ECE 451
Advanced Microwave Measurements
Nonideal Properties of Interconnects
Jose E. Schutt-Aine
Electrical & Computer Engineering
University of Illinois
jschutt@emlab.uiuc.edu
ECE 451 – Jose Schutt‐Aine
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Material Medium


 H
 E  
t

   E
 H  J 
t


  E   j H
or
 

  H  J  j E


J E
: conductivity of material medium (‐1m‐1)


 

 
  H   E  j E  E   j   j 1 
E

j 


 
since    1 

j



then


 
2
 E    1 
E

j 

2
ECE 451 – Jose Schutt‐Aine
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Wave in Material Medium



 
2
2
 E    1 
E  E

j 


 
2
2
    1 

j



2
 is complex propagation constant

  j  1 
   j
j
 associated with attenuation of wave
 associated with propagation of wave
ECE 451 – 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

 

 1 
  1
 
2 



1/2

 

 1 
  1
 
2 



Magnetic field
Complex intrinsic impedance
j

  j

Eo  z  j z
H  yˆ e e

ECE 451 – 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

ECE 451 – Jose Schutt‐Aine
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Skin Depth
e-1

Wave motion
I
V
C
z
t
For perfect conductor,  = 0 and current only flows on the surface
ECE 451 – Jose Schutt‐Aine
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DC Resistance
l
Rdc 
 wt
l: conductor length
: conductivity
ECE 451 – Jose Schutt‐Aine
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AC Resistance
l
l
l  f
Rac 


 w  w 2 /  w 
l: conductor length
: conductivity
f: frequency
ECE 451 – 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 451 – Jose Schutt‐Aine
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Frequency-Dependent Resistance
Resistance changes with
2
2
f
 I
 I
 CL 2
2
z
t
1  f
Rac 
w

1
Rdc 
 wt
Resistance is ~ constant when  >t
ECE 451 – Jose Schutt‐Aine
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Reference Plane Current
Rac , ground
l  f

6h 
ECE 451 – 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 451 – 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
 Eo  J o  Eo 
 y /
 jy /

The voltage measured over a section of conductor of length D is:
V  Eo D 
Jo D

ECE 451 – 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
 J o w w
where  
1

is the bulk resistivity of the conductor
Z skin  Rskin  jX skin
with
Rskin  X skin
D

 f 
w
 Skin effect has reactive (inductive) component ECE 451 – 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 451 – 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 451 – Jose Schutt‐Aine
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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 451 – 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 451 – 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 451 – 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
19
Dielectric Constants
Material
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
r
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 451 – 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 451 – Jose Schutt‐Aine
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AC Variations
Reverses in polarity cause incremental changes in the static conductivity s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 451 – Jose Schutt‐Aine
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Dielectric Properties
Good Dielectrics:
e
1
 '


e  

J cd  j ' 1  j
 E  j ' E
 ' 

Good Conductors:
e
1
 '


e  

J cd  j ' 1  j
 E  eE
 ' 

ECE 451 – Jose Schutt‐Aine
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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 451 – Jose Schutt‐Aine
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Kramers-Kronig Relations
e is a relaxation time constant:
 rs'  2
e  '
 r  2
'
'



 r'     r'   r r2
1   e 
 r"   
ECE 451 – Jose Schutt‐Aine
'
'



 r r   e
1   e 
2
25
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 451 – Jose Schutt‐Aine
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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 451 – Jose Schutt‐Aine
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