Download Lecture 07

ECE 546
Lecture - 07
Nonideal Conductors and
Dielectrics
Spring 2014
Jose E. Schutt-Aine
Electrical & Computer Engineering
University of Illinois
[email protected]
ECE 546 – Jose Schutt-Aine
1
Material Medium
 H
 E  
t
 E   j H
or
 E
 H  J 
t
 H  J  j E
J sE
s: conductivity of material medium (W-1m-1)

s 
  H  s E  j E  E s  j   j 1 
E

j 


s 
since    1 

j



then

s 
 E    1 
E

j 

2
ECE 546 – Jose Schutt-Aine
2
2
Wave in Material Medium

s 
2
 E    1 
E


E

j 


s 
 2   2  1 

j



2
2
 is complex propagation constant
s
  j  1 
   j
j
: associated with attenuation of wave
: associated with propagation of wave
ECE 546 – Jose Schutt-Aine
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Wave in Material Medium
ˆ o e  z  xE
ˆ o e  z e  j z decaying exponential
Solution: E  xE
 
 
 
2
 
2
1/2

s 

 1 
  1
2 
 



1/2

s 

 1 
  1
2 
 



Magnetic field
H  yˆ
Eo

Complex intrinsic
impedance
j

s  j
e  z e  j  z
ECE 546 – Jose Schutt-Aine
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Skin Depth
ˆ o e  z  xE
ˆ o e  z e  j z  Wave decay
E  xE
The decay of electromagnetic wave propagating into
a conductor is measured in terms of the skin depth
Definition: skin depth  is distance over which
amplitude of wave drops by 1/e.

1

For good conductors:  
2
s
ECE 546 – Jose Schutt-Aine
5
Skin Depth
e-1

Wave motion
I
V
C
z
t
For perfect conductor,  = 0 and current only
flows on the surface
ECE 546 – Jose Schutt-Aine
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DC Resistance
l
Rdc 
s wt
l: conductor length
s: conductivity
ECE 546 – Jose Schutt-Aine
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AC Resistance
l
l
l  f
Rac 


s w s w 2 / s w s
l: conductor length
s: conductivity
f: frequency
ECE 546 – Jose Schutt-Aine
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Frequency-Dependent Resistance
J  1

J e
0
 z
dz 
J

 J
Approximation is to assume that all the current
is flowing uniformly within a skin depth
ECE 546 – Jose Schutt-Aine
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Frequency-Dependent Resistance
Resistance
changes2with
2
f
 I
 I
 CL 2
2
z
t
1  f
Rac 
w s
1
Rdc 
s wt
Resistance is ~ constant when  >t
ECE 546 – Jose Schutt-Aine
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Reference Plane Current
Rac , ground 
l  f
6h s
ECE 546 – Jose Schutt-Aine
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Skin Effect in Microstrip
r
H. A. Wheeler, "Formulas for the skin effect," Proc. IRE, vol. 30, pp. 412-424,1942
ECE 546 – Jose Schutt-Aine
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Skin Effect in Microstrip
Current density varies as
J  J o e y / e jy /
Note that the phase of the current density varies as
a function of y

J o w
I   J o we e
dy 
1 j
0
Jo
s Eo  J o  Eo 
 y /
 jy /
s
The voltage measured over a section of conductor
of length D is:
V  Eo D 
Jo D
s
ECE 546 – Jose Schutt-Aine
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Skin Effect in Microstrip
The skin effect impedance is
Z skin
V J o D 1  j  D
 
 1  j   f 
I
s J o w w
where  
1
s
is the bulk resistivity of the conductor
Z skin  Rskin  jX skin
with
Rskin  X skin 
D
 f s
w
 Skin effect has reactive (inductive) component
ECE 546 – Jose Schutt-Aine
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Internal Inductance
The internal inductance can be calculated
directly from the ac resistance
Linternal 
Rac


Rskin

 Skin effect resistance goes up with frequency
 Skin effect inductance goes down with frequency
ECE 546 – Jose Schutt-Aine
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Surface Roughness
Copper surfaces are rough to facilitate adhesion to
dielectric during PCB manufacturing
When the tooth height is comparable to the skin
depth, roughness effects cannot be ignored
Surface roughness will increase ohmic losses
ECE 546 – Jose Schutt-Aine
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Hammerstad Model
 K H Rs
RH  f   
 Rdc
f
when   t
when   t

