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Divisionally analytical reconstruction of the magnetic field around an electromagnetic velocity probe
Sensors and Actuators A 150 (2009) 12–23
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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
Divisionally analytical reconstruction of the magnetic field around an electromagnetic velocity probe X. Fu, L. Hu, J. Zou ∗ , H.Y. Yang, X.D. Ruan, C.Y. Wang The State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou, China
a r t i c l e
i n f o
Article history: Received 31 July 2008 Received in revised form 19 November 2008 Accepted 28 November 2008 Available online 11 December 2008 Keywords: Electromagnetic velocity probe EVP Dry calibration Laplace’s equation Magnetic field Separation of variables Analytical solution Reconstruction
a b s t r a c t Determination of the magnetic field around an electromagnetic velocity probe is a key work for the dry calibration of the probe. In the paper, a novel divisionally analytical reconstruction approach to determine the magnetic field is introduced. It is especially suitable to be used in the dry calibration due to following advantages: no numerous measurements needed, unnecessary to know the inner magnetic exciting structure of the probe, high accuracy and convenient for further calculations. It is a measurement and calculation joint approach, in which the magnetic field is calculated from the measured boundary conditions through solving a Laplace’s equation of magnetic scalar potential. To obtain an analytical solution of the Laplace’s equation, the complex calculation geometry around the probe is divided into two simple ones using an auxiliary surface, and then the Laplace’s equation in the each generated geometry is analytically solved through a method of separation of variables. The divisionally analytical solution is pre-requisite since a pure analytical solution can bring convenience for further calculations in the dry calibration and such a solution is still lacked due to the complexity of the calculation geometry. A magnetic scanning device used to measure the distributions of the normal component of magnetic field on the surface of probe and the auxiliary surface is also introduced. The measured data provide the required boundary conditions for the determination of the magnetic field. Finally, the novel approach is validated through reconstructing the magnetic field around an actual electromagnetic velocity probe and comparing the results with the experimentally measured data. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Electromagnetic velocity probes (EVPs) are based on Faraday’s law of electromagnetic induction. As shown in Fig. 1, a magnetic field is built around a probe by its inner magnetic exciting unit and a voltage signal will be induced between the electrodes by conductive liquid moving through the built magnetic field. The relation of the induced voltage U, flow velocity v and magnetic flux density B is [1]
v(B × W )d
U =
(1)
Here, is the measuring volume, and W is the weight function, which expresses how the induced voltages contribute to the output signal of electrodes from different positions and is also termed by Bevir [2] as the “virtual currents j”. According to above relation, the sensitivity of the probe (U/¯v) can be mathematically calculated if the distributions of magnetic
∗ Corresponding author. Tel.: +86 571 87953395; fax: +86 571 87953395. E-mail address: [email protected] (J. Zou). 0924-4247/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2008.11.032
field, weight function and flow velocity are all known. This calculation approach to determine the sensitivity of EVP is called as dry calibration, which is very economical compared with liquid calibration methods using flow rigs as no actual liquid flow needed. It is clear that the determination of the magnetic field in the measuring volume is a key work for the dry calibration. At present, measurements and calculations are commonly used approaches to determine the magnetic field in the study of EVP. Magnetic measuring apparatus such as gauss meter can measure the magnetic flux density at a point accurately. However, directly measuring the distribution of the magnetic field around an EVP is still cumbersome, because all the x, y and z-directional components at numerous points need to be measured in a 3dimensional measuring volume. Low efficiency, complex position apparatus and additional location errors are all unavoidable disadvantages. Calculations from the inner exciting structure according to Maxwell’s equations are also usually used to determine the magnetic field. There are various numerical and analytical calculation approaches. Numerical calculations of magnetic field from the exciting units are commonly used approaches in study of electromagnetics and its applications now, which also includes electromagnetic flow measurements [3–11]. However, the exciting units of EVP are composed of coils, cores, air gap and sometimes
X. Fu et al. / Sensors and Actuators A 150 (2009) 12–23
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pole. The radius of the larger cylinder is r2 and length is L2 . The radius of the small cylinder is r1 and length is L1 . The surfaces of the large cylinder are remote enough from the probe until the normal magnetic flux density on the surfaces can be ignored. Thus, the calculation geometry equals to the large cylinder subtracted by the small cylinder of EVP and its installing pole. Since the magnetic field induced by the moving liquid is negligible compared with the magnetic field built by the inner exciting unit of EVP [14], there are no currents in the calculation geometry. Magnetic scalar potential can be used to calculate the magnetic field in the geometry. The potential obeys Laplace’s equation [15].
∇2
=0
(2)
Thus, we can determine the distribution of magnetic flux density B according to the following relation if the Laplace’s equation can be solved. B = −∇
Fig. 1. Electromagnetic velocity probe.
magnetic yokes. It is difficult to accurately model the size of coils, structure of air gap, assembly errors and material properties such as B–H curves for every EVP. Thus, these numerical models are not accurate enough and even voided when the inner structures of probe are unknown. Another obvious shortcoming of the numerical approaches is lacking of convenience for further joint calculations in the dry calibration. Therefore, the numerical approaches are usually used by the manufactures of EVP during design and optimization of the magnetic exciting performance of probes. They require knowing the general distributions of magnetic fields both in and out of probes, but relative low accuracies are acceptable. Calculations based on simplifying the complex exciting units to a circular [12] or square [13] circuit which have analytical solutions of the excited fields are analytical approaches. Obviously, the analytical mathematic models are convenient for further calculations but lack of accuracy because of the oversimplification. Thus, this method is used in prediction of the effect on the sensitivity of EVP by a nearby wall [12,13], which requires relative low accuracy in the determination of magnetic field but good convenience of further joint calculations with weight function and velocity profile mathematic models. In a word, the existing pure measurement and calculation approaches are not suitable for the determination of the magnetic field in the dry calibration of EVP. A suitable approach should have characteristics: no numerous measurements needed, unnecessary to know the inner magnetic exciting structure of the probe, high accuracy and convenient for further calculations. In following sections, a novel measurement and calculation joint approach to determine the magnetic field around an EVP is introduced and experimentally validated. This approach is especially suitable to be used in the dry calibration.
(3)
Here, the constant is the magnetic permeability of the media (air usually) in the measuring volume. Boundary conditions are necessary to solve the Laplace’s equation. Since the distribution of potential on the surface of EVP and its installing pole is immeasurable, solving the equation under Dirichlet boundary conditions is unrealizable. Thus, Neumann boundary conditions are used, as the distribution of the normal derivative of potential (normal magnetic flux density) on the surface can be measured by a magnetic scanning apparatus introduced in following sections. Thus, the boundary conditions on the surface of EVP including its installing pole are
∂ ∂
=− =r1
Bn(0−L1 )
,
0 ≤ z ≤ L1 ;
∂ ∂z
=− z=L1
0 ≤ ≤ r1
(4)
Here, Bn(0−L1 ) expresses the distribution of normal magnetic flux density on the side face of EVP including its installing pole, Bz expresses the distribution on the end face of EVP. As mentioned above, the large cylinder is built to be large enough, thus the remain-
2. Laplace’s equation in terms of magnetic scalar potential The measuring volume out of an EVP which is also called as calculation geometry in following calculations is shown in Fig. 2. For convenience of mathematic description, a large cylinder is built to enclose a small cylinder composed of the EVP and its installing
Bz ,
Fig. 2. Calculation geometry out of EVP.
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X. Fu et al. / Sensors and Actuators A 150 (2009) 12–23
where
ing boundary conditions are
∂ = 0, 0 ≤ z ≤ L2 ; ∂ =r2 ∂ ∂ = 0 r ≤ ≤ r 1 2 ∂z ∂z z=0
Bn =
=0
0 ≤ ≤ r2
(5)
z=L2
3. Divisionally analytical solution of the Laplace’s equation Direct analytical solution of the Laplace’s Eq. (2) is till unreachable due to the complexity of the calculation geometry. At present, numerical methods [16–18] are commonly used approaches for this kind of PDE solution problems. A semi-analytical solution approach also has been introduced by Zhang and Hemp [12,19] to solve a similar problem of weight function. However, they all lack convenience of further calculations if used in dry calibration. The obtained distribution of magnetic field B should be further cross multiplied with the distribution of weight function W and point multiplied with the distribution of flow velocity v, and finally a volume integral of the multiplication result should be calculated in the measuring volume to access the sensitivity of the probe. These are all further calculations in dry calibration mentioned in this paper. Obviously, multiplications and integral should be done numerically in a 3D volume if numerical approaches used. It will waste many calculation times compared with an analytical approach, since many analytical simplifications can be done during the further calculations if the distributions of W, B and v are all expressed in analytical mathematical expressions. In this section, the complex calculation geometry is divided into two simple ones using an auxiliary surface, and then the Laplace’s equation in the each generated geometry is analytically solved through a method of separation of variables. The cost of the pure analytical solution is an additional requirement of the boundary condition on the auxiliary cylinder surface, which can be determined through measurement introduced in following sections. 3.1. Division using an extended cylindrical surface An extended cylindrical surface is used to divide the complex calculation geometry into two simple ones, i.e. a cylinder and an annulus. The outlines of the division surface are showed in Fig. 3a using broken lines. The generated cylinder and annulus are showed in Fig. 3b as geometry I and Fig. 3c as geometry II, respectively. Potential is defined as potential I in geometry I and II in geometry II. Function Bn(L1 −L2 ) is predefined to express the distribution of normal magnetic flux density on the extended surface. Thus the boundary conditions to solve Laplace’s equations of potential II in geometry II are sufficient and listed as I in geometry I and follows Geometry I :
0 ≤ ≤ r1 ;
∇2
I
∂ I ∂
= 0;
=r1
∂ I ∂z
Bn =− ,
=− z=L1
Bz ,
∇2
II
0 ≤ z ≤ L2 ;
∂ II ∂z
∂ II ∂
=− =r1
= 0, z=0
I (, , z)
∂ II ∂z
=0 z=L2
∂ II ∂
= U(, , z) + F(, , z)
∇ U = C; 2
∂U ∂z
∂U ∂
0 ≤ ≤ r1
=r2
r1 ≤ ≤ r2
(7)
∂U ∂z
Bz =− , = 0,
L1 ≤ z ≤ L2
∂F ∂
0 ≤ ≤ r1
= 0, z=L2
(10)
=r1
∂F ∂z
∇ 2 F = −C;
∂F ∂z
= 0, z=L1
=− =r1
Bn
= 0, z=L2
L1 ≤ z ≤ L 2
(11)
The constant C can be determined according to Gauss theorem.
