2 bobine cuplate

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1

Modeling and Analysis of Coupled Inductors in Power Converters

Guangyong Zhu†

Richtek Technology Corp.

1006 East Yager Ln, Unit 101C, Austin, TX 78753

Brent McDonald, Member, IEEE

Texas Instruments

12500 TI Blvd., MS8709, Dallas, TX 75243

Kunrong Wang, Sr. Member, IEEE

Dell, Inc.

One Dell Way, Round Rock, TX 78682

Abstract-This paper describes a new approach to the analysis of switched mode power converters

utilizing coupled inductors and presents a novel canonical circuit model for N-winding coupled

inductors. Waveform and ripple of the winding current in a coupled inductor converter can be

easily determined using the developed model similar to those obtained in an uncoupled inductor

converter. Influence of coupling coefficient on converter steady state and transient performance

is readily predicted by the proposed model. It is found that in an N-phase coupled inductor

converter, the voltage waveforms driving the leakage inductors are no longer the phase node

voltages, but are the modified voltages with a frequency N times the original switching

frequency. In addition, their magnitudes also vary with the coupling coefficient among the

coupled windings. Through coupling, a converter is capable of responding faster to load transient

depending on the coupling coefficient and control mechanism, and that dependency is

analytically revealed in the paper. Finally, a two-phase buck regulator is experimentally tested to

verify the proposed model.

Key words: coupled inductors, coupled magnetics, coupled inductor modeling, multiphase dc-dc

converters, symmetrical transformer model.

This paper was presented at IEEE APEC, Feb. 15-19, 2009, Washington DC.

† corresponding author: Dr. Guangyong Zhu, 1006 East Yager Ln, Unit 101C, Austin, TX 78753. Tel.: 512-719-

5272, fax: 512-719-4556, email: [email protected].



2

I. INTRODUCTION

Power engineers are facing increasing challenges to design small, low cost, efficient and

reliable switching power supplies. It is even more challenging in today’s computing industry

with the ever increasing processor speed as power supplies are required to support not only the

high processor current, but also its fast slew rate.

To meet these demands, new circuit topologies and improved control mechanisms have been

proposed and studied, among which inter-leaved multi-phase coupled inductor converters

attracted special attention. As of today, there are a few such solutions commercially available on

the market.

Some studies [1-10, 16-19] reported that a converter utilizing coupled inductors can reduce

phase current and output voltage ripple, respond faster to load transients, and reduce the amount

of output decoupling capacitance. These benefits make it an appealing technique in the

computing industry. However, a comprehensive explanation is still unavailable or unconvincing

theoretically; some questions remain unanswered, such as: Is it universally true that a coupled

inductor converter always respond faster to load transient than its uncoupled counterpart? Are

there any special conditions that must be satisfied to optimize performance? To truly take

advantage of the coupled inductor technique, a clear understanding is needed of its limitations, of

the effect of the coupling coefficient on the current distribution in each winding and on the

overall regulator steady-state and transient performance, as well as of the specification of the

coupling coefficient in order to meet certain performance requirements.

The concept of magnetic coupling has been around for a long time. Its general operation

principles are well known and basic to the electrical engineering society. It is the special

application in switching power converters where periodic (as during steady state) or random (as



3

during transient) driving voltage pulses can be applied to the coupled inductors that makes it

difficult for many power engineers and designers to analyze. Advanced simulation tools are able

to demonstrate changes to overall converter performance caused by different couplings, but are

unable to reveal mathematically the inherent correlation between them. A valid circuit model for

coupled inductors suitable for power converter analysis becomes a necessity.

Over the years, different transformer models for coupled inductors or magnetic structures

have been published [2, 3, 11-15], which attempted to establish certain equivalent circuit models

suitable for analyzing switched mode power converters. One model directly links each of the

parameters to a particular type of magnetic fluxes which are usually difficult to measure or

calculate [11]. Discussion and construction of reluctance models for some special core magnetic

structures with simulation and experimental results can be found in [2], [3] and [13]. An

improved reluctance model which simplifies circuit simulation was given in [12]. In [14], a

cantilever transformer model was proposed, in which all parameters can be directly measured.

