IEEE
TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL.
29,
NO.
2,
MARCWAPRIL
1993
315
Design Considerations for High-Frequency
Coaxial Winding Power Transformers
Mark
S.
Rauls, Donald
W.
Novotny,
Fellow, IEEE,
and Deepakraj
M.
Divan,
Senior Member, IEEE
Abstract-The use of coaxial windings
to
create
low
loss,
low
leakage reactance, power transformers for use in high frequency
soft switched dddc and resonant converters has been demon-
strated in
[l].
This paper examines some of the important loss
aspects
of the design
of
coaxial winding transformers such
as
the iduence of skin effect on winding resistance, the variation
of core
loss
caused by nonuniform core flux density, and the
choice of the principle dimensions and aspect ratios for maximum
efficiency. Experimental measurements on a
50
kVA
50
kHz unit,
are included to confirm portions of the analytical results and
suggested design procedures.
I. INTRODUCTION
NE
OF
THE
MORE
important concerns in high fre-
0
quency power conversion is the question of magnetic
component design, particularly for higher power levels. The
question of how to realize high-power high-frequency trans-
formers has been particularly daunting. It is extremely im-
portant to obtain very
low
leakage inductance values, while
simultaneously ensuring that the leakage flux is not concen-
trated in a small section of the core used. Further, transformer
designs which feature an easily calculated leakage inductance
are appealing for use in soft switching circuits in which the
leakage inductance is a useful circuit component.
The use
of
coaxial transformers for high-frequency, high-
power converters was proposed in
[
11.
The coaxial transformer
concept has been used with considerable success in various
converters including a
50
kHz,
50
kW dual active bridge dc/dc
converter and a
600
watt,
1
MHz dual resonant dc/dc converter
[2,
31.
The design of such coaxial transformers is considerably
different from that of conventional transformer structures and
needs to be better understood. This paper investigates the
losses associated with the coaxial transformer configuration
shown in Fig.
1.
The transformer consists of an outer conducting tube which
forms one of the windings.
As
can be seen, the inner winding
is wound completely inside the outer conductor. Integral turns
ratios are possible by using multiple turns on the inner wind-
ing. The magnetic core is outside the outer conductor. Since
Paper IF‘CSD 92-95, approved by the Industrial Power Converter Committee
of
the
IEEE Industry Applications Society
for
presentation at the
1991
IAS
Annual Meeting, Dearborn, MI, September 28-October
4.
This work
was supported by the Wisconsin Electric Machines and Power Electronic
Consortium (WEMPEC).
M.
S.
Rauls was with the Department
of
Electrical and Computer Engi-
neering, University
of
Wisconsin-Madison, Madison,
WI
53706. He is now
with EMD Associates, Winona, MN 55987.
D.
W. Novotny and D. M. Divan
are
with the Department
of
Electrical
and Computer Engineering, University
of
Wisconsin-Madison, Madison,
WI
53706.
IEEE
Log
Number
9207098.
Toroi
outer
core
inding
T\.
/indins
J
\/
Fig.
1.
Coaxial transformer arrangement.
the magnetic configuration of a coaxial winding transformer
is significantly different from conventional transformers, a
number of unique design issues exist. Some of these are
explained in the following sections.
11.
FLUX
DISTRIBUTION
AND CORE
LOSS
When toroidal cores are used, the core flux distribution is
well defined and controlled. Essentially none of the leakage
flux finds its way into the core. The core flux distribution is,
however, nonuniformly spread across the radius of the core,
and indicates that core loss depends on the ratio
of
the toroidal
core inner
(rei)
and outer
(reo)
radii.
Ferrite manufactures often specify core loss as a log-log
plot of total power
loss
per unit volume versus flux density
at specific frequencies. In general, loss data
is
obtained using
toroids whose ratio of
rc,
to
r,i
is sufficiently close to unity
so
that the flux density is approximately uniform in the core
under test.
To
extend the core loss data to “fat” toroids where the flux
density cannot be assumed uniform a closed form expression is
curve fitted to the manufacture’s core
loss
plot for the material
and frequency of interest. In general, the core loss per unit
volume
(Cl,)
is of the form
where
B
is the spatial varying flux density,
n
is an exponent
usually near
2.5,
and
K,
is the core
loss
unitdscaling constant.
The actual flux density in the core will vary as
where
KB
is the flux density unitdscaling constant, and
r
is
the radial distance from the coaxial center. Substitution of
(2)
0093-9994/93$03.00
0
1993
IEEE
r--
376
IEEE
TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL.
29,
NO.
2,
MARCWAPRIL
1993
Fig.
3.
Examples
of
1
:
1
cylindncal
symmetry
windings
Solid
Tube.
Litz
Inner
Inner
Winding
Winding
-
Fig.
2.
Core
loss ratio
(R,i)
for
(1
<
(rc,,/rc;)
<
5).
into
(1)
gives
Ch
=
Kc
(
+)n
(3)
which is the core loss per unit volume as a function of position
in the core. Integration of
(3)
over the core volume gives the
total core loss. The total core
loss
resulting from nonuniform
flux distribution is compared to a calculation using uniform
flux distribution to show the effect
on
total core loss.
The
constraint for comparison is that the total flux and core volume
be the same for both calculations.
The ratio
(Rcl)
of total core loss for nonuniform flux
distribution to the total core
loss
for uniform flux distribution is
2(1-
g(1-
Rc1
=
(4)
for
n
#
2.
When
n
=
2,
(4)
involves an additional logarithm,
but the effect is the same. Notice that
Rcl
only depends on the
core radii and the material dependent exponent
n.
For “thin”
toroids where the ratio of
T,,
to
T,,
is close to one the total
core loss ratio of
(4)
will also be close to one. However, the
ratio is actually less than one as the toroid becomes more “fat.”
This result indicates that core losses are actually lower when
nonuniform flux distribution is taken into account. The effect
is small for toroids of practical dimensions and indicates that
using the “approximation” of uniform flux density for core
loss
calculations is a good one. This is contrary to intuition.
Qualitatively, the actual flux density is minimum at
T,,
and
maximum at
~,i
which results in the majority of core volume
being less lossy, and a small portion of volume being more
lossy than the uniform approximation predicts. The core loss
ratio
(4)
is plotted in Fig.
2
for several values of
n
and can be
used to improve the uniform core lois approximation.
The above discussion and (4) were derived by ignoring
core saturation (i.e., constant permeability was assumed). With
core saturation included, the plots of Fig.
2
would become
more flat because saturation has the effect of evening out any
(2
-
n)(ln
(?)y(l+
2)
applications for coaxial lines, these transformers are required
to operate at high current levels. One of the concerns this
raises is the current distribution in the inner and particularly
the outer conductor and the effect on winding losses.
A.
Losses in the Outer Winding with Sinusoidal Excitation
The current distribution
in
the outer tube is identical to
a coaxial transmission line when the transformer windings
have one
turn
cylindrical symmetry and the net current in the
outer and inner windings is equal and opposite. Two examples
of
this symmetry are shown in Fig.
3.
In a real coaxial
transformer the currents will not be exactly equal because
whichever winding is used as the primary will also carry the
magnetizing current. However, this effect is small and can be
neglected because the coaxial transformer requires very little
magnetizing current (e.g., typically less than
1%
full load).
When multiple turns
of
litz wire are used for the inner winding
there is no longer cylindrical current distribution in the inner
winding and there will be some proximity effect at the outer
tube. However, if the windings are centered, as in Fig.
1,
the
effect on the outer tube current distribution should be small,
and the coaxial transmission line approximation should give
good results. Future work will examine the validity of this
approximation.
Assuming cylindrical symmetry as in Fig.
3,
and sinusoidal
current flow, the current distribution in the outer winding is
described by Bessel’s differential equation of order zero
(5)
d2J 1dJ
W~O
-
+
-
-j-J=
0
dr2 Tdr
p
where
J
is the current density (magnitude and phase), and
T
is the radial distance from the axis of symmetry. The
general solution of
(5)
involves a linear combination of the
modified Bessel functions
Io
and
KO
and is worked out in
many textbooks [4],
[5].
Boundary conditions on the outer tube
are that there is no magnetic field at the outer surface
(rto),
because the net current enclosed is zero, and that the magnetic
field resulting from the current in the inner winding at the
inner surface
(rtZ)
is tangential. Using the solution to
(5)
and
an approach similar to one presented in [5], the ac resistance
of the outer winding per meter
(R,c-o)
is found to be
fcLo~e~~(~o(~o)K1(~o)
2bo
(11
(.OW1
(bo)
nonuniform flux distribution. Hence, with saturation-included
in the analysis, the actual core loss will look even more like
the core loss approximation which assumes uniform flux.
