Kt C Noise

kT/C noise is the noise floor a sampling capacitor cannot design its way below no matter how quiet the switch closing it happens to be. Every switch has some resistance, and resistance generates thermal noise; when that noisy switch connects a capacitor to a voltage and then opens, the capacitor is left holding not the clean voltage alone but that voltage plus a random sample of the switch's own thermal noise, frozen in place the instant the switch disconnects.

The derivation's most counterintuitive step is that the switch's own resistance cancels out of the final answer entirely. While a switch is closed, its on-resistance R and the sampling capacitor C form a simple RC low-pass filter, and that resistor generates thermal noise with a flat power spectral density of 4kTR, where k is Boltzmann's constant and T is absolute temperature. The RC combination also sets the noise bandwidth available to that same resistor — a larger R means more noise power per hertz, but it also means a narrower bandwidth over which that noise can appear. Integrating the noise density across the RC-limited bandwidth, the resistance term cancels completely, leaving a total sampled noise power of exactly kT/C — a result with no R in it at all, which is why the effect is named for the capacitor alone.

This is a genuinely different kind of noise from the continuous, time-varying thermal noise an amplifier's input sees while actively operating. While the switch stays closed, the noise is ordinary resistor noise, continuously present and averaged over time by whatever is reading the node. The moment the switch opens, one single instantaneous value of that noise is captured and held on the capacitor — not an average, a single frozen sample — and it stays exactly that value until the capacitor is reset or resampled. A faster-settling, lower-resistance switch does not reduce this error at all, because reducing R only widens the bandwidth proportionally, leaving the integrated kT/C total unchanged; the only lever that actually reduces this specific noise is making the capacitor itself larger.

The square-root relationship is what makes kT/C noise expensive to fight once it becomes the dominant error. Noise voltage scales as the square root of kT/C, not linearly with capacitance, so doubling the sampling capacitor only buys a factor of √2 reduction in noise — while quadrupling the die area and the switching power needed to charge and discharge that capacitor every cycle. This is the standard, unavoidable tradeoff at the bottom of practically every switched-capacitor circuit's noise budget: a successive-approximation ADC's capacitive DAC, a switched-capacitor filter's sampling stage, and a CCD or CMOS image sensor's pixel reset all trace their noise floor back to this same kT/C limit, and all pay the same area-and-power cost to push it down.

This effect sits at the bottom of the entire sampled-data noise stack this series has been building, and it is the one layer none of the others can remove. Autozeroing cancels offset and low-frequency flicker noise; careful switch sizing reduces charge injection and clock feedthrough; cascoding and layout technique manage device mismatch and gain limits. None of those techniques touch kT/C noise, because it is not a device imperfection or a circuit artifact at all — it is the thermodynamic noise floor of sampling anything with a finite-temperature resistor and a finite capacitor, present even in an imaginary, perfectly matched, perfectly laid-out circuit with zero offset and zero charge injection.

kT/C noise is the floor underneath every other noise and offset mechanism a sampled-data circuit fights, because it comes from thermodynamics rather than from any particular transistor's imperfection. Every technique this cluster of articles has covered reduces something layered on top of that floor; only a bigger capacitor, paid for in area, power, and speed, actually lowers the floor itself.