However, all these circuit models are either too complicated to use or unable to provide enough

analytical insight and design guidelines to help power engineers in designing power supplies

using the coupled inductor technique.

The purpose of this article is to clarify the operation of coupled inductor converters by

presenting a simple analytical modeling approach for coupled inductors based on the basic

circuit principle, which leads to a simple canonical symmetrical circuit model. This model

enables power engineers with a basic understanding of switching power converters to analyze

and design coupled inductor converters more effectively [20].



4

II. ANALYSIS OF COUPLED INDUCTORS AND DERIVATION OF A

SYMMETRICAL MODEL

For ease of derivation and understanding, only coupled inductors with two windings will be

discussed in this section. The methodology presented here and the derived conclusions can be

easily extended to multi-winding coupled inductors, which will be elaborated in section IV.

A. Review of the Transformer Model

Fig. 1(a) shows two inversely coupled inductors where vL1, i1 and vL2, i2 are voltages and

currents across each winding, respectively. Its transformer model can be derived mathematically

and can be expressed by the following equation according to the well-known basic circuit theory:

dt

diL

dt

diMv

dt

diM

dt

diLv

L

L

22

12

2111

+−=

−=

, (1)

where L1, L2 and M are the self inductance of each winding and the mutual inductance between

them, respectively.

The expression can be rearranged as:

dt

di

LL

MLv

L

Mv

vL

M

dt

di

LL

MLv

LL

LL

2

21

2

21

1

2

2

2

1

21

2

11

)1(

)1(

−+−=

−−=

, (2)

or,

dt

diLv

L

Lkv

vL

Lk

dt

diLv

kLL

LkL

221

1

22

2

2

1111

+−=

−=

, (3)



5

where k 121

≤=LL

M is the well-defined coupling coefficient and Lk1 =(1-k

2)L1 and Lk2 =(1-k

2)L2

are commonly referred to as the leakage inductances.

If we assume L1=L2=L, as is usually true in a two-phase dc-dc converter, eq. (3) is simplified

as:

dt

dikLkvv

kvdt

dikLv

LL

LL

22

12

212

1

)1(

)1(

−+−=

−−=

. (4)

From eq. (4) an equivalent circuit can be constructed as shown in Fig. 1(b), where LM=k2L is

commonly referred to as the magnetizing inductance. Note that Fig. 1(b) is one form of the

transformer models for two coupled inductors, where the leakage inductor appears only on one

side. The leakage inductor shown in the figure has the inductance seen from that side when the

other side is shorted.

(a) (b)

Fig. 1. Two inversely coupled inductors (a), and their equivalent

transformer circuit model if L1=L2=L (b).

B. Derivation of a New, Symmetrical Model

The model derived above, though mathematically accurate, is not easily used in analyzing

power converter operations, as it is asymmetrical to each inductor winding with the presence of



6

the magnetizing inductor, LM, and with the leakage inductor only on one winding. It is known

that the leakage inductance can be distributed to the two sides, but it still fails to make the model

symmetrical because of the presence of a magnetizing inductor on one side, whose impact on the

converter operation cannot be neglected [4].

As will be seen later, a symmetrical model developed in this paper can simplify the analysis

and understanding of coupled inductors in power converter applications under steady state and

dynamic load transient conditions. Specifically, the model is able to provide additional insight to

the impact of leakage inductance and coupling coefficient (or mutual inductances in case of

multiple coupled inductors) on individual winding current, overall output current ripple, and

converter response to sudden load transient. Therefore, it offers an additional tool to allow power

supply engineers to optimize the design and to analyze the performance of coupled inductor

voltage regulators.