Bz ds SendI
C=
Bn(L1 −L2 ) ds =
VI
SsideI
(12)
VI
Here, VI is the volume of geometry I, SendI is the area of the end face, SsideI is the area of the side face. Particular solutions u(, , z) and f(, , z) are given to change the Poisson’s Eqs. (10) and (11) to Laplace’s equations. ˜ + u; U=U
u=
∇ U˜ = 0;
˜ ∂U ∂z
C (z − L2 )2 2
2
z=L1
(13)
Bz + C(L2 − L1 ), =−
˜ ∂U ∂
0 ≤ ≤ r1 ;
=0
C f = − 2 4
∇ F = 0;
∂F˜ ∂z
∂F˜ ∂
=− =r1
˜ ∂U ∂z
= 0, z=L2
L1 ≤ z ≤ L2
(14)
=r1
F = F˜ + f ;
= 0,
z=L1
(9)
(6)
Bn ,
(8)
3.2.1. Solution of I To apply the method of separation of variables, homogeneous boundary conditions are necessary to determine eigenvalues. However, there are non-homogeneous boundary conditions both on side and end faces of geometry I as solving I (, , z) via Eq. (6). The approach used here is breaking the function I (, , z) up into the sum of two simpler functions, U(, , z) and F(, , z). U(, , z) has homogeneous boundary conditions on the side face. F(, , z) has homogeneous boundary conditions on the end face.
2˜
= 0;
0 ≤ z ≤ L1 L1 ≤ z ≤ L2
3.2. Analytical solution by the method of separation of variables
= 0, z=L2
L1 ≤ z ≤ L2
Geometry II :
∂ I ∂z
Bn Bn
(15)
= 0, z=L1
1 Bn + Cr1 , 2
∂F˜ ∂z
= 0,
0 ≤ ≤ r1 ;
z=L2
L1 ≤ z ≤ L2
(16)
The Laplace’s Eqs. (14) and (16) can be directly solved by the method of separation of variables.
X. Fu et al. / Sensors and Actuators A 150 (2009) 12–23
15
Fig. 3. (a) Division using an extended cylindrical surface. (b) Geometry I: a simple cylinder. (c) Geometry II: a simple annulus.
˜ The solution of U(, , z) is ˜ ˜c + U(, , z) = U
×
×
×
×
AmnI · cos m · Jm
m=0 n=1
(m)
n (z − L1 ) + BmnI · cos m · Jm r1 n (z − L1 ) r1
(m)
(m) n
(m)
n r1
(m)
n r1
+CmnI · sin m · Jm
n (z − L1 ) + DmnI · sin m · Jm r1
−
(m)
−
where
∞ ∞
(m)
(m)
· exp
fmn = exp
· exp
εm =
· exp
r1
2
AmnI = emn
−
×
0
0 (m)
n r1
⎧ ⎨1
1 ⎩2 0
r1
2
−
×
0
0
+ C(L2 − L1 ) · cos m · Jm
(18)
r1
r1
2
B z −
×
0
0
r1 r1
(m)
for
when
m=0
for CmnI and DmnI
AmnI and BmnI
(24)
(25)
A = (m + 1/2 + 2n) /2, B = 4m2 , C = 7B2 + 83B − 9, D = 83B3 + 207B2 − 3537, . . .; n = 1, 2, 3. . . ¯ n2 is the modulus of the Bessel function [21]. N
r1 2 Jm
(m)
n r1
d =
r12
2
1−
m
2
(m)
n
(m)
2 Jm ( n )
(26)
∞
∞ ∞
+
+ C(L2 − L1 ) · sin m · Jm
(EmI cos m + FmI sin m)m
m=1
GmnI · cos
m=0 n=1
+HmnI · cos
2
B z −
0
n r1
m= / 0 m=0
(20)
n
(z − L1 ) · cos m · Im L2 − L1
n
(z − L1 ) · sin m · Im L2 − L1
n
L2 − L1
n
L2 − L1
(27)
The expressions of EmI , FmI , GmnI , HmnI are
0
when
(19)
dd
DmnI = fmn
(23)
˜ The solution of F(, , z) is
(m) n
+ C(L2 − L1 ) · cos m · Jm
dd
· emn
when
˜ F(, , z) = F˜c +
CmnI = emn
(L2 − L1 )
(m)
n (L2 − L1 ) r1
0
(m) n
×
B z
(22)
C B+3 D − − −··· 3 8A 15(4A)5 6(4A)
=A−
¯ n2 = N
(m)
(m)
BmnI = fmn
(m)
n r1
tion of first kind and its derivative. n can be accessed through a method of look-up table or calculated through [20]. n
dd
2
1 − exp2
( r ), i.e. the intersecting points of are positive zeros of Jm n 1 (x) are ordinary Bessel funcJm−1 ( n r1 ) and Jm+1 ( n r1 ). Jm (x) and Jm
(17)
B z
·
εm
(m) n r1
(m) n
The expressions of AmnI , BmnI , CmnI and DmnI are
2 · Nn
· exp
(z − L1 )
r1
emn =
n r1
n r1
+ C(L2 − L1 ) · sin m · Jm
EmI =
dd
(21)
FmI =
εm
2 L2 0
L1
− Bn + 12 Cr1 cos mdzd m−1
εm
1 2 L2(L2 −BnL1 ) mr 1 0
L1
− + 2 Cr1 sin mdzd
(L2 − L1 ) mr1 m−1
(28) (29)
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X. Fu et al. / Sensors and Actuators A 150 (2009) 12–23
2
L2
GmnI = gmn
B n −
L1
0
2
· cos
n
z · cos mdzd L2 − L1 (30)
L2
HmnI = gmn
B n −
L1
0
1 Cr1 2
+
+
1 Cr1 2
· cos
n
z · sin mdzd L2 − L1 (31)
where gmn =
εm =
εm (n /L − L )(r ) (n 2 /2)Im 2 1 1
⎧ ⎨1 ⎩
1 2 0
(32)
m= / 0
when when
m=0
for
EmI and GmnI
when
m=0
for
FmI and HmnI
(33)
(x) are virtual Bessel function of first kind and its Im (x) and Im derivative. Finally, I (, , z) can be calculated according to I (, , z)
˜ ˜ = U(, , z) + F(, , z) + u(, , z) + f (, , z)
(34)
3.2.2. Solution of II Since there are homogeneous boundary conditions on the end faces of geometry II, the Laplace’s Eq. (7) can be directly solved by the method of separation of variables. The solution is II (, , z) =
c+
(AmII cos m + BmII sin m)m
+(CmII cos m + DmII sin m) ∞ ∞
+
m=0 n=1
n
n
z · sin m · Im ( ) +GmnII · cos L2 L2
AmII = −
BmII = −
CmII = −
2 L2 0
0
2 L2 0
0
n
n
z · sin m · Km ( ) L2 L2
Bn cos mdzd
εm
2 L2 0
0
2 L2
L2
Bn cos 0
0
εm
2m
⎧ ⎨1 ⎩
1 2 0
when when when
m=0
m=0
m= / 0
AmII CmII EmnII and FmnII
(35)
˜ c , F˜c and c 3.2.3. A note of the constant items U ˜ c , F˜c and c in above solutions are unavailThe constant items U able using the given Neumann boundary conditions. However, this will not affect using above solutions to calculate the magnetic field, because the gradient of the constant items will be equal to zero during the calculation.
