Generalized Steady-state Analysis of Multiphase Interleaved Boost

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


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


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


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


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


qD0 if inverse coupled and odd NOFF


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


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


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


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


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).


Generalized Steady-state Analysis of Multiphase Interleaved Boost

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