CSE245: Computer-Aided Circuit Simulation and Verification

1
CSE245 Computer-Aided Circuit Simulation and
Verification

• Lecture 1 Introduction and Formulation
• Spring 2008
• Chung-Kuan Cheng

2
Administration

• CK Cheng, CSE 2130, tel. [phone],
  ckcheng_at_ucsd.edu
• Lectures 1230 150pm TTH WLH2205
• Textbooks
• Electronic Circuit and System Simulation Methods
• T.L. Pillage, R.A. Rohrer, C. Visweswariah,
  McGraw-Hill
• Interconnect Analysis and Synthesis
• CK Cheng, J. Lillis, S. Lin, N. Chang, John
  Wiley Sons
• Grading
• Homework and Projects 60
• Project Presentation 20
• Final Report 20

3
CSE245 Course Outline

• Formulation (2-3 lectures)
• RLC Linear, Nonlinear Components,Transistors,
  Diodes
• Incident Matrix
• Nodal Analysis, Modified Nodal Analysis
• K Matrix
• Linear System (3-4 lectures)
• S domain analysis, Impulse Response
• Taylors expansion
• Moments, Passivity, Stability, Realizability
• Symbolic analysis, Y-Delta, BDD analysis
• Matrix Solver (3-4 lectures)
• LU, KLU, reordering
• Mutigrid, PCG, GMRES

4
CSE245 Course Outline (Cont)

• Integration (3-4 lectures)
• Forward Euler, Backward Euler, Trapezoidal Rule
• Explicit and Implicit Method, Prediction and
  Correction
• Equivalent Circuit
• Errors Local error, Local Truncation Error,
  Global Error
• A-Stable
• Alternating Direction Implicit Method
• Nonlinear System (2-3 lectures)
• Newton Raphson, Line Search
• Transmission Line, S-Parameter (2-3 lectures)
• FDTD equivalent circuit, convolution
• Frequency dependent components
• Sensitivity
• Mechanical, Thermal, Bio Analysis

5
Motivation

• Why
• Whole Circuit Analysis, Interconnect Dominance
• What
• Power, Clock, Interconnect Coupling
• Where
• Matrix Solvers, Integration Methods
• RLC Reduction, Transmission Lines, S Parameters
• Parallel Processing
• Thermal, Mechanical, Biological Analysis

6
Circuit Simulation

• Types of analysis
• DC Analysis
• DC Transfer curves
• Transient Analysis
• AC Analysis, Noise, Distortions, Sensitivity

7
Program Structure (a closer look)
Input and setup
Models

• Numerical Techniques
• Formulation of circuit equations
• Solution of ordinary differential equations
• Solution of nonlinear equations
• Solution of linear equations

Output
8
Lecture 1 Formulation

• Derive from KCL/KVL
• Sparse Tableau Analysis (IBM)
• Nodal Analysis, Modified Nodal Analysis (SPICE)

some slides borrowed from Berkeley EE219 Course
9
Conservation Laws

• Determined by the topology of the circuit
• Kirchhoffs Current Law (KCL) The algebraic sum
  of all the currents flowing out of (or into) any
  circuit node is zero.
• No Current Source Cut
• Kirchhoffs Voltage Law (KVL) Every circuit node
  has a unique voltage with respect to the
  reference node. The voltage across a branch vb is
  equal to the difference between the positive and
  negative referenced voltages of the nodes on
  which it is incident
• No voltage source loop

10
Branch Constitutive Equations (BCE)

• Ideal elements

Element Branch Eqn Variable parameter
Resistor v Ri -
Capacitor i Cdv/dt -
Inductor v Ldi/dt -
Voltage Source v vs i ?
Current Source i is v ?
VCVS vs AV vc i ?
VCCS is GT vc v ?
CCVS vs RT ic i ?
CCCS is AI ic v ?
11
Formulation of Circuit Equations

• Unknowns
• B branch currents (i)
• N node voltages (e)
• B branch voltages (v)
• Equations
• NB Conservation Laws
• B Constitutive Equations
• 2BN equations, 2BN unknowns gt unique solution

