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(Open Access) High-Frequency Power Transformers With Foil Windings: Maximum Interleaving and Optimal Design (2015) | Ernesto L. Barrios

Abstract— Foil conductors and primary and secondary

interleaving are normally used to minimize winding losses in

high-frequency transformers used for high-current power

applications. However, winding interleaving complicates the

transformer assembly, since taps are required to connect the

winding sections, and also complicates the transformer design,

since it introduces a new tradeoff between minimizing losses and

reducing the construction difficulty. This paper presents a novel

interleaving technique, named maximum interleaving, that makes

it possible to minimize the winding losses as well as the

construction difficulty. An analytical design methodology is also

proposed in order to obtain free-cooled transformers with a high

efficiency, low volume and, therefore, a high power density. For

the purpose of evaluating the advantages of the proposed

maximum interleaving technique, the methodology is applied to

design a transformer positioned in the 5 kW-50 kHz intermediate

high-frequency resonant stage of a commercial PV inverter. The

proposed design achieves a transformer power density of

28 W/cm

with an efficiency of 99.8%. Finally, a prototype of the

maximum-interleaved transformer is assembled and validated

satisfactorily through experimental tests.

Index Terms— Foil windings, high-frequency, maximum

interleaving, optimization, transformer design.

I. INTRODUCTION

or applications requiring galvanic isolation, the line-

frequency transformer has traditionally been the

heaviest, most expensive and least efficient component part of

an electronic power converter. Nowadays, there is a large

number of medium power applications (1-25 kVA) with a

limited weight and space, such as electric traction systems,

distributed generation systems (PV panels and mini-wind

turbines) and power supplies, in which cost and efficiency are

paramount. In these applications, one of the most widely

adopted solutions for achieving considerable reductions in

weight and volume whilst significantly increasing efficiency,

yet still maintaining the required galvanic isolation, is to

Manuscript received September 15, 2014; accepted November 03, 2014.

However, permission to use this material for any other purposes must be

obtained from the IEEE by sending a request to pubs-permission@ieee.org.

This work was supported in part by the Spanish Ministry of Economy and

Competitiveness under Grants DPI2010-21671-C02-01 and

DPI2013-42853-R and by the Public University of Navarre.

The authors are with the Department of Electrical and Electronic

Engineering, Public University of Navarre, Pamplona, Spain (e-mail:

ernesto.barrios@unavarra.es; andoni.urtasun@unavarra.es;

alfredo.ursua@unavarra.es; luisma@unavarra.es; pablo.sanchis@unavarra.es).

increase the transformer operating frequency to the range of

1-150 kHz [1]–[4].

In order to reduce winding loss, the primary and secondary

windings of high-frequency (HF) transformers are usually

sectioned and interleaved, and special wire geometries, such

as litz and foil conductors, are used [5]–[7]. Compared to litz

wire, a foil conductor is preferred due to its higher width-to-

thickness ratios, which provide lower DC resistances and

higher fill factors, its better heat conduction which facilitates

heat transfer to the environment, and its lower cost [1], [8].

However, the conventional procedure for interleaving foil

windings complicates the transformer assembly since taps are

required to connect in series the beginning and end of each

winding section. Although special techniques have been

proposed to interleave windings and minimize both losses and

assembly difficulty [8], [9], they have limitations since they

are only applicable either to transformers with turns ratios that

are close to the unity or to planar transformers.

Furthermore, the design of HF transformers needs to

address the interdependence between core sizing, foil

thickness sizing and winding interleaving, which complicates

the design process. The problem is usually solved by means of

an iterative process that depends on the designer's experience

and may entail the omission of some of the complex

electromagnetic, thermal and construction interdependencies

existing amongst the design parameters [5]–[7], [10]–[14].

To address the challenge of optimal HF transformer design,

this work proposes firstly an innovative technique, termed

maximum interleaving, which makes it possible to minimize

losses whilst minimizing the number of taps required and,

consequently, the construction difficulty. Then, a novel

non-iterative analytical methodology to optimally design HF

transformers is proposed, leading to the direct resolution of the

design problem as a non-linear optimization problem of only

four design variables. The methodology comprehensively

formulates the complex multi-physical phenomena present in

the operation of a HF transformer and the dependencies that

arise between these phenomena in the transformer design

process.

