Tuckey Inductor design DC link inverter.pdf

A Minimum Loss Inductor Design for an Actively

Clamped Resonant DC Link Inverter

A.M. Tuckey

Eindhoven University of Technology

Eindhoven, the Netherlands.

Tuckey @ieee.org

D.J. Patterson

Northern Territory Universit

Darwin, Australia.

Dean.Patterson @ntu.edu.au

11. INDUCTANCE

Given nominal system values such as the dc supply volt-

age and the load or motor voltage and current, the design of

an ACRLI entails choosing the resonant link frequency fR, the

clamp factor K, the main and clamping switching devices, and

the resonant circuit's characteristic impedance Z,. This lat-

ter value affects the loss in all circuit components by vary-

ing amounts. [46] show analytical loss formulae for all ma-

jor components as a function of either Z, or LR, with [6] de-

tailing formulae derivation, the use of a MOSFET clamping

device, and mathematical expressions for all important current

and voltage waveforms. [6] shows that for a machine drive sys-

tem

Ahstrucf-The Actively Clamped Resonant DC Link Inverter (ACRLI)

is a well established soft-switching topology, however its resonant inductor

is problematic, and is often cited as its primary stumbling block.

This paper presents the design and fabrication of a very-low-loss high

frequency ACRLI optimised inductor. The low-loss design was obtained

by minimising the hysteretic and eddy current loss of the inductor's ferrite

cores, and the conduction loss of the Litz wire windings. The inductor was

built, including custom Litz wire, with a rfsnlting loss of less than 16watts,

or 0.16% of the total lOkW inverter throughput. This represents a signifi-

cant improvement in the current state of the art in ACRLI inductor design

and extends the applicability of the ACRLI topology.

I. INTRODUCTION

T (ACRLI) shown in Fig. 1 is a well established soft-

switching topology [ 1-51; typical resonant dc link waveforms

are shown in Fig. 2.

The two components fundamental to the operation an

ACRLI are the resonant inductor L, and the resonant capac-

itor C,. Of these two components the inductor is problematic

since it passes resonant current in addition to the machine cur-

rent, and is often cited as the primary stumbling block of the

ACRLI; for a medium to high power ACRLI it will be large

and may add significantly to the overall loss of the inverter.

This paper details proper selection of inductor value and

presents minimum loss design for ACRLI inductors through

a worked example. The inductor was fabricated, including

specially designed Litz wire, and a measurement of its loss is

shown.

HE Actively

Clamped Resonant DC Link

Inverter

iLR(t) =I, + -

"s sm(2nfRt)

(1)

where 2, = and I, is the machine current. It should

be noted that sin(2nfRt) is the ac or resonant current, and is

independent of the machine current, notwithstanding transients

due to current controller action.

A. Inductor Size

For an efficient ACRLI, L, can be physically large-the

peak energy stored by the inductor can be used as a measure

of this. Using (1) the peak stored energy is

& LR-i

--L '(' f+- 2) 2 ,

--+

Time

onecycle

-E--

'LR -

Fig. 2. Typical ACRLI resonant dc link voltage and resonant inductor current.

VCR "-

Fig. 1. The Actively Clamped Resonant DC Link Inverter.

0-7803-6401-5/00/$10.00 0 2000 IEEE

3119

where fx is the peak machine current. Minimising ELR with

respect to Z,

results in [7]

60 -

40 8

30 2

- vs

fx '

'Ominimum srored energy - -

(3)

B. ACRLILoss

Using loss formulae published in [6] a graph of ACRLI loss

versus LR can be constructed. To construct such a graph it is

assumed that the inductor's Q, Q,,,

,,,,.,,,,,,,,...... ....'""

..qg;:.,,i;- - - . -i

- - -. - -

and the ratio of the induc-

*a--

-;____

tor's ac to dc resistance, ICAc,

are constant for the range of LR

considered.

Fig. 3 shows loss and ELR versus LR for the lOkW EV

ACRLI system studied in this paper with the nominal values

shown in Table I, assuming Q,, = 200 and kAc = 20. These

assumed values are validated in Section VIII. Full details of

the ACRLI design including switching component selection are

presented in [6].

Fig. 3 shows that for minimum stored energy LR should

have a value of about 2.5pH, whereas a minimum loss de-

sign dictates a value of about 4.SpH. For both these values

of LR the circulating resonant current is significant and must

be supported by the dc supply capacitors in parallel with V,,

the clamp capacitor and the resonant components, and incurs a

continuous loss independent of machine current. These com-

ponents must be chosen appropriately-quite a task for the dc

supply capacitor and the clamp capacitor; Fig. 4 quantifies this.

