2 Eddy Current Theory

2.1 Eddy Current Method

2.2 Impedance Measurements

2.3 Impedance Diagrams

2.4 Test Coil Impedance

2.5 Field Distributions

2.1 Eddy Current Method

Eddy Current Penetration Depth

0 ( ) i tyE F x e E e

0 ( ) i tzH F x e H e

δ standard penetration depth

/ /( ) x i xF x e e

aluminum (σ = 26.7 106 S/m or 46 %IACS)

-0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3Depth [mm]

Re

F

f = 0.05 MHz f = 0.2 MHz f = 1 MHz

f = 0.05 MHz f = 0.2 MHz f = 1 MHz

-0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3Depth [mm]

| F |

1

f

Eddy Currents, Lenz’s Law

conducting specimen

eddy currents

probe coil

magnetic field

s p s( )d

Vdt

p p H J

s p s( )t

E H H

s sJ E

p p pN I

s sI V

s s sI s s H J

secondary(eddy) current

(excitation) currentprimarymagnetic flux

primary

magnetic fluxsecondary

p p s( )d

V Ndt

pprobe

p( , , , , ... )

VZ

I

2.2 Impedance Measurements

Impedance Measurements

pI p

e

( )( )

( )

VK Z

I

Ie VpZp

Ve

Ze

VpZp

Voltage divider:

Current generator:

Iep p

Ve e p

( )( )

( )

V ZK

V Z Z

Ve p

V

( )

1 ( )

KZ Z

K

Resonance

Ve

R

L VoC

0

0.2

0.4

0.6

0.8

1

0 1 2 3Normalized Frequency,

Tra

nsfe

r F

unct

ion,

| K

|

Q = 2

Q = 5

Q = 10

p 2( )

1

i LZ

LC

po

e p

( )( )( )

( ) ( )

ZVK

V R Z

2/

( )1 /

i L RK

i L R LC

2 2( )

1 /

iQ

Ki

Q

1

LC

C RQ R R C

L L

o 21

14Q

Wheatstone Bridge

32 2

e 1 2 4 3

( )( )

( )

ZV ZK G

V Z Z Z Z

Ve

V2

Z1 Z4

Z2 Z3

+

_ G32

21 4

0 ifZZ

VZ Z

1 4 0Z Z R

*2 cZ i L R

3 c cZ i L R

R0 reference resistance

Lc reference (dummy) coil inductance

Rc reference coil resistance

L* complex probe coil inductance

2 3 (1 )Z Z

probe coil reference coil

3 3

0 3 0 3

(1 )( )

(1 )

Z ZK G

R Z R Z

0( ) ( )K G K

3 00 2

0 3( )

( )

Z RK

R Z

Impedance Bandwidth

3 c cZ i L R

R0 = 100 Ω, Rc = 10 Ω

0( ) ( )K G K

3 00 2

0 3( )

( )

Z RK

R Z

0 1 2 30

0.1

0.2

0.3

0.4

0.5

Frequency [MHz]

Tra

nsfe

r F

unct

ion,

| K

0 |

Lc = 100 µH Lc = 20 µH

Lc = 10 µH

c 00 2

c 0

/( )

1 ( / )

L RK

L R

3 cZ i L

0p

c

R

L

0 p1

( )2

K

02

c

2 R

L

0 1,22

( )5

K

01

c2

R

L

2

14

2 1

c 2 1

62 or 120%

5rB

B

( , , , ,...)

2.3 Impedance Diagrams

Examples of Impedance Diagrams

Im(Z)

Re(Z)

L

C

Im(Z)

Re(Z)0

Ω-

Ω+

∞

L

C

R 0

Ω-

Ω+

∞R

Im(Z)

Re(Z)

R

L

C

0 Ω

∞ R

Im(Z)

Re(Z)

