 Inductive bridge with a parallel variable arm

Consider a bridge driven across opposite nodes, with upper-path arms $Z_V=r_V\parallel i\omega L_V$ and $i\omega L_1$, lower-path arms $R_1$ and $R+i\omega L$, and a null detector between the path midpoints. Equality of midpoint voltages gives $Z_V(R+i\omega L)=i\omega L_1{R}_1$. At nonzero frequency, substitution and comparison of real and imaginary parts yield the displayed resistance and inductance. The formulas depend on the variable arm being parallel; treating it as a series arm gives a different circuit.