Simulation and Optimization of Conductor Structural Parameters of Free-space Hall-Effect Current Sensor

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The 5th IEEE International Workshop on Applied Measurements for Power Systems will be held September 24-26, 2014 in Aachen, Germany
Simulation and Optimization of Conductor Structural
Parameters of Free-space Hall-Effect Current Sensor
Jianhai Qiu, Ji-Gou Liu and Quan Zhang
ChenYang Technologies GmbH & Co. KG
Markt Schwabener Str. 8, 85464 Finsing, Germany
Email: jigou.liu@chenyang-ism.com
Jianzhong Lin
University of Shanghai for Science and Technology, USST
Jungong Rd. 516, 200093 Shanghai, China
Email: Linjz@usst.edu.cn
Abstract –In this paper, simulation is done to optimize the
conductor structural parameters of a new free space current
sensor by using Ansoft Maxwell software. These parameters are
conductor shape, gap between conductor and Hall-effect sensor,
cross-section geometry and length of conductor pins, respectively.
The optimized parameters are implemented in a new free-space
current sensor with using a Hall-effect element. Experimental
results show that the sensitivity of the new designed current
sensor is relatively stable, the linearity can be controlled within
±0.6%, and the total measuring error is less than 3.0%, which is
lower than that of the integrated current sensor, for instance
ACS756.
Keywords –Free-Space Hall Effect Current Sensor, Conductor
Sructural Optimization, Magnetic field simulation
I. INTRODUCTION
In industry, increasing numbers of occasions need to trace the
current for monitoring, controlling and protecting devices,
such as automotive, household appliances, generators, and
motor drives. Therefore current sensors find more and more
applications.
There are many current sensors, including Hall-effect
current sensors, current transformers, shunt resistors etc.
Compared with the current transformers and shunt resistors,
Hall-effect current sensors are preferred because of their wide
measuring range, better linearity and accuracy, good isolation
between the input and output [1].The traditional Hall current
sensors are divided as open loop current sensor and closed
loop current sensor, in which a magnetic core with a high
permeability is used to concentrate the magnetic flux
generated by the primary current so that the sensors can get a
high sensitivity [2,3,4]. However, in some applications where
space is limited and sensitivity and accuracy are not
dominantly concerned, such as a low cost servo driver, free-
space current sensors can be used thanks to its simple structure,
small size and low cost.
As shown in Fig. 1, free-space current sensor is a coreless
Hall-effect current sensor, and composed only of a primary
current carrying conductor, a Hall-effect element and an
amplifier circuit. The primary current carrying conductor
generates a magnetic field which varies with the current. The
magnetic field can be detected by the Hall element, the output
of which is processed by the amplifier circuit. In this way one
can monitor the current passing though the conductor.
Fig. 1. Structure of free-Space Hall-effect current sensor
Nevertheless, there are several problems should be noted
when designing a free-space Hall current sensor. Firstly, free-
space current sensor is highly sensitive to the position error
between the Hall sensor and current conductor. Secondly, the
Hall element cannot get a high magnetic field generated by the
primary current under test. The sensitivity and the accuracy of
the sensor are highly limited by these facts [1].
The goal of this paper is to improve the sensitivity and
linearity of free-space current sensor by optimizing the
structural parameters, such as conductor shape, gap between
conductor and Hall-effect element, cross-section geometry,
length of conductor pins. Optimization is done by magnetic
field simulation by using Ansoft Maxwell software. The
simulation model is proved by experiments. The optimized
parameters are applied to a new free-space current sensor with
using a Hall-effect element.
II. SIMULATION AND OPTIMIZATION
The goal of magnetic field simulation by using Ansoft
Maxwell is to optimize the magnetic flux density that the Hall
element senses under different conditions. Fig. 2 shows the
simulation model, the main characteristics of which are as
follows:

1) Current excitation loads on the two ends of the
electric conductor.
2) Adaptive meshing, the number of the elements is not
more than 30,000.
3) The nonlinear residual of the model is controlled
within 0.01.
Fig. 2. Simulation model
A. Conductor Shape
According to the Biot-Savart’s Law, the magnetic flux density
increases with the increasing current. Based on the Ampere
Circuital Theorem, for a straight conductor of circular-section,
the magnetic flux density around it can be expressed as:

