Constructing Circuit Transfer Functions directly from the State Equations, Using Convolutions

Given the Nodal Admittance Matrix (NAM) of a linear circuit this article provides means to generate any desired transfer function, or s-expanded polynomial, in a very efficient way. There are three features that are used in combinations to facilitate this task. The first feature is the use of the state space and characteristic equation. The second feature is tuple-based array manipulations. In these manipulations multiplications and divisions are replaced with convolutions and deconvolutions. The last feature to work with is the newly developed UaL decomposition for tuple-based matrices [7]. The UaL decomposition is proven to be computationally more efficient compared to the traditional LU factorization. It is shown that the growth of the entries in both U and L matrices are gradual and it follow a geometrical pattern as the decomposition progresses. Another important feature of the UaL decomposition is that it does not generate nonzero remainders, or simply, it is an “error free” procedure. The method is applied to the Branch Admittance Matrix (BAM) representation of an RC (or RL) circuit. Finally, it is shown that working in the tuple-based format puts the U and L matrix entries into s-expanded form, such as P(s)=ans
ⁿ
+an-1s
^n-I
+...+a0, which in turn produces s-expanded transfer functions.