TABLE OF CONTENTS This chapter is an introduction to state variables and state equations as they apply in circuit analysis. The state transition matrix is defined, and the state-space to transfer function equivalence is presented. Several examples are presented to illustrate their application.
As we know, when we apply Kirchoff's Current Law (KCL) or Kirchoff's Voltage Law (KVL) in networks that contain energy-storing devices, we obtain integro-differential equations. Also, when a network contains just one such device (capacitor or inductor), it is said to be a first-order circuit. If it contains two such devices, it is said to be second-order circuit, and so on. Thus, a first order linear, time-invariant circuit can be described by a differential equation of the form
A second order circuit can be described by a second-order differential equation of the same form as (5.1) where the highest order is a second derivative.
An nth-order differential equation can be resolved to n first-order simultaneous differential equations with a set of auxiliary variables called state variables. The resulting first-order differen-tial equations are called state-space equations, or simply state equations. These equations can be obtained either from the nth-order differential equation, or directly from the network, provided that the state variables are chosen appropriately. The state variable method offers the advantage that it can also be used with non-linear and time-varying devices. However, our discussion will be limited to linear, time-invariant circuits.
State equations can also be...
