Part 3 showed how complicated a single transistor’s noise becomes. An op-amp contains many such devices, so modelling every one is hopeless. This lecture introduces the standard op-amp noise model — the three equivalent input-referred sources that manufacturers specify on datasheets — and uses it to compute the output noise of a real circuit and to explain how to select an amplifier for a given source impedance.
An op-amp is generally a 5-terminal device (V+, V−, VSS, VDD, VOUT). Each terminal forms a port with respect to ground, so in principle noise can be fed in through all of them, including the power rails. For this analysis the PSRR is assumed infinite, so noise leaking through the supplies is ignored — bringing the port count down to three.
An inverting amplifier can be treated as a 2-port (input referenced to ground, and output), but other applications — such as a difference amplifier — need independent inputs at both terminals. In the general case the op-amp must be analysed as a 3-port device, so we can no longer represent all its noise with just a single input voltage source and a parallel current source.
The standard op-amp noise model therefore uses three noise sources around an otherwise noiseless op-amp:
Why two current sources but only one voltage source? Each input can see a different source impedance, so each needs its own current noise source to create its own voltage. The op-amp, however, essentially works in differential mode — it amplifies (V+ − V−) — so even separate voltage noise sources at the two terminals would act, once transferred to the output, as a single one.
Datasheets typically list a single input current noise value in even though the model has two sources. By symmetry of the underlying differential-pair input stage, īn+ and īn− have equal magnitudes (unless the datasheet gives separate values).
An op-amp is a sophisticated device with many internal resistors, capacitors and BJTs/MOSFETs, each of which can generate thermal, shot and flicker noise. For example, a BJT-based op-amp has two independent bias currents flowing into the two input BJTs, and that alone generates shot noise at the input. Because modelling all of this is impractical, we use the input-referred voltage and current noise values measured or calculated by the manufacturer.
These are specified as a curve (e.g. voltage noise density versus frequency, which rises at low frequency because of 1/f noise) and/or as typical values at specific frequencies; in the TLV2772 datasheet example used in the lecture the current noise is given only as a single value. The input-referred sources are usually assumed to be uncorrelated — a good approximation for a simple input differential pair, though in op-amps with input bias-current cancellation the two input current noises can be significantly correlated.
The intrinsic sources are not the whole story: the external biasing and gain resistors add their own thermal noise, and the current noise terms create extra voltage noise as they pass through those resistors. (The lecture points to TI application note SLVA043B for a step-by-step treatment.)
Problem. An inverting op-amp amplifier has a signal source of impedance RS feeding input resistor R1, with feedback resistor Rf. Derive the total output noise.
Set-up. Replace the op-amp with its input-referred noise model and add a thermal noise source to each resistor. The signal source is grounded for the noise calculation (superposition). Two gains matter:
Noise entering the inverting path sees GI = −Rf/(RS+R1).
Noise entering the non-inverting terminal sees GN = 1 + Rf/(RS+R1).
Note that both depend on the source impedance, so the effective gain can change significantly if RS is large.
External resistors. The thermal noise of RS and R1 is in series with the inverting path, so each is amplified by GI; the thermal noise of Rf is felt directly at the output:
Intrinsic sources. The voltage noise v̄n sits at the non-inverting terminal and is amplified by GN. The current noise īn−: because of negative feedback the inverting and non-inverting inputs are a virtual short, so no current flows through (RS+R1) and all of it flows through Rf. The current noise īn+, on the other hand, sees no resistance on the non-inverting leg and so has no effect at the output (if there were a resistor there, it would).
Total. All the output noise sources are added, and with all resistors at the same temperature the thermal terms combine neatly:
Here v̄n2 and īn−2 are the datasheet spectral densities (V2/Hz and A2/Hz), which is how they enter the Δf expressions above.
Referring the intrinsic noise back to the input (dividing by GI2) gives a quantity that makes the effect of source impedance explicit:
The second term grows with the square of the source resistance: the SNR drops as RS increases.
The term above hints at a general rule. Plotting the total amplifier input-referred noise density against source resistance shows three regimes:
The lecture illustrates this with an Analog Devices example: the OP27 (voltage noise 3 nV/√Hz, current noise 1 pA/√Hz, 25 °C), neglecting the noise of the gain-setting resistors:
| Contribution (nV/√Hz, referred to input) | R = 0 | R = 3 kΩ | R = 300 kΩ |
|---|---|---|---|
| Amplifier voltage noise | 3 | 3 | 3 |
| Amplifier current noise flowing in R | 0 | 3 | 300 |
| Johnson noise of R | 0 | 7 | 70 |
Highlighted cells mark the dominant noise source for each resistance, as in the slide (source: Analog Devices).
The lecture’s general noise guidance for component selection:
Homework (from the lecture). A sensor is modelled as an ideal voltage source vS (VS,RMS = 25 µV) with source resistance RS = 1 kΩ, driving a non-inverting amplifier with R1 = 1 kΩ, RF = 10 kΩ and CF = 1 nF. The op-amp has en = 10 nV/√Hz and in = 0.1 pA/√Hz (white; capacitors noiseless; T = 300 K; all noise sources uncorrelated). Tasks include: the transfer function, DC gain and corner frequency; the effective noise bandwidth; the SNR of the source and of the output in dB; drawing and referring every noise source to the input; the input-referred noise and noise figure; which source dominates; and finally repeating the analysis for a 100 kΩ source (e.g. a dry ECG electrode) — does the noise figure get better or worse, and why do the SNR and noise-figure answers point in different directions?
Important take-homes: