The transition matrix for linear circuits

The state transition matrix Φ(t)=e^At plays an important role in the state variable analysis of linear time-invariant circuits. In this paper, we give a method to compute the equivalent matrix for a set of differential-algebraic equations. Specifically, the algorithm is illustrated for the modified nodal analysis (MNA). A benefit of using MNA formulation is that equation formulation is straightforward and computer-aided analysis of large circuits is simplified. Another benefit is that inconsistent initial conditions in the analysis of switched circuits is automatically and correctly handled by the algorithm. Comparison with the state variable approach is made, and application to the simulation of linear circuits is given. The method is based on circuit theory concepts accessible to all electrical engineers. In addition to the transition matrix, a numerical method to compute the zero state response of linear circuits for a restricted set of inputs is given. The transition matrix along with the zero state response results in a special algorithm that is used to compute the time response of lumped linear time-invariant circuits at equally spaced intervals of time. This method is compared with the solution of linear circuits by SPICE-like simulators