Preparation of Papers in a Two-Column Format for the 21st Annual Conference of the IEEE Industrial Electronics Society FACTA UNIVERSITATIS Series: Electronics and Energetics Vol. 30, N o 1, March 2017, pp. 27 - 38 DOI: 10.2298/FUEE1701027Z CALCULATION MODEL FOR THE INDUCED VOLTAGE IN RECTANGULAR COILS ABOVE CONDUCTIVE PLATES  Siquan Zhang 1 , Nathan Ida 2 1 Department of Electrical and Automation, Shanghai Maritime University, Shanghai, 201306, China 2 Department of Electrical and Computer Engineering, The University of Akron, Akron, OH, 44325-3904, USA Abstract. Electromagnetic NDT methods and in particular eddy currents play an important role in nondestructive testing of conducting materials. In testing conductive structures, rectangular coils are often more useful than circular coils. A particular configuration consists of two rectangular coils located above the conductive plates, one placed parallel to the plates serving as an excitation coil and the other perpendicular to the plates serving as a sensing coil. In this work we derive analytical expressions for the induced voltage variations in the pick-up coil. Then the influences of the plate thickness, the exciting frequency and the moving speed of the conductor on the induced voltage variation are analyzed. The analytical calculation results are verified using the finite element method. Key words: Eddy current testing, Conductive plates, Rectangular coil, Induced voltage, Finite element method. 1. INTRODUCTION Eddy current testing (ECT) techniques are widely used in testing of conductive structures with advantages of high sensitivity when testing for surface flaws [1-3]. In standard eddy current testing a circular coil carrying current is used to test the conductive specimen. The alternating current in the coil generates an alternating magnetic field, which interacts with the test specimen and generates eddy currents. However, rectangular coils are more useful than circular coils, because the rectangular coil is not axisymmetric, hence it affects the field inside the medium resulting in higher sensitivity to sub-surface flaws [4]. In spite of these advantages, rectangular coils have been seldom discussed in the literature. In this paper, we analyse a model with two rectangular coils, one serving as the exciting coil and the other is the pick-up coil, both located above the conductive plates. The conductive materials’ characteristics or parameters of flaws can be evaluated  Received August 17, 2016 Corresponding author: Nathan Ida Department of Electrical and Computer Engineering, The University of Akron, Akron, OH, 44325-3904, USA (E-mail: ida@uakron.edu) 28 S. ZHANG, N. IDA from the induced voltage variation in the pick-up coil. The validity of the theoretical analysis is confirmed by the finite element method (FEM). 2. THEORETICAL ANALYSIS 2.1. Analytical model Fig. 1 shows two rectangular single-turn coils located above multi-layer conductive plates. The exciting coil is parallel to the surface of the conductor which coincides with the z = 0 plane. The dimensions of the exciting coil are 2a1, 2b1 and a lift-off z0. An AC harmonic current tjIe  flows in the coil. The pick-up coil is parallel to the yz plane and perpendicular to the conductor, it has dimensions of 2a2, 2b2 and a lift-off z0+w2. The thickness, conductivity and permeability of the two layer conductive plate are assumed to be di, σi and μi (i =1, 2) and the conductive media are assumed to be linear, isotropic and homogeneous. 11 ,  22 ,  I 1a 1b 1 d 2 a 2 b 0 x c 20 wz  0 z 0 yo x y z v 2 d Fig. 1 Filamentary rectangular coils above a multi-layer conductor To simplify the analysis, the solution region is divided into region 0, 1 and 2. In Region 0 (z > 0), the incident magnetic flux density Bi generated by the exciting current and the reflected magnetic flux density Br generated by inducted eddy currents exist simultaneously. The incident magnetic flux density Bi can be expressed by the vector potential Ai as: JAi 0 (1) ii AB  (2) The reflected magnetic flux density Br satisfies the following: 0 rB (3) 0 2  r B (4) Region 1 )0(  zd is the top conductive plate. The magnetic flux density B1 in this region satisfies the following: 2 1 1 1 1 1 1 1 0 B B v j B y           (5) 1 0B  (6) Calculation Model for the Induced Voltage in Rectangular Coils above Conductive Plates 29 Region 2 )( dz  is the lower conductive plate. The magnetic flux density B2 in this region satisfies: 0222 2 222 2     Bj y B vB  (7) 02  B (8) To solve these equations, the double Fourier transform and its inverse are introduced:         dxdyezyxBzb yxj )( ),,(),,(   (9)            ddezbzyxB yxj )( 2 ),,( 4 1 ),,( (10) where ξ and η are the integration variables. 