Welcome to Signal Chain Basics, a series of articles describing the operation of the analog signal chain. Over the course of this series, topics will include analog signal processing and the devices required to support them. Your comments are always welcome; they could even be the topic of a future discussion. (You can also see a short video introduction here. )
Operational Amplifier—The Basic Building Block
The basic building block of the signal chain is the operational amplifier (op amp) (Figure 1). In the simplest form this is a device with a differential input of infinite input impedance, and a voltage-controlled voltage source with a gain approaching infinity. These features alone would be of little value. However, through the use of various feedback techniques, this becomes a very valuable device.
Figure 1: An ideal op amp
The transfer function of the ideal op amp is seen from the circuit to be:
With a very large value for Aol (open-loop gain) this circuit is of minimal value. A survey of data sheets will reveal that the absolute value of Aol is not tightly controlled in production. Adding negative feedback, as shown in Figure 2 is the solution to the problem.
Figure 2: The ideal op amp with feedback
Since there can be no current flow at the input pins, the current through Ri must equal the current through Rf. This can be expressed as:
Combining these two terms, setting V2=0, and assuming the open-loop gain is very large results in the standard closed-loop gain (Acl) equation:
Notice, from the first equation, that the op amp amplifies the difference between the input voltages. As long as the op amp is operating in a linear mode, the input pins are at the same voltage (Figure 3).
Figure 3: Standard circuit drawing for the inverting op amp configuration.
For the non-inverting configuration, the gain equation results are slightly different (Figure 4).
Fig. 4: The non-inverting op amp configuration
With a very large value of Aol, the gain expression reduces to:
The complete development of the transfer functions can be found in the associated sidebar below. This development also treats the case where Aol is less than infinite.
Starting with this basic building block a large number of analog computing circuits can be configured. Three basic concepts developed here will be used many times in future articles: gain expression for very large Aol, gain expression of restricted Aol, and the concept that the op amp drives the output such as to keep the input pins at the same voltage.
Sidebar: Development of closed-loop gain expression
For the non-inverting configuration:
When Aol is a very large number the expression reduces to:
This is the closed loop gain expression for the ideal condition where Aol is very large. When the open loop gain is less than ideal the actual closed loop gain expression becomes:
.
Since Aol will always be less than infinite there will always be some error in the gain expression. The value of Aol is usually large enough that this error can be ignored.
A similar development for the inverting case:
When Aol is a very large number the expression reduces to:
This is the closed loop gain expression for the ideal condition where Aol is very large. When the open loop gain is less than ideal the actual closed loop gain expression becomes:
Notice that the Acl here is the non-inverting Acl.
About the author
William P. (Bill) Klein is a Senior Applications Engineer with the High Performance Analog group at Texas Instruments. Bill joined TI through its acquisition of Burr-Brown in August 2000. His experience as an analog circuit designer covers over 40 years in fields ranging from mineral exploration to medical nuclear imaging. One current role Bill has is hosting the Analog e-LAB Web Cast, presenting real world solutions to real world problems in analog circuit design. In addition to a BSEE from Arizona State University and registration as a Professional Engineer in the State of Arizona, he has authored numerous magazine articles, application notes and conference papers
