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

2. 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.

3. Example

4. Example

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

6. Example
y = x3 has a unique solution for every y.

7. n-dimensional case f(x) = y or, f1(x1,x2, ...,xn) = y1
fn(x1,x2, ...,xn) = yn

8. 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.

9. 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.

10. 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||→ ∞

11. 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 - ∞)

12. 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.

13. 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.

14. Now back to the numerical solution of: f(x) = y
Given y, find x (assuming it exists)

15. 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).
Given x = g(x)
Fixed-Point Iteration: xk+1 = g(xk)
Repeat until ||xk+1 - xk|| < ε

16. Examples

17. Examples (cont.)
Multiple solutions

18. 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*

19. Parallel Chord Method

20. Parallel Chord Method
xk+1= xk + A-l(y - f(xk) Or A(xk+1- xk) = (y - f(xk)

21. Parallel Chord Method
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.

22. Parallel Chord Algorithm

23. Parallel Chord Example (nonconvergence)

24. Newton or Newton-Raphson Method

25. Newton's (or Newton-Raphson) Method

26. Instead of solving:
It is sometimes more convenient to solve directly for x:

27. Newton's (or Newton-Raphson) Method
The slope changes with every iteration.

28. Convergence Properties of Newton's Method
scalar case
xk+1 =
quadratic convergence.

29. 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)

30. 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

31. 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.

32. Newton's Method
Has quadratic convergence if xk is "close enough" to the solution and J(x*) is nonsingular.

33. Convergence Problems of Newton’s Method

34. Convergence Problems of Newton’s Method

40. (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.

43. 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, i2k and obtain linearized circuit tableau equations, then use MNA., or any other formulation method.

47. Linearization

48. Stamp

50. Stamp

54. Stamp

56. Example

57. Example (cont.)
MNA Stamp

58. Other Methods of Linearization
Secant method
i = av + b

59. Other Methods of Linearization (cont.)
Line through origin
a = , b = 0
i = av

60. Multiport or Multiterminal y = f(x)
Taylor Series
Two-Port or Three-Terminal Elements

61. Multiport or Multiterminal: y = f(x) (cont.)
Transistor Models:
NPN Bipolar Junction Transistor Model:Ebers-Moll Model:
Reference: VLACH & SINGHAL, Chap. 11

62. 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)

63. Multiport or Multiterminal y = f(x) (cont.)
PNP Bipolar Junction Transistor Model:

64. Multiport or Multiterminal y = f(x) (cont.)
V-I Characteristics of MOSFETs - NMOS
linear
saturation

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

66. VGS ≥ Vtn , VDS ≥ (VGS-VTn)
Saturation Region:
VGS ≥ Vtn , VDS ≥ (VGS-VTn)
Cut-off:
Note that in DC, the current in the gate IG = 0.

67. PMOS

68. e.g., diodes, bipolar junction transistors
Modified Newton's Method for DC Analysis of Electronic Circuit with Exponential Nonlinearities
e.g., diodes, bipolar junction transistors

69. Possibility of overflow if v becomes "large."

70. Voltage-Current Iteration
- Prevents overflow and improves convergence

72. DC Input-Output Characteristics
Driving-Point Characteristics: Iin vs. Vin
Transfer Characteristics: Vout vs. Vin