� � �

Abstract—This paper deals with the use of coupledinductors for the DC|DC converters. A complete analysis ofthis topology is carried out and its advantages aredemonstrated by comparing it to the DC/DC converter withuncoupled inductor. In this paper, we show how the coupledinductors allow the I²R losses reduction and theminimization of the phase and the output current ripples.Different architectures and designs of coupled inductors aregiven. The benefits and the drawbacks of each one areanalyzed basing on three criteria: minimization of theharmonic currents, ability to filter the differential currentsand the converter volume-weight. Then, simulation resultsof a medium power DC drive using a coupled inductorconverter are given and discussed emphasizing thetheoretical results and showing the advantages of thestudied topology for this power range.

Index Terms—coupled inductors, DC/DC converter,symmetrical inductor, I²R losses, DC drive, Cascaderegulation.

I. INTRODUCTION

Different techniques are used to improve the efficiencyand the performance of the DC/DC converter. Amongthese techniques, we set the multi-interleaved convertersusing uncoupled inductors and the multi-interleavedconverters using coupled inductors.

The power systems operating with interleavedconverters present many advantages. One advantage isthat interleaved converters minimize the ripple outputcurrent and help increase the frequency of the outputcurrent from f to p*f where f is the switching frequencyand p is the number of interleaved phases. This advantagecan be obtained with an interleaving achieved throughindependent inductors or coupled inductors. The firstresult can be synthesized from the expressions of theripple current for a DC/DC converter and an interleavedDC/DC converter given successively as follows [1], [3],[5], [6]:

(1 ) . ..

V i nL f

α α−Δ Ι = (1) ������������������������������������������������������������

(1 . ) . ..p V in

L fα α−Δ Ι = ��������������������������������������������������������������������

Where, Vin is representing the input voltage, L isrepresenting the inductance, and � is representing theduty cycle.

Interleaved converters using coupled inductors presentanother advantage in addition to the previous one: it canminimize the ripple phase currents and increase itsfrequency from f to p*f. This advantage cannot beobtained with uncoupled inductors.

We present in the following parts the analysis ofdifferent configurations of coupled circuit and thesimulation results.

II. COUPLED INDUCTORS

In this part, we show the advantages and disadvantagesof different structures of coupled inductors through amathematical analysis using Fourrier Series.

The four studied configurations of coupled inductorsare [2], [4]:

• Symmetrical cascade configuration• Symmetrical shunt configuration• Cyclic cascade configuration• Cyclic shunt configuration

These configurations are represented by the Fig1, 2, 3and 4.

ANALYSIS AND COMPARISON OF MULTIPHASE DC/DC CONVERTERSWITH COUPLED INDUCTORS IN MEDIUM POWER APPLICATIONS

Sobhi Barg1, Afef Ben Abdelghani Bennani2

1University of El Manar – LSE-ENIT,2University of Carthage- LSE-INSAT,

�

Fig. 1. Symmetrical cascade configuration.�

�

Fig. 2. Symmetrical shunt configuration

2011 8th International Multi-Conference on Systems, Signals & Devices

978-1-4577-0411-6/11/$26.00 ©2011 IEEE

Sobhi.barg@gmail.com Afef.BenAbdelghani@insat.rnu.tn

� �

The parameters given in the following in the nomenclature list below:

TABLE I DEFINED PARAMETERS

L1,Lp Symmetrical inductance of ordTR Total rating Kd

Constant pdd

IK I=Ip Common mode current Id Differential mode current

In order to determine the advantages aof each topology, we define the followin

• The symmetrical inductance Lrole consists in conditioning the ripple cu

• The rating Lp/L1 which ccouplers effect: the ability of the coudifferential currents [4].

• The total rating TR which ivolume and weight of the inter-phases co

A. Mathematical development of compa

To determine the symmetrical inducand order p, we need to use the FourrThe DSF help determine the differentiand the common mode currents.

The phase voltages (VAS, VBS…) arperiodic signals. Its representation by thof two components: a sine componcomponent. The sinusoidal component fundamental sinusoidal component components correspond to the harmonand odd rank. The fundamental com

Fig. 3. Cyclic cascade configuration

Fig. 4. Cyclic shunt configuration

�

parts are defined

der 1 and p

and disadvantagesng criteria: Lp of order p: its urrents [4]. characterizes the upler to filter the

is related to the onverter [2].

araison criteria

ctance of order 1 rier Series (DSF). ial mode currents

re non-sinusoidal he DSF is the sum nent and a DC is composed of a and the other

nics of even rank mponent and the

harmonics of rank p and mucommon mode components. sinusoidal differential mode coFor each of these two msymmetrical inductance of theto determine the equivalent cirAny periodic function f(t) whi

period 2T πω

= can be decomp

0

1

2( ) cos( )2

nn

af t a n tTπ∞

=

�= + +��

�The waveform of the phase

by Fig.5:

Fig. 5. Waveform of the phase voltage

VAS is a rectangular signal andthe duty cycle so its DSF is wr

1

sin( ) 2AS

nV t E E

nα α

+∞

=

= + �

So, the p voltage supplied bycan be written as follows:

(1

1

sin( ) 2

nV t E E

nα α

α

+∞

=

= + �

( )1

sin( ) 2 cp

n

nV t E E

nαπ

α ααπ

+∞

=

= + ⋅�

We synthesize that for k=1..p,follows:

( )1

sin( ) 2 ck

n

nV t E E

nαπ

α ααπ

+∞

=

= + ⋅�

The sum of the voltages Vp is

(1 1 1

sin ( )

p p

k nk k n

nV t p E V

nα

ααπ

∞

= = =

�= + �

�� � �

Where . So, using of the sum, we can write:

�

��

ultiple of p correspond to The other harmonics are

omponents. modes, we determine the e coupler. We use the DSF rcuit of each mode [1], [9]. ich has a pulsation w and a

posed into DSF as follows:

2sin(nb n tTπ �+ �

� (3)

e voltage VAS is represented

e of the converter

d If E is its module and � is ritten as follows:

( ) cos( )n

nwtn

απαπ

⋅ (4)

y the cells of the converter

( ) cos( )n

nwtαπ

απ⋅

(5)

( ) 2cos 1nwt n ppπ� �

+ −� �� �

, the voltage is written as

( ) 2cos 1nwt n kkπ� �+ −� �

� � (6)

then:� �

) ( ) 2cos 1 (7)n

nwt n kp

απ παπ

�� �+ − �� �

� ��the commutative property

� � �

( ) ( )1 1 1

sin 2( ) cos 1p p

k nk n k

nV t p E V nwt n k

n pαπ πα

απ

∞

= = =

� �� �= + + −� �� �

� �� �� � �

(9)

Using the addition and difference trigonometric formula of the sine function, we get:

( )( ) ( )

( ) ( )1 1 1

2cos cos 1sin

( )2sin sin 1

p p

k nk n k

nwt n kpn

V t p E Vn

nwt n kp

παπ

ααπ π

∞

= = =

� �� �� �⋅ −� �� �� �

� �� �� �= + � �� �� �� �� �� �− ⋅ −� �� �� �� �� �� �� �� �

� � �

(9)

If we consider:

( ) ( )sincosn n

nB V nwt

nαπ

απ= ⋅ ⋅

(10)

( ) ( )sinsinn n

nC V nw t

nα π

α π= ⋅ ⋅

(11)

Then the previous equation is going equal to:

( )

( )

1

1 1

1

2cos 1( )

2sin 1

p

np

kk

pk n

nk

B n kp

V t p EC n k

p

π

απ

∞=

= =

=

� �� �−� �� �� �� �= + � �� �� �+ −� �� �� �� �

�� �

�

(12)

We determine the real and imaginary parts of the exponential function to calculate this sum:

21

0

nkp ip

ke X iY

π� �− � �� �

=

= +�

(13)

• For n=p and multiple of p, we obtain:

� �����

���� ��

�� is the sum of an arithmetic sequence

where ��������� is the common ration .

21

0

nkp ip

ke p

π� �− � �� �

=

=� where ����� 0

X pY

==

(14)

• If n is non multiple of p:

� �����

���� ��

�� ��is the sum of a geometric sequence, where

�����

���� is the common ration which is equal to :

221

20

1 0

1

nki pnk pp ip

nkik p

ee

e

ππ

π

� �� �� �− � �� �

� �� �� �=� �

−= =

−� where

00

XY

==

(15)

As a conclusion, we can get the two results:

• for n=p and multiple p, we obtain:

11 1

( ) ( )p

k nk n

V t p E p B p V tα∞

= =

= + ⋅ = ⋅� � (16)

• for n non-multiple p:

1( ) 0

p

kk

V t=

=� (17)

So, the equivalent circuit of the coupler in both common and differential modes can be represented as follows:

We can synthesize from these two models that the role of the symmetrical inductance of order p consists in conditioning the ripple current of the common mode at the frequency p*f while the role of the symmetrical inductance of order 1 consists in conditioning the differential currents at the frequency f.