In order to derive such a model, eq. (3) is re-arranged as:

21

1

222

2

2

11

11

LLk

LLk

vvL

Lk

dt

diL

vL

Lkv

dt

diL

+=

+=

. (5)

In the case of L1=L2=L, eq. (5) will be simplified to eq. (6) below, where Lk=(1-k2)L is the

leakage inductance.

212

211

LLk

LLk

vkvdt

diL

kvvdt

diL

+=

+=

. (6)

From eqs. (5) and (6), one can easily come up with a generalized symmetrical circuit model

for Fig. 1(a), as shown in Fig. 2.



7

2

2

1Lv

L

Lk

1

1

2Lv

L

Lk

(a) (b)

Fig. 2. A generalized symmetrical circuit model of two inversely coupled

inductors (a), and the simplified model if L1=L2=L (b).

It is also easy to verify that, by changing the polarity of the two controlled voltage sources,

the model shown in Fig. 2 also applies to the case with two non-inversely coupled inductors.

One of the distinctive features of the equivalent circuit model for two coupled inductors

given in Fig. 2 is that it does not contain the magnetizing inductor which, to most of us, is a

fundamental element of the well-known transformer model. Instead, the coupling effect is

reflected in the controlled voltage sources and the associated coupling coefficient k. In addition,

an inductor with a leakage inductance Lk is shown on each inductor winding.

The benefits of the derived model in Fig. 2 in analyzing coupled inductor converters will be

elaborated in the following sections.

III. APPLICATION OF THE DERIVED COUPLED INDUCTOR MODEL IN A TWO-

PHASE BUCK CONVERTER

Fig. 3 shows a two-phase buck converter utilizing inversely coupled inductors. By replacing

the coupled inductors with the symmetrical equivalent circuit model in Fig. 2(b), an equivalent

two-phase buck converter can be obtained, as shown in Fig. 4.

If we denote vph1 and vph2 as the phase node voltage of each phase (reference to ground),

respectively, then from Fig. 4, vL1 and vL2 can be expressed as:



8

Fig. 3. A two-phase coupled inductor buck converter.

Fig. 4. An equivalent circuit of the two-phase coupled inductor buck converter.

022

011

Vvv

Vvv

phL

phL

−=

−=. (7)

Therefore, voltages (reference to ground) at nodes x1 and x2 can be derived as:

021122

021211

)(

)(

kVvkvkvvv

kVkvvkvvv

phphLphx

phphLphx

−+=+=

−+=+=. (8)

Notice that x1 and x2 are two fictitious nodes emerged from the derivation of the model and

therefore, they are not physically accessible. However, interpretation of the result from eq. (8) is

very important in understanding the operation of the coupled inductor converter shown in Fig. 3

since vx1 and vx2 serve as the switching node voltages (to the leakage inductance) similar to those



9

in an uncoupled two phase converter. As a result, the current waveform through the leakage

inductor can be easily determined once the waveforms of vx1 and vx2 are obtained.

A. Steady-State Performance Analysis Using the Derived Model

Based on eq. (8), steady-state waveforms of the actual phase node voltages, vph1 and vph2, the

fictitious node voltages, vx1 and vx2, as well as the current waveforms generated in the leakage

inductors, i1 and i2, are shown in Fig. 5. It can be seen from the figure that, analogous to the

uncoupled inductor converter where an inductor is driven by the switching pulse, vph1 or vph2, the

leakage inductor of each phase in a coupled inductor converter is now driven by a new series of

periodic switching pulses, vx1 or vx2. However, because of the coupling, the fundamental

frequency of this new periodic pulse becomes twice of the original switching frequency fs =1/Ts.

In addition, the magnitude of this new periodic pulse also varies with the coupling coefficient of

the two inductors.

