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2008-09-30 Institutionen för fysik Jonas Larsson jonas.larsson@physics.umu.se

COMSOL Multiphysics labs for Electrodynamics C

Eddy currents and related phenomena Introduction Eddy currents are in practice very common since the basic ingredients are just a conductor and a time-varying B-field. For a qualitative understanding of eddy currents it is useful start from the following idea (stated as Lentz law by Griffriths but this is indeed a formulation useful for a qualitative discussion of Faraday’s law):

Nature hates changes in the B-flux When there is a time varying B-field crossing a conductor then the change in B-flux may be diminished by a conduction current (the eddy current) that creates a B-field with the right properties. A quantitative analysis includes Faraday’s law / t∇ × = −∂ ∂E B where the time varying B-field creates an E-field and thus a conduction current σE in the conductor. There are several related phenomena associated with conductors and Faraday’s law including

• Magnetic diffusion • Eddy currents • Skin depth • Inductive heating

1. Magnetic diffusion Consider a spherical shell of conductive material and place it in the center of a short solenoidal coil. This is an axi-symmetric configuration and the geometry is similar to the following one

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The picture should be rotated around the z-axis. The large semicircle disk represents our spherical “universe”. The cylindrical shell (i.e. the rectangle in the picture) is the coil. In the middle of the coil is the spherical conducting shell. We make the following experiment

0

0 for 0

for 0

tI

I t

<= >

Here I denotes the current in the coil. We thus start from a situation without current in the coil and thus without any E- or B-field. Then at 0t = we turn on a constant current. What happens? A magnetic field is created but this field will not directly penetrate the conducting spherical shell but will rather diffuse through it which takes some time. Why? Currents are created in the shell in order to make the changes in B-flux as small as possible. However, these currents will decay in time (and heat the shell) so eventually the B-field cannot be stopped but will diffuse through the shell. Let us use CMPH to model this experiment. We get an initial value problem for Maxwell’s equations. We start at 0t = with vanishing fields ( 0= =E B ) but there is an

external current in the coil, which we in the model take as a current density ˆe eJφ φ=J φφφφ , that

creates an electromagnetic field. The Maxwell equations decouple into two independent

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parts. One part involves the electromagnetic fields ( ), ,r zE B Eφ and these fields remains

zero in our case also for 0t > . The other part ( ), ,r zB E Bφ will evolve in time. It is

practical to use the vector potential ˆAφ φ=A φφφφ as the single dependent field variable in the

PDE. The electromagnetic fields are then obtained as (from = ∇ ×B A and /V t= −∇ − ∂ ∂E A ):

r

AB

zφ∂

= −∂

, A

Etφ

φ∂

= −∂

, ( )1zB rA

r r φ∂=∂

The PDE for Aφ is obtained from one of Maxwell’s equations

t

∂∇ × = +∂D

H J

We obtain from this equation (we write it in terms of the one-component vector field ˆAφ φ=A φφφφ rather than the scalar field Aφ to get notations similar to CMPH)

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0 20

1 er

r t tφ φ

φ φσ ε εµ µ

∂ ∂ ∇ × ∇ × = − + − ∂ ∂

A AA J

The quasi static approximation of this equation (meaning that the interactions may be considered as instantaneous, i.e. without the effect of time retardation) now means that the last term may be dropped and we write the resulting equation in a similar form as CMPH

0

1 e

rtφ

φ φσµ µ

∂ + ∇ × ∇ × = ∂

AA J

This is similar to a PDE for diffusion. We find this equation in the following application mode:

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Note that we have choosen the application mode with

• Axisymmetry (2D) • Quasi-Statics Magnetic (Magnetic because the Coulomb electric field is zero.

There will appear no charge density in this problem since the currents will always be parallel to the conductor boundary and there will be no pile up of charge)

• Azimuthal induction currents because this is implied by the symmetry of our experiment (this is also consistent with the statement in the previous sentence claiming currents parallel to the boundary)

• Transient analysis (we will solve an initial value problem) Observe the picture of the field structure. Simulate the above experiment using this application mode. We like to see how long time it takes for the magnetic field to penetrate into the curvature center of the spherical shell. This may be seen in plots like (I used in postprocessing Cross-Section Plot Parameters)

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Figure 1 The constant external current eJφ may be given any value without changing the shape of

this curve. For figure 1 it was given (a very large) value so that the B-field in the center become approximately 1 tesla after sufficiently long time. Exercise 1: a) Consider the experiment above for two cases, one with conductivity

610σ = of the shell and the other with 710σ = (SI-units). The shell is of nonmagnetic material. Choose eJφ so that the B-field becomes approximately 1 tesla after sufficient

long time. b) For 610σ = take the relative permeability 10rµ = Exercise 2: a) Replace the conducting shell with a cylinder with a rather narrow cylindrical hole drilled along the axis. (The axi-symmetry of the experiment must not be destroyed). Give for one value of sigma a similar graph as figure 1. b) Look a little more carefully on the start of this curve. Submit one figure on this. 2. Time-harmonic analysis in CMPH CMPH apply in the usual way complex numbers to represent time-harmonic variations. This formalism is also considered in the textbook we use (Griffiths in Chapter 9).