RH  f 
when   t
 Lexternal  2 f

LH  f   
RH  f t 
L

when   t
 external
2 f t
ECE 546 – Jose Schutt-Aine
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Hammerstad Model
2

2
 hRMS  
K H  1  arctan 1.4 
 


 
 
hRMS: root mean square value of surface roughness
height
: skin depth
f=t:
frequency where the skin depth is equal to
the thickness of the conductor
ECE 546 – Jose Schutt-Aine
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Hemispherical Model
 K hemi Rs
Rhemi  f   
 Rdc
f
when   t
when   t

Rhemi  f 
when   t
 Lexternal  2 f

Lhemi  f   
Rhemi  f t 
L
when   t
external 

2 f t
ECE 546 – Jose Schutt-Aine
19
Hemispherical Model
K hemi
Ks 
 1 when K s  1

 K s when K s  1
Re   3 / 4k 2   1   1     o / 4  Atile  Abase 
 o / 4  Atile
2
j
3 1    / r  1  j 
 1    kr 
3
1   / 2r  1  j 
 1  
2
1


4
j
/
k
r  1/ 1  j  
3
2j
 kr 
3
1   2 j / k 2 r  1/ 1  j  
ECE 546 – Jose Schutt-Aine
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Huray Model
 K Huray Rs
RHuray  f   
 Rdc
f
when   t
when   t

RHuray  f 
when   t
 Lexternal 
2 f

LHuray  f   
RHuray  f t 
L

when   t
 external
2 f t

ECE 546 – Jose Schutt-Aine
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Huray Model
K Huray 
Pflat  PN _ spheres
Pflat
2 3
1





  Re   H o
 1  1 
2
k
2
n
n1
N
PN _ spheres
Pflat   o / 4  Atile
   o /  o '
2
j
3 1    / r  1  j 
 1    kr 
3
1   / 2r  1  j 
Ho: magnitude of
applied H field.
2
2
j
3 1   4 j / k r  1/ 1  j  
 1    kr 
3
1   2 j / k 2 r  1/ 1  j  
ECE 546 – Jose Schutt-Aine
22
Dielectrics and Polarization
No field
Applied
field
Field causes the formation of dipoles  polarization
Bound surface charge density –qsp on upper
surface and +qsp on lower surface of the slab.
ECE 546 – Jose Schutt-Aine
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Dielectrics and Polarization
D   o Ea  P   o Ea   o  e Ea   o 1   e  Ea   s Ea
P : polarization vector
D : electric flux density
Ea : applied electric field
 e : electric susceptibility
 o : free-space permittivity
 s : static permittivity
ECE 546 – Jose Schutt-Aine
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Dielectric Constant
Material
Air
Styrofoam
Paraffin
Teflon
Plywood
RT/duroid 5880
Polyethylene
RT/duroid 5870
Glass-reinforced teflon (microfiber)
Teflon quartz (woven)
Glass-reinforced teflon (woven)
Cross-linked polystyrene (unreinforced)
Polyphenelene oxide (PPO)
Glass-reinf orced polystyrene
Amber
Rubber
Plexiglas
ECE 546 – Jose Schutt-Aine
r
1.0006
1.03
2.1
2.1
2.1
2.20
2.26
2.35
2.32-2.40
2.47
2.4-2.62
2.56
2.55
2.62
3
3
3.4
25
Dielectric Constants
Material
r
Lucite
Fused silica
Nylon (solid)
Quartz
Bakelite
Formica
Lead glass
Mica
Beryllium oxide (BeO)
Marble
Flint glass
Ferrite (FqO,)
Silicon (Si)
Gallium arsenide (GaAs)
Ammonia (liquid)
Glycerin
Water
3.6
3.78
3.8
3.8
4.8
5
6
6
6.8-7.0
8
10
12-16
12
13
22
50
81
ECE 546 – Jose Schutt-Aine
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AC Variations
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.
ECE 546 – Jose Schutt-Aine
27
AC Variations
Reverses in polarity cause incremental changes in
the static conductivity ss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.
ECE 546 – Jose Schutt-Aine
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Complex Permittivity
 r'  1 
N eQ 2 2
o   2 

