Optimizing Power Transformer and Inductor

Optimizing Power Transformer and
Inductor Performance With Custom Designs
S
pecifying nonstandard
magnetic components
(transformers and coils)
from specialty vendors
requires some knowledge of the
ways their physical and electrical
characteristics are described and
of how those characteristics affect
each other and the size and cost of
the component. (figure 1)
The first stage of the design of
the magnetic core evolves from the
electrical considerations, although
there is some flexibility in selecting
alternatives. A variety of core types
have been optimized for specific
kinds of applications, and each has
drawbacks and advantages relative
to circuit type and winding technique.
That said, once input, output,
and size characteristics have been
established, the design comes down
to evaluating various sources of
thermal losses, as they relate to
frequency, flux density, saturation
flux, and permeability.[1]
Voltage Effects
Core loss, due to hysteresis in the
core material, is a function of the
voltage applied across the primary
winding of a transformer. The operating flux density, B, can be calculated using Equation 1. Once the
frequency and flux density level are
known, core loss can be estimated
from suppliers’ core-loss curves.
Bmax =
E 10
4 fNAe
8
Equation 1
E 10 8
In equationN1,= E is the applied volt4 fBmax Ae
l
R=
(Ω)
A
age, f is the operating frequency,
N is the number of turns, and Ae
Is the cross-sectional area of of the
core. The size of a core is expressed
in terms of its “turn-area product,”
roughly, the cross-sectional area of
the core times the number of wire
turns wound around it.
Transformer input voltage affects
core size and cost. As the voltage,
increases, it affects the requirements
for turns spacing and the type f
dielectric material that can be used
between windings.
Frequency Effects
Equation 2, a simple rearrangement of Equation 1 yields the
number of turns. It is easy to see
that for a given value of flux, B, and
cross-sectional area, any increase
in frequency requires fewer turns.
To take advantageEof this
10 8 benefit a
Bmax
= to choose a core
designer may
opt
4 fNAe
with a smaller cross-section.
N=
E 10 8
4 fBmax Ae
Equation 2
A smaller cross-sectional area
means a smaller core and bobbin,
l is where
so its cost is less, but
R = this(Ω)
A the picture.
the ac frequency enters
Higher frequencies are accompanied
by higher core losses. Offsetting this
fig. 1. Custom magnetics come in a
variety of form factors, with unique
requirements for magnetic cores and
coil materials require tradeoffs between dimensions and performance.
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may require the use of a highergrade, more expensive core materials. Also, higher frequencies magnify
the effect of parasitic loss effects
from distributed capacitance, leakage inductance and skin effect in the
windings. To offset this, the transformer design may require the use
of more expensive Litz wire and a
more complex winding pattern. Litz
wire consists of multiple fine strands
of insulated solid wire twisted together to form a single conductor.)
I2R losses
determined by weight, base price of
copper plus adders. The adders are
highest for smaller gauge wire and
diminish as the gauge increases. The
adders for square wire are more than
round wire.
Wire Diameters and cost
The diameter of copper wire or
wire gauge is specified in terms
of American Wire Gauge (AWG)
which ranges from 0.162” (6 AWG)
to 0.00124” (48 AWG). Windings
of fine wire require coil finishing
as an additional assembly process.
Finishing adds to the labor cost. On
the other hand, heavier gauges are
is gapped the tolerance is typically
reduced to about +10% and in some
materials to less than +5%.
To a small extent, the measured
AL value of a core will depend on the
coil used for measurement. Ferromagnetic inductance measurements
are subject to variations depending
on the measuring method and the
magnetizing conditions.[6]
For any given design, variations
in material permeability, assembly,
and the gapping process contribute
to deviation in inductance. In many
applications, the higher the inductance the better. In the case oh high
inductances, a minimum inductance
-- with some margin -- may
be specified. The thing to
keep in mind is that pecifying a tight tolerance on inductance (+10%) can have
negative economic consequences in production. A
tolerance of less than +5%
always incurs significant
cost penalties.
The required diameters of the
conductors (typically copper magnet
wire) that make up the
primary and secondary
windings are determined by
the RMS currents. Higher
currents affect cost by contributing to the required
size of the core, winding
Aw
Aw
Aw
cost and packaging.
The current carrying
capacity of copper wire
Core
Core
is commonly specified by
the manufacturer based
Resistance
on 1000 circular mils per
Fig.2. Magnetic core shapes offer tradeoffs in terms of
Considerations
ampere (c.m./A). That’s a
size, possible winding densities and other parameters.
At low frequencies, the
baseline. In practice, lower
dc resistance (DCR) is
current densities may be specified,
wound one winding at time on slow- the most important parameter to
depending on the application and
moving equipment. This results in a be considered in the winding of a
winding construction. In fact, curhigher labor rate per turn.
magnetic device. The resistance per
rent densities as low as 200 c.m./A
unit length of the wire is inversely
may be adequate.[7] Real-world de- Inductance Effects
proportional to the winding-window
sign decisions may take into account
The effect of a winding’s inducarea. (Fig 2) A larger window area
voltage drop, insulation temperature tance (L) on current capacity can
yields less resistance per length
limit, thickness, thermal conductivonly be defined approximately. This of wire, given the same number of
ity, air convection and temperature
AL value is normally stated in terms turns.[2]
must be taken into account.
of inductance per turn (N) squared
The wire winding pattern is the
As a rough guide, solid wire is
in nano-henries (nH) for a given
principal determinant of the DCR of
typically used for magnetics operatcore:
the copper in a winding. Fully utilizing at below 100 kHz with RMS curing the winding space to maximize
rent levels under 20 A. Copper foil
L = M2xAL: (nH)
the cross- sectional area of wire is
is a common choice for low-voltage,
Equation 3
the objective.[2]
high-current windings.