The compromise between large and small values of LR can

be seen-a lower value reduces the peak energy storage of the

inductor and increases the size of C,. This increase in C, re-

duces the dv/dr imposed on the switching devices and there-

fore reduces their switching loss. However, lower values of LR

increase the circulating current in the inverter which increases

stress on the electrolytic capacitors and the peak current in the

clamp device. For the inverter considered herein a value of

SpH was chosen.

LR (pH)

Clamp switching loss -- _. - ..

Clamp conduction loss

Inductor loss - - - - --

Total loss -

Main conduction loss "__" --

- ~..

Main switching loss .... ...... .

l.-

__I"__

Fig. 3. ACRLI loss and inductor size versus LR.

-..

--._

LR (PH)

relative to minimum value ___

- I .

Peak circulating current - - - - - .

Increase in

Increase in total loss relative to minimum loss

Fig. 4. Inductor size, inverter loss, and circulating current versus LR.

Therefore, to allow a small safety margin the design criteria

were set as follows:

Design current IMAx = 17SA,

Inductance LR = 5.OpH,

Maximum flux density b = 250mT'.

111. INDUCTOR DESIGN CRITERIA

Three criteria are needed for an inductor design, namely, the

absolute maximum current which the inductor must endure, its

inductance, and the desired peak flux density in the inductor

core. It can be shown that setting LR = 5.00pH and V, = 96V

yields Z, = 1.90Q. Substituting these values and I, = 104A

(required for a power rating of 1OkW) into (1) results in the

absolutemaximum value of inductor current

IV. INDUCTOR

CORES

With the nominal switching frequency being SOkHz, an in-

ductance of 5.OpH, and a peak current of 15SA, a high fre-

quency core material was needed for the inductor. The alterna-

tives were powered iron, with its distributed and non-adjustable

air-gap, and air-gapped ferrites, with the latter being the most

popular in the power range considered, and the core material

used. At higher powers, where the current level increases, the

required air-gap increases to the point where the use of air-

cored inductors may be the most energy and space efficient [7].

For an inductor with effectivecross-sectional area of the core

equal to 15SA.

TABLE I

NOMINALACRLI VALUES FOR THE 10 KILOWATT EV SYSTEM,

Parameter

Value

DC supply voltage ( Vs)

Maximum power lOkW

Resonant link frequency UR) 5OkHz

Clamp factor (K)

96V

1.5

'8 was based on ferrite core data-book recommendations [13].

3120

A,, the number of turns is given by

(4)

This shows the compromise between core cross-sectional area

and number of turns. For high current inductors the number

of turns should be kept small to minimise the overall length

of the conductors and allow large Litz wire conductors to be

used. Also, the window area of the core should be large to

facilitate this large diameter wire. This necessitates the use of

cores with large cross-sectional areas and large window areas,

with “U” type cores being entirely suitable. Neosid U100/57

ferrite cores with A, = 645mm2, grade F5A and dimensions

shown in Fig. 5 were used for L,.

L 25,0-i----~

--L 25.0 4

JIl.0 ~~~

Fig. 5. Dimensions of the U100/57 core (sizes are in mm)

waveform is approximately sinusoidal with a dc offset, (6) re-

duces to (5) with fsin eq = fR, and with a change in the definition

of 8. In this case 8 refers to the peak ac flux density, not the

peak flux density including the dc component [12]. Thus the

formula can be written as

V. CORELOSS

A. Hysteresis Loss

Hysteretic loss in magnetic material has been the subject of

much research for over a century with the first work presented

by C. P. Steinmetz in 1892 [SI. The Steinmetz equation [%lo],

gives the specific hysteresis power loss in a core in W/m3 as

bAAC

representing the peak ac flux density. In fact recent studies

show that this is not completely accurate, and that the dc offset

in the flux density does increase the hysteresis loss associated

with the ac flux density waveform [ll,15,161.

where f is the frequency in Hz, 8 is the peak flux density in

tesla, z is the temperature in “C,and C,, X, Y and CT-CT2 are

constants of the particular core material considered.

There is abundant literature on ferrites made by Philips

[9,11,12] but little available for F5A. It can be deduced that

F5A has properties between 3C80 and 3C85 [6,13];estimated

values shown in Table I1

B. Eddy-Current Loss

Most of the literature only addresses hysteresis core loss in

ferrites because eddy current loss is assumed to be minimal.

