generalized steady-state analysis of multiphase interleaved boost
TRANSCRIPT
2005-38
Generalized Steady-state Analysis of MultiphaseInterleaved Boost Converter with Coupled Inductors
**Department of Electrical &Computer Engineering,
University of Wisconsin-Madison,1415, Johnson Drive, Madison,
WI, 53706,USA
WisconsinElectricMachines &PowerElectronicsConsortium
University of Wisconsin-MadisonCollege of Engineering
Wisconsin Power Electronics Research Center2559D Engineering Hall1415 Engineering Drive
Madison, WI 53706-1691
© 2005 Confidential
Research Report
H. B. Shin, J. G. Park, S. K. Chung,H. W. Lee* and T. A. Lipo**
Division of Electrical &Electronic Engineering,
Engineering Research Institute,Gyeongsang National
University, 900, Gazwa-dong,Jinju, Gyeongnam, 660-701,
Republic of Korea
*Division of Electrical & ElectronicEngineering,
Kyungnam University, 449,Wolyoung-dong, Masan, Gyeongnam,
631-701,Republic of Korea
Generalised steady-state analysis of multiphaseinterleaved boost converter with coupled inductors
H.-B. Shin, J.-G. Park, S.-K. Chung, H.-W. Lee and T.A. Lipo
Abstract: The generalised steady-state analysis of the multi-phase interleaved boost converter withcoupled inductors operated in continuous inductor current mode is addressed. The analyticalexpressions for efficiency, inductor and input currents, and output voltage are derived from thetransformed average state–space model. Generalised expressions for the input and inductor currentripples and the output voltage ripple are also derived for various inductor couplings and thecharacteristics are analysed according to the inductor couplings. The steady-state performance isverified experimentally.
1 Introduction
Recently, the interleaved boost converter (IBC) has beenstudied for applications to power-factor-correction circuitsand as the interface between fuel cells, photovoltaic arrays,or battery sources and the DC bus of AC inverters [1–3].The IBC is composed of several identical boost convertersconnected in parallel. The converters are controlled byinterleaved switching signals, which have the same switchingfrequency and the same phase shift. By virtue of parallelingthe converters, the input current can be shared among thecells or phases, so that high reliability and efficiency inpower electronic systems can be obtained. In addition, it ispossible to improve the system characteristics such asmaintenance, repair, fault tolerance, and low heat dissipa-tion. As a consequence of the interleaving operation, theIBC exhibits both lower current ripple at the input side andlower voltage ripple at the output side. Therefore, the sizeand losses of the filtering stages can be reduced, and theswitching and conduction losses as well as EMI levels canbe significantly decreased [4–9].
However, more phases in the IBC increase the numberof components, such as inductors, and active and passiveswitches. The dimension of state and control inputalso becomes higher and it is more difficult to analyse andinvestigate the operating characteristics at both steadyand transient states. The multiphase IBC has been modelledand analysed in [5] by using a signal flow graph at thesteady state. Some useful expressions such as conversionratio and efficiency are also presented. The graphical modelof the IBC becomes very complex as the number of phasesincreases and the interleaved switching pattern is alsorestrictive. It was shown that increasing the number of
phases in the IBC could significantly reduce the inputcurrent ripple [6]. The output voltage ripple was calculatedfor the buck converter case in [7]. The conditions of the dutycycles for minimising the current ripples were found. Forthe two-phase IBC, some converter performance expres-sions and transfer functions have been derived when theinductors are coupled [8]. In addition, good current sharingcould be naturally achieved by operating the two-phase IBCwith direct coupled inductors in the discontinuous inductorcurrent [9].
In this paper, a generalized average state–space model ofthe multiphase IBC with coupled inductors is developed byusing Lunze’s transformation, which enables considerstionof the inductor currents of the multiphase as the common-mode and differential-mode currents, separately. Thecommon-mode current is useful for deriving the steady-statecharacteristics and the converter is assumed to operate incontinuous inductor current mode. Generalised and explicitexpressions for converter performance, such as efficiency,input and inductor current ripples, and output voltageripple, are derived and characterised according to theinductor couplings. The generalised analysis for converterperformance is verified through the experimental results.
2 Generalised average state equation ofmultiphase IBC with coupled inductors
Figure 1 shows the multi-phase IBC with coupled inductorsin which 2N boost converters are connected in parallel.
Vg
r, Ll
D1
C RS1 S2 S2N
D2
D2N-1
igiL,1
iL,2
iL,2N-1
io
vo
+
−
Lm
r, Ll D2NiL,2N
Fig. 1 2N-phase interleaved boost converter with coupled inductors
H.-B. Shin, J.-G. Park and S.-K. Chung are with the Division of Electrical &Electronic Engineering, Engineering Research Institute, Gyeongsang NationalUniversity, 900, Gazwa-dong, Jinju, Gyeongnam, 660-701, Republic of Korea
H.-W. Lee is with the Division of Electrical & Electronic Engineering,Kyungnam University, 449, Wolyoung-dong, Masan, Gyeongnam, 631-701,Republic of Korea
T.A. Lipo is with the Department of Electrical & Computer Engineering,University of Wisconsin-Madison, 1415, Johnson Drive, Madison, WI, 53706,USA
r IEE, 2005
IEE Proceedings online no. 20045052
doi:10.1049/ip-epa:20045052
Paper first received 5th June and in final revised form 29th November 2004.Originally published online: 8th April 2005
584 IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005
Each converter consists of a coupled inductor, active switchand diode. It is assumed that the parallel converters aresymmetrical and operate in the continuous inductor currentmode. Considering the polarity of the coupling inductors asshown in Fig. 1, the dynamics of the inductor currents canbe written as:
LldiL;2k�1
dtþ Lm
XN
j¼1
diL;2j�1dt
þ rLm
XN
j¼1
diL;2j
dt
¼� riL;2k�1 � s2k�10vo þ Vg
LldiL;2k
dtþ rLm
XN
j¼1
diL;2j�1dt
þ Lm
XN
j¼1
diL;2j
dt
¼� riL;2k � s2k0vo þ Vg
ð1Þ
where s0k ¼ 1� sk, k¼ 1,2,?,N, and
Ll, Lm: leakage and mutual inductances of the coupledinductor
r: effective series resistance (ESR) of the coupled inductor
sk: Switching function of active switch in the kth converter
Vg; vo: input and output voltages.