om
2

 o 

2 2
 d
  
 m
2


d


2 
NQ
m

 r"  e 
2
 om  2
 d 
2 2
  o        
 m 

s
o 
m
N e : dipole density
m : mass
Q : dipole charge
d : damping coefficient
 o : free space permittivity
o: natural frequency
s : spring (tension) factor
 : applied frequency
ECE 546 – Jose Schutt-Aine
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Complex Permittivity
 H  J i  J c  j E  J i  s s E  j  ' j " E
  H  J i  s s   " E  j ' E  J i  s e E  j ' E
s e  equivalent conductivity  s s   "  s s  s a
s a  alternating field conductivity   "
s s  static field conductivity
se: total conductivity composed of the static
portion ss and the alternative part sa caused by the
rotation of the dipoles
ECE 546 – Jose Schutt-Aine
30
Complex Permittivity
J t  J i  J ce  J de  J i  s e E  j ' E
J i : total electric current density
J t : impressed (source) electric current density
J ce : effective electric conduction current density
J de : effective displacement electric current density
se 

J t  J i  s e E  j ' E  J i  j ' 1  j
 E  J i  j ' 1  j tan  e  E
 ' 

se ss sa ss sa
tan  e  effective electric loss tangent 



 '
 '
 '  '
ECE 546 – Jose Schutt-Aine
31
Complex Permittivity
ss  "
 e"
tan  e 

 tan  s  tan  a  '
 '  '
e
ss
tan  s  static electric loss tangent 
 '
sa "
tan  a  alternating electric loss tangent 
 '
 ' 
se 

J cd  J ce  J de  s e E  j ' E  j ' 1  j
 E  j ' 1  j tan  e  E
 ' 

ECE 546 – Jose Schutt-Aine
32
Dielectric Properties
Good Dielectrics:
se
 '
se 

J cd  j ' 1  j
E
 ' 

Good Conductors:
se
 '
1
j ' E
1
se 

J cd  j ' 1  j
 E seE
 ' 

ECE 546 – Jose Schutt-Aine
33
Dielectric Properties
Good Dielectrics:
se
 '
se 

J cd  j ' 1  j
E
 ' 

Good Conductors:
se
 '
1
j ' E
1
se 

J cd  j ' 1  j
 E seE
 ' 

ECE 546 – Jose Schutt-Aine
34
Kramers-Kronig Relations
There is a relation between the real and
imaginary parts of the complex permittivity:
 r'    1 
2

 '  r"  '
   ' 
2
0
 r"   

2
'

2 1   r  '

  ' 
0
2

2
d '
d '
Debye Equation
'
'



 r ( )   r' ( )  j r" ( )   r'   rs r
1  j e
ECE 546 – Jose Schutt-Aine
35
Kramers-Kronig Relations
e is a relaxation time constant:
 rs'  2
e  '
 r  2
'
'



 r'     r'   rs r 2
1   e 
 r"   
ECE 546 – Jose Schutt-Aine
'
'



