CR · Operational amplifier

The Inverting Amplifier and the Virtual Ground

The single choice of tying the non-inverting input to ground turns the inverting input into a fixed 0V reference point. This unit shows how to use that virtual ground to set the gain neatly by a ratio of resistors.

A virtual ground turns gain into a resistor ratio

In the figure the + input is tied to ground, and the input Vin enters the − input through Rin. Change the feedback ratio Rf/Rin and the output becomes −(Rf/Rin) times the input, with the length of the output arrow showing its size. Set the ratio so that the output is flipped in sign and exactly 4 times as large, that is, the gain is −4 and the arrow reaches the target mark.

Feedback ratio Rf/RinRf/Rin = 1.0

Voltage gain G

G = −Rf/Rin = -1.0

Far from −4

Tie the non-inverting input to ground

The non-inverting input V+ is connected straight to 0V ground. An ideal op-amp with negative feedback makes its two inputs equal (virtual short), so the inverting input V− also sits at 0V. But V− is not wired to a real ground; the op-amp merely holds it at 0V by adjusting its output, which is why it is called a virtual ground. Its potential is 0, but unlike a real ground it cannot swallow current freely.

The current has only the feedback to go to

The input voltage Vin drives current through the input resistor Rin into the virtual ground. Since V− is at 0V, the whole of Vin appears across Rin, and the input current is Vin/Rin. That current cannot enter the op-amp's input (its input resistance is infinite), so it can only flow through the feedback resistor Rf. The same current flows through Rf, and since one end of Rf is at 0V, the output is 0 − (Vin/Rin)·Rf = −(Rf/Rin)·Vin.

The sign flip and summing come for free

The negative sign in the gain −Rf/Rin means the output swings opposite to the input: when the input goes up, the output goes down. The real power of the virtual ground shows up with several inputs. Bring each input through its own resistor to the same virtual-ground node, and the currents add at that node, making the output the weighted sum −(R_f/R_1·V_1 + R_f/R_2·V_2 + …). This is the analog summing amplifier.

ObserveV+ = 0 → V− = 0

V+ is grounded, so V− is 0 V too.

Choosei = Vin / ?

Thanks to the virtual ground, the whole input voltage falls across Rin.

Fill inG = ?

The gain is the negative of the resistor ratio.

On your ownRf = Rin → G = ?

With equal resistors, only the sign flips.

Back to the first screen

As you changed the feedback ratio, the output grew to −(Rf/Rin) times the input and reached exactly −4 at Rf/Rin = 4. The single choice of tying the non-inverting input to ground held the inverting input at a 0V virtual ground, and the input current, with nowhere else to go, flowed through the feedback path and set the gain cleanly by the resistor ratio. The flipped sign was no penalty; it was the seed from which the same structure grows into a summing amplifier.

The inverting amplifier ties the non-inverting input to ground so that the virtual short holds the inverting input at 0V. That node is called the virtual ground. The input current Vin/Rin flows entirely through the feedback resistor Rf, so the output is Vout = −(Rf/Rin)·Vin and the gain is −Rf/Rin. The sign is inverted, and bringing several inputs to that one node makes a summing amplifier that outputs a weighted sum.