If sk ¼ 1, the corresponding active switch Sk is turned on. Ifsk¼ 0, Sk is turned off. For the sake of generality, thecoupling parameter r has been introduced in (1). Dependingon the winding orientation of the coupling inductor, r isdefined as
r ¼ 1 if direct coupling�1 if inverse coupling
�
When the inductors are not coupled, the value of themutual inductance Lm will be zero. In matrix form, (1) canbe written as
LdiL
dt¼ �riL � Svo þ GVg ð2Þ
where iL ¼ iL;1 iL;2 � � � iL;2N½ �T , r ¼ rI2N�2N ,
L ¼
Ll þ Lm rLm Lm � � � rLm
rLm Ll þ Lm rLm � � � Lm
..
. ... ..
. ... ..
.
Lm rLm Lm � � � rLm
rLm Lm rLm � � � Ll þ Lm
266666664
377777775;
S ¼
s01s02
..
.
s02N
266664
377775 G ¼
1
1
..
.
1
266664
377775
The capacitor or output voltage equation can be expressedas
dvo
dt¼ 1
C
X2N
k¼1s0kiL;k �
vo
R
!¼ � 1
RCvo þ
1
CST iL ð3Þ
where R and C denote the load resistance and output filtercapacitance, respectively. The state equations in (2) and (3)describe the dynamics of the 2N-phase IBC with coupledinductors.
As the number of phases increases, the system dimensionalong with the coupling becomes higher. Consequently, it is
more difficult to analyse the steady-state performance suchas the current and voltage ripples as well as dynamicperformance. However, the following state transformationallows simplification of the analysis:
i ¼ GiL ð4Þwhere G is Lunze’s transformation for linear symmetrically-coupled systems such as [12]:
G ¼ 1
2N
ð2N � 1Þ �1 � � � � � � �1�1 ð2N � 1Þ � � � � � � �1... ..
. ...
� � � ...
�1 �1 � � � ð2N � 1Þ �11 1 � � � � � � 1
2666664
3777775
ð5ÞThe inverse transformation is given, for reference, by
G�1 ¼
1 0 � � � � � � 10 1 � � � � � � 1
..
. ... ..
.� � � ..
.
0 0 � � � 1 1�1 �1 � � � �1 1
266664
377775 ð6Þ
The transformation in (4) replaces 2N-phase inductorcurrents by an average current and (2N�1) deviatedcurrents from their average, which will be considered asthe common-mode current and the differential-modecurrents, respectively, in the following.
The average or common-mode current i2N is defined as
i2N ¼1
2N
X2N
j¼1iL;j ð7Þ
The differential-mode currents are also defined as
ik ¼ iL;k � i2N ð8Þwhere k¼ 1,2,?,2N�1. Substituting (4) into (2) yields
di
dt¼ A11i þ A12vo þ B1Vg ð9Þ
whereA11
¼ �GL�1rG�1
¼ rLl
�1þ b 0 b 0
�b �1 �b 0
b 0 �1þ b 0
�b 0 �b �1... ..
. ... ..
.
�b �b �b 0
b b b 0
0 0 0 0
2666666666666664
b . . . 0 b 0
�b . . . 0 �b 0
b . . . 0 b 0
�b . . . 0 �b 0
..
. ... ..
. ... ..
.
�b . . . �1 �b 0
b . . . 0 �1þ b 0
0 . . . 0 0 � L1
Lc
3777777777777775
IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005 585
A12 ¼ �GL�1S
¼ 1
Ll
�s02k�1 þ bPNj¼1
s02j�1 þg2N
P2N
j¼1s0j for odd rows
�s02k þ bPNj¼1
s02j þg2N
P2N
j¼1s0j for even rows
LlLc
�12N
P2N
j¼1s0j
!for 2N th row
26666666664
37777777775
B1 ¼ GL�1G ¼ 1
Lc
0
0
..
.
