The introduction to operational amplifiers in Chapter 11 began with a discussion of the ideal op amp, and then explored the behavior of circuits containing nonideal op amps. Key points and topics are outlined here.

• Ideal operational amplifiers are assumed to have infinite gain and zero input current, and circuits containing these amplifiers were analyzed using two primary assumptions:
  1. The differential input voltage is zero: v_id = 0.
  2. The input currents are zero: i₊ = 0 and i_- = 0.
• Assumptions 1 and 2, combined with Kirchhoff's voltage and current laws, are used to analyze the ideal behavior of circuit building blocks based on operational amplifiers. Constant feedback with resistive voltage dividers is used in the inverting and noninverting amplifier configurations, the voltage follower, the difference amplifier, the summing amplifier,and the instrumentation amplifier, whereas frequency-dependent feedback is used in the integrator, low-pass filter, and differentiator circuits.
• Infinite gain and input resistance are the explicit characteristics that lead to Assumptions 1 and 2. However, many additional properties are implicit characteristics of ideal operational amplifiers; these assumptions are seldom clearly stated, though. They are
  • Infinite common-mode rejection
  • Infinite power supply rejection
  • Infinite output voltage range
  • Infinite output current capability
  • Infinite open-loop bandwidth
  • Infinite slew rate
  • Zero output resistance
  • Zero input-bias currents
  • Zero input-offset voltage
• The effect of removing the various ideal operational amplifier assumptions was explored in detail. Expressions were developed for the gain, gain error, input resistance, and output resistance of the closed-loop inverting and noninverting amplifiers, and it was found that the loop gain Aβ plays an important role in determining the value of these closed-loop amplifier parameters.
• The dc error sources, including offset voltage, bias current, and offset current, all limit the dc accuracy of op amp circuits. Real op amps also have limited output voltage and current ranges as well as a finite rate of change of the output voltage called the slew rate. Circuit design options are constrained by these factors.
• The frequency response of basic single-pole operational amplifiers is characterized by two parameters: the open-loop gain A_o and the gain-bandwidth product ω_T. Analysis of the gain and bandwidth of the inverting and noninverting amplifier configurations demonstrated the direct trade-off between the closed-loop gain and the closed-loop bandwidth of these amplifiers. The gain-bandwidth product is constant, and the closed-loop gain must be reduced in order to increase the bandwidth, or vice versa.
• The bandwidth of multistage amplifiers is less than the bandwidth of any of the single amplifiers acting alone. An expression was developed for the bandwidth of a cascade of N identical amplifiers and was cast in terms of the bandwidth shrinkage factor.
• A comprehensive example of the design of a multistage amplifier was presented in which a computer spreadsheet was used to explore the design space. The influence of resistor tolerances on this design was also explored.
• Simplified macro models are often used for simulation of circuits containing op amps.
• Simple macro models can be constructed in SPICE using controlled sources, and most SPICE libraries contain comprehensive macro models for a wide range of commercial operational amplifiers.