ECE 552 Numerical Circuit Analysis Chapter Six NONLINEAR DC ANALYSIS OR: Solution of Nonlinear Algebraic Equations Copyright © I. Hajj 2012 All rights.

ECE 552Numerical Circuit Analysis

Chapter Six

NONLINEAR DC ANALYSIS

OR: Solution of Nonlinear Algebraic Equations

Copyright © I. Hajj 2012 All rights reserved

Nonlinear Algebraic Equations

• A system of linear equations Ax = b has a unique solution, unless A is singular.

• However, a system of nonlinear equations f(x) =y may have one solution, multiple finite solutions, no solution, or infinite number of solutions.

Example

Example

Example

y = x2 has two solutions for y > 0 one solution for y = 0 no solution for y < 0

Example

y = x3 has a unique solution for every y.

n-dimensional case

• f(x) = y

or,

f1(x1,x2, ...,xn) = y1

f2(x1,x2, ...,xn) = y2

:

fn(x1,x2, ...,xn) = yn

Problem

• Given y ε Rn, find x* ε Rn, if it exists, such that f(x*) = y.

• Some Theorems on the Existence and Uniqueness of Solutions of Nonlinear Resistive Networks.

Definition

• Given a mapping f(.): Rn → Rn.

• f ε C1 means f is continuously differentiable; and

f is C1 diffeomorphism means that the inverse

function f-1 exists, and is also of class C1.

Palais's Theorem

• The necessary and sufficient conditions that the mapping

f(.): Rn → Rn to be a C1 diffeomorphism of Rn onto itself are:

(i) f is of class C1

(iii) lim ||f(x)|| → ∞ as ||x|| → ∞

• For existence and uniqueness of solution, can allow det [J] = 0 at

isolated points as long as it does not change signs and

lim||f(x)||→ ∞ when ||x||→ ∞

Circuit-Theoretic Theorems

• Theorem 2 (Duffin): In a network consisting of independent voltage and current sources, and voltage-controlled two-terminal resistors (i = g(v)), there exists at least one solution provided that each resistor's v-i characteristic function g(v) is continuous in v and satisfies:

g(v) → + ∞ (or - ∞) as v → + ∞ (or - ∞)

Circuit-Theoretic Theorems (cont.)

• Theorem 3 (Duffin): In a network consisting of independent voltage and current sources and voltage-controlled resistors (i = g(v)), there exists at most one solution provided that each resistor's v-i characteristic function g is strictly monotone increasing:

For existence and uniqueness, both Theorems 2 and 3 should be satisfied.

Circuit-Theoretic Theorems (cont.)

• Theorem 3 (Desoer and Katzenelson): A sufficient condition for the existence of a unique solution for a network consisting of time-varying voltage-controlled and current-controlled resistors characterized by continuous (not necessarily strictly) monotone increasing functions, and independent voltage and current sources, is that the resistor network formed by short-circuiting all voltage sources and open-circuiting all current sources has a tree (or forest) such that all tree branches correspond to current-controlled elements and all links correspond to voltage-controlled elements.

Now back to the numerical solution of:

f(x) = y

Given y, find x (assuming it exists)

Fixed-Point Iteration

x = g(x)

•A given problem can be recast into fixed-point problem, where x = g(x) is a suitably chosen function whose solutions are the solution of f(x) = y.

•For example, f(x) = y can be written as x = f (x) - y + x = g(x).

•Givenx = g(x)•Fixed-Point Iteration: xk+1 = g(xk)

Repeat until ||xk+1 - xk|| < ε

Examples

Examples (cont.)

Multiple solutions

Contraction mapping theorem

• Suppose g: D ε Rn → Rn maps a closed set D0 ε D into

itself and ||g(x) - g(y)|| ≤ α ||x-y||, x, y ε D0 for some

α < 1

Then for any x0 ε D0, the sequence xk+1 = g(xk), k = 0, 1,2, ..., converges to a unique x* of g in D0.

• Proof:

||x*-xk|| = ||g(x*) – g(xk -1)||

≤ α ||x*-xk-1||

≤ αk ||x*-x0||

Since α < 1, α k → 0 and ||x* - xk||→ 0 or xk → x*

Parallel Chord Method


xk+1= xk + A-l(y - f(xk)

Or A(xk+1- xk) = (y - f(xk)


A remains constant; it is usually chosen to be = where

“Jacobian matrix”

Instead of computing A-1, the following equation is solved:

A(xk+l - xk) = y-f(xk) or A ∆xk = ∆yk

At every iteration, ∆yk changes, while A (and its LU factors) remain unchanged.