0
1
26666664
37777775
The parameters in the above equations are defined as
Lc ¼Ll þ Nð1þ rÞLm
Ld ¼Ll þ Nð1� rÞLm
b ¼ð1� rÞ Lm
Ld
g ¼ Ll
Ld
ð10Þ
The equation of capacitor voltage in (3) can be rewritten as
dvo
dt¼ A21i þ A22vo ð11Þ
where
A21 ¼1
CSTG�1
¼ 1
Cs01 � s02N � � � s02N�1 � s02N
P2N
j¼1s0j
" #
A22 ¼�1
RC
Combining (9) and (11) yields the generalised state equationof the multiphase IBC
_x ¼ Axþ Bu ð12Þ
where x ¼ i1 � � � i2N vo½ �T , u ¼ Vg� �
, A ¼A11 A12
A21 A22
� �, and B ¼ B1
0
� �. It is noted that the switching
function sj in the system matrix is a time-varying function.To derive the averaged state–space model, the averaging
operator is defined as [13]:
yðtÞ � 1
Ts
Zt
t�Ts
yðtÞdt ð13Þ
The duty cycle function d(t) can then be defined as
dðtÞ ¼ sðtÞ ¼ 1
Ts
Zt
t�Ts
sðtÞdt ð14Þ
The generalised average state equation of the multiphaseIBC with coupled inductors can be expressed, from (12) and(13), as:
_x ¼ Axþ Bu ð15Þ
where d 0k ¼ 1� dk, x ¼ i1 � � � i2N vo
� �T, u ¼ Vg
� �, A
¼ A11 A12
A21 A22
� �; B ¼ B1
0
� �
A12¼1
Ll
�d 02k�1 þ bPNj¼1
d 02j�1 þg2N
P2N
j¼1d 0j for odd rows
�d 02k þ bPNj¼1
d 02j þg2N
P2N
j¼1d 0j for even rows
LlLc
�12N
P2N
j¼1d 0j
!for 2N th row
26666666664
37777777775
A21 ¼1
Cd 01 � d 02N � � � d 02N�1 � d 02N
P2N
j¼1d 0j
" #
It is noted that the averaged state vector is composedof a common-mode current, (2N�1) differential-mode currents, and the output voltage. Using thegeneralised average model in (15), the steady-statecharacteristics of the multiphase IBC will be analysed inthe following.
3 Steady-state analysis
The steady-state solution for (15) is derived in theAppendix. The output voltage in the steady state isrewritten as:
Vo ¼1
D2N
X2N
j¼1D0j
!Vg ð16Þ
where D2N ¼ rRþ
P2N
j¼1D0j
2. The inductor current in the k-th
converter can be written from (8), (68), and (69), as
IL;k ¼Vg
D2N
1
Rþ 1
r
X2N
j¼1D02j � D0k
X2N
j¼1D0j
!( )ð17Þ
where k¼ 1, y, 2N. The input current Ig is composed ofthe common-mode current only such as
Ig ¼X2N
j¼1IL;j
¼2NI2N
¼ Vg
r2N � 1
D2N
X2N
j¼1D0j
!28<:
9=; ð18Þ
Efficiency of the multiphase IBC, Z, can also be derived as:
Z ¼ V 20 =RVgIg
¼ rR
P2N
j¼1D0j
!2
D2N 2ND2N �P2N
j¼1D0j
!28<:
9=; ð19Þ
For the case of identical duty cycles (Dj¼Dfor j¼ 1,?,2N ), the steady-state performance can be
586 IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005
summarised as:
Vo ¼2ND0
r=Rþ 2ND02Vg ð20Þ
Ik ¼ 0; k ¼ 1; � � � ; 2N � 1 ð21Þ
IL;k ¼ I2N ¼Vg
R1
r=Rþ 2ND02ð22Þ
Z ¼ 1
1þ r=2ND02Rð23Þ
Note that the steady-state solutions are not dependent onwhether the inductors are directly or inversely coupled ornot. If the boost converters connected in parallel aresymmetrical, the differential-mode currents disappear andthe inductor currents are identical with the common-modecurrent. The multiphase IBC with coupled inductors has thesame analytical expressions for the steady-state performanceregardless of the inductor coupling.
3.1 Inductor and input current ripplesThe generalised and analytical expressions for current andvoltage ripples are derived under the following assumptions:
(i) Switching elements of the converter are ideal.
(ii) The inductor resistances are negligible.
(iii) The effective series resistance of capacitors and straycapacitance are negligible.
(iv) The converters in parallel are operated in thecontinuous inductor current mode.
The control signals have the same duty cycle D in thesteady state. Figure 2 shows the steady-state currentwaveforms of the four-phase IBC, as an example, wherethe inductors are not coupled. The active switches areswitched with sequence of S1; S2;� � �;S2N during the PWM(pulse-width modulation) period Ts. For the inverselycoupled inductor, the active switch connected to theinductor with positive or negative polarity is alternatelyswitched, as shown in Fig. 1. The control signals are equallyshifted with a value of t ð¼ Ts=2NÞ. There are 2N (for thisexample, 2N¼ 4) repetitive sub-periods in a PWM periodand the inductor currents have the same waveforms withphase shift of t, as shown in Fig. 2a. Only one active switchis switched during a sub-period t and the following relationis thus satisfied:
2N ¼ NON þ NOFF þ 1 ð24Þwhere NON and NOFF denote the number of active switchesthat are always in ON and OFF states during the sub-period, respectively. As shown in Fig. 2a, it is convenient toconsider only the sub-period for deriving the current andvoltage ripples in the steady state. The ON-duration of anactive switch during a PWM period, TON ð¼ D TsÞ, can beexpressed as
TON ¼ NON � tþ tON ð25Þwhere tON represents the ON-duration of an active switch intransition during the sub-period. Let the new duty cycle q bedefined in the sub-period as
q ¼ tON
tð26Þ
The duty cycle q can also be represented as
q ¼ 2N D� truncð2N DÞ ð27Þwhere truncð�Þ denotes the integer part and 2N is thenumber of phases. In other words, q is equal to the
fractional part of the multiplication of phase number andduty cycle. Then, substituting (26) into (25) yields:
2ND ¼ NONþq ð28Þ
2ND0 ¼ NOFF þ q0 ð29Þwhere q0 ¼ 1�q. These relations are useful in deriving thecurrent and voltage ripples.