R,c-o
=
Real
111.
SKIN
EFFECT
AND
WINDING
LOSSES
where
The windings of a coaxial transformer are essentially a
coaxial cable and therefore much of the basic theory associated
(7)
I
RAULS: DESIGN CONSIDERATIONS FOR HIGH-FREQUENCY COAXIAL TRANSFORMERS
Me-\+
+
+{
A--
:::}
0.51
'
I
0
10
20
30
40
50
Frequency
[kHz]
Measured and theoretical resistance
of
coaxial pipe.
Fig.
4.
and
If the inner winding is also a solid hollow cylinder, the
ac resistance of the inner winding can be found using the
same method with different boundary conditions. Boundary
conditions for the inner tube are that there is no magnetic
field at the inner surface because there is no enclosed current,
and that the magnetic field at the outer surface of the inner
tube is tangential. The resulting resistance per meter is
\-
The theoretical results of
(6)
and
(9)
were compared to
resistance measurements of a coaxial conductor in the lab
under sinusoidal excitation. Two
1.5
meter solid copper pipes
were placed concentric to one another similar to Fig.
3.
At one
end the pipes were soldered together and at the other end the
pipes were driven by a current source maintained at
90
A,,,.
In this configuration the pipes
are
essentially two resistors in
series, but carrying equal and opposite current. The highest
frequency used was
50
kHz
so
that transmission line effects
would not be present. The effective ac resistance of the coaxial
pipe was found by measuring the input power and current in
the pipes on the LeCroy digital oscilloscope and then solving
for the resistance using
(10)
The theoretical and measured resistances of the coaxial pipe
versus frequency are plotted in Fig.
4.
The measured values
agree well with the theory, but are slightly higher than the
theory predicts which is believed to be caused by contact and
solder joint resistances.
A useful design curve can be obtained by plotting
(6)
normalized to the dc equivalent resistance at one skin depth
(S),
versus tube thickness normalized to one
S.
The curve is
plotted in Fig.
5
for several ratios of the outer tube's inner
radius normalized
to
skin depth
(Rt;/S).
The results of Fig.
5
indicate that there is actually an
optimum wall thickness for the outer tube in order to minimize
the resistance af the outer tube at a particular frequency. The
curve with
Rt;/6
=
0.2 does not represent tube dimensions of
pave
cms
R,,
=
-0.
Wall
lhickness normalized
lo
one skin depth
ac resistance
of
outer winding versus tube thickness.
Fig.
5.
.-
02
04
06
0.8
in
1.2
1.4
1.6
Tube
th~ckness normalized
10
one skm depth
377
y
1.151
I
Fig.
6.
Distribution
of
current density magnitude, outer tube.
practical size, but shows that the curve is somewhat dependent
on the inner radius to skin depth ratio. With
Rti/S
in the
range
(4
<
Rtz/S
<
00)
the curve lies essentially where the
Rt;/S
=
9.0
curve is in the figure and represents practical tube
dimensions and frequencies of interest. As the tube thickness
approaches
4
S
the ac resistance is equivalent to the dc
resistance of a tube one skin depth thick with the constraint
that the inner radius is the same. The minimum resistance
occurs when the wall thickness is
1.55
S.
The resistance for
a tube which has a wall thickness significantly less than
S
is
approximately the dc resistance of the tube.
It is useful to look at the current density distribution to
understand the minimum resistance phenomenon of Fig.
5.
The current density magnitude is plotted in Fig.
6
for an
outer winding tube thickness of
1.55
S
carrying a net current
of 200 A,,,. The current density magnitude exhibits the
exponential shape and crowding of the current towards the
inner surface of the outer conductor which is expected for a
coaxial transmission line. The current density phase is plotted
in Fig.
7.
The phase difference between the current at the two
surfaces of the tube is
90".
Tubes with thicker walls would
experience a phase shift greater than 90" which would result
in some current flowing in the opposite direction of the net
current flow, which results in increased resistance even though
the tube is thicker.
B.
Losses in the Outer Winding with Nonsinusoidal Current
The above analysis examined the ac resistance of the
outer tube for sinusoidal excitation which is not all that
useful by itself for power electronic applications. The usual
situation involves a transformer driven by a square wave
voltage, with a triangular current waveform resulting from
inductive switching. In terms of outer winding losses, a Fourier
series representation can be used on the current waveform
in conjunction with the ac resistance at each harmonic to
I--
I
378
IEEE
TRANSACTIONS ON INDUSTRY APPLICATIONS,
VOL.
