Wien bridge

An alternating-current electrical circuit comparing a parallel resistor-capacitor arm with a series resistor-capacitor arm. Let the upper-left arm have electrical impedance $Z_1=R_1/(1+i\omega R_1{C}_1)$, the lower-left arm $Z_2=R_2+1/(i\omega C_2)$, and the upper-right and lower-right arms have electrical resistances $R_3,R_4$. For input between the upper and lower nodes and an unloaded difference measurement between the arm junctions, voltage divider balance requires $Z_2/Z_1=R_4/R_3$. Separating real and imaginary parts givesThe balance angular frequency is positive for positive components. Equal $R_1=R_2=R$ and $C_1=C_2=C$ require $\omega =1/(RC)$ and $R_4=2R_3$. Reversing which reactive arm is upper changes the divider ratios, so the connections must always be specified.

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.