The differential-mode currents can be rewritten from (9)as
_i2k�1 ¼Vo
Ll�s02k�1 þ b
XN
j¼1s02j�1 þ
g2N
X2N
j¼1s0j
!ð30Þ
_i2k ¼Vo
Ll�s02k þ b
XN
j¼1s02j þ
g2N
X2N
j¼1s0j
!ð31Þ
The common-mode current can also be expressed as
_i2N ¼1
LcVg �
Vo
2N
X2N
j¼1s0j
!ð32Þ
3.1.1 Input current ripple: Since the input currentis the sum of inductor currents as shown in Fig. 1, it can beexpressed with the common-mode current as:
ig ¼X2N
k¼1iL;k ¼ 2Ni2N ð33Þ
Therefore, the slope of input current can be obtained, from(20) and (32), as
digdt¼ 2NVg
Lc1� 1
2ND0X2N
j¼1s0j
( )ð34Þ
The current slope depends on the states of 2N switchingfunctions. The input current in the steady state has aperiodic waveform with a period t. Hence, it is necessary toknow the number of switches in the ON or OFF stateduring a sub-period. It is convenient to consider thesub-period in which only one active switch is transferredfrom the ON to the OFF state. Figure 3 shows four
TON
i1 i2 i3 i4
τON
i
tτ
i4
ig
t
kTs (k+1)Tst
io
a
b
c Ts
Fig. 2 Current waveforms of four-phase IBC with decoupledinductora Inductor currentsb Input currentc Output current
IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005 587
different switching functions of an active switch inone phase corresponding to various duty cycles for thefour-phase IBC example. The number of an active switch,which is always in the ON state during the sub-period,is different according to the value of the duty cycle.For example, if toDTso2t, NON¼ 1. The active switchesin other phases have the same duty cycle but they have aphase shift of t from each other. Therefore, it can be shownthat (NON+1) switches are in the ON state during qt, asshown in Fig. 4, and from (34) the input current has theslope:
dig
dt¼ 2NVg
Lc1� NOFF
2ND0
� �¼ Vg
Lc
q0
D0ð35Þ
Hence, the magnitude of the input current ripple can beexpressed as
Dig ¼1
2
Vg
Lc
qq0
D0Ts
2N
¼ VgTs
2Ll
1
1þ Nð1þ rÞLm=Llð Þqq0
2ND0ð36Þ
Note that the input current ripple depends on theinductance Lc, which has a different value according tothe inductor couplings. For the case of both decoupled(Lm¼ 0) and inversely coupled (r¼�1) inductors, theinput current ripple has the same expression and it is relatedto the leakage inductance. Fig. 5a shows the input currentripple according to the number of phases. Increasing thenumber of phases can reduce the input current ripple overallbut it may make the current ripple larger at certain dutycycles. For the case of a directly coupled (r¼ 1) inductor,increasing the number of phases can considerably reducethe current ripple magnitude, as shown in Fig. 5b. There-fore, direct coupling is the best choice in view of reducingthe input current ripple. When the inductors are highlycoupled or have small leakage for the same number ofphases, the current ripple can be much reduced.
The condition of continuous input current can be derivedfrom the following inequality:
Ig4Dig ð37Þ
which gives the condition, from (22), (33), and (36),
K4KcritðDÞ ð38Þ
where
K ¼ 2Ll
RTs; KcritðDÞ ¼
1
1þ Nð1þ rÞLm=Ll
D0q0q2N
ð39Þ
Figure 6 shows Kcrit for several numbers of phases. Thecontinuous range for input current becomes wider at theduty cycle where the input current ripple is minimised, asshown in Fig. 5. For the case of the directly coupledinductor, Kcrit depends on the mutual inductance. The rangeof the continuous input current mode is much wider thanthat of the decoupled or inversely coupled inductor. Whenthe inductors are highly coupled, the range of thecontinuous input current mode expands.
3.1.2 Inductor current ripple: Since the inductorcurrent ripples have the same waveform in the steady state,it is sufficient to consider the (2k�1)th inductor currentripple only. The slope of the (2k�1)th inductor current canbe written from (8) and (9) as:
diL;2k�1dt
¼ Vg
Lcþ Vo
Ll
�s02k�1 þ1
2Ng� Ll
Lc
� �X2N
j¼1s0jbXN
j¼1s02j�1
( ) ð40Þ
τ
q τ
NON + 1 NON
t
Fig. 4 Number of active switches in ON state during a sub-period
sk
sk
NON = 0
sk
sk
Ts
qτ
qτ
NON = 1
qτ
qττ
NON = 2
NON = 3
t
t
t
t
Fig. 3 Switching function of active switch corresponding to variousduty ratios for four-phase IBC
0
0.2
0.4
0.6
0.8
1.0
∆ig
×
2N = 1
2N = 2
2N = 4
2N = 6
2Ll
VgT
s0 0.2 0.4 0.6 0.8
0
0.04
0.08
0.12
0.16
0.20
D
(i) 2N = 2, Lm /Ll = 1/0.5
2N = 2, Lm /Ll = 1/0.3
2N = 4, Lm /Ll = 1/0.5
2N = 4, Lm /Ll = 1/0.3
(ii)
(iii)
(iv)
(i)
(ii)
(iii)
(iv)
1.
