ARID ZONE JOURNAL OF ENGINEERING, TECHNOLOGY & ENVIRONMENT
AZOJETE - Centre for Satellite Technology Development Special Issue:

Space Science and Technology for Sustainable Development
AZOJETE, June, 2019. Vol. 15(SP.i2):14-20

Published by the Faculty of Engineering, University of Maidiguri, Maidiguri, Nigeria.

Print ISSN: 1596-2490, Electronic ISSN: 2545-5818
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14

ORIGINAL RESEARCH ARTICLE
APPLICATION OF FIREFLY OPTIMIZATION ALGORITHM TO THE DESIGN AND
CHARACTERIZATION OF HELMHOLTZ COILS FOR MAGNETIC FIELD SENSOR

CALIBRATION
U. A. Sadiq, O. W. Oluyombo, S. O. Onuh

1Centre for Satellite Technology Development, National Space Research and Development Agency, Obasanjo Space
Centre, Airport Road, Abuja

2Engineering and Space System department, National Space Research and Development Agency, Obasanjo Space
Centre, Airport Road, Abuja

3Director Centre for Satellite Technology Development, National Space Research and Development Agency, Obasanjo
Space Centre, Airport Road, Abuja

ARTICLE INFORMATION

Submitted: 11/12/2018
Revised: 12/01/2019
Accepted: 04/06/2019

ABSTRACT
This paper presents the application of Firefly Optimization
Algorithm (FOA) to the design and characterization of Helmholtz
coils to create homogenous magnetic field needed for calibrating
magnetic field sensors and used for other experiments.
Magnetometer is used to monitor and record the earth’s magnetic
field data at the geomagnetic observatory. Earth’s magnetic field
data from ground-based magnetometer observatories are
important for studying geomagnetic storm. The absence of earth’s
magnetic field data observatories results in a complex mysterious
phenomenon of geomagnetic storm and remains as unexplained
one. Fluxgate magnetometer is contributing to the ongoing
extensive research work dedicated to the explanation of some of
the complex phenomena related to geomagnetic storm and solar
terrestrial system. In order to examine magnetic field sensing of a
fluxgate sensor, a large area with uniform magnetic field is required.
The advantage of having a large area is to allow easy access of the
sensor during magnetic field measurements. A laboratory design
and characterization of high quality Helmholtz coils is a better
choice when Helmholtz coil with larger areas that are available in
the market are very expensive
© 2019 Faculty of Engineering, University of Maiduguri, Nigeria. All rights reserved.

1.0 Introduction
Helmholtz coil is a parallel pair of two similar circular or square shaped coils, spaced one radius
apart with coils wound in series such that the current passes in the same direction in each coil
(Abbott, 2015; Alao et al., 2018). Helmholtz coils are used for Earth’s magnetic field nullification,
magnetic field sensors calibration and other experiments where a controllable amount of
uniform magnetic field is required (Abbott, 2015). The basic principle of Helmholtz coils is that
the coils produces uniform magnetic field at the center (Abbott, 2015; Daron et al., 2015) of the
coils, which is equivalent to the radius of the coil. This arrangement of coil was invented by
German physicist, Hermann von Helmholtz, over a century ago (Bhatt et al., 2010). The intensity
of magnetic field is directly proportional to the number of turns, radius of the coil and the
excitation current through the coils (Abbott, 2015). According to the definition of Helmholtz coil,
there are some uncertainties even if the mean radius of the coils is equals to the separation

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Arid Zone Journal of Engineering, Technology and Environment, June, 2019; Vol. 15(sp.i2):14-20. ISSN 1596-2490;
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distance between the coils (Yang et al., 2018). An ideal Helmholtz coil with ampere-turn (Ni) can
be realized with the number of layers, number of turns per layer, coil excitation current and
radius of the coil (Yang et al., 2018). Biot-Savart law is employed for the magnetic field
calculation, while the coil turns of each Helmholtz coil are concentric rings as shown in Figure 1.
Thus, the magnetic field strength at the center of Helmholtz coils due to current-carrying
element is as follows (Bhatt et al., 2010):

B = N .
μ0.IC
R .