According to Fig. 5, the slopes of inductor ripple current, SR1, SF and SR2, and the peak-to-

peak inductor current ripple, ∆Ip1, in each phase as well as the induced current ripple from

coupling, ∆Ip2, are given by eqs. (9) and (10), respectively:

−

+−=

−−=

+−=

−

+−=

+−=

Lk

VkkVS

Lk

V

L

VkS

Lk

VkV

L

VkVS

inR

k

F

in

k

inR

)1(

)1(

)1(

)1(

)1(

)1()1(

2

02

00

2

001

, (9)

s

k

in

sRp

s

k

in

sRp

DTL

VkkVDTSI

DTL

VkVDTSI

022

0

11

)1(

)1(

+−==∆

+−==∆

. (10)



10

In eq. (10) and Fig. 5, D=V0/Vin is the duty cycle, which is assumed to be less than 0.5 during

steady state operation of the two-phase converter discussed in this section. This means there is no

phase node voltage overlapping between phases in steady state.

Fig. 5. Steady-state voltage and current waveforms from the converter circuit in Fig. 4.

Unlike in an uncoupled converter where the inductor winding current ripple (and also output

voltage ripple) in each phase is determined by the self inductance L, eq. (10) shows that in a

coupled inductor converter, the steady state current ripple in each phase is affected by the

coupling coefficient k and is inversely proportional to the leakage inductance Lk. It is therefore



11

clear that a smaller Lk will result in larger output voltage and winding current ripple. Even

though meeting the output voltage ripple specification in today’s power regulators for modern

microprocessors has not been a big challenge in general [19], the self inductance, coupling

coefficient and switching frequency still need to be properly selected to limit the winding current

and output voltage ripple.

Fig. 5 also reveals quantitatively how the coupling coefficient affects the inductor current

waveforms with two unequal inductor current peaks or valleys in a switching cycle in each phase.

To obtain an equal peak or valley current, coefficient k has to be close to 1 according to Fig. 5

and eq. (10), which will affect the winding current and output voltage ripple.

The converter in Fig. 3 turns to the traditional uncoupled inductor converter if k=0. The new

model shown in Fig. 2 is still valid in this scenario as now the controlled voltage sources are both

shorted and voltages at nodes x1 and x2 become equal to that at nodes ph1 and ph2, respectively,

as can also be verified in eq. (8), and Figs. 4 and 5.

In an inter-leaved two-phase converter, the total output peak-to-peak current ripple with

coupled (∆Icoupled-total) and uncoupled (∆Idis-total with inductance Ldis) inductors is given as:

s

k

in

pptotalcoupled DTL

VkDIII

)1()21(21

+−=∆+∆=∆ − , (11)

s

dis

in

totaldis DTL

VDI )21( −=∆ −

. (12)

Since the output voltage ripple is proportional to the total output current ripple, it can be

concluded from eqs. (11) and (12) that converters utilizing coupled inductors always generate

higher total output current ripple and output voltage ripple considering the following two special

conditions: Ldis=L and Ldis= Lk =(1-k2)L.

B. Transient Performance Analysis Using the Derived Model



12

In today’s computer systems, a regulator powering a micro-processor is frequently subject to

a substantial load application with a slew rate as high as hundreds of amperes per microsecond

following a processor’s exit from a deeper sleep state. To ensure that the voltage does not fall

below a minimum level required by the processor for its normal operation, the regulator must be

able to respond quickly to deliver the required current or energy before the output decoupling

capacitors are over-discharged.

The most effective solution to speeding up the regulator response to a sudden load

application is to increase the duty cycle of the phase immediately following the event while

pulling in the remaining phases. Fig. 6 depicts a scenario in a two-phase regulator where an event

of load application happens at t=0. The arrows on Vph1 and Vph2 waveforms show that the duty

cycle of the phase node voltage Vph1 increases (pulse pushed out from the dotted line) and Vph2

goes from 0 V to Vin earlier than it is normally scheduled (pulse pulled in from the dotted line),

resulting in a short period of pulse overlapping between t1 and t2. It should be pointed out that

duty cycle increase and phase node pulse overlapping can take place over several switching

cycles depending on the controller behavior. However, only the first phase node pulse

overlapping between the phases is illustrated in the figure to demonstrate the effectiveness of the

proposed coupled inductor model and to show the pattern of energy transfer from input to the

load.

