Inductor Design –
with Continous/DC Current
The circuit design process will/should specify:
L:
required inductance at full load
I dc :
average DC current
ΔI :
peak current variation at full load
Ploss :
maximum allowed loss
iL
2
2
I rms
= I dc
+
1 2
ΔI
12
I max
ΔI
I dc
I min
Ton
t
T
For the continuous current case the AC current swing is small, so the AC flux density is also small.
Consequently, most of the loss is DC copper winding loss unless the frequency is very high.
ΔB ≈ ΔI
Bmax
Ip
Winding loss is related to the wire guage and length (number of turns). Both of these parameters are
constrained by the core geometry and thermal considerations. In the absence of a detailed thermal analysis, it
is common to specify a certain maximum rms current density,
I rms
≤ J max
Abw
4I rms
dbw ≥
π J max
J max ≈ 400 − 600 A/cm2 (rms)
Typical current
densities assumed
in the design of
power inductors
It is important to emphasize that this is just a rule-of-thumb expedient. The actual maximum allowed rms current
density depends on the particular thermal environment. We will consider a better treatment of thermal issues later.
The next step is to choose the core material and geometry, and the correct number of turns in the winding.
The core selection must satisfy several constraints:
•  Must achieve the correct inductance
•  Keep the loss below a specified level
•  Keep the inductor size as small as possible without saturating the core
Most manufacturers now provide some software tools to help select the correct core for a given application and
circuit conditions.
© Bob York
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Typical Core Geometries
Some Basic Core shapes
C or U core
Pot core
Toroid
E-core
There are many variants on the above, here are a few examples
PQ
EQ
RM
EP
ETD
© Bob York
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Core Comparison
Magnetics Inc. Magnetic Cores for Switching Power Supplies , www.mag-inc.com
© Bob York
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Single-Layer Toroid
The inductance is given by:
b
A
L = µe N 2 e = N 2 AL
le
w
a
To keep the flux density below
some specified maximum, the
peak current must satisfy:
I max ≤
w=b−a
h
I.D.
NBmax Ae
L
O.D.
φ
The number of turns that will fit on the toroid is
constrained by the inner diameter and wire size:
a ? dw
θ = 2π − φ :
θ = 2π =
θ
2( a − d w )sin
> dw
2N
2N
The copper loss is given by:
For a single-layer winding:
PCu = ρCu
N ( MLT ) 2
I rms
Abw
Nd w ≤ 2π a
dw
dbw
MLT = 2(h + d w ) + 2( w + d w )
(Note: DC resistance is justified in the continuous
current case where the AC current variation is small)
The core loss can be estimated from the Steinmetz equation:
Pcore = kcore f n ΔB mVe
© Bob York
ΔB ≈ Δ I
Bmax
I max
Abw =
winding angle
π 2
dbw
4
dbw :
bare wire diameter
Abw :
bare wire area
dw :
diameter with insulation
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Single-Layer Toroid: Core Selection
Note that the first two constraints on the previous page scale in
opposite ways with the number of turns
Ae ≥
LI max
NBmax
I.D. ≥
Nd w
π
The product of these two numbers gives
The R.H.S can be computed
from the circuit specifications
and an estimate of the peak flux
density, which can be obtained
from manufacturer data on
specific core materials
LI d
Ae × I.D. ≥ max w
π Bmax
This gives simple guidance for choosing the correct core
dimensions: we must use a toroid with an area-diameter product
that satisfies the above constraint.
Once the core is selected, all the geometrical parameters will be
known, so we just need to calculate the number of turns:
N=
Often the manufacturers will provide several choices for AL for a given core
dimension. The remaining constraint on copper loss can help us choose:
L
AL
AL >
ρCu ( MLT )
PCu Abw L
2
I rms
This is a simple method for core selection that closely parallels the area-product method
to be developed later
© Bob York
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Note that toroid dimensions are not entirely
independent: the inner diameter must scale with the
outer diameter, and the height is also constrained.
Let s parameterize the core dimensions as follows:
ξ≡
I.D.
O.D.
γ≡
h
w
I.D./O/D.
There are often many possible core sizes and
materials that can be selected for a given design. Is
there a best design?
Catalog cores (chart at right):
γ ≈1− 2
ξ ≈ 0.5 − 0.6
0.8
8
0.7
7
0.6
6
0.5
0.4
I.D./O.D. Micrometals
h/w: Ferroxcube
0.3
k1
4
3
h/w: Micrometals
0.2
2
0.1
1
0.0
1
With these definitions the various geometrical parameters
can all be expressed in terms of the outer radius b:
(1! " ) 2 b 2 ! e
a = ξ b h = γ (1 − ξ )b Ae = !!
