Surface crack detection using flanged parallel-plate waveguide

1. Surface Crack Detection Using a Flanged
Parallel-Plate Waveguide
Hyun H. Park , Yong H. Cho , and Hyo J. Eom
Radio Technology Department
Electronics and Telecommunications Research Institute
161, Kajong Dong, Yusung Gu, Taejon, 305-600, Korea
Phone: 82-42-860-1324
Fax: 82-42-860-5199
E-mail: hyunho@etri.re.kr
Department of Electrical Engineering
Korea Advanced Institute of Science and Technology
373-1, Kusong Dong, Yusong Gu, Taejon, 305-701, Korea
Phone: 82-42-869-3436
Fax: 82-42-869-8036
E-mail: hjeom@ee.kaist.ac.kr
Abstract A anged parallel-plate waveguide is used for the detection of a two-
dimensional surface crack on a metal plate. The Fourier transform and the mode-
matching technique are used to obtain simultaneous equations, which are solved to
represent the crack characteristic signal. The characteristics of crack, such as crack
position, width, depth, and material in the crack, are discussed in terms of the crack
characteristic signal.
1

2. 1 Introduction
The detection of a surface crack in metals using a microwave signal has been of prac-
tical interest in diagnosing metal fatigue of aircraft fuselage and steel bridge. An
extensive study to investigate the crack characteristic signal has been performed in
1] utilizing an open-ended rectangular waveguide. In this letter, we will consider
a problem of surface crack detection using a anged parallel-plate waveguide. An
assumption of a parallel-plate waveguide facilitates the problem formulation and the-
oretical analysis yet retains all important detection characteristics of a rectangular
waveguide with a TE10 excitation. In the next section, we will present a theoret-
ical method for the V-shaped crack detection by using the Fourier transform and
mode-matching technique.
2 Field Analysis
An electromagnetic wave with TM-polarization is incident from the parallel-plate
waveguide on the surface crack as shown in Fig. 1. Here, e i!t time-harmonic varia-
tion is assumed throughout. In region (I) (a waveguide interior, jxj < b, y > 0), the
incident and re ected H- elds are
Hi
z(x;y) = cosbs(x + b)e i sy (1)
Hr
z(x;y) =
1X
m=0
m cosbm(x + b)ei my (2)
where bs = (s )=(2b), s =
q
k2
0 b2s, and k0 = !p 0 0 = 2 = is the free-space
wavenumber. In region (II) (a gap between the waveguide and the metal surface,
h < y < 0), the scattered eld is
Hg
z(x;y) = 1
2
Z 1
1
~A( )cos (y + h) + ~B( )cos( y)]e i xd (3)
where =
q
k2
1 2 and k1 = !p 0 1. In region (III) (a crack interior, jx x0j < a,
h d < y < h), the H- eld inside the crack is
Hc
z(x;y) =
1X
m=0
h
cm cos m(y + h) + dm sin m(y + h)
i
cosam(x x0 + a) (4)
2

3. where am = (m )=(2a), m =
q
k2
2 a2m, and k2 = !p 0 2. In region (IV) (a crack
interior, jx x0j < a; h d c < y < h d), the H- eld is
Hv
z(r; ) =
1X
m=0
pmGm(r; ) (5)
Gm(r; ) = J m(k2r)cos m( + 0) (6)
where J m( ) is the Bessel function of a fractional order, m = m =(2 0), 0 =
tan 1(a=c), r =
q
(x x0)2 + (y + h + d + c)2, and = tan 1(x x0)=(y+h+d+c).
To determine the unknown coe cients m, cm, dm, and pm, we enforce the boundary
conditions on Ex and Hz at y = 0, h, and h d. Applying the Fourier transform
2] to the Ex continuities at y = 0 and h, we represent ~A( ) and ~B( ) in series of
m and dm. Substituting ~A( ) and ~B( ) into Hz continuities, multiplying them by
cosbq(x + b), and performing integration from b to b, we obtain the simultaneous
equations for m, cm, and dm. Similarly, from the Hz and Ex continuities at y =
h d, we obtain
"q sq + ib3 1
2 0
sIsq(b) = "q q + ib3 1
2 0
1X
m=0
m mImq(b)
1X
m=0
pmK1
mq (7)
iab2 1
2 0
sJsp(b;a;x0) = iab2 1
2 0
1X
m=0
m mJmp(b;a;x0) +
1X
m=0
pm(Lmp K2
mp) (8)
where where sq is the Kronecker delta, "0 = 2, "1 = "2 = = 1,
I (u) =
Z 1
1
2
tan( h)F ( u)F ( u)d (9)
J (u;v;x0) =
Z 1
1
2
sin( h)F ( u)F ( v)e i x0
d (10)
Fm(v) = ( 1)meiv e iv
v2 (m =2)2 (11)
Kmq = a 1
4 2
1X
n=0
cos( nd)G0
mn( h d) n sin( nd)Gmn( h d)
"n
8
<
:
bJnq(a;b; x0) ( = 1)
aInq(a) ( = 2)
(12)
Lmp = p cos( pd)Gmp( h d) + sin( pd)G0
mp( h d)
2a p
(13)
Gmn(y) =
Z x0+a
x0 a
Gm(r; )cosan(x x0 + a)dx (14)
3