In Fig. 6, the voltage waveforms of the two fictitious nodes, vx1 and vx2, are first determined

from eq. (8). Waveforms of current in each inductor winding are then derived based on these

waveforms. As can be seen, the inductor current in each winding ramps up at a slew rate of ST

which is much higher than its steady state slew rate SR1 or SR2 in eq. (9) whenever there is phase

node voltage overlapping during a transient. This slew rate is given by:



13

Lk

VV

L

VVkS in

k

in

T)1(

)1( 00

−

−=

−+= (13)

For comparison purposes, the inductor current waveform in each phase in an uncoupled

inductor converter is also depicted in the figure in dotted lines. Its slew rate during inductor

charging period is given by:

Fig. 6. Transient voltage and current waveforms in a coupled inductor converter

during a sudden load application.



14

dis

in

RL

VVS 0−

= . (14)

Since ST > SR no matter Ldis=L or Ldis= Lk, it is obvious that, depending on the coupling

coefficient k, the current in each winding is able to ramp up much faster in a coupled inductor

regulator if the controller allows phase overlapping. This means that a two-phase coupled

inductor converter is inherently capable of transferring more current or energy to the load when

needed. The difference between the solid and dotted lines in the i1 or i2 waveform in Fig. 6 shows

how much higher the phase current in a coupled inductor converter can build up than that in an

uncoupled converter. Conceivably, this difference will only get larger if more than one phase

node voltage overlapping occurs.

On the other hand, when a microprocessor suddenly reduces its current drawn from the

regulator (an event of sudden load release), the extra energy built up and stored in the inductors

right before the load release will transfer to the output capacitors and cause the output voltage to

rise. To avoid overshooting above a microprocessor’s acceptable maximum voltage level,

presence of larger capacitance may be required at the regulator output. The regulator controller

should also turn off the highside mosfets on all phases to prevent delivering extra energy to the

microprocessor.

Coupled inductors also perform favorably in this situation when all phase node voltages

remain at 0 V. Unlike in an uncoupled inductor converter where the inductor current decreases at

a slew rate of disLtv )(0

, each winding current in the coupled inductor converter decreases at a

slew rate of Lktv )1()(0 − , which means that it decreases faster and causes less overshoot on the

output voltage.



15

That conclusion can be explained by examining the proposed coupled inductor model in Fig.

4. Now that the phase nodes ph1 and ph2 are grounded, the two controlled voltage sources kvL1

and kvL2 are actually receiving part of the energy being released from the leakage inductor from

each winding as current i1 and i2 are flowing through them (note vL1 =vL2=-v0(t)). As a result, the

output capacitors do not need to absorb all the magnetic energy stored in the leakage inductors,

and therefore less output voltage overshoot will be expected. This interesting mechanism does

not exist in uncoupled inductor regulators.

IV. EXTENSION OF THE MODEL TO MULTI-COUPLED INDUCTORS

The concept of the symmetrical coupled inductor model as well as the process of derivation

can be easily extended to multiple coupled inductors so that the performance of multiphase

interleaved coupled inductor regulators can be analyzed and understood.

In order to derive closed-form expressions and to explain the methodology of analyzing

coupled inductor converters using the proposed model, the following assumptions are made for

the coupled inductors with N windings:

a. Each coupled winding has an equal inductance L,

b. The mutual inductance between any two coupled windings is equal to M=kL. This

assumption implies the coupled flux generated by one winding is equally distributed to

the rest of the N-1 windings, or 1

1

−≤

Nk , and,

c. There is no phase node voltage overlapping during steady state, which implies D<1/N, or

a high step down conversion from input to output for high number of phases N. This is

usually the case in low output voltage, high current and fast load transient applications

such as in powering today’s computer processors.