This is only true for cores with small cross-sectionalareas, and,

since the cores used had a large cross-sectional area, the eddy

current loss was accounted for. The specific eddy current core

loss in w/m3is [9, IO]

A. 1 Accounting for Non-sinusoidal Current Waveforms

For sinusoidal currents the hysteretic core loss can be calcu-

lated using (5). Albach et al. and Reinert et al. show that for

arbitrary (non-sinusoidal) current waveforms, (5) can be modi-

fied to account for the fact that hysteresis loss in core materials

is most accurately described by a function of dB/dt [12,14]:

where A, is the cross-sectional area of the core in m2, f is

the excitation frequency, equal to fR in this case, and p is the

resistivity of the ferrite; pFsA(lOOoC) = 0.286Q m. Once again

this formula is written specifically for sinusoidal ac waveforms

and for the particular current waveform considered here BAc

represents the peak ac flux density.

where T is the period of the current waveform and fsineq is

the frequency of an equivalent sinusoid representing the cur-

rent waveform. For the case considered here, where the current

TABLE I1

C. Inductor Configurations

There are many ways of configuring or stacking U-cores.

Two configurations were considered-those shown in Fig. 6.

Table 111 shows the flux density, hysteresis core loss

TYPICAL

FERRITE SPECIFIC POWER LOSS PARAMETERS.

eddy-

current core loss Peddy

and the total core loss P,,, for both con-

figurations for various numbers of turns. To stay within the

8 < 250mT constraint at least 6 turns on 2 U-cores, or at least

3 turns on 4 U-cores were required, with more turns resulting

in lower core loss. There is a tradeoff between core loss and

conduction loss, which will be addressed after conduction loss

is discussed.

3121

It can be shown that for Litz

wire transformers and some Litz wire inductors

current with rms amplitude lAc.

where w is the frequency of the sinusoidal current, n is the

number or strands in the Litz wire, N is the number of inductor

turns, d, is the diameter of the copper in each strand, pcu is

the resistivity of the copper conductor, b, is the breadth of thc

window area of the core, and C is a factor accounting for field

distribution in multi-winding transformers (normally equal to

1) [10,21].

For the ACRLI it is possible to replace w with an effective

frequency wef to account for the non-sinusoidal current wave-

form (1);

(a) lWo U cores.

(b) Four U cores.

Fig. 6. Inductor core configurations considered.

Loss

Skin EfSect: The skin depth, 6 = l/m,where B is

the conductivity and p is the permeability of the conductor,

and f is the frequency of the sinusoidal current, is used to

estimate the impact of skin effect on the effective conductiv-

ity of a round conductor. As a general rule if the skin depth

is larger than the radius of the conductor, skin effect will not

cause a significant reduction in the conductivity of the con-

ductor. For a frequency of 5OkHz the skin depth of copper

is 0.3625”. Therefore if the diameter of the conductors, in-

dividual, or within a fully-transposed Litz wire construction,

were to be kept helow 0.725 mm (21AWG) skin effect would

not be significant.

Proximiry EfSect: Proximity effect is a more significantcon-

duction reducing factor in high frequency magnetic systems.

Since Litz wire has been used extensively to combat the prob-

lems of proximity effect in magnetic systems a number of com-

mentators have been exploring its advantages and carrying out

optimisation through analysis and modelling [ 17-20]. How-

ever, only recently has there been literature on optimum con-

ductor size and number of conductors in Litz wire cables, with

the most up to date material presented by Sullivan [21,22].

Conduction loss is represented by

VI. CONDUCTION

where IAc is the rms ac current, and IDC is the dc offset [21].

Substituting I,, = 104A and IAc = 35.7A results in oerl =

102krs-I (16.2kHz).

A. Full Bobbin Design

For the minimum loss in a transformer or an inductor the

bobbin should be full of wire. When using Litz wire some of

the cross-sectional area of the wire is occupied by insulation

since each strand must be insulated from each other strand.

Also air (due to non-perfect packing) and filler materials oc-

cupy some percentage of the area. Additionally each strand is

longer than the length of the Litz wire due to the twisting of

the strands. To account for these factors, which affect the dc

resistance of the Litz wire, the dc resistance factor k,, is used.

This factor is the ratio of the dc resistance of the Litz wire to the

dc resistance of a single solid conductor of the same diameter.

Therefore

PLOSS = kAC~,2CRDC (9)

where kAc is the factar relating dc resistance, R,,, to the ac

resistance to account for conduction loss, given a sinusoidal

TABLE 111

CORE LOSS FOR U~OOl57 CORES IN VARIOUS CONFIOURATIONS.

where dLirz

is the diameter of the hypothetical solid copper con-

ductor (same diameter as the Litz wire), F,, is the factor ac-

counting for the extra strand length due to twisting, and d, is

the diameter of the copper in one strand used in the Litz wire.