(38) ]
(40)
FmnII = hmn Im (
n 2 r2 ) · L2 n2
( GmnII = −hmn Km
0
n
2 r2 ) · L2 n 2
2
L2
0 2
Bn cos L2
n zcos mdzd L2
Bn cos 0
0
(45)
for BmII DmII GmnII and HmnII
0
(n /L )(r )K (n /L )(r ) − K (n /L )(r )I (n /L )(r ) Im 2 2 m 2 1 2 2 m 2 1 m
(37)
− (1/r1 )
n
z sin mdzd (43) L2
hmn =
]
Bn cos mdzd
2m L2 m · r1 m−1 [(1/r2 )
εm
2m
2
(36)
Bn sin mdzd
L2 m · r1 m−1 [1 − (r2 /r1 )
(x) are virtual Bessel function of second kind and Km (x) and Km its derivative, respectively.
2m L2 m · r1 m−1 [1 − (r2 /r1 ) ]
εm
2 n 2
Bn sin mdzd 0 (39) m−1 L2 m · r1 [(1/r2 )2m − (1/r1 )2m ] 2 L2 2 n n −hmn Km r2 · Bn cos zcos mdzd L2 L2 n2 0 0
DmII = − EmnII =
εm
·
where
εm =
The expressions of AmII , BmII , CmII , DmII , EmnII , FmnII , GmnII and HmnII are
L2
r2
(44)
n
n
z · cos m · Km ( ) L2 L2
+ HmnII · cos
n
−m
n
n
EmnII · cos z · cos m · Im ( ) L2 L2
+FmnII · cos
HmnII = hmn Im
−
∞ m=1
Fig. 4. Magnetic scanning apparatus.
(41)
n
z sin mdzd(42) L2
Fig. 5. Probe of gauss meter.
X. Fu et al. / Sensors and Actuators A 150 (2009) 12–23
17
Fig. 6. The measured EVP.
4. Measuring device of the boundary condition
series port to control the auto scanning procedure using a scanning program.
A magnetic scanning apparatus shown in Fig. 4 is designed to measure the distributions of normal magnetic flux density on the auxiliary cylinder surface, the side and end faces of the probe including its installing pole, i.e. Bn(L1 −L2 ) , Bn(0−L1 ) and Bz . They are necessary boundary conditions to solve above Laplace’s equations. A probe A of gauss meter is fit on a beam B and droved by a scanning mechanism composed of a dovetail slot C, a rectangular block D, an electrical rotary unit E, a manual lifting unit F, two electrical moving units G and H. Probe A is controlled by a gauss meter I. Units E, G and H are controlled by a motion controller J. Computer K is connected with the gauss meter I and the controller J through
4.1. The gauss meter The magnetic sensor is a hall-effect gauss meter. As shown in Fig. 5, the diameter of the hall-effect area is about 1 mm and locates at the end of probe A. Only the average of the normal magnetic flux density passing through this small area is measured. The beam B is used to reinforce and protect the probe A. The material of the stem of probe A and beam B is aluminum, whose magnetic permeability is very close to air, and thus will not distort the magnetic field. All above designing ensure a high measuring precision of mag-
Fig. 7. Measurement of Bn(0−L1 ) and Bn(L1 −L2 ) .
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X. Fu et al. / Sensors and Actuators A 150 (2009) 12–23
Fig. 8. Measurement of Bz (, ).
netic field. The maximum measuring error for DC magnetic field is ±0.20% of reading and for AC magnetic field is ±2.0% of reading. 4.2. The scanning mechanism A total of four freedoms of motions can be realized by the scanning mechanism, which are a rotation motion realized by electrical rotary unit E, three linear motions in x, y, z directions realized by electrical moving units G, H and manual lifting unit F. Electrical rotary unit E and moving unit G together drive the probe A of gauss meter to measure the distributions of normal magnetic flux density on the auxiliary cylinder surface and the side face of EVP including its installing pole. Electrical moving unit G and H together drive the probe A of gauss meter to measure the distribution of normal magnetic flux density on the end face of EVP. The manual lifting unit F is used to adjust the coaxiality between rotary unit E and EVP before measurements on the side face and the auxiliary cylinder surface, and to adjust the height of probe A to keep the probe in same plane with the end face of EVP before measurement on it. The dovetail slot C is used to adjust the diameter of scanned cylinder surface. All the motions are droved by inner stepper motors, which are controlled by the controller J. The motion range of unit E is 0–360◦ , G and H are 300 mm. The maximum location error of unit E is 0.005◦ , G and H are 5 m. 4.3. The scanning program The auto scanning procedure is under the control of a scanning program. The control parameters include positions of the measuring points, sampling times at every point, delay time between two measurements and so on. The measured data at every point will be listed in a file automatically by the program.
mentioned in the Section 4.1, the measuring error of gauss meter for AC magnetic field is much larger than for DC magnetic field. Thus, a DC exiting current that equal to the amplitude of the rectangular square exciting current (250 mA) is used to ensure the measuring precision. This change will not influent the determination of the sensitivity of EVP when the method is used in dry calibration. 5.2. Measurement of the boundary conditions 5.2.1. Measurements of Bn(0−L1 ) and Bn(L1 −L2 ) Electrical rotary unit E and moving unit G together drive the probe A of gauss meter to measure the distributions of normal magnetic flux density on the auxiliary cylinder surface and the side face of EVP including its installing pole. As shown in Fig. 7a, the measured EVP is fixed on an installing pole whose material is aluminum. The auxiliary cylinder surface and the side face of EVP including its installing pole together construct a measured cylinder surface whose outline is shown in the figure using broken lines. During the measurement, the hall-effect area of probe A should overlap with the section at every measuring point on the measured cylinder surface. The measurement starts from a point on the side face of the stalling pole where the measured magnetic flux density is close to zero, goes on loop and loop along the axis of EVP, and stops at a point on the auxiliary cylinder surface where the measured data are close to zero too. For the actual EVP measured in this paper, the length of the measured cylinder surface is 410 mm and diameter is 60 mm. The measured data on the measured cylinder surface are shown in Fig. 7b. It provides the required boundary condition Bn . The part z = 0–210 mm of the measured data is separately shown in Fig. 7c.
5. Validation on an actual EVP 5.1. The actual EVP and its exciting current The actual EVP measured in this paper is shown in Fig. 6a. It is a cylinder whose diameter is 60 mm and length is 80 mm. Two electrodes locate on its end face. The inner exciting unit of the EVP is shown in Fig. 6b. A flat shape and the magnetic yoke ensure enough strength of magnetic field near the electrodes. After the exciting unit is assembled into EVP, the inner space of EVP will be filled with glue for sealing. Thus, the exact inner structure parameters are unknown. A rectangular square exciting current is used by the EVP as measuring liquid velocity, for the purpose of generating an alternating magnetic field to avoid polarization on the electrodes. However, as
Fig. 9. Measurement of earth magnetic field.
X. Fu et al. / Sensors and Actuators A 150 (2009) 12–23
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It expresses the required boundary condition Bn(0−L1 ) on the side face of EVP including its installing pole. The part z = 210–410 mm is separately shown in Fig. 7d. It expresses the required boundary condition Bn(L1 −L2 ) on the auxiliary cylinder surface. During above measurement, the step of linear motion along the axis of EVP is 2 mm, and rotary motion is 3◦ . Five measurements are done in 0.5 s at every point. Thus, a total of 24,720 points are measured, which costs about 4 h. 5.2.2. Measurement of Bz Electrical moving units G and H together drive the probe A of gauss meter to measure the distribution of normal magnetic flux
˜ Fig. 11. Convergence characteristic of the series expansion for F(, , z).
˜ Fig. 10. Convergence characteristic of the series expansion for U(, , z).
density on the end face. As shown in Fig. 8a, the probe A of gauss meter is very close and parallel to the end face of EVP. The measurement starts from the center of end face, goes on loop and loop along the radius and stops at the edge of end face. The measuring points construe a disc whose outline is shown in the figure using a broken line. The diameter of the measured disc is 60 mm. The step of linear motion on the direction of radius is 1 mm, and rotary motion is 3◦ . Five measurements are done in 0.5 s at every point. Thus, a total of 3600 points are measured, which costs about three quarters of an hour. The measured distribution of normal magnetic flux density on the end face of EVP is shown in Fig. 8b.
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X. Fu et al. / Sensors and Actuators A 150 (2009) 12–23
Fig. 12. Convergence characteristic of the series expansion for
5.2.3. Measurement of earth magnetic field Earth magnetic field will destruct the rules given by Eq. (5), thus it should be measured and excluded from the excited magnetic field. The components on every direction are measured and shown in Fig. 9. 5.3. Calculation of the mathematic model
II (,
, z).
(iv) The convergences of the series expansions in the Eqs. (17) of ˜ ˜ U(, , z), (27) of F(, , z) and (35) of II (, , z) are key characteristics for the calculation. If there are numerous significant terms needed in an expansion, the entire formalism is of no use at all. Thus, the convergence characteristics are especially studied under the measured boundary conditions and the results at several points are shown by figures as examples. The char˜ acteristics of U(, , z) at points (0 mm, 0◦ , 212 mm), (15 mm,
A function is built and run in the software Matlab according to above built mathematic model to calculate the magnetic field around the actual EVP. There are several notes of the calculation: (i) The values for the size parameters of the calculation geometry are as follows: r1 = 30 mm, r2 = 150 mm, L1 = 210 mm and L2 = 410 mm. The value of r1 is determined according to the size of the actual EVP. The values of r2 , L1 and L2 are determined through measurement using the gauss meter. Enough large values are selected for r2 , L1 and L2 to ensure the measured data of normal magnetic flux density on the outer surface of the calculation geometry are close to zero. (m) (ii) A table of n with different values of variables m and n is built by the software Mathematica beforehand. During the cal(m) ˜ culation of U(, , z), the value of n is accessed through the method of look-up table. (iii) Trapezoidal method is used for the integrations of above built mathematic model.
