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.

d

rp2rp1

c

xp0

rd2rd1

xd0

z

ρozp1

zp2zd1zd2

x

y

12 0

tube

coils

(a) driver-pickup probe in 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:

a z z bW W×∇W = e + e (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.

2 2 0a aW k W∇ + = (2)

2 2 0b bW k W∇ + = (3)

Where, 20j rk = − ωµ µ σ .

With Separation Method, the solutions to Eqs.(2) and (3) are formulated as

j j(| | )e e dzas s mm

W D K r ωϕ ωα α∞∞

−∞=−∞

= ∑∫ (4)

j j(| | )e e dzaec ec mm

W C I r ωϕ ωα α∞∞

−∞=−∞

= ∑∫ (5)

j j1 1 1 1 1[ ( ) ( )]e e dza a m a m

mW C I r D K r ωϕ ωα α α

∞∞

−∞=−∞

= +∑∫ (6)

j j1 1 1 1 1[ ( ) ( )]e e dzb b m b m

mW C I r D K r ωϕ ωα α α

∞∞

−∞=−∞

= +∑∫ (7)

j j2 2 (| | )e e dza a m

mW D K r ωϕ ωα α

∞∞

−∞=−∞

= ∑∫ (8)

Where, Im and Km are modified Bessel functions of the first and second kind of order m,

respectively. 2 21 kα α= + . 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.

2 1 2 1

2 1 2 1 2 1

( ) 0

( ) ( ) ( ) 0z z z

B BH H H H

⋅ − = − =

× − = − + − =ρ ρ ρ

ρ ϕ ϕ ϕ

e B Be H H e e

(9)

Further, six equations are obtained and solved, and coefficients are expressed as

1 2

3 4ec s

G S G TC DG S G T

−= −−

(10)

21 2

1 21

ec sa

H C H DCT+

= αα

(11)

21 2

1 21

ec sa

L C L DDT+

= αα

(12)

' '1 1 12 1 2 12 1 1 12 2 2 12 1( ) ( )b a aC M K QM K C M K Q M K D= + + + (13)

' '1 1 12 1 2 12 1 1 12 2 2 12 1( ) ( )b a aD M I QM I C M I Q M I D= − + − + (14)

212 1 12 1

1 2021

a ab

I C K DDK+

= αα

(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.

{}

2 1 2 2

2 1 1 1

( ) ( )12 1 2 1 2 150

( ) ( )

1j ( , ) ( , ) 2( ) d p d p

p d p d

z z z zair c d d p p p p

z z z z

M K I r r I r r z z e e

e e d

∞ − − − −−

− − − −

⎡= − + −⎣

⎤+ − ⎦

∫ α α

α α

ω αα

α (16)

Where, 0

2 1 2 1 2 1 2 1

j( )( )( )( )

d pc

d d d d p p p p

Kz z r r z z r r

π=

− − − −ωµ Ν Ν

, 2

1

' ' '2 1 1( , ) ( )

x

xI x x x J x dx= ∫ .

∆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

^

0 020

1j ( ) ( ) d dF

ds p p dsSM z

IΔ = ⋅ × − ×∫∫ω ρ ϕ

µ-­‐ E B E Bρ (17)

Where, ^ρ is unit vector in the ρ direction and SF is the inner surface of the tube. dsE and dsB

stand for the driver source electric field intensity and magnetic flux density, respectively, in the

absence of the sample. 0pE and 0pB 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

220 0

20

jj [ ] d dpa padas dasd

W WW WM d zr z z r z zI

∞ π=−∞ −π

∂ ∂∂ ∂−Δ = − ⋅∂ ∂ ∂ ∂ ∂ ∂∫ ∫ ρ

ωω ϕµ

(18)

Further, the closed form to ∆M is represented as

2cos 06

00

2 1 2 1

(2 )j 2 ( ) ( )

( , ) ( , ) ( , )d

mc m

m

d d p p

M K A I x

M r r M r r R m

∞ ∞

=

−Δ = ∑∫δω α α

αα α α α α α

(19)