a
b[
[
∆ig
×2L
l
VgT
s[
[
Fig. 5 Input current ripple according to number of phasesa Decoupled and inversely coupled inductorsb Directly coupled inductor
588 IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005
Using (20) with r¼ 0, (40) can be simplified as
diL;2k�1dt
¼ Vg
Lcþ Vg
D0
� 1
Lls02k�1 þ
Lm
LdLc
XN
j¼1s02j�1 þ r
XN
j¼1s02j
!( ) ð41Þ
The inductor current slope depends on the correspondingswitch function for the case of the decoupled inductor(Lm¼ 0). When the inductor is coupled, the states of otherswitch functions also affect the current slope. Therefore, theinductor current ripple will be derived according to theinductor couplings.
1 Decoupled inductor:The inductor current is affected by the state of the
corresponding active switch only. The inductor currentslope during DTs can be written, from (41), as:
diL;2k�1dt
¼ Vg
Lcð42Þ
Since Lc ¼ Ll for the case of the decoupled inductor, themagnitude of the inductor current ripple can therefore beexpressed as:
DiL ¼Vg
2LcDTs ¼
VgTs
2LlD ð43Þ
2 Directly coupled inductor:When the inductor is coupled, the inductor current is not
only affected by the state of the corresponding activeswitch but it is also dependent on the other switch functions.When the inductor is directly coupled, the slope of the kth
inductor current in (41) can be rewritten as:
diL;k
dt¼ Vg
Lcþ Vg
D0� 1
Lls0k þ
Lm
LdLc
X2N
j¼1s0j
( )ð44Þ
For convenience sake, the index (2k�1) is replaced by k in(44). Since the right-hand side of (44) contains s0j, the
switches in the OFF state contribute to the inductor currentslope. Therefore, to calculate the magnitude of currentripple, it is necessary to investigate the number of switchesin the OFF state during the ON time of a correspondingswitch, DTs. Using (24) and the number of active switches inthe ON state during a sub-period in Fig. 4, the number ofswitches in the OFF state during DTs can be calculated, asshown in Fig. 7. Hence, the magnitude of current ripple canbe expressed from (44) as:
DiL ¼1
2
Vg
LcDTs
þ 1
2
Vg
D0Lm
LdLcNON NOFF qtþ ðNOFF þ 1Þ q0tð Þf
þ NOFF � qtg
ð45Þ
Using (28) and (29), (45) can be simplified as
DiL ¼1
2
Vg
LcDTs þ
Vg
D0Lm
LdLc4N2DD0 � qq0� Ts
2N
�
¼ VgTs
2Ll
D1þ 2NLm=Ll
� 1þ Lm
Ll2N � qq0
2NDD0
� �� �
ð46Þ3 Inversely coupled inductor:When the inductors are inversely coupled, (41) can be
written as:
diL;2k�1dt
¼ Vg
Lc
þ Vg
D0� 1
Lls02k�1 þ
Lm
LdLc
XN
j¼1ðs02j�1 � s02jÞ
( )
ð47ÞTherefore, the inductor current slope depends on thenumber of OFF switches during DTs. If NOFF is an oddnumber, the summation in the right-hand side of (47) is zeroduring every q0t period, so that the current ripple magnitudecan be expressed as:
DiL ¼VgTs
2LlD 1� Lm
Ll þ 2NLm
q0
2NDD0
� �ð48Þ
If NOFF is an even number, the summation in the right-handside of (47) is zero during every qt period, so that thecurrent ripple magnitude can also be derived as
DiL ¼VgTs
2LlD 1� Lm
Ll þ 2NLm
q2NDD0
� �ð49Þ
0 0.2 0.4 0.6 0.80
0.05
0.10
0.15
D
Kcr
it2N = 1
2N = 2
2N = 4
0
0.01
0.02
Kcr
it
(i)
(ii)
(iii)
(iv)(ii)
(i)
(iii)
(iv)
0 0.2 0.4 0.6 0.8D
1.0
1.0
2N = 2, Lm /Ll = 1/0.5
2N = 2, Lm /Ll = 1/0.3
2N = 4, Lm /Ll = 1/0.5
2N = 4, Lm /Ll = 1/0.3
a
b
Fig. 6 Kcrit(D):a Decoupled and inversely coupled inductorsb Directly coupled inductor
q τ t
τ
DTs
NON sub-periods
NOFFNOFF + 1 NOFF NOFF
NOFF + 1
q 'τ
Fig. 7 Number of active switches in OFF state during DTs
IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005 589
Figure 8 shows the magnitudes of inductor current ripplefor various inductor couplings. For the decoupled inductor,the magnitude of the inductor current ripple is proportionalto the duty cycle regardless of the number of phase as givenin (43), and it is shown by a, dotted line in Fig. 8. It can beseen that increasing the number of phases makes theinductor current ripple larger and the inversely coupledinductor has smaller current ripple than the directly coupledinductor.
