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Derive an expression for inductance measurement using Hay bridge
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Hay Bridge is used to measure high value inductances.

The circuit diagram is shown below.

When bridge is balance,

Z1 Zx=Z2 Z3Z2=R2Z3=R3Zx=Rx +jωLx

From equation 1,

(R1jωC1)(Rx+jωLx)=R2R3

R1Rx+jωLxR1jRxωC1+LxC1=R3 R2

(R1Rx+LxC1)+j(ωLxR1RxωC1)=R2R3

Equating real and imaginary terms,

R1Rx+LxC1=R2R3

ωLxR1 RxωC1=0

Lx=Rxω2R1C1

R1 Rx+Rxω2R1C2=R2 R3Rx R1+ 1ω2R1C12=R2 R3Rx ×ω2 R12C12+1ω2R1C12=R2 R3 Rx=ω2C12R1R2R31+ω2R12C12Lx=ω2C22R1R2R3(1+ω2R12C12) (ω2R1C1)Lx=C1R2R3(1+ω2R12C12)

A w appears in the expression for Lx, this bridge is frequency sensitive.

Quality Factor (Q) for capacitance is

Q=1ωR1C1Or  ωR1C1=1/QLx=C1R2R31+1/Q2

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