Maxwell’s Inductance Bridge

The Maxwell’s Inductance Bridge circuit is used for medium inductances and can be arranged to yield results of considerable precision. As shown in Figure (A) 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 an unknown impedance Z₁ is placed in one arm, then its positive phase angle φ₁ can be compensated for in either of the following two ways:

(i) A known impedance with an equal positive phase angle may be used in either of the adjacent arms (so that φ₁ = φ₃or φ₂= φ₄ ), 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.

(ii) Or an impedance with an equal negative phase angle (i.e. capacitance) may be used in opposite arm (so that φ₁ + φ₃ = 0). Such a network is known as Maxwell-Wien bridge (Figure (C)) or Maxwell’s L/C bridge.

Hence, we conclude that an 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 impedances must be matched.

As shown in Figure (A),

Z₁ = R₁ + jX₁ = R₁ + jωL₁ ……………….. unknown;

Z₄ = R₄ + jX₄ = R₄ + jωL₄ …………………known

R₂, R₃ = known pure resistances; D = detector

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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 resistances 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 Z_{1 .}Z₃ = Z₂ . Z₄

∴ (R₁ + jωL₁)R₃ = (R₄ + jωL₄)R₂

Equating the real and imaginary parts on both sides, we have
R₁ . R₃ = R₂ . R₄ or R₁ / R₄ = R₂ / R₃

(i.e. products of the resistances of opposite arms are equal).

and ωL1R3 = ωL4R2

L₁ = L₄ (R₂/R₃)

We can also write that L₁ = L₄ (R₁/R₄)

Hence, the unknown self-inductance can be measured in terms of the known inductance L₄ and the two resistors. Resistive and reactive terms balance independently and the conditions are independent of frequency. This bridge is often used for measuring the iron losses of the transformers at audio frequency.

The balance condition is shown vectorially in Figure (B). The currents I₄ and I₃ are in phase with I₁ and I₂. This is, obviously, brought about by adjusting the impedances of different branches, so that these currents lag behind the applied voltage V by the same amount. At balance, the voltage drop V₁ across branch 1 is equal to that across branch 4 and I₃ = I₄. Similarly, voltage drop V2 across branch 2 is equal to that across branch 3 and I₁ = I₂.