ECE 598 JS Lecture 07 Nonideal Conductors andjsa.ece.illinois.edu/ece598js/Lect_07.pdf · For good conductors: = ... Dielectrics and Polarization Field causes the formation of dipoles

Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 20121

ECE 598 JS Lecture ‐ 07Nonideal Conductors

and DielectricsSpring 2012

Jose E. Schutt-AineElectrical & Computer Engineering

University of [email protected]


σ: conductivity of material medium (Ω‐1m‐1)

HEt

μ∂∇× = −

∂

EH Jt

ε∂∇× = +

∂J Eσ=

Material Medium

E j Hωμ∇× = −

H J j Eωε∇× = +

( ) 1H E j E E j j Ejσσ ωε σ ωε ωεωε

⎛ ⎞∇× = + = + = +⎜ ⎟

⎝ ⎠

2 2 1E Ejσω μεωε

⎛ ⎞∇ = − +⎜ ⎟

⎝ ⎠1

jσε εωε

⎛ ⎞→ +⎜ ⎟

⎝ ⎠

or

since then


Wave in Material Medium2 2 21E E E

jσω με γωε

⎛ ⎞∇ = − + =⎜ ⎟

⎝ ⎠

γ is complex propagation constant

2 2 1jσγ ω μεωε

⎛ ⎞= − +⎜ ⎟

⎝ ⎠

1j jjσγ ω με α βωε

= + = +

α: associated with attenuation of wave

β: associated with propagation of wave


Wave in Material Medium

ˆ ˆz z j zo oE xE e xE e eγ α β− − −= =

1/22

1 12

με σα ωωε

⎡ ⎤⎛ ⎞⎢ ⎥= + −⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

decaying exponentialSolution:

1/22

1 12

με σβ ωωε

⎡ ⎤⎛ ⎞⎢ ⎥= + +⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

ˆ z j zoEH y e eα β

η− −=

jj

ωμησ ωε

=+

Magnetic field

Complex intrinsic impedance


Skin Depth

1δα

=

The decay of electromagnetic wave propagating into a conductor is measured in terms of the skin depth

2δωμσ

=For good conductors:

ˆ ˆz z j zo oE xE e xE e eγ α β− − −= = Wave decay

Definition: skin depth δ is distance over which amplitude of wave drops by 1/e.


Skin Depth

I VCz t

∂ ∂− =

∂ ∂δ

e-1Wave motion

For perfect conductor, δ = 0 and current only flows on the surface


DC Resistance

dclRwtσ

= l: conductor lengthσ: conductivity


AC Resistance

2 /acl l l fRw ww

πμσ δ σσ ωμσ

= = =

l: conductor lengthσ: conductivityf: frequency


Frequency-Dependent Resistance

1Jδ =

0

z JJ e dz Jα δα

∞− = =∫

Approximation is to assume that all the current is flowing uniformly within a skin depth


2 2

2 2

I ICLz t

∂ ∂=

∂ ∂ 1ac

fRw

π μσ

=

1dcR

wtσ=

Resistance is ~ constant when δ >t

Resistance changes with f

Frequency-Dependent Resistance


, 6ac groundl fRh

πμσ

≈

Reference Plane Current


εr

H. A. Wheeler, "Formulas for the skin effect," Proc. IRE, vol. 30, pp. 412-424,1942

Skin Effect in Microstrip


/ /y jyoJ J e eδ δ− −=

/ /

0 1y jy o

oJ wI J we e dy

jδ δ δ∞

− −= =+∫

oo o o

JE J Eσσ

= ⇒ =

oo

J DV E Dσ

= =

Current density varies as

Note that the phase of the current density varies as a function of y

The voltage measured over a section of conductor of length D is:



( ) ( )11o

skino

jJ DV DZ j fI J w w

π μρσ δ

+= = = +

1ρσ

=

skin skinDR X fw

π μσ= =

The skin effect impedance is

where


is the bulk resistivity of the conductor

skin skin skinZ R jX= +

with

Skin effect has reactive (inductive) component


internalac skinR RL

ω ω= =

The internal inductance can be calculated directly from the ac resistance

Internal Inductance

Skin effect resistance goes up with frequency

Skin effect inductance goes down with frequency


When the tooth height is comparable to the skin depth, roughness effects cannot be ignored

Surface Roughness

Copper surfaces are rough to facilitate adhesion to dielectric during PCB manufacturing

Surface roughness will increase ohmic losses


Hammerstad Model

( ) whenwhen

H sH

dc

K R f tR fR t

δδ

⎧ <= ⎨

≥⎩

( )

( )