R2

L

C

0 Ω

∞ R1 R1+R2

R1

Magnetic Coupling

12 21

22 11

2 2 21 22( )d

V Ndt

1 1 11 12( )d

V Ndt

1 11 12 1

2 21 22 2

V L L Ii

V L L I

12 21 11 22L L L L

221 11

1

NL L

N 1

12 222

NL L

N

1 1121 11

1

I L

N 2 22

12 222

I L

N

1 1111

1

I L

N 2 22

222

I L

N

I1

N1 N2 V2

11

V1

I2

2212 21,

V1 V2L , L , L11 12 22

I1 I2

Probe Coil Impedance

e 22222n

e 22 e 22

R i LLZ i

R i L R i L

2 222 e 222 2

n 2 2 2 2 2 2e e22 22

(1 )LL R

Z iR L R L

V2V1

I1 I2

L , L , L11 12 22 Re

2 2 e 12 1 22 2V I R i L I i L I

122 1

e 22

i LI I

R i L

1 11 1 12 2V i L I i L I

2 212

1 11 1e 22

( )L

V i L IR i L

2 212

coil 11e 22

LZ i L

R i L

222n

22e

LZ i

R i L

1 11 12 1

2 12 22 2

V L L Ii

V L L I

1coil

1

VZ

I

coiln

11(1 )

ZZ i

L

coil ref [1 ( , , )]Z Z

ref 11Z i L

2 211 2212L L L

( )

Impedance Diagram

22 eL R /

2n n 2

Re 1

R Z

22

n n 2Im 1

1X Z

n n0 0

lim 0 and lim 1R X

2n nlim 0 and lim 1R X

2 2

n n( 1) and ( 1) 12 2

R X

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5

Normalized Resistance

Nor

mal

ized

Rea

ctan

ce

κ = 0.6 κ = 0.8 κ = 0.9

Re=10

Re=5

Re=30

22 e e3 H, = 1 MHz, / 10%L f R R lift-off trajectories are straight:

n n1X R

conductivity trajectories are semi-circles

2 22 22n n 1

2 2R X

Electric Noise versus Lift-off Variation

0.32

0.34

0.36

0.38

0.40

0.42

0.28 0.3 0.32 0.34 0.36 0.38

“Horizontal” Impedance Component“V

erti

cal”

Im

peda

nce

Com

pone

nt0.32

0.34

0.36

0.38

0.40

0.42

0.28 0.3 0.32 0.34 0.36 0.38

Normalized Resistance

Nor

mal

ized

Rea

ctan

ce lift-offlift-off

“physical” coordinates rotated coordinates

nZ nZ

Conductivity Sensitivity, Gauge Factor

22 e e3 H, = 1 MHz, 10 , 1L f R R

nnorm

e e/

ZF

R R

n

abse e/

ZF

R R

0 (1 )R R F /

/

R RF

0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 0.2 0.4 0.6 0.8 1

Frequency [MHz]

Gau

ge F

acto

r, F

absolute

normal0.32

0.34

0.36

0.38

0.40

0.42

0.28 0.3 0.32 0.34 0.36 0.38

Normalized Resistance

Nor

mal

ized

Rea

ctan

ce lift-off

nZ

nZ

Conductivity and Lift-off Trajectories

lift-off trajectories are not straightconductivity trajectories are not semi-circles

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5

Normalized Resistance

Nor

mal

ized

Rea

ctan

ce

κ

lift-off

conductivity

eL

RA

( )

e ( )

LR

A

( , ) finite probe size

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5

Normalized Resistance

Nor

mal

ized

Rea

ctan

ce

κlift-off

conductivity

2.4 Test Coil Impedance

Air-core Probe Coils

single turn L = a L = 3 a

center 2

IH

a

24 rI d

dr

H e e

a coil radius

L coil length

encd IH s

center/lim

L a

N IH

L

2

axis 2 2 3/ 22( )

I aH

a z

Infinitely Long Solenoid Coil

encd IH s

sJ n I

1 2( ) ( ) 0z zL H r L H r

for outside loops (r1,2 > a)

0zH

1 2( ) ( ) 0z zL H r L H r

for inside loops (r1,2 < a)

constantzH

1 2 s( ) ( )z zL H r L H r L J

1 s( )zI

H r J n I NL

for encircling loops (r1 < a < r2)

inside loop outside loopencircling

2a

L

+ Js_ Js

z

Magnetic Field of an Infinite Solenoid with Conducting Core

in the air gap (b < r < a) Hz = Js

in the core (0 < r < b) Hz = H1 J0(k r)

Jn nth-order Bessel function of the first kinds

10( )

JH

J k b

+ Js_ Js

2 a

2 b

z

0s

0

( )