4
5

3 2
(1)

Where: R is the radius of each turn of a coil, IC is the coil current, and μ0 is the permeability of
air. When calculating the diameter of wire for the design of Helmholtz coils, the power supply for
driving the coils must be taken into consideration as this provides the current needed to
generate the required magnetic field (Bhatt et al., 2010; Yang et al., 2018). The schematic
describing the configuration of Helmholtz coils is shown in Figure 1.

Figure 1: The Schematic Diagram of Helmholtz Coils (Bhatt et al, 2010)
As shown in Figure 1, each Helmholtz coil produces equal amount of magnetic field parallel to
the axis of the coils (Bhatt et al., 2010), while the two coils produce uniform magnetic field, which
is accomplished by a controllable amount of excitation current (Abbott, 2015). The uniform field
within the coils is due to the summation of fields parallel to the axis and the difference of vertical
component fields (Abbott, 2015; Yang et al., 2018). Helmholtz coil design is simple and there are
many commercially available designs for all kinds of applications (Yang et al., 2018). In order to
examine the magnetic field sensing of a fluxgate sensor, a large area with uniform magnetic field
is required (Butta et al., 2010). Helmholtz coils with large area will allow easy access of the sensor
during measurements (Butta et al., 2010; Alao et al., 2018). A laboratory design and
characterization of Helmholtz coils is of great advantage when large area Helmholtz coil is
desired and the commercial Helmholtz coils are very expensive (Alao et al., 2018).
Moreover, in order to calibrate magnetic field sensors, a high magnetic field with low rise in
temperature and power consumption are required. When total power consumption and
magnetic field of the Helmholtz coils are the objectives function, Helmholtz coils design needs
to satisfy the design limits and geometric constraints. In order to find the optimal performance
of Helmholtz coils in this study, a systematic design approach is employed, where the modified
multi-objective Firefly Optimization Algorithm (FOA) is used to simultaneously obtain the
parameters and predicts the magnetic field inside the Helmholtz coils. Such a multi-objective
optimization approach is powerful in dealing with design problems in electro-magnetic with a

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Sadiq et al.: Application of Firefly Optimization Algorithm to the Design and Characterization of Helmholtz Coils for
Magnetic Field Sensor Calibration. AZOJETE, 15(sp.i2):14-20. ISSN 1596-2490; e-ISSN 2545-5818, www.azojete.com.ng

16

large number of design variables and multiple objectives under complex nonlinear constraints
(Yang, 2013).

2.0 Methodology
In order to implement the design of Helmholtz coils, the FOA was written in Matlab environment
and the results were obtained by running the developed program on a 1.50 GHz Intel® core™
Duo CPU Windows 7 Ultimate 32-bit personal computer (Alao et al., 2018). In this study, the
primary aim of the Helmholtz coils design was to create uniform magnetic field in the test area.
The characterization was carried out by using a MatLab program for plotting the magnetic field
inside the Helmholtz coils.

2.1 Helmholtz Coils Design Approach Using FOA
The aim of systematic design approach applied to Helmholtz coils design in this study is to
simultaneously find the optimal number of coil turns per layer, number of coil layers, wire size,
coil radius, and the excitation current requires for creating uniform magnetic field at the center
of the Helmholtz coils. This is achieved by using Helmholtz coils equation to modify firefly
optimization algorithm. The algorithm starts by placing the fireflies in random locations. The
location of a firefly corresponds to the values of the parameters (number of coil turns per layer,
number of coil layers, wire size, coil radius, and the excitation current) for the objective function
(magnetic field) to be solved.
Initialize the number of fireflies, n, biggest attraction β0, absorption coefficient of light intensity
γ, step size factor α, and maximum number of iterations or generations tmax.
Initialize the positions of fireflies randomly, namely initializing design variables of the Helmholtz
coils parameters (number of coil turns per layer, number of coil layers, wire size, coil radius, and
the excitation current), the values of objective function (magnetic field) of fireflies are set as their
maximum brightness of fluorescence I0.
Calculate relative brightness and attractiveness of fireflies, which belong to the population. The
direction of movement depends on the relative brightness of fireflies. Here (Yang, 2013):

I = I0 × e−γrij (2)
β = β0 × e−γrij (3)

