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Inverting Operational Amplifier: Gain Formula, Virtual Ground
An inverting operational amplifier is an op amp circuit that produces an output with opposite polarity from the input. In the ideal voltage-amplifier form, the gain is set by two external resistors:
Vout = -Vin x Rf / Rin
Av = -Rf / Rin
The negative sign is the first thing to notice. It means a positive input produces a negative output, and a negative input produces a positive output. The circuit is called an inverting amplifier because the output waveform is 180 degrees out of phase with the input waveform.
The second important idea is virtual ground. In a standard inverting op amp circuit, the non-inverting input is tied to a reference voltage, often ground. Negative feedback then drives the output until the inverting input sits very close to that same reference voltage. The node behaves like ground for analysis, but it is not a physical ground connection.
Inverting Operational Amplifier Circuit
A basic inverting amplifier uses these parts and nodes:
| Circuit Item | Role |
|---|---|
Vin | Input signal applied through the input resistor. |
Rin | Input resistor that converts the input voltage into current. |
Rf | Feedback resistor from output to the inverting input. |
Non-inverting input (+) | Connected to the reference voltage, commonly ground. |
Inverting input (-) | Summing node where input current and feedback current meet. |
Vout | Output voltage created by the op amp and feedback network. |
In the textbook version, the non-inverting input is connected to ground. Because the op amp output is fed back through Rf, the circuit uses negative feedback. The output moves until the inverting input is held very close to the non-inverting input voltage.
That is the basis of the virtual-ground model.
How the Inverting Op Amp Gain Formula Works
The easiest way to derive the gain formula is to follow the current.
- The non-inverting input is connected to ground or another fixed reference.
- Negative feedback makes the inverting input sit near that same reference.
- The ideal op amp input draws almost no current.
- Therefore, the current through
Rinmust continue throughRf. - The output voltage moves in the opposite direction to create that feedback current.
For a grounded reference node:
Input current:
Iin = Vin / Rin
Feedback current:
If = -Vout / Rf
Ideal op amp input current:
Iin = If
Therefore:
Vin / Rin = -Vout / Rf
Vout = -Vin x Rf / Rin
Av = Vout / Vin = -Rf / Rin
This is the standard inverting op amp gain formula:
Av = -Rf / Rin
Only the resistor ratio sets the ideal voltage gain. If Rf is larger than Rin, the circuit amplifies. If Rf equals Rin, the circuit inverts the signal without changing its amplitude. If Rf is smaller than Rin, the circuit attenuates while still inverting polarity.
Example: Gain of -10
If the resistor values are:
Rin = 10 kOhm
Rf = 100 kOhm
Then:
Av = -Rf / Rin
Av = -100 kOhm / 10 kOhm
Av = -10
So a +0.2 V input ideally produces:
Vout = -0.2 V x 10
Vout = -2.0 V
A -0.2 V input ideally produces +2.0 V. The amplitude changes by the resistor ratio, and the polarity is reversed.
Unity-Gain Inverter
When Rf equals Rin, the gain becomes -1.
Av = -Rf / Rin = -1
Vout = -Vin
This version is called a unity-gain inverter. It is useful when the signal amplitude should stay the same but the polarity must be reversed.
Do not confuse it with a voltage follower. A voltage follower is non-inverting. A unity-gain inverter has the same output amplitude as the input, but the waveform is flipped around the reference voltage.
Virtual Ground Is Not a Physical Ground
Virtual ground means the inverting input is held near the same voltage as the non-inverting input by negative feedback. In a dual-supply circuit where the non-inverting input is tied to ground, the inverting node is very close to 0 V.
But the node is not actually connected to ground. It is not a power return path, and it should not be used as a normal ground point for other parts of the circuit.
The virtual-ground idea depends on three conditions:
| Condition | Meaning |
|---|---|
| Negative feedback is present | The output must feed back to the inverting input through Rf. |
| The op amp is operating linearly | The output must not be stuck at a supply rail. |
| The reference voltage is valid | The non-inverting input sets the voltage that the inverting input follows. |
If those conditions are not true, the virtual-ground assumption breaks down.
Single-Supply Virtual Ground
In a single-supply circuit, ground is not always the best reference. For example, a 3.3 V or 5 V system may use a mid-supply reference so the signal can swing above and below that reference while staying inside the supply range.
In that case, the non-inverting input may be connected to Vref instead of 0 V. The inverting input then becomes a virtual copy of Vref, not a virtual copy of ground.
The level-shifted form is:
Vout = Vref - (Vin - Vref) x Rf / Rin
This is the same inverting idea, but centered around the reference voltage.
Why the Output Is Negative
When Vin is positive, current flows through Rin toward the virtual-ground node. Since the ideal op amp input draws almost no current, that current must flow through Rf. The op amp output moves negative to pull the same current through the feedback resistor.
When Vin is negative, the current direction reverses. The op amp output moves positive to keep the inverting input near the reference voltage.
Common Misunderstandings
Virtual Ground Is Not a Real Ground
The inverting node is only held near the reference voltage by feedback. It should not be treated as a physical ground connection.
The Negative Sign Means Phase Inversion
The minus sign in Av = -Rf/Rin does not mean the circuit only produces negative voltages. It means the output polarity is opposite to the input polarity relative to the reference.
The Formula Assumes an Ideal Op Amp
The formula is the ideal first-order result. Real op amps have limits, but the basic derivation still starts with the same current-flow model: input current through Rin, feedback current through Rf, and very small current into the op amp input.
FAQ
What is an inverting operational amplifier?
An inverting operational amplifier is an op amp circuit that amplifies an input signal while reversing its polarity. In the ideal resistor-feedback form, its gain is Av = -Rf/Rin.
Why is the inverting op amp output negative?
The output is negative for a positive input because negative feedback drives the inverting input near the reference node. To balance the input current through Rin, the output must move in the opposite direction through Rf.
What is virtual ground in an inverting amplifier?
Virtual ground means the inverting input sits near the same voltage as the grounded or referenced non-inverting input due to negative feedback. It is not a physical ground connection.
What happens if Rf equals Rin?
The gain becomes -1, so the circuit acts as a unity-gain inverter. The output has the same amplitude as the input but opposite polarity.
Can an inverting op amp use a reference voltage instead of ground?
Yes. In single-supply circuits, the non-inverting input is often connected to a reference voltage such as mid-supply. The inverting input then sits near that reference voltage, and the output swings around it.
Conclusion
An inverting operational amplifier is built around one simple feedback idea: the op amp output moves until the inverting input sits near the reference voltage. Because the ideal op amp input draws almost no current, the current through Rin flows through Rf, giving the gain formula:
Av = -Rf / Rin
The resistor ratio sets the gain magnitude. The negative sign shows polarity inversion. Virtual ground explains why the current-flow derivation works, but it should always be understood as a feedback condition rather than a real ground connection.
Sources Used
- Electronics Tutorials: Inverting Operational Amplifier
- Analog Devices: Inverting Op Amp
- Analog Devices: So, What Exactly Is a Virtual Ground?
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