( )zJ k r

H JJ k b

2 2( )k H 0 2k i 1 i

k

22

21

0zk Hr rr

2 2s

02 ( ) ( )

b

zH r r dr a b J

z zB dA H dA

Magnetic Flux of an Infinite Solenoid with Conducting Core

+ Js_ Js

2 a

2 b

z

0s

0

( )( )

( )zJ k r

H r JJ k b

( )z zH H r

szH J

0zH

2 2s 0

0 0

2[ ( ) ]

( )

bJ J k r r dr a b

J k b

0 1( ) ( )J d J

1 2 2s

0

2 ( )[ ]

( )

b J k bJ a b

k J k b

1

0

2 ( )( )

( )

Jg

J

2 2s [1 ( )]J a b g k b

For an empty solenoid (b = 0):

Normalized impedance:

1 1 1, ,s LJ n I V i V NV n LV

1 2 2 2 2

s [1 ( )]LV V

Z n L i a b g k b n LI J

2 2e eZ i a n L i X

22

2is called fill-factor ( lift-off)

b

a

2n

e1 [1 ( )]

ZZ i g k b

X

2 2s [1 ( )]J a b g k b

Impedance of an Infinite Solenoid with Conducting Core

Resistance and Reactance of an Infinite Solenoid with Conducting Core

2n n n1 [1 ( )]Z i g k b R i X

0 Re ( ) 1g k b 0.4 Im ( ) 0g k b

2n Im ( )R g k b 2

n 1 [1 Re ( )]X g k b

n n1 RX m Re ( ) 1

Im ( )

g k bm

g k b

1 ik

(1 )

bk b i

22 2

bi

0.01 0.1 1 10 100 1000-0.4

-0.20.0

0.20.4

0.6

0.81.0

1.2

Normalized Radius, b/δ

g-fu

ncti

on

real part

imaginary part

Effect of Changing Coil Radius

a (changes)

b (constant)

lift-off

b

a

Normalized Resistance

Nor

mal

ized

Rea

ctan

ce

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

b/δ = 1

3

5

10

20

2

κ = 1

0.9

0.8

0.7

a

lift-off

2n 1 [1 ( )]Z i g k b

Effect of Changing Core Radius

b (changing)

a (constant)

lift-off

2n 1 n 2 n1 R RX m m

b

a

n 1 21 0

1, where ( )

2a

a

Normalized Resistance

Nor

mal

ized

Rea

ctan

ce

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

100400

9

25

n = 4

κ = 1

0.9

0.8

0.7

b

lift-off

2n 1 [1 ( )]Z i g k b

Permeability

Normalized Resistance

Nor

mal

ized

Rea

ctan

ce

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2

ωn = 0.6

1.5

1

2

3

1

µr = 4

µ ω

0.8b

a

n1

2 2r 0 0 s

02 ( ) ( )

b

zH r r dr a b J

2n r1 [1 ( )]Z i g bk

1 20 r

1( )

2a

a

Solid Rod versus Tube

2 2 20 3 r 0 0 s2 ( ) ( )

b

zc

c H H r r dr a b J

1 0 2 0( ) ( )zH H J k r H Y k r

1 0 2 0 sBC1: ( ) ( )H J k b H Y k b J

1 0 2 0 3BC2: ( ) ( )H J k c H Y k c H

1 1 2 1 3BC3: ( ) ( )2

k cH J k c H Y k c H

b

a

1 1 2 1[ ( ) ( )] ( )k

H J k c H Y k c E c

H J E

zHE

r

20 3 ( )2i H c E c c

solid rod

BC1: continuity of Hz at r = b

tube

BC1: continuity of Hz at r = b

BC2: continuity of Hz at r = c

BC3: continuity of Eφ at r = c

b

a

c

( )z zH H r

szH J

0zH

3zH H

Solid Rod versus Tube

b

a

c

1,b c

a b

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6Normalized Resistance

very thin

solid rod

tube

Nor

mal

ized

Rea

ctan

ce

thick tube

σ1

σ2

σ1

σ2

Wall Thickness

b

a

c

1,b c

a b

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6

η = 0solid rod

b/ = 3

b/ = 2

Normalized Resistance

Nor

mal

ized

Rea

ctan

ce

b/ = 5

b/ = 10

b/ = 20 η 1thin tube

η = 0.2η = 0.4η = 0.6η = 0.8

Wall Thickness versus Fill Factor

b

a

c

,b c

a b

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6Normalized Resistance

Nor

mal

ized

Rea

ctan

ce

solid rodκ = 0.95, η = 0

solid rodκ = 1, η = 0

thin tubeκ = 1, η = 0.99

thin tubeκ = 0.95, η = 0.99

Clad Rod

b

a

c

2 2core core clad clad 0 s

02 ( ) 2 ( ) ( )