16

With these assumptions, the relationship among all winding voltages, vLj(t), and currents, ij(t)

(j=1, 2, …, N), can be expressed in the form of an N×N dimension symmetrical matrix given

below in (15). It should be pointed out that for this inductor network to work properly in a buck

regulator, inverse coupling has to be maintained between any two windings.

⋅=

)(

)(

)(

)(

)(

)(

2

1

2

1

ti

ti

ti

dt

d

tv

tv

tv

NLN

L

L

MML or,

⋅=

−

)(

)(

)(

)(

)(

)(

2

1

12

1

tv

tv

tv

ti

ti

ti

dt

d

LN

L

L

N

MML , (15)

where

−−

−−

−−

=

LMM

MLM

MML

L

MOMM

L

L

L , and

[ ]

−−

−−

−−

−−+=−

kNkk

kkNk

kkkN

LkNk

)2(1

)2(1

)2(1

)1(1)1(

11

L

MOMM

L

L

L .

Define:

[ ]

LkN

kNkLk

)2(1

)1(1)1(

−−

−−+∆ , (16)

as the equivalent leakage inductance of the multiple coupled inductors with N windings. The

voltage across the leakage inductance of each winding can be expressed in terms of voltages

across every winding as follows:

jmNj

tvkN

ktv

dt

tdiL

N

m

LmLj

j

k

≠=

−−+= ∑

=

,,,2,1

)()2(1

)()(

1

L

, (17)

Equation (17) describes a canonical model for N–winding coupled inductors. Based on this

equation, a canonical symmetrical equivalent circuit model can be obtained as shown in Fig. 7.



17

It is noticed that the second term on the right side of eq. (17) and the controlled voltage

sources in Fig. 7 reflect the influence of the quality of coupling (denoted by the coupling

coefficient k) among all inductor windings. k also affects the equivalent leakage inductance of

the coupled inductors. It is easy to prove that when N=2, eq. (17) and Fig. 7 are simplified to the

previously discussed two-winding coupled inductor model, as in eq. (6) and Fig. 2.

∑=−−

N

m

Lm tvkN

k

2

)()2(1

+

−−∑

=

N

m

LmL tvtvkN

k

3

1 )()()2(1

∑−

=−−

1

1

)()2(1

N

m

Lm tvkN

k

Fig. 7. A canonical symmetrical equivalent circuit model for multiple coupled inductors.

Fig. 8 shows an equivalent symmetrical circuit when the derived canonical model is applied

to an N-phase interleaved buck converter. One can easily obtain the voltage at any fictitious node

xj, vxj(t) (j=1, 2, …, N), in terms of all phase node voltages as:

jmNj

kN

kVNtv

kN

ktvtv

N

m

phmphjxj

≠=

−−

−−

−−+= ∑

=

,,,2,1

)2(1

)1()(

)2(1)()( 0

1

L

. (18)

Similar to previous discussions in a two-phase converter with coupled inductors, steady state

and transient current waveforms on each phase in an N-phase converter can be accurately



18

determined and converter performance can be predicted once the voltage waveform on each

fictitious node is derived according to eq. (18).

+

−−∑

=

N

m

LmL tvtvkN

k

3

1 )()()2(1

∑=−−

N

m

Lm tvkN

k

2

)()2(1

∑−

=−−

1

1

)()2(1

N

m

Lm tvkN

k

Fig. 8. An N-phase interleaved buck converter with the canonical symmetrical coupled inductor

model.

