OVERVIEW

There are many reasons for considering the time and frequency domain relationships of a second-order system in the study of operational amplifiers. One is that many operational amplifier configurations can be modeled with reasonable accuracy assuming just a second-order system. Such a procedure represents a reasonable compromise between complexity and accuracy of the model. Another reason is that these relationships allow us to predict frequency domain performance from the simpler to measure time domain performance.

C.1 GENERAL SECOND-ORDER SYSTEM IN THE FREQUENCY DOMAIN

The general transfer function of a low-pass, second-order system in the frequency domain using voltage variables is

where

The roots of Eq. (C.1-1) are illustrated in Fig. C.1-1.

Figure C.1-1: Pole locations of a general second-order system.

The magnitude of the frequency response can be found from Eq. (C.1-1) as

However, Eq. (C.1-2) may be generalized by normalizing the amplitude with respect to A ₀ and the radian frequency by ? _n to give

A plot of Eq. (C.1-3) in dB versus log ?/ ? _n is shown in Fig. C.1-2, where ? or Q is used as a parameter. By taking the derivative of Eq. (C.1-3) with respect to ?/ ? _n and setting it to zero, the peak of A(j ?/ ? ₀)/A ₀ can be found as

Figure C.1-2: Gain magnitude response for various values of for a second-order, low-pass system.

when ? < 0.707.

The second-order function of Eq. (C.1-1) is...