Magneticpermeabilitymeasurementmethodforparticlematerials

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Magnetic permeability measurement method for particle materials
Conference Paper · May 2018
DOI: 10.1109/I2MTC.2018.8409702
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Magnetic Permeability Measurement Method for Particle Materials
Ziqiang Cui, Weiyang Zhang and Huaxiang Wang
Tianjin Key Laboratory of Process Measurement and Control,
School of Electrical and Information Engineering,
Tianjin University, Tianjin 300072, China
Email: cuiziqiang@tju.edu.cn
Abstract—Permeability measurement is mainly used to determine the permeability of magnetic material, which is generally
required to be shaped prior to test. In many chemical processes,
the distribution and density of particle materials play a key
role in determining the reaction efficiency. However, particles
do not induce as much eddy current as the thin film/ discshaped planar material does. Therefore, it is not convenient to
employ the aforementioned method to perform permeability
measurement. In this paper, a method for measuring the
equivalent permeability of particle mixtures is proposed. The
measurement system consists of a four-tap solenoid coil sensor
and a LCR meter. Whats more, the relationship between
the measured inductance and the permeability of particles is
defined. The proposed method is verified numerically by using
Comsol Multiphysics software and by experiments. Experimental results show that the relative error of the developed system
is below 5.35%.
1. Introduction
The magnetic permeability is a basic physical parameter. There are a number of methods for determining the
permeability, e.g. the inductance method, resonance method
and probe method. These measurement methods dedicate to
a specific type of samples, and usually require the preparation of the samples, i.e. casting into special shapes. For
instance, reflection measurements can be used to measure
the permeability spectra of magnetic thin film [1]–[5] and
other small specimen with an accuracy of 10% [6]. The
eddy current measurement can be used to measure the
permeability for magnetic plates, with a typical error less
than 7.5%, primarily due to the liftoff effect [7]. It also
applies to the metallic tubes/pipes measurements [8]. The
inductance method can measure the effective permeability
for metamaterial unit of planar spiral structure or magnetic
cylinder materials [9].
Recently, there are fast-growing demands for the permeability measurement instruments/methods in the nanomaterial, biomedical applications. Therefore, the emerging
measuring techniques/instruments usually focus on specific
applications. A differentially excited loop method was employed to measure the permeability of magnetic thin film,
with an error less than 8% [10]. An induced voltage method
‹,(((
was introduced to measure the permeability of ring-shaped
samples, with the deviations of 5% [11]. A cavity method
was designed to measure the permeability of F e3 O4 and
silver nano-powder, in which the samples have to be casted
into molds prior to measurement [12]. A Gauss-meter probe
was used for Magneto-Rheological Elastomers (MREs), in
which the cylindrical shape samples are required [13]. The
permeability of nanoferrite powders can be obtained by the
in-waveguide transmission reflection technique, which needs
the disc-shaped planar samples [14].
In the case of multi-phase flow measurement, the effective magnetic permeability may represent the concentration
of the ferrite material within its mixtures with the liquid
and/or gas phases. However, the aforementioned measuring
instruments cannot be directly employed in the circumference. The reasons include, 1) the multi-phase flows are
usually fast-changing dynamic processes, thus require that
the measuring circuit should be of fast response and large
dynamic range, and 2) the test samples are in a circular vessel/pipe, and usually not applicable for sample preparation.
In this paper, the design and implementation of a permeability measurement system is presented, which can measure
both the permeability and conductivity of particle mixtures.
The mixtures of Al2 O3 /N iO/F e and air are evaluated with
the proposed method. Experimental results show that the
proposed method can provide permeability measurements
of high stability and precision.
2. Method
2.1. Sensor design
As shown in Figure. 1, the toroidal core is commonly
used in determining the permeability of the block materials.
Here, Lw is the inductance of the air-core coil and Le is
the inductance filled with the test samples. Re is composed
of the resistance of wire, i.e. Rw , and the resistance that
represents the core losses. The core losses will increase as
the applied frequency.
Effective permeability is derived from the inductance
measurement result using the following equations:
μe =
lLe
μ0 N 2 S
(1)
where La is the inductance between taps 2 and 3.
The inductance of the solenoid coil is
L a = μ0 n 2 l a S
Figure 1: Toroidal inductor and the equivalent circuits.
