Modeling and evaluation of bobbin probe radial offset for eddy
current nondestructive testing of metallic tubes
Mengbao Fan, Binghua Cao
(China University of Mining and Technology, Xuzhou 221116, Jiangsu Province, China)
Abstract: Eddy current testing has been widely used to evaluate metallic tubes. In engineering, bobbin probes are
often used and required to work along the axis of the tube under test. However, alignment work can not be
performed in many cases, and probes probably deviate from the tube axis in the detection process, namely probe
offset. Based on the second-order potential (SOVP) theory, this work presents the analytical formulation for the
off-centered driver-pickup bobbin probe. Following that, simulations are carried out to investigate radial offset
impact on induced eddy current distribution in a conducting tube. Finally, the resulting signal errors are evaluated.
This research may provide some help to improve accuracy and reliability for eddy current nondestructive testing.
Keywords: eddy current testing, conducting tube, probe radial offset, modeling
1 Introduction
Heat exchanger tubes are used in a variety of industries for transferring heat to the fluid
circulating outside the tube. A steam generator (SG) is a typical heat exchanger used in nuclear
power plants. Over long period of service, corrosion may occur, and the consequent wall-thinning
might lead to a catastrophe. Maintaining structural integrity of SG tubes to a proper level is one of
the fundamental tasks for safe and economic operation of nuclear power plants with pressurized
water reactors. This requires in turn detection and evaluation of degradations that have occurred in
the SG tubes under operation.
Eddy current testing (ECT) has been so far regarded as one of the most effective methods for
evaluation of SG tubes [1-3]. Chen[1] et al establish a prediction model based on thin-skin theory for
fast and accurate calculation of minor defect in SG tubes. For high sensitivity, Kobayashi[2] et al
design a new coil with flux guide made of iron-nickel alloy, which leads to output voltage of
detector coil increased more than 100 times. Xin[3] et al propose a novel design of rotating
magnetic field eddy-current probe for evaluation of SG tubes. Simulation results demonstrate that
the proposed probe has compact configuration and higher speed compared to traditional bobbin
coil and rotating coils. Generally, bobbin probes are used and required to move along axis of the
conductive tubes for integrity detection. However, axes of the tube and coil are not always
perfectly aligned and correction of the impedance for the coil radial offset is necessary.
Unfortunately, little attention has until lately been paid to the problem above. Theodoulidis TP
[4]
introduces SOVP to analyze the wobble effect of bobbin probes. Following that, Skarlatos A
[5]
with his group again uses SOVP to compute the impedance change of bobbin coil in a metallic
tube with eccentric tube wall. Accurate measurements of inner radius and electromagnetic
properties of oil-well casings are essential in assessment of their condition. Based on the
Theodoulidis’s formulation, Vasic
[6-7]
reduces the effect of the coil wobble and makes
measurements more accurate by correction of the sampled signals.
In this paper, we will extend Theodoulidis’s model to driver-pickup bobbin coils, which is
more popular in practice. Then, radial offset effect on induced eddy current distribution and probe
signal will be investigated and evaluated by simulation.
2 Theory model
2.1 Formulation of Second Order Vector Potential
The geometry covered in this work is shown in Fig.1.
z
y
rd2
rd1
xd0
zd2 zd1
zp2
zp1
rp2
rp1
2
xp0
1
x
0
coils
ρ
o
d
c
(a) driver-pickup probe in tube
tube
(b) cross section
Fig.1 Offcentered driver-pickup probe located inside a metallic tube
The tube is considered as infinitely long, with inner radius d and wall thickness c. The
involved material is linear, isotropic and homogeneous with relative magnetic permeability ur and
electrical conductivity σ. As driver-pickup coils are usually wounded on the same skeleton, radial
offset distances are identical. The driver coil has height ld, inner radius rd1, outer radius rd2, and
turns Nd. The pickup coil has height lp, inner radius rp1, outer radius rp2, and turns Np.
Note that for applications to ECT, exciting frequency usually ranges from a few Hz to several
MHz. As a result, displacement current is negligible. According to the Maxwell’s equations,
ignoring stray capacitance and skin effect[8,9], the governing equation of SOVP in cylindrical
coordinates is expressed as:
W = Wa ez + ez ×∇Wb
(1)
Where, ez stands for unit vector of z direction, both of Wa and Wb are scalar potential and solutions
of the following Helmholtz equations.