In transformers with high-voltage
Magnet wire is available with
However, introducing a mechaniisolation requirements between
rectangular, round or square cross
cal gap between core halves of a two windings nearly 1 cm of window
sections. In general, the rectangular
piece ferrite core makes AL adjustbreadth can be lost, and the inand square types have larger diamable. Because the air gap can be
creased separation creates higher
eters and allow for a higher density
ground to any length, any value of
leakage inductance between primary
of material in a given area. Broadly
AL can be provided within the limits and secondary windings.
speaking, the diameter of the windpermitted by the core.[6]
Winding resistance, R, is a funcing wire can vary in thickness from a
The AL tolerance of an un-gapped tion of the cross-sectional area and
few microns to several centimeters.
ferrite core is typically about +25%
the length of the conductor. (EquaThe price of magnet wire is
or higher. However, once the core
tion 4 )
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Bmax =
E 10 8
4 fNAe
“The first stage of the design of the magnetic core evolves from
E 10
the electrical
considerations, although there is some flexibility in
N=
A
selecting4 fBalternatives.
A variety of core types have been optimized
for specific kinds of applications, and each has drawbacks and
advantages relative to circuit type and winding technique.”
8
max
e
l
(Ω)
A
Equation 4
R=
where resistivity of copper, ρ, is
1.724 × 10-6 Ω-cm at 20 °C; l is the
length and A is the cross-sectional
area of the wire.
Precision Considerations
The resistivity of copper varies
with temperature by a coefficient
of 0.00393 /°C. Using EQ 4, the accuracy of the calculated winding resistance is only about 10 to 20%. The
manufacturing tolerances of copper
wire are between 10 to 20 %. Errors
in measurement of dc resistance can
be off by as much as 50 % from the
calculated value. Finally, variations
in tension during winding and in
per-unit-length resistance cause
variations in winding resistance.[1]
AC Resistance
At higher frequencies, ac resistance becomes important. A
winding’s ac resistance frequently
exceeds its dc resistance and can
overwhelm the design if not managed properly.
A copper-foil winding typically has
far greater window utilization and
typically has the lowest DCR of any
other alternative.
Specifying a tight tolerance on dc
resistance should not be considered
until nominal values have been established from measurements made
on prototype units. Coils that fail dc
resistance in production cannot be
reworked and must be scrapped.
Specifying a dc resistance tolerance of +20% incurs no significant
cost penalty. However, specifying a
tolerance of +10% may have some
economic consequences, more so on
smaller-gauge wire.
A tight tolerance on dc resistance
should be specified only after careful
consideration. In most cases specifying a maximum DC resistance is
sufficient.[1]
Leakage Inductance
Leakage inductance reduces
transformer performance. It is
proportional to the height of the
winding and inversely proportional
to the width of the winding. It is also
inversely proportional to the square
of the number of interface sections
of the windings.[2]
It cannot be accurately calculated
because manufacturing dimensions
cannot be maintained consistently.
Therefore nominal values are established only after measurement and
design verification has been made on
of prototypes, units.
Coils that fail leakage inductance
cannot be reworked and must be
scrapped. A maximum specification
of leakage inductances is adequate
for most circuit requirements.
Where a circuit requires tight control a +25% from a nominal value
can be adequately specified.[1]
Capacitance
There are two types of capacitance requirements associated with
transformers: inter-winding and
distributed. Inter-winding capacitance results in capacitive loading to
ground and in noise being coupled
from primary to secondary. Distributed capacitance creates series and
parallel resonance with the mutual
inductance of the transformer.
In a filter inductor, distributed
capacitance, also creates resonance,
but it also passes high frequency
components of the switching frequency to the output.
In high voltage applications distributed capacitance affects the high
frequency performance of transformers. An unfortunate tradeoff is
that winding techniques that reduce
leakage inductance increase winding
capacitance.
In production, there are variations
in both capacitance parameters
because physical dimensions cannot
be consistently maintained. There
are also variations in dielectrics
constants of the insulation between
conductors which effect capacitance.
Coils that fail capacitance requirements cannot be reworked and
must be scrapped. Nominal values
of capacitance parameters cannot
be accurately predicted and can be
established only after measurement
and design verification of prototype
units. Once capacitance parameters
have been characterized and nominal values established a tolerance of
about + 25% can be maintained without economic consequences.[1] n
REFERENCES
Flanagan William M. “Handbook of
Transformer Design and Applications” 2nd
Edition. Chapter 15 and 16, ISBN 0-07021291-0.
Fu Keung Wong, B. Eng. and M. Phil.
School of Microelectronics Engineering,
Factulty of Engineering and Information
Technology, Griffith University, Brisbane
Australia, March 2004. “High Frequency
Transformer for Switching Mode Power
Supplies”, Chapter 2.
3. L.H. Dixon, Section 3 -- Windings,
Unitrode/TI Magnetics Design Handbook,
2000, Topic 3, TI Literature No. SLUP132.
4. Dixon, Lloyd H, Magnetics Design
Handbook, Section 5, Inductor and
Flyback Transformer Design, Texas Instruments, 2001.
5. Magnetics, A Division of Spang and
Company, Ferrite Catalog Section 4 Power
Design, 2001, FC-601-11H.
Ferroxcube Data Handbook 2009, Soft
Ferrites, page 22.
7. Chryssis, George, “High-Frequency
Switching Power Supplies Theory & Design” 1989, 1984 by McGraw-Hill, Chapter
5, ISBN 0-07-010951-6
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