The total output peak-to-peak current ripple, ∆Icoupled-total, is the sum of the phase current

ripple, ∆Iripple-coupled, when a particular phase is turned on and the induced current ripple on all

other windings, and can be calculated from eq. (17):

[ ]

[ ] s

in

coupledripple DTLkNk

VNkkNVVI

)1(1)1(

)1()2(1)( 00

−−+

−−−−−=∆ −

, (19)

[ ]

[ ] sin

s

k

in

coupledrippletotalcoupled

DTLkN

VND

DTL

VNVVkN

kV

II

)1(1

)1(

)2()()2(1

000

−−

−=

−−−−−

+−

+∆=∆ −−. (20)

When N=2, eq. (19) is simplified to ∆Ip1 in eq. (10) and eq. (20) is simplified to eq. (11).



19

For comparison purposes, the phase current ripple, ∆Iripple-dis, and the total output peak-to-

peak current ripple, ∆Idis-total, in an uncoupled inductor converter are also given below in eqs. (21)

and (22):

s

dis

indisripple DT

L

VVI 0−

=∆ − (21)

s

dis

ins

dis

intotaldis DT

L

VNDDT

L

VV

D

NDI

)1(

1

1 0 −=

−

−

−=∆ − . (22)

Since 1

1

−≤

Nk , it can be proven from eqs. (20) and (22) that ∆Icoupled-total is always greater

than ∆Idis-total under two specific conditions of Ldis=L and Ldis=Lk, where Lk is now defined in eq.

(16). Therefore it can be concluded that the output voltage ripple in a coupled inductor converter

is always greater than that in an uncoupled inductor converter if Ldis=L and Ldis=Lk, as the output

voltage ripple is proportional to the total output current ripple.

When k=0 and L=Ldis, eq. (20) is simplified to eq. (22), as is expected.

During transient when all phases are turned on (vLj(t)=Vin-V0, j=1, …, N) at load applications

or turned off (vLj(t)=-V0 ) at load releases, the ramp-up and ramp-down slew rates of the current

in each winding can be obtained from eq. (17) as:

[ ]LkN

VV

L

VVNkN

kVV

S in

k

inin

T)1(1

))(1()2(1

)(0

00

−−

−=

−−−−

+−

= , (23)

[ ]LkN

VSD

)1(1

0

−−

−= . (24)

By comparing eqs. (14) and (23), it can be easily found out that ST >SR for Ldis=L or Ldis=Lk,

where Lk is defined in eq. (16). Again it shows that during load applications the phase current

ramp-up slew rate in coupled inductor regulators is always higher than that in uncoupled inductor



20

regulators when phase overlapping is allowed; therefore faster energy transfer during load

applications can be achieved in coupled inductor regulators. Similarly, during load releases the

phase current always ramps down faster in coupled inductor regulators (at a slew rate determined

by eq. (24)) than that in uncoupled inductor regulators, which implies less overshoot on the

output voltage in coupled inductor regulators. This observation has been discussed extensively in

Section III for the two-phase regulator.

Based on the above discussions, Table I summaries if an N-phase coupled inductor converter

performs better or worse than its uncoupled counterpart in terms of steady state phase current

ripple, output voltage ripple (proportional to total output current ripple) and response speed to

load transient (assume phase overlapping is allowed) under the specific condition of Ldis=L or

Ldis=Lk. It is shown that, in both cases, coupled inductor regulators always perform inferior in

terms of output voltage ripple but have the capability of responding faster to load transient if

phase node voltage overlapping is allowed. However, the phase current ripple, ∆Iripple-coupled,

obtained in eq. (19) in coupled inductor regulators may be higher or lower than ∆Iripple-dis given in

eq. (21) in uncoupled inductor regulators depending on the inductance (Ldis=L or Ldis=Lk) and the

coupling coefficient, k.

TABLE I: Performance Advantages of Coupled over Uncoupled Inductor Converters

Uncoupled with Ldis=L Uncoupled with

Ldis=Lk Condition

Category D

Dk

−<

1 1

1

1 −<<

− Nk

D

D 1

1

−<

Nk

Steady state phase

current ripple better worse better

Steady state output

voltage ripple worse worse worse

Transient response better better better

V. EXPERIMENTAL VERIFICATION



21

A two-phase regulator test board shown in Fig. 9 using the ISL6266 controller was used to

verify the validity of the derived model. The measured data were then compared with the results

predicted by the model.