Where, 0

2 1 2 1 2 1 2 1

j( )( )( )( )

d pc

d d d d p p p p

Kz z r r z z r r

π=

− − − −ωµ Ν Ν

, 2

1

' ' '2 1 1( , ) ( )

x

xM x x x I x dx= ∫ ,

cos 2 2 1 1 2 1 1 2( ) cos ( ) cos ( ) cos ( ) cos ( )p d p d p d p dA z z z z z z z z= − + − − − − −α α α α α ,

1 2

3 4

( , ) G S G TR mG S G T

−=−

α , is the generalized reflection coefficient [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 jec = − − (∇× )ωσ ωJ A= W (20)

By expanding Eq.(20) in cylindrical coordinate systems, the components of Jec becomes

21 11j ( )ec a bW WJ

z∂ ∂= − −∂ ∂ ∂ρ ω

ρ ϕ ρ (21)

21 11j )ec a bW WJ

z∂ ∂= ( +∂ ∂ ∂ϕ ωρ ρ ϕ

(22)

2 21 1 12 2 2

1 1j ( )ec b b bz

W W WJ ∂ ∂ ∂= − + +∂ ∂ ∂

ωρ ρ ρ ρ ϕ

(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

driver-pickup probe tube properties

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 (b) 0.5 mm radial offset

(c) 1.0 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.

0.01 0.02 0.03 0.04 0.05 0.06 0.070.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

20 kHz100 kHz200 kHz400 kHz

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.

normalized resistance

norm

aliz

ed re

acta

nce

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 ZΔ can be divided into two parts, tubeZΔ and offsetZΔ . tubeZΔ

means the impedance change without probe radial offset. offsetZΔ is obtained by subtracting

tubeZΔ from ZΔ , denoting the impedance change due to probe radial offset. The relationship

between ZΔ , tubeZΔ and tubeZΔ is described as

tube offsetZ Z ZΔ = Δ + Δ (24)

After this, the error produced by probe radial offset is defined by:

offset

tube

100%ZeZ

Δ= ×

Δ (25)

With fixed excitation frequency 100 kHz, signal errors vary with the radial offset distance is

depicted in Fig.5.

0 0.5 1 1.5 2 2.5 3-20

0

20

40

60

80

radial offset/ mm

sig

nal e

rror /

%

eReXeZ

eθ

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.

0 100 200 300 400-20

0

20

40

60

80

100

exciting frequency /kHz

sign

al e

rror /

%

eReXeZ

eθ

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

( )ij m i jI I b= α , ( )ij m i jK K b= α ,

01

'11 01 1 112

1

OG K b I K= −ααα

, 01

2 '12 01 1 1

OG m K b PK= −α

α α,

01

'13 01 1 112

1

OG I b I I= −ααα

, 01

2 '14 01 1 1

OG m I b PI= −α

α α,

01

2 '1 1 1 11 01 2H b I K I O= −

αα

α,

01

2 '2 1 1 11 01 2H b K K K O= −

αα

α,

01

2 '1 01 1 1 1 11L I O b I I= −

αα

α,

01

2 '2 01 1 1 1 11L K O b K I= −

αα

α,

1 12mM I=α

, 1 22 2

bMk m

=α

,

1 2 2 1S O P O P= − , 1 11 2 11T OK O I= − ,

2 2'

1 1 1 11 12 1 2 1 1 2k mO b I I b Q= + +αα Λ α Λα

, 2 2

'2 1 1 11 12 1 2 1 2 2

k mO b K K b Q= + +αα Λ α Λα

,

2 22 21 1

1 11 12 1 1 1 2 1 2k m bP m I I bb Q= + +αα Γ α Γ

α,

2 22 21 1

2 11 12 1 1 1 2 2 2k m bP m K K bb Q= + +αα Γ α Γ

α,

'02'

1 2 1 12 1 1202

KQ b I I

K= − +

αα α α

α,

'02'

2 2 1 12 1 1202

KQ b K K

K= − +

αα α α

α,

1 12 11 11 12K I K I= −Λ , ' '2 11 12 11 12K I I K= −Λ ,

' '1 12 11 11 12K I K I= −Γ , ' ' ' '

2 11 12 11 12K I I K= −Γ .

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