ECE 546
Lecture 03
Resistance, Capacitance, Inductance
Spring 2014
Jose E. Schutt-Aine
Electrical & Computer Engineering
University of Illinois
jschutt@emlab.uiuc.edu
ECE 546 – Jose Schutt-Aine
1
What is Extraction?
.
-
+
-
+
-
-
-
+
+
+
+
-
-
+
Process in which a complex arrangement of
conductors and dielectric is converted into a netlist
of elements in a form that is amenable to circuit
simulation.
Need Field Solvers
ECE 546 – Jose Schutt-Aine
2
Electromagnetic Modeling Tools
We need electromagnetic modeling tools to analyze:
Transmission line propagation
Reflections from discontinuities
Crosstalk between interconnects
Simultaneous switching noise
So we can provide:
Improved design of interconnects
Robust design guidelines
Faster, more cost effective design cycles
ECE 546 – Jose Schutt-Aine
3
Field Solvers – History
1960s
Conformal mapping techniques
Finite difference methods (2-D Laplace eq.)
Variational methods
1970s
Boundary element method
Finite element method (2-D)
Partial element equivalent circuit (3-D)
1980s
Time domain methods (3-D)
Finite element method (3-D)
Moment method (3-D)
rPEEC method (3-D)
1990s
Adapting methods to parallel computers
Including methods in CAD tools
ECE 546 – Jose Schutt-Aine
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Categories of Field Solvers
• Method of Moments (MOM)
• Application to 2-D Interconnects
• Closed-Form Green’s Function
• Full-Wave and FDTD
• Parallel FDTD
• Applications
ECE 546 – Jose Schutt-Aine
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Capacitance
Relation: Q = Cv
Q: charge stored by capacitor
C: capacitance
v: voltage across capacitor
i: current into capacitor
dv dQ
i (t )  C

dt dt
1 t
v(t )   i ( )d
C 0
ECE 546 – Jose Schutt-Aine
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Capacitance
A : area
 oA
C= d
o : permittivity
For more complex capacitance geometries, need to use
numerical methods
ECE 546 – Jose Schutt-Aine
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Capacitance Calculation
 (r )   g (r , r ') (r ')dr '
 (r )  potential (known)
g (r , r ')  Green ' s function (known)
 (r ')  charge distribution (unknown)
Once the charge distribution is known, the total charge Q can
be determined. If the potential =V, we have
Q=CV
To determine the charge distribution, use the moment method
ECE 546 – Jose Schutt-Aine
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METHOD OF MOMENTS
Operator equation
L(f) = g
L = integral or differential operator
f = unknown function
g = known function
Expand unknown function f
f   nf n
n
ECE 546 – Jose Schutt-Aine
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METHOD OF MOMENTS
in terms of basis functions fn, with unknown coefficients  to get
n
  L (f
n
n
)g
n
Finally, take the scalar or inner product with testing of weighting
functions wm:

n
wm , Lf n  wm , g
n
Matrix equation
 lmn  n    g m 
ECE 546 – Jose Schutt-Aine
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METHOD OF MOMENTS
 w1 , Lf1

lmn    w2 , Lf1
 ..........
 1 
 n    2 
 . 
w1 , Lf 2
w2 , Lf 2
.............
...

...
...
 w1 , g

g

 m   w2 , g
 .




Solution for weight coefficients
1



l
 n   nm   g m 
ECE 546 – Jose Schutt-Aine
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Moment Method Solution
D  
E  
  x ', y ', z '
  x, y, z   
dx ' dy ' dz '
4 R

  

R
2

L  

 x  x '   y  y '   z  z '
2
2
2
Green’s function G: LG = d
ECE 546 – Jose Schutt-Aine
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Basis Functions
Subdomain bases



1 xn   x  xn 
P  xn   
2
2
1
otherwise

xn
x2
x1



xn   x  xn 
1  x
T  xn   
2
2
0
otherwise
x1
xn
x2
Testing functions often (not always) chosen same as basis
function.
ECE 546 – Jose Schutt-Aine
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Conducting Plate
  x ', y ', z '
  x, y, z    dx '  dy '
4 R
a
a
a
a
 = charge density on plate
ECE 546 – Jose Schutt-Aine
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Conducting Plate
Setting  = V on plate
R
a
V
 x  x '   y  y '
2
a
  x ', y ', z '
a
 dx '  dy '
a
2
4
 x  x '   y  y '
2
for |x| < a;
|y| < a
a
Capacitance of plate:
2
a
q 1
C    dx '  dy '  x ', y ', z '
V V a a
ECE 546 – Jose Schutt-Aine
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Conducting Plate
Basis function Pn

