Equal-ratio RC bridge (${\omega }_0^{-2}=R_x^2{C}_x(C-C_x)$)

A bridge driven between top and bottom nodes has equal resistors on its left arms, upper-right electrical impedance $Z=R_x/(1+i\omega R_x{C}_x)$ and lower-right electrical impedance $W=R_v+1/(i\omega C)$. A zero detector electric current requires $Z=W$. Equating real parts and imaginary parts gives $R_v=R_x(1-C_x/C)$ and ${\omega }_0^{-2}=R_x^2{C}_x(C-C_x)$, requiring $0<C_x<C$. Its open-circuit right-minus-left junction voltage is $V_0(W-Z)/[2(W+Z)]$, whereas a connected ideal ammeter has zero terminal voltage at every angular frequency. This distinction separates a null-current measurement from an unloaded voltage measurement.