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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 Illinoisjesa@illinois.edu
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 20122
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
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 20123
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
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 20124
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
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 20125
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.
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 20126
Skin Depth
I VCz t
∂ ∂− =
∂ ∂δ
e-1Wave motion
For perfect conductor, δ = 0 and current only flows on the surface
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 20127
DC Resistance
dclRwtσ
= l: conductor lengthσ: conductivity
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 20128
AC Resistance
2 /acl l l fRw ww
πμσ δ σσ ωμσ
= = =
l: conductor lengthσ: conductivityf: frequency
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 20129
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
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201210
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
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201211
, 6ac groundl fRh
πμσ
≈
Reference Plane Current
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201212
εr
H. A. Wheeler, "Formulas for the skin effect," Proc. IRE, vol. 30, pp. 412-424,1942
Skin Effect in Microstrip
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201213
/ /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:
Skin Effect in Microstrip
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201214
( ) ( )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
Skin Effect in Microstrip
is the bulk resistivity of the conductor
skin skin skinZ R jX= +
with
Skin effect has reactive (inductive) component
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201215
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
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201216
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
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201217
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
=
=
⎧+ <⎪
⎪= ⎨⎪ + ≥⎪⎩
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201218
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
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201219
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
=
=
⎧+ <⎪
⎪= ⎨⎪ + ≥⎪⎩
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201220
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− −
= −+ −
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201221
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
=
=
⎧+ <⎪
⎪= ⎨⎪ + ≥⎪⎩
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201222
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.
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201223
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.
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201224
( )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
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201225
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
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201226
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
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201227
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
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201228
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
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201229
( )
( )
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
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201230
( )' "i c i sH J J j E J E j j E∇× = + + = + + −ωε σ ω ε ε
Complex Permittivity
( )" ' '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
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201231
't i ce de i eJ J J J J E j Eσ ωε= + + = + +
Complex Permittivity
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
+= = = = +
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201232
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⎛ ⎞= + = + = − = −⎜ ⎟⎝ ⎠
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201233
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:
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201234
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:
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201235
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
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201236
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
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201237
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
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 201238
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
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