13.4.1 Inverting amplifier configuration

Inverting Amplifier Configuration:

In an inverting amplifier configuration, the output voltage from the operational amplifier (op-amp) is fed back into the inverting input. This forms a closed-loop circuit with negative feedback, allowing for control over the amplifier's gain.

Since negative feedback stabilises the output, this configuration makes it possible to set the op-amp's gain to much lower values, which is useful for controlling the amplification of signals.

Circuit Diagram Explanation:

• The inverting amplifier circuit contains a resistor $R_{\text{in}}$ at the input and a feedback resistor $R_{\text{f}}$ connected between the output and the inverting input.
• Power supply connections (not shown in the diagram) are assumed to be present in most amplifier circuits.

Virtual Earth Concept and Derivation:

1. Virtual Earth Analysis:

• In this configuration, the non-inverting input is grounded ($0$ V). Due to the properties of an ideal op-amp, the open-loop gain (denoted as $A_{\text{OL}}$) is assumed to be infinite.
• This leads to the condition where the voltage at the inverting input is effectively $0$ V, called a virtual earth (or virtual ground). Although not physically connected to earth, this point is close to $0$ V due to the infinite open-loop gain.

2. Transfer Function Derivation:

• Using Kirchhoff's Current Law (KCL), which states that the total current entering a junction is equal to the current leaving the junction, we can determine the relationship between input and output voltages.
• Because of the virtual earth, the input current through $R_{\text{in}}$ (from $V_{\text{in}}$) to the inverting input) is equal to the feedback current through $R_{\text{f}}$ (from the inverting input to $V_{\text{out}}$). Using Ohm's law, we have:

$$I_{R_{\text{in}}} = \frac{V_{\text{in}}}{R_{\text{in}}}$$ $$I_{R_{\text{f}}} = \frac{V_{\text{out}}}{R_{\text{f}}}$$

Since $I_{R_{\text{in}}} = I_{R_{\text{f}}}$, we can write:

$$\frac{V_{\text{in}}}{R_{\text{in}}} = \frac{-V_{\text{out}}}{R_{\text{f}}}$$

Rearranging, the gain (transfer function) of the inverting amplifier is given by:

$$\frac{V_{\text{out}}}{V_{\text{in}}} = -\frac{R_{\text{f}}}{R_{\text{in}}}$$

This negative sign indicates inversion of the input signal's polarity.

Graph of Input and Output Voltages:

The graph of the output voltage $( V_{\text{out}} )$ versus time shows that the output is an inverted (mirrored along the x-axis) and amplified version of the input voltage $( V_{\text{in}} )$.

• Inverted Polarity: The negative sign in the gain equation reflects that the output waveform is 180° out of phase with the input.
• Reduced Distortion: The inverting configuration provides lower distortion compared to non-inverting configurations, as it benefits from stabilisation due to the virtual earth.