By measuring the impedance of the coil filled with air or
ferrite particles, i.e. Ze and Zw , one can obtain the complex
permeability of the particle material with the following
formula:
μr =
μe =
l(Re − Rw )
μ0 N 2 ωS
(2)
where l is the average magnetic path length of toroidal
core, S is the cross-sectional area of toroidal core, N
is the number of turns, ω is the angular frequency, and
μ0 = 4π × 10−7 H/m.
The toroidal core is usually used for estimating the
permeability of the inductor/transoformer cores. It requires
sample preparation prior to permeability measurement.
However, this is not convenient for use as a container for the
particles and its mixtures. Instead, a 4-tap solenoid sensor is
designed and implemented for this task, as shown in Figure.
2.
(5)
Z e − Zw
+1
jωμ0 n2 la S
(6)
2.2. Equivalent circuit model
In the permeability measurement, the applied frequency
may ranges between 100 Hz and 10 M Hz , depending on
specific applications. Figure.3 shows a simplified equivalent
circuit model of the single layer solenoid coil [15]. The
stray capacitances should be taken into account, especially
when the applied frequency is higher than 100 kHz . The
inductance La together with the wire resistance RS1 and
stray capacitance CS1 , i.e. between taps 2 and 3, make up
the impedance under test. By applying a constant current
excitation, the rest components in the equivalent circuit
actually have no effect on the induced potential ξ between
taps 2 and 3.
RS0
LS0
RS1
1
La
2
CS0
RS0
LS0
3
CS1
4
CS0
Figure 3: Simplified equivalent circuit model of the single
layer solenoid coil.
To obtain accurate value of La , the two parameters, e.g.
RS1 and CS1 , should be determined at first. The impedance
between taps 2 and 3 is
Figure 2: Principle of the four-tap solenoid coil sensor.
The total length of the solenoid coil is l, and the length
between the two inner taps is la . The radius of the coil is R.
The number of turns in unit length is n. By applying an ac
current signal through taps 1 and 4, an ac electromagnetic
field is set up. Assuming that coil length l is infinite and
the coil is filled with air, the magnetic field intensity B is:
μ0 nI,
inside the coil,
B0 =
(3)
0,
outside the coil.
The induced potential ξ between taps 2 and 3 can be
calculated by:
ξ = −nla
dΦ
= jωLa I
dt
(4)
Zx
=
Rx + jX
=
(RS1 + jωLa )//
=
1
jωCS1
RS1 + jωLa
1 + jωRS1 CS1 − ω 2 La Cs1
(7)
The wire resistance can be directly measured while the
applied frequency is maintained at 0 Hz . In this case, the
impedance Zx = RS1 . The stray capacitance CS1 and the
self-inductance La will resonate, i.e. at the self resonance
frequency (SRF). The inferred frequency response is shown
in Figure. 4. Above the SRF, the capacitive reactance becomes the dominant part of the impedance. The SRF of the
solenoid coil can be obtained with an impedance analyzer.
And the stray capacitance CS1 can be solved by assuming
the reactance X = 0 in Eq. (7).
By taking into account of the edge effect, the magnetic
flux density of a finite solenoid can be approximated by
stacking together a number of circular loops coaxially. The
magnetic flux density at a point on the z -axis of finite
solenoid coil can be calculated by
μ0 nI
B(z) =
2
(l/2) − z
+
(z + l/2)2 + R2
(9)
It can be found that the ratio of solenoid length to
its diameter, i.e. l/D, plays a key role in determining the
factor k . Figure. 6 shows the distributions of B(z) of two
different solenoid coils. The ratios of l/D are 5.69 and 1.73,
respectively.
Figure 4: Frequency response of reactance X .
(z − l/2)2 + R2
(l/2) + z
1
2.3. Correction factor
Figure 5: Distribution of magnetic flux density inside a finite
solenoid coil.
la
2
− l2a
Φ(z)
dz
Φ0
0.6
l/D=5.69
l/D=1.73
0.4
0.2
0
−6
−4
−2
0
l/R
2
4
6
Figure 6: Magnetic flux density along z -axis.
Note that the magnetic flux density B(z) is normalized
by B0 in Figure. 6. At the both ends of the solenoid coils,
B(z)/B0 falls sharply to nearly zero, while it reaches the
peak value at its center part. It can be expected that the
solenoid coil with larger l/D ratio can provide a longer
part of uniformly distributed magnetic field at its center,
and get close to the case of an infinite coil. Therefore,
the two inner taps, i.e. taps 2 and 3, are introduced for
potential measurement, which can significantly reduce the
impact caused by edge effect.