( )

when2

when2

Hexternal

HH t

externalt

R fL t

fL f

R fL t

fδ

δ

δπ

δπ

=

=

⎧+ <⎪

⎪= ⎨⎪ + ≥⎪⎩


Hammerstad Model

221 arctan 1.4 RMSH

hKπ δ

⎡ ⎤⎛ ⎞= + ⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

hRMS: root mean square value of surface roughness height

δ: skin depthfδ=t: frequency where the skin depth is equal to

the thickness of the conductor


Hemispherical Model

( ) whenwhen

hemi shemi

dc

K R f tR fR t

δδ

⎧ <= ⎨

≥⎩

( )

( )

( )

when2

when2

hemiexternal

hemihemi t

externalt

R fL t

fL f

R fL t

fδ

δ

δπ

δπ

=

=

⎧+ <⎪

⎪= ⎨⎪ + ≥⎪⎩


Hemispherical Model

1 when 1when 1

shemi

s s

KK

K K≤⎧

= ⎨ >⎩

( ) ( )( )( )

( )( )3 1 / 121

3 1 / 2 1r jj krr j

δα

δ− +

= −+ +

( ) ( ) ( )( ) ( )( )( )

2Re 3 / 4 1 1 / 4/ 4

o tile bases

o tile

k A AK

Aη π α β μ ωδ

μ ωδ

⎡ ⎤+ + −⎣ ⎦=

( ) ( ) ( ) ( )( )( ) ( )( )

23

2

1 4 / 1/ 1213 1 2 / 1/ 1

j k r jj krj k r j

δβ

δ− −

= −+ −


Huray Model

( ) whenwhen

Huray sHuray

dc

K R f tR fR t

δδ

⎧ <= ⎨

≥⎩

( )

( )

( )

when2

when2

Hurayexternal

HurayHuray t

externalt

R fL t

fL f

R fL t

fδ

δ

δπ

δπ

=

=

⎧+ <⎪

⎪= ⎨⎪ + ≥⎪⎩


Huray Model_flat N spheres

Hurayflat

P PK

P+

=

( ) ( )( )( )

( )( )3 1 / 121

3 1 / 2 1r jj krr j

δα

δ− +

= −+ +

( )/ 4flat o tileP Aμ ωδ=

( ) ( ) ( ) ( )( )( ) ( )( )

23

2

1 4 / 1/ 1213 1 2 / 1/ 1

j k r jj krj k r j

δβ

δ− −

= −+ −

( ) ( )2_ 2

1

1 3Re 1 12

N

N spheres onn

P Hkπη α β

=

⎡ ⎤= − +⎢ ⎥⎣ ⎦∑

/ 'o oη μ ε ε=

Ho: magnitude of applied H field.


Dielectrics and Polarization

Field causes the formation of dipoles polarization

No field Applied field

Bound surface charge density –qsp on upper surface and +qsp on lower surface of the slab.


( )1o a o a o e a o e a s aD E P E E E E= + = + = + =ε ε ε χ ε χ ε

sε

D

oε

eχaE

: electric flux density

: applied electric field: electric susceptibility: free‐space permittivity: static permittivity

Dielectrics and Polarization

P : polarization vector


Material ε r

AirStyrofoamParaffinTeflonPlywoodRT/duroid 5880PolyethyleneRT/duroid 5870Glass-reinforced teflon (microfiber)Teflon quartz (woven)Glass-reinforced teflon (woven)Cross-linked polystyrene (unreinforced)Polyphenelene oxide (PPO)Glass-reinf orced polystyreneAmberRubberPlexiglas

1.00061.032.12.12.12.202.262.352.32-2.402.472.4-2.622.562.552.62333.4

Dielectric Constant


Material ε r

LuciteFused silicaNylon (solid)QuartzBakeliteFormicaLead glassMicaBeryllium oxide (BeO)MarbleFlint glassFerrite (FqO,)Silicon (Si)Gallium arsenide (GaAs)Ammonia (liquid)GlycerinWater

3.63.783.83.84.85666.8-7.081012-161213225081

Dielectric Constants


When a material is subjected to an applied electric field, the centroids of the positive and negative charges are displaced relative to each other forming a linear dipole.

When the applied fields begin to alternate in polarity, the permittivities are affected and become functions of the frequency of the alternating fields.

AC Variations


Reverses in polarity cause incremental changes in the static conductivity σs heating of materials using microwaves (e.g. food cooking)

When an electric field is applied, it is assumed that the positive charge remains stationary and the negative charge moves relative to the positive along a platform that exhibits a friction (damping) coefficient d.

