Crossover from square-root to linear capacitive bridging bandwidth scaling

The crossover from square-root to linear capacitive bridging bandwidth scaling occurs when excess error allowance becomes comparable with the resistive error floor itself. In square-root collapse of capacitive bridging bandwidth near the DC error floor, put ${\eta }_0=g/(1+g)$, $g=R_s/R_{\mathrm{in}}$, $\Delta =\eta -{\eta }_0>0$, and ${\tau }_s=R_s{C}_{\mathrm{in}}$. For positive ${\eta }_0$, introduce the dimensionless ratio $\xi =\Delta /{\eta }_0$. The exact bandwidth can then be writtenThis single expression distinguishes the two regimes without exchanging small-parameter limits silently.

When $\Delta \ll {\eta }_0$ and the tolerance is away from one, the factor $\sqrt{\xi (2+\xi )}$ is approximately $\sqrt{2\xi }$: bandwidth scales as $\sqrt{\Delta }$. When ${\eta }_0\ll \Delta \ll 1$, that factor is approximately $\xi$, while the denominator is close to one: bandwidth is approximately $(1+g)\Delta /(2\pi {\tau }_s)$. The exactly zero-floor model has the corresponding small-tolerance linear behavior.

For an original dimensionless comparison, take ${\eta }_0=10^{-4}$. At $\xi =0.01$, replacing $\sqrt{\xi (2+\xi )}$ by $\sqrt{2\xi }$ changes the numerator by only about $0.25\%$. At $\xi =100$, replacing it by $\xi$ changes the numerator by about $1\%$; here the total tolerance is $0.0101$, so the denominator differs little from one. The tolerance is small in both cases, but a different approximation is useful because its size relative to the floor differs.

In component terms, increasing $R_{\mathrm{in}}$ moves the crossover by lowering ${\eta }_0$; changing only $C_{\mathrm{in}}$ rescales the frequency axis through ${\tau }_s$. A sweep of receiving electrical resistance and input capacitance can therefore separate a floor effect from a frequency-scale effect. The resistive/shunt-capacitive manufacturer model motivates the apparatus assumptions.^[ref] The specification remains a full complex relative gain error, including phase shift, rather than only an amplitude or half-power threshold.