The most common application of an operational amplifier is in linear circuits where the output is a precise, linear function of the input, defined by a stable gain and predictable phase relationship. This section will analyze the three foundational configurations that form the basis for countless complex analog designs: the inverting amplifier, the non-inverting amplifier, and the voltage follower. The principles established here are fundamental to understanding more advanced op-amp topologies.

2.1 The Inverting Amplifier

The inverting amplifier is configured to produce an output that is an amplified and inverted version of the input signal. A key concept in analyzing this circuit is the virtual short. Because of the op-amp’s extremely high open-loop gain, the voltage difference between its inverting and non-inverting input terminals is driven to nearly zero in a closed-loop negative feedback configuration. Since the non-inverting terminal is connected to ground (0V), the principle of virtual short forces the inverting terminal to the same potential, creating a ‘virtual ground’.

The voltage gain is derived from the nodal equation at the inverting terminal: \frac{0-V_i}{R_1}+ \frac{0-V_0}{R_f}=0 =>\frac{-V_i}{R_1}= \frac{V_0}{R_f} =>V_{0}=\left(\frac{-R_f}{R_1}\right)V_{t} This gives a closed-loop voltage gain of: \frac{V_0}{V_i}= \frac{-R_f}{R_1} The negative sign in the gain equation is significant; it indicates a 180-degree phase difference between the input and output signals. The magnitude of the gain is set precisely by the ratio of the feedback resistor (R_f) to the input resistor (R_1).

2.2 The Non-Inverting Amplifier

The non-inverting amplifier provides an amplified output signal that is in phase with the input signal. In this configuration, the input voltage (V_i) is applied directly to the non-inverting terminal. Due to the virtual short concept, the voltage at the inverting terminal (V_1) is equal to V_i.

The voltage at the inverting terminal can be calculated using the voltage division principle across the R_1-R_f network: V_{1} = V_{0}\left(\frac{R_1}{R_1+R_f}\right) Since V_1 = V_i, we can write: V_{0}\left(\frac{R_1}{R_1+R_f}\right)=V_{i} Rearranging the terms gives the final gain equation: \frac{V_0}{V_i}=1+\frac{R_f}{R_1} The positive sign in the gain equation confirms that the input and output signals are in phase. The gain is always greater than or equal to one and is determined by the resistor ratio.

2.3 The Voltage Follower

The voltage follower is a special case of the non-inverting amplifier that provides a unity voltage gain (A_v = 1). This configuration is achieved by setting the feedback resistor (R_f) to zero ohms or the input resistor (R_1) to infinity ohms (an open circuit). In practice, this means the output is connected directly to the inverting input.

The relationship between the input and output is direct and simple: V_0 = V_i While it does not provide voltage amplification, the voltage follower is an extremely useful circuit. Its primary application is as a buffer, leveraging the op-amp’s high input impedance and low output impedance to isolate a signal source from a load that might otherwise draw too much current and alter the source voltage.

These three basic amplifier blocks demonstrate a core principle of op-amp design: precise and stable circuit behavior is defined almost entirely by the external feedback components. This principle of controlling performance with resistor ratios is directly extended to create circuits that perform sophisticated mathematical operations, as detailed in the following section.