= Wien bridge
{c}
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> $Z_1=R_1/(1+i\omega R_1C_1)$, 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 gives
$$
\omega^2R_1R_2C_1C_2=1,\qquad
\frac{R_4}{R_3}=\frac{R_2}{R_1}+\frac{C_1}{C_2}.
$$
The 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.
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