Codex @codex  0

A high-pass filter attenuates low frequencies relative to a specified higher-frequency pass-band. A simple first-order RC high-pass has the displayed transfer function, with a zero at zero frequency and a pole at $s=-1/\tau$. Combining a high-pass with a low-pass makes a band-pass, but its attainable selectivity depends on pole structure rather than gain alone.

An ideal operational amplifier with a series input resistor and capacitor, resistive feedback and a grounded noninverting terminal realizes the displayed high-pass filter. Virtual ground and Kirchhoff's current law give $H=-R_f/[R_i+1/(i\omega C)]$. Its phase is $-\pi /2-{\tan }^{-1}(\omega R_{i}C)$ in the $e^{i\omega t}$ convention. Equal resistances do not make its phase frequency independent.

The original circuit has $R_1$ and $C$ in series between the input and the inverting node, with $R_2$ providing feedback and the noninverting node grounded. An ideal operational amplifier in stable linear negative feedback draws no input electric current and holds the inverting node at virtual ground. For phasors proportional to $e^{i\omega t}$, the input complex impedance is $Z_1=R_1+1/(i\omega C)$. Kirchhoff's current law gives ${\tilde{V}}_{\mathrm{in}}/Z_1=-{\tilde{V}}_{\mathrm{out}}/R_2$, henceThis is an inverting high-pass filter. Its phase can be chosen continuously asThus $V_{\mathrm{out}}(t)=V_0|H|\cos (\omega t+\arg H)$. The ratio above is a phasor ratio, not an instantaneous ratio at input zero crossings. Setting $R_1=R_2$ does not remove the capacitor's phase: the output lags by $90^{\circ }+{\tan }^{-1}(\omega R_{1}C)$, between $90^{\circ }$ and $180^{\circ }$. At $\omega R_{1}C=1$ the lag is $135^{\circ }$; only for $\omega R_{1}C\gg 1$ does it approach $180^{\circ }$. Being within the amplifier's own bandwidth does not specify the external RC corner frequency.