Where: I0 is the maximum fluorescence brightness of the firefly, namely the fluorescence
brightness itself (r = 0), which depends on the value of the objective function. β0 is the
maximum attractiveness, namely the attractiveness of the light source (r = 0). γ is the absorption
coefficient of the light intensity. The fluorescence will gradually weaken according to the
increasing distance and the absorption of media. The absorption coefficient of light intensity is
set to reflect this feature. rij is the spatial distance between firefly i and j.
Update the spatial positions of fireflies. Random perturbations are injected to the firefly with the
best position. The updated equation is:

xi = xi + β × xj − xi + α × rand − 0.5 (4)
where xi, xj represent the spatial positions of firefly i and j, respectively. α is the step size factor.
rand is random factor distributed uniformly in [0,1].
Recalculate the brightness of fireflies according to the updated positions.

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Arid Zone Journal of Engineering, Technology and Environment, June, 2019; Vol. 15(sp.i2):14-20. ISSN 1596-2490;
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Corresponding author’s e-mail address: sadiqumar2001@googlemail.com

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Return to Step 3 until the search precision is met or the maximum number of generations is
achieved.
The optimization of Helmholtz coils design problem is formulated as non-linear programming
problem, expressing the objective function and constraint function in terms of the specified
independent variable.

2.2 Design Variables
In the design optimization of Helmholtz coils in this study, if large number of variables is
selected, the problem will become complicated. The following quantities are chosen as
Helmholtz coils design variables for optimization.
Number of turns per layer (x1)
Number of coil layers in the winding (x2)
Wire diameter (x3), mm
Radius of coil winding (x4), m
Coil excitation current (x5), A
The problem can be stated in mathematical terms as follows:

X = x1, x2, … x5 =

Number of turns per layer
Number of layers
Wire diameter
Coil radius
Coil current

T

(5)

Where x1, x2, x3, x4, and x5 are the set of independent design variables with their lower and
upper bounds as xmin and xmax. The design variables are presented in Table 1.

Table 1: Helmholtz Coils Design Variables and Ranges
Variables Range Unit
Number of turns per layer 8 – 60 -
Number of layers 6 – 65 -
Wire diameter 0.135 – 1.5 mm
Coil radius 0.01 – 0.35 m
Coil current 5 – 250 mA

2.3 Constraints
g(x) is the set of constraints imposed on the design to make it feasible and practically acceptable,
the constraints that have been used in this study are:
Total number of winding turns
Total magnetic field
The geometric constraints and operating limits are represented in Equation (6) and the geometry
constraints and operating limits are shown in Table 2.

g x ≤ 0 ⇔
Total winding turns
Total magnetic field (6)

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Sadiq et al.: Application of Firefly Optimization Algorithm to the Design and Characterization of Helmholtz Coils for
Magnetic Field Sensor Calibration. AZOJETE, 15(sp.i2):14-20. ISSN 1596-2490; e-ISSN 2545-5818, www.azojete.com.ng

18

Table 2: Helmholtz Coils Design Constraints
Constraints Range Unit
Total winding turns ≤ 1600 -
Magnetic field ≤ 750 μT
The objective function optimization problem is to optimize the magnetic field. In the
computation of F(x) and g(x) as well as calculating the optimal performance of the Helmholtz
coils, the optimization design problem of the Helmholtz coils is formulated by defining the
magnetic field at the center of the Helmholtz coils as the objective function:

Maximize B z =
μ0 .Ic . R2

2
.

z +
R
2

2
+ R2

− 3
2

+ z −
R
2

2
+ R2

−3/2
(7)

3.0 Results and Discussion
To verify the effectiveness of the proposed FOA for optimizing the Helmholtz coils design
problem, the optimization procedure starts with the design variables. In the design optimization
problem, five independent variables are chosen, such as number of coil turns per layer, number
of coil layers, wire size, coil radius, and the excitation current during the iteration. The modified
FOA was written in Matlab environment and the results were obtained by running the developed
program on a 1.50 GHz Intel® core™ Duo CPU Windows 7 Ultimate 32-bit personal computer.
Equation (7) is used in the firefly optimization algorithm to calculate the optimum magnetic field
inside the Helmholtz coils and to find the coils parameters. The algorithm predicts the total
number of windings needed to produce a magnetic field of 356.1µT. The number of windings is
a function of the radius of the raw coil form. The results obtained by the proposed FOA method
are presented in Figure 2.

