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Operational amplifiers (commonly called op amps) are a ubiquitous building block for designing electronic circuits. Today, these devices are fabricated as small integrated circuits, but the concept started long ago using vacuum tubes. There is a 1946 patent for an early use of op amp concepts (Reference 1), although that name was not used at the time. Raggazinni is often credited with coining the term “operational amplifier” in 1947 (Reference 2).

I encountered op amps back in the 20th century while taking a college lab course in analog computing. Analog circuitry was used to simulate systems by connecting summing amplifiers, differentiators, and integrators via complex patch panels. The use of analog computing was on the way out, being displaced by digital computers, so I can’t say I got much out of the computing part of the class. I did learn a lot about op amp circuits and control systems, though, which are still valuable today.

**Ideal operational amplifier**

To understand basic op amp functions, we use the concept of the “ideal op amp.” The ideal op amp is a voltage-controlled voltage source as shown in **Figure 1**, with these attributes:

- Infinite gain (A
_{v}) and infinite bandwidth - Zero output impedance
- Infinite input impedance (zero input current)

**Figure 1** The ideal op amp is a voltage-controlled voltage source with infinite input impedance and zero output impedance.

An important fourth attribute is usually included but it is only valid if there is negative feedback applied to the op amp:

- Zero volts between the two inputs

In case you are wondering how this op amp gets its power, there are two power-supply connections (positive and negative) for the device, often neglected when discussing the circuit design (but absolutely essential when wiring up a real circuit). Typically, bipolar power is supplied, +/-15V, which supports a healthy signal swing.

The cool thing about op amps is that for many non-critical applications the op amp performance (gain, bandwidth, impedance, etc.) is so good compared to the circuit requirements that they really do act like ideal op amps. They are easy to design with and have become an essential building block for electronic systems.

**Non-inverting amplifier**

The first common op amp configuration we will look at is the non-inverting amplifier (**Figure 2**). I always wonder why we don’t call this the “regular amplifier” configuration or maybe just “amplifier.”

**Figure 2** The non-inverting amplifier uses two resistors to provide negative feedback to the op amp.

In this configuration, we see that we have feedback from the output back to the inverting input. This negative feedback means that attribute #4 is invoked, and the two inputs will always have zero voltage across them (i.e., they are at the same voltage). Because no current can flow into the inputs, the voltage present at the non-inverting input is determined by the voltage divider formed by R_{1} and R_{2}.

Rearranging to obtain the gain of the amplifier,

Notice that the voltage gain of the circuit does not depend on the gain of the op amp. We are assuming that if the op amp gain is *really big*, then enough feedback will be applied to the non-inverting input to produce the desired function.

Let us check that assumption about the two op amp inputs having zero voltage between them. Suppose the non-inverting input is a few millivolts higher than the inverting input. The huge voltage gain of the op amp would cause the output to increase, which would feedback via the resistor divider to the *inverting* input. An increased voltage on the inverting input will cause the op amp output to decrease until both inputs have the same voltage. Thus, the high gain of the op amp plus negative feedback keeps the input voltages the same.

**Buffer amplifiers
**A special case of the non-inverting amplifier is the buffer amplifier (also called unity-gain amplifier or voltage follower), having a voltage gain of one (

**Figure 3** The buffer amplifier provides infinite input impedance and zero output impedance.

**Inverting amplifier
**Another common op-amp circuit is the inverting amplifier (

**Figure 4** The inverting amplifier produces the negative value of the input, scaled by the ratio of the two resistors.

This circuit is analyzed by noting that both inputs of the op amp will be at 0V. The non-inverting input is connected to ground and the inverting input will be driven to the same voltage via feedback through the resistors. We also note that current (*i*) flows through both resistors because no current enters the inverting input of the op amp.

Rearranging to obtain the gain of the amplifier,

The minus sign in the gain is important and must be considered in applying the circuit. In some situations, it may not matter, you just may need to amplify the input signal without regard to changes in polarity. In other cases, the polarity may be critical, and your signal could end up being upside down.

**Differential amplifier**

The inverting amplifier and non-inverting amplifier can be combined to create a differential amplifier (also called a difference amplifier), as shown in **Figure 5**.

**Figure 5** The differential amplifier produces an output voltage that is the difference between the two inputs.

Applying superposition, we can combine the gain equations of the inverting and non-inverting amplifier configurations.

Substituting v_{1} for v_{in}, the inverting gain remains unchanged:

The v_{2} input has an additional voltage divider made up of R_{3} and R_{4}, so the gain equation becomes:

Combining the two equations gives:

If we set R_{1} = R_{3} and R_{2} = R_{4}, the equation reduces to:

We assumed that we have ideal op amps, but we didn’t say anything about the resistors. The gain of these circuits will depend on the actual values of the resistors and therefore their tolerance. This is especially true with the differential amplifier where we are relying on matched resistor values.

These common op-amp circuits are useful for amplifying various analog signals. The ideal op amp model helps us understand how these circuits operate. For more detailed information on op amp circuits, refer to the excellent material in references 3 and 4 below.

*— Bob Witte is President of Signal Blue LLC, a technology consulting company.*