1


Pn  xm , yn   1

0

s
s
xm 
 x  xm 
2
2
s
s
yn 
 y  yn 
2
2
otherwise
Representation of unknown charge
N
  x, y     n f n
n 1
ECE 546 – Jose Schutt-Aine
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Conducting Plate
Matrix equation:
N
V   lmn f n
n 1
Matrix element:
lmn 
 dx '  dy '
xm
yn
1
4
 xm  x '   yn  x '
2
2
1 N
1
C    n sn   lmn
sn
V n1
mn
ECE 546 – Jose Schutt-Aine
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Parallel Plates
Using N unknowns per plate, we get 2N  2N matrix equation:
[l tt ] [l tb ] 

[l]   bt
bb

[l ] [l ]

Subscript ‘t’ for top and ‘b’ for bottom plate, respectively.
ECE 546 – Jose Schutt-Aine
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Parallel Plates
Matrix equation becomes  l mn

Solution:
tt
 ltbmn   nt    g mt 
tt
tb 1 

    l  l   g nt 

mn 

t
m
charge on top plate
Capacitance C 
2V
1
t



n sn
2V top
Using
s=4b2 and
all elements of
2

mn
C  2b  l
tt
g  V
t

tb 1
l
mn
ECE 546 – Jose Schutt-Aine
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Inductance
Relation:  = Li
: flux stored by inductor
L: inductance
i: current through inductor
v: voltage across inductor
di d 
v(t )  L 
dt dt
1 t
i (t )   v( )d
L 0
ECE 546 – Jose Schutt-Aine
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Inductance
Magnetic Flux
   B  dS
Total flux linked
Inductance =
Current
S
ECE 546 – Jose Schutt-Aine
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2-D Isomorphism
Electrostatics
Magnetostatics
 zˆ  nˆ  tVi  0
nˆ    ritVi   
qs
o
 zˆ  nˆ  t Azi  0
 1

nˆ   t Azi    o J z
 ri

CV  Q
LI  
Consequence: 2D inductance can be calculated from 2D capacitance formulas
ECE 546 – Jose Schutt-Aine
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2-D N-line LC Extractor using MOM
w
s
t
h
r
• Symmetric signal traces
• Uniform spacing
• Lossless lines
• Uses MOM for solution
ECE 546 – Jose Schutt-Aine
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Output from MoM Extractor
Capacitance (pF/m)
118.02299 -8.86533
-8.86533 119.04875
-0.03030 -8.86185
-0.00011 -0.03029
-0.00000 -0.00011
-0.03030
-8.86185
119.04876
-8.86185
-0.03030
-0.00011
-0.03029
-8.86185
119.04875
-8.86533
-0.00000
-0.00011
-0.03030
-8.86533
118.02299
Inductance (nH/m)
312.71680 23.42397 1.83394 0.14361 0.01128
23.42397 311.76042 23.34917 1.82812 0.14361
1.83394 23.34917 311.75461 23.34917 1.83394
0.14361 1.82812 23.34917 311.76042 23.42397
0.01128 0.14361 1.83394 23.42397 312.71680
ECE 546 – Jose Schutt-Aine
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RLGC: Formulation Method
Electrical and Computer Engineering
y
z
x

(x,y,z)
y=dn
y=dn-1
y=dn-2
y=dm
y=dm-1
y=dm-2
3

n
n-1

(x o ,y o ,z o )
2
1
t
t
2
r=1
70 m
1
P.E.C.
m
150 m
350 m
r=3.2
100 m
r=4.3
200 m
P.E.C.
m-1
Closed-Form Spatial Green's Function
* Computes 2-D and 3-D capacitance matrix in multilayered dielectric
* Method is applicable to arbitrary polygon-shaped conductors
* Computationally efficient
•
Reference
– K. S. Oh, D. B. Kuznetsov and J. E. Schutt-Aine, "Capacitance Computations in a Multilayered Dielectric
Medium Using Closed-Form Spatial Green's Functions," IEEE Trans. Microwave Theory Tech., vol. MTT42, pp. 1443-1453, August 1994.
ECE 546 – Jose Schutt-Aine
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Multilayer Green's Function
Optional Top Ground Plane
y=d Nd
Nd, o
Nd
y=d Nd-1
2, o
1, o
y=d 2
2
1
y=d 1
y
x
y=0
Bottom Ground Plane
ECE 546 – Jose Schutt-Aine
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Extraction Program: RLGC
RLGC computes the four transmission line parameters, viz., the capacitance matrix C, the
inductance matrix L, the conductance matrix G, and the resistance matrix R, of a multiconductor
transmission line in a multilayered dielectric medium. RLGC features the following list of
functions:
h
ECE 546 – Jose Schutt-Aine
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RLGC – Multilayer Extractor
•
Features
–
–
–
–
–
–
–
–
•
Handling of dielectric layers with no ground plane, either top or bottom ground plane
(microstrip cases), or both top and bottom ground planes (stripline cases)
Static solutions are obtained using the Method of Moment (MoM) in conjunction with the
recently-developed closed-form Green’s functions: one of the most accurate and efficient
methods for static analysis
Modeling of dielectric losses as well as conductor losses (including ground plane losses
The resistance matrix R is computed based on the current distribution - more accurate than
the use of any closed-form formulae
Both the proximity effect and the skin effect are modeled in the resistance matrix R.
Computes the potential distribution
Handling of an arbitrary number of dielectric layers as well as an arbitrary number of
conductors.
The cross section of a conductor can be arbitrary or even be infinitely thin
Reference
–
K. S. Oh, D. B. Kuznetsov and J. E. Schutt-Aine, "Capacitance Computations in a
Multilayered Dielectric Medium Using Closed-Form Spatial Green's Functions," IEEE
Trans. Microwave Theory Tech., vol. MTT-42, pp. 1443-1453, August 1994.
ECE 546 – Jose Schutt-Aine
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RLGC – General Topology
Three conductors in a layered medium. All conductor
dimensions and spacing are identical. The loss tangents of
the lower and upper dielectric layers are 0.004 and
0.001respectively, the conductivity of each line is 5.8e7 S/m,
and the operating frequency is 1 GHz
ECE 546 – Jose Schutt-Aine
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3-Line Capacitance Results
Capacitance Matrix (pF/m)
142.09 21.765 0.8920
21.733 93.529 18.098



0.8900 18.097 87.962 

Delabare et al.
145.33 23.630 1.4124
22.512 93.774 17.870



1.3244 17.876 87.876 

RLGC Method
ECE 546 – Jose Schutt-Aine
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Modeling Vias
l1
Possible P.E.C.
Nd
y=dn
y=dn-1
2
1
l3
l2
y=d1
y
x
y=0
Possible P.E.C.
Medium
Via
1.6
2
4
l2
4.02
50
l1
3.2
1
50
l3
4
4.02
Via in multilayer medium
Equivalent circuit
ECE 546 – Jose Schutt-Aine
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Modeling Discontinuities
l1
l
w1
w
ce
ce
l2
w2
Open
l1
w1
l2
l1
l
Bend
l3
l1
l2
l3
l2
l1
l1
l2
w1
w2
ce
w2
l2
ce
Step
T-Junction
ECE 546 – Jose Schutt-Aine
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3D Inductance Calculation
Loop Inductance
Lloop
 1
1
   B  da     A  da
I I a
I a


QUESTION: Can we associate inductance with piece
of conductor rather than a loop? PEEC Method
ECE 546 – Jose Schutt-Aine
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Partial Inductance (PEEC) Approach
QUESTION: Can we associate inductance
with piece of conductor rather than a loop?
I
I

a
I
I
ak
1
 dli  dl j
Lloop   
dai da j
  
i1 j1 ai a j aia j li l j 4 ri  rj
4
4
DEFINITION OF PARTIAL INDUCTANCE
dli  dl j
1 
Lpij 
dai da j
  
ai a j 4 a a l l ri  rj
i j i j
4
4
Lloop    sij L pij
i1 j1
ECE 546 – Jose Schutt-Aine
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Circuit Element K
[K]=[L]-1
• Better locality property
• Leads to sparser matrix
• Diagonally dominant
• Allows truncation of far off-diagonal elements
• Better suited for on-chip inductance analysis
ECE 546 – Jose Schutt-Aine
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Locality of K Matrix
l
h
1
w
11.4
 4.26

[ L]   2.54

1.79
1.38
4.26
11.4
4.26
2.54
1.79
2
3
4
5
s
2.54
4.26
11.4
4.26
2.54
1.79
2.54
4.26
11.4
4.26
1.38 
1.79 


