1.7 System matrix solution
IThe analog simulator use a computer equivalent to Gaussian
elimination to solve system matrix with linear elements. With the introduction
of nonlinear elements the solver uses a nonlinear technique known as the
Newton-Raphson equation. The nonlinear elements in the circuits
provide transcendental equations to be resolved with an iterative guessing
technique. The Newton-Raphson algorithm, a method of successive apoproximations,
is an iterative approach solving a set of nonlinear equations. This algorithm
starts the iterative process with an initial guess and finds the solution
following the formula:
Xn+1 = Xn - F(Xn)/F`(Xn)
The iterations terminate when the difference between Xn+1
= Xn tends to zero (or required precision).
Let us take an example of a diode connected in parallel
with a resistor and a current source. This circuit is characterized by
a nodal equation and its derivative as follows:
I(Vd) = 0 = -5 + Vd/2 + 1pA*(exp(40*Vd) - 1)
I`(Vd) = 0 + 0.5 + 40pA*exp(40*Vd)
Exercise:
Calculate the 12 iterative steps for this nodal equation
starting with Vd=1.