Moreover, given the fact that winding interleaving may not

be advisable in some applications, such as when a high

leakage inductance or a low interwinding capacitance is

required or in high voltage applications, the scope of the

design methodology proposed in this paper is extended to

High-Frequency Power Transformers with Foil

Windings: Maximum Interleaving and Optimal

Design

Ernesto L. Barrios, Student Member, IEEE, Andoni Urtasun, Student Member, IEEE, Alfredo Ursúa,

Member, IEEE, Luis Marroyo, Member, IEEE, and Pablo Sanchis, Senior Member, IEEE

include a non-interleaved approach. This is then used as a

reference point to assess the advantages of the maximum

interleaving technique.

The paper is organized as follows. Section II presents the

models for the calculation of the core loss, foil winding loss

and transformer thermal resistance for use in the design

process. Section III explains the maximum interleaving and the

winding loss calculation when this technique is applied.

Section IV discusses the transformer design problem and

proposes the optimal design methodology. With this

methodology two theoretical designs for a PV application are

obtained in Section V, one with maximum-interleaved and the

other with non-interleaved windings. When comparing both

designs, the maximum interleaving design achieves the best

performance in terms of power density and efficiency. A

prototype for the maximum interleaving design is then built in

Section VI, paying particular attention to the assembly process

and the use of low-cost standardized materials. The prototype

is finally validated by means of experimental tests with

satisfactory results.

II. T

RANSFORMER MODELING

A. Core Loss and Geometry

In the transformer design, the calculation of the core loss is

made in practice through empirical formula based on the

Steinmetz equation [15]. For typical HF power transformer

applications, in which the voltage applied is rectangular in

form, with or without zero voltage periods, the modified

Steinmetz equation (MSE) [16], the improved generalized

Steinmetz equation (iGSE) [17], and the improved iGSE

GSE) [18] have been shown to be accurate core loss

empirical models [18]–[20]. Each model improves the

accuracy of the previous one but with an increasing

complexity. Thus, the MSE is generally used to calculate the

core loss in design processes due to its good trade-off between

accuracy and simplicity [12]. Its expression is [16]:

( )

cTopeTopeT

eqmc

VcccBffCP ⋅+⋅−⋅⋅⋅⋅⋅=

−

ττ

(1)

where f is the applied voltage waveform frequency, B

is the

magnetic induction amplitude, τ

ope

is the operating temperature

of the magnetic material, V

is the magnetic core volume, C

x, and y, and c

, c

and c

are the losses and temperature

coefficients for the material, respectively, as provided by the

manufacturers in their datasheets, and f

is the equivalent

frequency:

∫













⋅

π∆

. (2)

Fig. 1 shows the geometry of the double U and double E

cores commonly used in power applications. Traditionally, the

core geometry is characterized by five characteristic

dimensions: mean length turn of a winding that completely

fills the window (MLT

), equivalent volume including the core

and windings (V

), effective cross-sectional area of the core

), window area (A

), and core volume (V

). The latter three

dimensions only depend on the type of core whilst the first

two also depend on the place where the windings are wound.

Table I shows, for the two most common winding and core

types, the five characteristic dimensions based on the three

non-dimensional coefficients c

, c

and the dimensional

factor a defined in Fig. 1. In the fourth column, the

characteristic dimensions are expressed based on the

characteristic coefficients mlt

, v

, a

, v

and the

dimensional factor a in order to facilitate its use in the design

process.

(a) (b)

Fig. 1. Characteristic dimensions MLT

, A

, and V

, dimensional factor

a and form coefficient c1, c2, and c3, for the main power cores: (a)

double U and (b) double E.

By applying these generic form expressions, the core loss

can be expressed as follows:

BaKP ⋅⋅

(3)

where coefficient K

is:

( )

TTTc

mmag

cvf

CkK

+⋅−

⋅⋅⋅

⋅⋅=

ττ

(4)

where k

mag

is the ratio between the losses for a non-sinusoidal

magnetic induction and those for a sinusoidal one, and can be

expressed as a function of the length of the zero-voltage

period in rad, θ, as follows [19]:

−−−













−⋅













xyx

mag

. (5)

B. Foil Winding Losses

When operating at high frequencies, the amplitude and non-

uniformity of the current density distribution in the winding

cross-sectional area increases due to the well-known skin and

c1·a

MLT

c2·a

c3·a

c1·a

MLT

c2·a

c3·a

TABLE I . CHARACTERISTIC DIMENSIONS FOR MAIN POWER CORES:

DOUBLE E AND DOUBLE U

Core

characteristic

dimensions

Only one leg

double U

Shell type

double E

Generic

form

MLT

2·(2c1+c3+1)·a

mlt·a

2·(c1+1)·(c2+2)

·(c3+c1)·a

2·(c1+1)·(c2+1)

·(c3+2·c1)·a

·a

c3·a

·a

c1·c2·a

·a

2c3(c1+c2+2)a

2c3(c1+c2+5/4)a

·a

proximity effects. This can lead to an increase in winding loss.