Fig. 13. Location of the compared lines.
X. Fu et al. / Sensors and Actuators A 150 (2009) 12–23
Fig. 14. The compared results of the calculated and measured data.
21
22
X. Fu et al. / Sensors and Actuators A 150 (2009) 12–23
0◦ , 212 mm) and (30 mm, 0◦ , 212 mm) are shown in Fig. 10a–c. ˜ The characteristics of F(, , z) at points (0 mm, 0◦ , 212 mm), (15 mm, 0◦ , 212 mm) and (30 mm, 0◦ , 212 mm) are shown in Fig. 11a–c. The characteristics of II (, , z) at points (30 mm, 0◦ , 212 mm), (70 mm, 0◦ , 212 mm) and (100 mm, 0◦ , 212 mm) are shown in Fig. 12a–c. All of the examples show that the values of significant terms decay to zero quickly as the values of the variables m and n increase. This means a good convergence characteristic of the built mathematic model. 5.4. Experimental validation The calculated and experimentally measured distributions of the z-directional component of the magnetic filed on several lines are compared to validate the approach. This is for the convenience of experimental measurement and also due to that the z-directional component is the major effective component during flow velocity measurement using EVP. Six comparison lines whose lengths are all equal to 200 mm are shown in Fig. 13. Lines 1 and 2 locate at z = 212 mm, i.e. the distance from the end face of EVP is 2 mm. Lines 3 and 4 locate at z = 220 mm, i.e. the distance is 10 mm. Lines 5 and 6 locate at z = 260 mm, i.e. the distance is 50 mm. Lines 1, 3, 5 are on the x–z plane and parallel to the x-axis. Lines 2, 4, 6 are on the y–z plane and parallel to the y-axis. The distance between two measuring points on every line is 1 mm. The compared results of the calculated and measured data on the lines 1–6 are shown in Fig. 14. Since the divisionally analytical solution method used, the calculated result on every line is composed of several parts. In every figure, the two side parts where x(or y) = −30 to −100 mm and x(or y) = 30–100 mm express the result in geometry II, which are calculated according to the Eq. (35) of II (, , z). The middle part where x(or y) = −30 to 30 mm expresses the result in geometry I. According to the Eq. (34) of I (, , z), it is ˜ ˜ a sum of the functions U(, , z), F(, , z), u(, , z) and f(, , z). These four component functions are further calculated according to Eqs. (17), (27), (13) and (15), respectively. Thus, although only the summed results ∂1 /∂z are compared with measured data, results ˜ ˜ of the component functions ∂U/∂z, ∂F/∂z, ∂u/∂z and ∂f/∂z are also shown in the figures. The calculated results show good accuracy but not perfect compared with the experimentally measured data. Small relative errors and some discontinuities of the calculated data at the connecting points of geometry I and II are still observable. This is because the measuring errors of the boundary conditions are avoidless although the mathematic model has shown a good calculation characteristic in above study. During the measurement, the possible errors caused by the gauss meter and electrical motion device are small as mentioned in Section 4. The measuring error is mainly due to an imprecise manual relative location of the EVP and the probe of gauss meter. Using some special devices for example optic localizers can further improve the accuracy. 6. Conclusion and further work In the paper, a novel divisionally analytical reconstruction approach to determine the magnetic field around an EVP is introduced, including its principle, mathematic model, measuring device and experimental validation on an actual EVP. Its principle is calculating the magnetic field from the measured boundary conditions according to a mathematic model. The mathematic model is built through solving the Laplace’s equation of magnetic scalar potential using a divisionally analytical method. The complex calculation geometry is divided into two simple ones using an auxiliary surface, and then the Laplace’s equation in the each generated geometry is analytically solved by a method of separation of vari-
ables. A magnetic scanning device is designed to determine the distributions of the normal component of magnetic field on the auxiliary surface and the surface of probe including its installing pole, and hence to provide the boundary conditions required by the built mathematic model. Finally, the novel approach is experimentally validated through a calculation of the magnetic field around an actual EVP. The measured boundary conditions, the calculation characteristics of the mathematic model, the comparison between the calculated and experimentally measured results on several lines, are all given and discussed. The approach introduced in this paper is suitable to be used in the dry calibration of EVP due to its advantages: no numerous measurements needed unnecessary to know the inner magnetic exciting structure of the probe, high accuracy and convenient for further calculations. In the paper, the boundary conditions are simply measured at adequately dense and uniformly distributed points, and this causes a waste of the measuring time (about 5 h needed). Since series expansions are used in the mathematic model, sampling theory is expected to be introduced to access a more economical distribution of the measuring points in further study. Methods of surface fitting are also possibly used to access a nonuniform distribution of the measuring points. In further work, more suitable integration methods are also necessary to be studied and then replace the simple trapezoidal method used in this paper. In the paper, an extended cylindrical surface is used to divide the calculation geometry. However, this is not the unique division way. Other division ways such as using a plane which overlaps the end face of EVP are also feasible. Obviously, different mathematic models and distributions of boundary condition measuring points are required by different division ways. Comparisons between various division ways will be done in further study to determine the best division way for different kinds of EVP. Finally, the reconstruction approach to determine the magnetic field studied in this paper can also be used for other electromagnetic devices and has a lot of applications in physics, chemistry and engineering. The divisionally analytical solution method of the Laplace’s equation in the complex geometry is also a worth considering method for other similar solution problems in complex domains. Acknowledgments The authors are grateful to the financial support of the National Basic Research Program (973) of China (No. 2006CB705400), the National Key Technologies Research & Development Program of China (No. 2005DFBA0008) and Key Industrial Program of Zhejiang Province of China (No. 2008C11G2010077). References [1] J.A. Shercliff, The Theory of Electromagnetic Flow-Measurement, Ist ed., Cambridge University Press, Cambridge, UK, 1962. [2] M.K. Bevir, Theory of induced voltage electromagnetic flow measurement, IEEE Transactions on Magnetics MAG-6 (2) (1970) 315–320. [3] A. Michalsk, J. Starzynski, Optimal design of the coils of an electromagnetic flowmeter, IEEE Transactions on Magnetics 34 (5) (1998) 2563–2566. [4] A. Michalsk, Optimal shape of an electromagnetic flow gauge, IEEE Transactions on Magnetics 24 (1) (1998) 565–568. [5] A. Michalsk, J. Starzynski, S. Wincenciak, 3-D approach to designing the excitation coil of an electromagnetic flowmeter, IEEE Transactions on Instrumentation and Measurement 51 (4) (2002) 833–839. [6] A. Michalsk, S. Wincenciak, Method of optimization of primary transducer for electromagnetic flow meter, IEEE Instrumentation and Measurement Technology Conference (1996) 1350–1353. [7] J.Z. Wang, Numerical simulation modelling for velocity measurement of electromagnetic flow meter, International Symposium on Instrumentation Science and Technology, Journal of Physics: Conference Series 48 (2006) 36–40. [8] J.Z. Wang, C.L. Gong, Sensing induced voltage of electromagnetic flow meter with multi-electrodes, in: Proceedings of the 2006 IEEE International Conference on Information Acquisition, 2006, pp. 1031–1036.
X. Fu et al. / Sensors and Actuators A 150 (2009) 12–23 [9] H.F. Wu, Y. Yan, H.P. Wu, Research on electromagnetic flow meter modeling based on ANSYS, Chinese Journal of Scientific Instrument 29 (2) (2008) 372–376. [10] X.T. Qiao, L.J. Xu, F. Dong, Design of exciting coils for multi-electrode electromagnetic flowmeter, Chinese Journal of Scientific Instrument 23 (2) (2002) 867–869. [11] Y. Jin, Q. Zhou, X. Fu, The development of an insertion electromagnetic flowmeter, Chinese Journal of Industrial Instrumentation and Automation 14 (1) (2006) 63–64. [12] X.Z. Zhang, J. Hemp, Measurement of pipe flow by an electromagnetic probe, ISA Transactions (33) (1994) 181–184. [13] J. Hemp, Theory of a simple electromagnetic velocity probe with prediction of the effect on sensitivity of a nearby wall, Measurement Science and Technology (6) (1995) 376–382. [14] W.C. Cai, Z.Y. Ma, G.F. Qu, S.L. Wang, Electromagnetic Flowmeters, Ist ed., Petrochemical Press, Beijing, China, 2004. [15] N. Ida, J.P.A. Bastos, Electromagnetics and Calculation of Fields, 2nd ed., Springer Press, London, England, 1997. [16] G.D. Smith, Numerical Solution of Partial Differential Equations, 3rd ed., Oxford University Press, Oxford, UK, 1985. [17] G. Evans, J. Blackledge, P. Yardley, Numerical Methods for Partial Differential Equations, Ist ed., Springer Press, London, England, 2000. [18] K.W. Morton, D.F. Mayers, Numerical Solution of Partial Differential Equations, 2nd ed., Cambridge University Press, Cambridge, UK, 2005.
