3D Optimization of Ferrite Inductor
Considering Hysteresis Loss
Hokkaido University:
Muroran Institute of Technology:
Taiyo Yuden Co.:
Kyoto University：
Fujitsu Ltd.：
○ T. Sato, H. Igarashi
K. Watanabe
K. Kawano, M. Suzuki
T. Matsuo, K. Mifune
U. Uehara, A. Furuya
1. Introduction
A)
B)
Background
Purpose
2. Coupled analysis
3. 3D shape optimization of Inductor
4. Conclusion
2
Background

1.
2.
3.
Introductions
Coupled analysis
3D optimizations
Dc-dc converters
They are used for electric and electronic devices,
Their efficiency must be improved.

The energy losses in dc-dc converters are caused in
 control ICs
 FETs and diodes
 Inductors
The purpose of this work is that the efficiency of
the dc-dc converters is improved by reducing
the inductor losses.
3
1.
2.
3.
Purpose

Introductions
Coupled analysis
3D optimizations
The losses in the inductors are




Copper loss due to the winding resistance,
Hysteresis loss due to the magnetic cores.
The inductors must be designed to reduce them simultaneously
under the condition that the inductance satisfies the specifications
for inductance.
Design optimization based on the finite element analysis is
effective to solve this problems
However, optimizations of 3D inductor models are
computationally prohibitive mainly due to time
consuming mesh generations.
 We present the 3D shape optimization of the
inductor on the basis of nonconforming voxel FEM.

4
(Introduction. B): Purpose

1.
2.
3.
Introductions
Coupled analysis
3D optimizations
In the present method, the voxels at the material
boundaries are subdivided into fine voxels.


The fine voxels at the boundaries improve the
accuracy in the voxel FEM.
This method realizes fast generation of FE
meshes.
The multiobjective optimization of the inductors is
performed to minimize the copper and hysteresis.
5
1. Introduction
2. Coupled analysis
A)
B)
C)
D)
E)
Dc-dc converter model
Inductance under dc bias
Hysteresis loss
Inductor model for test
Analysis results
3. 3D shape optimization of Inductor
4. Conclusion
6
1.
2.
3.
Dc-dc converter model

Introductions
Coupled analysis
3D optimizations
Before the optimizations, we evaluate the inductance
and hysteresis loss under the dc bias condition by
performing the coupled analysis of inductors and dc-dc
converters.

A boost circuit shown in Fig 1 is analyzed.



The steady-state mode is evaluated.
The FET and diode are treated as ideal devices,
except for their on-resistance.
The hysteresis loss is evaluated using the Steinmetz
formula in the post processing.
L=1μH
Vout=5 V
Vin=3.3V
RIC
Cin
=2.2 μF
Control logic
RFB
Cout
=1MΩ =4.7 μF
Ro
PWM signal,
3.5MHz
Fig. 1 Boost circuit.
7
1.
2.
3.
Inductance under dc bias
Introductions
Coupled analysis
3D optimizations
1.
(Circuit analysis)：Assuming the value of inductance and winding resistance
in the inductor, the circuit analysis of the boost circuit is carried out.
2.
(Nonlinear FEM)： The operating points in FEs corresponding to the dc bias
are computed based on the nonlinear analysis using the dc property.
3.
（Coupled analysis）：The reluctivity for the operating points is determined
from the ac property. The coupled analysis is performed over one cycle
around the operating point, in which magnetic field is analyzed by the
linear FE equation using the determined reluctivity.
L=1μH
Vout=5 V
Cin
=2.2 μF
RFB
Cout
=1MΩ =4.7 μF
PWM signal,
3.5MHz
0.4
relative permeability
RIC
Control logic
Ro
0.3
B (T)
Vin=3.3V
0.2
0.1
0
0
200
400
H (A/m)
600
800
250
Reluctivity for
operating points
200
150
100
50
0
0
dc property
20000
40000
H (A/m)
60000
ac property
 K
b T
 u
bu   A  0 

,
Z   i  V 
8
1.
2.
3.
Hysteresis loss

Introductions
Coupled analysis
3D optimizations
The hysteresis loss in the inductor, Wh (W), is estimated
based on the Steinmetz formula using the coupled analysis
results.
Wh  K h f
V
e e
Be

