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
Jan. 24, 2003
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Cheng & Zhu, UCSD @ 2003
History of SPICE
 SPICE -- Simulation Program with
Integrated Circuit Emphasis
 1969, CANCER developed by Laurence
Nagel 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
since then
Jan. 24, 2003
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Cheng & Zhu, UCSD @ 2003
RCL circuit
R
L
C
Vs
1  v2   0 
c 0 v2   0
0 l   i    1  r   i   v 

 1  
 1   s 
1 

0

v

v2   0


2
c

  vs 

 i   1


r  i1   
 1  
 
l 
l
 l
Jan. 24, 2003
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Cheng & Zhu, UCSD @ 2003
RCL circuit (II)
 General Circuit Equation

Y  AY  BU
 Consider homogeneous form first

Y  AY
Y  e Y0
At
2 2
and
k k
At A t
At
e I 
 ... 
 ...
1!
2!
k!
At
Q: How to Compute Ak ?
Jan. 24, 2003
5
Cheng & Zhu, UCSD @ 2003
Solving RCL Equation
 Assume A has non-degenerate eigenvalues
1 , 2 ,..., k and corresponding linearly
independent eigenvectors 1 ,  2 ,...,  k , then A
can be decomposed as
1
A   
1 0
0 
where
2

 

0 
Jan. 24, 2003
0
  and    ,  ,...,  
1
2
k
 

 k 


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Cheng & Zhu, UCSD @ 2003
Solving RCL Equation (II)
 What’s the implication then?
1
A   
1 t
e  e 
At
1 2
1
A   A   
2
2
where e t
 To compute the eigenvalues:
e 1

0




 0
2




k 
  e 
0 
e 2 
 
0


det( I  A)  n  cn 1n 1  ...  c0
 (  p0 )(2  p1  p2 )...(...)  0
real
eigenvalue
Jan. 24, 2003
Conjugative
Complex
eigenvalue
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Cheng & Zhu, UCSD @ 2003
Solving RCL Equation (III)
 0

In the previous example v2   e At X  e  1/ l
0
i 
 1
1/ c 
t
r / l 
v2 (0)
 i (0) 
1 

0 1
1
1  
     
Let c=r=l=1, we have A  

 1  1
0
1  3 j
 1  3 j


1

j 
2
2



where
3  1 3 j
1  3 j
1


 
2
2


e
0
At
1
e  

hence
 
0 e 
 
Jan. 24, 2003
8
1

   1 3 j

2

1
0


 
1

1 3 j 

2

Cheng & Zhu, UCSD @ 2003
Solving RCL Equation (IV)
 What if matrix A has degenerated
eigenvalues? Jordan decomposition !
1
A   J
J is in the Jordan Canonical form
And still
Jan. 24, 2003
e At   1e Jt 
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Cheng & Zhu, UCSD @ 2003
Jordan Decomposition
 1 
J 

0



t
1
0

1

e




e Jt  

t   


0 1  0  
0
tet 
t 
e 
similarly
 1 0 
J   0  1 
 0 0  
Jan. 24, 2003
 t
e
1 0 0  1 0 

e Jt  0 1 0   0  1 t     0
0
0 0 1  0 0  


10
te
t
e t
0
t 2 t 
e 
2! 
tet 
e t 


Cheng & Zhu, UCSD @ 2003
Agenda
 RCL Network
 Sparse Tableau Analysis
 Modified Nodal Analysis
Jan. 24, 2003
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Cheng & Zhu, UCSD @ 2003
Equation Formulation
 KCL
 Converge of node current
 KVL
 Closure of loop voltage
 Brach equations
 I, R relations
Jan. 24, 2003
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Types of elements
 Resistor
 Capacitor
dQ Q(v) dv
dv
i

  c (v )
dt
v dt
dt
 Inductor
d  (i ) di
v
dt

i

dt
 l (i )
di
dt
 L is even dependent on frequency due to skin
effect, etc…
 Controlled Sources
 VCVS, VCCS, CCVS, CCCS
Jan. 24, 2003
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Cheng & Zhu, UCSD @ 2003
Cut-set analysis
1
3
5
1. Construct a spanning tree
2. Take as much capacitor
branches as tree branches as
possible
2
4
6
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
Jan. 24, 2003
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KCL Formulation
 
