Maxwell's Inductance Bridge

Maxwell Bridge

A Maxwell bridge (in long form, a Maxwell-Wien bridge) is a type of Wheatstone bridge used to measure an unknown inductance(usually of low Q value) in terms of calibrated resistance and capacitance. It is a real product bridge.

It uses the principle that the positive phase angle of an inductive impedance can be compensated by the negative phase angle of a capacitive impedance when put in the opposite arm and the circuit is at resonance; i.e., no potential difference across the detector and hence no current flowing through it. The unknown inductance then becomes known in terms of this capacitance.

With reference to the picture, in a typical application  and  are known fixed entities, and  and  are known variable entities.  and  are adjusted until the bridge is balanced.

 and  can then be calculated based on the values of the other components:

To avoid the difficulties associated with determining the precise value of a variable capacitance, sometimes a fixed-value capacitor will be installed and more than one resistor will be made variable. It cannot be used for the measurement of high Q values. It is also unsuited for the coils with low Q values, less than one, because of balance convergence problem. Its use is limited to the measurement of low Q values from 1 to 10.

The additional complexity of using a Maxwell bridge over simpler bridge types is warranted in circumstances where either the mutual inductance between the load and the known bridge entities, or stray electromagnetic interference, distorts the measurement results. The capacitive reactance in the bridge will exactly oppose the inductive reactance of the load when the bridge is balanced, allowing the load's resistance and reactance to be reliably determined.

Derivation Of Equation Of Maxwell 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 Z₁ is placed in one arm, then its positive phase angle ?₁ 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 ?₁ = ?₂ or ?₁ = ?₄), 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 ?₁ + ?₃ = 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 theThe 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 ?₁ 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 ?₁ = ?₂ or ?₁ = ?₄), 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 ?₁ + ?₃ = 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.

 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 + j?L₁)R₃ = (R₄ + j?L₄)R₂

Equation the real and imaginary parts on both sides, we have
R₁R₃ = R₂R₄ or R₁/R₄ = R₂/R₃

(i.e. products of the resistances of opposite arms are equal).
And
?L₁R₃ = ?L₄R₂
Or L₁ =L₄R₂/R₃
We can also write that L₁ = L_�4R₁/R₄

Hence, the unknown self-inductance can be measured in term 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 2. The current I₄ and I₃ are in phase with I₁ and I₂. 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 V₁n 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.

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