NATIONAL
INSTITUTE OF INDUSTRIAL ENGINEERING
PGDIE-42
INDUSTRIAL
ENGINEERING CONCEPTS
ASSIGNMENT
Designing
& Fabrication of E- Core Inductor and Reduction of Hysteresis Losses
Submitted to
Submitted by
Professor, NITIE, Mumbai Rahul Awadhiya, 71
Filter
inductor design constraints
Objective:
Design inductor having a given
inductance L,
which carries worst-case current Imax
without saturating,
and which has a given winding
resistance R, or, equivalently, exhibits a worst-case copper loss.
Pcu= Irms2R
Example:
filter inductor in CCM buck converter
Constraint:
maximum flux density
Given a peak winding current Imax,
it is desired to operate the core flux density at a peak value Bmax. The
value of Bmax is chosen to be less than the worst-case saturation flux
density Bsat of the core material.
From solution of magnetic
circuit:
Let I = Imax and B
= Bmax :
This is constraint #1. The turns
ratio n and air gap length lg are unknown.
ni = BAcRg
nImax = Bmax AcRg
= Bmax lg
Constraint:
inductance
Must obtain specified inductance L.
We know that the inductance is
This is constraint #2. The turns
ratio n, core area Ac, and air gap length
Lg are unknown.
L = n2
Rg =Ac n2lg
Constraint:
winding area
core window area WA
wire bare area AW
Wire must fit through core window
(i.e., hole in center of core) nAW
Total area of copper in window: KuWA
Third design constraint: KuWA ≥ nAW
The window utilization factor Ku also called the “fill factor”
Ku is the fraction
of the core window area that is filled by copper
Mechanisms that cause Ku to
be less than 1:
• Round wire does not pack
perfectly, which reduces Ku by a factor of 0.7 to 0.55 depending on
winding technique
• Insulation reduces Ku by
a factor of 0.95 to 0.65, depending on wire size and type of insulation
• Bobbin uses some window area
• Additional insulation may be
required between windings
Typical values of Ku :
0.5 for simple low-voltage
inductor
0.25 to 0.3 for off-line
transformer
0.05 to 0.2 for high-voltage
transformer (multiple kV)
0.65 for low-voltage foil-winding
inductor
Winding
resistance
The resistance of the winding is
R = qlb AW,
where q is the resistivity of
the conductor material, lb is the length ofthe wire, and AW is
the wire bare area. The resistivity of copper at room temperature is 1.724x10–6 Ω-cm. The length
of the wire comprising an n-turn winding can be expressed as
lb = n (MLT)
where (MLT) is the
mean-length-per-turn of the winding. The meanlength- per-turn is a function of
the core geometry. The above equations can be combined to obtain the fourth
constraint:
R =n (MLT)
AW
The
core geometrical constant Kg
The four constraints:
R =n (MLT)
AW
These equations involve the
quantities
Ac, WA, and
MLT, which are functions of the core geometry,
Imax, Bmax ,
μ0, L, Ku, R, and r, which are
given specifications or other known quantities, and n, lg, and AW,
which are unknowns.
Eliminate the three unknowns,
leading to a single equation involving the remaining quantities
Core
geometrical constant Kg
Elimination of n, lg,
and AW leads to
• Right-hand side: specifications
or other known quantities
• Left-hand side: function of
only core geometry
So we must choose a core whose
geometry satisfies the above equation.
The core geometrical constant Kg
is defined as
Kg =Ac2WA(MLT)
Discussion
Kg =Ac2WA
(MLT) ≥ ρ L2I max2 Bmax2 RKu
Kg is a
figure-of-merit that describes the effective electrical size of magnetic cores,
in applications where the following quantities are specified:
• Copper loss
• Maximum flux density
How specifications affect the
core size:
A smaller core can be used by
increasing Bmax use core material
having higher BsatR allow more
copper loss
How the core geometry affects
electrical capabilities:
A larger Kg can be
obtained by increase of Ac more
iron core material, or WA larger
window and more copper
A
step-by-step procedure
The following quantities are
specified, using the units noted:
Wire resistivity
Peak winding current Imax (A)
Inductance L (H)
Winding resistance R
Winding fill factor Ku
Core maximum flux density Bmax
(T)
The core dimensions are expressed
in cm:
Core cross-sectional area Ac (cm2)
Core window area WA (cm2)
Mean length per turn MLT (cm)
The use of centimeters rather
than meters requires that appropriate factors be added to the design equations.
Determine
core size
Kg ≥ ρ L2I max2 Bmax2 RKu 108
(cm5)
Choose a core which is large
enough to satisfy this inequality.
Note the values of Ac, WA, and MLT
for this core.
Determine
air gap length
lg =LI max2 Bmax2 Ac 104 (m)
with Ac expressed in cm2. μ0 = 4πx10–7
H/m.
