EE 616 Computer Aided Analysis of Electronic Networks Lecture 5

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EE 616 Computer Aided Analysis of Electronic
NetworksLecture 5

• Instructor Dr. J. A. Starzyk, Professor
• School of EECS
• Ohio University
• Athens, OH, 45701

09/19/2005
Note materials in this lecture are from the
notes of EE219A UC-berkeley http//www-
cad.eecs.berkeley.edu/nardi/EE219A/contents.html
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Outline

• Nonlinear problems
• Iterative Methods
• Newtons Method
• Derivation of Newton
• Quadratic Convergence
• Examples
• Convergence Testing
• Multidimensonal Newton Method
• Basic Algorithm
• Quadratic convergence
• Application to circuits

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DC Analysis of Nonlinear Circuits - Example
Need to Solve
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Nonlinear Equations

• Given g(V)I
• It can be expressed as f(V)g(V)-I
• ? Solve g(V)I equivalent to solve f(V)0

Hard to find analytical solution for f(x)0
Solve iteratively
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Nonlinear Equations Iterative Methods

• Start from an initial value x0
• Generate a sequence of iterate xn-1, xn, xn1
  which hopefully converges to the solution x
• Iterates are generated according to an iteration
  function F xn1F(xn)

• Ask
• When does it converge to correct solution ?
• What is the convergence rate ?

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Newton-Raphson (NR) Method

• Consists of linearizing the system.
• Want to solve f(x)0 ? Replace f(x) with its
  linearized version and solve.
• Note at each step need to evaluate f and f

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Newton-Raphson Method Graphical View
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Newton-Raphson Method Algorithm
Define iteration
Do k 0 to .?
until convergence

• How about convergence?
• An iteration x(k) is said to converge with
  order q if there exists a vector norm such that
  for each k ? N

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Newton-Raphson Method Convergence
But
by Newton definition
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Newton-Raphson Method Convergence
Subtracting
Dividing
Convergence is quadratic
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Newton-Raphson Method Convergence
Local Convergence Theorem
If
Then Newtons method converges given a
sufficiently close initial guess (and
convergence is quadratic)
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Newton-Raphson Method Convergence
Example 1
Convergence is quadratic
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Newton-Raphson Method Convergence
Example 2
Note not bounded away from zero
Convergence is linear
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Newton-Raphson Method Convergence
Example 1, 2
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Newton-Raphson Method Convergence
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Newton-Raphson Method Convergence Check
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Newton-Raphson Method Convergence Check
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Newton-Raphson Method Convergence
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Newton-Raphson Method Local Convergence
Convergence Depends on a Good Initial Guess
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Newton-Raphson Method Local Convergence
Convergence Depends on a Good Initial Guess
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Nonlinear Problems Multidimensional Example
Nodal Analysis
Nonlinear Resistors
Two coupled nonlinear equations in two unknowns
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Multidimensional Newton Method
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Multidimensional Newton Method Computational
Aspects

• Each iteration requires
• Evaluation of F(xk)
• Computation of J(xk)
• Solution of a linear system of algebraic
  equations whose coefficient matrix is J(xk) and
  whose RHS is -F(xk)

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Multidimensional Newton Method Algorithm
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Multidimensional Newton Method Convergence
Local Convergence Theorem
If
Then Newtons method converges given a
sufficiently close initial guess (and
convergence is quadratic)
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Application of NR to Circuit EquationsCompanion
Network

• Applying NR to the system of equations we find
  that at iteration k1
• all the coefficients of KCL, KVL and of BCE of
  the linear elements remain unchanged with respect
  to iteration k
• Nonlinear elements are represented by a
  linearization of BCE around iteration k
• ? This system of equations can be interpreted as
  the STA of a linear circuit (companion network)
  whose elements are specified by the linearized
  BCE.

