Op Amp Noise Gain
# Op-Amp Noise Gain: Follow the Amplifier’s Voltage Noise
Noise gain is the closed-loop gain from an op-amp’s equivalent input voltage-noise source to its output. In a simple resistive-feedback circuit it equals the noninverting gain, 1 + Rf/Rg. It can differ from the gain seen by a signal applied through an inverting input resistor.
This distinction matters for output-noise calculations and for understanding the feedback loop of a voltage-feedback op-amp.
## 1. Trace the two paths
*Conceptual voltage-feedback op-amp with resistive feedback. The noise generator represents the amplifier’s input-referred voltage noise. Supply rails, bias currents, resistor thermal noise and input current-noise sources are omitted.*
With an ideal low-impedance signal source, the inverting signal gain is Gs = −Rf/Rin. To find the voltage-noise path, set the independent signal voltage to zero, making Rin a connection to ground. The equivalent voltage-noise source then drives a noninverting configuration, so NG = 1 + Rf/Rin while loop gain is high.
For Rin = 1 kilohm and Rf = 10 kilohms, the signal gain is −10 V/V but noise gain is +11 V/V. Noise density is a magnitude quantity: the sign of signal inversion is not a negative noise contribution.
If a generator has significant series impedance, include it when finding the source-zeroed impedance to ground. A current source is set to zero by opening it, rather than shorting it. The circuit left after zeroing sources determines the feedback network.
## 2. Quantify the voltage-noise contribution
*Ideal low-frequency examples. Both bars use the same gain scale. The numbers represent only the op-amp’s voltage-noise contribution and do not claim the amplifier’s total noise or integrated RMS noise.*
For flat input voltage-noise density en and approximately constant noise gain in the passband, output density is eout = en × |NG|. A 5 nV/√Hz op-amp therefore contributes 10 nV/√Hz in a unity inverter, where NG = 2, and 55 nV/√Hz at inverting gain −10, where NG = 11.
To obtain RMS noise, integrate the actual output power density. If a flat 55 nV/√Hz contribution is filtered by a unity-gain rectangular bandwidth of 20 kHz, its RMS value is about 7.78 µV. Use equivalent noise bandwidth for a shaped response, and include the frequency dependence of both en and the closed-loop voltage-noise transfer when necessary.
## 3. Connect noise gain to feedback
Let β(f) be the fraction of output voltage fed back to the inverting input after independent signal sources are zeroed. The ideal noise-gain function is NG(f) = 1/β(f). For an impedance Zf from output to the inverting node and Zg from that node to ground, NG(f) = 1 + Zf/Zg.
With finite open-loop voltage gain A(f), the input voltage-noise transfer is A(f)/[1 + A(f)β(f)]. It approaches 1/β only where loop gain is sufficiently large. Capacitors, input capacitance, source impedance and loading can make the response strongly frequency dependent.
Loop gain is Aβ. Its magnitude and phase determine stability; a plot of noise gain against open-loop gain helps locate crossover but does not by itself prove adequate phase margin. Respect the device’s specified minimum stable gain and verify the actual feedback network. Current-feedback op-amps require their own feedback-impedance and stability treatment.
## 4. Keep the other noise paths separate
| Noise source | Typical analysis path | Common error |
|---|---|---|
| Op-amp input voltage noise | Closed-loop voltage-noise transfer | Using inverting signal gain instead |
| Inverting input current noise | Current-to-output transfer through feedback | Multiplying current density by a voltage gain |
| Input resistor thermal noise | Transfer from that resistor to output | Treating every resistor as if it sits at the noninverting input |
| Noninverting-source resistance noise | Its voltage/current paths and loading | Ignoring source impedance or double-counting it |
For a simple ideal inverting stage, an inverting-input noise current produces output voltage through Rf at low frequency. That is a transimpedance in ohms, unlike dimensionless voltage noise gain. Independent contributions add in mean-square units after each has been propagated to the output.
A useful verification records signal gain, voltage-noise gain, loop response, source impedance and integration band separately. Include resistor thermal noise, amplifier current noise and any noise correlations relevant to the device model. The complete output-noise budget comes from those source-specific paths.
## References
- [Analog Devices: Ask the Applications Engineer — Noise Gain](https://www.analog.com/en/resources/analog-dialogue/articles/op-amp-issues-noise-gain.html) — inverting signal gain versus voltage-noise gain.
- [Analog Devices: Noise Gain vs. Signal Gain](https://www.analog.com/en/resources/analog-dialogue/raqs/raq-issue-91.html) — feedback and stability implications.
- [Texas Instruments: FDA Stability and Simulating Phase Margin](https://www.ti.com/video/5501001382001) — signal gain, noise gain and loop-gain analysis.