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[19] X.Z. Zhang, J. Hemp, Calculation of the virtual current around an electromagnetic velocity probe using the alternating method of Schwarz, Flow Measurement and Instrumentation 5 (3) (1994) 146–149. [20] K.M. Liang, Mathematic and Physical Methods, 3rd ed., Higher Education Press, Beijing, China, 1998. [21] Z.L. Xu, Modern Mathematics Handbook-Classical Mathematics, Ist ed., Huazhong University of Science and Technology Press, Wuhan, China, 2000.
Biographies X. Fu is currently a professor and director in State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, China. His main research areas are flow measurement, computational fluid dynamic in hydraulic components, fluid power transmission, MEMS and vibration/noise control in hydraulic valves. He has published more than 100 papers. L. Hu received the MS degree in mechanical engineering from Zhejiang University, China, in 2004. He is currently working toward the PhD degree in area of flow measurement at Zhejiang University, China. J. Zou received the PhD degree from the Zhejiang University, China, in 2006. From then on his research focused on flow measurement and applied fluid mechanics. He is author or coauthor of about 20 journal and conference papers and several patents.
Contents lists available at ScienceDirect
Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
Divisionally analytical reconstruction of the magnetic field around an electromagnetic velocity probe X. Fu, L. Hu, J. Zou ∗ , H.Y. Yang, X.D. Ruan, C.Y. Wang The State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou, China
a r t i c l e
i n f o
Article history: Received 31 July 2008 Received in revised form 19 November 2008 Accepted 28 November 2008 Available online 11 December 2008 Keywords: Electromagnetic velocity probe EVP Dry calibration Laplace’s equation Magnetic field Separation of variables Analytical solution Reconstruction
a b s t r a c t Determination of the magnetic field around an electromagnetic velocity probe is a key work for the dry calibration of the probe. In the paper, a novel divisionally analytical reconstruction approach to determine the magnetic field is introduced. It is especially suitable to be used in the dry calibration due to following advantages: no numerous measurements needed, unnecessary to know the inner magnetic exciting structure of the probe, high accuracy and convenient for further calculations. It is a measurement and calculation joint approach, in which the magnetic field is calculated from the measured boundary conditions through solving a Laplace’s equation of magnetic scalar potential. To obtain an analytical solution of the Laplace’s equation, the complex calculation geometry around the probe is divided into two simple ones using an auxiliary surface, and then the Laplace’s equation in the each generated geometry is analytically solved through a method of separation of variables. The divisionally analytical solution is pre-requisite since a pure analytical solution can bring convenience for further calculations in the dry calibration and such a solution is still lacked due to the complexity of the calculation geometry. A magnetic scanning device used to measure the distributions of the normal component of magnetic field on the surface of probe and the auxiliary surface is also introduced. The measured data provide the required boundary conditions for the determination of the magnetic field. Finally, the novel approach is validated through reconstructing the magnetic field around an actual electromagnetic velocity probe and comparing the results with the experimentally measured data. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Electromagnetic velocity probes (EVPs) are based on Faraday’s law of electromagnetic induction. As shown in Fig. 1, a magnetic field is built around a probe by its inner magnetic exciting unit and a voltage signal will be induced between the electrodes by conductive liquid moving through the built magnetic field. The relation of the induced voltage U, flow velocity v and magnetic flux density B is [1]
v(B × W )d
U =
(1)
Here, is the measuring volume, and W is the weight function, which expresses how the induced voltages contribute to the output signal of electrodes from different positions and is also termed by Bevir [2] as the “virtual currents j”. According to above relation, the sensitivity of the probe (U/¯v) can be mathematically calculated if the distributions of magnetic
∗ Corresponding author. Tel.: +86 571 87953395; fax: +86 571 87953395. E-mail address: [email protected] (J. Zou). 0924-4247/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2008.11.032
field, weight function and flow velocity are all known. This calculation approach to determine the sensitivity of EVP is called as dry calibration, which is very economical compared with liquid calibration methods using flow rigs as no actual liquid flow needed. It is clear that the determination of the magnetic field in the measuring volume is a key work for the dry calibration. At present, measurements and calculations are commonly used approaches to determine the magnetic field in the study of EVP. Magnetic measuring apparatus such as gauss meter can measure the magnetic flux density at a point accurately. However, directly measuring the distribution of the magnetic field around an EVP is still cumbersome, because all the x, y and z-directional components at numerous points need to be measured in a 3dimensional measuring volume. Low efficiency, complex position apparatus and additional location errors are all unavoidable disadvantages. Calculations from the inner exciting structure according to Maxwell’s equations are also usually used to determine the magnetic field. There are various numerical and analytical calculation approaches. Numerical calculations of magnetic field from the exciting units are commonly used approaches in study of electromagnetics and its applications now, which also includes electromagnetic flow measurements [3–11]. However, the exciting units of EVP are composed of coils, cores, air gap and sometimes
X. Fu et al. / Sensors and Actuators A 150 (2009) 12–23
13
pole. The radius of the larger cylinder is r2 and length is L2 . The radius of the small cylinder is r1 and length is L1 . The surfaces of the large cylinder are remote enough from the probe until the normal magnetic flux density on the surfaces can be ignored. Thus, the calculation geometry equals to the large cylinder subtracted by the small cylinder of EVP and its installing pole. Since the magnetic field induced by the moving liquid is negligible compared with the magnetic field built by the inner exciting unit of EVP [14], there are no currents in the calculation geometry. Magnetic scalar potential can be used to calculate the magnetic field in the geometry. The potential obeys Laplace’s equation [15].
∇2
=0
(2)
Thus, we can determine the distribution of magnetic flux density B according to the following relation if the Laplace’s equation can be solved. B = −∇
Fig. 1. Electromagnetic velocity probe.
magnetic yokes. It is difficult to accurately model the size of coils, structure of air gap, assembly errors and material properties such as B–H curves for every EVP. Thus, these numerical models are not accurate enough and even voided when the inner structures of probe are unknown. Another obvious shortcoming of the numerical approaches is lacking of convenience for further joint calculations in the dry calibration. Therefore, the numerical approaches are usually used by the manufactures of EVP during design and optimization of the magnetic exciting performance of probes. They require knowing the general distributions of magnetic fields both in and out of probes, but relative low accuracies are acceptable. Calculations based on simplifying the complex exciting units to a circular [12] or square [13] circuit which have analytical solutions of the excited fields are analytical approaches. Obviously, the analytical mathematic models are convenient for further calculations but lack of accuracy because of the oversimplification. Thus, this method is used in prediction of the effect on the sensitivity of EVP by a nearby wall [12,13], which requires relative low accuracy in the determination of magnetic field but good convenience of further joint calculations with weight function and velocity profile mathematic models. In a word, the existing pure measurement and calculation approaches are not suitable for the determination of the magnetic field in the dry calibration of EVP. A suitable approach should have characteristics: no numerous measurements needed, unnecessary to know the inner magnetic exciting structure of the probe, high accuracy and convenient for further calculations. In following sections, a novel measurement and calculation joint approach to determine the magnetic field around an EVP is introduced and experimentally validated. This approach is especially suitable to be used in the dry calibration.
(3)
Here, the constant is the magnetic permeability of the media (air usually) in the measuring volume. Boundary conditions are necessary to solve the Laplace’s equation. Since the distribution of potential on the surface of EVP and its installing pole is immeasurable, solving the equation under Dirichlet boundary conditions is unrealizable. Thus, Neumann boundary conditions are used, as the distribution of the normal derivative of potential (normal magnetic flux density) on the surface can be measured by a magnetic scanning apparatus introduced in following sections. Thus, the boundary conditions on the surface of EVP including its installing pole are
∂ ∂
=− =r1
Bn(0−L1 )
,
0 ≤ z ≤ L1 ;
∂ ∂z
=− z=L1
0 ≤ ≤ r1
(4)
Here, Bn(0−L1 ) expresses the distribution of normal magnetic flux density on the side face of EVP including its installing pole, Bz expresses the distribution on the end face of EVP. As mentioned above, the large cylinder is built to be large enough, thus the remain-
2. Laplace’s equation in terms of magnetic scalar potential The measuring volume out of an EVP which is also called as calculation geometry in following calculations is shown in Fig. 2. For convenience of mathematic description, a large cylinder is built to enclose a small cylinder composed of the EVP and its installing
Bz ,
Fig. 2. Calculation geometry out of EVP.
