The self-inductance of a coil with a high-quality factor (Q > 10) is measured using the Hays bridge, a particular type of AC bridge circuit. It is a modified version of Maxwell’s bridge that works well for determining the quality factor of medium coils (1 to 10). We will describe Hays bridge’s construction, theory, phasor diagram, benefits, and drawbacks in this article.

What is Self-Inductance?

A coil’s or circuit’s ability to resist changes in the current flowing through it is known as self-inductance. So It is expressed in henries (H) and is influenced by the number of turns, the size, shape, and permeability of the core material of the coil. According to Lenz’s law, self-inductance generates an electromotive force (emf) that opposes the change in current.

What is Quality Factor?

A dimensionless parameter called the quality factor describes how well a coil or circuit resonates at a specific frequency. It is also referred to as the Q factor or the merit figure. So It is calculated by dividing the coil’s reactance at the resonant frequency by its resistance. Lower energy losses and sharper resonance are correlated with higher Q factors. The ratio of energy dissipated to stored per cycle is another way to state the Q factor.

Construction of Hays Bridge

The schematic diagram of Hay bridge is shown below:

Four arms make up the bridge: AB, BC, CD, and DA. An unknown inductor L1 is connected in series with a resistor R1 in the arm AB. A standard capacitor C4 and a resistor R4 are connected in series on the arm CD. So Pure resistors R3 and R2 are found in the arms BC and DA, respectively. Between points B and D, a detector or galvanometer is connected to show the balance condition. To power the bridge, an AC source is connected between points A and C.

Theory of Hays Bridge (hays bridge derivation)

When the voltage drops across AB and CD and the voltage drops across BC and DA are equal and opposite, Hay’s bridge’s balance condition is met. This indicates that the detector has zero deflection and no current flowing through it.

Using Kirchhoff’s voltage law, we can write the balance condition as:

Z1Z4 = Z2Z3

where Z1, Z2, Z3, and Z4 are the impedances of the four arms.

Substituting the values of impedances, we get:

(R1 – jX1)(R4 + jX4) = R2R3

So where X1 = 1/ωC1 and X4 = ωL4 are the reactances of the inductor and capacitor, respectively.

So Expanding and equating the real and imaginary parts, we get:

R1R4 – X1X4 = R2R3

R1X4 + R4X1 = 0

Solving for L1 and R1, we get:

L1 = R2R3C4/(1 + ω^2R42C4^2)

R1 = ω^2R2R3R4C42/(1 + ω^2R42C4^2)

So The quality factor of the coil is given by:

Q = ωL1/R1 = 1/ωR4C4

These equations show that L1 and R1 depend on the frequency of the source ω. Therefore, to measure them accurately, we need to know the exact value of ω. However, for high Q factor coils, we can neglect the term 1/ω^2R42C4^2 in the denominators and simplify the equations as:

L1 ≈ R2R3C4

R1 ≈ ω^2R2R3R4C42

Q ≈ 1/ωR4C4

Phasor Diagram of Hay Bridge

Because capacitor C4 is present in arm CD, currents I1 and I2 are not in phase. As can be seen, the current I2 trails I1 by an angle. Due to the fact that the voltage drops E1 and E2 are crossing pure resistors R1 and R2, respectively, they are equal in size and phase. Due to the fact that voltage drops E3, and E4 are across pure resistors R3 and R4, respectively, they are also equal in magnitude and phase. Because it is across the capacitor C4, the voltage drop E5 is perpendicular to E4. Due to its location across the inductor L1, voltage drop E6 is perpendicular to E1. According to the phasor diagram, E6 + E5 = E3 + E4 = E.

Advantages

• So For calculating the unknown inductance and resistance of high Q factor coils, Hay’s bridge offers straightforward expressions.

• By adjusting C4, Hay’s bridge can determine the Q factor over a large range.

• Compared to Maxwell’s bridge, Hay’s bridge requires a lower value of R4. So This lessens the error brought on by leakage resistance and stray capacitance.

Disadvantages

• A standard capacitor with high accuracy and stability is necessary for the Hay’s bridge.

• Hay’s bridge cannot be used to measure coils with a low Q factor (Q 10). So Maxwell’s bridge or other bridges are preferred for these coils.

• A frequency meter is necessary to precisely measure L1 and R1 for Hay’s bridge.

Applications

There are many uses for Hays bridge, including:

• other bridges or measurement tools for self-inductance or Q factor calibration.

• coils with a high Q factor’s self-inductance in laboratories or in the workplace.

• creating oscillators or filters with coils with a high Q factor.

• investigating how frequency affects the Q factor or self-inductance.