2.54

4.26 
11.4 
 103
 34.1

[ K ]   7.80

 4.31
 3.76
ECE 546 – Jose Schutt-Aine
34.1
114
31.6
6.67
4.31
7.80
31.6
115
31.6
7.80
4.31
6.67
31.6
114
34.1
3.76
4.31


7.80

34.1
103 
36
Package Inductance & Capacitance
Component
Capacitance
(pF)
68 pin plastic DIP pin†
4
68 pin ceramic DIP pin††
7
68 pin SMT chip carrier †
2
Inductance
(nH)
35
20
7
† No ground plane; capacitance is dominated by wire-to-wire
component.
†† With ground plane; capacitance and inductance are determined by
the distance between the lead frame and the ground plane, and the lead
length.
ECE 546 – Jose Schutt-Aine
37
Package Inductance & Capacitance
Component
Capacitance Inductance
(pF)
(nH)
68 pin PGA pin††
2
7
256 pin PGA pin††
5
15
Wire bond
1
1
Solder bump
0.5
0.1
† No ground plane; capacitance is dominated by wire-towire component.
†† With ground plane; capacitance and inductance are
determined by the distance between the lead frame and
the ground plane, and the lead length.
ECE 546 – Jose Schutt-Aine
38
Metallic Conductors

Area
th
Leng
Re s ist an ce : R
Length
R
  Area
Submicron level:
W=0.25 microns
R=422 /mm
Package level:
W=3 mils
R=0.0045 /mm
ECE 546 – Jose Schutt-Aine
39
Metallic Conductors
Metal
Conductivity
 -1 m1 10-7)
Silver
Copper
Gold
Aluminum
Tungsten
Brass
Solder
Lead
Mercury
ECE 546 – Jose Schutt-Aine
6.1
5.8
3.5
1.8
1.8
1.5
0.7
0.5
0.1
40
Dielectrics
Dielectrics contain charges that are tightly
bound to the nuclei
Charges can move a fraction of an atomic
distance away from equilibrium position
Electron orbits can be distorted when an
electric field is applied
+
+
-
E
ECE 546 – Jose Schutt-Aine
41
Dielectrics
Charge density within volume is zero
Surface charge density is nonzero
sp
D=o(1+e)E=E
ECE 546 – Jose Schutt-Aine
42
Dielectric Materials
v
1
LC
Conductivity (-1-m-1)
2.2
0.0016
10-10-10-14
0.510-17
Material
Germanium
Silicon
Glass
Quartz
tan d 


ECE 546 – Jose Schutt-Aine
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Dielectric Materials
v
1
LC
Material
r
v(cm/s)
Polyimide
Silicon dioxide
Epoxy glass (FR4)
Alumina (ceramic)
2.5-3.5
3.9
5.0
9.5
16-19
15
13
10
ECE 546 – Jose Schutt-Aine
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Conductivity of Dielectric Materials
r j i
Material
Conductivity ( -1 m-1)
Germanium
2.2
Silicon
0.0016
Glass
10-10 - 10-14
Quartz
0.5 x 10-17

Loss TANGENT : tand 
ECE 546 – Jose Schutt-Aine
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Combining Field and Circuit Solutions
Field Solution
Network
Description
 Bypass extraction
procedure through the use
of Y, Z, or S parameters
(frequency domain)
ECE 546 – Jose Schutt-Aine
Macromodel
Generation
Circuit
Simulation
46
Full-Wave Methods
 E  
B
t
 H  J 
 D  
B  0
Faraday’s Law of Induction
D
t
Ampère’s Law
Gauss’ Law for electric field
Gauss’ Law for magnetic field
FDTD: Discretize equations and solve
with appropriate boundary conditions
ECE 546 – Jose Schutt-Aine
47
Finite Difference Time Domain (FDTD)
z
E y
H z
y
x
E x
E x
Ez
Ez
E y
H x
H y
H y
E y
H x
Ez
E x
E x
H z
E y
Space Discretization
ECE 546 – Jose Schutt-Aine
48
FDTD – Yee Algorithm
E  i, j , k   E
n
x
z
Ey
y
Hz
x
Ex
Ex
c t

