Apply the gain equations for the inverting and non-inverting op-amp amplifier and explain the role of negative feedback

What this dot point is asking

SEAB wants you to apply the gain equations for the inverting and non-inverting op-amp amplifier and to explain the role of negative feedback. The central insight is that connecting a feedback resistor from the output back to the inverting input tames the op-amp's enormous open-loop gain into a controlled, predictable gain set entirely by the resistor values.

The answer

Negative feedback

On its own, an op-amp has a huge but imprecise open-loop gain. To make a useful amplifier, a fraction of the output is fed back to the inverting ($-$) input through a feedback resistor. Because it returns to the inverting input, this is negative feedback: it opposes change. Negative feedback trades away the unwanted excess gain in exchange for:

• A stable, predictable gain set by the resistor values, not by the op-amp's variable open-loop gain.
• Less distortion, so the output is a more faithful, larger copy of the input.

This is why nearly every linear amplifier uses negative feedback.

The inverting amplifier

In the inverting amplifier, the input signal is applied through an input resistor $R_{in}$ to the inverting input, and the non-inverting input is held at $0\ \text{V}$. The gain is:

The minus sign means the output is inverted: a positive input gives a negative output, and the waveform is turned upside down. The size of the gain is the ratio of the feedback resistor to the input resistor.

The non-inverting amplifier

In the non-inverting amplifier, the input signal is applied to the non-inverting ($+$) input, and the feedback network ($R_f$ and a resistor $R_{in}$ to $0\ \text{V}$) sets the gain. The gain is:

There is no minus sign, so the output is in phase with the input (not inverted), and the gain is always at least one.

Choosing resistors for a wanted gain

To design for a particular gain, pick the ratio of the resistors. For an inverting amplifier of gain $-10$, make $R_f$ ten times $R_{in}$ (for example $100\ \text{k}\Omega$ and $10\ \text{k}\Omega$). For a non-inverting amplifier of gain $6$, make $R_f$ five times $R_{in}$, since the formula adds one. The actual values are usually in the kilohm to megohm range.

Examples in context

Example 1. A sensor pre-amplifier. A thermocouple produces only a few millivolts. A non-inverting amplifier with a gain of, say, $100$ boosts this to a useful level for a meter or analogue-to-digital converter, while negative feedback keeps the gain accurate and steady over temperature. The resistor ratio, not the op-amp, fixes the gain a designer can rely on.

Example 2. An audio tone stage. An inverting amplifier sets the volume of a stage by the ratio of two resistors, and the inversion is harmless for audio because the ear cannot hear the phase flip. Swapping the input resistor for a potentiometer turns the fixed gain into an adjustable volume control, a common building block in audio equipment.

Try this

• Cue. An inverting amplifier has $R_f = 47\ \text{k}\Omega$ and $R_{in} = 4.7\ \text{k}\Omega$. Find the gain. $A_v = -R_f/R_{in} = -47/4.7 = -10$.

• Cue. A non-inverting amplifier has $R_f = 90\ \text{k}\Omega$ and $R_{in} = 10\ \text{k}\Omega$. Find the gain. $A_v = 1 + R_f/R_{in} = 1 + 9 = 10$.

• Cue. State one advantage of using negative feedback in an amplifier. The gain becomes stable and predictable, set by the resistor values rather than the op-amp's variable open-loop gain, with less distortion.