Ai  0
c12
c24
c34
c45
c46
 i12 i13 i24
1 1 0
0 1 1

0  1 0

0 0 0
0 0 0
Jan. 24, 2003
i34
0
0
1
0
0
i35 i45
0 0 0
0 0 0
1 0 0
1 1 0
0 0 1
1
3
5
2
4
6
i12 
 
i46 i56  i13 
0 
0  i24   
0
0  i34   
   0 
0  i35   
   0
 1 i45   
0



1
i
 46 
i56 
15
#nodes-1 lines
#braches columns
Cheng & Zhu, UCSD @ 2003
KCL Formulation (II)
 Permute the columns to achieve a
systematic form
1
 
~ 
I|Ai 0
i12 i24 i34 i45 i46 i13 i23
c12
c24
c34
c45
c46
1

0
0

0
0

Jan. 24, 2003
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0 1
0 1
0 1
0 0
1 0
0
0
1
1
0
i12 
2


i56
i24  0
0  i34   
0

0  i45   
   0 
0  i46   
   0 
 1  i13 
0



1  i23
 
i56 
16
3
5
4
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Cheng & Zhu, UCSD @ 2003
KVL Formulation
AT e  V
1
1

0

0
0

0
0

0
0 0 0
1 1 0
1 0 0
0 1 0
0 1 1
0 0 1
0 0 0
0 0 1
0
 v12 
v 
0
13
 v12   
0   v24 
 
 v24
0   v34 
v34  
0   v35 
 v45   
v
0
v46   45 
v 
1
 46 

1
v56 
Remove the
equations for
tree braches
and systemize
1
3
5
2
4
6
 v12 
v 
 24 
v34 
l13  1  1 1 0 0 1 0 0  
v
l35  0 0  1  1 0 0 1 0  45   0
v46 
l56  0 0 0 1  1 0 0 1  
 v13 
v 
 35 
v56 
BV  0

~
B B
Jan. 24, 2003
17
I

Cheng & Zhu, UCSD @ 2003
Cut & Loop relation
A e V
BV  0
T
In the previous example
1
1
~ 
A   1

0
 0
BA T  0
~T
~
[ B I ][ IA]  0
~T
B A 0
~T
~
B  A
Jan. 24, 2003
0 0
0 0 
1 0

1  1
0 1 
 1  1 1 0 0 
~ 
B   0 0  1  1 0 
 0 0 0 1  1
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Cheng & Zhu, UCSD @ 2003
Sparse Tableau Analysis (STA)
 n=#nodes, b=#branches
b
(n-1) KCL
b
KVL
b branch
relations
A
0

 K i
b
0
I
Kv
n-1
0 i 
T  
 A  v 
0  e
0


b  0
 
n-1 
 S 
b
Due to independent
sources
Totally 2b+n-1 variables, 2b+n-1 equations
Jan. 24, 2003
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Cheng & Zhu, UCSD @ 2003
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 &
programming techniques
Jan. 24, 2003
20
Cheng & Zhu, UCSD @ 2003
Agenda
 RCL Network
 Sparse Tableau Analysis
 Modified Nodal Analysis
Jan. 24, 2003
21
Cheng & Zhu, UCSD @ 2003
Nodal Analysis
 Derivation
From STA:
(3) x Ki-1 
(4) x A 
Using (a) 

e

Ai  0 (1)

v  AT e (2)

 
Kii  Kvv  S
(3)



1
1
i  K i K v v  K i S (4)



1
1
Ai  AK i K v v  AK i S

1
T
1
AK i K v A e  AK i S
(5)
(6)
Tree trunk voltages
Substitute
with node voltages (to a given reference),
we get the nodal analysis equations.
Jan. 24, 2003
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Cheng & Zhu, UCSD @ 2003
Nodal Analysis (II)


1
AK K v A e  AKi S
1
i
T
1 1
0 1

A  0  1

0 0
0 0
y12
K i1 
y13
y24
y34
y35
y 45
Kv  I
y 46
y56
 y12  y13
 y
13

1
T
AKi K v A e    y13



Jan. 24, 2003
y13
y13  y24
 y13
23
 y13
 y13
y13  y34  y35
y35
0
1
0
0
0
1
3
5
2
4
6
0
0
1
0
0
0
0
1
1
0
y35
y35  y45  y56
 y56
0
0
0
1
0
0 0
0 0 
0 0