Figure 2: Optimization Results obtained from MATLAM Program

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Arid Zone Journal of Engineering, Technology and Environment, June, 2019; Vol. 15(sp.i2):14-20. ISSN 1596-2490;
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Corresponding author’s e-mail address: sadiqumar2001@googlemail.com

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As shown in Figure 2, magnetic field of 356.1µT will be generated with an excitation current of
68.30 mA by each of the Helmholtz coils when wound with 23 turns per layer and 22 layers of
coil. As obtained from the modified FOA result, each coil had a diameter of 8.9 cm with 506 turns
each of copper wire and copper wire size of 0.62 mm diameter. Table 3 shows the optimal
design variables for the proposed Helmholtz coils design.

Table 3: FOA Results for the Helmholtz Coils Simulations.
Input variables Output

variable
Helmholtz
Coils

Coil turns
per layer

Number of
layers

Wire
diameter
(mm)

Coil radius
(mm)

Coil
current
(mA)

Magnetic
field (μT)

FOA Design 23 22 0.62 89.23 68.30 356.1

As shown in Table 3, the width of each coil is 14.26 mm while the height of each coil is 13.64 mm.
Moreove, the mean radius of each coil is 96.05mm with inner radius of 89.23 mm and outer
radius of 102.87 mm, respectively. The designed ampere-turn is Ni = 68.3 mA * 506, where
number of layers is 22 and number of turns per layer is 23. In order to characterize the designed
Helmholtz coils, these parameters were put in MatLab-program. Magnetic field plot arising from
the contributions of the modified FOA designed coils is shown in Figure 3.

-0.15 -0.1 -0.05 0 0.05 0.1 0.15
0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Coils Axial Distance [m]

M
ag

ne
tic

 F
ie

ld
 [m

T]

Figure 3: Magnetic Field as a function of Distance obtained from FOA Design.

From this plot, good agreement between the predicted and simulated magnetic fields can be
found in the Helmholtz coils. It is observed that the total magnetic field at the center point of
the Helmholtz coils is eventually the vector summation of the two coil winding pairs. The field
distribution is homogenous, especially at the center of the coils where the axial distance is
equivalent to the radius of the coil. Beyond this distance, the field uniformity tends to be
deteriorated. This result indicates that one can increase the axial distance to improve the field
uniformity by increasing the coil radius.

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Sadiq et al.: Application of Firefly Optimization Algorithm to the Design and Characterization of Helmholtz Coils for
Magnetic Field Sensor Calibration. AZOJETE, 15(sp.i2):14-20. ISSN 1596-2490; e-ISSN 2545-5818, www.azojete.com.ng

20

Moreover, the magnetic field in the negative axis on the graph represents the contribution of the
left coil while the magnetic field in the positive axis on the graph represents the contribution of
the right coil. The magnetic field inside the Helmholtz coils is homogenous within 0.08 m axial
distance. This is close to the 0.089 m coil radius predicted by firefly algorithm. Therefore, any
deviation from the optimum coil configuration would cause the graph to be in different form.

4.0 Conclusion
The optimization algorithm and MatLab program have enabled the predictions of the Helmholtz
coils design parameters and the visualization of the magnetic field coverage inside the chosen
design. This is done by using Equation (7) to modify the firefly optimization algorithm and also
using MatLab program to plot Equation (7) as a function of distance along the axial length of the
Helmholtz coils. It is observed that there is 10% difference between the radius of the coil and the
magnetic field coverage along the axial distance of the Helmholtz coils (Figure 3). This validates
the Helmholtz coils principle; the separation distance between the Helmholtz coils equals the
radius of each coil. This therefore shows the effectiveness of firefly optimization algorithm for
Helmholtz coils design.

References
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Instruments, 86:1-10.
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Bhatt, V., Rautela, RS., Sharma1, P., Tiwari, DC. and Khushu, S. 2010. Design and development of
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Yang, J., Wang, Y., Pan, J. and Wang, Y. 2018. Parametric tuning of the Helmholtz coil and its
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