Where,
∇ 2Wa + k 2Wa = 0
(2)
∇ 2Wb + k 2Wb = 0
(3)
k 2 = − jωµ0 µrσ .
With Separation Method, the solutions to Eqs.(2) and (3) are formulated as
∞
∞
Was = ∫
−∞
Waec = ∫
∞
Wb1 = ∫
∞
Ds K m (| α | r )e jωϕ e jω z dα
(4)
∑ Cec I m (| α | r )e jωϕ e jω z dα
(5)
m =−∞
∞
−∞
Wa1 = ∫
∑
∞
m =−∞
∞
∑ [Ca1I m (α1r )+ Da1Km (α1r )]e jωϕ e jω z dα
−∞
(6)
m=−∞
∞
∑ [Cb1I m (α1r ) + Db1K m (α1r )]e jωϕ e jω z dα
−∞
(7)
m=−∞
Wa 2 = ∫
∞
−∞
∞
∑
m =−∞
Da 2 K m (| α | r )e jωϕ e jω z dα
(8)
Where, Im and Km are modified Bessel functions of the first and second kind of order m,
respectively. α1 = α 2 + k 2 . Was and Waec are excitation source and eddy current field in the
region R0, respectively.
Ds characterizes the coil excitation field, independent of tube properties. The unknown
coefficients Cec, Ca1, Da1, Cb1, Db1 and Da2 can be determined by applying boundary conditions at
the positions ρ=d and ρ=d+c, respectively.
eρ ⋅ ( B2 − B1 ) = B2 ρ − B1ρ = 0
eρ × ( H 2 − H1 ) = ( H 2ϕ − H1ϕ )eϕ + ( H 2 z − H1z )ez = 0
Further, six equations are obtained and solved, and coefficients are expressed as
(9)
G1S − G2T
Ds
G3 S − G4T
(10)
Ca1 =
α 2 H1Cec + H 2 Ds
T
α12
(11)
Da1 =
α 2 L1Cec + L2 Ds
T
α12
(12)
Cec = −
Cb1 = (M1K12 + Q1M 2 K12' )Ca1 + (M1K12 + Q2 M 2 K12' )Da1
(13)
Db1 = −(M1I12 + Q1M 2 I12' )Ca1 − (M1I12 + Q2 M 2 I12' )Da1
(14)
Db1 =
α 2 I12Ca1 + K12 Da1
K02
α12
(15)
Where, all symbols are given in appendix.
2.2 Mutual impedance of driver-pickup probe
Currently, using coil as magnetic field sensor is still the most popular. The total mutual
impedance of driver-pickup probe is the sum of Mair (probe in air) and ∆M due to the tube. The
closed form of Mair[8,9] is.
∞
{
1
I (rd 2 , rd 1 ) I (rp 2 , rp1 ) ⎡2( z p 2 − z p1 ) + α −1 e
⎣
α
−α ( z − z )
−α ( z − z )
+ e p 2 d 1 −e p 1 d 1 ⎤ d α
⎦
jω M air = Kc ∫
0
5
−α ( zd 2 − z p1 )
−e
−α ( zd 2 − z p 2 )
(16)
}
Where, K c =
jωµ0 πΝ d Ν p
( zd 2 − zd 1 )(rd 2 − rd 1 )( z p 2 − z p1 )(rp 2 − rp1 )
,
x2
I ( x2 , x1 ) = ∫ x' J1 ( x ' )dx ' .
x1
∆M can be calculated using a formula derived by Auld[10]. For driver-pickup bobbin coils and
a tube, as shown in Fig.1, ∆M gives
jωΔM =
1
µ0 I 2
∫∫
SF
^
(-­‐ ρ ) ⋅ ( Eds × B p 0 − E p 0 × Bds )ρ dϕ dz
(17)
^
Where,
ρ is unit vector in the ρ direction and SF is the inner surface of the tube. Eds and Bds
stand for the driver source electric field intensity and magnetic flux density, respectively, in the
absence of the sample.
E p 0 and B p 0 are the same quantities in the region 0, when pickup coil
serves as the exciter with conducting tubes present.
