Proceedings of PowerMEMS 2008+ microEMS2008, Sendai, Japan, November 9-12, (2008)
COMPARATIVE STUDY OF ELECTROMAGNETIC COUPLING
ARCHITECTURES FOR VIBRATION ENERGY HARVESTING DEVICES
D. Spreemann1, B. Folkmer1, Y. Manoli1, 2
Institute for Micromachining and Information Technology, HSG-IMIT,
Villingen-Schwenningen, Germany
2
Department of Microsystems Engineering, University of Freiburg, IMTEK, Freiburg, Germany
1
Abstract: In this paper five different commonly used electromagnetic coupling architectures in vibration
energy harvesting devices are optimized in a construction volume of 1 cm³ using semi–analytical magnetic
field calculations. The optimization goal is the maximum voltage as well as the maximum power output.
Quantitative comparison of the architectures is presented. It is shown that for optimized conditions the
maximum possible output voltage can be increased by a factor of two and the output power can be increased
by a factor of three only by choosing a proper architecture. Experimental results obtained with a setup to
measure the magnetic flux gradient are used to verify the simulation models.
Key words: Vibration energy harvesting, Electromagnetic coupling architecture, Optimization
1. INTRODUCTION
The effect of electromagnetic induction can be
used to convert energy of different physical domains.
In vibration energy harvesting applications mechanical
energy (movement of an inertial mass) needs to be
converted into electrical energy. Following Faraday’s
law of induction this can be obtained by producing a
magnetic flux change in a closed loop of wire. In
vibration transducers this is mostly realized by moving
a magnet relative to a coil. This can be realized in
various kinds. For this reason a lot of different
architectures have been established and applied by
numerous research facilities in the recent years. To the
best of the authors knowledge a detailed realistic
comparison on how much voltage and power can be
obtained for the different architectures in a certain
construction volume has not been published so far. In
this paper we present a comparative study of five
commonly used architectures (Fig. 1). They can be
divided into two groups namely “Magnet in–line coil”
and “Magnet across coil” architectures. Optimization
calculation is applied to each architecture in order to
optimize the voltage and power output.
Primarily, boundary conditions based on
mesoscale vibration transducers are defined which are
valid for all architectures. Afterwards each architecture
is optimized in a construction volume of 1 cm³. For the
optimization approach semi–analytical static magnetic
field calculations are used. The investigated
architectures consist of a permanent magnet
(cylindrical or rectangular) and a cylindrical coil. The
coil material is assumed to have no influence on the
static magnetic field distribution. The optimal
a)
Architecture I
Architecture II
D
D
Di,coil
hcoil
Di,coil
hcoil
Coil
h
N
Coil
h
Magnet
N
Magnet
S
S
Construction volume
Construction volume
Architecture III
D
Di,coil
Spacer
Coil
Magnet
h
hcoil
S
N
N
S
Construction volume
b)
Architecture IV
h
h
Magnet
hcoil
hcoil
a
a
Architecture V
Di,coil
Magnet
N S
N S
N
S
Coil
Di,coil
b
b
S N
S N
Coil
Construction volume
S
N
Construction volume
Fig. 1: a) “Magnet in–line coil” architectures and b)
“Magnet across coil” architectures
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Proceedings of PowerMEMS 2008+ microEMS2008, Sendai, Japan, November 9-12, (2008)
dimension for each architecture is simply obtained by
varying the geometric parameters of coil and magnet
in the construction volume. In a final step the optimal
voltage and power levels are compared to each other.
With the presented results the designer of
electromagnetic vibration transducers gets a guideline
which architecture to choose for practical voltage and
power levels, respectively.
Table 1: Overall boundary conditions applied for
optimization calculations
2.
BOUNDARY
CONDITIONS
OPTIMIZATION PROCEDURE
Br
ρ
Symbol
Vconstr
G
Zmax
AND
kco
Dco
R’
The first goal of the optimization is to maximize
the voltage and power output of the architectures (in
the following abbreviated arch). Because the
quantitative comparison of each arch is another
intention of the presented work, it is important to
declare overall fixed boundary conditions (Tab. 1). For
the presented results values similar to mesoscale
vibration transducers have been used. The construction
volume Vconstr (cylinder resp. cuboid) contains the coil
and the magnet at its equilibrium point. In the first step
of the optimization procedure (Fig. 2) two geometry
vectors for the inner coil diameter (Di,coil) and the coil
height (hcoil) are defined. With the fixed gap between
coil and magnet the geometry of the magnet follows.