However, in CMPH we use the factor i te ω to express time-harmonic variation while in

Griffiths i te ω− appears, we here follow the conventions in CMPH. The components of all

field variables (B, H, A, fJ ) etc in CMPH are compex valued scalar fields that varies in space but are independent of time. For example let us consider the real physical x-

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component of the B-field ( ),xB tr . This is represented by a complex scalar field ( )xB r

by

( ) ( ), Re i tx xB t B e ω = r r

Any time harmonic real valued scalar field has a unique complex representation defined in this way. Example: Take ( ) ( ) ( ), cosxB t f tω ϕ= +r r , any time harmonic real field may be

written in this way (only real numbers appears in this expression). The complex

representation of ( ),xB tr is ( ) ( ) ixB f e ϕ=r r .

We use complex notations in Maxwell’s equations and get

fρ∇ ⋅ =D iω∇ × = −E B 0∇ ⋅ =B

f iω∇ × = +H J D We consider for simplicity linear and isotropic media where µ=B H and ε=D E We also may represent the fields by potentials as = ∇×B A and i Vω= − − ∇E A where 0∇ ⋅ =A Above the complex fields ( )B r , ( )H r , ( )fJ r etc appears and they are related to the

real physical fields ( ),tB r , ( ),tH r , ( ),f tJ r etc as previously stated, i.e.

( ) ( ), Re i tt e ω = B r B r

and so on. About post processing in time-harmonic case: In the time-harmonic case the fields are

complex valued. The pictures we see are just the real part like ( )Re B r which is the B-

field at time t=0. We can get pictures of ( )Re ie φ B r instead (corresponding to some

other time) by giving the value of φ on the main page of Plot parameters (on Post menu). Another way to get a fuller picture of the fields is to use the Animate feature in Plot parameters. Then a sequence of the fields for different φ (and thus different times ) are shown. (This seems to be true unless you are using the parametric solver, in that case CMPH uses the varied parameter in animate). 3. Skin depth If a conductor is situated in a time harmonic magnetic field then the B-field inside the conductor decrease with the distance to the boundary because of shielding due to induced currents (the B-fields varies harmonically in time and nature tries prevent changes in the

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B-flux). The distance it takes to reduce the amplitude of the B-field by a factor 1/e (about a third) is called the skin depth and below we derive (in a particular situation) the analytical result (see analytical derivation in the end of this section)

2δ

ωµσ=

This shows that the B-field is more effectively excluded from the conductor with increased frequency (why is this expected?).

We will assume the symmetry 0z

∂ ≡∂

and take the external oscillating magnetic field as

( ) ( ) ( )0 0ˆ ˆ, cos Re i tt B t B e ωω= =B r x x

so in the complex representation we get 0ˆB=B x . A vector potential resulting in this B-

field is 0 ˆB y=A z . The E-field is then 0 ˆi B yω= −E z (from i Vω= − − ∇E A and 0V = ).

We make the following experiment. Let us put a conducting rod (long in the z-direction) with a rectangular cross-section (the same cross-section for all z) in this external oscillating field. The Maxwell equations decouple into two independent parts. One part

involves the electromagnetic fields ( ), ,x y zE E B and these fields are zero in our case. The

other part ( ), ,x y zB B E may be expressed in terms of the vector potential ˆz zA=A z as the

single dependent field variable in the PDE. The electromagnetic fields are then obtained as (from = ∇ ×B A and /V t= −∇ − ∂ ∂E A ):

zx

AB

y

∂=∂

, zy

AB

x

∂= −∂

, z zE i Aω= −

The PDE for zA is obtained from one of Maxwell’s equations iω∇ × = +H J D We obtain from this equation (we write it in terms of the one-component vector field

ˆz zA=A z rather than the scalar field zA to get notations similar to CMPH)

20

0

1 ezz z r z

r tσ ε ε ω

µ µ ∂ ∇ × ∇ × = − + + ∂

AA J A

The quasi static approximation of this equation (meaning that the interactions may be considered as instantaneous, i.e. without the effect of time retardation) now means that the last term may be dropped and we write the resulting equation in a similar form as CMPH

