The Inverting Amplifier and Virtual Ground

A quick recap, then one new wire

The last guide introduced the operational amplifier as a high-gain differential amplifier: it looks at the tiny difference between its two inputs (the + and the -) and multiplies it by an enormous open-loop gain, often 100,000 or more. On its own that is almost useless — the slightest difference slams the output against a supply rail. The trick that tames it is negative feedback: feed a slice of the output back to the inverting (-) input, like a thermostat forever nudging the room back to the set temperature.

With that feedback in place, the two golden rules of the ideal op-amp hold: (1) no current flows into either input — they are infinitely high impedance, perfect listeners that draw nothing; and (2) the output does whatever it must so that the two inputs sit at the same voltage. Rule 2 is not magic — it is just the thermostat doing its job. If V- drifts below V+, the huge gain shoves the output up; that pushes V- back up through the feedback wire until the difference is again essentially zero. Keep both rules in hand; everything below is just bookkeeping around them.

The virtual ground: a node held at zero that draws nothing

Here is the one new move. Take the non-inverting (+) input and tie it straight to ground (0 V). Now apply the golden rules. Rule 2 says the op-amp will work to make V- equal V+, and V+ is 0 V — so the minus input is dragged to 0 V too. It is not wired to ground; the op-amp is holding it there through feedback. This near-magical node is the virtual ground: it sits at 0 V like a real ground, yet because of rule 1 no current actually flows into the input pin there.

Picture the op-amp as a tireless attendant whose only job is to keep two water levels matched. You hold one level at the ground line; the attendant frantically pumps the output up or down so the other level never strays from it. From outside, that second node behaves like solid ground — anything you connect to it sees 0 V — yet it quietly accepts no current of its own. This is the virtual short between the inputs: zero volts across them, but no current through them. That contradiction (looks shorted, but no current) is exactly what makes op-amp circuits so easy to analyze.

Deriving the inverting amplifier: gain = -Rf / Rin

Now build the inverting amplifier. Tie the + input to ground. Feed the signal in through a resistor Rin to the - input. From the - input, run a feedback resistor Rf back to the output. The - input is the summing node — and we just learned it is a virtual ground, pinned at 0 V. With that one fact the analysis is two lines of arithmetic.

         Rin            Rf
  Vin o--[====]----+----[====]----o Vout
                   |
                   |  (-)  summing node = virtual ground, 0 V
                  |\
                  | >-------------o Vout
                  |/
                   |  (+)
                  GND

  Iin = Vin / Rin   flows in through Rin
  No current into (-)  ->  the SAME current flows on through Rf
  Vout = 0 - Iin x Rf = -(Vin/Rin) x Rf
  Gain = Vout / Vin = -Rf / Rin

Follow the current. The left end of Rin is at Vin and its right end is at 0 V (the virtual ground), so by Ohm's law the current through it is Iin = Vin / Rin. Rule 1 says none of that current can turn into the op-amp input — so every bit of it must carry straight on through Rf. The left end of Rf is at 0 V, so its right end (the output) must be Vout = 0 - Iin times Rf. Substitute and the input cancels: Vout = -(Vin / Rin) times Rf, giving the famous closed-loop gain of -Rf / Rin. The minus sign is the 'inverting' part: a positive input gives a negative output, flipped upside down.

Put numbers on it. Choose Rin = 10 kΩ and Rf = 100 kΩ, so the gain is -100/10 = -10. Apply Vin = +0.5 V. The input current is Iin = 0.5 V / 10 kΩ = 50 uA. That same 50 uA flows through Rf, and 50 uA times 100 kΩ = 5 V, so Vout = -5 V. A half-volt in becomes minus five volts out — exactly ten times bigger and flipped negative. Notice we never once needed the op-amp's open-loop gain in that calculation; the resistors did all the talking.

Designing one, step by step

Suppose a sensor gives a signal that swings up to 0.2 V, and a later stage wants to see up to 2 V — and you do not mind the inverted sign (or you will flip it back in a second stage). You need a gain of about -10. Here is the recipe.

1. Set the input resistor from the source's needs. The signal source has to drive Rin, so Rin IS the amplifier's input resistance. Too small and it loads the source down; too large and stray currents and noise creep in. A few kΩ to tens of kΩ is the comfortable middle — pick Rin = 10 kΩ.
2. Set the feedback resistor from the gain you want. Gain = -Rf/Rin, and you want magnitude 10, so Rf = 10 times Rin = 100 kΩ. Want gain -4.7 instead? Rf = 4.7 times 10 kΩ = 47 kΩ. The whole gain lives in this one ratio.
3. Sanity-check the output swing. The peak input is 0.2 V, so the peak output is 0.2 times 10 = 2 V — well inside a typical supply, good. If gain times peak input had exceeded what the op-amp can swing to, the output would clip flat at the rail; then you would lower the gain or raise the supply.
4. Add a small bias-balancing resistor at the + input (optional but tidy). Instead of tying + straight to ground, tie it through a resistor equal to Rin parallel Rf (here about 9.1 kΩ). On a real op-amp both inputs draw a tiny bias current; matching the resistance each input sees cancels most of the offset that would otherwise create. Ideal theory ignores it; good practice does not.

The revolution: gain from a ratio, and a glimpse of the family

Step back and see what just happened. A real op-amp's open-loop gain is huge but sloppy — it varies enormously from part to part, drifts with temperature, and sags at high frequency. Yet our amplifier's gain came out as exactly -Rf/Rin, depending on nothing but two resistors. Negative feedback trades away the wild, untrustworthy open-loop gain in exchange for a precise, stable, predictable gain set by components you can buy to 1% or better. That trade — sacrifice raw gain you have in excess to buy precision you need — is the central idea of all of analog electronics.

The same virtual-ground bookkeeping unlocks a whole family, which the next guides explore. Move the input to the + pin instead and you get the non-inverting amplifier, gain 1 + Rf/Rin (so Rf = 90 kΩ with Rin = 10 kΩ gives 1 + 90/10 = 10, positive this time) — guide 3, along with its extreme case the unity-gain buffer. Tie several input resistors to the one virtual ground and their currents add, giving the summing amplifier (a mixer); a matched pair of dividers gives the difference amplifier that subtracts — guide 4. Swap Rf for a capacitor and the same node math yields the integrator and its mirror the differentiator — guide 5.

Honest limits, and the op-amp with no feedback

The virtual ground is only as good as the loop. The op-amp can hold V- at 0 V only while it has gain to spare and time to react. Demand too much speed and you hit slew-rate and bandwidth limits (a later rung's topic); demand an output beyond the supply rails and it simply clips. And the virtual ground is 'virtual', not perfect: it actually sits a hair off zero by roughly Vout divided by the open-loop gain — microvolts when the loop gain is large, but not exactly zero. Treat -Rf/Rin as an excellent approximation, true to the precision your resistors and your op-amp allow, not as an unbreakable law.

There is also an honest downside specific to this topology: the inverting amp's input resistance is just Rin, because the source stares straight into a resistor whose far end is a virtual ground. A 10 kΩ input may load a delicate sensor too heavily. That is precisely why the non-inverting amp of the next guide exists — its input goes to the + pin, which draws essentially nothing, giving an enormous input resistance. Each topology is a deliberate trade; knowing the trade is the design.