In the first-order approximation, non-ideal operational amplifiers (or real OpAmps) can be modeled using an input resistor $R_{in}$, an output resistor $R_{out}$, and a dependent voltage source with a gain of $A$ (Fig. 1). In a non-ideal OpAmp at least one of the above three quantities are positive and finite. For instance, the OpAmp shown in Fig. 2 is also non-ideal because the gain $A$ is finite. An ideal OpAmp has $R_{in}=\infty$, $R_{out}=0$, and $A=\infty$.

OpAmps are unually represented in simplified form, as shown in Fig. 3. Notice that, excluding the power lines that are normally necessary to power the dependent voltage source, the OpAmp is a 4-terminal device and, according to KCL the algebraic sum of the currents flowing out through the 4-terminals is equal to 0 . Quite often, to simlify notations, we do not show the ground electrode in when representig OpAmps (see the right diagram in Fig. 3). Unfortunately, this representation is somewhat misleading because it suggests that the algrbraic sum of the currents flowing out of the 3 terminals is zero, which is generally not true (because we are missing the 4th, "hidden" electrode).

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 it's equivalent circuit shown in Table -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 $A=\infty$ (i.e. in the case if ideal OpAmps) the above equation simplifies to $$\begin{equation}\frac{V_{out}}{V_{in}}=-\frac{R_2}{R_1} \end{equation}$$ which is the same as the gain of an inverter with ideal OpAmp.

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 if ideal OpAmps) we get $$\begin{equation}\frac{V_{out}}{V_{in}}=1+\frac{R_2}{R_1} \end{equation}$$ which is the same as the gain of a follower with ideal OpAmp.

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 claculators, anolog-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.