In addition, the correction factor k can further reduce
the error caused by edge effect. Therefore, Eq.(6) changes
into:
μr =
k(Ze − Zw )
+1
jωμ0 n2 la S
(10)
3. Results and Discussions
The correction factor can be expressed as:
k=
0.8
Bz/B0
The finite solenoid is different with the infinite one in
the magnetic field distribution, i.e. Eq. (3). For a finite
solenoid coil, its magnetic field is not uniformly distributed,
primarily due to the edge effect. Therefore, a correction
factor is introduced to account for the edge effect in the
finite solenoid coil sensor. In addition, it is possible to
employ an enough long solenoid coil to generate a satisfied
magnetic field between the centre part of the solenoid coil,
i.e. between taps 2 and 3.
Figure.5 shows the distribution of magnetic flux density
inside a finite solenoid coil. It can be found that the magnetic flux density distribution of the solenoid coil is axial
symmetrical. The magnetic flux density decreases gradually
at both ends of the coil, which is a phenomena of the edge
effect.
(8)
where Φ(z) is the magnetic flux within the sensor along
z -axis, and Φ0 = B0 S is the magnetic flux in an infinite
solenoid coil.
3.1. Numerical validation
The validation of the method is verified by numerical
simulation. As shown in Figure. 7, a solenoid coil of 96mm
in length is employed in the simulation. The inner and outer
diameters of the coil are 16mm and 17mm, respectively.
The diameter of the wire is 0.5mm. The magnitude and
frequency of the applied signal are 1A and 10kHz , respectively.The total number of turns is 192.
i.e. taps 2 and 3 as indicated in Figure. 2, and then the
inductance Le can be obtained by Eq.(3). Figure.9 shows
the comparison between the set values and the calculated
results.
96mm
Ø17mm
Ø16mm
Figure 7: Single-layer solenoid.
The simulations are carried out with Comsol Multiphysics software. Considering individual regions in the
solenoid coil sensor that filled with the material of constant electrical conductivity and magnetic permeability,
Maxwell’s equations for the induction problem can be
rewritten as follows:
∇2 A + jωμσA = μJ
Figure 9: Given and calculated permeability results.
(11)
ρ
∇ ϕ=−
(12)
ε
where σ is the conductivity, ρ is the charge density, A is
the vector potential, and ∇ × A = B.
The partial differential equations can be solved by applying the appropriate boundary conditions, which express
the particular application and sensor geometry. Then, the
magnetic flux density along the z -axis can be obtained
from the simulation results. Figure. 8 shows the comparison
between the simulated result and the result calculated from
Eq.(9).
2
As shown in Figure.9, the calculated permeability increase linearly as the given permeability. However, the
permeability of the core, μc , does not equal to the material
permeability. This is caused by demagnetizing field effect
and can be corrected by demagnetizing factor N ∗ , which
depends on the geometry of the core [16]. Eq.(13) and
Eq.(14) define the relationships between them. The effect
of pipe wall is disregarded here.
μc =
N* 1+
μr
*
N (μ
r
− 1)
Dc 2
2lc
2 (ln( D ) − 1)
lc
c
(13)
(14)
Where lc and Dc are the length and diameter of the cylindrical core, respectively.
3.2. Experiments
Figure 8: Simulated and calculated results of B(z).
It can be found that the calculated result perfectly matches the simulation, validating Eq.(9). In order to validate
the proposed method, the complex permeability has been
calculated with the Eqs. (1) and (2). In addition, one can
obtain the induction potential from the two inner taps,
The proposed method is also verified by experiments.
The experiments are carried out with a commercial LCR
meter, i.e. ZM2371. The LCR meter can measure the inductance between 0.001nH and 99.9999GH at a basic
accuracy of 0.08%. Its frequency range is between 1mHz
and 100kHz .
In the experiments, four-terminal configuration is employed for the LCR meter. The ports Hc and Lc are connecting to taps 1 and 4, respectively, through which a constant
current of different frequencies is applied. The electric potential is measured between taps 2 and 3, i.e. by connecting
them to the ports Hp and Lp. The measured parameters are
Ls and Rs in series. Prior to the measurements, short and
open circuit calibrations are performed. The measured data
can be read from the LCR meter either from the instrument
display or via USB port.
Z(e/w) = Rs + jωLs
(15)
The correction factor k is 0.8758 for the sensor indicated
in Figure.7, which is obtained by numerical simulation.
