cse 245: computer aided circuit simulation and verification
DESCRIPTION
CSE 245: Computer Aided Circuit Simulation and Verification. Winter 2003 Lecture 1: Formulation. Instructor: Prof. Chung-Kuan Cheng. Agenda. RCL Network Sparse Tableau Analysis Modified Nodal Analysis. History of SPICE. SPICE -- Simulation Program with Integrated Circuit Emphasis - PowerPoint PPT PresentationTRANSCRIPT
CSE 245: Computer Aided Circuit Simulation and Verification
Instructor:Prof. Chung-Kuan Cheng
Winter 2003
Lecture 1: Formulation
Jan. 24, 2003 Cheng & Zhu, UCSD @ 20032
Agenda RCL Network Sparse Tableau Analysis Modified Nodal Analysis
Jan. 24, 2003 Cheng & Zhu, UCSD @ 20033
History of SPICE SPICE -- Simulation Program with Integrat
ed Circuit Emphasis 1969, CANCER developed by Laurence Nag
el on Prof. Ron Roher’s class 1970~1972, CANCER program May 1972, SPICE-I release July ’75, SPICE 2A, …, 2G Aug 1982, SPICE 3 (in C language) No new progress on software package sinc
e then
Jan. 24, 2003 Cheng & Zhu, UCSD @ 20034
RCL circuit
Vs
R L
C
svi
v
ri
v
l
c 0
1
10
0
0
1
2
1
2
l
vi
v
l
r
l
ci
vs
0
1
10
1
2
1
2
Jan. 24, 2003 Cheng & Zhu, UCSD @ 20035
RCL circuit (II) General Circuit Equation
Consider homogeneous form first
BUAYY
AYY
0YeY At
...!
...!2!1
22
k
tAtAAtIe
kkAt
Q: How to Compute Ak ?
and
Jan. 24, 2003 Cheng & Zhu, UCSD @ 20036
Assume A has non-degenerate eigenvalues
and corresponding linearly independent eigenvectors , then A can be decomposed as
where and
Solving RCL Equation
1A
k ,...,, 21
k ,...,, 21
k
0
0
00
2
1
k ,...,, 21
Jan. 24, 2003 Cheng & Zhu, UCSD @ 20037
What’s the implication then?
To compute the eigenvalues:
Solving RCL Equation (II)
1A
01
1 ...)det( ccAI nn
n
0)...(...))(( 212
0 ppp
realeigenvalue Conjugative
Complexeigenvalue
212A 212A
tAt ee 1 where
ke
e
e
e t
0
0
002
1
Jan. 24, 2003 Cheng & Zhu, UCSD @ 20038
Solving RCL Equation (III)
In the previous example
)0(
)0(
1
2//1
/10
01
2
i
veXe
i
v tlrl
c
At
0
0
11
10 11A
2
31
2
31
j
j
where
1
2
31
12
31
3 j
jj
2
31
2
3111
1 jj
hence
e
ee At
0
01
Let c=r=l=1, we have
Jan. 24, 2003 Cheng & Zhu, UCSD @ 20039
What if matrix A has degenerated eigenvalues? Jordan decomposition !
Solving RCL Equation (IV)
JA 1
J is in the Jordan Canonical form
And still JtAt ee 1
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200310
Jordan Decomposition
0
1J
t
ttJt
e
teete
00
1
10
01
00
10
01
J
t
tt
ttt
Jt
e
tee
et
tee
te
00
0!2
00
10
01
100
010
001
2
similarly
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200311
Agenda RCL Network Sparse Tableau Analysis Modified Nodal Analysis
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200312
Equation Formulation KCL
Converge of node current KVL
Closure of loop voltage Brach equations
I, R relations
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200313
Types of elements Resistor Capacitor
Inductor
L is even dependent on frequency due to skin effect, etc…
Controlled Sources VCVS, VCCS, CCVS, CCCS
dt
dvvc
dt
dv
v
vQ
dt
dQi )(
)(
dt
diil
dt
di
i
i
dt
dv )(
)(
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200314
Cut-set analysis
1. Construct a spanning tree
2. Take as much capacitor branches as tree branches as possible
3. Derive the fundamental cut-set, in which each cut truncates exactly one tree branch
4. Write KCL equations for each cut
5. Write KVL equations for each tree link
6. Write the constitution equation for each branch
1
2
3
4
5
6
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200315
0
0
0
0
0
11000000
10110000
00011010
00000110
00000011
56
46
45
35
34
24
13
12
46
45
34
24
12
5646453534241312
i
i
i
i
i
i
i
i
c
c
c
c
ciiiiiiii
KCL Formulation
0
iA
1
2
3
4
5
6
#nodes-1 lines
#braches columns
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200316
KCL Formulation (II) Permute the columns to achieve a
systematic form
0~
|
iAI
0
0
0
0
0
100
110
011
001
001
10000
01000
00100
00010
00001
56
23
13
46
45
34
24
12
46
45
34
24
12
i
i
i
i
i
i
i
i
c
c
c
c
c5623134645342412 iiiiiiii
1
2
3
4
5
6
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200317
KVL Formulation
56
46
45
35
34
24
13
12
46
45
34
24
12
11000
10000
01000
01100
00100
00010
00111
00001
v
v
v
v
v
v
v
v
v
v
v
v
v
VeAT
0
10011000
01001100
00100111
56
35
13
46
45
34
24
12
56
35
13
v
v
v
v
v
v
v
v
l
l
l
0BV
Remove the equations for tree braches and systemize
IBB~
1
2
3
4
5
6
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200318
Cut & Loop relation
0
BV
VeAT
0TBA
0]~
][~[ TAIIB
0~
TAB
TAB~~
100
110
011
001
001
~A
11000
01100
00111~B
In the previous example
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200319
Sparse Tableau Analysis (STA) n=#nodes, b=#branches
e
v
i
KK
AI
A
vi
T
0
0
00(n-1) KCL
b KVL
b branch relations
b
b
n-1
S
0
0
Totally 2b+n-1 variables, 2b+n-1 equations
b b n-1
Due to independentsources
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200320
STA (II) Advantages
Covers any circuit Easy to assemble Very sparse
Ki, Kv, I each has exactly b non-zeros. A and AT each has at most 2b non-zeros.