After extensive work, Eq.(17) becomes
jωΔM =
− jω
µ0 I 2
∞
π
∫−∞ ∫−π
2
∂ 2Wdas ∂W pa 0 ∂ W pa 0 ∂Wdas
−
⋅
]ρ = d d dϕ dz
∂r ∂z ∂z
∂r ∂z
∂z
[
(18)
Further, the closed form to ∆M is represented as
∞ ∞
jωΔM = 2 K c ∫ ∑
(2 − δ m )
α
0 m=0
6
Acos (α )I m2 (α x0 )
(19)
M (α rd 2 , α rd 1 ) M (α rp 2 , α rp1 ) R(α , m)dα
Where, K c =
jωµ0 πΝ d Ν p
( zd 2 − zd 1 )(rd 2 − rd 1 )( z p 2 − z p1 )(rp 2 − rp1 )
,
x2
M ( x2 , x1 ) = ∫ x' I1 ( x ' )dx ' ,
x1
Acos (α ) =cos α ( z p 2 − zd 2 ) + cos α ( z p1 − zd 1 ) −cos α ( z p 2 − zd 1 ) −cos α ( z p1 − zd 2 ) ,
R(α , m) =
G1S − G2T
, is the generalized reflection coefficient
G3 S − G4T
[9]
, which contains the tube
properties (wall thickness c, permeability u and conductivity σ), irrespective of driver-pickup
properties. When radial offset distance equals to zero, Eq.(19) reduces to the expression from
Dodd and Deed.
2.3 Induced eddy current density in tube
In the region of tube, the induced eddy current density can be derived by
J ec = − jωσ A = − jω(∇×W )
(20)
By expanding Eq.(20) in cylindrical coordinate systems, the components of Jec becomes
J ρ ec = − jω (
Jϕ ec = jω(
Jz
ec
1 ∂Wa1 ∂ 2Wb1
−
)
ρ ∂ϕ ∂ρ∂z
∂Wa1 1 ∂ 2Wb1
+
)
∂ρ ρ ∂ϕ∂z
∂ 2Wb1 1 ∂Wb1 1 ∂ 2Wb1
= − jω (
+
+
)
∂ρ 2 ρ ∂ρ ρ 2 ∂ϕ 2
(21)
(22)
(23)
When there is no radial offset for probe, induced eddy current J has only ϕ component. Probe
radial offset brings ρ and z components, and of course, changes ϕ component of J.
3 Simulations and discussions
3.1 Probe and tube parameters
Tab. 1 Parameters of the probe and tube
tube properties
driver-pickup probe
aluminum
Inner radius r1/mm
3.0
raltive permeability ur
1
Outer radius r2/mm
5.0
conductivity σ/(MS/m)
38
height l/mm
4.0
inner radius d/mm
8
turns N
200
wall thickness c/mm
2
3.2 Distribution of induced eddy current density
The distribution of induced eddy current density plays a key role in ECT. In conventional
ECT of tubes, the eddy current density has a uniform distribution in the circumferential direction.
In contrast, probe radial offset destroys the circumferential uniformity, and makes eddy current
density gather closer to probe. Fig.2 shows the effect of radial offset in the x direction.
(a) no radial offset
(c) 1.0 mm radial offset
(b) 0.5 mm radial offset
(d) 2.0 mm radial offset
Fig.2 Illustration of radial offset effect on distribution of induced eddy current density
From Fig.3, it can been seen that increasing exciting frequency gather induced eddy current
in smaller local region for the same radial offset distance. As a result, high frequency perhaps
makes radial offset influence more obvious.
(a) 5 kHz
(b) 50 kHz
Fig.3 Radial offset effect on distribution of induced eddy current density under different frequencies
3.2 Impedance variation with radial offset distance
we have shown how the distribution of induced eddy current density varies with increase of
radial offset distance. In this section, we further investigate the influence of radial offset on probe
signal—coil impedance by simulations. The radial offset distance changes in the range of 0 mm to
3 mm. For ease of comparison, all the coil impedances are initialized. Fig. 4 shows the normalized
impedance diagrams at different frequencies.
0.96
normalized reactance
0.94
0.92
20 kHz
100 kHz
200 kHz
400 kHz
0.9
0.88
0.86
0.84
0.82
0.01
0.02
0.03
0.04
0.05
0.06
0.07
normalized resistance
Fig. 4 Impedance diagram due to the radial offset
From Fig.4, it is found that:
(1) The trajectory of impedance change is approximately a straight line, which is similar to
that of liftoff. The resistance increases and reactance decreases when radial offset becomes larger.
(2) The higher the excitation frequency, the trajectory is closer to the reactance axis.
3.3 Error evaluation
It is absolutely definite that probe radial offset have some influence on impedance, thus
degrading the measurement accuracy. Next, errors resulting from probe radial offset are evaluated
quantitatively.