The applied frequency is 1kHz . The measurements were
performed on air, Al2 O3 and F e particles. The measured relative permeability of air, Al2 O3 and F e particles are 0.965,
0.98 and 2.15, respectively, according to Eq.(10). However,
the relative permeability of air should be 1, indicating a system error of 3.5%. There is a calculation error of the factor
k , because of approximatively treatment of Eq.(8), and/or
the instrumental error. Although there is a difference of
B distribution between solenoid filled with air or magnetic
powders, which is caused by difference of their permeability,
they can be approximatively compensated by normalizing
each measurement with the measured permeability of air. In
this way, the instrumental error can be eliminated at least,
and system error can be relieved.
A number of mixtures of Al2 O3 and F e are prepared
for further test, i.e. each with different volume ratio. The
equivalent permeability μe of the mixtures may be estimated
using the Maxwell model, i.e.
μ1 + 2 + 2φ(μ1 − 1)
μe =
μ1 + 2 − φ(μ1 − 1)
(16)
where μ1 = 2.21 is the relative permeability of F e particles,i.e. after compensation, and φ is the volume ratio of F e
particles.
Figure.10 presents the comparison between the measured
permeability and calculated permeability of the mixtures. It
can be found that the measured values are close to that
estimated by using Maxwell model. The maximum relative
difference is 5.35%, which appears at φ = 75%.
In addition, the distribution of the ferrite particles play a
key role in affecting the efficiency of the chemical reaction.
Here, uniform spatial distribution of permeability for
particles is mainly discussed.
What impacts effective permeability for mixtures more
is particle concentration instead of spatial distribution. In
experiment, it can be treat as approximation uniform spatial
distribution whose effect is weak enough. Complex relative
permeability is usually simply permeability for short. In this
paper, permeability will be short for that.
3.3. Repeatability
A repeatability index Up is introduced to evaluate the
measurement system, as defined in Eq.(17).
2.2
Relative permeability
By applying a constant current excitation, one can obtain
the equivalent inductance by resorting to Eq.(4). The applied
magnetic flux density is less than 260mT , which is much
lower than the saturation magnetic flux density of the test
samples. According to Eq.(6), the calculation of the complex
permeability requires the measurements of Ze and Zw . Here,
Ze and Zw can be calculated by:
2
1.8
1.6
1.4
1.2
1kHz
calculated
1
0.8
0
20
40
60
80
Volume ratio of Fe particles(%)
100
Figure 10: Variation curve of given permeability and calculated results.
Up = ks = s
1
n−1
n
(xi −
i=1
1
n
2
n
xi )
(17)
i=1
where s is standard deviation calculated by Bessel formula,
and k is corresponding to fiducial probability.
For k = 2, P is 95.4%. The repeatability results are
obtained for air and F e powders when measured at different
frequencies, as listed in Table.(1).
TABLE 1: Repeatability of the permeability measurements
for (P = 95.4%).
Frequency (kHz )
Air
Fe powder
0.01
0.0374
0.02
0.1
0.0035
0.002
1
7.90E-04
8.31E-04
10
7.07E-04
0.0022
100
0.0018
0.0186
4. Conclusions and future works
In this paper, a method for estimating the permeability
of particle mixtures is proposed and the details of the sensor
design are presented. By introducing the four-tap configuration, the solenoid coil sensor can provide nearly uniformly
distributed magnetic field at its central section, which helps
reducing the impact of edge effect. As compared with the
conventional method, the proposed method is flexible and
reliable, and does not require to preparation of a toroidal or
other shape cores. These features are especially useful for
test of particle mixtures, i.e. either solid-solid, liquid-solid
or gas-solid.
The sensor can be configured to work with commercial instruments, i.e. LCR meter or impedance analyzer, to
perform the permeability measurements. In the experiments,
the measured permeability of air is 0.965, showing a system
error of 3.5%. And this can be compensated to achieve high
accuracy. Mixtures of Al2 O3 and F e particles of different
volume ratios are tested, showing a maximum relative error
of 5.35% when compared with values estimated by using
Maxwell model.
Note that the applied magnetic flux density is much
lower than the saturation magnetic flux density. Therefore,
the measured permeability can be recognized as the initial
permeability. Future works will focus on the test of the
proposed sensor with Agilent 4294A impedance analyzer
and development of a hand-held instrument.
Acknowledgment
The authors would like to thank the financial supports
from Natural Science Foundation of China (Grant Nos.
61671319 and 61627803).
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