Formula & Calculator
Inverting Op-Amp Gain
The closed-loop voltage gain of an inverting operational amplifier configuration.
Interpretation
Inverting op‑amp gain: V_out = −(R_f/R_in)·V_in inverts the input signal and scales it by the feedback‑to‑input resistance ratio.
The input impedance is approximately R_in.
Example: R_f=10kΩ, R_in=1kΩ, V_in=0.5V → V_out = −10 × 0.5 = −5V.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| V_out | Output voltage | V |
| R_f | Feedback resistor | Ω |
| R_in | Input resistor | Ω |
| V_in | Input voltage | V |
What it means
The inverting operational amplifier configuration has the output voltage V_out = −(R_f / R_in) · V_in. The negative sign indicates phase inversion. The gain is set by the ratio of the feedback resistor R_f to the input resistor R_in. The input impedance is approximately R_in. This configuration is widely used for summing amplifiers, integrators, and active filters. Because the non‑inverting input is grounded, the inverting input is a virtual ground (V− ≈ 0). The gain can be made less than, equal to, or greater than one (in magnitude). Example: With R_f=10kΩ and R_in=1kΩ, the gain is −10. A 0.5V input gives V_out = −10 * 0.5 = −5V. This is used for signal inversion and amplification.
Worked example
Inverting Op‑Amp – Practical Example
Real‑World| Parameter | Value |
|---|---|
| Rf | 100 kΩ |
| Rin | 10 kΩ |
| Vin | 1 V |
| Formula | Vout = −(Rf/Rin)·Vin |
Common mistakes
Watch unit consistency and the assumptions behind the formula; misapplying it outside its valid conditions is the most frequent error.Applications
Inverting op‑amp gain V_out = −(R_f/R_in)·V_in inverts and scales the input signal. This is a standard building block for analog signal processing. Engineers use it for signal inversion, scaling, and summing. By selecting resistor values, they set the desired gain. This circuit is fundamental in analog electronics.
- Signal inversion and phase shifting
- Precision scaling and attenuator circuits
- Summing amplifiers and analog computation
- Active filter implementation
- Educational understanding of op‑amp applications
Frequently Asked Questions
The voltage gain of an inverting op‑amp is V_out = −(R_f / R_in)·V_in. The negative sign indicates a 180° phase inversion.
The input impedance is approximately R_in (since the inverting input is at virtual ground). This is lower than the non‑inverting configuration.
Choose R_in based on the desired input impedance, then set R_f = −Gain × R_in. For example, for a gain of −10 with R_in = 1 kΩ, R_f = 10 kΩ.
Common errors: 1) forgetting the negative sign, 2) using the wrong feedback resistor, 3) not considering the input bias current, 4) applying the formula to a non‑inverting circuit.
The closed‑loop gain is approximately −R_f/R_in as long as the open‑loop gain is much larger than the closed‑loop gain. At high frequencies, the gain rolls off.
Used in summing amplifiers, active filters, and signal inversion.
The inverting input is at virtual ground because the op‑amp forces the differential voltage to zero (assuming feedback). This simplifies analysis.
The bandwidth is determined by the gain‑bandwidth product (GBW) of the op‑amp: BW = GBW / |Gain|. Higher gain reduces bandwidth.