2.2. Incident magnetic flux density The single filamentary rectangular coil consists of four finite length wires, as shown in Fig.1. By solving (1), the vector potential generated at an arbitrary point ),,( zyxp by a source point )',','( zyx in the coil can be written as:  v R dvzyxJ zyxA ')',','( 4 ),,( 0   (11) where J is the current density in the coil, v is the coil segment carrying current, R is the distance of ),,( zyxp to the source point )',','( zyx as follow: 222 )'()'()'( zzyyxxR  (12) Performing the Fourier transform on (11), the expression of the vector potential in the region z < z0 is obtained as: '} 1 {)',','( 4 ),,( )(0 dvdxdye R zyxJza v yxj                  v zzyxj dveezyxJ ' 1 )',','( 2 22 0 22 )''(0    (13) Similarly, the components of the incident magnetic flux density are obtained by performing the Fourier transform on (2): z a ajb y zx     , z x y aj z a b     , xyz ajajb   (14) As shown in Fig. 1, the wire parallel to the x axis satisfies IzyxJ )',','( , 0' yy  and z  z0 < 0. Substituting these into (13), the x component of the vector potential becomes:     v zzyxj x dveezyxJa ' 1 )',','( 2 22 0 22 )''(0        0 0 0 22 0 ' 2 ' 22 )( 0 x x xjyj zz dxee eI     0 22 0 22 )( 0 2 yj zz e eI          )sin(2 0 x 22 )( 00 22 00)sin(      zzyj eexI (15) 30 S. ZHANG, N. IDA Similarly, the wire parallel to the y axis satisfies IzyxJ )',','( , 0' xx  and z  z0 < 0, substituting into (13), the y components of the vector potential becomes:     v zzyxj y dveezyxJa ' 1 )',','( 2 22 0 22 )''(0        0 0 0 22 0 ' 2 ' 22 )( 0 y y yjxj zz dyee eI     0 22 0 22 )( 0 2 xj zz e eI          )sin(2 0 y 0 22 0 22 )( 00 )sin( xj zz e eyI        (16) The x components of the magnetic flux density can be obtained by substituting (15) and (16) into (14) as follows: )},,(),,({ 01120112 1 zbaazbaa zz a b yy y ix        22 0 )(110 )sin()sin(2      zz e baIj (17) Similarly, the y and z components of the magnetic flux density can be obtained as: 22 0 )(110 )sin()sin(2      zz iy e baIj b (18) 22 0 )(11 22 0 )sin()sin(2      zz iz e baI b (19) The general solution for the z component of the incident magnetic flux density in region 0 is: 22    z iziz eCb (20) where the coefficients Ciz are: 22 0 )sin()sin(2 11 22 0      z iz e baI C (21) 2.3. Reflected magnetic flux density Performing the Fourier transform on (4), the reflected magnetic flux density in region 0 can be expressed as: 0)( 22 2 2    r r b z b  (22) In similar fashion, performing the Fourier transform on (5) and (7), the magnetic flux density in region 1 and 2 can be expressed as: 0)( 11111 22 2 1 2    bjvj z b  (23) 0)( 22222 22 2 2 2    bjvj z b  (24) The normal component of B and the tangential components of H must be continuous on the z = 0 and z = -d planes. Calculation Model for the Induced Voltage in Rectangular Coils above Conductive Plates 31 Applying the continuity of Bz, we obtain zrziz bbb 1 (z = 0) (25) zz bb 21  (z = d) (26) Applying the continuity of Hx, we obtain 1 1 0 )(  xrxix bbb   (z = 0) (27) 2 2 1 1  xx bb  (z = d) (28) Applying the continuity of Hy, we obtain 1 1 0 )(  yryiy bbb   (z = 0) (29) Due to the fact that 0 J , the current density Jz does not exist in regions 1 and 2, and we get: xy bb 11   (30) xy bb 22   (31) The following equations are obtained from (3) z b bj ry rz     (32) z b bj rx rz     (33) Following similar steps, the following equations are obtained from (6) and (8): 0111     z b bjbj z yx  (34) 0222     z b bjbj z yx  (35) The coefficient of the reflected magnetic flux density is obtained by solving the above equations: izd d rz C Penn Penn D 1 1 2 2 )1(1 )1()1(      (36) where  cos ,  sin , 22   , m 12 1   , P m m    1 1 , n   1 10 , 1111 22 1  jvj  (37) Let      d d Penn Penn 1 1 2 2 )1(1 )1()1( (38) 32 S. ZHANG, N. IDA The coefficient of the reflected magnetic flux density becomes:    22 0)sin()sin(2 11 22 0    z rz ebaI D (39) rzrx D j D    22      22 0)sin()sin(2 110    z ebaIj (40) The x component of the reflected magnetic flux density becomes: 22    z rxrx eDb    22 0 )( 110 )sin()sin(2    zz ebaIj (41) The x component of the reflected magnetic flux density in region 0 is obtained by performing the inverse Fourier transform on (41):            )sin()sin( 2 11 2 0 baIj B rx   )( 0zze   ( )j x y e d d       (42) Fig. 2 shows two multi-turn rectangular coils obtained by extending the two single- turn coils shown in Fig. 1 in width and length respectively. The coil parallel to the surface of the conductor is the excitation coil and the coil perpendicular to the conductor is the pick-up coil. The turns of the excitation and pick-up coil are N1 and N2 respectively. The lower surfaces of the two rectangular coils are level with each other. 11 ,  22 ,  1 d c v o 1 z 1 h 1 w 1 a 2 a 2 b 1 b 2 w 2 h z x y 2 d Fig. 2 Configuration of two multi-turn rectangular coils The reflected magnetic flux density generated by the multi-turn rectangular exciting coil shown in Fig. 2 is obtained by integrating (42) with respect to the width and length as follows:     11 1 1 0 0 11 1 hz z w rx total rx dpBdz hw N B 0 1 2 1 1 2 j IN w h           }])sin[()])sin[({ 1 0 11  w dppbpa  1 1 0 1 0 { } z h z z e dz      ( )z j x y e e d d                 ddeee k hw INj yxjhzzzz )()()(1 11 2 10 ][ 2 111           (43) where Calculation Model for the Induced Voltage in Rectangular Coils above Conductive Plates 33 ][ 1 )( 0 111 11 1 0   hzz hz z z eedze     (44) 1 1 1 1 0 sin[( ) )]sin[( ) ] w k a p b p dp    )(2 )sin(])(sin[ 11111      bawba )(2 ])(sin[)sin( 11111      wbaba (45) Fig. 3 shows a comparison of the variation of the reflected magnetic flux density’s x component as calculated from (43) and as simulated using Maxwell 3D respectively. The results of the simulation are obtained by subtracting the x component of the magnetic flux density without the conductor from the x component of the magnetic flux density with the conductor. The points shown belong to the line between (-16,0,5) and (16,0,5) which is located below the exciting coil and above the conductive plate. It can be seen that the analytical calculation results agree with the simulated results very well. -20 -15 -10 -5 0 5 10 15 20 -40 -30 -20 -10 0 10 20 30 40 Position along x axis (mm)  B x （ G a u s s ） FEM Fourier transform Fig. 3 Variations of the x component the magnetic flux density calculated from the analytical and FEM simulation 3. INDUCED VOLTAGE IN PICKUP COIL 3.1. Magnetic flux penetrating through the pick-up coil To obtain the reflected magnetic flux penetrating through the multi-turn rectangular pickup coil shown in Fig. 2, we first derive the reflected magnetic flux penetrating through the single-turn rectangular coil with lengths 2a2, 2b2, and assume it is located at (c, 0, zc), where zc = z1 + w2 + a2. The reflected magnetic flux penetrating through the single-turn coil is obtained by integrating (43) on the area of coil as: 34 S. ZHANG, N. IDA 2 2 2 2 c c z a b total r rx x cz a b dz B dy           1 11 2 10 2 k hw INj        cj e  ][ )( 111  hzz ee   }{ 2 2 dze az az zc c         dddye b b jy  2 2 }{ 22 1 11 2 10     k hw INj        ][ )( 111  hzz ee   cze  cj e   ][ 22 aa ee    2sin( )b d d  (46) Then the reflected magnetic flux penetrating through the multi-turn rectangular pickup coil is obtained by integrating (46) with respect to the width and length of pickup coil as follows: 2 2 2 / 2 2 0 / 2 2 2 w c h r c h N dp dc w h        22 1 2211 2 210     k hwhw NINj        ][ )()( 111  hzzzz cc ee   }{ 2/ 2/ 2 2 dce hc hc cj          2 22 0 )()( ][{ w papa ee   dddppb })](sin[ 2  cj e hkk hwhw NINj              ) 2 sin( 2 2 22 21 2211 2 210 1 2 2 1 2 2 1 (2 ) (2 ) [ ] z w a z w a h e e d d              (47) where   2 22 0 2 )()( 2 )](sin[][ w papa dppbeek   22 )()( 22 )()( 22 ])[(cos])[(sin 22222222        awawawaw eewbeewb 22 22 ])[sin(])[cos( 2222        aaaa eebeeb (48) 3.2. Induced voltage in the rectangular pickup coil The relationship between the magnetic flux penetrating through the pickup coil and induced voltage is:   j dt d V  (49) Therefore, the induced voltage can be derived as: 0 1 2 1 2 2 2 2 2 1 1 2 2 2 sin( ) 2 j cIN N k k h V e w h w h               1 2 2 1 2 2 1 (2 ) (2 ) [ ] z w a z w a h e e d d              (50) 4. RESULTS The induced voltage variation of the