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For a semi - infinite (thick) conductor the solution to the above equation is
J /J • e-e stn(wt-e) X O
(2.12a)
where Jx/J 0 is the ratio of eddy current density Jx at depth x to the surface density J 0 ,and e • 2.718 is the base of natural logarithms. e is given by x/ó where Ó = (TTf~<J) - 1/2, the standard depth of penetration (see next section).
Equation 2 . 12(a) ean be separated into two components :
J 1 J X O
a: e -x/ ó (2.12b)
which describes the exponential decrease in eddy current density with depth,and
(2.12c)
denoting the increasing time or phase lag of the sinusoidal signa! with depth.
~ 2.4.1 Standard Depth of Penetration
Figure 2.6 illustrates the change in eddy current densit y in a semi-infinite conductor. Eddy current density decreases exponentially with depth.
Fig. 2.6: With Depth
A. sin ( w t) 't'o
\1111111 0.2 0.4 0 .6 0.8 1.0
( . 37 )
S= so~. mm f J.lr
Eddy Current and Magnetic Flux Distribution in a Thit;k Plate
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The depth at which eddy current density has decreased to 1/e or 36.8% of the surface density is ca11ed the standard depth of penetration. The w¿rd 'standard' denotes plane wave e1ectromagnetic field excitation within the test samp1e (conditions which are rare1y achieved in practice). The standard depth of penetration is given by
or o = 2/p/f'f). r
, mm (2.13a)
inches (2.13b)
where p . is electrical resistivity in microhm-centimetres, f is test frequency in hertz, and ~r is re1ative magnetic permeability (dimension1ess)*.
The skin depth equation is strict1y true only for infinitely thick material and planar magnetic fields. Using the standard depth, o , calculated from the above equation makes it a material/test parameter rather than a true mea~ure of penetration.
Depth of Penetration in Finite Thickness Samples
Sensitivity to defects depends on eddy current density at defect location. Although eddy currents penetrate deeper than one standard depth of penetration they decrease rapidly with depth. At two standard depths of penetration (2 6 ), eddy current density has decreased to (1/e)2 or 13 .5% of the surface density. At three depths (3o) the eddy current density is down to only 5% of the surface density. However, one shou1d keep in mind these va1ues only apply to thick samples (thickness, t >S e ) and planar magnetic excitation fields. Planar field conditions require large diameter probes (diameter > 10t) in plate testing or long coi1s (length >5t) intube testing. Real test coils will rarely meet these requirements since they wou1d possess low defect sensitivity. For thin plate or tube samples, current density drops off less than calculated from equation 2.12(b) as shown in Figure 2.7. For salid cylinders the overriding factor is a decrease to zero at the centre resulting from geometry effects as shown in Fig . 2.7(c) and discussed in Section 7.3.1.
One should also note, that the magnetic flux is attenuated across the sample, but not completely. Althou~h the currents are restricted to flow within specimen boundaries, the magnetic field extends into the air space beyond. This allows the inspection of multi-layer components separated b y an air space.
~See Chapter 9 for a description of electrical and magneti c propert ies. .~ r = ~t. , incremental permeability, at zero biasing magnetization flux.
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2. 4. 3
lx
Jo
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t 1.0 8 =o .01
.a .! = 1 o 8 •
. 6 t
.4 8 = 1 .5
t
.2 8 = 2.0
o o .2 .4 .6 .a 1. o
~
(a) PLATE (LARGE COIL. O >IOt)
--- EQUATION 2 12 (b) -ACTUAL
o .2 .4 . 6 .8 t .o
(e) ROO ( ENC 1 RCLI NG CO 1 L. -t > 5 r 0 )
r' -= 0. 8 ro
o.~~--~--~~--~ o .2 .4 .6 .8 1.0
X
t (b) TUBE ( LONG ENCIRCL ING COIL . ~ '~t)
J0 = EOOY CURRENT OENS ITY AT SURFACE
Jx OR lr = EDDY CURRENT DENSITY AT LOCAT ION x OR r BELOW THE SURFACE
1><:1 I><J VZl27ZilW - '
PLATE GEOIIETRY
t:::::::-::: - - ( L. fr Ir. )
TUBE ANO ROO GEOIIETRY ( r 1 =O FOR ROO)
Fig. 2.7: Eddy Current Distribution With Depth in Various Samples
The sensitivity to a subsurface defect depends on t h e edd~ current density at that depth, it is therefore important to know th~ effective d~pth of penetration. The effective depth of penetration is arbitrarily defined as the depth at which eddy current density decreases to 5% of the surface density. Fo r large probes and thick samples, this depth is about three standard depths of penetration. Unfortunately, for most components and practica! probe sizes, thi s depth will be less than 3 5 , the eddy currents being attenuated more than predicted by the skin depth equation. The effect of probe diameter on the decrease in eddy current density or defect sensitivity with depth is discussed in Section 5 . 3.1 .
Standard Phase Lag
The signa! produced by a flaw depends on both amplitude and phase of the currents being obstructed. A small surface defect and large interna! defect can have a similar effect on the magnitude of test coil impedance. However, because of the increasing phase lag with depth, there will be a characteristic difference in the test coi! impedance vector. This effect allows location and extent of a defect to be determined.