0  1
1 1 
  v12 
 v 
  24 
 v34 
 
 y56  v45 
y46  y56  v46 
Cheng & Zhu, UCSD @ 2003
Modified Nodal Analysis
 General Form
Independent
current source
Node Conductance
matrix
KCL
 Yn
Y AT
 2 2
A2  Vn   J 2 
 



Z 2   I 2  W2 
Due to non-conductive
elements
Independent
voltage source
 Yn can be easily derived
 Add extra rows/columns for each non-conductive elements
using templates
Jan. 24, 2003
24
Cheng & Zhu, UCSD @ 2003
1
3
2
4
5
MNA (II)
 Fill Yn matrix according to incidence matrix
n1
A
n2
n3
n4
n5
n6
i12 i13 i24 i34 i35 i45 i46 i56
 1  1 0 0 0 0 0 0 
 1 0 1 0 0 0 0 0 


 0 1 0 1 1 0 0 0 


0
0
0
1
0

1

1
0


 0 0 0 0 1 1 0  1


 0 0 0 0 0 0 1 1 
6
Yn  AYe A
T
Choose n6 as reference node
 y12  y13
 y
12

Yn    y13

 0
 0
Jan. 24, 2003
 y12
y12  y 24
0
 y 24
0
 y13
0
y13  y34  y35
 y34
 y35
25
0
 y 24
 y34
y 24  y34  y 45  y 46
 y 45
0


0


 y35

 y 45

y35  y 45  y56 
Cheng & Zhu, UCSD @ 2003
MNA Templates
Add to the right-hand
side of the equation
j
Independent
current source
j  i j 
 
j'  ij 
ij
j'
j
Independent
voltage source
vj
ij
vg
j 

j' 
m  1 1
v j'
ij
1
 1

1








 
 
 
v g 
 
j'
Jan. 24, 2003
26
Cheng & Zhu, UCSD @ 2003
MNA Templates (II)
vj
CCVS
j
ij
CCCS
k'
j
k
j'
Jan. 24, 2003
vk
vk '
ij
ik



1
j' 

1
k 


 1
k' 

m  1 1  1


m  2 
1 1  r

ri j
j'
ij
j 
k
ik
v j'
1
N  N  N c N c i j
N 
N 

N c 
i j
N c
m 1
k'
27


1  1
1 
 1 
 



Cheng & Zhu, UCSD @ 2003
MNA Templates (III)
VCVS
j
v j v j ' vk vk ' ik
1 
j 

j' 

1


k 
 

k' 




m  1 1  1

k
+
vj
ik
v j
-
VCCS
j'
k
j
k
vj
+
vj
k
k'
g mv j
-
j'
Jan. 24, 2003
 gm
 g
 m
v j'
 gm 
g m 
k'
28
Cheng & Zhu, UCSD @ 2003
MNA Templates (IV)
Mutual inductance
j
i1
M
v j v j ' vk vk '
i2
j 
1
1
j ' 
k 

'
k 
m  1 1  1
 jL1

m  2 
1  1  jM
k
+
+
vj
vk
-
L1
L2
j'
k
-
'
Operational Amplifier
j
j'
Jan. 24, 2003
i
i1
vj
v j'






m  1 1  1
j
j'
k
k'
k
k'
29
vk
vk '
i2



1 

1 
 jM 

 jL2 
i



1

 1

Cheng & Zhu, UCSD @ 2003
MNA Example
4
C4
G2
n1
G3
n3
n2
C5
vg
n4
+
n1
v7 =  v6
v6
n2
2
1
_
n3
3
5
ig
n0
6
n4
7 i vcvs
n0
Circuit Topology
MNA
Equations
Jan. 24, 2003
 G2
 G
 2
 0

 0
 1

 0
 G2
0
0
G2  G3  jC4
 G3
 jC 4
 G3
G3  jC5
0
 jC4
0
jC 4
0
0
0
0

1
30
1 0  vn1   0 
0 0  vn 2   0 
0 0  v n 3   0 
    
0 1  vn 4   0 
0 0   i g  v g 
   
0 0 ivcvs   0 
Cheng & Zhu, UCSD @ 2003
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
Jan. 24, 2003
31
Cheng & Zhu, UCSD @ 2003