 rs r  e
1   e 
2
36
Dielectric Materials
Material
Air
Alcohol (ethyl)
Aluminum oxide
Bakelite
Carbon dioxide
Germanium
Glass
Ice
Mica
Nylon
Paper
Plexiglas
Polystyrene
Porcelain
r ’
1.0006
25
8.8
4.74
1.001
16
4*7
4.2
5.4
3.5
3
3.45
2.56
6
tan
0.1
6 x 10-4
22x10-3
1 x 10-3
0.1
6x10-4
2x10-2
8 x 10-3
4 x 10-2
5x10-5
14x10-3
ECE 546 – Jose Schutt-Aine
37
Dielectric Materials
Material
Pyrex glass
Quartz (fused)
Rubber
Silica (fused)
Silicon
Snow
Sodium chloride
Soil (dry)
Styrofoam
Teflon
Titanium dioxide
Water (distilled)
Water (sea)
Water (dehydrated)
Wood (dry)
r ’
4
3.8
2.5-3
3.8
11.8
3.3
5.9
2.8
1.03
2.1
100
80
81
1
1.5-4
tan
6x10-4
7.5x10-4
2 x 10-3
7.5 x 10-4
0.5
1x10-4
7 x 10-2
1x10-4
3x10-4
15 x 10-4
4x10-2
4.64
0
1x10-2
ECE 546 – Jose Schutt-Aine
38
PCB Stackup
Source: H. Barnes et al, "ATE Interconnect Performance to 43 Gps Using Advanced PCB Materials", DesignCon 2008
ECE 546 – Jose Schutt-Aine
39
Differential Signaling
Differential signaling is widely used in the industry today. High-speed
serial interfaces such as PCI-E, XAUI, OC768, and CEI use differential
signaling for transmitting and receiving data in point-to-point topology
between a driver (TX) and receiver (RX) connected by a differential pair.
The skew (time delay) between the two traces of the
differential pair should be zero. Any skew between the
two traces causes the differential signal to convert into
a common signal.
ECE 546 – Jose Schutt-Aine
40
Fiber Weave Effect
Fiberglass weave pattern causes signals to propagate at
different speeds in differential pairs
Source: S. McMorrow, C. Heard, "The Impact of PCB Laminate Weave on the
Electrical Performance of Differential Signaling at Multi-Gigabit Data Rates",
DesignCon 2005.
ECE 546 – Jose Schutt-Aine
41
Fiber Weave Effect
Source: Lambert Simonovich, "Practical Fiber Weave Effect Modeling",
White Paper-Issue 3, March 2, 2012.
ECE 546 – Jose Schutt-Aine
42
Fiber Weave Effect
Source: S. Hall and H. Heck , Advanced Signal Integrity for High-Speed Digital
Designs, J. Wiley, IEEE , 2009.
ECE 546 – Jose Schutt-Aine
43
Fiber Weave Effect
Group delay variation
Source: S. McMorrow, C. Heard, "The Impact of PCB Laminate Weave on the
Electrical Performance of Differential Signaling at Multi-Gigabit Data Rates",
DesignCon 2005.
ECE 546 – Jose Schutt-Aine
44
Fiber Weave Effect
Group delay variation: effect of angle
Source: S. McMorrow, C. Heard, "The Impact of PCB Laminate Weave on the
Electrical Performance of Differential Signaling at Multi-Gigabit Data Rates",
DesignCon 2005.
ECE 546 – Jose Schutt-Aine
45
Fiber Weave Effect
Straight traces
Source: S. Hall and H. Heck , Advanced Signal Integrity for High-Speed Digital
Designs, J. Wiley, IEEE , 2009.
ECE 546 – Jose Schutt-Aine
46
Fiber Weave Effect
45o traces
Source: S. Hall and H. Heck , Advanced Signal Integrity for High-Speed Digital
Designs, J. Wiley, IEEE , 2009.
ECE 546 – Jose Schutt-Aine
47
Fiber Weave Effect
straight
traces
zig-zag
traces
Source: PCB Dielectric Material Selection and Fiber Weave Effect on High-Speed Channel Routing, Altera Application Note
AN-528-1.1, January 2011.
ECE 546 – Jose Schutt-Aine
48
Fiber Weave Effect
Skew on
straight
traces
Source: PCB Dielectric Material Selection and Fiber Weave Effect on High-Speed Channel Routing, Altera Application Note
AN-528-1.1, January 2011.
ECE 546 – Jose Schutt-Aine
49
Fiber Weave Effect
Skew on
zig-zag
traces
Source: PCB Dielectric Material Selection and Fiber Weave Effect on High-Speed Channel Routing, Altera Application Note
AN-528-1.1, January 2011.
ECE 546 – Jose Schutt-Aine
50
Fiber Weave Effect
• Mitigation Techniques
Use wider widths to achieve impedance targets.
 Specify a denser weave (2116, 2113, 7268, 1652)
compared to a sparse weave (106, 1080).
 Move to a better substrate such as Nelco 4000-13
 Perform floor planning such that routing is at an angle
rather than orthogonal.
 Make use of zig-zag routing
ECE 546 – Jose Schutt-Aine
51