4. and G0
mn(y) = @Gmn(y)=@y. It is possible to solve (7) and (8) for the modal coe -
cients, m and pm. The crack characteristic signal at y = yd is calculated as
Vdetector / je ik0yd 0eik0ydj2: (15)
Fig. 2(a) shows the behavior of the crack characteristic signal at yd = 9:45 cm
versus the waveguide displacement x0. It is seen that our theoretical results with a
rectangular crack agree well with 1]. Note that the experiment in 1] is measured
with a K-band rectangular waveguide (a1 = 10:67 mm and a2 = 4:32 mm). In our
computations, the wavenumber k0 is replaced with
q
k2
0 ( =a1)2 to make comparison
with the experimental result using the TE10 mode in a rectangular waveguide. The
detector voltage of a V-shape crack is very similar to that of a rectangular crack. In
Fig. 2(b), the e ects of a dielectric gap between the waveguide and the metal surface
are shown in terms of the detector voltage. The resonant dips are observed at almost
every g=2, where g = 2 =k1. This means that the presence of a dielectric gap may
result in an error in detecting a crack location. The variations of the gap material
and the shape of a crack a ect the behavior of the detector voltage very little. Hence,
it is concluded that the detection of a crack is possible with this method, although a
non-rectangular crack is covered with a dielectric layer and lled with a rust powder.
3 Conclusion
A simple yet rigorous solution for a surface crack detection using a anged parallel-
plate waveguide is obtained. The behaviors of the crack characteristic signal of a
rectangular and V-shaped crack are represented. The comparison to other measure-
ment indicates that our simple series solutions are accurate enough to use for most
practical applications. It is also possible to extend our approach to a long-surface
multiple-crack detection.
4

5. References
1] C.Y. Yeh and R. Zoughi, A novel microwave method for detection of long surface
cracks in metals," IEEE Trans. Instrum. Meas., vol. 43, no. 5, pp. 719-725, Oct.
1994.
2] K.H. Park, H.J. Eom, and Y. Yamaguchi, An analytic series solution for E-
plane T-junction in parallel-plate waveguide," IEEE Trans. Microwave Theory
Tech., vol. 42, no. 2, pp. 356-358, Feb. 1994.
5

6. y
x
d
h
xo
2b
c
2a
e1
e2
Fig. 1 : Geometry of an open-ended parallel-plate waveguide and a metal surface
with a crack.
6

7. 0 0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Detectorvoltage[mV]
Waveguide dispacement, [mm]xo
rectangular crack
V-shaped crack
o o o experiment [1]
(a)
Detectorvoltage[mV]
0 5 10 15
0
0.2
0.4
0.6
0.8
1
1.2
Waveguide dispacement, [mm]xo
rectangular crack
V-shaped crack
=e2 e0
= (6.25+0.05i)e2 e0
=e2 e0
e2 e0= (6.25+0.05i)
(b)
Fig. 2 : Crack characteristic signal at 24 GHz for a rectangular crack (d =
1:03 mm; c = 0) and a V-shaped crack (d = 0; c = 1:03 mm) with b = 2:16 mm; a =
0:42 mm; yd = 9:45 cm, (a) h = 0, and 1 = 2 = 0, (b) h = 0:16 mm, and
1 = (3 + 0:2i) 0.
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