Circuits Theory Examples NewtonRaphson Method Formula for onedimensional

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Circuits Theory Examples Newton-Raphson Method

Circuits Theory Examples Newton-Raphson Method

Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged

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

Multidimensional case: where: JACOBIAN MATRIX

ALGORITHM STARTING POINT STEP 0 STEP 1 STEP 2 STEP 3 Calculate Solve the

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

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

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

• Stop condition parameter

• Stop condition parameter

Numerical EXAMPLES Example 1

Numerical EXAMPLES Example 1

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

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

Starting point (first approximation): Calculate:

Starting point (first approximation): Calculate:

where:

where:

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

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

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

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

Gauss elimination computer scheme STEP 1 ELIMINATE y 1 from b i c: c

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

Multiply by and add to 1 c

Multiply by and add to 1 c

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

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

Elimination scheme repeat for equations 2 b i 2 c: (2 a) (2 b)

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

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

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

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

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

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

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)

Example 2 Nonlinear circuit having two variables (node voltages)

e 1 e 2

e 1 e 2

Data:

Data:

Nodal equations: 1 2

Nodal equations: 1 2

Jacobian matrix:

Jacobian matrix:

We choose starting vector: Calculate:

We choose starting vector: Calculate:

Applying N-R scheme: where: hence:

Applying N-R scheme: where: hence:

STOP CRITERIA not satisfied: k=k+1:

STOP CRITERIA not satisfied: k=k+1:

Second NR iteration where: hence:

Second NR iteration where: hence:

for k=7: where: hence:

for k=7: where: hence:

Because:

Because:

Briefly about: Iterative models of nonlinear elements

Briefly about: Iterative models of nonlinear elements

Iterative NR model of nonlinear resistor (voltage controled)

Iterative NR model of nonlinear resistor (voltage controled)

From NR method: circuit

From NR method: circuit

Model iterowany opornika (6)

Model iterowany opornika (6)

Example 3 Newton-Raphson Iterative model method

Example 3 Newton-Raphson Iterative model method

e 1 e 2

e 1 e 2

Data:

Data:

Scheme for (k+1) iteration 1 2

Scheme for (k+1) iteration 1 2

1 1 2

1 1 2

1 2 2

1 2 2

1 2

1 2

1 2

1 2

• For starting vector: • We calculate parameters of the models:

• For starting vector: • We calculate parameters of the models:

• For nonlinear element g 6:

• For nonlinear element g 6:

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

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

Second step Solution for k=2

Second step Solution for k=2

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

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

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

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

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

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

vs v (t) C

vs v (t) C

vc(tk)

vc(tk)

Example with nonlinear capacitor • FEM

Example with nonlinear capacitor • FEM

FEM steps

FEM steps

BEM step 1

BEM step 1

Using N-R method with starting point

Using N-R method with starting point

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

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

Using N-R method with starting point

Using N-R method with starting point