The following equation should be satisfied for thecontinuous inductor current mode:
IL;k4DiL;k ð50ÞHence, the condition for the continuous inductor currentcan be written, from the average inductor current in (22)with r¼ 0, as:
K4Kcrit;LðDÞ ð51Þwhere
K ¼ 2Ll
RTs; ð52Þ
Kcrit;LðDÞ
¼
2NDD02 if decoupled
2NDD02 � LmLlþ2NLm
qq0D0 if directly coupled
2NDD02 � LmLlþ2NLm
qD0 if inverse coupled and odd NOFF
2NDD02 � LmLlþ2NLm
q0D0 if inverse coupled and even NOFF
8>>>><>>>>:
ð53ÞFigure 9 shows Kcrit;L for several phase numbers. The rangeof the continuous inductor current is narrower as the
number of phase increases and the inversely coupledinductor has a wider continuous range.
3.2 Output voltage rippleThe converter output current has a repetitive waveformwith a sub-period as shown in Fig. 2c and it is the sum ofthe inductor currents in the phases whose active switches arein the OFF state. With reference to Fig. 7, NOFF and ðNOFFþ1Þ switches are connected to the output capacitor duringqt and q0t, respectively, and their average currents aredepicted in Fig. 10. The output current during the sub-period is also shown in Fig. 10 for the decoupled inductors.When the inductors are coupled, the shape of the outputcurrent waveform may be different from the one shown inFig. 10. However, it is not necessary to know the correctshape of the output current waveform in order to calculate
2N = 2
2N = 4
2N = 6
2N = 1
1
0.3
Lm
Ll=
1
0.3
Lm
Ll=
0 0.2 0.4 0.6 0.8D
0
0.2
0.4
0.6
0.8
1.0
0
0.2
0.4
0.6
0.8
1.0
2N = 2
2N = 4
2N = 6
2N = 1
1.0
a
0 0.2 0.4 0.6 0.8D
1.0
b
∆iL
×2L
l
VgT
s[
[
∆iL
×2L
l
VgT
s[
[
Fig. 8 Ripple magnitude of inductor current according to numberof phasea Directly coupledb Inversely coupled inductor
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
D
Kcr
it,L
2N = 4
2N = 2
1.0
Fig. 9 Kcrit,L(D)F decoupledF directly coupled- � - � inversely coupled
Q
�
t
io
q � q ' �
Io
(NOFF +1)IL,k
NOFF IL,k
Fig. 10 Converter output currents for calculating output voltageripple
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
0.8
1.0
D
Vo
�v o
2N = 1
2N = 2
2N = 4
1.0
Fig. 11 Relative output voltage ripple ð�Ts=2RCÞ
590 IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005
the output voltage ripple. Instead, the charge Q can beobtained with the average current instead of the actualcurrent as:
Q ¼ ðNOFF þ 1ÞIL;k � Io� �
q0t ð54Þ
where
Io ¼Vo
Rð55Þ
Substituting (22) and (29) into (54) yields
Q ¼ Ts
4N 2
Vo
Rqq0
D0ð56Þ
Therefore, the output voltage ripple can be written as
Dvo ¼Q2C
¼ Ts
2RCVo
N2
qq0
D0
ð57Þ
The voltage ripple in (57) becomes the same equationas in the conventional boost converter [13] for the caseof the single-phase IBC (2N¼ 1) and it is valid whenthe converter operates fully in the range of the continuousinductor current mode. Note that the inductor couplingsmay not affect the magnitude of the output voltageripple.
Figure 11 shows the output voltage ripple of themultiphase IBC in the continuous inductor current moderegardless of the inductor couplings. It can be seen that thevoltage ripple has also a minimum value at the duty cycleswhere the current ripple is minimised (q¼ 0). Also, themaximum voltage ripple is much smaller as the IBC hasmore phases.
k = 0.725 k = 0.725
k = 0.65
k = 0.65
k = 0.725 k = 0.725
core
Fig. 12 Coupling coefficients between cores of four-phaseinductors
.5 ms
.5 ms
10.0 V
2.00 V
2.00 V
2.00 V
4
3
2.0 A
2.0 A
10 �s
10 �s
4
3
4
3
3
4
a
b
Le Croy
Le Croy
Fig. 13 Experimental results for single-phase IBCa Start-up responses (upper trace: output voltage, 10V/div., lowertrace: inductor current, 2A/div.)b Steady-state responses (upper trace: output voltage ripple, 2V/div.,lower trace: inductor current ripple: 1A/div.)
.5 ms
.5 ms
10.0 V
2.00 V
4
2.0 A
.5 ms2.00 V
3
2.0 A
.5 ms2.00 V
1
2.0 A
.5 ms
.5 ms
10.0 V
2.00 V2.0 A
.5 ms2.00 V2.0 A
.5 ms2.00 V2.0 A
.5 ms
.5 ms
10.0 V
2.0 A
.5 ms2.00 V
2.00 V
2.0 A
.5 ms2.00 V2.0 A
4
1
4
1
3
4
1
2
3
3
4
1
2
3
4
1
2
a
b
c
2
3
2
2
Le Croy
Le Croy
Le Croy
Fig. 14 Start-up experimental results for four-phase IBCUpper trace: output voltage, 10V/div., middle traces: inductorcurrents, 2A/div., lower trace: input current, 2A/div.a Decoupledb Inversely coupledc Directly coupled inductors
IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005 591
4 Experimental results
The generalised analysis in the previous Sections has beenverified through experimental results for single-, two- andfour-phase IBC. The parameters used in the experiment areas follows: rk � 0:24O (including the current-sensingresistance), C¼ 22mF, ESR of C¼ 0.3O, R¼ 21.6O,Vg¼ 7V, Ts ¼ 40 m sec. The coupling coefficients of four-phase inductors were measured and are given in Fig. 12,
where the coefficient k is defined asffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2
m=L1L2
p. The self
inductances of the coupled inductors are LkE600mH,k¼ 1,?,4, and the inductances of the decoupled inductorsare LkE200mH.