12
Equation Formulation - KCL
Law State Equation
A i 0
Node 1 Node 2
N equations
Branches
Kirchhoffs Current Law (KCL)
13
Equation Formulation - KVL
R3
1
2
Is5
R1
R4
G2v3
0
Law State Equation
v - AT e 0
vi voltage across branch i ei voltage at node
i
B equations
Kirchhoffs Voltage Law (KVL)
14
Equation Formulation - BCE
Law State Equation
Kvv Kii is
B equations
15
Equation FormulationNode-Branch Incidence Matrix
A
branches
n o d e s
(1, -1, 0)
16
Equation Assembly (Stamping Procedures)

• Different ways of combining Conservation Laws and
  Branch Constitutive Equations
• Sparse Table Analysis (STA)
• Nodal Analysis (NA)
• Modified Nodal Analysis (MNA)

17
Sparse Tableau Analysis (STA)

1. Write KCL Ai0 (N eqns)
2. Write KVL v - ATe0 (B eqns)
3. Write BCE Kii KvvS (B eqns)

N2B eqns N2B unknowns
N nodes B branches
Sparse Tableau
18
Sparse Tableau Analysis (STA)

• Advantages
• It can be applied to any circuit
• Eqns can be assembled directly from input data
• Coefficient Matrix is very sparse
• Disadvantages
• Sophisticated programming techniques and data
• structures are required for time and memory
• efficiency

19
Nodal Analysis (NA)

• 1. Write KCL
• Ai0 (N equations, B unknowns)
• 2. Use BCE to relate branch currents to branch
  voltages
• if(v) (B equations ? B unknowns)
• Use KVL to relate branch voltages to node
  voltages
• vh(e) (B equations ? N unknowns)

N eqns N unknowns N nodes
Yneins
Nodal Matrix
20
Nodal Analysis - Example
R3

1. KCL Ai0
2. BCE Kvv i is ? i is - Kvv ? A Kvv A
   is
3. KVL v ATe ? A KvATe A is

Yne ins
Yn AKvAT Ins Ais
21
Nodal Analysis

• Example shows how NA may be derived from STA
• Better Method Yn may be obtained by direct
  inspection (stamping procedure)
• Each element has an associated stamp
• Yn is the composition of all the elements stamps

22
Nodal Analysis Resistor Stamp
Spice input format Rk N N- Rkvalue
What if a resistor is connected to
ground? . Only contributes to the diagonal
KCL at node N KCL at node N-
23
Nodal Analysis VCCS Stamp
Spice input format Gk N N- NC NC-
Gkvalue
KCL at node N KCL at node N-
24
Nodal Analysis Current source Stamp
Spice input format Ik N N- Ikvalue
N N-
N N-
Ik
25
Nodal Analysis (NA)

• Advantages
• Yn is often diagonally dominant and symmetric
• Eqns can be assembled directly from input data
• Yn has non-zero diagonal entries
• Yn is sparse (not as sparse as STA) and smaller
  than STA NxN compared to (N2B)x(N2B)
• Limitations
• Conserved quantity must be a function of node
  variable
• Cannot handle floating voltage sources, VCVS,
  CCCS, CCVS

26
Modified Nodal Analysis (MNA)
How do we deal with independent voltage sources?
Ekl
k l

-
l
k
ikl

• ikl cannot be explicitly expressed in terms of
  node voltages ? it has to be added as unknown
  (new column)
• ek and el are not independent variables anymore ?
  a constraint has to be added (new row)

27
MNA Voltage Source Stamp
Spice input format Vk N N- Ekvalue
0 0 1
0 0 -1
1 -1 0
N
28
Modified Nodal Analysis (MNA)

• How do we deal with independent voltage sources?
• Augmented nodal matrix

In general
Some branch currents
29
MNA General rules

• A branch current is always introduced as an
  additional variable for a voltage source or an
  inductor
• For current sources, resistors, conductors and
  capacitors, the branch current is introduced only
  if
• Any circuit element depends on that branch
  current
• That branch current is requested as output

30
MNA CCCS and CCVS Stamp
31
MNA An example
Step 1 Write KCL
(1) (2) (3) (4)
32
MNA An example
Step 2 Use branch equations to eliminate as many
branch currents as possible
(1) (2) (3) (4)
Step 3 Write down unused branch equations
(b6) (b7)
33
MNA An example
Step 4 Use KVL to eliminate branch voltages from
previous equations
(1) (2) (3) (4) (b6) (b7)
34
MNA An example
35
Modified Nodal Analysis (MNA)

• Advantages
• MNA can be applied to any circuit
• Eqns can be assembled directly from input data
• MNA matrix is close to Yn
• Limitations
• Sometimes we have zeros on the main diagonal