Wien bridge band-pass response ($H=-[3+i(u-u^{-1}){]}^{-1}$)

Use the equal-component Wien bridge with upper-left parallel arm and lower-left series arm. Let $A$ be the left arm junction, $C$ the upper input node, and $V_i$ the upper-minus-lower input electric potential difference. With $u=\omega RC$, the unloaded transfer ratio $H=(V_A-V_C)/V_i$ is the displayed expression. Its band-pass filter magnitude is $[9+(u-u^{-1}{)}^2{]}^{-1/2}$, maximal at $u=1$ with $H=-1/3$. One continuous wave phase is $\varphi =-\pi -\arctan [(u-u^{-1})/3]$, tending from $-\pi /2$ at low positive frequency to $-3\pi /2$ at high frequency. Reversing the measured output polarity multiplies $H$ by $-1$ and shifts its phase by $\pi$ without changing its magnitude.