The air gap length is given in
meters.
The value expressed above is
approximate, and neglects fringing flux and other non-idealities.
AL
Core manufacturers sell gapped
cores. Rather than specifying the air gap length, the equivalent quantity AL is
used.
AL is equal to the inductance, in
mH, obtained with a winding of 1000 turns.
When AL is specified, it is the
core manufacturer’s responsibility to obtain the correct gap length.
The required AL is given by:
AL =10Bmax2 Ac2
LI max2 (mH/1000 turns)
L = AL n2 10– 9 (Henries)
Units:
Ac cm2, L Henries, Bmax Tesla.
Determine
number of turns n
LI max4
n=10
Bmax A c
Evaluate
wire size
n = KuWA
Aw d
Select wire with bare copper area
AW less than or equal to this value.
An American Wire Gauge table is
included in Appendix D.
As a check, the winding
resistance can be computed
Aw ≤ KuwA (Ω)
n
Multiple-winding
magnetics design using the Kg method
– Copper loss dominates the total
loss (i.e. core loss is ignored), or
– The maximum flux density Bmax
is a specification rather than
a quantity to be optimized
To do this, we must
– Find how to allocate the window
area between the windings
– Generalize the step-by-step
design procedure
Copper
loss in winding j
Copper loss (not accounting for
proximity loss) is
Pcu, j = I R j
Resistance of winding j is
R j ≥ lj
A W, j
With
l j = n j (MLT ) length of wire,
winding j
A W, j ≤ WAK u wire area,
winding j
nj
Hence
Pcu, j = n j i j ñ(MLT )
WAK uá j
Step-by-step
design procedure:
Coupled inductor
The following quantities are
specified, using the units noted:
Wire resistivity
(cm2)
Peak magnetizing current IM, max
(A) (referred to winding 1)
Desired turns ratios n2/n1. n3/n2. etc.
Magnetizing inductance LM (H) (referred to winding 1)
Allowed copper loss Pcu (W)
Winding fill factor Ku
Core maximum flux density Bmax (T)
The core dimensions are expressed
in cm:
Core cross-sectional area Ac (cm2)
Core window area WA (cm2)
Mean
length per turn MLT (cm)
Determine core size
Kg ≥ ρLM2 I tot2 I M,max2 108 (cm5)
Bmax2 PcuKu
Choose a core that satisfies this
inequality. Note the values of Ac, WA,
and MLT for this core.
The resistivity ρ of copper wire
is 1.724 · 10–6 Ω cm at room
temperature, and 2.3 · 10–6Ωcm at
100°C
Determine air gap
length
lg =µ0LMI M,max2 104 (m)
Bmax2 Ac
(value neglects fringing flux,
and a longer gap may be required)
The permeability of free space is
μ0 = 4π · 10–7 H/m
Determine number of
turns
For winding 1: n1 =LMIM,max 104
BmaxAc
For other windings, use the
desired turns ratios:
Evaluate wire sizes
Aw1 ≤ α1KuWA
n1
Aw2 ≤ α2KuWA
n2
Specifications
Input voltage
Vg = 200V
Output (full load) 20
V at 5 A
Switching frequency 150 kHz
Magnetizing current ripple 20% of
dc magnetizing current
Duty cycle
D = 0.4
Turns ratio
n2/n1 = 0.15
Copper loss 1.5 W
Fill factor Ku = 0.3
Maximum flux density Bmax = 0.25
T
Basic
converter calculations
Components of magnetizing
current, referred to primary:
diM = 20% IM
= 0.25 A
I M,max = IM + ΔiM = 1.5 A
Choose magnetizing inductance
LM =VgDTs = 1.07 mH
2ΔiM
RMS
winding currents:
Choose
core size
Choose
air gap and turns
Comparison
of core and copper loss
Copper loss is 1.5 W
– does not include proximity
losses, which could substantially increase total copper loss
• Core loss is 0.25 W
– Core loss is small because
ripple and ΔB are small
– It is not a bad approximation
to ignore core losses for ferrite in CCM filter inductors
– Could consider use of a less
expensive core material having higher
core loss
– Neglecting core loss is a
reasonable approximation for this application
• Design is dominated by copper
loss
– The dominant constraint on flux
density is saturation of the core,
rather than core loss
Summary
of key points
1. A variety of magnetic devices
are commonly used in switching
converters. These devices differ
in their core flux density
variations, as well as in the
magnitudes of the ac winding
currents. When the flux density
variations are small, core loss can
be neglected. Alternatively, a
low-frequency material can be used,
having higher saturation flux
density.
2. The core geometrical constant
Kg is a measure of the magnetic
size of a core, for applications
in which copper loss is dominant.
In the Kg design method, flux
density and total copper loss are
specified.
References