In order to reduce this increase as far as possible foil

conductors are used. The calculation of the foil winding loss

has aroused great interest since Dowell’s work [21] and its

generalization in [22] until nowadays [23]. The power loss in a

foil winding section of p layers filling the full window height

with thickness h is calculated in this work by means of the

expression proposed by Snelling in [24] based on the

approximation of the analysis made by Dowell in [21] for

h ≤ δ:

























−

+⋅

⋅

rmsdcw

(6)

where I

rms

is the rms winding current, δ is the skin depth, and

is the dc resistance. The expressions for these latter two

parameters are:

µσπ

⋅⋅⋅

, (7)

NMLT

⋅

(8)

where f

is the current waveform frequency, MLT

is the mean

length turn of the winding, N is the number of turns, σ is the

material conductivity, μ is the magnetic permeability and w

the foil height. When the foil thickness is greater than the skin

depth, this expression overestimates the winding loss.

C. Thermal Modeling

It is possible to use both theoretical and empirical models to

estimate the thermal resistance of the transformer in a steady

state. A wide range of theoretical thermal models are

available, depending on the heat transfer mechanisms

considered and their interpretation. Empirical models achieve

a similar accuracy to theoretical ones, but with greater

simplicity [6], [11], and are therefore generally preferred for

use in the design process. In fact, the value of the transformer

thermal resistance R

is probably the most uncertain

parameter in the entire transformer design [13]. Studies made

by the magnetic material manufacturers show that it is

possible to establish an empirical relationship between the

thermal resistance of the transformer and the volume of its

core [25], [26]. Based on the data included by the

manufacturers in their application notes for various double E

and double U ferrite cores, for a temperature increase of 50 ºC,

and including the core volume in its generic form as shown in

Table I, the empirical formula for the thermal resistance R

a naturally-cooled transformer is as follows:

56.1

52.0

0457.0

⋅

(9)

where R

is expressed in ºC/W, v

is a non-dimensional factor

and a is expressed in meters. Very similar expressions are

commonly used in design processes for soft magnetic

materials with relatively high thermal conductivity such as

ferrites, nanocrystalline, and amorphous iron alloys [7].

III. M

AXIMUM INTERLEAVING: OPTIMAL WINDING

DISTRIBUTION

A. Conventional Interleaving

Thanks to the presence of a secondary winding, it is

possible to reduce the amplitude of the magnetomotive force

) in the window responsible for the proximity effect. With

this end, primary and secondary windings are usually divided

into sections and interleaved. The interleaving reduces the

number of layers per section p and, thus, as can be seen from

applying (6), also the losses in the interleaved winding.

However, this entails considerable construction difficulty as,

in order to series-connect the last turn of one section with the

start of the following section, taps need to be made. As a

result, the maximum feasible interleaving is limited by this

construction complexity [5], [8].

Considering now an example in which four turns for the

primary and eight for the secondary are required with primary

current value i, Fig. 2 shows three different possible winding

configurations and their resultant f

distributions. As

indicated in Fig. 2 (a), one option is to make a 4-8 winding

arrangement, i.e. no interleaving, having a primary section

with four turns (p=4) and a secondary section with eight

(p=8). Consequently, maximum values are obtained for the f

and proximity effect losses, and no taps are required, thereby

the winding process difficulty is minimum.

As shown in Fig. 2 (b), another option is to divide the

windings into two groups with a 2-4 configuration. This is an

intermediate interleaving comprising two primary sections

with two turns each (p=2) and two secondary sections with

four turns each (p=4). In this case, in comparison with the first

option above, the maximum f

has been halved, leading to

lower losses. However, the construction process is more

complicated, consisting in making two turns with the insulated

primary foil and then cutting it. The insulated secondary foil is

then wrapped around the primary and cut. The process is then

repeated, but when starting the second primary section, the

end of the first section needs to be connected to the beginning

of this second section. This connection is called a tap. The

same procedure is followed for tapping the secondary sections.

This interleaving level is an acceptable trade-off between

losses and construction difficulty [5] and, therefore,

manufacturers do not usually continue increasing the

interleaving.