Kh [W/(m3∙s)]＝9945.7, α＝2.26 : hysteresis coefficients
f : switching frequency, Ve : volume of element e
Be：amplitude of flux density in element e

Kh and α are measured under the condition that f is 3.5MHz.
L=1μH
Vout=5 V
Vin=3.3V
RIC
Cin
=2.2 μF
Control logic
RFB
Cout
=1MΩ =4.7 μF
Ro
PWM signal,
3.5MHz
Fig. 1 Boost circuit.
9
1.
2.
3.
Inductor model
Introductions
Coupled analysis
3D optimizations
To test the validity of the analysis, the efficiency of the
boost circuit in the steady-state mode is computed.
 The inductor shown in Fig. 2 is used in the circuit.

Coil, 6 turn
Ferrite core
unit: (mm)
1.2
0.55
z

Inductance：1.09(μH)

Winding resistance:0.14(Ω)
3.0
1.2
y
0.275
x
x
Fig. 2 Original model
10
Nonconforming connections
1.
2.
3.
Introductions
Coupled analysis
3D optimizations
In the FE analysis, the unknowns assigned to
nonconforming edges (slave edge) are interpolated by
those assigned to the conforming edges (master edge).
 Slave unknown, As, is expressed by the linear combination
of the master unknowns.

As 
 
Aim N im  dl
i 1
Aim：master unknowns
es
N im：interpolated function
Master
l：tangential vector of slave
Slaves
11
1.
2.
3.
Analysis results
The simulation results and measurement data are shown in Fig. 3.


The simulation results are in good agreement with the
measurement results. The efficiency in the heavy load
conditions obtained by the analysis and measurement is almost
identical.
There exist small discrepancies for the light load conditions.
This would be due to the fact that turn-on and off times in FET
are not taken into account.
The inductance and hysteresis loss, as well as the
copper loss, under the bias condition are correctly
considered
Analysis
Measurement
100
80
60
40
20
0
efficiency (%)

Introductions
Coupled analysis
3D optimizations
0
0.1
0.2 0.3 0.4 0.5
load current (A)
Fig. 3 Efficiency.
0.6
12
1.
2.
3.
Analysis results
Introductions
Coupled analysis
3D optimizations
Fig. 4 shows the computed hysteresis and copper losses.
 Hysteresis loss is dominant in the light loads, whereas the
copper loss is dominant in the heavy loads.

The inductor current is shown in Fig. 5.
 The amplitudes of the current are about 320mA being
independent of the load resistance.
30
Copper loss
180
20
120
10
60
0
0
0
0.1 0.2 0.3 0.4 0.5 0.6
load current (A)
Fig. 4 Hysteresis. and Copper losses
inductor current (A)
Hysteresis loss (mW)
Hysteresis loss
Copper loss (mW)

50Ω
1.6
16Ω
10Ω
1.2
320mA
0.8
0.4
0
0
2
4
6
8
time (μs)
10
12
14
Fig. 5 Inductor current
13
1.
2.
3.
Analysis for optimization
If the hysteresis loss is evaluated using the coupled analysis, long
computational time is necessary in the optimizations.


The inductor current is independent
of the load resistance.
We use Steinmetz formula.
inductor current (A)

Introductions
Coupled analysis
3D optimizations
0.5
320mA
0.25
0
0
2
4
6
8
time (μs)
Wh  K h f

e
12
14
Ve Be

The hysteresis loss is estimated only by the
magnetostatic analysis.
1. The operating points in FEs for a bias
current are computed.
2. Linearized FE equation around the
operating point is solved, in which the
inductor current is set to 160mA.
From the resultant flux density, the
hysteresis loss can be evaluated,
in addition to the inductance.
inductor current (A)

10
0.5
160mA
0.25
Bias current
0
0
Wh  K h f
2
4

e
6
8
time (μs)
Ve Be
10
12
14

14
1. Introduction
2. Coupled analysis
3. 3D shape optimization of Inductor
A)
B)
C)
Optimized inductor model
Objectives
Optimization results
4. Conclusion
15
1.
2.
3.
Optimized inductor model

Introductions
Coupled analysis
3D optimizations
The inductor shown in Fig. 6 is optimized.

