Maxwells inductance bridge solution and equation evaluation

     The bridge circuit is used for medium inductance and can be arranged to yield results of considerable precision. As shown in figure 1, in the two arms, there are two pure resistances so that for balance relations, the phase balance depends on the remaining two arms. If a coil of unknown impedance Z₁ is placed in one arm, then its positive phase angle ɸ₁ can be compensated for in either of the following two ways:

1. A known impedance with an equal positive phase angle may be used in either of the adjacent arms (so that ɸ₁ = ɸ₂ or ɸ₁ = ɸ₄), remaining two arms have zero phase angles (being pure resistances). Such a network is known as Maxwell’s a.c. bridge or L₁/L₄ bridge.
2. Or an impedance with an equal negative phase angle (i.e. capacitance) may be used in opposite arm (so that ɸ₁ + ɸ₃ = 0). Such a network is known as Maxwell-Wien bridge or Maxwell’s L/C bridge.

Hence, we conclude that inductive impedance may be measured in terms of another inductive impedance (of equal time constant) in either adjacent arm (Maxwell Bridge) or the unknown inductive impedance may be measured in terms of a combination of resistance and capacitance (of equal time constant) in the opposite arm (Maxwell-Wien bridge). It is important, however, that in each case the time constants of the two impedance must be matched.As shown in figure 1.

Z₁ = R₁ + jX₁ = R₁ + jωL₁…….unknown;

Z₄ = R₄ + jX₄ = R₄ + jωL₄….…known;

R₂,R₄ = known pure resistances; D = detector

The inductance L₄ is a variable self-inductance of constant resistance, its inductance being of the same order as L₁. The bridge is balanced by varying L₄ and one of the resistance R₂ or R₃. Alternatively, R₂ and R₃ can be kept constant and the resistance of one of the other two arms can be varied by connecting an additional resistance in that arm.
The balance condition is that Z₁Z₃ = Z₂Z₄

(R₁ + jωL₁)R₃ = (R₄ + jωL₄)R₂

Equation the real and imaginary parts on both sides, we have

Z₁ = R₁ + jX₁ = R₁ + jωL₁…….unknown;

Z₄ = R₄ + jX₄ = R₄ + jωL₄….…known;

R₂,R₄ = known pure resistances; D = detector

The inductance L₄ is a variable self-inductance of constant resistance, its inductance being of the same order as L₁. The bridge is balanced by varying L₄ and one of the resistance R₂ or R₃. Alternatively, R₂ and R₃ can be kept constant and the resistance of one of the other two arms can be varied by connecting an additional resistance in that arm.
The balance condition is that Z₁Z₃ = Z₂Z₄

(R₁ + jωL₁)R₃ = (R₄ + jωL₄)R₂