When the number of layers per section for the primary and

secondary is minimized, the windings are fully interleaved and

the f

and the proximity effect losses are also minimized.

However, this design creates the greatest construction

difficulty, requiring the higher number of cuts and taps. As

indicated in Fig. 2 (c), in the example studied, the windings

are fully interleaved when four groups are formed with a 1-2

configuration. Consequently, six taps are needed; three taps to

series-connect the primary turns and another three to

series-connect the end of each secondary section with the

beginning of the next.

(a) (b) (c)

Fig. 2. Magnetomotive force in the cross section of a transformer window

for three different winding interleaving arrangements: (a) non-interleaved,

(b) intermediate interleaving, and (c) fully interleaved.

It should be pointed out that the interleaving affects the

parasitic elements of the transformer. The greater the

interleaving, the lower the energy stored in the stray magnetic

field and the greater the energy stored in the stray electric field

between the primary and secondary windings. In this way, the

greater the interleaving, the lower the leakage inductance and

the greater the capacity between the primary and secondary

windings [7]. Consequently, depending on the application, the

interleaving may not be advisable. For instance, when

electromagnetic interference (EMI) needs to be minimized,

when leakage inductance is used as a filter or resonant

components [27], [28], or in high voltage applications in

which primary and secondary windings are grouped into

separate chambers due to isolation concern, a non-interleaved

structure is preferred.

B. Maximum Interleaving of Foil Windings

In order to maximize the reduction of the proximity effect

yet without a complicated assembly, a new winding

interleaving technique, named maximum interleaving, is

proposed in this paper. With this technique, the f

distribution is the same as for the conventional fully-

interleaved winding configuration, however, due to the

construction process proposed, the number of taps required is

reduced to the minimum technically necessary. In order to

make it easier to understand the proposed technique, this is

firstly applied to the example above (see Fig. 3) and it is then

generically described.

In the first step, three insulated foils are wound around the

magnetic core’s central leg to make four turns. The first foil

has the primary winding thickness, while the second and third

foils have the secondary thickness. Finally, and as detailed in

Fig. 3, the beginning of the third foil is soldered to the end of

the second foil through a tap, so that the second and third foils

are series connected and eight turns for the secondary winding

are completed. In so doing, the same minimum losses as for

conventional full interleaving are achieved yet with a much

simpler construction, given the fact that only one tap is needed

instead of the conventional six.

The maximum interleaving technique proposed in this paper

is generically described below. It can be easily applied to any

transformer ratio, even a non-integer number. The winding

with the least number of turns (N

) is termed A whilst the one

with the greatest number of turns (N

) is named B. One foil

conductor is taken for winding A, and p foil conductors for

winding B, with p equal to:

(a) (b)

Fig. 3. Maximum winding interleaving of a 4 primary and 8 secondary

turns foil transformer: (a) cross section of the magnetic core central leg,

and (b) window cross section (Section A-A’).

























round

(10)

where generic transformation ratio n, taken as the quotient

between the number of turns for windings A and B, equal to

for step-up transformers and N

for step-down

transformers. The conductor for winding A and the p

conductors for B are insulated from each other and then placed

one on top of the other in order to proceed with the winding.

From here onwards, two cases can be differentiated:

- Decimal part of 1/n ≥ 0.5: Fig 4 shows the cross section of

the central leg of a double E core with a generic winding

distribution for this case. As can be seen, the conductors are

wound jointly and continuously, with the conductor of the

winding with the least number of turns positioned inside,

until z number of turns has been reached:













floorz

. (11)

Fig. 4. Cross section of a double E core with maximum interleaving when

1/n is rounded upwards. Winding A in blue and B in green.

Turn z indicates the end of the winding of the p-p’ external

conductors of winding B, where p’ is:

pzN

−

⋅−

. (12)

1׀1 5 ׀ 2 ׀ 2 6 ׀

3׀ 3 7 ׀4׀ 4 8

TAP

Section A-A’

A’

MAGNETIC CORE

1 2 3 4 ׀ 1 2 3 4 5 6 7 8

1 2 ׀

1 2 3 4

׀ 3 4

׀ 5 6 7 8

1׀1 2 ׀ 2 ׀ 3

4 ׀

3׀ 5 6 ׀ 4׀

7 8

TAPS

…

p’

p’+1

p’

…

In this way, p-p’ foils of winding B are cut at turn z as

indicated by an x in Fig. 4. Then, the remaining turns are

wound until N

is reached, however in this case p’

conductors are used for winding B. Once the winding has

been completed, the next step is to use taps to connect in

series the ends of the various layers of the winding B. The

end of layer 1 is connected to the start of layer 2, and so on,

until the end of layer p-1 is connected to the start of layer p.

