What is Hays Bridge : Construction, Phasor Diagram & Its Applications

Before we discuss the Hays bridge, we have to know about the Maxwell bridge limitations to understand how this bridge is used in numerous applications. The main function of the Maxwell Bridge is to measure the average QF (quality factor) in the coils (1<Q<10). This type of bridge is not useful for calculating high QF like Q>10. To overcome the limitation of Maxwell’s bride, Hays bridge is used. This article discusses an overview of Hay’s bridge.

What is Hays Bridge?

Definition: A bridge circuit that is used to measure the resistance & inductance of coils with high Q-factor is known as Hays Bridge. This is the modification of Maxwell’s bridge. So this bridge is used to determine the high-quality factor in the circuit.


The connection of hays bridge circuits can be done by connecting the capacitor and resistor in series with each other. So that the voltage drop across the resistance & capacitance will be changed. In Maxwell Bridge, the connection of the resistance & capacitance can be done parallel. Therefore, the magnitude of a voltage supply throughout the resistor & capacitor will be the same.

Construction of Hays Bridge

The construction of Hays Bridge is shown below. In the following circuit, ‘L1’ inductor is unknown and it is arranged with Resistance ‘R1’ in-between ab arm. The comparison of this inductor can be done with the capacitor ‘C4’ which is connected with ‘R4’ resistance in the cd arm. Similarly, the remaining resistances like R2 & R3 are connected in the arms ad & bc.


To make the bridge in a balanced condition, both the ‘R4’ resistance and ‘C4’ capacitor are adjusted. Once the circuit is in a balanced condition, then there is no flow of current throughout the detector. Here, the detector is placed in between b & d. The potential drop across the ad & cd arm is equivalent. In the same way, the potential drop across the ab & bc arm is equivalent.

Hays Bridge Theory

In the above circuit, inductor ‘L1’ is unknown inductor including ‘R1’ resistance


R2, R3, R4 are known as non-inductive resistance.

‘C4’ is a standard capacitor

The load impedances of the above bridge are

Z1 = R1-j/ωc1

Z2 = R2

Z3 = R3

Z4 = R4 + jωL4

When the circuit is balanced

Z1Z4 = Z2Z3

Substitute the load impedances in the above equations

(R1-j/ωc1)*(R4 + jωL4) = R2*R3

Here, 1/C1 = L1 and L4 = 1/C4

R1R4+R1jωL4 – jR4/ωc1+ jωL4/ωc1 = R2*R3

R1R4+L1/C4+jωL1R4-jR1/ωc4 = R2*R3

Once the real & imaginary terms are separated then we can get the following

R1R4+(L1/C4) = R2*R3

jωL1R4-(jR1/ωc4) = R2*R3

By solving the above equations we can get

L1 = R2R3C4/(1+ ω2R42C42)

R1= ω2C42R2R3R4/ω2R42C42

The QF of the coil is

Q = ωL1/R1 = 1/ ω2R4C4

The unknown capacitance & inductance equation mainly includes frequency term. Therefore to find the unknown inductance value, the supply frequency must be known.

Here, the frequency doesn’t play an essential role in the high QF

Q = 1/ ω2R4C4

Substituting this value in the L1

L1 = R2R3C4/1+ (1/Q)2

For a high value of ‘Q’, the 1/Q can be ignored and thus the equation will be

L1 = R2R3C4

Hays Bridge Phasor Diagram

In the following phasor diagram of Hays bridge, e1, e2, e3, and e4 are null points. Once the current flows through arm ‘bd’ then e1=e2 and e3=e4. Here ‘i1’ is the reference axis in the phasor diagram and this axis leads ‘i2’ with some angle due to the capacitor connected in between arm ‘cd’. Mark the resultant of the null point’s e1&e2 to e. The phase angle between the electrical resistance (r4) & capacitor (c4) is 90° shown in the figure.



The advantages of hays bridge are

  • This bridge is used for the unknown inductances to provide a simple expression. It is appropriate for the coil that has a high Q factor than the 10 ohms.
  • For the Q factor, this bridge provides a simple equation.
  • It uses a small resistance value to determine the quality factor.


The disadvantages of hays bridge are

  • It is not applicable for the measurement of the coil which has less than 10 ohms Q factor.
  • The balanced equation of the bridge depends on operating frequency and thus the frequency change will influence the measurements.
  • The Q factor is used to determine the main relationship between the energy which is stored & dissipated within the circuit.

Applications of Hays Bridge

The applications are

  • This bridge is used to determine the self-inductance of the circuit.
  • This is used to overcome the drawback of Maxwell’s bridge. The
  • This bridge circuit is used to measure the high QF (quality factor) in the circuit.

Thus, this is all about an overview of Hay’s bridge. The quality factor can be measured by using Maxwell as well as Hay’s bridge but Maxwell is used to calculating medium QF (Q < 10) whereas the Hay’s is used to calculate high-quality factor (Q > 10). So to overcome Maxwell’s limitation, this bridge circuit is used. Here is a question for you, what is the difference between Maxwell’s & Hay’s Bridge?

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