29, NO.
2,
MARCWAPRIL 1993
bnagnitizing
Inner
winding
Winding
P
Fig.
7.
Variation
of
current phase in outer tube.
determine the total loss. If the designer uses a tube thickness
of 1.55
S
to minimize the ac resistance at the fundamental,
the remaining harmonic currents will essentially see the dc
equivalent resistance for one skin depth at each harmonic. The
resulting ac resistance of the outer tube with a triangular cur-
rent waveform is
25%
greater than when the current waveform
is sinusoidal for the same rms current.
Fig.
8.
Equivalent circuit showing asymmetrical leakage.
Rcn
C.
Losses in the Inner Winding(s)
Single Turn Solid Cylindrical Winding:
The ac resistance
for this type of winding was introduced above and is de-
scribed by
(9).
A
solid tube inner winding exhibits the same
minimum resistance phenomenon as the outer tube winding.
By constructing a curve similar to the one in Fig.
5,
one can
show that the minimum resistance occurs at a wall thickness
of 1.55
6.
This is the same optimum thickness as the outer
tube winding and the shape of the curve is identical to Fig.
5.
Single Tum Litz Winding:
Litz wire is typically used over
solid or stranded wire when high frequency and high current
capacity is needed in transformers and inductors. The ac
resistance of litz wire can be found using the manufacture's
data sheets and is typically close to the dc equivalent of the
individual strands in parallel when the gauge on the individual
strands is appropriate for the frequency of interest. If the litz
wire is centered in the outer tube there will be no difference
in the current distribution and losses compared to the case of
an isolated litz wire carrying an equal amount of current.
Multitum Litz Winding:
When there are multitums used as
the inner winding there is no longer coaxial symmetry between
the inner and outer winding and the proximity effect needs to
be
looked at for precise loss calculations. However, proximity
effect is minimized by the litz wire, and good results can be
obtained by ignoring the fact that the windings are not centered
(i.e., assume uniform current distribution in the litz conductor
and compute the ac resistance as if there were a single tum
present).
IV.
OPTIMUM
EFFICIENCY DESIGNS
Transformer designs
are
essentially governed by the amount
of core needed to avoid saturation and the amount of winding
cross sectional area needed to support the full load current.
It has been shown
[l]
that one of the main advantages to
the coaxial transformer design is its low leakage reactance
which is also precisely controllable and hence a useful circuit
component. The leakage reactance is primarily confined to
the inner winding side of the transformer as shown in the
Fig.
9.
Radii
for
the coaxial transformer.
asymmetrical circuit of Fig.
8
[3].
This essentially adds an
additional constraint to the design if one intends to use the
leakage reactance as an actual circuit component rather than
simply trying to minimize it.
A.
Design
Procedure
As
an initial design, the desired current capacity and choice
of conductor type with appropriate fill factor determines the
equivalent outer radius
(rin)
of the inner winding as shown
in Fig.
9.
The leakage inductance on the inner winding side
was shown to be
in
[I],
where
rti
is the inner radius of the primary tube and
N,
is the number
of
inner winding turns. This relation is
exact for a
1
:
1
tums ratio, but is an approximation when the
inner winding has multiple tums or when there is significant
skin effect. The exact leakage reactance under these conditions
needs to be addressed in future work. By knowing the desired
leakage reactance and selecting a transformer length, one can
determine
rti
from (11). The results in the previous section
on winding losses indicate that the outer tube resistance will
be minimized when the wall thickness is
1.55
6.
However,
if the outer winding is to provide structural integrity to the
transformer, the outer tube wall thickness may need to be
several
6
at multi-kHz frequencies. With the outer tube wall
thickness determined, the inner radius of the toroidal cores can
be determined after allowing necessary space for insulation
between the core and outer tube winding.
If the leakage reactance is not a desired component and is
to be minimized, the space between the outer tube and the
inner winding(s) should be minimized. Under this condition
the crowding of the current towards the inner surface of the
outer conductor could become a problem in the presence of
RAULS: DESIGN CONSIDERATIONS FOR HIGH-FREQUENCY COAXIAL TRANSFORMERS
379
Fig.
10.
Transformer
length
and width definitions
significant skin effect since the outer tube will have signifi-
cantly less cross-sectional area compared to an assumed litz
wire for the inner winding. This problem could be solved
by using coaxial litz cable
so
that both the outer and inner
windings are formed of litz wire.