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Phase lag is derived .from equation 2.12(c) for infinitely thick material. It , represents a phase angle lag of x/ ~ radians between the sinusoidal eddy cu rrent s at ' t -he surface a n d t h os e be 1 o w t he s u r fa e e • I t i s de no t e d by t he s y m b o 1 p (beta) and is given by:
6 = x/ o radians (2.14a)
or B = x/6 x 57 de~reAG
where x is distance below the surfa ce in the same uni ts as .S •
cf>o s.i 10 ( w t )
f f t ¡ ¡ ¡ l ¡ 1----+---__,¡,.. ____ {3 ( DEG,REES)
X
8 i 1f x 57 , OEGREES
~ 2.8: Eddy Current Phase Lag Variation With Depth in Thick Samples
When x is equal to one standard depth of penetration, phase lag is 57° or one radian. This means that the eddy curren t s f lowing below the s ur face, at one standard depth of penetration, lag the surface currents by 57u. At two standard depths ~f penetra ti on they lag the surface currents by 114u. This is illustrated in Figure 2.8 • .,.
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2.4 . 4 Phase Lag in Fini te Thick n ess Samples
t
Fo r t hin samples, eddy. current phase decreases slightly less rapidly with depth than stated above. See Figu r e t .9( a), (b) and (e) for the plots of phase lag with depth for a plate , tube, and cyl i nde r, respectively. The phase lag illustrated in these plots does not change si gnificantly with coil di4meter or length. For thi ck samples and practical p robe si zes, equation 2.14 is sufficiently accurate.
/3x
!3r
o·~--------------· ,.....__
20" ' ' ........
40"
so•
80'
2D-
40"
so•
80'
' ........ '\ ...............
ro 8 •2
' ....... t '\ '--=1.0
' 8
' '
( a ) FLATE
t = 2.0 8
~ = 1.0 8
r • = 1 4 8 . ~. 1 .o !.•- 2 o 8 - .
100' !..• . 2 1 o .2 . 4 .6 .8 1 .O
8 .
(e) ROO
.!.. = 0.7 40 ' 8
!3r so-
e o· ~=O 8
1 oo• r,
o .2 .4 .6 . 8 1 .o
( b) TUBE
¡o-¡ [
I><J 1>'<1 vzaazz zzzr
PLATE GEOMETRY
rtt t,
z:::::::m, ,Jt l. ¡., TUBE ANO ROO ( r 1 = 0 ) GEOMETRY
P,, p r = PHASE "WITH OEPTH x, OR r ,RELATIVE TO SURF ACE CURRENT
ACTUAL CURVES
CALCULA lEO . EQUAT ION 2.1 4 ( b)
Fig. 2.9: Eddy Cu rrent Phase Lag in Various Samples
Phase lag can be visualized as a shift in time of the sinusoidally va ryi ng current flowing below the sÚrface. This was illustrated in Figure 2 . 5. Phase lag plays a key ro le in the analysis of eddy current test signals . It will be used thro ughout the manual to link theo r y and observations. It should not be misin terpreted o r confused with the phase angle between voltage and current in AC theory. Both the voltage and current (and magnetic field) have this phase shift or lag with depth .
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2. 5 S UMMARY
Eddy current testing is based on inducing electrical currents in the material being inspected and observing th~ interaction between these currents and the material.
This process occurs as follows: When a periodically varying magnetic field intersects an electrical conductor, eddy currents are induced according to Faraday's and Ohm's Laws. The induced currents _(known as eddy currents because of their circulatory paths) generate their own magnetic field which opposes the excitation field. The equilibrium field is reduced resulting in a change of coi! impedance. By monitoring coil impedance, the electrical, magnetic and geometric pro~ perties of the sample can be measured. Eddy currents are closed loops of induced current circulating in planes perpendicular to the magnetic flux. They normally travel parallel to the coil's winding and parallel to the surface. Eddy current flow is limited to the area of the inducing magnetic field.
Depth of penetration into a material depends on its electrical resistivity, magnetic permeability and on test frequency. The basic equation of ET is the standard depth of penetration given by
o = solp/ f u r , mm (2.13a)
where p is electrical resistivity, microhm-centimetres; f is test frequency, hertz;
and Ur is relative magnetic permeability , dimensionless.
It states that in thick materials eddy current density decreases to 37 % of the surface density a t a depth of one standard depth of penetration. In most eddy current tests, especially with surface probes, the actual eddy current density (at a depth equal to that calculated by equation 2.13a) is much less than 37%.
Eddy currents also lag in phase with depth into the material. Phase lag depends on the same material properties that govern depth of penetration and is given by
B = x / o = X
so l p / f u r
, radi ans
where x is distance below surface, mm.
( 2.14a)
Phase lag is the parameter that makes it possible to determine the depth of a defect. I t also allows discrimination between defect signals and false indicati ons. It is the key parameter in eddy current testing.
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2 . 6 WORKED EXAMPLES
2.6.1 Standard Depth of Penetration and Phase Lag
PROBLEM: (a) Calculate the standard depth of penetxation in a thick 304 SS sample, at a test frequen c y o f 100 kHz.
(b) Determine the eddy current phase lag at a depth of 1.5 mm in 304 SS at 100 kHz.
SOLUTION: 304 SS properties: P '"'72 mi crohm -centimetres
and lJ - 1 r
(a) from equati on 2. 1 3(a),
ó • 50~ 72
100 X 10 3
X 1
1 / 2 S O ( 7. 2 x 1 O- 4 ) • 1. 3 mm
Therefore the standard depth of penetration is 1.3 mm .
(b) from equation 2.14(b),
B = x /8 x 57
= l . S x 5 7 = 64 degrees 1. 3
Theref ore the phase lag is 64 degrees.