Figure 13 shows the experimental waveforms of thesingle-phase IBC when the duty cycle D is 0.67. Forcomparison of the output voltage ripple, a small outputcapacitor is used. It can be seen that the peak-to-peakripple magnitudes of output voltage and inductor currentare about 1.2V and 0.8A. Figure 14 compares theexperimental results of four-phase IBC for variousinductor couplings during startup and the steady-state performance are compared in Fig. 15. The inductorcurrents of the first and second phases are measured.
It can be seen that the magnitudes of the output voltageripple are considerably reduced by increasing the number ofphases but they are similar regardless of the inductorcouplings. The input current ripple is also considerablyreduced and the directly coupled inductor has the bestperformance of minimising the input current ripple. Theinductor current ripple with the coupled inductors hassmaller magnitude than with the decoupled inductor. Theexperimental results for the steady-state performance fortwo-phase IBC are shown in Fig. 16. The output voltageripples are similar regardless of the inductor couplings. Theinversely coupled inductor has the smallest inductor currentripple among the inductor couplings and the directlycoupled inductor is the best with respect to the inputcurrent ripple.
Based on the generalised steady-state analysis and theexperimental results, we can, therefore, summarise asfollows. The output voltage ripple does not depend on theinductor coupling methods and it can be much reduced byincreasing the number of phases. The input current ripplecan be considerably improved by increasing the number ofphases and the directly coupled inductor has the bestperformance in view of the input current ripple. The
0.50 V0.5 A
10 �s
0.50 V0.5 A
10 �s
0.50 V0.5 A
10 �s
0.50 V0.5 A
10 �s
1.00 V1.0 A
10 �s
1.00 V1.0 A
10 �s
2.00 V10 �s
2.00 V10 �s
0.50 V0.5 A
10 �s
0.50 V0.5 A
10 �s
1.00 V1.0 A
10 �s
2.00 V10 �s
4
1
2
3
4
1
2
3
4
1
3
4
2
3
4
2
3
4
2
3
a
b
c
2
Le Croy
Le Croy
Le Croy
Fig. 15 Experimental results of four-phase IBCUpper trace: output voltage ripple, 2V/div., middle trances: inductorcurrent ripples, 0.5A/div., lower trace: input current ripple, 1A/div.a Decoupledb Inversely coupledc Directly coupled inductors
0.50 V0.5 A
10 �s
0.50 V0.5 A
10 �s
1.00 V1.0 A
10 �s
2.00 V10 �s
0.50 V0.5 A
10 �s
0.50 V0.5 A
10 �s
1.00 V1.0 A
10 �s
2.00 V10 �s
0.50 V0.5 A
10 �s
0.50 V0.5 A
10 �s
1.00 V1.0 A
10 �s
2.00 V10 �s
4
1
2
4
1
2
4
1
2
4
2
3
4
2
3
4
2
3
a
b
c
3
3
3
Le Croy
Le Croy
Le Croy
Fig. 16 Experimental results of two-phase IBCUpper trace: output voltage ripple, 2V/div., middle trances: inductorcurrent ripples, 0.5A/div., lower trace: input current ripple, 1A/div.a Decoupledb Inversely coupledc Directly coupled inductors
592 IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005
inductor current ripple is affected by the inductor couplingand it has the smallest magnitude in case of the inverselycoupled inductor. However, as the number of phasesincreases, the inductor coupling has little effect on theinductor current ripple.
5 Conclusions
The multiphase interleaved boost converter operated in thecontinuous inductor current mode has been analysed forvarious inductor couplings in the steady state. Theanalytical expressions for efficiency, inductor and inputcurrents, and output voltage were derived from thetransformed average state–space model and it has beenshown that the steady-state solutions are not dependent onthe inductor coupling. Generalised expressions for the inputand inductor current ripple were also derived for variousinductor couplings and their characteristics were analysed.In addition, the equation of the output voltage ripple wasderived and it has been shown that the voltage ripple is notdependent on the inductor coupling. The experimentalresults have verified the steady-state performance of theinterleaved converter analysed herein according to theinductor coupling.
6 Acknowledgment
This work was financially supported by MOCIE throughthe IERC programme.
