Noise sources & bandwidth trade-offs
Treat electronic noise as a spectral quantity before integrating it into an RMS voltage or current. Fix a one-sided PSD convention, derive resistor thermal noise, shot noise and transistor channel noise models, introduce flicker noise and input-referred noise, propagate independent sources through their own transfer functions, define equivalent noise bandwidth, derive kT/C for an RC network, and connect bandwidth, gm, resistance, device area and capacitance to quantitative noise trade-offs.
- Distinguish noise power spectral density, amplitude spectral density and integrated RMS noise with correct units.
- Use a one-sided PSD convention consistently and convert resistor thermal noise between Thevenin voltage and Norton current forms.
- State and apply white shot-noise PSD Si=2qI and the common MOS channel thermal-noise model Si≈4kTγgm under their stated validity ranges.
- Represent flicker noise as a low-frequency device/model-dependent PSD proportional to 1/f^α and explain why device area and bias matter.
- Propagate each independent noise source through its own transfer function and sum uncorrelated output PSDs.
- Define input-referred noise and explain why it is a representation of the total circuit noise, not a new physical source.
- Define equivalent noise bandwidth and derive Bn=(π/2)fc for a first-order low-pass.
- Derive the classical kT/C integrated noise of a resistor driving a capacitor.
- Explain why widening bandwidth raises integrated white noise approximately as the square root of bandwidth.
- Analyze design trade-offs among gm, bias current, resistor value, device area, capacitance, gain and bandwidth without claiming one parameter improves everything.
Noise is not a single RMS number until a bandwidth and transfer function have been specified. AE-04 closes by treating noise spectrally first, then integrating it through the real amplifier response.
05.1Fix the convention before writing a noise formula
This chapter uses one-sided power spectral densities defined for f≥0 in hertz. For voltage noise, Sv has units V²/Hz; its amplitude spectral density is √Sv in V/√Hz. The mean-square noise in a band is the PSD integrated over that band:
A quoted “10 nV/√Hz” is a spectral density amplitude, not an RMS voltage until bandwidth and filtering are specified.
05.2Resistor thermal noise
Figure 5.1 — Resistor thermal noise: Thevenin or Norton
The one question is: how can the same resistor noise PSD be represented at the same two terminals?
With a one-sided PSD convention, the series model uses Sv=4kBTR and the parallel model uses Si=4kBT/R. They are two representations of the same resistor thermal noise in the classical hf≪kBT regime.
In the classical regime hf≪kBT, an ideal resistor at thermodynamic temperature T has white open-circuit voltage-noise PSD and short-circuit current-noise PSD
The two forms are related by the same Thevenin–Norton transformation as deterministic sources. They must not be added as independent sources: they represent the same physical resistor noise.
05.3Shot noise and transistor channel noise are different mechanisms
For independent Poisson carrier-transfer events with average DC current I, the one-sided shot-noise current PSD is
This model is central to junction-device currents, but correlations, high-frequency transport and non-Poisson statistics can modify it.
A common low-frequency strong-inversion MOS saturation model writes drain-channel thermal noise as
For an ideal long-channel device γ is commonly approximated by 2/3 in saturation; short-channel technologies can have larger/model-dependent effective γ. The formula is a compact small-signal model, not a universal microscopic constant.
05.4Flicker noise dominates toward low frequency
Many devices show a low-frequency PSD approximately proportional to 1/f^α, with α near 1 over a useful range. For MOS input-referred gate-voltage noise, a common compact form is
The exact definition and units of Kf are model/process dependent. The robust design lesson is that larger device area often reduces MOS flicker-noise density while increasing capacitance, so the trade is not free.
The 1/f corner is the frequency where the extrapolated flicker and white-noise PSDs are equal. It is not the same quantity as an amplifier −3 dB cutoff.
05.5Each source has its own transfer to the output
For mutually uncorrelated noise sources xk with one-sided PSDs Sxk and transfer functions Hk from that source to the selected output,
Noise voltages should therefore not be added directly unless they are already referred to the same node through the correct gains. RMS values of independent sources combine by root-sum-square because their powers/variances add.
If two sources are correlated, cross-spectral terms must be retained:
05.6Input-referred noise is a representation of the whole circuit
If the signal transfer from input voltage to output is Av(f), and the total output-noise PSD is Sv,out(f), define the equivalent input-referred voltage-noise PSD by
This does not create a new physical source. It asks what fictitious input noise, passed through the noiseless signal gain, would produce the same output PSD. Input-referred noise makes stages easier to compare when they have different gains.
For a MOS transconductor dominated by its own channel thermal noise, the simple input-referred white voltage-noise density is
Increasing gm lowers this particular input-referred density, but obtaining more gm generally costs current, device area/capacitance, or both.
05.7Equivalent noise bandwidth converts a real filter into a white-noise bandwidth
For white input PSD S0 passing through H(f), define the one-sided equivalent noise bandwidth
Then the integrated output variance is S0|H(0)|²Bn. For a first-order low-pass
Thus the noise bandwidth of a one-pole low-pass is 1.571 times its −3 dB bandwidth.
05.8The RC kT/C result
Figure 5.2 — The same RC sets bandwidth and integrates noise
The one question is: why does the total noise on C become kBT/C instead of remaining proportional to R?
R produces the 4kBTR voltage-noise PSD, but R also sets fc=1/(2πRC). Integrating the filtered PSD to infinite frequency cancels R and gives ⟨v²⟩=kBT/C.
The resistor contributes Sv,R=4kBTR. The voltage across C is filtered by
Therefore
R controls how quickly equilibrium is reached and the bandwidth over which noise is admitted, but the final equilibrium mean-square capacitor voltage is independent of R in this ideal classical model.
05.9Noise–bandwidth design trade-offs
There is no universal “lowest-noise” transistor size or bias without bandwidth, power, source impedance and signal requirements.
05.10Mastery exercises
- At 300 K, find the voltage-noise amplitude density √(4kTR) of a 10 kΩ resistor.
- A white input density is 5 nV/√Hz and an ideal brick-wall measurement bandwidth is 100 kHz. Find the RMS noise.
- A first-order low-pass has fc=20 kHz. Find its equivalent noise bandwidth.
- A 2 pF capacitor equilibrates through a resistor at 300 K. Find √(kBT/C).
- For a MOS transistor with gm=10 mS and γ=2/3 at 300 K, estimate its channel-noise input-referred amplitude density √(4kTγ/gm).
- Why can increasing MOS area reduce 1/f noise while simultaneously hurting high-frequency response?
Solutions
1. √(4kTR)≈12.9 nV/√Hz at 300 K for R=10 kΩ.
2. vrms=5 nV/√Hz·√100000≈1.58 µV.
3. Bn=(π/2)fc≈31.4 kHz.
4. √(kBT/C)≈45.5 µV for C=2 pF at 300 K.
5. √(4kTγ/gm)≈1.05 nV/√Hz for gm=10 mS and γ=2/3.
6. Larger WL often averages/traps over larger area and lowers the compact 1/f term, but also raises gate/parasitic capacitances and therefore input loading/poles.
AE-04 is now conceptually closed: device charge storage → poles/zeros → Miller → cascode → noise integrated through bandwidth.