Virtual short and virtual ground

A virtual short is the condition $v_{+}\approx v_{-}$ that an op-amp in Negative feedback forces between its two inputs even though no wire connects them. When the non-inverting input happens to be grounded, the inverting input is dragged to $0\text{V}$ as well — that special case is a virtual ground: a node sitting at ground potential without being wired to ground.

Why the inputs are forced equal

This is golden rule 2 of the Ideal op-amp model, stated as a circuit fact. The output obeys $v_O=A(v_{+}-v_{-})$ with open-loop gain $A\sim 10^5$ (ideally $\infty$). The output is physically constrained to live between the supply rails — a few volts, not infinity. Solve for the input difference:

$v_{+}-v_{-}=\frac{v_O}{A}\stackrel{A\to \infty }{\to }0$

With a $\pm 13\text{V}$ output and $A=10^5$, the actual difference is at most $13/10^5\approx 130\mu \text{V}$ — for analysis we call it zero. The negative-feedback loop is what enforces it: if $v_{-}$ drifts below $v_{+}$, the output rises, and through the feedback resistor that rise pushes $v_{-}$ back up until the two inputs match. The op-amp is a servo that drives its own input error to zero. It is “virtual” because the equality is maintained by feedback action, not by a physical connection — pull the feedback away and it vanishes instantly.

One prerequisite is built into the derivation: it assumes the output is genuinely living between the rails. If the op-amp saturates — output pinned at a supply rail — feedback can no longer move $v_O$ to correct the input error, so $v_{+}-v_{-}$ is no longer driven to zero and the virtual short fails. The virtual short therefore requires negative feedback and an unsaturated (linear-region) output; see Op-amp output saturation.

Why this is the whole game

The virtual short lets us treat the inverting node as a known voltage without drawing any current there (golden rule 1 says no current enters the inputs). That combination is extraordinarily powerful: we can write KCL at the inverting node knowing its voltage, and every input branch behaves as if its far end were terminated at a fixed potential it cannot disturb.

The virtual ground is the cleaner special case. Ground $v_{+}$, and the inverting node is held at $0\text{V}$ but draws no current. In the Inverting amplifier (op-amp) the source current $v_I/R_1$ has nowhere to go but through the feedback resistor — that single observation gives the gain. In the Summing amplifier each input branch sees the same $0\text{V}$ node, so the branches do not interact: you can superpose currents without superposing voltages, and each weight is set by its own resistor alone. The same trick derives the Difference amplifier, the Instrumentation amplifier, the Op-amp integrator and the Op-amp differentiator. Once you see the virtual short, every Chapter 21–24 derivation is a one-line KCL.