Disadvantages Sophisticated data structures & programmin
g techniques
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200321
Agenda RCL Network Sparse Tableau Analysis Modified Nodal Analysis
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200322
Nodal Analysis Derivation
SvKiK
eAv
iA
vi
T
0
SAKeAKAK
SAKvKAKiA
SKvKKi
iT
vi
ivi
ivi
11
11
11
From STA:
(1)
(2)
(3)
(3) x Ki-1
(4) x A
(4)
Using (a)
(5)
(6)
Tree trunk voltages
Substitute with node voltages (to a given reference), we get the nodal analysis equations.
e
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200323
Nodal Analysis (II)
SAKeAKAK iT
vi
11
1iK
12y13y
24y34y
35y45y
46y56y
IKv
11000000
10110000
00011010
00000110
00000011
A
46
45
34
24
12
564656
5656453535
353534131313
13241313
13131312
1
v
v
v
v
v
yyy
yyyyy
yyyyyy
yyyy
yyyy
eAKAK Tvi
1
2
3
4
5
6
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200324
2
2
2222
2
W
J
I
V
ZAY
AY nT
n
Modified Nodal Analysis General Form
Node Conductance matrix
KCL
Independent current source
Independent voltage source
Due to non-conductive elements
Yn can be easily derived
Add extra rows/columns for each non-conductive elements using templates
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200325
11000000
10110000
01101000
00011010
00000101
00000011
MNA (II) Fill Yn matrix according to incidence matrix
5645354535
45464534243424
353435341313
24241212
13121312
00
0
0
00
00
yyyyy
yyyyyyy
yyyyyy
yyyy
yyyy
Yn
5646453534241312 iiiiiiii
6
5
4
3
2
1
n
n
n
n
n
n
A
Choose n6 as reference node
1
2
3
4
5
6
Ten AAYY
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200326
MNA Templates
j
j
i
i'j
j
j
'j
ji
11
1
1
1
'
m
j
j
'jj vv
gv
Independent current source
Independent voltage source
j
'j
gv
Add to the right-hand side of the equation
jiji
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200327
MNA Templates (II)
CCVS
'j
ji
j k
'k
ji
CCCS
11
1
1
jcc iNNNN
1
m
N
N
N
N
c
c
r11
11
1
1
1
1kjkkjj iivvvv ''
2
1
'
'
m
m
k
k
j
j
'j
ji
j k
'k
jri
ki
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200328
MNA Templates (III)
'j
j k
k
jv
VCVS
'j
jv
j k
'k
jmvg
VCCS
11
1
1
kkkjj ivvvv ''
1
'
'
m
k
k
j
j+
-
jv
ki
+
-
mm
mm
gg
gg'k
k
'jj vv
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200329
2
1
11
11
1
1
1
1
LjMj
MjLj
MNA Templates (IV)
j
'j
k
'k
+ +
- -
jv kv
1L 2L
1i 2i
j
'j
k
'k
i
'
'
k
k
j
j
M'' kkjj vvvv 2i
'
'
k
k
j
j
2
1
m
m
Mutual inductance
Operational Amplifier
11
1
1
'' kkjj vvvv i
1m
1i
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200330
MNA Example
n 1n 2
n 3 n 4G 2 G 3
C 4
C 5v g
n 0
+
_
v 6 v 7= v 6
Circuit Topology
0
0
0
0
0
00100
000001
1000
0000
00
0100
4
3
2
1
44
533
434322
22
g
vcvs
g
n
n
n
n
v
i
i
v
v
v
v
CjCj
CjGG
CjGCjGGG
GG
n 1 n 2 n 3 n 4
n 0
2 3
4
5 6 71i g
i vcvs
MNA
Equations
Jan. 24, 2003 Cheng & Zhu, UCSD @ 200331
MNA Summary Advantages
Covers any circuits Can be assembled directly from input data.
Matrix form is close to Yn
Disadvantages We may have zeros on the main diagonal. Principle minors could be singular