# Hay’s Bridge

**Definition:** The **Hay’s bridge** is used for **determining** the **self-inductance** of the **circuit**. The bridge is the **advanced** **form** of **Maxwell’s bridge**. The Maxwell’s bridge is only appropriate for measuring the medium quality factor. Hence, for **measuring** the **high-quality factor** the **Hays bridge** is used in the circuit.

In **Hay’s bridge**, the **capacitor** is **connected** in **series** with the **resistance**, the voltage drop across the capacitance and resistance are varied. And in Maxwell bridge, the **capacitance** is **connected** in **parallel** with the **resistance**. Thus, the magnitude of a voltage pass through the resistance and capacitor is equal.

## Construction of Hay’s Bridge

The unknown inductor **L _{1} **is placed in the arm

**ab**along with the resistance

**R**This unknown inductor is compared with the standard capacitor

_{1}.**C**connected across the arm

_{4}**cd**. The resistance

**R**is connected in series with the capacitor

_{4}**C**. The other two non-inductive resistor

_{4}**R**and

_{2}**R**are connected in the arm

_{3}**ad**and

**bc**respectively.

The **C _{4}** and

**R**are adjusted for making the bridge in the balanced condition. When the bridge is in a balanced condition, no current flows through the detector which is connected to point

_{4}**b**and

**c**respectively. The potential drops across the arm

**ad**and

**cd**are equal and similarly, the potential across the arm

**ab**and

**bc**are equal.

## Hay’s Bridge Theory

Let,

L_{1} – unknown inductance having a resistance R_{1}

R_{2}, R_{3}, R_{4} – known non-inductive resistance.

C_{4} – standard capacitor

Separating the real and imaginary term, we obtain

Solving the above equation, we have

The quality factor of the coil is

The equation of the unknown inductance and capacitance consists frequency term. Thus for finding the value of unknown inductance the frequency of the supply must be known.

For the high-quality factor, the frequency does not play an important role.

Substituting the value of Q in the equation of unknown inductance, we get

For greater value of Q the 1/Q is neglected and hence the equation become

## Phasor Diagram of Hay’s Bridge

The phasor diagram of the Hay’s bridge is shown in the figure below. The magnitude and the phase of the **E _{3}** and

**E**are equal and hence they are overlapping each other and draw on the horizontal axis. The current

_{4}**I**flow through the purely resistive arm

_{1}**b**

**d.**The current

**I**and the potential E

_{1}_{3}= I

_{3}R

_{3}are in the same phase and represented on the horizontal axis.

The current passes through the arm **ab** produces a potential drop **I _{1}R_{1} **which is also in the same phase of

**I**. The total voltage drop across the arm

_{1}**ab**is determined by adding the voltage

**I**and

_{1}R_{1}**ωI**.

_{1}L_{1}The voltage drops across the arm **ab** and **ad** are equal. The voltage drop** E _{1}** and

**E**are equal in magnitude and phase and hence overlap each other. The current

_{2}**I**and

_{2}**E**are in the same phase as shown in the figure above.

_{2}The current **I _{2} **flows through the arms

**cd**and produces the

**I**voltage drops across the resistance and

_{2}R_{4}**I**voltage drops across the capacitor

_{2}/ωC_{4}**C**. The capacitance

_{4}**C**lags by the currents 90º.

_{4}The voltage drops across the resistance **C _{4}** and

**R**gives the total voltage drops across the arm

_{4}**cd**. The sum of the voltage

**E**and

_{1}**E**or

_{3}**E**and

_{2}**E**gives the voltage drops

_{4}**E**.

### Advantages of Hay’s Bridge

The following are the advantages of Hay’s Bridge.

- The Hays bridges give a simple expression for the unknown inductances and are suitable for the coil having the quality factor greater than the 10 ohms.
- It gives a simple equation for quality factor.
- The Hay’s bridge uses small value resistance for determining the Q factor.

### Disadvantages of Hay’s Bridge

The only disadvantage of this type of bridge is that it is not suitable for the measurement of the coil having the quality factor less than 10 ohms.

**Note:** The quality factor is a parameter which determines the relation between the stored energy and the energy dissipated in the circuit.