14
X. Fu et al. / Sensors and Actuators A 150 (2009) 12–23
where
ing boundary conditions are
∂ = 0, 0 ≤ z ≤ L2 ; ∂ =r2 ∂ ∂ = 0 r ≤ ≤ r 1 2 ∂z ∂z z=0
Bn =
=0
0 ≤ ≤ r2
(5)
z=L2
3. Divisionally analytical solution of the Laplace’s equation Direct analytical solution of the Laplace’s Eq. (2) is till unreachable due to the complexity of the calculation geometry. At present, numerical methods [16–18] are commonly used approaches for this kind of PDE solution problems. A semi-analytical solution approach also has been introduced by Zhang and Hemp [12,19] to solve a similar problem of weight function. However, they all lack convenience of further calculations if used in dry calibration. The obtained distribution of magnetic field B should be further cross multiplied with the distribution of weight function W and point multiplied with the distribution of flow velocity v, and finally a volume integral of the multiplication result should be calculated in the measuring volume to access the sensitivity of the probe. These are all further calculations in dry calibration mentioned in this paper. Obviously, multiplications and integral should be done numerically in a 3D volume if numerical approaches used. It will waste many calculation times compared with an analytical approach, since many analytical simplifications can be done during the further calculations if the distributions of W, B and v are all expressed in analytical mathematical expressions. In this section, the complex calculation geometry is divided into two simple ones using an auxiliary surface, and then the Laplace’s equation in the each generated geometry is analytically solved through a method of separation of variables. The cost of the pure analytical solution is an additional requirement of the boundary condition on the auxiliary cylinder surface, which can be determined through measurement introduced in following sections. 3.1. Division using an extended cylindrical surface An extended cylindrical surface is used to divide the complex calculation geometry into two simple ones, i.e. a cylinder and an annulus. The outlines of the division surface are showed in Fig. 3a using broken lines. The generated cylinder and annulus are showed in Fig. 3b as geometry I and Fig. 3c as geometry II, respectively. Potential is defined as potential I in geometry I and II in geometry II. Function Bn(L1 −L2 ) is predefined to express the distribution of normal magnetic flux density on the extended surface. Thus the boundary conditions to solve Laplace’s equations of potential II in geometry II are sufficient and listed as I in geometry I and follows Geometry I :
0 ≤ ≤ r1 ;
∇2
I
∂ I ∂
= 0;
=r1
∂ I ∂z
Bn =− ,
=− z=L1
Bz ,
∇2
II
0 ≤ z ≤ L2 ;
∂ II ∂z
∂ II ∂
=− =r1
= 0, z=0
I (, , z)
∂ II ∂z
=0 z=L2
∂ II ∂
= U(, , z) + F(, , z)
∇ U = C; 2
∂U ∂z
∂U ∂
0 ≤ ≤ r1
=r2
r1 ≤ ≤ r2
(7)
∂U ∂z
Bz =− , = 0,
L1 ≤ z ≤ L2
∂F ∂
0 ≤ ≤ r1
= 0, z=L2
(10)
=r1
∂F ∂z
∇ 2 F = −C;
∂F ∂z
= 0, z=L1
=− =r1
Bn
= 0, z=L2
L1 ≤ z ≤ L 2
(11)
The constant C can be determined according to Gauss theorem.
Bz ds SendI
C=
Bn(L1 −L2 ) ds =
VI
SsideI
(12)
VI
Here, VI is the volume of geometry I, SendI is the area of the end face, SsideI is the area of the side face. Particular solutions u(, , z) and f(, , z) are given to change the Poisson’s Eqs. (10) and (11) to Laplace’s equations. ˜ + u; U=U
u=
∇ U˜ = 0;
˜ ∂U ∂z
C (z − L2 )2 2
2
z=L1
(13)
Bz + C(L2 − L1 ), =−
˜ ∂U ∂
0 ≤ ≤ r1 ;
=0
C f = − 2 4
∇ F = 0;
∂F˜ ∂z
∂F˜ ∂
=− =r1
˜ ∂U ∂z
= 0, z=L2
L1 ≤ z ≤ L2
(14)
=r1
F = F˜ + f ;
= 0,
z=L1
(9)
(6)
Bn ,
(8)
3.2.1. Solution of I To apply the method of separation of variables, homogeneous boundary conditions are necessary to determine eigenvalues. However, there are non-homogeneous boundary conditions both on side and end faces of geometry I as solving I (, , z) via Eq. (6). The approach used here is breaking the function I (, , z) up into the sum of two simpler functions, U(, , z) and F(, , z). U(, , z) has homogeneous boundary conditions on the side face. F(, , z) has homogeneous boundary conditions on the end face.
2˜
= 0;
0 ≤ z ≤ L1 L1 ≤ z ≤ L2
3.2. Analytical solution by the method of separation of variables
= 0, z=L2
L1 ≤ z ≤ L2
Geometry II :
∂ I ∂z
Bn Bn
(15)
= 0, z=L1
1 Bn + Cr1 , 2
∂F˜ ∂z
= 0,
0 ≤ ≤ r1 ;
z=L2
L1 ≤ z ≤ L2
(16)
The Laplace’s Eqs. (14) and (16) can be directly solved by the method of separation of variables.
X. Fu et al. / Sensors and Actuators A 150 (2009) 12–23
15
Fig. 3. (a) Division using an extended cylindrical surface. (b) Geometry I: a simple cylinder. (c) Geometry II: a simple annulus.
˜ The solution of U(, , z) is ˜ ˜c + U(, , z) = U
×
×
×
×
AmnI · cos m · Jm
m=0 n=1
(m)
n (z − L1 ) + BmnI · cos m · Jm r1 n (z − L1 ) r1
(m)
(m) n
(m)
n r1
(m)
n r1
+CmnI · sin m · Jm
n (z − L1 ) + DmnI · sin m · Jm r1
−
(m)
−
where
∞ ∞
(m)
(m)
· exp
fmn = exp
· exp
εm =
· exp
r1
2
AmnI = emn
−
×
0
0 (m)
n r1
⎧ ⎨1
1 ⎩2 0
r1
2
−
×
0
0
+ C(L2 − L1 ) · cos m · Jm
(18)
r1
r1
2
B z −
×
0
0
r1 r1
(m)
for
when
m=0
for CmnI and DmnI
AmnI and BmnI
(24)
(25)
A = (m + 1/2 + 2n) /2, B = 4m2 , C = 7B2 + 83B − 9, D = 83B3 + 207B2 − 3537, . . .; n = 1, 2, 3. . . ¯ n2 is the modulus of the Bessel function [21]. N
r1 2 Jm
(m)
n r1
d =
r12
2
1−
m
2
(m)
n
(m)
2 Jm ( n )
(26)
∞
∞ ∞
+
+ C(L2 − L1 ) · sin m · Jm
(EmI cos m + FmI sin m)m
m=1
GmnI · cos
m=0 n=1
+HmnI · cos
2
B z −
0
n r1
m= / 0 m=0
(20)
n
(z − L1 ) · cos m · Im L2 − L1
n
(z − L1 ) · sin m · Im L2 − L1
n
L2 − L1
n
L2 − L1
(27)
The expressions of EmI , FmI , GmnI , HmnI are
0
when
(19)
dd
DmnI = fmn
(23)
˜ The solution of F(, , z) is
(m) n
+ C(L2 − L1 ) · cos m · Jm
dd
· emn
when
˜ F(, , z) = F˜c +
CmnI = emn
(L2 − L1 )
(m)
n (L2 − L1 ) r1
0
(m) n
×
B z
(22)
C B+3 D − − −··· 3 8A 15(4A)5 6(4A)
=A−
¯ n2 = N
(m)
(m)
BmnI = fmn
(m)
n r1
tion of first kind and its derivative. n can be accessed through a method of look-up table or calculated through [20]. n
dd
2
1 − exp2
( r ), i.e. the intersecting points of are positive zeros of Jm n 1 (x) are ordinary Bessel funcJm−1 ( n r1 ) and Jm+1 ( n r1 ). Jm (x) and Jm
(17)
B z
·
εm
(m) n r1
(m) n
The expressions of AmnI , BmnI , CmnI and DmnI are
2 · Nn
· exp
(z − L1 )
r1
emn =
n r1
n r1
+ C(L2 − L1 ) · sin m · Jm
EmI =
dd
(21)
FmI =
εm
2 L2 0
L1
− Bn + 12 Cr1 cos mdzd m−1
εm
1 2 L2(L2 −BnL1 ) mr 1 0
L1
− + 2 Cr1 sin mdzd
(L2 − L1 ) mr1 m−1
(28) (29)
16
X. Fu et al. / Sensors and Actuators A 150 (2009) 12–23
2
L2
GmnI = gmn
B n −
L1
0
2
· cos
n
z · cos mdzd L2 − L1 (30)
L2
HmnI = gmn
B n −
L1
0
1 Cr1 2
+
+
1 Cr1 2
· cos
n
z · sin mdzd L2 − L1 (31)
where gmn =
εm =
εm (n /L − L )(r ) (n 2 /2)Im 2 1 1
⎧ ⎨1 ⎩
1 2 0
(32)
m= / 0
when when
m=0
for
EmI and GmnI
when
m=0
for
FmI and HmnI
(33)
(x) are virtual Bessel function of first kind and its Im (x) and Im derivative. Finally, I (, , z) can be calculated according to I (, , z)
˜ ˜ = U(, , z) + F(, , z) + u(, , z) + f (, , z)
(34)
3.2.2. Solution of II Since there are homogeneous boundary conditions on the end faces of geometry II, the Laplace’s Eq. (7) can be directly solved by the method of separation of variables. The solution is II (, , z) =
c+
(AmII cos m + BmII sin m)m
+(CmII cos m + DmII sin m) ∞ ∞
+
m=0 n=1
n
n
z · sin m · Im ( ) +GmnII · cos L2 L2
AmII = −
BmII = −
CmII = −
2 L2 0
0
2 L2 0
0
n
n
z · sin m · Km ( ) L2 L2
Bn cos mdzd
εm
2 L2 0
0
2 L2
L2
Bn cos 0
0
εm
2m
⎧ ⎨1 ⎩
1 2 0
when when when
m=0
m=0
m= / 0
AmII CmII EmnII and FmnII
(35)
˜ c , F˜c and c 3.2.3. A note of the constant items U ˜ c , F˜c and c in above solutions are unavailThe constant items U able using the given Neumann boundary conditions. However, this will not affect using above solutions to calculate the magnetic field, because the gradient of the constant items will be equal to zero during the calculation.