For the total rating, noted TR, its objective is the evaluation of the weight and volume of a given converter. It can be defined by the product of the rms values of the voltage and current. So, the total rating TR of a multi-phase coupled circuit is defined as follows:

TR 1 1 2 2 .......V I V I Vp Ip= ⋅ + ⋅ + + ⋅ (18)

This parameter will be function of the common mode current, the value of the inductance Lp and a constant

dK where pdd

IK I= : Ip is representing the common

mode current and Id is representing to the differential mode current. This parameter can be expressed by the symmetrical inductance of order p and order p and dK . Using the tow models of a circuit in the common and differential modes given previously, the total rating can be expressed as follows:

�

Fig. 6. Equivalent circuit of coupler of order 1

�

Fig. 7. Equivalent circuit of coupler of order p

� � �

( ) ( )( )( ) ( )

1 1 1

1 1 1

... (19)

² ² ... ² ²

as a asp p asp p asp p

a b p p p

TR V I V I V I V I

TR w L I I I w p L I

= ⋅ + ⋅ + + ⋅ + ⋅

= ⋅ + + + + ⋅ ⋅ ⋅

We know that:

d d pI K I= ⋅ (20)

So we can write:

1 1 1...a b p d pdI I I I K I= = = = = ⋅ (21)

The total rating’s expression can be written as follows:

( ) ( )( )

1

1

² ² ²² ²

d p p p

p d p

TR w L K I jw p L ITR w p I L K L

= ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅

= ⋅ ⋅ ⋅ ⋅ + (22)

Finally, we can define the TR as follows:

1

1² 1p

p dLTR w p I KL

−� �� �= ⋅ ⋅ ⋅ +� �� �� �� �� � (23)

B. Study case: the Symmetrical cascade configuration

If we apply the two fundamental results presented in the Fig.6 and Fig.7, the voltage can be written as follows:

3 (24)as a b c dV jlw I jMw I jMw I jMw I= ⋅ − ⋅ − ⋅ − ⋅

( )3 (25)as a b c dV jw l I M I M I M I= ⋅ − ⋅ − ⋅ − ⋅

Where:

��� and M represent respectively the inductance of the coupler and the mutual inductance of the coupler.

We consider VAS1 the voltage of order 1 and VAS4 the voltage of order 4. First, we start with the order 1. The output current is the sum of the phase currents which is equal to zero. So, we write:

b c d s a aI I I I I I+ + = − = − (26)

Substituting this result in the previous equation, we have:

( )1 13a s aV jw l M I= − ⋅ (27)

For a supply of order 4, the phase currents are equal and

the out current is the sum of these currents.

( )1 43as aV jw l M I= − ⋅ (28)

If we generalize these results for p phases, we obtain

these expressions of the symmetrical inductances of order

1 and p:

( )1 1L p l M= − − (30)

( ) ( )1 1L p l M= − ⋅ − (31)

The following table resumes the study of the four configurations of fig.1 and from it, we can compare between them by using the criteria defined in paragraph I

TABLE II SYNTHESIZE OF THE CHARACTERISTIC CRITERIA

Coupled

circuit Lp Lp/L1 TR

Fig.1 ( 1)( )pL M

− ×−

( 1)( )( 1)p L Mp L M− −− +

Fig.2 ² ²

( 1)( )L M

p L M−

− +3

( 1)( )L M

p L M−

− +

Fig.3 2( )L M−

( )2cos

L M

L Mpπ� �� �−� � �

� �� �

−

Fig.4 ² ²

2( )L M

L M−+

2cos

( )

L Mp

L M

π� �� �+� �� �� �� �

+

For the output current quality, the best configuration is the symmetrical cascade one: it has the highest rating Lp/L1 and the harmonic currents will be the lowest. It has also the highest value of the inductance Lp, and the ripple current will be the lowest. But, for the volume, basing on the forth criterion and referring to the equation (30), we can conclude that that this criterion is proportional inversely to the third one. That means, if we ameliorate the rating Lp/L1, we lose the performance of the TR and vice versa. So the symmetrical cascade has the biggest volume and weight.

III. APPLICATION IN THE MEDIUM POWER

The configurations of DC/DC converter presented previously are simulated for driving a medium power DC machine using PSIM Software. The characteristics of the motor are: (power is 268 kW, speed is 1820 r/mn, nominal current is 475A and the nominal voltage is 600 V). We used the cascade regulation for the closed loop

� �

control of the machine: The followingthe structure of closed loop control converter [7], [8].

This study is done with a different numbstart with a double-phase converter in oradvantages of the coupled inductoindependent inductors. Then, we presenof the four configurations to determinand disadvantages of each one. The following figures (fig.9 and fiwaveforms of the ripple phase currentconverter. We notice that ripple phareduced from 42A with independent inwith coupled inductors. In addition, the callow obtain phase currents equal, inphase, with a frequency equal to twice th

In this part, we will be interested influence of the different coupled circucurrents of the converter. Indeed, we vpresented by the table. To determine current, we use the second criterion. Wthe current ripple is conditioning by symmetrical inductance of order p.

To determine the ripple output current, wcriterion. We remember that the cconditioning by the value of the symmeof order p.

To determine the capacity of each cirdifferential currents, we use the harmoncharacter is conditioned by the third critratio.