Design variables g：five parameters
g  r1
r2
ch
w cr T
cr is the diameter of the winding wire, which
takes 0.1 or 0.15mm.
y
80
unit: (mm)
z
80
Ferrite core
w
Coil
0.6
r2
ch
r1
r1
r2 w
80
x
r1
r2
w
80
x
Fig. 6 Optimized inductor model (1/8)
16
1.
2.
3.
Objectives

Introductions
Coupled analysis
3D optimizations
Two objective functions and constraints are defined.
Objectives:
constraints:
L  LT 2
LT
2
f1  R,
 0.03,
f 2  Wh ,
Ll  0.7 LT ,
r2  w  1.5mm
R (Ω): winding resistance，Wh (mW): Hysteresis loss.
LT=1μH: specified inductance.
L (μH): inductance value when dc bias is 0.2A.
Ll (μH) : inductance value when dc bias is 2.7A.
y
80
unit: (mm)
z
80
Ferrite core
w
Coil
0.6
r2
ch
r1
r1
r2 w
80
x
r1
r2
w
80
x
17
1.
2.
3.
optimization results

The optimization is performed using the real-ceded Generic
Algorithm and Strength Pareto Evolutionally Algorithm 2.
The optimization is performed over 200 generations.


The Pareto solutions are shown in Fig. 6 in which the red and
blue makers denote the solutions with cr=0.15 and 0.1 mm,
respectively.
All Pareto solutions are superior to the original inductor.
hysretesis loss (mW)

Introductions
Coupled analysis
3D optimizations
Optimized (0.15)
12
Optimized (0.1)
Original
(A)
10
8
6
(B)
4
2
0.1
0.12
0.14
0.16
winding resistance (Ω)
Fig. 8 Pareto solutions
0.18
18
1.
2.
3.
optimization results



The resultant inductor shapes are shown below.
Shape(A) has large cross-section of the coil. The cross-section on the
cylindrical core is so small that the magnetic induction is strong here
solutions (0.15) solutions (0.1) test model
and hysteresis loss is large.
Shape (C) has small and large crosssections of the coil and cylindrical core,
respectively.
Shape (B) is in between (A) and (C). This
seems to be optimum in this case.
(A)
1.45mm
Upper core
volume:0.78mm3
3.75mm
L: 0.976μH, Ll: 0.96μH,
R: 10.5mΩ, Wh: 9.08mW
Num. turn: 3
(B)
1.37mm
hysretesis loss (mW)

Introductions
Coupled analysis
3D optimizations
12
10
8
(A)
6
(C)
(B)
4
2
Upper core
volume:0.38mm3
0.1
0.12
0.14
0.16
winding resistance (Ω)
(C)
1.15mm
Upper core
volume:0.28mm3
3mm
1.5mm
L: 0.97μH, Ll: 0.94μH,
R: 12.07mΩ, Wh: 6.67mW
Num. turn: 4
0.18
L: 0.99μH, Ll: 0.98μH,
R: 16.1mΩ, Wh: 4.8mW
Num. turn: 4
19
1.
2.
3.
optimization results
The circuit efficiency is computed for the shape (B).


Because both losses in the inductor are reduced,
the efficiency in the all loads is improved.
It can be concluded that the present method can
find the inductor to improve the efficiency of the
converter.
Solution (B)
Original model (A)
94
efficiency (%)

Introductions
Coupled analysis
3D optimizations
92
90
88
86
84
0
0.1
0.2
0.3
0.4
load current (A)
0.5
0.6
Fig. 9 Efficiency after optimization.
20
1. Introduction
2. Coupled analysis
3. 3D shape optimization of Inductor
4. Conclusion
21
Conclusions

The efficiency in the dc-dc converters must be improved.

The hysteresis and copper losses in
the inductors should be minimized.
Multiobjective optimization of 3D inductor shape is
performed based on the nonconforming voxel FEM.
 The hysteresis loss and winding resistance can be
reduced under the restriction that the inductance
satisfies the specifications.
 The circuit efficiency is improved using the resultant
inductor.

Future works

Hysteresis modeling is introduced for the accurate analysis.
22