= 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 $\eta_0=g/(1+g)$, $g=R_s/R_{\rm in}$, $\Delta=\eta-\eta_0>0$, and $\tau_s=R_sC_{\rm in}$. For positive $\eta_0$, introduce the dimensionless ratio $\xi=\Delta/\eta_0$. The exact bandwidth can then be written
$$
f_\eta=\frac{1+g}{2\pi\tau_s}\,
\frac{\eta_0\sqrt{\xi(2+\xi)}}{\sqrt{1-\eta_0^2(1+\xi)^2}}.
$$
This 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 $\eta_0\ll\Delta\ll1$, 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 $R_{\rm in}$ moves the crossover by lowering $\eta_0$; changing only $C_{\rm in}$ 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.<MT-040: Op Amp Input Impedance by Analog Devices>[Printed pp. 1–2, input-path models and source-impedance dependence]{ref} The specification remains a full <complex relative gain error>, including <phase shift>, rather than only an <amplitude> or half-power threshold.
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