## Maxwell’s Inductance Bridge

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 Z_{1} is placed in one arm, then its positive phase angle ɸ_{1} can be compensated for in either of the following two ways:

- A known impedance with an equal positive phase angle may be used in either of the adjacent arms (so that
**ɸ**, remaining two arms have zero phase angles (being pure resistances). Such a network is known as Maxwell’s a.c. bridge or_{1}= ɸ_{2}or ɸ_{1}= ɸ_{4})**L**bridge._{1}/L_{4} - Or an impedance with an equal negative phase angle (i.e. capacitance) may be used in opposite arm (so that
**ɸ**). Such a network is known as Maxwell-Wien bridge or Maxwell’s L/C bridge._{1}+ ɸ_{3 }= 0

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.

**Z _{1} = R_{1} + jX_{1} = R_{1} + jωL_{1}…….unknown;**

**Z _{4} = R_{4} + jX_{4} = R_{4} + jωL_{4}….…known;**

R_{2},R_{4} = known pure resistances; D = detector

The inductance L_{4} is a variable self-inductance of constant resistance, its inductance being of the same order as L_{1}. The bridge is balanced by varying L_{4} and one of the resistance R_{2} or R_{3}. Alternatively, R_{2} and R_{3} 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}Z_{3} = Z_{2}Z_{4}**

**(R _{1} + jωL_{1})R_{3} = (R_{4} + jωL_{4})R_{2}**

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

**R _{1}R_{3} = R_{2}R_{4} or R_{1}/R_{4} = R_{2}/R_{3}**

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

And

**ωL _{1}R_{3} = ωL_{4}R_{2}**

Or **L _{1} =L_{4}R_{2}/R_{3}**

We can also write that **L _{1} = L_{4}R_{1}/R_{4}**

Hence, the unknown self-inductance can be measured in term of the known inductance L_{4} 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 2. The current I_{4} and I_{3} are in phase with I_{1} and I_{2}. This is, obviously, brought about by adjusting the impedance of different branches, so these currents lag behind the applied voltage V by the same amount. At balance, the voltage drop V_{1}n across branch 1 is equal to that across branch 4 and I_{3} = I_{4}. Similarly, voltage drop V_{2} across branch 2 is equal to that across branch 3 and I_{1} = I_{2}.