IEEE Robotics & Automation Magazine - September 2013 - 29

(Figure 6), h and c (r) , where r = b/2 , are dependent variables for the wire thickness and diameter of the device,
respectively. These dependencies enable the magnetic ﬂux
to be maintained constant at each point of the magnetic
core. We approximated these values to a constant size in
this prototype.
2
C (r) = bd + b ,
2r
+ b2 .
bd
h=
D

ND =

R=

VEMF | sec = N t $ 2r 2 f $ B max (bd + b 2),

VEMF | sec 1
Pow u =
$
,
2
4R

(2)

h

f
a
b
d

D

c(r)

(3)

(4)

Figure 6. Secondary cylindrical magnetic core unit cross section.
How the laminations were shaped for the secondary magnetic
core section is shown. The external diameter (D), the column
diameter (b), the wings for coupling with the primary core (a),
and length of the device (L) are given data. The wing diameter
(h (d)) and the winding area (c (r)) are dependent variables
(r = b/2) . The inner diameter (d) is the independent parameter.
The magnetic ﬂux is f.

(5)
(6)

Pow_u
Veff_load

10

leff_load
N_t/100

5

0

Mass [hg]
Pow_s
0

0.01 0.02 0.03 0.04 0.05 0.06 d

Figure 7. Output power, voltage, current, number of turns, mass, and
power/mass ratio are plotted as a function of the inner diameter d,
considering the external diameter, the axial length, and the diameter
of the wire (0.5 mm) as given. Arrow: the working point chosen,
whereas the related obtained values can be found in Table 1.

(7)

The general law governing the relation between the
number of turns (N D), the current (i), and the magnetic
ﬂux are represented by (7). The reluctance (0) calculated by using (8) is an intrinsic value of the magnetic
core, depending on its shape. Air reluctance is calculated
by approximating the air gap size:

0 tot = 20 air/water + 0 sec + 0 pr .

(10)

L

where t is the resistivity, f is the frequency, B max is the
maximum magnetic ﬁeld for the given material, and l t is
the turn length. Relevant characteristics, plotted in Figure 7, were evaluated to choose the dimension of the inner
diameter. We chose the maximum power/mass ratio as
working point.
The primary coil consists of a number of subcores (six)
that surround the secondary coil. The primary coil was
designed to ensure an adequate magnetic ﬂux in the secondary coil, by exploiting (1)-(6) and the following relations:
(Ni) D = 0{.

4 tN D l t
.
2
rl f

(1)

The voltage (VEMF ; sec) and inner resistance (R) are calculated using (4) to estimate the output power (Pow u) estimated
in (6) and the current transferred to the secondary coil:
R = N t $ l t $ t 1 = 2N t $ t D + d2+ 2b ,
Sf
df

(9)

Resistance was estimated in

The maximum number of turns (N t) was calculated using
(3) by varying the inner diameter (d) and considering the following values: external diameter (D) 50 mm, diameter of iron
strips (b) 1 mm (r = b/2), length (L) of the device 60 mm,
and copper wire diameter (d f ) 0.5 mm:
L - 2a2 ((bd + b 2) / (2b + d))
Nt = 2 $
.
d 22

W (L - 2A w )
.
d 2f

(8)

N D is calculated by using (9) considering the free space
around the magnetic core where W is the width of winding, L is the length, and A w are "wing" dimensions (refer
to a in Figure 6):

Table 1. Working values obtained from the model.
Parameters

Values (IU)

N t , number of turns

560

Veff load in voltage

7.63 (V)

R, inner resistor

7.07 (X)

Ieff load, current

1.08 (A)

Pow u , output power *

8.23 (W)

M, mass *

0.22 (kg)

Pow s , power/mass ratio *

36 (W/kg)

Marked (*) parameters are not dependent on the wire diameter
(df). Other parameters are strictly related to df. The wire diameter
was chosen considering the current and voltage desired for the
application.

september 2013

*

IEEE ROBOTICS & AUTOMATION MAGAZINE

*

29
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