(38) ]
(40)
FmnII = hmn Im (
n 2 r2 ) · L2 n2
( GmnII = −hmn Km
0
n
2 r2 ) · L2 n 2
2
L2
0 2
Bn cos L2
n zcos mdzd L2
Bn cos 0
0
(45)
for BmII DmII GmnII and HmnII
0
(n /L )(r )K (n /L )(r ) − K (n /L )(r )I (n /L )(r ) Im 2 2 m 2 1 2 2 m 2 1 m
(37)
− (1/r1 )
n
z sin mdzd (43) L2
hmn =
]
Bn cos mdzd
2m L2 m · r1 m−1 [(1/r2 )
εm
2m
2
(36)
Bn sin mdzd
L2 m · r1 m−1 [1 − (r2 /r1 )
(x) are virtual Bessel function of second kind and Km (x) and Km its derivative, respectively.
2m L2 m · r1 m−1 [1 − (r2 /r1 ) ]
εm
2 n 2
Bn sin mdzd 0 (39) m−1 L2 m · r1 [(1/r2 )2m − (1/r1 )2m ] 2 L2 2 n n −hmn Km r2 · Bn cos zcos mdzd L2 L2 n2 0 0
DmII = − EmnII =
εm
·
where
εm =
The expressions of AmII , BmII , CmII , DmII , EmnII , FmnII , GmnII and HmnII are
L2
r2
(44)
n
n
z · cos m · Km ( ) L2 L2
+ HmnII · cos
n
−m
n
n
EmnII · cos z · cos m · Im ( ) L2 L2
+FmnII · cos
HmnII = hmn Im
−
∞ m=1
Fig. 4. Magnetic scanning apparatus.
(41)
n
z sin mdzd(42) L2
Fig. 5. Probe of gauss meter.
X. Fu et al. / Sensors and Actuators A 150 (2009) 12–23
17
Fig. 6. The measured EVP.
4. Measuring device of the boundary condition
series port to control the auto scanning procedure using a scanning program.
A magnetic scanning apparatus shown in Fig. 4 is designed to measure the distributions of normal magnetic flux density on the auxiliary cylinder surface, the side and end faces of the probe including its installing pole, i.e. Bn(L1 −L2 ) , Bn(0−L1 ) and Bz . They are necessary boundary conditions to solve above Laplace’s equations. A probe A of gauss meter is fit on a beam B and droved by a scanning mechanism composed of a dovetail slot C, a rectangular block D, an electrical rotary unit E, a manual lifting unit F, two electrical moving units G and H. Probe A is controlled by a gauss meter I. Units E, G and H are controlled by a motion controller J. Computer K is connected with the gauss meter I and the controller J through
4.1. The gauss meter The magnetic sensor is a hall-effect gauss meter. As shown in Fig. 5, the diameter of the hall-effect area is about 1 mm and locates at the end of probe A. Only the average of the normal magnetic flux density passing through this small area is measured. The beam B is used to reinforce and protect the probe A. The material of the stem of probe A and beam B is aluminum, whose magnetic permeability is very close to air, and thus will not distort the magnetic field. All above designing ensure a high measuring precision of mag-
Fig. 7. Measurement of Bn(0−L1 ) and Bn(L1 −L2 ) .
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X. Fu et al. / Sensors and Actuators A 150 (2009) 12–23
Fig. 8. Measurement of Bz (, ).
netic field. The maximum measuring error for DC magnetic field is ±0.20% of reading and for AC magnetic field is ±2.0% of reading. 4.2. The scanning mechanism A total of four freedoms of motions can be realized by the scanning mechanism, which are a rotation motion realized by electrical rotary unit E, three linear motions in x, y, z directions realized by electrical moving units G, H and manual lifting unit F. Electrical rotary unit E and moving unit G together drive the probe A of gauss meter to measure the distributions of normal magnetic flux density on the auxiliary cylinder surface and the side face of EVP including its installing pole. Electrical moving unit G and H together drive the probe A of gauss meter to measure the distribution of normal magnetic flux density on the end face of EVP. The manual lifting unit F is used to adjust the coaxiality between rotary unit E and EVP before measurements on the side face and the auxiliary cylinder surface, and to adjust the height of probe A to keep the probe in same plane with the end face of EVP before measurement on it. The dovetail slot C is used to adjust the diameter of scanned cylinder surface. All the motions are droved by inner stepper motors, which are controlled by the controller J. The motion range of unit E is 0–360◦ , G and H are 300 mm. The maximum location error of unit E is 0.005◦ , G and H are 5 m. 4.3. The scanning program The auto scanning procedure is under the control of a scanning program. The control parameters include positions of the measuring points, sampling times at every point, delay time between two measurements and so on. The measured data at every point will be listed in a file automatically by the program.
mentioned in the Section 4.1, the measuring error of gauss meter for AC magnetic field is much larger than for DC magnetic field. Thus, a DC exiting current that equal to the amplitude of the rectangular square exciting current (250 mA) is used to ensure the measuring precision. This change will not influent the determination of the sensitivity of EVP when the method is used in dry calibration. 5.2. Measurement of the boundary conditions 5.2.1. Measurements of Bn(0−L1 ) and Bn(L1 −L2 ) Electrical rotary unit E and moving unit G together drive the probe A of gauss meter to measure the distributions of normal magnetic flux density on the auxiliary cylinder surface and the side face of EVP including its installing pole. As shown in Fig. 7a, the measured EVP is fixed on an installing pole whose material is aluminum. The auxiliary cylinder surface and the side face of EVP including its installing pole together construct a measured cylinder surface whose outline is shown in the figure using broken lines. During the measurement, the hall-effect area of probe A should overlap with the section at every measuring point on the measured cylinder surface. The measurement starts from a point on the side face of the stalling pole where the measured magnetic flux density is close to zero, goes on loop and loop along the axis of EVP, and stops at a point on the auxiliary cylinder surface where the measured data are close to zero too. For the actual EVP measured in this paper, the length of the measured cylinder surface is 410 mm and diameter is 60 mm. The measured data on the measured cylinder surface are shown in Fig. 7b. It provides the required boundary condition Bn . The part z = 0–210 mm of the measured data is separately shown in Fig. 7c.
5. Validation on an actual EVP 5.1. The actual EVP and its exciting current The actual EVP measured in this paper is shown in Fig. 6a. It is a cylinder whose diameter is 60 mm and length is 80 mm. Two electrodes locate on its end face. The inner exciting unit of the EVP is shown in Fig. 6b. A flat shape and the magnetic yoke ensure enough strength of magnetic field near the electrodes. After the exciting unit is assembled into EVP, the inner space of EVP will be filled with glue for sealing. Thus, the exact inner structure parameters are unknown. A rectangular square exciting current is used by the EVP as measuring liquid velocity, for the purpose of generating an alternating magnetic field to avoid polarization on the electrodes. However, as
Fig. 9. Measurement of earth magnetic field.
X. Fu et al. / Sensors and Actuators A 150 (2009) 12–23
19
It expresses the required boundary condition Bn(0−L1 ) on the side face of EVP including its installing pole. The part z = 210–410 mm is separately shown in Fig. 7d. It expresses the required boundary condition Bn(L1 −L2 ) on the auxiliary cylinder surface. During above measurement, the step of linear motion along the axis of EVP is 2 mm, and rotary motion is 3◦ . Five measurements are done in 0.5 s at every point. Thus, a total of 24,720 points are measured, which costs about 4 h. 5.2.2. Measurement of Bz Electrical moving units G and H together drive the probe A of gauss meter to measure the distribution of normal magnetic flux
˜ Fig. 11. Convergence characteristic of the series expansion for F(, , z).
˜ Fig. 10. Convergence characteristic of the series expansion for U(, , z).
density on the end face. As shown in Fig. 8a, the probe A of gauss meter is very close and parallel to the end face of EVP. The measurement starts from the center of end face, goes on loop and loop along the radius and stops at the edge of end face. The measuring points construe a disc whose outline is shown in the figure using a broken line. The diameter of the measured disc is 60 mm. The step of linear motion on the direction of radius is 1 mm, and rotary motion is 3◦ . Five measurements are done in 0.5 s at every point. Thus, a total of 3600 points are measured, which costs about three quarters of an hour. The measured distribution of normal magnetic flux density on the end face of EVP is shown in Fig. 8b.
