Introduction

Electric circuits composed of discrete elements in which the voltages and currents are described in continuous time are sometimes called analog circuits, to distinguish them from circuits used in digital signal processors. We now model some analog circuits by writing ordinary differential equations in the state-variable form just as we modeled mechanical systems in Chapter 2. We apply the basic principles of circuit analysis, which are Kirchhoff's voltage and current laws, Ohm's law, Faraday's law, and the law relating the voltage across a capacitor to the current through it. We consider only circuits composed of the following elements: voltage sources, current sources, resistors, capacitors, inductors, and operational amplifiers. Our treatment here is limited to the type of circuits commonly encountered in the signal-processing parts of control systems, as opposed to those in power supplies or in other highpowered devices.

In Section 3.2 we confine our attention to passive circuits. These circuits contain no energy sources (such as operational amplifiers or batteries) except for the voltage or current sources that drive them. We can apply nearly all the basic circuit laws in this simple setting, and we can demonstrate an important principle of control engineering pertaining to interconnected elements, called the loading effect.

In Section 3.3 we model active circuits having operational amplifiers. This extends significantly our capability to design dynamic circuits for control system purposes. We also study simple instrumentation circuits for measuring dynamic variables.

Introduction

We now develop methods for calculating the dynamic response of a linear system that is modeled by its transfer functions. The response depends upon the physical parameters of the system, the input function, and any nonzero initial conditions. We are particularly interested in relating the dynamic features of the response to the physical properties of the system. If the input is a step function, the response may fluctuate temporarily and eventually reach a constant value. The nature of the fluctuations – the length of time during which they persist, and whether they cause the response to overshoot its final value excessively or to oscillate with both positive and negative values – are dynamic features of vital importance. The initial values of the response and its derivatives and the final value of the response are also important characteristics that depend on the physical parameters. In simple systems with simple inputs – for example, a firstor second-order system with a step input – the dynamic features of the response are directly related to simple combinations of the parameter values. But in higher-order systems these important relationships are less obvious because the significant dynamic features of the response depend on complicated combinations of the parameter values. We must now employ a mixture of analysis tools, computer calculations, and approximation techniques to determine which of our system parameters have the most influence on the significant dynamic features of the response.