 (1)
Where  is the permeability of vacuum, r is the distance
from an arbitrary point outside the conductor to the centerline
of the conductor.
Apparently, if two or more electric conductors are under test,
the target point has a larger magnetic field. Therefore the
conductor could have a pattern of polygon. Considering other
factors such as ease installation and small sensor size, a U-
shaped conductor is preferred, as shown in Fig. 3.
Fig. 3. Conductor Structure under Simulation
In Fig. 3 L is the distance from the end of the pins to the
sensing center of the Hall element, defined as the length of the
pins, and d is the distance of the gap between the Hall element
and the conductor.
Fig. 4. Magnetic flux density generated by a U-shaped conductor and a straight
conductor at d=2mm and I=50A.
Fig. 4 shows the magnetic flux density generated by a U-
shaped conductor and a straight conductor at the same gap
d=2mm and under supplying the same current I=50A. It can be
found that the magnetic flux density generated by the U-shaped
conductor is much higher than that by the straight conductor.
The improvement factor is about 2.6. Therefore the sensitivity
under using U-shaped conductor is higher than that of using
straight conductor.
B. Gap size between Hall Element and Conductor
Based on the scheme shown in Fig. 3 and (1), it can be
concluded that the smaller the gap d is, the larger the magnetic
field at the target point where the Hall element locates.
However, for the new free-space current sensor that this paper
concerns, the current carrying conductor should be electrically
isolated from the Hall element with a galvanic isolation of 2.5
kV, therefore the gap size needs to be optimized.
According to the Paschen’s Law, for uniform electric field,
the air gap breakdown voltage can be estimated by using the
following empirical equation [5]:
 (kV) (2)
with  as the relative air density, used to characterize the
change in barometric pressure P, and d the gap size in cm.
Apparently, equation (2) is symmetrically respect to the two
variables, . Supposing the relative air density  equal
1, the breakdown voltage is the function of the gap size d, as
shown in Fig. 5.
Fig. 5. Relationship between the breakdown voltage and the gap d (under δ = 1)
0
1
2
3
4
5
6
0% 1000% 2000% 3000% 4000%
Flux Density B (mT)
Solve Region
U-shaped
Straight

It is important to note that this is obtained on the basis of a
uniform electric field. If the excitation electric field is non-
uniform, the air gap breakdown voltage will decrease. The
extent of the decline is associated with the uniformity of
electric field. In fact, most of the electric field is non-uniform.
In order to guarantee a galvanic isolation of 2.5 kV, a margin
coefficient can be used to keep the air from breakdown by a
non-uniform electric field. For a margin coefficient of 1.6, the
gap size d can be 0.86mm, see Fig. 5.
A gap size of 2 mm is used in following analyses as it
makes the experiments easy to realize.
C. Cross-section Geometry of Conductor
The magnetic field generated by an electric conductor depends
on its cross-section [6, 7]. In order to maximize the magnetic
flux density where the Hall element positions it is necessary to
optimize the conductor cross-section.
Fig. 6 shows five kinds of cross-sectional geometry, four of
them have a same cross-section area of 12.56 (see Fig.6 a,
b, c, d), another one has a square cross-section whose area is
16(see Fig. 6 e).
Fig. 6. Cross-section geometry of electric conductor
Fig. 7 and Table 1 show the results of the magnetic field
simulation of the cross-sections when d=2mm, I=50A,
L=2.05mm.
Fig. 7. Magnetic field simulation results with different cross-section
geometries
TABLE 1. MAXIMUM MAGNETIC FLUX DENSITY B
Cross-section
(a)
(b)
(c)
(d)
(e)
B (mT)
5.33
5.41
4.93
5.53
5.25
From the simulation results one can conclude that:
 The larger the current density or the smaller the
distance between the Hall element and conductor
centerline, the bigger the magnetic flux density.
 The magnetic field of a rectangular cross-section
conductor which is slightly narrower than a circular-
section conductor (see Fig. 6 d) is the largest.
Nevertheless, the difference is not significant.
Considering easy manufacturing, installation and low
production cost etc., a circular cross-section conductor (see
Fig. 6 a) is appreciated. The section radius depends on the
measuring range of the current sensor.
D. Conductor Length
For a free-space current sensor, the magnetic field generated by
the external input pins could affect the flux density of the target
point. In order to ignore this influence the conductor length
should be optimized by simulation. In the simulation the length
L changes from 6mm to 24mm, under d=2mm, the conductor
diameter 2R2=Ø1.3mm. The simulation results are given as
relative deviation:

 (3)
As shown in Fig. 8, the relative deviation decreases with
increasing length of the conductor pins. This is because the
influence of the magnetic field generated by the part near the
end of the conductor pins weakens gradually as L increases.
Fig. 8. Relative deviation of the magnetic field simulation results under
different length L
It can be concluded that the influence is really small when
the conductor length L ≥ 24mm.
III. EXPERIMENTAL VERIFICATION
The simulation model is verified by using the simulation
results of a straight cable H07V-K 1*25 mm2 and a circular-
section, U-shaped conductor with diameter Ø1.3mm, in
comparison with their experimental results. The Gauss meter
CYHT201, whose accuracy is ±2.0% for DC magnetic field
measurement [8], is used to measure the magnetic field
density of target point.
2
3
4
5
0% 1000% 2000% 3000% 4000%
Flux Density B (mT)
Solve Region
a
b
c
d
e
-1,00%
-0,80%
-0,60%
-0,40%
-0,20%
0,00%
0 5 10
Relative Deviation
Primary Current I (A)
L from 6 to 12mm
L from 12 to 18mm
L from 18 to 24mm

A. Sraight Cable H07V-K 1*25 mm2
Fig. 9 shows a measuring system of current flowing in a
straight cable. Experiments are repeat done for ten times as the
position of Gauss meter probe changes along the axis of cable.
The magnetic field density as function of current is shown in
Fig. 10.
Fig. 9. Measuring system for straight cable
Fig. 10. Measured magnetic field density for straight cable as function of
current
As the cable is straight, magnetic field can be calculated by (1).
In this example, r is the sum of the cable radius and the
distance from the top of the Hall probe of Gauss meter
CYHT201 to the center of the Hall chip, r=7.15mm, see Fig.
9. The relative deviations are given as:





 

 

 
 (4)
Where Bs, Bm and Bt are the simulated, measured and
theoretical values of the magnetic flux density of the straight
cable, respectively.
Fig. 11 shows the relative error between the simulation,
measured and theoretical values according to (4). Bm is the
average of each measured magnetic field density. All relative
deviations are not higher than ±3% when current changes from
5A to 50A. And the relative deviations between simulation
and theoretical values are quite small, less than ±0.4%.
Fig. 11. Relative deviation between the simulation, measured and theoretical
values
B. U-shaped Conductor of Diameter Ø1.3mm
Firstly, it should be noted that the cross-section area of the
conductor used in this experiment is relatively small. It heats
remarkably when the current goes high. Due to the thermal
drift of the Hall probe of the Gauss meter, the measured values
of magnetic flux density will deviate from the true values. The
deviations increase with the increasing current.
Fig. 12. Magnetic field generated by electric conductor in which a
bidirectional current flows
As shown in Fig. 12, the measuring error due to thermal
drift is seen as an interferential magnetic field. Its magnetic
flux density, db(I) , varies with the current I. The current flows
into the terminal (Fig. 12a) and flows out from the terminal
(Fig. 12b). According to the Ampere’s rule, the magnetic flux
density is given as:

 (5)
By calculating the modulus on the both sides of (5) as:

 (6)
The modulus of the magnetic flux density at point O can be
determined by the average value:
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
0 5 10 15 20 25 30 35 40 45 50
Flux Density B (mT)
Primary Current I (A)
1 2
3 4
5 6
7 8
910


