Numerical Integration

Abstract

In general terms, the transient simulation problem can be stated as follows: Given a system of differential-algebraic circuit equations and their solution at times t₀⋯, t _n-1 , t _n , it is desired to solve the system of equations for its solution at time t _{n + 1} . Assume the system of equations contains capacitors characterized by

$$ q = q\,(V) $$

((5.1))

such that

$$ I = dq/dt = (dq/dV) (dV/dt) = q'V' $$

((5.2))

and/or inductors where

$$ \phi = \phi (I) $$

((5.3))

such that

$$ V = d\phi /dt = (d\phi /dI)(dI/dt) = \phi ' I' $$

((5.4))

and where the capacitor charges, q(V), and the inductor fluxes, ø(I) are known functions of voltage and current. Were an algebraic algebraic approximation to the derivatives V’ and I’ available, the system of equations could be converted to a purely algebraic form solvable via Newton-Raphson and/or Gaussian elimination methods.