How Negative Feedback Sets Inverting Amplifier Voltage Gain

Negative feedback is a fundamental concept in electronics that stabilizes operational amplifiers (op-amps) and sets their precise closed-loop voltage gain. By feeding a portion of the output signal back to the inverting input, the circuit counteracts the op-amp's inherently massive and unpredictable open-loop gain. This article explains how negative feedback creates a virtual ground, governs current flow through external resistors, and ultimately defines the exact amplification of an inverting amplifier using a simple ratio of two resistance values.

The Problem with Open-Loop Gain

An operational amplifier on its own has an extremely high open-loop voltage gain (\(A_{OL}\)), often exceeding 100,000. While high gain sounds beneficial, it is highly unstable. It fluctuates significantly with changes in temperature, power supply variations, and manufacturing tolerances. Without control, even a tiny input voltage will cause the op-amp output to immediately saturate, clipping the signal at the power supply limits. To make an op-amp useful for precise linear amplification, this raw gain must be tamed.

How Negative Feedback Works

In an inverting amplifier configuration, the non-inverting input (+) is connected directly to ground (0V). The input signal (\(V_{in}\)) is applied to the inverting input (-) through an input resistor (\(R_{in}\)). A feedback resistor (\(R_f\)) is connected between the output (\(V_{out}\)) and the inverting input.

This connection via \(R_f\) creates a negative feedback loop. When the input voltage goes positive, the inverting nature of the op-amp causes the output voltage to go negative. Because the output is tied back to the inverting input through \(R_f\), this negative output voltage pulls down on the positive input voltage, opposing the original change.

The Virtual Ground Concept

To understand how this opposition sets the precise gain, you must understand the concept of a "virtual ground." Because the op-amp's open-loop gain is nearly infinite, the voltage difference between its two input terminals must be driven to nearly zero for the output to remain within a normal, non-saturated range.

Since the non-inverting input (+) is physically connected to ground (0V), the negative feedback loop forces the op-amp to adjust its output voltage dynamically so that the inverting input (-) is also maintained at 0V. It is not physically connected to ground, so no current flows into the ground from this node, which is why it is called a virtual ground.

Current Flow and the Resistor Ratio

The ideal op-amp also has infinite input impedance, meaning zero electrical current flows directly into the inverting or non-inverting input pins. Knowing that the inverting input is at 0V (virtual ground) and draws no current, the behavior of the circuit becomes entirely dependent on Ohm's Law and Kirchhoff's Current Law.

  1. Input Current (\(I_{in}\)): The voltage across the input resistor (\(R_{in}\)) is the difference between the input voltage (\(V_{in}\)) and the virtual ground (0V). Therefore, the current flowing through \(R_{in}\) is: \(I_{in} = \frac{V_{in} - 0}{R_{in}} = \frac{V_{in}}{R_{in}}\)

  2. Feedback Current (\(I_f\)): Because this input current cannot enter the op-amp's input pin, 100% of it must be diverted through the feedback resistor (\(R_f\)) toward the output. The voltage across \(R_f\) is the difference between the virtual ground (0V) and the output voltage (\(V_{out}\)). Therefore, the feedback current is: \(I_f = \frac{0 - V_{out}}{R_f} = \frac{-V_{out}}{R_f}\)

Deriving the Closed-Loop Gain Formula

Since the input current must equal the feedback current (\(I_{in} = I_f\)), we can set the two equations equal to each other:

\(\frac{V_{in}}{R_{in}} = \frac{-V_{out}}{R_f}\)

Closed-loop voltage gain (\(A_{CL}\)) is defined as the ratio of the output voltage to the input voltage (\(\frac{V_{out}}{V_{in}}\)). By rearranging the equation above, we get the standard formula for the gain of an inverting amplifier:

\(A_{CL} = \frac{V_{out}}{V_{in}} = -\frac{R_f}{R_{in}}\)

Why This Makes the Gain Precise

The formula \(-\frac{R_f}{R_{in}}\) reveals why negative feedback is so powerful in electronics. Notice that the op-amp's erratic open-loop gain (\(A_{OL}\)) does not appear in the final equation at all. As long as the open-loop gain remains exceptionally high, the negative feedback loop completely masks the internal variations of the silicon chip.

Instead of relying on the unstable internal properties of the transistor network inside the op-amp, the closed-loop voltage gain is dictated entirely by the ratio of two passive external components: \(R_f\) and \(R_{in}\). Because engineers can easily purchase resistors with tight tolerances (such as 0.1% or 0.01%) and high thermal stability, they can use negative feedback to set the inverting amplifier's voltage gain to an exact, unchanging value. The negative sign in the equation simply indicates that the output signal is 180 degrees out of phase with the input signal.