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Application of NR to Circuit EquationsCompanion
Network

• General procedure the NR method applied to a
  nonlinear circuit (whose eqns are formulated in
  the STA form) produces at each iteration the STA
  eqns of a linear resistive circuit obtained by
  linearizing the BCE of the nonlinear elements and
  leaving all the other BCE unmodified
• After the linear circuit is produced, there is no
  need to stick to STA, but other methods (such as
  MNA) may be used to assemble the circuit eqns

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Application of NR to Circuit EquationsCompanion
Network MNA templates
Note G0 and Id depend on the iteration count k
? G0G0(k) and IdId(k)
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Application of NR to Circuit EquationsCompanion
Network MNA templates
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Modeling a MOSFET(MOS Level 1, linear regime)
d
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Modeling a MOSFET(MOS Level 1, linear regime)
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DC Analysis Flow Diagram
For each state variable in the system
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Implications

• Device model equations must be continuous with
  continuous derivatives and derivative calculation
  must be accurate derivative of function
• (not all models do this - Poor diode models and
  breakdown models dont - be sure models are
  decent - beware of user-supplied models)
• Watch out for floating nodes (If a node becomes
  disconnected, then J(x) is singular)
• Give good initial guess for x(0)
• Most model computations produce errors in
  function values and derivatives.
• Want to have convergence criteria x(k1) -
  x(k) lt ? such that ? gt than model errors.

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Summary

• Nonlinear problems
• Iterative Methods
• Newtons Method
• Derivation of Newton
• Quadratic Convergence
• Examples
• Convergence Testing
• Multidimensonal Newton Method
• Basic Algorithm
• Quadratic convergence
• Application to circuits

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Improving convergence

• Improve Models (80 of problems)
• Improve Algorithms (20 of problems)
• Focus on new algorithms
• Limiting Schemes
• Continuations Schemes

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Outline

• Limiting Schemes
• Direction Corrupting
• Non corrupting (Damped Newton)
• Globally Convergent if Jacobian is Nonsingular
• Difficulty with Singular Jacobians
• Continuation Schemes
• Source stepping
• More General Continuation Scheme
• Improving Efficiency
• Better first guess for each continuation step

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Multidimensional Newton MethodConvergence
Problems Local Minimum
Local Minimum
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Multidimensional Newton MethodConvergence
Problems Nearly singular
f(x)
Must Somehow Limit the changes in X
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Multidimensional Newton MethodConvergence
Problems - Overflow
f(x)
X
Must Somehow Limit the changes in X
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Newton Method with Limiting
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Newton Method with LimitingLimiting Methods

• Direction Corrupting

• NonCorrupting

Heuristics, No Guarantee of Global Convergence
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Newton Method with LimitingDamped Newton Scheme
General Damping Scheme
Key Idea Line Search
Method Performs a one-dimensional search in
Newton Direction
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Newton Method with LimitingDamped Newton
Convergence Theorem
If
Then
Every Step reduces F-- Global Convergence!
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Newton Method with LimitingDamped Newton
Nested Iteration
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Newton Method with LimitingDamped Newton
Singular Jacobian Problem
X
Damped Newton Methods push iterates to local
minimums Finds the points where Jacobian is
Singular
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Newton with Continuation schemes Basic Concepts
- General setting

• Newton converges given a close initial guess
• ? Idea Generate a sequence of problems, s.t. a
  problem is a good initial guess for the following
  one

? Starts the continuation
?Ends the continuation
?Hard to insure!
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Newton with Continuation schemes Basic Concepts
Template Algorithm
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Newton with Continuation schemes Basic Concepts
Source Stepping Example
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Newton with Continuation schemes Basic Concepts
Source Stepping Example
Diode
Source Stepping Does Not Alter Jacobian
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Newton with Continuation schemes Jacobian
Altering Scheme
Observations
Problem is easy to solve and Jacobian definitely
nonsingular.
Back to the original problem and original Jacobian
(1),1)
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Summary

• Newtons Method works fine
• given a close enough initial guess
• In case Newton does not converge
• Limiting Schemes
• Direction Corrupting
• Non corrupting (Damped Newton)
• Globally Convergent if Jacobian is Nonsingular
• Difficulty with Singular Jacobians
• Continuation Schemes
• Source stepping
• More General Continuation Scheme
• Improving Efficiency
• Better first guess for each continuation step