( )20

0

1 er z z z

r

iωσ ε ε ωµ µ

− + ∇ × ∇ × =

A A J

In order to model this with CMPH we choose the following application mode

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Note that we have choosen the application mode with

• 2D symmetry • Quasistatic-Magnetic (there will appear no charge density in this problem because

the currents will always be parallel to the conductor boundary there will be no pile up of charge)

• Perpendicular Induction Currents because this is implied by the symmetry of our experiment (this is also consistent with the statement in the previous sentence claiming currents parallel to the boundary)

• Time-harmonic analysis (we find steady state solutions and transients are not included in the analysis)

Exercise 3: a) Let our universe be a cylinder which in 2D is represented by a circle with radius 1 and center in the origin. Use the boundary conditions to get the external fields above as the solution of the PDE in CMPH. (b) Place a small rectangle (0.2 0.1× ) with center in the middle of the circle. Physically this is a long rod with rectangular cross section of a conducting material (characterised conductivity σ ). Variables we like to vary is frequency and conductivity. Therefore, define these in Constants (Options menu). Consider a conducting but nonmagnetic body. Take for example conductivity as 610 (SI-unit) and vary the frequency (0:50:500). Use parametric solver. Consider B-fieldlines (contour plots of Az give these, remember?) You is expected to see the skin effect where at higher frequency it is more difficult for the B-field to penetrate a conductor. Use Animation to see how the B-fieldlines change with frequency. To present the result in the report use Cross-Section Plot Parameter (on Post

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menu). Consider the absolute value of the B-field (in CMPH: Magnetic flux density, norm) along a vertical line through the center of the rectangle from the lower side to the upper side of it. Analytical result for the skin effect

Consider the planar symmetry 0z

∂ ≡∂

perpendicular current case. Now also assume that

0y

∂ =∂

. The equation now becomes (where 0rµ µ µ= and with 20 rε ε ω neglected)

( ) ( ) ( )1 ez z

d dj A x A x J x

dx dxωσ

µ − =

(19)

Consider some homogeneous, isotropic material with constant µ and look for solutions

of the form ( ) 0xA x A eγ= . We get from (19) with 0eJ = the equation 2 jγ ωσµ= with

solutions

( )

( )

1

2

12

12

j

j

ωσµγ

ωσµγ

= +

= − +

These are associated with skindepth δ (see textbook) as

2δ

ωµσ=

4. Eddy currents There are of course currents in the conductors in the examples above but they may both

be represented by one-component vector fields ( ˆEφ φσ=J φφφφ and ˆz zEσ=J z ). We now

like to get such currents in the xy-plane as a two component vector fields ˆ ˆx yE Eσ σ= +J x y . We consider the experiment with an long conducting rod inside a long

solenoid with a time-harmonic current in the coil. The cross-section is independent of the

z-coordinate thus assuming the symmetry 0z

∂ ≡∂

. In this case only the field components

( ), ,x y zE E B appears while the other components vanishes. The cross-section of the

system may look like

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The grey is the conducting rod. To use CMPH for modeling this experiment we choose the application mode

Note that we have choosen the application mode with

• 2D symmetry • Quasistatic-Electromagnetic (Electromagnetic since there will appear a

nonvanishing Coulomb electric field C V= −∇E . This is because the eddy current

will not be exactly parallel to the conductor boundary but there will be some pile up of charge and thus a time-varying surface charge density)

• In-Plane Induction Currents because this is implied by the symmetry of our experiment

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• Time-harmonic analysis (we find steady state solutions and transients are not included in the analysis)

Exercise 4: Use CMPH to model this experiment and create a picture of the eddy currents like the one below. (I used 610 as conductivity and frequency 60. The vector plot is for the total current and the contourplot for zB ). It is also interesting to consider plots for the scalar potential V since this the source of V is the surface charge. Use animate to see how the fields oscillates. Hint: The geometry is seen in the figure below. The circle represent the solenoid. Use surface currents in boundary conditions to model the current in the solenoid. The solenoid is also the outer boundary and its current is modelled in the boundary conditions (see hint above)

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We zoom and get a better picture of the eddy currents:

How to report the results of this laboration

• The report is a word document with some pictures that you send to me by e-mail as an attachment (jonas.larsson@physics.umu.se).

• Always fill in the Subject in the mail. For example write elfen lab2. Otherwise your mail may easily be treated as Spam

• For Exercise 1a you submit two pictures similar to Figure 1 and for Exercise 1b one such picture

• For Exercise 2 you submit one submit two such pictures (one for part a and one for part b)

• For Exercise 3 you submit one crossectional plot showing the skin effect (it contains curves for several frequencies)

• For Exercise 4 submit any plots similar to the one above. You may use some different geometry of the conductor.