rectangular pick-up coil is now calculated by considering the influencing factors based on the expressions derived in the previous section. The parameters of the coils and the conductive plates are given in Tables 1 and 2 respectively. Calculation Model for the Induced Voltage in Rectangular Coils above Conductive Plates 35 Table 1 Parameters of the rectangular coil Exciting coil Pick-up coil a1 (mm) 12 a2 (mm) 3 b1 (mm) 12 b2 (mm) 5 z1 (mm) 1 z1 (mm) 1 w1 (mm) 2 w2 (mm) 5 h1 (mm) 8 h2 (mm) 2 turns 500 c (mm) 6 turns 300 Table 2 Parameters of the conductive plate Top layer σ1 (S/m) 3.8×10 7 μr1 1 Lower layer σ2 (S/m) 5.8×10 7 μr2 1 Fig. 4 shows the induced voltage due to the conductive plates as a function of the excitation frequency. The thickness of the top-layer conductor is 200 μm and the thickness of the lower-layer is semi-infinite and both conductors are stationary. c is the distance from the center of the pick-up coil to the z axis. It can be seen from Fig. 4 that the variation of the induced voltage increases with frequency. At any given exciting frequency, the pick-up coil with larger distance to the z axis has a higher induced voltage. 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 Frequency (kHz) R e a l p a rt o f in d u c e d v o lt a g e ( v ) c = 3 mm c = 6 mm c = 9 mm Fig. 4 Induced voltage in the pickup coil as a function of exciting frequency Fig. 5 compares the induced voltage calculated from the analytical method and FEM simulation. The analytical results are calculated as the square root of the sum of squares of the real and imaginary parts of the induced voltage. The results of the FEM are the effective values of the induced voltage obtained in pick-up coil, simulated with a time-dependent formulation. 36 S. ZHANG, N. IDA 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 Frequency (kHz) In d u c e d v o lt a g e i n p ic k -u p c o il ( v ) Analytical method FEM Fig. 5 Comparation of the induced voltage variation in rectangular pick-up coil from analytical and FEM at different excitation frequency The induced voltages in the coil for different thicknesses of the top-layer conductor are shown in Fig. 6. The excitation frequencies are fixed at 0.5, 2, and 5 kHz respectively, and the conductor is stationary. The distance from the center of the pick-up coil to the z axis is fixed at 9 mm. The induced voltage variation initially increases with the thickness, then, at a specific thickness, the induced voltage reaches a maximum, followed by a decreases with increasing thickness. As can be seen from Fig. 6, the higher excitation frequency produces a higher maximum at a smaller thickness, but the induced voltage decreases faster with increasing excitation frequency. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Thickness of top-layer conductor (mm) R e a l p a rt o f in d u c e d v o lt a g e ( v ) 0.5 kHz 2 kHz 5 kHz Fig. 6 Induced voltage in pickup coil as a function of top-layer conductor thickness The speed characteristics are shown in Fig. 7. The induced voltage variations are calculated at speeds from v = 0 to 50 m/s. The excitation frequency is fixed at 2 kHz. Fig. 7 shows the differences of the coils induced voltage at different speeds of the conductor relative to the Calculation Model for the Induced Voltage in Rectangular Coils above Conductive Plates 37 coils’ induced voltage when the conductor is stationary. The rectangular coils’ induced voltage variation keeps increasing with the moving speed of conductor, the maximum variation of induced voltage is achieved with the top-layer conductor of thickness 200 μm. 0 5 10 15 20 25 30 35 40 45 50 -35 -30 -25 -20 -15 -10 -5 0 Moving speed of conductor (m /s) R e a l p a rt o f  v ( m v ) 50 m 100 m 200 m 1000 m Fig. 7 Induced voltage of pickup coil at different speed of conductor 5. CONCLUSION A closed-form expression for the induced voltage between a pair of rectangular coils above a multi-layered conductive plate has been derived using a 2D Fourier transform method. The excitation coil is parallel to the plates and the pickup coil is perpendicular to the conductor. We discussed the influencing factors on the induced voltage, such as the excitation frequency, the thickness of the top-layer conductor and the speed of the conductor. 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