Virtual earth in an operational-amplifier circuit

An operational amplifier produces an output that depends on the potential difference between its two inputs. In a circuit with negative feedback, the output acts through the feedback connection to keep that difference very small during normal linear operation.

Because the amplifier’s open-loop gain is very large, only a very small input potential difference is needed to produce an output within its normal range. If the non-inverting input is connected to earth, its potential is zero, so the inverting input is also at approximately zero potential. The inverting input is then a virtual earth.

Currents at a virtual earth

A virtual earth can be treated as being at zero potential when finding the voltage across each component connected to it. Current can still flow through those components: zero potential at the node does not mean zero current there.

For an ideal operational amplifier, a separate assumption is that no current enters either input. Currents at the virtual-earth node must therefore balance through the external circuit. Zero input current follows from the ideal-amplifier assumption, not from the node being at zero potential.

For example, suppose a

1,\text{k}\Omega

resistor connects a point at

+2,\text{V}

to the virtual-earth node, and a

2,\text{k}\Omega

feedback resistor connects the node to an output at

-4,\text{V}

. Taking the node potential as zero, the first resistor carries

2,\text{mA}

into the node. The feedback resistor carries

2,\text{mA}

out of the node towards the output. The currents balance without any current entering the ideal amplifier input.

Inverting operational amplifier

An inverting amplifier produces an output voltage with the opposite polarity to its input. Its voltage gain is determined by two resistors: an input resistor and a feedback resistor.

The input voltage $V_{\mathrm{in}}$ is connected through the input resistor $R_{\mathrm{in}}$ to the operational amplifier’s inverting input. The feedback resistor $R_{\mathrm{f}}$ connects the output back to that same input. The non-inverting input is connected to earth, so it is at zero potential.

The virtual earth

The connection from the output through the feedback resistor provides negative feedback. While the amplifier operates linearly, this feedback keeps the inverting input at approximately the same potential as the non-inverting input: approximately zero volts.

The inverting input is therefore called a virtual earth. It is approximately at earth potential, but it is not directly connected to earth.

Inverting amplifier gain

For an ideal operational amplifier, no current enters the inverting input. The current through $R_{\mathrm{in}}$ must therefore also pass through $R_{\mathrm{f}}$ .

Taking current towards the virtual-earth node through the input resistor as positive, the potential difference across that resistor is $V_{\mathrm{in}}-0$ . Across the feedback resistor, in the same current direction, it is $0-V_{\mathrm{out}}$ . Thus:

I=\frac{V_{\mathrm{in}}}{R_{\mathrm{in}}}=\frac{0-V_{\mathrm{out}}}{R_{\mathrm{f}}}

Equating the two current expressions and rearranging gives the voltage gain:

\frac{V_{\mathrm{out}}}{V_{\mathrm{in}}}=-\frac{R_{\mathrm{f}}}{R_{\mathrm{in}}}

The minus sign shows that the output has the opposite polarity to the input. The magnitude of the gain is the resistance ratio $R_{\mathrm{f}}/R_{\mathrm{in}}$ . For example, if the feedback resistance is three times the input resistance, the gain is $$$-3$$$ .

The relationship can be rearranged to find the output voltage:

V_{\mathrm{out}}=-\frac{R_{\mathrm{f}}}{R_{\mathrm{in}}}V_{\mathrm{in}}

It can also be rearranged to find the input voltage or either resistance when the other relevant quantities are known. These relationships apply while negative feedback maintains linear operation; they do not apply when the output is saturated.