An operational amplifier (or OpAmp) is a voltage amplifier with a differential input, a single-ended output, and very high gain. Operational amplifiers are widely used in electronics for signal amplification, filter design (for instance to avoid using bulky inductors), analog calculators, analog-to-digital converters, differentiators, integrators, etc.

The equivalent circuit of an OpAmp is shown in Fig. 1, where $R_{in}$ is the input resistance, $R_{out}$ is the output resistance, and $A$ is the gain. Practical (or non-ideal) OpAmps have finite input resistance, non-zero output resistance, and/or the dependent voltage source has a finite gain. In contrast, ideal OpAmps have $R_{in}=\infty$, $R_{out}=0$, and $A=\infty$.

Sometimes, only one of the three parameters describing the OpAmp is finite. For instance, the input resistance of the OpAmp represented in Fig. 3 is infinite, the output resistance in zero, but the gain is finite. In this case, the OpAmp is still called non-ideal.

OpAmps are usually represented in simplified form, as shown in Fig. 3, where the power supply is assumed to be part of the device itself. Since the OpAmp is a four-terminal device, the algebraic sum of the currents flowing out through its four terminals must equal zero in accordance with (Kirchhoff's current law).

However, to simplify notations, the ground electrode is often omitted when representing OpAmps (see the right-side diagram in Fig. 3). This representation is somewhat misleading, as it may suggest that the algebraic sum of the currents flowing out of the three visible terminals is zero, which is generally not the case. Consequently, even when an OpAmp is depicted with only three electrodes, it is implicitly understood that a fourth terminal (the ground electrode) is present to satisfy Kirchhoff's current law.

Table 1 shows the values of the input resistance, output resistance and gain for a few OpAmps available commercially. In addition to these parameters manufacturers also specify parameters such maxim voltage, operating range of temperatures, cutoff frequency, transient response time, input noise, etc.

To solve problems with non-ideal OpAmps it is always recommended to replace the OpAmp with its equivalent circuit shown in Fig. 1. Then, one can use nodal analysis to compute the sought variables.

A standard configuration for OpAmps is the inverting amplifier shown in Fig. 4. If the input and output resistances of the OpAmp are $R_{in}$ and $R_{out}$ respectively, and the gain is $A$, it is relatively simple to show that $$\begin{equation}\frac{V_{out}}{V_{in}}=-\frac{1}{\frac{{\color{blue}R_{out}}+R_2}{{\color{blue}A} R_2 - {\color{blue}R_{out}}} \left(1+\frac{R_1}{R_2}+\frac{R_1}{{\color{blue}R_{in}}}\right) + \frac{R_1}{R_2}} \end{equation}$$ In the limit when $R_{in}=\infty$ and $R_{out}=0$, the above equation simplifies to $$\begin{equation}\frac{V_{out}}{V_{in}}=-\frac{1}{\frac{1}{{\color{blue}A}} \left(1+\frac{R_1}{R_2}\right) + \frac{R_1}{R_2}} \end{equation}$$

In the limit when $R_{in}=\infty$, $R_{out}=0$ and $A=\infty$ (this is the case of ideal OpAmps) the above equation simplifies to $$\begin{equation}\frac{V_{out}}{V_{in}}=-\frac{R_2}{R_1} \end{equation}$$ which is the gain of an inverter with ideal OpAmp (see next section in this WebBook).

Another standard configuration for OpAmps is the non-inverting amplifier (follower) shown in Fig. 5. If the input and output resistances of the OpAmp are $R_{in}$ and $R_{out}$ respectively, and the gain is $A$, one can show that $$\begin{equation}\frac{V_{out}}{V_{in}}=\frac{\frac{1}{{\color{blue}R_{in}}}+A \frac{\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{{\color{blue}R_{in}}}}{\frac{{\color{blue}R_{out}}}{R_2}-A} } {\left(\frac{1}{{\color{blue}R_{in}}} + \frac{1}{R_2}\right) \frac{\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{{\color{blue}R_{in}}}}{\frac{1}{R_2}-\frac{A}{{\color{blue}R_{out}}}} - \frac{1}{R_2}} \end{equation}$$ If $R_{in}=\infty$ (this approximation works well for OpAmps made with FETs), the above equation simplifies to $$\begin{equation}\frac{V_{out}}{V_{in}}= \frac{1}{ \frac{1+\frac{{\color{blue}R_{out}}}{R_2}}{A}+ \frac{1}{\beta}}\end{equation}$$ where we denoted $\beta=1+\frac{R_2}{R_1}$. If also $R_{out}=0$, the above equation simplifies to $$\begin{equation}\frac{V_{out}}{V_{in}}= \frac{1}{ \frac{1}{A}+ \frac{1}{\beta}}\end{equation}$$ Finally, if $R_{in}=\infty$, $R_{out}=0$ and $A=\infty$ (i.e. in the case of ideal OpAmps) we get $$\begin{equation}\frac{V_{out}}{V_{in}}=1+\frac{R_2}{R_1} \end{equation}$$ which is the gain of a follower with ideal OpAmp (see next section in this WebBook).

Ideal OpAmps have many important applications in electronics, audio and video-frequency pre-amplifiers and buffers, differential amplifiers, precision rectifiers, peak detectors, voltage and current regulators, analog calculators, analog-to-digital and digital-to-analog converters, clippers and clampers. When used in combination with inductors and capacitors one can build integrators, differentiators, filters, and oscillators.