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X. Fu et al. / Sensors and Actuators A 150 (2009) 12–23
Fig. 12. Convergence characteristic of the series expansion for
5.2.3. Measurement of earth magnetic field Earth magnetic field will destruct the rules given by Eq. (5), thus it should be measured and excluded from the excited magnetic field. The components on every direction are measured and shown in Fig. 9. 5.3. Calculation of the mathematic model
II (,
, z).
(iv) The convergences of the series expansions in the Eqs. (17) of ˜ ˜ U(, , z), (27) of F(, , z) and (35) of II (, , z) are key characteristics for the calculation. If there are numerous significant terms needed in an expansion, the entire formalism is of no use at all. Thus, the convergence characteristics are especially studied under the measured boundary conditions and the results at several points are shown by figures as examples. The char˜ acteristics of U(, , z) at points (0 mm, 0◦ , 212 mm), (15 mm,
A function is built and run in the software Matlab according to above built mathematic model to calculate the magnetic field around the actual EVP. There are several notes of the calculation: (i) The values for the size parameters of the calculation geometry are as follows: r1 = 30 mm, r2 = 150 mm, L1 = 210 mm and L2 = 410 mm. The value of r1 is determined according to the size of the actual EVP. The values of r2 , L1 and L2 are determined through measurement using the gauss meter. Enough large values are selected for r2 , L1 and L2 to ensure the measured data of normal magnetic flux density on the outer surface of the calculation geometry are close to zero. (m) (ii) A table of n with different values of variables m and n is built by the software Mathematica beforehand. During the cal(m) ˜ culation of U(, , z), the value of n is accessed through the method of look-up table. (iii) Trapezoidal method is used for the integrations of above built mathematic model.
Fig. 13. Location of the compared lines.
X. Fu et al. / Sensors and Actuators A 150 (2009) 12–23
Fig. 14. The compared results of the calculated and measured data.
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X. Fu et al. / Sensors and Actuators A 150 (2009) 12–23
0◦ , 212 mm) and (30 mm, 0◦ , 212 mm) are shown in Fig. 10a–c. ˜ The characteristics of F(, , z) at points (0 mm, 0◦ , 212 mm), (15 mm, 0◦ , 212 mm) and (30 mm, 0◦ , 212 mm) are shown in Fig. 11a–c. The characteristics of II (, , z) at points (30 mm, 0◦ , 212 mm), (70 mm, 0◦ , 212 mm) and (100 mm, 0◦ , 212 mm) are shown in Fig. 12a–c. All of the examples show that the values of significant terms decay to zero quickly as the values of the variables m and n increase. This means a good convergence characteristic of the built mathematic model. 5.4. Experimental validation The calculated and experimentally measured distributions of the z-directional component of the magnetic filed on several lines are compared to validate the approach. This is for the convenience of experimental measurement and also due to that the z-directional component is the major effective component during flow velocity measurement using EVP. Six comparison lines whose lengths are all equal to 200 mm are shown in Fig. 13. Lines 1 and 2 locate at z = 212 mm, i.e. the distance from the end face of EVP is 2 mm. Lines 3 and 4 locate at z = 220 mm, i.e. the distance is 10 mm. Lines 5 and 6 locate at z = 260 mm, i.e. the distance is 50 mm. Lines 1, 3, 5 are on the x–z plane and parallel to the x-axis. Lines 2, 4, 6 are on the y–z plane and parallel to the y-axis. The distance between two measuring points on every line is 1 mm. The compared results of the calculated and measured data on the lines 1–6 are shown in Fig. 14. Since the divisionally analytical solution method used, the calculated result on every line is composed of several parts. In every figure, the two side parts where x(or y) = −30 to −100 mm and x(or y) = 30–100 mm express the result in geometry II, which are calculated according to the Eq. (35) of II (, , z). The middle part where x(or y) = −30 to 30 mm expresses the result in geometry I. According to the Eq. (34) of I (, , z), it is ˜ ˜ a sum of the functions U(, , z), F(, , z), u(, , z) and f(, , z). These four component functions are further calculated according to Eqs. (17), (27), (13) and (15), respectively. Thus, although only the summed results ∂1 /∂z are compared with measured data, results ˜ ˜ of the component functions ∂U/∂z, ∂F/∂z, ∂u/∂z and ∂f/∂z are also shown in the figures. The calculated results show good accuracy but not perfect compared with the experimentally measured data. Small relative errors and some discontinuities of the calculated data at the connecting points of geometry I and II are still observable. This is because the measuring errors of the boundary conditions are avoidless although the mathematic model has shown a good calculation characteristic in above study. During the measurement, the possible errors caused by the gauss meter and electrical motion device are small as mentioned in Section 4. The measuring error is mainly due to an imprecise manual relative location of the EVP and the probe of gauss meter. Using some special devices for example optic localizers can further improve the accuracy. 6. Conclusion and further work In the paper, a novel divisionally analytical reconstruction approach to determine the magnetic field around an EVP is introduced, including its principle, mathematic model, measuring device and experimental validation on an actual EVP. Its principle is calculating the magnetic field from the measured boundary conditions according to a mathematic model. The mathematic model is built through solving the Laplace’s equation of magnetic scalar potential using a divisionally analytical method. The complex calculation geometry is divided into two simple ones using an auxiliary surface, and then the Laplace’s equation in the each generated geometry is analytically solved by a method of separation of vari-
ables. A magnetic scanning device is designed to determine the distributions of the normal component of magnetic field on the auxiliary surface and the surface of probe including its installing pole, and hence to provide the boundary conditions required by the built mathematic model. Finally, the novel approach is experimentally validated through a calculation of the magnetic field around an actual EVP. The measured boundary conditions, the calculation characteristics of the mathematic model, the comparison between the calculated and experimentally measured results on several lines, are all given and discussed. The approach introduced in this paper is suitable to be used in the dry calibration of EVP due to its advantages: no numerous measurements needed unnecessary to know the inner magnetic exciting structure of the probe, high accuracy and convenient for further calculations. In the paper, the boundary conditions are simply measured at adequately dense and uniformly distributed points, and this causes a waste of the measuring time (about 5 h needed). Since series expansions are used in the mathematic model, sampling theory is expected to be introduced to access a more economical distribution of the measuring points in further study. Methods of surface fitting are also possibly used to access a nonuniform distribution of the measuring points. In further work, more suitable integration methods are also necessary to be studied and then replace the simple trapezoidal method used in this paper. In the paper, an extended cylindrical surface is used to divide the calculation geometry. However, this is not the unique division way. Other division ways such as using a plane which overlaps the end face of EVP are also feasible. Obviously, different mathematic models and distributions of boundary condition measuring points are required by different division ways. Comparisons between various division ways will be done in further study to determine the best division way for different kinds of EVP. Finally, the reconstruction approach to determine the magnetic field studied in this paper can also be used for other electromagnetic devices and has a lot of applications in physics, chemistry and engineering. The divisionally analytical solution method of the Laplace’s equation in the complex geometry is also a worth considering method for other similar solution problems in complex domains. Acknowledgments The authors are grateful to the financial support of the National Basic Research Program (973) of China (No. 2006CB705400), the National Key Technologies Research & Development Program of China (No. 2005DFBA0008) and Key Industrial Program of Zhejiang Province of China (No. 2008C11G2010077). References [1] J.A. Shercliff, The Theory of Electromagnetic Flow-Measurement, Ist ed., Cambridge University Press, Cambridge, UK, 1962. [2] M.K. Bevir, Theory of induced voltage electromagnetic flow measurement, IEEE Transactions on Magnetics MAG-6 (2) (1970) 315–320. [3] A. Michalsk, J. Starzynski, Optimal design of the coils of an electromagnetic flowmeter, IEEE Transactions on Magnetics 34 (5) (1998) 2563–2566. [4] A. Michalsk, Optimal shape of an electromagnetic flow gauge, IEEE Transactions on Magnetics 24 (1) (1998) 565–568. [5] A. Michalsk, J. Starzynski, S. Wincenciak, 3-D approach to designing the excitation coil of an electromagnetic flowmeter, IEEE Transactions on Instrumentation and Measurement 51 (4) (2002) 833–839. [6] A. Michalsk, S. Wincenciak, Method of optimization of primary transducer for electromagnetic flow meter, IEEE Instrumentation and Measurement Technology Conference (1996) 1350–1353. [7] J.Z. Wang, Numerical simulation modelling for velocity measurement of electromagnetic flow meter, International Symposium on Instrumentation Science and Technology, Journal of Physics: Conference Series 48 (2006) 36–40. [8] J.Z. Wang, C.L. Gong, Sensing induced voltage of electromagnetic flow meter with multi-electrodes, in: Proceedings of the 2006 IEEE International Conference on Information Acquisition, 2006, pp. 1031–1036.
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Biographies X. Fu is currently a professor and director in State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, China. His main research areas are flow measurement, computational fluid dynamic in hydraulic components, fluid power transmission, MEMS and vibration/noise control in hydraulic valves. He has published more than 100 papers. L. Hu received the MS degree in mechanical engineering from Zhejiang University, China, in 2004. He is currently working toward the PhD degree in area of flow measurement at Zhejiang University, China. J. Zou received the PhD degree from the Zhejiang University, China, in 2006. From then on his research focused on flow measurement and applied fluid mechanics. He is author or coauthor of about 20 journal and conference papers and several patents.