Example for Nonlinear SystemsSolve the nonlinear system:We have:Jacobian matrix:Interactive step:1) Solve: to obtain 2) UpdateStarting point:Assuming we have,First iteration:From equation 1) From equation 2) , Second iteration:From equation 1) From equation 2) , Third iteration:From equation 1) From equation 2) Nodal FormulationAssume all resistors are voltage controlled, so Branch equations: KVL: KCL: Substituting for one obtainsThis is the generalized form of the nodal equations for a nonlinear network. To solve it by the N-R algorithm, formulate The Jacobian is obtained by applying the chain rule for differentiation:Here is a matrix which will be denoted by . Differentiation of KVL with respect to provides Sowhich can be determined directly from schematic using the stamp-like approach and replacing nonlinear components by their derivatives .Piecewise Linearization (Katzenelson method)Piecewise linear resistors: (a)Voltage controlled; (b) Current controlledEquations for the elements in the th segment arewhere and are equivalent sources (cut-off values) for -th segment. In the matrix form:The tableau equations can be written as follows:or in compact formThe subscript denotes the region in which the network operates.An error vectorThe solution is reached when we reduce to zero. The correction in the Newton-Raphson method is obtained by solvingand the new solution is provided none of the elements crossed to a new region. If this is true, is the desired final solution.If at least one element crossed into a new region, the full step is not taken and the formulaand are relative distance scaling factorswhere and are the boundary points.Fig. Obtaining the step-reducing coefficient.Since at the th step we are at the boundary of two regions, two equations are simultaneously valid:Substituting for, we haveExample 12.6.1. Consider the network with its nonlinearities described by the characteristics shown. Apply Katzentelsons algorithm by using the nodal formulation for the network. Fig. Piecewise linear network and its initial equivalent with additional sources.Let the initial nodal voltages be Denote the branch voltages by. The voltages are , In the regions shown in the figure, . This initial state of the network is shown in the figure. nonlinear resistor G1nonlinear resistor G2nonlinear resistor G3Fig. Piecewise linear resistors.Applying the steps of the algorithm:Step 2: Error vectorStep 3: SolveThe solution is Step 4: Nodal voltagesthe element voltages are All the increments are positive and all elements cross into new regions. The full step is not possible, so calculate the step-reduction coefficients. Step 5:Step 6: Scaling factorStep 7: Nodal voltageand the element voltages are Step 8: Error vectorStep 9: We are now in a new region in which , (remain unchanged) and ; the new nodal admittance matrixis used when returning to step3.This completes one iteration. The solution of the problem is obtained after 4 more iterations.