( 10V/div)( 10V/div)

Command iavg*(0.25 p.u./div)

0
0A
0A

Time (20 ms/div)
Input Current iin (2A/div)Time
DCM(b)Current
iin (2A/div)
Switchingsignal at heavy load
(0.9 (400p.u.).ms/div)

Fig. 15. DCM synchronous
switching.
SW 2 response.
is turned off exactly when(b) Proposed
DCM current
the current
reacheszero,
which achieves
Fig.
13. Current
step
response
by inputZCS.
current control with 2-kHz
cutoff frequency in CCM and DCM. The DCM current response almost
agrees with the conventional CCM current response.

Fig. 12 depicts the frequency characteristics of the closed-looptransfer function between average inductor current and its reference, when the proposed DCM nonlinearity compensation is
implemented, using the simplified model and the complete model that includes also the output voltage dynamic. It is understoodfrom Fig. 12 that a resonance occurs due to the second-order nature of the transfer function between the duty-ratio and the inductor
current [10]. Apart from that, the maximum gain difference between the simplified model and the complete model is below 0.5 dB.
Consequently, in the proposed DCM nonlinearity compensation, the output capacitor is required to be designed so that thefrequency of the output voltage dynamic is much higher than the cutoff frequency of the current control loop.
Fig. 13 illustrates the current step response obtained in experiment by the input current control with 2-kHz cutoff frequency at
the switching frequency of 20 kHz. It is concluded that the DCM current response almost agrees with the conventional CCM
current response. This confirms the validity of the proposed DCM nonlinearity compensation by experiment. Furthermore, theDCM nonlinearity compensation part is designed independently from the PI controller. Therefore, in order to further improve theperformance of the DCM current control, the PID controller or the IP controller, which have been researched thoroughly for CCM
operation, can be applied simply. Note that when the boost converter operates in CRM, the proposed DCM control can still beapplied. However, the DCM nonlinearity compensation is required to be deactivated when the boost converter operates in CCM.
Fig. 14 shows the load transient response obtained in experiment by the output voltage control with 0.2-kHz cutoff frequency.
The test is conducted with the load current step from 0.4 p.u. to 1 p.u. and vice versa. The overshoot voltage and the settling timeare 9 V and 1660 ms, the errors of which compared to the designed values are 4.3% and 1.6%, respectively. The stability of thevoltage control in the proposed DCM control is similar to that in the conventional CCM control because the nonlinearity in the
DCM current control is eliminated completely. Consequently, this implies that the bandwidth voltage control in DCM can beincreased at least as same as CCM in order to minimize the output capacitance because the DCM operation can also eliminate theright-half plan zero in CCM [20].
B. Discontinuous-Current-Mode Synchronous Switching Operation Waveforms, and Comparison of Efficiency andComputation Time
Fig. 15 illustrates the synchronous switching waveforms in DCM. It is concluded that SW2 is turned off exactly when the current
reaches zero, which achieves ZCS. Furthermore, before SW 2 is turned on, during the dead time, the current flows through the diode.

0.2%

98.5

Efficiency [%]TABLE IIICOMPUTATION TIME IN CLOCK CYCLES FOR CURRENT CONTROL LOOP.

Maximum Efficiency:98.7%

99

CCM
Computation
Conventional DCM Proposed DCM
Current Control Current Control Current Control
Time

98

PI controller

97.5

Duty-ratio
feedforward
Nonlinearity
compensation
Total

97
96.5
96
95.5

2.8%
DCM Synchronous Switching（ ）DCM Asynchronous Switching（ ）CCM Synchronous Switching（ ）1 p.u. of output power equals 360W.

9

9

9

16

0

0

0

39

22

25 (1 p.u.)48 ( 1.9 p.u. )31 (1.2 p.u.)

Therefore, SW2 turns on at the forward voltage of the diode,which is considered as ZVS. As a result, the synchronous
95
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
switching in DCM causes almost no switching loss in SW 2.
Ouput Power [p.u.]
Furthermore, this proposed DCM synchronous switching can be
Fig. 16. Efficiency comparison among CCM synchronous switching, realized without modifying the main circuit, which implies that
DCM synchronous switching and DCM asynchronous switching. The
efficiency is maintained at high value above 98% over wide load range. the conversion efficiency can be increased simply.
Fig. 16 illustrates the efficiency comparison among the CCM
synchronous switching, the DCM synchronous switching, and the
DCM asynchronous switching with the same inductor LHCR. The maximum efficiency of the DCM synchronous switching and theCCM synchronous switching are same as 98.7% because the circuit is designed in order to operate in CRM at full load. As the load
becomes lighter, the efficiency of the CCM synchronous switching decreases as explained in Fig. 1, whereas the efficiency of the
DCM synchronous switching is still maintained at high values over than 98%. In particular, at load of 0.125 p.u., the efficiency isimproved by 2.8% when the DCM synchronous switching is applied. This efficiency improvement especially benefits theapplication with a frequent variation between light load operation and full load operation. On the other hand, compared to the DCM
asynchronous switching, the efficiency of the DCM synchronous switching is higher by 0.2% at all load range. This efficiency
improvement is because the conduction loss of MOSFET is lower than that of a diode in the given switching device.
Table III shows the approximate computation time for all the arithmetic operations in the current control loop. Both division andsquare root digital calculations are time-consuming computation involving multiple clock cycles. In particular, when a STM32
Cortex-M4 microcontroller is applied to process 32-bit floating-point single-precision data, 14 clock cycles are required for adivision or square root calculation, whereas 1 clock cycle is required for an addition or subtraction and 3 clock cycles are required
for a multiplication [21]. In the conventional DCM current control, a division followed by a square root calculation is required tocompute the DCM duty ratio as shown in Fig. 9(a). On the other hand, only one division is required in both the CCM current control
as in Fig. 4 and the proposed DCM current control as in Fig. 9(b). Consequently, the computation time of the proposed DCM
current control is longer only 6 clock cycles than the CCM current control, whereas the conventional DCM current control requires
35% longer computation time than the proposed DCM current control. Therefore, the proposed DCM current control helps in
reducing the cost of the controller.
V. CONCLUSION
The approach to deal with the nonlinearity of DCM in this paper was to utilize the duty ratio at the previous calculation period.
This provided both the complete elimination of the DCM nonlinearity and the short computation time. Furthermore, the simple
DCM synchronous switching without any modifications of the main circuit was realized to achieve high efficiency above 98% overwide load range, which helped in reducing the power consumption under light load conditions. In comparison with the CCM
current control, the proposed DCM current control achieves the same current control bandwidth and improves the light load
efficiency by 2.8% at most. Furthermore, the proposed DCM current control not only reduces the computation time by 35% but
also increases the current control bandwidth by 47.5% compared to the conventional DCM current control. Consequently, theproposed DCM current control contributes into both the downsizing of the boost converter by the wide bandwidth current andvoltage control, and the high efficiency over wide load range as well as short computation time.
ACKNOWLEDGMENT
The authors would like to thank Dr. Hiroki Takahashi and Dr. Keisuke Kusaka, Power Electronics Laboratory, Nagaoka
University of Technology, Nagaoka, Japan, for constructive advice and collaboration.
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