 Square-root collapse of capacitive bridging bandwidth near the DC error floor

The square-root collapse of capacitive bridging bandwidth near the DC error floor describes how the usable frequency interval disappears when a resistive divider has almost exhausted a complex-error allowance. Use the model and exact formula in capacitive bridging complex-error bandwidth: $g=R_s/R_{\mathrm{in}}>0$, ${\tau }_s=R_s{C}_{\mathrm{in}}$, and the DC floor ${\eta }_0=g/(1+g)$. Write the available extra tolerance as $\Delta =\eta -{\eta }_0>0$. The exact bandwidth becomesAt fixed positive $g$, the leading numerator is linear in $\Delta$, so $f_{\eta }$ is proportional to $\sqrt{\Delta }$. Thus demanding four times less extra tolerance approximately halves the bandwidth; it does not reduce it by a factor of four.

For an original example, choose $R_s=20\mathrm{k\Omega }$, $R_{\mathrm{in}}=1\mathrm{M\Omega }$, and $C_{\mathrm{in}}=50\mathrm{pF}$. Then $g=0.02$, ${\tau }_s=1\mu \mathrm{s}$, and ${\eta }_0\approx 0.019608$. Taking $\Delta =0.00020$ gives about $456\mathrm{Hz}$ from the exact formula. Taking $\Delta =0.00005$ gives about $228\mathrm{Hz}$. The small deviation from an exact two-to-one ratio comes from the retained quadratic term and denominator.

The qualification fixed positive $g$ matters. If the receiving resistance is infinite, $g={\eta }_0=0$ andwhich is linear in small $\eta$. For a small but nonzero floor, the square-root approximation needs $\Delta$ small compared with ${\eta }_0$ as well as with the remaining distance to unit error. The crossover from square-root to linear capacitive bridging bandwidth scaling prevents using the fixed-floor asymptotic expression uniformly as $g$ also tends to zero.

This is a limit of the chosen passive input impedance model, not an empirical limit for every amplifier. The manufacturer's shunt-capacitance model motivates the resistive/capacitive paths; an actual bench comparison must include both amplitude and phase shift and the probe's added capacitance.^[ref]