7 References
1 Veerachary, M., Senjyu, T., and Uezato, K.: ‘Maximum powerpoint tracking of coupled inductor interleaved boost convertersupplied PV system’, IEE Proc.,-Electr. Power Appl., 2003, 150, (1),pp. 71–80
2 Newton, A., Green, T. C., and Andrew, D.: ‘AC/DC powerfactor correction using interleaved boost and Cuk converters’.Proc. IEE Power Electr. & Variable Speed Drives Conf., 2000,pp. 293–298
3 Miwa, B. A., Otten, D. M., and Schlecht, M. F.: ‘High efficiencypower factor correction using interleaving techniques’. Proc. IEEEAPEC’92, Boston, MA, USA, 1992, Vol. 1, pp. 557–568
4 Perreault J., D., and Kassakian, J.G.: ‘Distributed interleaving ofparalleled power converters’, IEEE Trans. Circuits Syst.I, Fundam.Theory Appl., 1997, 44, (8), pp. 728–734
5 Veerachary, M., Senjyu, T., and Uezato, K.: ‘Signal flow graphnonlinear modeling of interleaved converters’, IEE Proc.,-Electr.Power Appl., 2001, 148, (5), pp. 410–418
6 Chang, C., and Knights, M.A.: ‘Interleaving technique in distributedpower conversion systems’, IEEE Trans. Circuits Syst.I Fundam.Theory Appl., 1993, 42, (5), pp. 245–251
7 Dahono, P.A., Riyadi, S., Mudawari, A., and Haroen, Y.: ‘Outputripple analysis of multiphase DC-DC converter’. IEEE Int. Conf.on Power Electr. and Drive Systems (PEDS), HongKong, 1999,pp. 626–631
8 Veerachary, M.: ‘Analysis of interleaved dual boost converter withintegrated magnetics: signal flow graph approach’, IEE Proc.-Electr.Power Appl., 2003, 150, (4), pp. 407–416
9 Lee, P., Lee, Y., Cheng, D.K.W., and Liu, X.: ‘Steady-state analysis ofan interleaved boost converter with coupled inductors’, IEEE Trans.Ind. Electron., 2000, 47, (4), pp. 787–795
10 Giral, R., Martinez-Salamero, L., and Singer, S.: ‘Interleavedconverters operation based on CMC’, IEEE Trans. Power Electron.,1999, 14, (4), pp. 643–652
11 Giral, R., Martinez-Salamero, L., Leyva, R., and Maxie, J.:‘Sliding-mode control of interleaved boost converters’, IEEETrans. Circuits and Syst. I Fundam Theory Appl., 2000, 47, (9),pp. 1330–1339
12 Garg, A., Perreault, D.J., and Verghese, G.C.: ‘Feedback control ofparalleled symmetric systems, with applications to nonlinear dynamicsof paralleled power converters’. Proc. IEEE Int. Symp. on Circuitsand Systems (ISCAS), 1999, Vol. 5, pp. 192–197
13 Erickson, R.W., and Maksimovic, D.: ‘Fundamentals of powerelectronics’ (Kluwer Academic Publishers, 2001, 2nd edn.)
8 Appendix
From (15), the relationships between the steady-state valuescan be written as: for the (2k�1)th differential-mode
currents:
rI2k�1 ¼rbXN
j¼1I2j�1
þ �D02k�1 þ bXN
j¼1D2j�1 þ
g2N
X2N
j¼1D0j
!Vo
ð58Þ
for the 2kth differential-mode currents,
rI2k ¼� rbXN
j¼1I2j
þ �D02k þ bXN
j¼1D2j þ
g2N
X2N
j¼1D0j
!Vo
ð59Þ
for the common-mode or 2Nth current,
rI2N ¼Vg �Vo
2N
X2N
j¼1D0j ð60Þ
for the output voltage:
Vo
R¼X2N
j¼1D0j � D02N
� �Ij þ I2N
X2N
j¼1D0j ð61Þ
where Dj denotes the steady-state value of the dutycycle function dj(t). Solving the above algebraic equationsyields the common-mode and the kth differential-modecurrents, I2N and IK, and the output voltage Vo in the steadystate.
Adding (58) for k ¼ 1; � � � ;N yields
ð1� NbÞrXN
j¼1I2j�1 ¼ �ð1� NbÞ
XN
j¼1D02j�1
(
þ g2
X2N
j¼1D0j
)Vo
ð62Þ
Using 1� Nb ¼ g from (10), (62) becomes
XN
j¼1I2j�1 ¼
Vo
r�XN
j¼1D02j�1 þ
1
2
X2N
j¼1D0j
( )ð63Þ
Substituting (63) into (58) gives
I2k�1 ¼Vo
r�D02k�1 þ
1
2N
X2N
j¼1D0j
( )ð64Þ
The 2kth differential-mode current can be calculated in asimilar way:
I2k ¼Vo
r�D02k þ
1
2N
X2N
j¼1D0j
( )ð65Þ
IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005 593
Therefore, the differential-mode current can be expressedfrom (64) and (65) as
Ik ¼Vo
r�D0k þ
1
2N
X2N
j¼1D0j
( )ð66Þ
Substituting (60) and (66) into (61) yields the output voltagein the steady state:
Vo ¼1
D2N
X2N
j¼1D0j
!Vg ð67Þ
where D2N ¼ rRþ
P2N
j¼1D0j
2. The differential-mode and com-
mon-mode currents can be written, using (67), as:
Ik ¼Vg
r1
2ND2N
X2N
j¼1D0j
! X2N
j¼1ðD0j � D0kÞ
!ð68Þ
I2N ¼Vg
r1� 1
2ND2N
X2N
j¼1D0j
!28<:
9=; ð69Þ
Note that the differential-mode currents disappear and thecommon-mode current only exists if the duty cycles are allidentical (symmetrical).
594 IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005