The remaining constraint of the transformer design is to
determine the necessary flux cross-sectional area for the trans-
former core.
The
core cross sectional area
(A,)
is determined
by the transformer voltage
(VTms),
maximum flux density for
the core material being used
(&),
frequency of interest
(f),
and the number of turns
(N).
This relation is derived in many
textbooks
[6]
and is repeated here:
where the form factor
(Kf)
is
4.0
for square wave excitation
or
4.44
for
a
sinusoidal waveform.
An efficiency optimization can be done on the initial dimen-
sion choices by iterating on
the
transformer length until losses
are
minimized. Keeping the inner winding cross-sectional
window fixed and using constraints
(1
1) and (12), one can
compute the losses as a function of length to width ratio. See
Fig. 10 for clarification of the transformer length and width
definitions. Core loss will
be
minimized for geometries which
maximize cross sectional area per unit volume. This implies
using a core which has a thin profile and a small inner radius.
Winding losses will be minimized when the winding length is
minimized. The total minimum
loss
will occur for a geometry
somewhere in the middle of these extremes.
B.
Efficiency Optimization
for
a
50
kVA,
50
kHz
Unit
Theoretical losses
are
examined in the following optimiza-
tion example for a
50
kVA,
50
kHz transformer which was
built for experimental work done in
[l]
and
[2].
The desired
leakage reactance and current rating was used to select the
winding geometry in the original design. The core geometry
chosen was limited by practical considerations such as readily
available ferrite cores with an inner core radius appropriate for
the size of the outer tube winding. In the optimization process
the inner winding cross sectional area remains fixed and the
outer winding is allowed to vary
so
that as the length changes,
the leakage reactance remains fixed. As the transformer length
is varied the outer tube radius is determined using (1
1).
This
determines the core inner radius, and the core outer radius
is
adjusted
so
that the flux cross-sectional area remains fixed. The
losses are calculated at each iteration of the dimensions using
the core and winding loss results of the previous sections. Fig.
11
shows the resulting variation in losses as a function of the
transformer length to width ratio as defined in Fig. 10.
I..
,
.
.
.
J
2
4
6
8
10
12
14
Length
to
Width Ratio
T.,...
I
--/
2
4
6
8
10
12
i4
Length
to
Width Ratio
Fig.
11.
Winding,
core,
total
loss
versus length-to-width ratio.
The results of Fig. 11 indicate that a length-to-width ratio
near
6
provides the minimum loss using the initial design
constraints. The optimum transformer design is long and .thin
because the core
loss
dominates in this example, and core loss
will be minimum with long and thin cores. The length-to-width
ratio of the transformer built in the lab is
1.8.
The dashed
line in Fig. 11 shows where the experimental transformer
lies in comparison to the optimal point. The curve shows
that the transformer loss could be reduced by
30%
if the
optimal
loss
dimensions were used over the initial design
dimensions.
C.
Optimum Efficiency Length-to- Width Ratios
In the above optimization example it was shown that
the transformer profile for maximum efficiency was rather
long and thin. This is atypical since maximum efficiency
designs for conventional electric machines and transformers
tend to be cubical in shape. Without some means of forced
cooling, coaxial transformers will tend
to
have long-thin
profiles
so
that the transformer temperature rise is kept under
control. The windings are almost entirely surrounded by core
which makes it difficult to dissipate heat. When the current
density is constrained to maintain acceptable temperature rise,
the loss per unit volume for the windings will tend to
be
lower than the loss per unit volume of the core. Hence,
for maximum efficiency designs, the transformer profile will
favor minimum core volume which implies long-thin toroidal
cores to provide the necessary flux cross sectional area at the
minimum volume.
Heat removal from the transformer occurs primarily from
the surface, which is core material irrespective of the trans-
former profile. Consequently, a long-thin profile for the trans-
former
is
advantageous as surface area is maximized.
How-
ever, it should be noted that as the core is typically a poor
thermal conductor, short transformer designs may benefit
from additional heat removal along the primaryhecondary
conductors to the outside and to the end windings.
Transformer performance can be substantially improved if
forced convection heat transfer is implemented. Forcing a
coolant through the primary tube would allow a dramatic
increase in the current handling capability of the transformer,
and thus impact the power rating. This is not as easily
accomplished in conventional transformers because voltage
drop across the leakage inductance at the higher current levels
can become a significant limitation.