AC Variations


( )

( )

22 2

'2

22 2

1

eo

or

o

N Qm

dm

−= +

⎛ ⎞− + ⎜ ⎟⎝ ⎠

ω ωεε

ω ω ω ( )

2"

222 2

er

oo

dN Q m

m dm

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎛ ⎞− +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

ωε

εω ω ω

Complex Permittivity

eN

oε

Q

oω

m

d : damping coefficient: mass

: free space permittivity: dipole charge: dipole density

: natural frequencyω : applied frequency

osm

ω =

s : spring (tension) factor


( )' "i c i sH J J j E J E j j E∇× = + + = + + −ωε σ ω ε ε


( )" ' 'i s i eH J E j E J E j E∇× = + + + = + +σ ωε ωε σ ωε

equivalent conductivity "e s s a= = + = +σ σ ωε σ σ

alternating field conductivity "a = =σ ωε

static field conductivitysσ =

σe: total conductivity composed of the static portion σs and the alternative part σa caused by the rotation of the dipoles


't i ce de i eJ J J J J E j Eσ ωε= + + = + +


iJ

tJ

ceJ

deJ

: total electric current density

: impressed (source) electric current density

: effective electric conduction current density: effective displacement electric current density

( )' ' 1 ' 1 tan'

et i e i i eJ J E j E J j j E J j j Eσσ ωε ωε ωε δ

ωε⎛ ⎞= + + = + − = + −⎜ ⎟⎝ ⎠

tan effective electric loss tangent' ' ' '

e s a s ae

σ σ σ σ σδωε ωε ωε ωε

+= = = = +


Complex Permittivity"

'

"tan tan tan' '

s ee s a

e

σ εεδ δ δωε ε ε

= + = + =

tan static electric loss tangent'

ss

σδωε

= =

"

'tan alternating electric loss tangent'

aa

σ εδωε ε

= = =

( )' ' 1 ' 1 tan'

ecd ce de e eJ J J E j E j j E j j Eσσ ωε ωε ωε δ

ωε⎛ ⎞= + = + = − = −⎜ ⎟⎝ ⎠


Dielectric Properties

' 1 ''

ecdJ j j E j Eσωε ωε

ωε⎛ ⎞= −⎜ ⎟⎝ ⎠

1'

eσωε

' 1'

ecd eJ j j E Eσωε σ

ωε⎛ ⎞= −⎜ ⎟⎝ ⎠

1'

eσωε

Good Dielectrics:

Good Conductors:


Dielectric Properties

' 1 ''

ecdJ j j E j Eσωε ωε

ωε⎛ ⎞= −⎜ ⎟⎝ ⎠

1'

eσωε

' 1'

ecd eJ j j E Eσωε σ

ωε⎛ ⎞= −⎜ ⎟⎝ ⎠

1'

eσωε

Good Dielectrics:

Good Conductors:


Kramers-Kronig Relations

( ) ( )( )

"'

2 20

' '21 ''

rr d

∞

= +−∫

ω ε ωε ω ω

π ω ω

There is a relation between the real and imaginary parts of the complex permittivity:

Debye Equation

( ) ( )( )

'"

2 20

1 '2 ''

rr d

ε ωωε ω ωπ ω ω

∞ −=

−∫

' '' " '( ) ( ) ( )

1rs r

r r r re

jj

∞∞

−= − = +

+ε εε ω ε ω ε ω ε

ωτ


Kramers-Kronig Relations

'

'

22

rse

r∞

+=

+ετ τε

τe is a relaxation time constant:

( )( )

' '' '

21r r

r re

∞ ∞∞

−= +

+

ε εε ω εωτ ( ) ( )

( )

' '"

21r r e

re

∞ ∞−=

+

ε ε ωτε ω

ωτ


Material εr’ tanδAirAlcohol (ethyl)Aluminum oxideBakeliteCarbon dioxideGermaniumGlassIceMicaNylonPaperPlexiglasPolystyrenePorcelain

1.0006258.84.741.001164*74.25.43.533.452.566

0.16 x 10-4

22x10-3

1 x 10-3

0.16x10-4

2x10-2

8 x 10-3

4 x 10-2

5x10-5

14x10-3

Dielectric Materials


Material εr’ tanδPyrex glassQuartz (fused)RubberSilica (fused)SiliconSnowSodium chlorideSoil (dry)StyrofoamTeflonTitanium dioxideWater (distilled)Water (sea)Water (dehydrated)Wood (dry)

43.82.5-33.811.83.35.92.81.032.1100808111.5-4

6x10-4

7.5x10-4

2 x 10-3

7.5 x 10-4

0.51x10-4

7 x 10-2

1x10-4

3x10-4

15 x 10-4

4x10-2

4.6401x10-2

Dielectric Materials

ECE 598 JS Lecture 07 Nonideal Conductors andjsa.ece.illinois.edu/ece598js/Lect_07.pdf · For good conductors: = ... Dielectrics and Polarization Field causes the formation of dipoles

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