 (7)
Based on (5)-(7), the way to measure the magnetic flux
density can be optimized.
Fig. 13. Measuring system for U-shaped conductor
The measuring system for U-shaped conductor is shown in
Fig. 13. Fig. 14 shows the magnetic flux density at the target
point generated by a conductor of Ø1.3mm as function of
current. It can be seen that the green curve obtained by using
the above measuring method is very approximate to the
simulation result. The relative deviations between the
simulation and measured results of each experiment are not
higher than ±3%, see Fig.15.
Fig. 14. Magnetic flux density of target point as function of current
Fig. 15. Relative deviation between the simulation and measured values
IV. APPLICATION
The above simulation and optimization results are used to a
new designed free-space Hall-effect current sensor. And the
test model is shown in Fig. 16.
Fig. 16. Test system for development of new current sensor
Considering the current carrying capacity, a conductor of
diameter Ø1.3mm is preferred for the experimental range 0-
10A. Fig. 17 shows the sensitivity of the current sensor, which
has been tested for ten times, under d=2mm and L=24mm. The
red point represents the average sensitivity. The relative
deviations between the measured values and the average are
within ±1.0%, see Fig. 18.
Fig. 17. Sensitivity of the free-space current sensor
Fig. 18. Relative deviation between the measured sensitivities and the average
0
0,4
0,8
1,2
1,6
0 2 4 6 8 10 12
Flux Density B (mT)
Primary Current I (A)
Bs
Bₒ
Bma Bmb
-1,00%
0,00%
1,00%
2,00%
3,00%
0 2 4 6 8 10
Relative Deviation
Primary Current I (A)
1
2
3
4
5
14,85
14,95
15,05
15,15
15,25
0246810
Sensitivity (mV/A)
Test Number
Sensitivity
Average
-1,00%
-0,50%
0,00%
0,50%
1,00%
0 2 4 6 8 10
Relative Deviation
Test Number
Current Source

Fig. 19. Linearity error of the current sensor
Fig. 19 shows the linearity error of the current sensor. The
linearity is repeat measured for 10 times. The linearity error of
the current sensor is lower than ±0.6%.
TABLE 2. ACCURACY OF THE SENSOR
Accuracy at 25
Zero
Offset
error
Half-scale
Current
Linearity
Repeatability
Total
Output
error
±0.2%
±0.3%
±0.6%
±1.5%
±2.6%
Table 2 gives the accuracy of the free-space sensor at
ambient temperature of 25. The total output error is ±2.6%.
A basic accuracy of ±3.0% is realizable for mass-production. It
is a better accuracy than that of the integrated current sensor
ACS756, in which a ferrite core is built [9].
Furthermore, the above results are based on gap size d of
2mm. For mass-production a gap size of 1.0mm should be used.
In this case the accuracy will be further improved thanks to the
higher magnetic flux density generated by the conductor.
V. CONCLUSIONS
In this paper, simulations have been done to optimize the
structural parameters of the conductor in a new free-space Hall-
effect current sensor. The simulation model has been verified
by the experiments. And the parameters optimized are used in
the current sensor. From the results one can draw the following
conclusions:
 The magnetic flux density generated by a U-shaped
conductor is much higher than that by a straight
conductor.
 A gap size of about 1.0mm can guarantee the sensor
to have an electrical isolation voltage of 2.5kV.
 A circular cross-section conductor is appreciated
thanks to its low cost, easy availability and relatively
high magnetic flux density generated by it.
 The influence of the magnetic field generated by the
part near the end of the conductor pins is gradually
reduced by increasing length L. And it can be ignored
if L is greater than 24mm.
 The sensitivity of the new designed free-space current
sensor is relatively stable and the average sensitivity is
about 15mv/A.
 The linearity error of the new designed free-space
current sensor can be controlled within ±0.6%.
 A basic accuracy of ±3.0% is realizable for free-space
current sensors under using the optimized parameters
of the conductor.
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[4] Ho-Gi et al., “Coreless current sensor for automotive inverters
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2009.
[5] Z.-C. Zhou, “Breakdown characteristics of the air gap,” in High Voltage
Engineering, 3nd. Beijing, China:CEP Press,2007, pp. 38-44.
[6] J. T. Conway, “Trigonometric integrals for the magnetic field of the coil
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[8] ChenYang Technologies GmbH & Co. KG. Gaussmeter/Teslameter
CYHT201 [Online]. Availabe: http://www.chenyang.de.
[9] Allegro MicroSystems. LLC. (2006). Fully integrated, Hall effect-based
linear current sensor IC with 3 kVRMS voltage isolation and a low-
r es is ta nc e c ur re nt co nd uc to r [ On li ne ] . A va i la be :
http://www.allegromicro.com/en/Products/Current-Sensor-ICs/Fifty-To-
Two-Hundred-Amp-Integrated-Conductor-Sensor-ICs/ACS756.aspx
-0,60%
-0,40%
-0,20%
0,00%
0,20%
0,40%
0,60%
0 2 4 6 8 10 12
Linearity Error
Primary Current I (A)
1 2 3 4 5
6 7 8 9 10