State Space Analysis of Dynamic Systems and Circuits, Eigenvalues

Author ORCID Identifier

Reza Hashemian:https://orcid.org/0000-0002-0028-7079

Publication Title

Understanding Eigenvalues



Document Type



A circuit with reactive (C and L) elements or a dynamic system can be functionally described by its state variables in a state space. A standard form of representing such a system in a state space is through its system equations, which in matrix format are given as sX(s) = AX(s) = BU(s) and the output equations Y(s) =CX(s) = DU(s). where, X(s) represents the state variables, U(s) is a vector of the forcing functions, and Y(s) is the system output vector. The coefficient matrices A, B, C, and D fully describe the movement of the system in the state space, which is based on the initial state, denoted by X(0-), and the forcing function U(s). Now, we can solve for X(s) Xs =(sJ-A)-1 [BU(s)] +1 X s sJ A BU s X where, J is a unit diagonal matrix. One specific case is when the circuit or system is left to respond to its initial state X(0-), with no forcing function. In this situation the system moves based on its natural frequencies, also known as the eigenvalues. So, to determine the eigenvalues we need to solve for s AXs sX = or solve for the roots of the characteristic equation sJ = A= 0 Here in this chapter we present two approaches to construct the state space equations of an analog circuit. In the first approach the state matrix A is directly constructed from the circuit inspections. This is proven to be a very efficient technique to identify the state matrix without going through a traditional nodal analysis. The matrix A is basically constructed by making specific measurements on the resistive circuit, when all the reactive (L and C) components are removed. Or alternatively, A is constructed by conducting simulation on the resistive circuit alone. In the second approach, a method is presented through which the zeros of a transfer function are first converted into poles and then the poles are extracted through the circuit eigenvalues. In addition, a new Modified Nodal Procedure (MNP) is discussed in this chapter that allows the circuit solutions with all types of independent and controlled sources present. It shows how the presence of these sources are incorporated into the conductance matrix, providing a unified nodal (branch) solution to circuit problems. The results obtained are double checked with circuit simulators, such as SPICE.

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Analog circuits, Eigenvalues, Node analysis, Poles and zeros, State space analysis, Transfer functions


Department of Electrical Engineering

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