2.1. The Inverting Amplifier

The inverting amplifier is a circuit designed to amplify an input signal and invert its phase (change its sign).

In this configuration, the input signal (V_i) is applied to the inverting terminal through a resistor (R_1), while the non-inverting terminal is connected to ground (0V). Because of the “virtual short” rule, the voltage at the inverting terminal is also considered to be zero volts. The amount of amplification, or gain, is determined by the ratio of the feedback resistor (R_f) to the input resistor (R_1).

The formula for the voltage gain is: Gain = V₀ / Vᵢ = -R_f / R₁

Key Insight: This formula shows that the gain is entirely controlled by the external resistors you choose. This is a core principle of op-amp circuits: the high internal gain of the op-amp is leveraged so that the overall circuit’s behavior is determined almost entirely by stable, precise external components like resistors. The negative sign is crucial; it signifies that the output signal is always a 180-degree phase-inverted version of the input.

2.2. The Non-Inverting Amplifier

The non-inverting amplifier amplifies an input signal without inverting its phase.

In this circuit, the input voltage (V_i) is applied directly to the non-inverting terminal. Thanks to the “virtual short” rule, the voltage at the inverting terminal must also be equal to V_i. The output signal remains in phase with the input.

The formula for the voltage gain is: Gain = V₀ / Vᵢ = 1 + (R_f / R₁)

Key Insight: The gain of a non-inverting amplifier is always greater than 1. The positive sign in the formula indicates that the output signal is in phase with the input signal.

2.3. The Voltage Follower

The voltage follower is a special case of the non-inverting amplifier. Its purpose is not to amplify, but to produce an output voltage that is exactly equal to the input voltage.

This is achieved by connecting the output directly back to the inverting input. The input voltage (V_i) is applied to the non-inverting input. The result is a circuit with a gain of exactly 1.

V₀ = Vᵢ

Now that we understand amplification, let’s see how op-amps can be used to perform mathematical calculations.