In this regard the only exception is arch I where the
depth of immersion has to be taken into account too
[1]. Afterwards the windings and internal resistance of
the Coil can be calculated as:
N long =
N lat =
2 ⋅ hcoil
Dco π k co
Y
f
cm
Dco π k co
Rin = N long ⋅ N lat ⋅ 2π
Geometry
Construction volume
Gap (Coil/Magnet)
Maximum displacement
Magnet
Remanenz
Density of magnet
Coil
Copper fill factor
Wire diameter
Resistance per unit length
Other
Excitation amplitude
Excitation frequency
Mechanical damping
Value
Unit
1
0.5
1
cm³
mm
mm
1.1
7.6
T
g/cm³
0.6
40
13.6
1
µm
Ω/m
10
100
0.1
m/s²
Hz
N/m/s
ϕ = ∫ BdA
,
2 ⋅ (D − Di ) 2
Description
dϕ
(1)
,
(D − Di )
4
dx
⋅ R '.
The magnetic field (cylindrical and rectangular
magnets) is calculated using the scalar potential model
[2, 3]. Due to lack of space the formulas for the
magnetic field have been left out, but it is essential to
know that they contain elliptic integrals which have to
be solved numerically. These results are than used to
compute the total magnetic flux φ in the coil (at the
equilibrium point of the oscillating magnet) and its
gradient:
ϕ=
∑∑ ∫ Bd A
i, j
N long N lat Ai , j
,
(2)
K = (dϕ dx )equi .
Therein Ai,j indicates the area enclosed by the
respective Winding. With the internal resistance and
Fig. 2: Procedural structure of the optimization
approach
258
Proceedings of PowerMEMS 2008+ microEMS2008, Sendai, Japan, November 9-12, (2008)
0.5
2
10
6
8
D
2
4
6
Di,coil
8
U
10
max
[V]
3
2
1
2
4
D
6
8
6
1000
4
6
Di,coil
8
P
10
[mW]
3
4
2
2
1
10
2
4
i,coil
D
6
8
4
6
8
Di,coil
10
max
[V]
coil
d)
1.5
6
1
0.5
4
6
D
8
i,coil
10
6
4000
2000
4
6
8
Di,coil
10
P
max
6
4
4
6
D
8
4
6
i,coil
coil
h
coil
Rload,opt [Ω ]
2500
2
2
D
4
6
i,coil
Umax [V]
1
4
0.5
4
Di,coil
6
1500
4
1000
2
500
2
d)
D
4
i,coil
6
Pmax [mW]
3
8
1.5
6
2000
6
6
2
4
1
2
2
4
Di,coil
6
the magnetic flux gradient the optimal load resistance
can be determined [4]:
Rload ,opt = Rin +
K2
.
cm
(3)
This result is in turn necessary to determine the
electromagnetic damping coefficient:
[mW]
1.4
1.2
1
0.8
0.6
0.4
0.2
8
2
h
coil
8
4
U
coil
4
6000
h
c)
8
4
5
D
Fig. 7: Optimization result for “Magnet across coil”
architecture V
h
10
2
2
10
Rload,opt [Ω ]
15
6
h
coil
8
4
i,coil
K [V/m/s] b)
2
8
2
Fig. 4: Optimization result for “Magnet in–line
coil” architecture II
a)
1
2
6
8
6
2
4
6
4
c)
max
h
2
h
4
4000
2000
2
d)
8
3000
2
3
6
K [V/m/s] b)
5000
coil
2
6
coil
coil
5
4
h
2
coil
coil
10
h
4
6
4
[mW]
4
i,coil
a)
h
15
6
D
6
max
Fig. 6: Optimization result for “Magnet across coil”
architecture IV
Rload,opt [Ω ]
K [V/m/s] b)
c)
coil
10
i,coil
P
8
0.5
2
Fig. 3: Optimization result for “Magnet in–line
coil” architecture I.
a)
1
4
Di,coil
10
1.5
6
500
2
d)
2
4
0.5
i,coil
[V]
h
8
D
max
1000
2
h
1
2
6
U
8
1.5
4
4
Di,coil
10
coil
6
2
c)
2
h
h
[mW]
6
1
4
max
1500
4
2
2
2000
6
coil
6
h
coil
1.5
8
10
Di,coil
P
4
4
1000
6
d)
6
h
[V]
8
coil
max
2
10
6
coil
8
10
Di,coil
U
2000
8
8
h
6
c)
4
10
h
2
3000
coil
coil
h
4
6
Rload,opt [Ω ]
K [V/m/s] b)
a)
4000
h
12
10
8
6
4
2
6
coil
Rload,opt [Ω ]
K [V/m/s] b)
a)
ce =
K2
(Rin + Rload ,opt ).