H zn1/2  i, j , k   H zn1/2  i, j  1, k  

 y
c t
n 1/2
n 1/2

H
i
,
j
,
k

H


 i, j, k  1 

y
y
 z
Ez
Ez
Ey
n 1
x
Hx
Hy
Hy
Ey
Hx
Ez
Ex
Ex
Hz
Ey
H
n 1/2
x
c t n
n
E
i
,
j

1,
k

E
 i, j , k   H 



z
z  i, j , k  
 y
c t n1/2

E y  i, j , k  1  E yn  i, j , k  

 z
n 1/2
x
ECE 546 – Jose Schutt-Aine
49
2D-FDTD
Ey
Ex
Hz
y
x
1
 i  1 , j  t H n 1/2  i  1 , j  1   H n1/2  i  1 , j  1  
Enx  i  , j  En1
x
z
 2 
 2   o y  z
 2
 2
2
2 
1
 i, j  1   t  H n1/2  i  1 , j  1   H n1/2  i  1 , j  1  
Eny  i, j    En1
y
z
z


 2
 2
2
2   o x 
2
2 
1
1
1
1
t  n  1
n 1/2 
  E n  i  1 , j 
i

,
j


H
i

,
j


E
i

,
j
1
z
x
x
 2
 2

2
2   o y   2
2 
n1/2 
Hz
-
t  n 
1
1
n
E y  i 1, j    Ex  i, j   
 o x 
2
2 
ECE 546 – Jose Schutt-Aine
50
Absorbing Boundary Condition: 2D-PML Formulation
Simulation Medium
E x Hz
o

t
y
E y
H
o
 z
t
x
H z E x E y
o


t
y
x
PML Medium
y
E x
Hz
o
 E x 
t
y
E y
Hz
o
 E y  
t
x
H z
E x E y
o
  * Hz 

t
y
x
x
 *

 o o
No reflection from PML interface
ECE 546 – Jose Schutt-Aine
51
FDTD – Coupled Microstrips
s/ h=0 .8
.
0 .1 0 7
s/ h=1 .0
-0 .0 1
s/ h=1 .2
-0 .0 1 1
0 .1 0 6
C12 (nF/m)
-0 .0 1 2
C11 (nF/m)
0 .1 0 5
0 .1 0 4
-0 .0 1 3
-0 .0 1 4
s/ h=0 .8
-0 .0 1 5
s/ h=1 .
s/ h=1 .2
0 .1 0 3
-0 .0 1 6
w
0 .1 0 2
0
5
10
15
Frequency (GHz)
20
s
w
-0 .0 1 7
0
25
5
10
15
Frequency (GHz)
20
25
h
Parameters: w=0.3mm, h=0.25mm. Dielectric constant is 4.5.
s/ h=0 .8
360
s/ h=1 .2
100
355
s/ h=1 .0
95
s/ h=1 .2
350
L12 (nH/m)
L11 (nH/m)
s/ h=1 .0
105
s/ h=0 .8
345
340
90
85
80
75
70
335
0
5
10
15
Frequency (GHz)
20
25
65
0
ECE 546 – Jose Schutt-Aine
5
10
15
Fr equency ( GHz)
20
25
52
Parallel-FDTD - Motivations
PCB interconnect EM problems
1. Interconnect coupling / crosstalk
2. Signal distortion
3. Radiation (EMI)
4. etc
PCB modeling challenges:
1. Size of the problem - determined by the ratio of
board size and finest features
2. Simulations time
Solutions: Supercomputer
Mini-computer cluster
ECE 546 – Jose Schutt-Aine
Gflops/dollar
53
Parallel-FDTD
Scalability
Speed vs. problem size
ECE 546 – Jose Schutt-Aine
54
Parallel-FDTD
Physical layout
Parser/translator
Mesh generator
GDSII, Mentor Graphic
Parallel Mesh
Generator
EM modeling
Circuit simulation
ECE 546 – Jose Schutt-Aine
55
Parallel-FDTD
Max( computation/communication)
Maximize the ratio of volume to surface
Partition along the largest dimension
Node 1
ECE 546 – Jose Schutt-Aine
Node 2
Node 3
Node 4
56
Parallel-FDTD
A cellular phone PCB test board
• six layered test board
• over 4000 vias
• dielectric materials FR 4
• total thickness < 20 mils
• over 60 traces
• around 30 PEC fills
• around 300 component pads
ECE 546 – Jose Schutt-Aine
57
Parallel-FDTD
Comparison between measurement and simulation
S21
S11
ECE 546 – Jose Schutt-Aine
58
Parallel-FDTD
S31
•
S41
Reference
–
F. Liu, J. Schutt-Ainé and J. Chen, "Full Wave Analysis and Modeling of Multiconductor
Transmission Lines via 2-D-FDTD and Signal-Processing Techniques," IEEE Trans.
Microwave Theory Tech. vol. 50, pp. 570-577, February 2002.
ECE 546 – Jose Schutt-Aine
59