Nikkei Electronics Asia -- April 2009
Insights
Basic Building Block, Operations of Op Amps

The basic building block of a signal chain is the operational amplifier (op amp). In the simplest form this is a device with a differential input of infinite input impedance, and a voltage-controlled source with a gain approaching infinity.

These features alone would be of little value. However, through the use of various feedback techniques, an op amp can become a very valuable device. The transfer function of an ideal op amp is seen from the circuit to be:

V(out) = (V2-V1) Aol

With a very large value for Aol (open-loop gain) this circuit is of minimal value. A survey of data sheets reveals that the absolute value of Aol is not tightly controlled in production. Adding a negative feedback, as shown in the Fig, is a solution to the problem. 

Since there can be no current flow at the input pins, the current through Ri must equal to the current through Rf. This can be expressed as:

[V(in) - V1]/Ri = [V1-V(out)]/Rf

Combining these two terms, setting V2=0, and assuming the Aol is very large, results in the standard closed-loop gain (Acl) equation:

V(out)/V(in) = -(Rf/Ri) = Acl

From the first equation, 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. For the non-inverting configuration, the gain equation results are slightly different. With a very large value of Aol, the gain expression reduces to:

V(out)/V(in) = (Ri + Rf)/Ri = Acl

Starting with this basic building block a large number of analog computing circuits can be configured. Three basic concepts developed are frequently used: gain expression for very large Aol, gain expression of restricted Aol, and the concept that the op amp drives the output so as to keep the input pins at the same voltage.

Basic Operations

The high-gain circuit with a differential input received its name in the days of analog computers. Every mathematical operation required an amplifier to isolate one function from the next. In the simplest form, an op amp can be configured for inverting or non-inverting gain. The gain equations show that the inverting stage may have an Acl of less than one when Ri>Rf. When Ri=Rf, the gain is minus one (inverting). The non-inverting stage can never have a gain less than unity. When Ri is open, the circuit reduces to a unity-gain voltage follower. If a gain less than one is required, a voltage divider is placed before the amplifier. 

Since this is a linear system, the rules of linear superposition apply. Therefore, the next development is the summation of two or more signals. To reach these relationships, start by assuming V2=0 and write the equation for Vout as a function of V1. Then assume V1=0 and write the expression for V2. Combine the two terms to get the entire transfer function. More inputs can be added in parallel, and the total transfer function can be developed with this superposition technique. 

This ability to add voltages has more value than just the arithmetic. There are times in a design when it is necessary to accomplish a level shift. These circuits will do just that. With these variations, it is also possible to accomplish the complementary arithmetic operation, and subtraction. By using linear superposition, the general output expression for the difference amp is:

Vout = [(R1+R2)/R2] [R4/(R3+R4)] V1 - (R1/R2) V2

A widely used application is where the desired signal is riding on an interfering signal. The interference signal is termed the common mode voltage (Vcm) as it is common to both inputs. The desired signal is the differential mode voltage (Vdm). In this case, it is the sum of Vdm1 and Vdm2. If R1=R4 and R2=R3, Vout is given by:

Vout = R1/R2 (V1-V2)

The accuracy with which the interference signal is eliminated is based on two variables: the accuracy of the resistor match; and a parameter of the op amp called common mode rejection ratio (CMRR). With the assumption that the perfect op amp does exist, the calculation of output due to resistor mismatch is a simple spreadsheet exercise.

by Bill Klein, Senior Applications Engineer, High Performance Analog Group, Texas Instruments

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