CSE 245: Computer Aided Circuit Simulation and Verification

1
CSE 245 Computer Aided Circuit Simulation and
Verification
Winter 2003 Lecture 1 Formulation

• Instructor
• Prof. Chung-Kuan Cheng

2
Agenda

• RCL Network
• Sparse Tableau Analysis
• Modified Nodal Analysis

3
History of SPICE

• SPICE -- Simulation Program with Integrated
  Circuit Emphasis
• 1969, CANCER developed by Laurence Nagel on Prof.
  Ron Rohers class
• 19701972, 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

4
RCL circuit
5
RCL circuit (II)

• General Circuit Equation
• Consider homogeneous form first

and
Q How to Compute Ak ?
6
Solving RCL Equation

• Assume A has non-degenerate eigenvalues
  
• and corresponding linearly
  independent eigenvectors , then A
  can be decomposed as
• where and

7
Solving RCL Equation (II)

• Whats the implication then?
• To compute the eigenvalues

where
8
Solving RCL Equation (III)
In the previous example
Let crl1, we have
where
hence
9
Solving RCL Equation (IV)

• What if matrix A has degenerated eigenvalues?
  Jordan decomposition !

J is in the Jordan Canonical form
And still
10
Jordan Decomposition
similarly
11
Agenda

• RCL Network
• Sparse Tableau Analysis
• Modified Nodal Analysis

12
Equation Formulation

• KCL
• Converge of node current
• KVL
• Closure of loop voltage
• Brach equations
• I, R relations

13
Types of elements

• Resistor
• Capacitor
• Inductor
• L is even dependent on frequency due to skin
  effect, etc
• Controlled Sources
• VCVS, VCCS, CCVS, CCCS

14
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
15
KCL Formulation
nodes-1 lines braches columns
16
KCL Formulation (II)

• Permute the columns to achieve a systematic form

17
KVL Formulation
Remove the equations for tree braches and
systemize
18
Cut Loop relation
In the previous example
19
Sparse Tableau Analysis (STA)

• nnodes, bbranches

b
b
n-1
(n-1) KCL
b
b KVL
b
b branch relations
n-1
Due to independent sources
Totally 2bn-1 variables, 2bn-1 equations
20
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

21
Agenda

• RCL Network
• Sparse Tableau Analysis
• Modified Nodal Analysis

22
Nodal Analysis

• Derivation

(1)
From STA
(2)
(3)
(3) x Ki-1 ?
(4)
(4) x A ?
(5)
Using (a) ?
(6)
Tree trunk voltages
Substitute with node voltages (to a given
reference), we get the nodal analysis equations.
23
Nodal Analysis (II)
24
Modified Nodal Analysis

• General Form

Independent current source
Node Conductance matrix
KCL
Independent voltage source
Due to non-conductive elements

• Yn can be easily derived
• Add extra rows/columns for each non-conductive
  elements using templates

25
MNA (II)

• Fill Yn matrix according to incidence matrix

Choose n6 as reference node
26
MNA Templates
Add to the right-hand side of the equation
Independent current source
Independent voltage source
27
MNA Templates (II)
CCVS
CCCS
28
MNA Templates (III)
VCVS

-
VCCS

-
29
MNA Templates (IV)
Mutual inductance
M
Operational Amplifier
30
MNA Example
Circuit Topology
MNA Equations
31
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