If p’ is less than 1, then this means that when z turns have

been wound, less turns need to be given to winding B than

to winding A. In these cases, it is sufficient to equal p’ to 1

and make the remaining turns until N

has been reached.

Then, the remaining turns are made for winding A until

reaching N

. An example of this special case is shown in

Section VI.

- Decimal part of 1/n ≥ 0.5: Fig 5 shows the cross section of

the central leg of a double E core with a generic winding

distribution for this case. Likewise, the windings are made

jointly and continuously, however locating the p conductors

of the winding with the largest number of turns on the

inside. So N

turns are made around the central leg of the

core, and then the winding of B is completed by one more

turn with p’ conductors:

pNNp

⋅−='

. (13)

For this purpose, the p-p’ external conductors of winding B

are cut and ended after turn N

. Finally, the layers of

winding B must be series connected. As in the case above,

the end of layer 1 must be connected to the start of 2 and so

on until the end of layer p-1 has been connected to the

beginning of layer p.

If p’ is greater than p, then after turn N

an additional turn

should be given with p conductors of winding B, cutting the

2p-p’ external conductors and giving an additional turn with

the remaining p’-p conductors.

Fig. 5. Cross section of a double E core with maximum interleaving when

1/n is rounded downwards.

It can be concluded that the maximum interleaving serves to

minimize the proximity losses in the transformer windings

whilst it also minimizes the construction complexity, given the

fact that the number of taps required is reduced to p-1.

C. Total Winding Losses Calculation

1) Maximum-Interleaved Windings

The expression for the total power loss in the transformer

windings when implementing maximum interleaving is now

obtained for use in the novel transformer design process

that is proposed in Section IV. Firstly, the number of turns

N for any winding can be expressed according to the

magnetic induction amplitude B

with frequency f by means

of the rms voltage equation V

rms

induced in the said

winding:

4 aa

psh

rms

⋅

⋅⋅

(14)

where k

is the waveform factor, equal to 1 when a square

voltage is applied and to 1.11 when the voltage is

sinusoidal. For other voltage waveforms, k

have to be

specifically calculated as indicated in [7].

When the maximum interleaving is implemented, it can

be assumed that the mean length turn MLT

is equal for

both transformer windings, and equal to the mean length of

the core MLT

. The winding with a least number of turns,

named A, has a single layer of thickness h

in all its

sections, whilst the other, named B, has p layers of

thickness h

in most of its sections and p’ in the rest. To

simplify the design process it is considered that, across the

length of the winding, there is a constant number of layers

per section p. Introducing (14) in (6), including the generic

expression for the characteristic core dimensions developed

in Table I, expressing the foil height w

as the height of the

window c2·a multiplied by height fill factor k

, and

particularizing for each of the windings, the losses are

obtained for the winding with the least number of turns P

and for the winding with the greatest number of turns P













⋅

⋅⋅⋅⋅

⋅⋅

δσ

rms

ArmsA

VImlt

, (15)

( )













⋅−

⋅⋅⋅⋅⋅⋅⋅⋅

⋅⋅

δσ

pchsh

rmsBrmsB

aBackfk

VImlt

.(16)

By referring voltages and currents to the winding with the

least number of turns by means of transformation ratio n

defined in (10), introducing δ according to the current

frequency f

in (7), and adding both expressions, the total

losses in transformer windings P

w,t

are obtained:

( )

[ ]













⋅−⋅

+⋅++

⋅

323

nhK

mlt

(17)

where coefficients K

and K

have the following

expressions:

fackk

chsh

rmsA

⋅⋅⋅⋅⋅⋅

⋅

, (18)

2222

⋅⋅⋅

σµπ

. (19)

…

p’

p’+1

…

High-Frequency Power Transformers With Foil Windings: Maximum Interleaving and Optimal Design

Figures

Citations

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Semi-Empirical Model for Precise Analysis of Copper Losses in High-Frequency Transformers

References

Power Electronics

Power electronics

Effects of eddy currents in transformer windings

Transformer and inductor design handbook

Accurate prediction of ferrite core loss with nonsinusoidal waveforms using only Steinmetz parameters

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Frequently Asked Questions (1)

Q1. What are the contributions in this paper?