(4)
Next the relative amplitude of steady-state motion can
be calculated:
10
i,coil
Fig. 5: Optimization result for “Magnet in–line
coil” architecture III
Z=
259
mω 2Y
(k − mω ) + ((c
2 2
+ cm )ω )
2
e
,
(5)
Proceedings of PowerMEMS 2008+ microEMS2008, Sendai, Japan, November 9-12, (2008)
(a)
where k indicates the spring constant yielding the
resonance frequency defined in the boundary
conditions. Using Faraday’s law of induction the EMF
(electromotive force) is given as:
Vopt [V]
0
(6)
PRload , opt =
V Rload , opt
⎞
⎟,
⎟
⎠
Popt [mW]
(7)
IV
V
4
2
0
I
II
III
IV
V
IV
V
(d)
3
2
I
II
III
IV
V
2
1
0
I
II
III
Fig. 8: Comparison of maximal values of EMF
voltage (a), power at maximal voltage point (b),
maximal output power (c) and voltage at maximal
output power point (d)
Note that the inductance of the coil has been neglected
because the reactance we want to consider is much
smaller than the resistance. The optimization
procedure following from (1) – (7) is applied to every
pair of variables of the two geometry vectors. In order
to understand the advantage and disadvantage of each
arch the optimal load resistance and magnetic flux
gradient has been recorded beyond the maximum
voltage and power output.
RESULTS
III
4
0
.
3.
OPTIMIZATION
DISCUSSION
II
6
(c)
2
Rload ,opt
I
6
In order to obtain the output power Ohm’s law and
Kirchhoff’s law has to be applied to the simple
resistive load considered here:
⎛ Rload ,opt
V Rload , opt = ε ⋅ ⎜
⎜R
⎝ load ,opt + Rin
P at Vopt [mW]
dϕ
dB ⎞
dϕ dz
⎛ dA
= −⎜
B+
A⎟ = −
⋅ .
dt
dt ⎠
dz dt
⎝ dt
2
V at Popt [V]
ε =−
(d)
4
apparent that there exists a separate optimum for
voltage and power. Which point to choose depends on
whether high output voltage or power levels are
desired. As an essential summary highest voltage
levels are expected for arch II and greatest power with
arch IV. The values of voltage and power at the
optimum points are shown in Fig. 8.
ACKNOWLEDGEMENTS
AND
The authors want to acknowledge the funding of
the ZOFF III project “Energieeffiziente Autonome
Mikrosysteme” by the government of BadenWürttemberg
The results of the optimization for arch I – V are
shown in Fig. 3 – 7. In each figure a) illustrates the
magnetic flux gradient at equilibrium point, b) the
optimal load resistance, c) the maximal output voltage
and d) the maximal output power. As it can be seen for
example in Fig. 3a the magnetic flux gradient has an
optimum (Di,coil = 8 mm, hcoil = 5 mm), but for the EMF
the velocity (high inner displacement due to high
oscillating mass) has to be taken into account too.
That’s why the optimum for EMF voltage (Fig. 3c) is
shifted to higher values of Di,coil and smaller values of
hcoil respectively (bigger magnet!). Since a bigger
magnet implies a smaller coil and smaller resistance as
well the optimum of the output power is shifted to
even higher values of Di,coil and smaller values of hcoil
(Fig. 3d). This effect is obvious for arch III too. In this
regard the only difference for arch II, IV and V is that
the magnet size is not dependent on Di,coil. Therefore
the EMF voltage steadily increases for smaller Di,coil
which has been limited to 1 mm. The importance of
the mass is best observable for arch III [5]. Here the
greatest magnetic flux gradient can be achieved but
due to the comparatively small magnet the maximum
output power (and voltage) is nevertheless quite low
compared to arch II, for example. From all arch it is
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