The coupled inductor adopted on the test board is LC1740-R30R09A from NEC/Tokin,

which specifies a typical self inductance of 310 nH. The measured parameters (with a short

external wire connected to one terminal of each winding to fit a current probe) are L=353.5 nH,

k=0.622, and Lk=216.7 nH.

Fig. 9. Picture of the two-phase coupled inductor regulator test board.

A. Steady State Comparison

Fig. 10 shows the captured phase node voltage and inductor winding current waveforms of

the regulator under two different input voltages. Table II summaries the test condition, measured

current ripple and the calculated current ripple predicted by the model in these two cases. In Fig.

10(a), for example, the current ripple in each winding can be obtained according to eq. (10) as:

)A(61.510215107.216

)622.01(115.112622.0

)A(11.1010215107.216

)622.01(115.112

9

92

9

91

≈⋅⋅⋅

+⋅−⋅=∆

≈⋅⋅⋅

+⋅−=∆

−

−

−

−

p

p

I

I



22

(a) Vin=12 V (time scale: 500 ns/div)

(b) Vin=19.5 V (time scale: 500 ns/div)

Fig. 10. Experimental waveforms with 1.115V output voltage and 20A load.

5.30 A

Phase 1 switching node


iL1

iL2

9.85 A

10.50 A

6.00 A



iL1

iL2



23

TABLE II: Measured and Calculated Current Ripple in the Inductor Windings

∆Ip1 (A) ∆Ip2 (A) D

fs

(kHz) measured calculated measured calculated

Vin=12 V 10.1% 470 9.85 10.11 5.30 5.61

Vin=19.5 V 6.19% 476 10.50 10.61 6.00 6.19

B. Transient State Comparison

Fig. 11 shows the scope waveforms of the winding current with Vin=9 V, V0=1.15 V under a

sudden load application at the regulator output. The waveforms clearly show a distinctive and

sharp increase in the current of both inductor windings during a particular time period (71 ns)

when both phases are on (the slope is measured as 58.64 A/µs.) This observation agrees with the

theoretical analysis elaborated in section III that faster energy transfer can be achieved in

coupled inductor converters whenever phase node voltages overlap.

The slope of this current, ST, according to eq. (13), is given as:

76.58107.216

15.19)622.01(

9=

⋅

−+=

−TS (A/µs)

As can be seen, both steady state and transient experimental results match excellently with

the theoretical prediction from the proposed model.

VI. CONCLUSION

A new modeling approach for analyzing coupled inductors is proposed and a novel canonical

symmetrical circuit model is developed. The proposed model is easy to use and suitable for

analyzing switched-mode power converters utilizing coupled inductors. With the introduction of

fictitious nodes, the model states that coupling among inductor windings creates a new series of

periodic voltage waveforms on the fictitious nodes. It is this new series of periodic voltage

waveforms with N times the original switching frequency and varying magnitudes that actually

drives each leakage inductor of the coupled inductors. The symmetrical model successfully and

quantitatively reveals how model parameters affect the steady state and transient performance of



24

power converters. It is found that coupled inductors are able to respond faster to a load transient

if phase overlapping is allowed. Results from experimental measurement and theoretical

calculation show excellent agreement, demonstrating the validity and accuracy of the proposed

model.

Fig. 11. Experimental waveforms under a sudden load application

with 9 V input and 1.15 V output (time scale: 500 ns/div).

ACKNOWLEDGEMENT

The authors would like to thank Dr. Jia Wei and Dr. Kun Xing from Intersil Corp. for

providing the test board to support the validation of the derived model in this study.

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71 ns

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2 bobine cuplate

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