To calculate the total resistance of the winding both k,, and

kAc must be known. Sullivan shows that for wire in the range

of 30 to 60 AWG with conductor diameter d,,

I I

(T) I ’

I phys

I pfOi%

,4c

Two “U’ cores

the overall di-

ameter including the insulation, d,, is given by

Four “U” cores

0.095

19.8W

16SW

36.3W

where DR is an arbitrarily defined reference diameter (=

0.079mm). and IJ and fi are dimensionless constants, equal to

0.170

3122

1.12 and 0.97 respectively. Using (13) k,,

can be derived as

where FLp is the factor accounting for the serving (outer cov-

ering) area, the bundle packing, any filler area, strand packing

and the effect of twist on the diameter of the Litz wire (FLp

in-

corporates FW). Using this formula and (10) it can be shown

that the total resistance factor, kAc x k,,,

cI!1

1000

where

10000

100000

le+06

Number of strands (n)

N=6 ...........

N=4- -._.".I N=7 ______

N=5 ____.. N=8 x_._x_

N=3-

N=9

(16)

b, is the breadth of the bobbin, and Fp is the packing factor

[21]. Minimising (15) with respect to n, results in the optimal

number of strands in the Litz wire for a full bobbin being

Fig. I. Total resistance factor, kToT. versus number or strands, n. for three

through nine turns with a full bobbin.

Total

resistance

100000

factor

TOT)

18 --^I"

It was shown in Table 111 that only a few turns of wire were

required for the inductor; between three and nine depending

on the core configuration and desired peak flux density. To get

an indication of the possible minimum loss inductor configura-

tions it was instructive to graph k, against n for three through

nine turns, as shown in Fig. 7. Packing factor was calculated

for each number of turns.

It is evident from the graph that the required number of

strands in the optimal Litz wire was indeed large; in the or-

der of lo6 with a wire gauge of about 52AWG. This large

number of very fine strands is almost unmanufacturable. As

an indication of the manufacturing difficulty it is noted that the

maximum number of strands in a single Litz wire stocked by

a leading manufacturer is only 31,200 [23]. A second problem

with the minimum loss design as it stands is that the wire is

very fine-most manufacturers can only produce strands up to

48 or SOAWG [21].

Fig. 8 is a graph of lines of constant k, for a set number

of turns and packing factor on the plane of number of strands

versus wire gauge. The straight line on the graph is the point

at which the bobbin is completely filled, so one can only make

the inductor for points on or below this line. There are many

operating points with a partially filled bobbin and a much re-

duced number of strands and much thicker wire which have a

comparable total resistance factor to the filled bobbin configu-

ration, As an example consider the line representing k,, = 18.

When the bobbin is full the number of strands is approximately

10,000 and the wire size is 34AWG. This value of total resis-

tance factor can equally be met with the minimum wire gauge

(maximum wire size) of 30AWG and about 1,000 strands, or

a minimum number of 800 strands of 31AWG wire. The fact

that the straight line is above most of the curved lines indicates

it is favourable to design with an under-filled bobbin.

10000~

16 ___._.

14 ...........

12 " '.

1000 *

8 .. -- .. t,,,,,,,,'l 100

6 ^_I r__

28 30 32 34 36 38 40 42 44

4 I"__ _"

. ... . . . .

Wire Size (AWG)

Full Bobbin

I..

Fig. 8. Constant total resistance factor, kToT. for a set number of turns and

packing factor, on the number of strands versus wire gauge plane. The

indicates the final Litz-wire fabricated.

B. Under-filled Bobbin Design

The question of choosing the optimal strand diameter for

a fixed number of strands has been addressed by many au-

thors [lo,24-26]. Although these authors addressed the opti-

mal strand diameter of single stranded wire transformer wind-

ings (i.e. not Litz wire), their analysis can he adapted to multi-

stranded Litz wire windings simply by substituting the number

of turns with the product of the number of strands by the num-

ber of turns, n.N, giving the total number of strands. The result

is that for optimal performance with a fixed number of strands

k,,, = 1.5 and

Fig. 9 shows k,, for optimally sized wire for three through

nine turns versus the number of strands per Litz wire (these are

the lines with negative slope). It can be seen that a reasonable

value can he obtained for kToT with a much reduced number of

strands. The diagram doesn't show whether the bobbin is over-

3123

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