c b

cH r r dr H r r dr a b J

clad 1 0 clad 2 0 clad( ) ( )H H J k r H Y k r c r b

core 3 0 core( ) 0H H J k r r c

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6Normalized Resistance

Nor

mal

ized

Rea

ctan

ce

copper claddingon brass coresolid

copper rod

solidbrass rodbrass cladding

on copper core

d

master curve forsolid rod

d

thin wall

lower fill factor

clad

core, ,

b c

a b

(1 )d b c b

2D Axisymmetric Models

b

a

c

2ao

2ai

t

h

ℓ

short solenoid (2D)

↓long solenoid (1D)

↓thin-wall long solenoid (≈0D)

↓ coupled coils (0D)

pancake coil (2D)

o

i

1( ) ( )a

aI x J x dx

2 20

2 2 60o i

( )( )

( )

i N IZ f d

h a a

r 1( ) 2

r 1( ) 2( 1) [ ]h hf h e e e

2 2 2 2r 01 k i

Dodd and Deeds. J. Appl. Phys. (1968)

Flat Pancake Coil (2D)

0

0.05

0.1

0.15

0.2

0.1 1 10 100

Frequency [MHz]

(Nor

mal

) G

auge

Fac

tor

4 mm

2 mm

1 mm

coil diameter

o iM 2

1

2

a aa f

a

a0 = 1 mm, ai = 0.5 mm, h = 0.05 mm, = 1.5 %IACS, = 0

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2 0.25 0.3

Normalized Resistance

Nor

mal

ized

Rea

ctan

ce

0 mm

0.05 mm

0.1 mm

lift-off

frequency

fM

2.5 Field Distributions

Field Distributions

air-core pancake coil (ai = 0.5 mm, ao = 0.75 mm, h = 2 mm), in Ti-6Al-4V (σ = 1 %IACS)

10 Hz

10 kHz

1 MHz

10 MHz

1 mm

magnetic field

2 2r zH H H

electric field Eθ

(eddy current density)

Axial Penetration Depth air-core pancake coil (ai = 0.5 mm, ao = 0.75 mm, h = 2 mm) in Ti-6Al-4V

Axi

al P

enet

rati

on D

epth

, δ a

[m

m]

10-2

10-1

100

101

Frequency [MHz] 10-5 10-4 10-3 10-2 10-1 100 101 102

standard

actual1

f

ai

i o1

1/e point below the surface at ( )2

r a a a

1 22a a

Radial Spread air-core pancake coil (ai = 0.5 mm, ao = 0.75 mm, h = 2 mm) in Ti-6Al-4V

Rad

ial S

prea

d, a

s [m

m]

Frequency [MHz] 10-5 10-4 10-3 10-2 10-1 100 101 102

analytical

finite element

0.8

1.2

1.6

2.0

1.0

1.4

1.8

1/e point from the axis at the surface ( 0)z

2 o1.2a a

Radial Penetration Depth air-core pancake coil (ai = 0.5 mm, ao = 0.75 mm, h = 2 mm) in Ti-6Al-4V

Rad

ial P

enet

rati

on D

epth

, δr

[mm

]

10-2

10-1

100

101

Frequency [MHz] 10-5 10-4 10-3 10-2 10-1 100 101 102

standard

actual1

f

r s 2a a

2 o1.2a a

Lateral Resolution ferrite-core pancake coil (ai = 0.625 mm, ao = 1.25 mm, h = 3 mm) in Ti-6Al-4V

1.0

0

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

experimental

FE prediction

Rad

ial S

prea

d, a

s [m

m]

Frequency [MHz] 10-2 10-1 100 101