Parallel Chord Algorithm

Parallel Chord Example (nonconvergence)

Newton or Newton-Raphson Method

Newton's (or Newton-Raphson) Method

Instead of solving:

It is sometimes more convenient to solve directly for x:

Newton's (or Newton-Raphson) Method

The slope changes with every iteration.

Convergence Properties of Newton's Method

scalar case

xk+1 =

quadratic convergence.

n-dimensional case

• Applying Taylor Series expansion at xk:

• y = f(x*) = f(xk) + Jk(x* - xk) + R(x* - xk)

where x* is the solution

• If the derivative of Jk (i.e., second derivative of f) is bounded, then:

||R(xk - x*)|| ≤ α ||xk - x*||2

• Newton's Method:

xk+1 = xk + [Jk]-1 (y - f(xk))

or, [Jk](xk+1 - xk) = y - f(xk)

n-dimensional case (cont.)

From Taylor Series:

• y - f(xk) = Jk(x* - xk) + R(x* - xk)

• Jk(xk+l - xk) = Jk(x* - xk) + R(x* - xk)

• xk+l - xk = x* - xk + [Jk]-1R(x* - xk)

• xk+l - x* = [Jk]-lR(x* - xk)

• ||xk+1 - x*|| ≤ c ||x* - xk||2

Provided J(x*) is nonsingular

Rate of Convergence

• Define ek = x* - xk

• A method is said to converge with rate r if:

||ek+1|| = c ||ek||r

for some nonzero constant c

• If r = 1, and c<1, the convergence is linear.

• If r > 1, c>0, the convergence rate is superlinear.

• If r = 2, c>0, the convergence rate is quadratic.

Newton's Method

Has quadratic convergence if xk is

"close enough" to the solution

and J(x*) is nonsingular.

Convergence Problems of Newton’s Method

Convergence Problems of Newton’s Method

(6) Homotopy: λf(x) + (1 - λ)(x - x°) = λ y When λ = 0, x = x° (chosen)

Put λ = 0,...,1 and solve starting from the last solution. f(x) = y when λ = 1.

Application to Electronic Circuits: DC Analysis

• Capacitors are open and inductors are short-circuited. Why?

Tableau Equations:

To formulate in Newton’s method, linearize element

characteristic equations at iteration point v1k, i2

k and obtain linearized circuit tableau equations, then use MNA., or any other formulation method.

Linearization

Stamp

Stamp

Stamp

Example

Example (cont.)

MNA Stamp

Other Methods of Linearization

Secant method

i = av + b

Other Methods of Linearization (cont.)

Line through origin

i = av

a = , b = 0

Multiport or Multiterminal y = f(x)

Taylor Series

Two-Port or Three-Terminal Elements

Multiport or Multiterminal: y = f(x) (cont.)

Transistor Models:NPN Bipolar Junction Transistor Model:Ebers-Moll Model:

Reference: VLACH & SINGHAL, Chap. 11

Multiport or Multiterminal y = f(x) (cont.)

In addition, αfIes = αrIcs = Is

Characteristics are of the form:

ic = f1 (vbc,vbe)

ie = f2 (vbc,vbe)


PNP Bipolar Junction Transistor Model:


V-I Characteristics of MOSFETs - NMOS

saturationlinear


Linear Region: VGS ≥ VTn, VDS ≤ (VGS - VTn)

IDS = Kn [(VGS-VTn)VDS-(1/2)V2DS]

Kn = μn Cox , Cox = , μ n electron mobility

(Threshold Voltage)

Saturation Region:

VGS ≥ Vtn , VDS ≥ (VGS-VTn)

Note that in DC, the current in the gate IG = 0.

Cut-off:

PMOS

Modified Newton's Method for DC Analysis of Electronic Circuit with

Exponential Nonlinearities

e.g., diodes, bipolar junction transistors

• Possibility of overflow if v becomes "large."

Voltage-Current Iteration

- Prevents overflow and improves convergence

DC Input-Output Characteristics

Driving-Point Characteristics: Iin vs. Vin Transfer Characteristics: Vout vs. Vin

ECE 552 Numerical Circuit Analysis Chapter Six NONLINEAR DC ANALYSIS OR: Solution of Nonlinear Algebraic Equations Copyright © I. Hajj 2012 All rights.

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