The coil impedance change ΔZ can be divided into two parts, ΔZ tube and ΔZ offset . ΔZ tube
means the impedance change without probe radial offset. ΔZ offset is obtained by subtracting
ΔZ tube from ΔZ , denoting the impedance change due to probe radial offset. The relationship
between ΔZ , ΔZ tube and ΔZ tube is described as
ΔZ = ΔZ tube + ΔZ offset
(24)
After this, the error produced by probe radial offset is defined by:
e=
ΔZ offset
× 100%
ΔZ tube
(25)
With fixed excitation frequency 100 kHz, signal errors vary with the radial offset distance is
depicted in Fig.5.
80
eR
eX
signal error / %
60
eZ
40
e
θ
20
0
-20
0
0.5
1
1.5
2
radial offset/ mm
2.5
3
Fig.5 Variation of Signal errors with probe radial offset
It can be seen from Fig.5 that:
(1) With increase of the eccentricity, all the signal errors becomes larger;
(2) The phase error eθ is less than 6%, which indicates that phase rotation method may be
used to eliminate the radial offset effect;
(3) The errors does not exceed 5% when radial offset distance is smaller than 0.5 mm. In
some cases, the signal errors arising from small radial offset distance could be neglected.
Fig.5 demonstrates that radial offset effect is not negligible in many cases. Also, it is found
that when radial offset distance keeps constant, signal error is getting larger with increase of
excitation frequency, as shown in Fig.6. For relative low driving frequency such as lower than 100
kHz, probe signal is affected obviously by frequency fluctuation. Therefore, it is concluded that
signal error due to radial offset be reduced by decreasing exciting frequency.
100
signal error / %
eR
80
eX
60
eZ
e
θ
40
20
0
-20
0
100
200
300
exciting frequency /kHz
400
Fig. 6 Variation of Signal errors with excitation frequencies
4 Conclusions
Closed-form theoretical expression has been presented to predict mutual impedance of
driver-pickup bobbin probe with radial offset in a conducting tube. The probe radial offset effect
with distance and exciting frequency has been studied, and signal errors produced have been also
evaluated by simulation. The radial offset effect destroys the uniform distribution of induced eddy
current density, and make it accumulate toward the direction where the probe moves. An
approximate amplitude increase with distance has been observed. The larger the working
frequency, the more obvious the impact of radial offset on probe signal is.
Future work of the authors will focus on modeling and evaluation of probe radial offset effect
for conducting tubes in the presence of a defect.
Acknowledgements
This work is supported by the Fundamental Research Funds for the Central Universities
under grant 2012QNA30, Jiangsu Provincial Natural Science Foundation of China under grant
BK2012567 and the Priority Academic Program Development of Jiangsu Higher Education
Institutions.
Appendix
I ij = I m (α i b j ) , K ij = K m (α i b j ) ,
G1 =
G3 =
O1
α
2
1
O1
α
2
1
K01 −
α
α
O
b1 I11 K 01' , G2 = m2 1 K01 − b1 P1K 01' ,
α
α
α
I 01 −
α
α
O
b1 I11 I 01' , G4 = m2 1 I 01 − b1 P1I 01' ,
α
α
α
α 2 '
α
α1 b1 I 01 K11 − I 01O2 , H 2 = α12b1K 01' K11 − K01O2 ,
α
α
α
α
L1 = I 01O1 − α12b1 I 01' I11 , L2 = K01O1 − α12b1 K 01' I11 ,
α
α
αb
m
M1 = I12 , M 2 = 12 2 ,
α
k m
H1 =
S = O1P2 − O2 P1 , T = O1K11 − O2 I11 ,
O1 = b1αα1I11' +
P1 = α m2 I11 +
α
I12 Λ1 + b2α1Q1Λ2 , O2 = b1αα1K11' +
k 2 m2b1α1
P2 = α m K11 +
2
k 2 m2
α
α
K12 Λ1 + b2α1Q2 Λ2 ,
I12Γ 1 + α12b1b2Q1Γ 2 ,
k 2 m2b1α1
α
Q1 = −b2α1α I12' + α1
k 2 m2
K12Γ 1 + α12b1b2Q2 Γ 2 ,
α K 02'
α K 02'
I12 , Q2 = −b2α1α K12' + α1
K ,
α K 02
α K 02 12
Λ1 = K12 I11 − K11I12 , Λ2 = K11 I12' − I11 K12' ,
Γ 1 = K12 I11' − K11' I12 , Γ 2 = K11' I12' − I11' K12' .
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