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The inverting amplifier connects the input signal through a resistor R_in to the op-amp's inverting terminal, with a feedback resistor R_f from output to inverting input, while the non-inverting terminal is grounded. Negative feedback forces the inverting input to virtual ground (0 V), so the input current I_in = V_in / R_in flows entirely through R_f (since no current enters the op-amp input), producing V_out = -I_in * R_f = -(R_f / R_in) * V_in. The closed-loop gain is A_v = -R_f / R_in, with the negative sign indicating phase inversion. Input impedance equals R_in (not infinite, because the inverting terminal is at virtual ground, not floating), which is a key practical consideration — the source must drive current into R_in. This configuration is the basis for the summing amplifier (multiple input resistors to the inverting node) and the transimpedance amplifier (current input, voltage output). Practical limitations include finite open-loop gain (causing gain error), finite bandwidth (gain-bandwidth product limits usable frequency range), and output voltage swing limited by supply rails.
Derive the gain formula from first principles using the virtual ground and virtual open rules rather than memorizing it. Then re-derive it including finite open-loop gain A_OL to see how the ideal formula emerges as A_OL approaches infinity and to quantify the gain error for realistic op-amps. Compare input impedance to the non-inverting configuration to understand why the choice between topologies matters for high-impedance sources.
From your work on op-amp fundamentals, you know the two golden rules that make ideal op-amp analysis tractable: no current enters the input terminals (virtual open), and negative feedback drives the differential input voltage to zero (virtual short, or in this circuit, virtual ground). The inverting amplifier is the circuit where these rules produce a result that initially seems paradoxical: the inverting input is held at 0 V even though it is not physically connected to ground.
Here is how the feedback loop creates virtual ground. Suppose the input V_in is positive. Current flows through R_in toward the inverting terminal. If that terminal were truly floating, voltage would build up there and the output would swing negative. But the op-amp's output swings negative in response — and the feedback resistor R_f connects that negative output back to the inverting terminal, pulling its voltage back toward zero. The feedback loop continuously corrects any departure from 0 V at the inverting node. In the ideal case (infinite open-loop gain), the correction is perfect: the inverting input is held exactly at 0 V. This is virtual ground — not a physical connection, but an enforced potential.
Once you accept virtual ground, the gain derivation follows directly from Ohm's Law. The current through R_in is I = V_in / R_in (since the inverting node is at 0 V). Because no current enters the op-amp input terminal itself (virtual open), this entire current must flow through R_f. The voltage at the output is V_out = 0 − I × R_f = −(R_f / R_in) × V_in. The gain magnitude is R_f / R_in, and the negative sign reflects the phase inversion. Notice what sets the gain: not the op-amp itself, but the external resistor ratio. The op-amp's job is to provide enough open-loop gain that the feedback loop enforces virtual ground; the precision of the closed-loop gain depends on the resistors, not on the exact value of the op-amp's open-loop gain (as long as it is large).
The practical tradeoff to internalize is the input impedance penalty. In a non-inverting configuration, the signal feeds directly into the op-amp's input (near-infinite impedance). In the inverting configuration, the source sees R_in as its load, because the inverting terminal is held at virtual ground — any current into the node flows through R_in to the virtual ground, so the source must supply it. High-gain inverting designs require small R_in (to keep R_f manageable), which loads down high-impedance sources. This is why you'll encounter the non-inverting configuration for sensors and low-impedance-sensitive applications, while the inverting configuration is preferred when gain accuracy, signal summation (summing amplifier), or current-to-voltage conversion (transimpedance amplifier) are the priority.