Inverting Amplifier
The ideal op-amp on its own is really just a comparator — useful for detecting which input is bigger, but useless as a controllable amplifier, since its enormous open-loop gain slams the output to one rail or the other for almost any input difference. Wrap a single resistor from output back to the inverting input, and everything changes: the op-amp becomes a precise, predictable amplifier whose gain you choose by picking two resistor values.
From open-loop comparator to closed-loop amplifier
Used on its own, an op-amp's open-loop gain is so large that almost any realistic difference between its two inputs drives the output straight to one supply rail or the other. That behaviour is exactly what makes a bare op-amp useful as a comparator, but it's no use at all if what you actually want is an amplifier that produces an output proportional to the input, scaled by some chosen, controllable factor.
The fix is negative feedback: instead of leaving the output disconnected from the inputs, a fraction of the output signal is fed back into the inverting input through a feedback resistor, $R_f$. This turns the circuit from an open-loop system, where the output depends only on the raw input difference, into a closed-loop system, where the output feeds back and constrains itself. The simplest and most widely used closed-loop configuration built this way is the inverting amplifier.
The standard inverting amplifier circuit
The inverting amplifier circuit has a specific, standard layout:
- The input signal, $V_{in}$, is applied through an input resistor $R_{in}$ to the inverting input ($V_-$) of the op-amp.
- The non-inverting input ($V_+$) is connected directly to ground (0 V).
- A feedback resistor $R_f$ connects the output back to the same inverting input node.
- The output, $V_{out}$, is taken from the op-amp's output terminal in the usual way.
Because the inverting input is where both the input signal (via $R_{in}$) and the feedback (via $R_f$) arrive, this single node is the key to analysing the whole circuit.
The virtual earth concept
Two ideal assumptions about the op-amp, combined with the presence of negative feedback, lead to a powerful shortcut for analysing this circuit and every other one in this topic:
- The open-loop gain is so large (ideally infinite) that for the output to sit at any sensible, finite voltage, the difference between the two inputs, $V_+ - V_-$, must be essentially zero.
- Since $V_+$ is grounded here, at exactly 0 V, it follows that $V_-$ must also sit at essentially 0 V.
This is despite the fact that $V_-$ is not actually wired to ground anywhere — it only behaves as if it were, because negative feedback constantly adjusts the output to keep $V_-$ pinned close to $V_+$. This point is therefore called a virtual earth: virtually, but not physically, at 0 V.
The second ideal assumption needed is infinite input resistance: no current flows into either input terminal of the op-amp. Combining this with the virtual earth gives the standard technique used to analyse every op-amp circuit in this topic — virtual-earth analysis: treat the inverting input as sitting at a known, fixed voltage (0 V here), and treat no current as flowing into the op-amp itself, so all the current arriving at that node from one resistor must leave again through another.
The minus sign is not a mathematical inconvenience — it's a physically meaningful part of the result. It means the output is an inverted version of the input: a positive-going input produces a negative-going output and vice versa, a 180° phase shift for a sinusoidal input. This is exactly why the circuit is called the inverting amplifier. The magnitude of the gain, $R_f/R_{in}$, is set purely by the ratio of the two resistors, and can be made larger or smaller than 1 by choosing $R_f$ larger or smaller than $R_{in}$.
The same relationship works equally well in reverse: given a required gain and one resistor value, the other resistor can be found by rearranging $R_f = |{\rm gain}| \times R_{in}$. This is exactly how a real circuit is designed — the desired gain is decided first, then convenient resistor values satisfying that ratio are chosen.
Virtual-earth analysis — treating the difference between the op-amp's two inputs as zero, and the current flowing into either input as zero — is not a trick specific to the inverting amplifier. It is the standard method used to analyse every op-amp circuit in this topic, including the non-inverting, summing, and difference amplifiers that follow. Whenever you meet a new op-amp circuit, start by asking what voltage the feedback forces the inverting input to sit at, and what that implies for the currents flowing into and out of that node.
Key takeaways
- Negative feedback, via a feedback resistor $R_f$ from output to inverting input, turns an op-amp from an open-loop comparator into a closed-loop, controllable amplifier.
- In the inverting amplifier, $V_{in}$ reaches the inverting input through $R_{in}$, the non-inverting input is grounded, and $R_f$ feeds the output back to the inverting input.
- Because open-loop gain is enormous and feedback keeps the output finite, $V_- \approx V_+$; with $V_+$ grounded, $V_-$ sits at a "virtual earth" of essentially 0 V without being physically connected to ground.
- Infinite input resistance means no current flows into either input terminal, so the current through $R_{in}$ must equal the current through $R_f$.
- Combining these gives $\dfrac{V_{out}}{V_{in}} = -\dfrac{R_f}{R_{in}}$, derived from current continuity at the virtual earth node.
- The negative sign represents a 180° phase inversion between input and output — the source of the circuit's name.
- Gain magnitude, resistor values, and output voltage can all be found from this formula given any two of the relevant quantities.
- Virtual-earth analysis (zero input voltage difference, zero input current) is the general technique used throughout the rest of this topic, not just for this one circuit.