Maxwell bridge measure an unknown inductance in term of known capacitance. The two opposite ratio arms consists pure resistance, while one of the ratio arms has a resistance and capacitance in parallel and last arm of bridge has a resistance and unknown inductance in series. The positive phase angle of inductive impedance may be compensated by the negative phase angles of capacitive impedance put in the opposite arm.
Z1, Z2, Z3 and Z4 are the impedance of the arm AB, BC, CD, and DA respectively.
Let us first find the combined impedance of arm AB.
Z1 = R1||XC
Z3 = R3 + jωL3 and Z4 = R4
Balance condition is Z1Z3 = Z2Z4
Or, R1R3+jωL3R1 = R2R4 + jωCR1R2R4
Separating the real and imaginary, we get
R1R3 = R2R4 and L3R1 = CR1R2R4;
L3 = CR2R4