Inverting amplifier (op-amp)

One of the two basic op-amp Negative feedback topologies. The input signal $v_I$ drives the inverting input through a resistor $R_1$, a feedback resistor $R_2$ runs from the output back to that same inverting node, and the non-inverting input is grounded. Closed-loop gain is the negative resistor ratio $-R_2/R_1$, set entirely by external components.

The two basic feedback configurations: inverting (signal into $R_1$ at $v_{-}$) and non-inverting.

Deriving the gain

Three steps, using the two golden rules of the Ideal op-amp model plus KCL.

Step 1 — golden rule 2 (virtual ground). Negative feedback forces $v_{+}=v_{-}$. The non-inverting input is wired to ground, so $v_{+}=0$, and therefore $v_{-}=0\text{V}$. The inverting node sits at ground potential even though it is not connected to ground, a virtual ground.

Step 2 — golden rule 1 + KCL. No current flows into the inverting input ($i_{-}=0$). So at the inverting node, all the current arriving from $v_I$ through $R_1$ must continue on through $R_2$ toward the output, it has nowhere else to go. The current into the node from the source is $(v_I-v_{-})/R_1$; the current leaving toward the output through $R_2$ is $(v_{-}-v_O)/R_2$. Setting them equal with $v_{-}=0$:

$\frac{v_I-0}{R_1}=\frac{0-v_O}{R_2}$

Step 3 — solve for the gain. Rearranging,

$\boxed{\frac{v_O}{v_I}=-\frac{R_2}{R_1}}$

The gain depends only on the resistor ratio $R_2/R_1$: not on the op-amp’s Open-loop gain $A$, not on temperature, not on the device at all (to the accuracy discussed in Closed-loop gain). That insensitivity is the whole point of feedback. The minus sign says the output is $180^{\circ }$ out of phase with the input: a positive input swing produces a negative output swing. With $R_2=100\text{k}\Omega$ and $R_1=1\text{k}\Omega$ the gain is $-100$.

Inverting gain $-R_2/R_1$; non-inverting $1+R_2/R_1$, both depend only on resistor ratios.

Input resistance — the drawback

What resistance does the source see? Looking in from $v_I$, the current is $(v_I-v_{-})/R_1$ and $v_{-}=0$ (virtual ground), so the source sees exactly $R_1$ to ground:

$R_{in}=R_1$

This is finite, and it’s the inverting configuration’s main weakness. A high-impedance source (a sensor, a piezo element) connected through $R_1$ forms a Voltage divider with the source’s own output resistance and loses signal. You also can’t make $R_{in}$ large without making $R_1$ large, which forces $R_2$ huge to keep the gain. When a source must not be loaded, use the Non-inverting amplifier (op-amp) (infinite input resistance) or buffer it with a Voltage follower (op-amp). The same virtual ground that costs you here is what makes the Summing amplifier work.