#"#
$
5
I.D./O.D. Ferroxcube
10
! 2! ab = 2! " b
"#$
0
1000
100
O.D. [mm]
Aspect Ratio h/w
Single-Layer Toroid:
A Closer Look at Core Selection
MLT ! 2(! +1)(1! ! ) b
!#
#"##
$
k3
k2
Eliminating the number of turns from the equations on the previous page
gives these three constraints that relate permeability to core size:
2
⎛B ⎞ kk
µe ≤ ⎜ max ⎟ 1 2 b3
⎝ I max ⎠ L
To limit the peak flux density
© Bob York
2
k2 k32 L ⎛ ρCu I rms
µe ≥
⎜
k1 ⎜⎝ Abw PCu
2
⎞
⎟⎟ b
⎠
To keep the Ohmic loss below PCu
2
k L⎛ d ⎞ 1
µe ≥ 2 ⎜ w ⎟ 3
k1 ⎝ θ ξ ⎠ b
To insure that the required number of
turns will fit the core size
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Example Calculations
Circuit Spec:
Assumptions:
L = 500 µ H
I dc = 2 A
ξ = 0.5
γ =2
ΔI = 0.4 A
Bmax = 0.3T
J max = 400 A/cm 2
The shaded region is the range of allowed
permeability and core sizes that will satisfy the
constraints on number of turns and Bmax.
10000
Required ue (relative) at full load
The three constraints are plotted at right for a
particular design example:
PCu < Pcrit
1000
µe,crit
10
B<Bmax
PCu=Pcrit
N turns just fit I.D.
1
0
The minimum possible core size and associated
permeability and copper loss can be found as:
1/3
bmin
⎛ LI
d ⎞
= ⎜ max w ⎟
⎝ k1Bmax 2πξ ⎠
µe,crit
B
d
= k2 max w
I p 2πξ
PCu > Pcrit
100
Pcrit
2bmin
50
100
O.D. [mm]
⎛ LI
⎞
= k3 ⎜ max ⎟
⎝ k1Bmax ⎠
2/3
This approach gives a quick estimate of the core outer diamater that is required (2  bmin).
This is useful because most manufacturers list or label toroids in terms of the O.D.
Key tradeoff: reducing ohmic loss requires a larger core size and higher permeability
© Bob York
150
1/3
⎛ 2πξ ⎞
⎜
⎟
⎝ dw ⎠
2
⎛ ρCu I rms
⎜⎜
⎝ Abw
⎞
⎟⎟
⎠
Pcrit is the winding
loss that would be
incurred if the
minimum core size
is used
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Single-Layer Toroid: Optimization
The parameterization allows us to explore the possibility of optimal toroid geometries:
1
2 ⎤ −1/3
Core has minimum
⎡
bmin ∝ ⎣γ (1 − ξ ) ξ ⎦
ξ
=
γ = large as possible
outer radius when:
3
1/3
⎛ γ +1⎞⎛ ξ ⎞
PCu ∝ 2 ⎜ 2/3 ⎟ ⎜
⎟
1
−
ξ
γ
⎝
⎠
⎝
⎠
ξ = small as possible
Winding loss
minimized when:
1
ξ=
3
This suggest a possible design goal of:
γ =2
(Charts show that there are a range of
values around these design goals that
should yield god results)
γ =2
But these are not the only possible considerations. For example the designer might want to minimize the footprint
on a PC board. For a vertical toroid the device consumes an area of about 2hb, and this is minimized when
1/3
⎛
⎞
γ
APCB ∝ 2hb = 2 ⎜ 2
⎟
ξ
(1
−
ξ
)
⎝
⎠
ξ=
2
3
γ = small as possible
Or we could look for a compromise by minimizing the area-loss product:
1/3
⎞
⎛ γ +1⎞⎛
1
APCB PCu ∝ ⎜ 1/3 ⎟ ⎜
2 ⎟
γ
(1
−
ξ
)
ξ⎠
⎝
⎠⎝
ξ=
1
3
γ=
1
2
Cost is another important factor. Assuming cost is proportional to the volume of ferrite
material in the core, the cost-loss product varies as:
1/3
⎛
⎞
⎛
⎞⎛
⎞
γ
+
1
1
+
ξ
ξ
Cost × PCu ∝ 2 ⎜ 2/3 ⎟ ⎜
Cost ∝ Vcore = π h b2 − a 2
⎟⎜
⎟
(
)
⎝γ
⎠ ⎝ ξ ⎠⎝ 1 − ξ ⎠
ξ = 0.5
γ =2
Catalog data shows that available core geometries are mostly clustered near this latter design goal, but there
often other cores available for which the above considerations can help make a judicious choice
© Bob York
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Permability and Material Selection
Once a specific core geometry is chosen the design equations should be recalculated as needed, and this
will give a range of permeabilities required for the design.
Once the desired permeability is known, the core material can be chosen. Powder cores are available in
many different mixtures. Alternative a gapped ferrite core could be used.
Shown below are representative curves for available mixtures of Ferroxcube MPP material.
80%
Note that the effective permeability falls off at high currents. The design calls for a certain permeability at
full-load conditions. This means that the chosen material should have a low-field permeability that is
significantly higher, to insure the correct inductance at full-load. It is common to choose the material such
that the inductance has fallen to no less than 60-70% of its initial low-field value at full-load conditions.
© Bob York
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Core Loss
As mentioned, the core loss is typically small for inductors with
small AC current variations and low switchign speeds, but this
should be checked at the end of the design.
Coefficients in the Steinmetz equation can be estimated from
manufacturers data. Note that high-mu materials are typically
more lossy than low-mu materials
300µ
14µ
Losses are related to frequency and the
peak AC field, and can be read directly from
these charts, where the peak AC field is
B
ΔB
Bˆ =
≈ ΔI max
2
Ip
Note that these charts assume sinusoidal variations. Typical
switching waveforms are nonsinusoidal, so a more precise
analysis should examine the influence of harmonics.
© Bob York
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