Circuits Theory Examples Newton-Raphson Method

Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged, the limit is the solution of the equationf(x)=0.

Multidimensional case: where: JACOBIAN MATRIX

ALGORITHM STEP 0 STARTING POINT STEP 1Calculate STEP 2Solve the equation: STEP 3 find check STOP conditions If the current solution is not acceptable: GO TO 1

EXAMPLE of STOP PROCEDURE NoNoNoNo No k=k+1 GOTO 1 Yes Yes STOP

Stop condition parameter

Numerical EXAMPLES Example 1

Solve the following set of nonlinearequation using the Newton’s Method:

Starting point (first approximation): Calculate:

where:

(1a) (1b) (1c)

(1a) (1b) (1c) Let us assume (1a) (1b) (1c)

Gauss elimination computer scheme STEP STEP 1 ELIMINATE ELIMINATE y1 y1 y1 y1 from from b i cc:cc: Multiply by and add to 1b

Multiply by and add to 1c

New set : (2a) (2b) (2c) (2a) (2b) (2c)

(2a) (2b) (2c) Elimination scheme repeat for equations 2b i 2c: Multiply by add o 2c

(3a) (3b) (3c) (3a) (3b) (3c)

Back substitution part: Setting y 3 to 3b: Multiply by add to 3b

Because It is the first calculated approximation of the solution. Next iterations form a converged series:

Example 2 Nonlinear circuit having two variables (node voltages)

e1e1 e2e2

Data:

Nodal equations: 1 2

Jacobian matrix:

We choose starting vector: Calculate:

Applying N-R scheme: where: hence:

STOP CRITERIA not satisfied: k=k+1:

Second NR iteration where: hence:

for k=7: where: hence:

Because:

Briefly about: Iterative models of nonlinear elements

Iterative NR model of nonlinear resistor (voltage controled)

circuit  From NR method:

Model iterowany opornika (6)

Example 3 Newton-Raphson Newton-Raphson Iterative model method

e1e1 e2e2

Data:

Scheme for (k+1) iteration 1 2

1 1 2

2 1 2

1 2

1 2

For starting vector: We calculate parameters of the models:

For nonlinear element g 6 :

Linear equations for the first approximation: Solution for k=1

Second step Solution for k=2

Briefly about: Forward Euler Method (Explicit) Backward Euler Method (Implicit)

Forward Euler Method (Explicit) Backward Euler Method (Explicit)

Backward Euler Method (Explicit) is based on the following Taylor series expansion

v (t) C vsvs

v c (t k )

Example with nonlinear capacitor FEM

FEM steps

BEM step 1

Using N-R method with starting point

BEM step 2 after N-R procedure with new starting point

Using N-R method with starting point