A. Ripple output current

Fig.11 present the ripple current of theach coupled circuit. We notice that cascade circuit allows getting a more

Fig. 9. Phase currents of 2 phase DC/DC coninductors

Fig. 10. Phase currents of 2 phase DC/Duncoupled inductors

Fig. 8. Cascade regulation of the DC machine.

�

g figure describes of double-phases

ber of phases. We rder to present the or compared to nt the application

ne the advantages

ig.10) show the ts of the DC/DC se currents havenductors to 0.6A coupled inductors

n module and in he switching one.

in studying the uit on the output verify the results the ripple output

We remember that the value of the

we use the second current ripple is etrical inductance

rcuit to reduce the nic analysis. This terion: The Lp/L1

he converter with the symmetrical

e reduced ripple

current than the other confiresults coincide with the theore

Fig. 11. Ripple currents of fourtorque=1000 N.m, fo=50 hz. B. Harmonic currents

The capabilities of the four charmonic currents are deteranalysis of the output curre

nverter with coupled

DC converter with

�

a. Symmetrical cascade configuratio

d. Cyclic shunt configuration

b. Symmetrical shunt configuratio

c. Cyclic cascade configuration

igurations. The simulation etical one.

r-phase DC/DC converter, load

oupled circuits to filter the rmined through a spectral ent of each configuration.

�

on

�

on

�

� � �

Fig.12 shows the harmonic currents of each coupled circuit.

.

Fig.12. Harmonic currents of four-phase DC/DC converter, load torque=1000 N.m, fo=50 hz.

We can notice that the symmetrical cascade and cyclic cascade configurations are, as showed on Table 1, the best for the minimization of the amplitude of the differential currents differential currents. For example, the harmonic currents obtained with the first and third

coupled circuit is equal to 1.33Aand 0.6A respectively, but this is equal to 1.8A with the other configurations.

IV. CONCLUSION

We presented in this work the advantages of using the coupled inductors in the structure of DC/DC converter. We approved that coupled inductors allow reduce the ripple phase currents and output current��We compared also between four configurations� of coupled circuit basing on the ripple current, the harmonic current and the volume. Finally, we achieved by the simulation results which are coincided with the theoretical one. The different configurations of DC/DC converter are used to drive a DC machine of 268 kW. We approved, on one hand, that the symmetrical cascade circuit can improve more the efficiency related to the ripple current and the capacity to filter the harmonic currents of the DC/DC converter than the other circuits. On the other hand, this configuration leads to augment the volume and the weight of the converter.

V. ACKNOWLEDGMENT

This work was supported by the Tunisian Ministry ofHigh Education and Scientific Research.

REFERENCES

��� ����� ������ ���������� ������ �������� ���������� ��� ���������������� ��� ���� ������ ���� ������������� ����� ���!���"� #��"��$%&�'(�(���

�'� %$� )�*� +�,"� ��-$� %,� .��� �������� ��� /��*���� �0� ����1%��������� &���0������� 0��� ����������� +�2��� ����������� ���+�����3�%����(145(67658(19�:4�;�(�((�::4��

�6� <�)���������������%�������������������=��,��00������*3�+�2��������������&��������*�<���*"'((>����=�����?�@@@���2������������������

�8� ��������"� &� �*���"� A������"� ��#=���B"C�C����"� <�1<�D���������"� �/��*��� ��� ��00B������ �������� �E����������������B��� ���� ���� ��������������� ������ ����F�����������B��3#��%�&�������"��/&%������"�#� � ������������

�9� /��� �% -$� G�#/"� � /������������ ��� ��������������� ������=���������������0��������������������������B��H����3���F������������������E���������B����&�������"�$����=���'((5��

�>� �&��1C�� @�"� I�1������ #�"� <��1��*��� D���"� ����� ����������2���� �������� %��������� ��� ��,1����� &*��� �0� /������ ����3�%����&J/$�/�&%-$��-$�%$�G�&%/#��#��&J-$%��"�A-#��99"�$-���"�</$G/JI�'((5��

�4� <��1+��� #����"� ������ �����"� I��� ��������K"� ������#�=��"� � �������� ������������ L� �������� �������M ��N� L��������� ���=��"3� &�����H��� ��� �E���B�����"� ����B� )B����B������H����6>�(���

�5� ������ ����H��"� JB�������� ������H���3� D���� �������E%��B������ ��� ����� ��� A��� MD�%)1A�N"� �B��������� �B�����������H��"�����'((9��

�:� J�=���� ��������O� ��B���H���� ���B������� ����� ���B���������� ���*������H����P� %�$� '15(8�1�>891Q� ��� ��,�@�����"� ����::6���K��������

�

a. symmetrical cascade configuration ��� � �����

�

b. symmetrical shunt configuration ��� � �����

�

c. cyclic cascade configuration ��� � ����

