Analog Intuition

Part 2 · Output Noise

Why Vrms

The waveform is a random draw; the bell curve is the physics.

Scope + distribution — fixed 10 ms (1 ms/div)

100 kΩ @ 300 K · en ≈ 40.7 nV/√Hz · Johnson (flat PSD)

Pk-pk (this capture) Lottery — changes every trigger
Mean (this capture) Near 0 V for AC noise
σ (accumulated) Stabilizes as triggers add samples
σ (band spec)
Triggers 0 Fixed 10 ms · 1 ms/div

Fixed 20 µV/div on both panels (shared voltage axis). Blue bars = accumulated histogram; green curve = Gaussian at σ; purple lines = ±1σ, ±3σ, ±3.3σ (labels on both sides). Callout shows Each band line reads: ±nσ = voltage span = % of accumulated samples inside that window. Reset histogram clears the bell.

Try it: hit Trigger several times — the scope trace redraws but the bell barely moves. Then switch the integration band: trace and σ both change. Vrms is always “over a band,” not one scope grab.

Pk-pk is not a spec number

Part 1 showed pk-pk jumping shot-to-shot. Datasheets quote integrated noise in Vrms — the standard deviation σ of the zero-mean Gaussian that models the fuzz.

For Gaussian noise, most samples fall in a bell curve. σ is stable once you have enough data; pk-pk depends on how lucky you were in one capture.

σ is Vrms for zero-mean noise

With mean ≈ 0, RMS and σ are the same number: \(V_\text{rms} = \sqrt{\frac{1}{N}\sum x_i^2} = \sigma\). That is what ADC dynamic range, headroom, and datasheet plots converge on.

Bandwidth is part of the answer

Flat Johnson noise: \(V_\text{rms} = e_n \sqrt{f_H - f_L}\). Widen the band → more noise power → larger σ. Part 4 will draw the nV/√Hz curve; here you only pick the integration limits.