---
title: "Bridge for high-Q inductance measurement"
subject: "Instrumentation I"
chapter: 2
key: "semester_4/instrumentation_i:high-q-inductance-bridge"
source: https://www.bctengineer.com/q/semester_4/instrumentation_i/high-q-inductance-bridge
site: BCT Engineering
---

# Bridge for high-Q inductance measurement

Instrumentation I · Chapter 2: Theory of measurement

- Asked in: year not given, 8 marks
- Video: [Hay's Bridge | Hay's Bridge for Inductance Measurement | AC Bridges | Inductance Measurements](https://www.youtube.com/watch?v=FdDf_SiV5-k) — School of Physics
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/instrumentation_i/high-q-inductance-bridge
- Whole chapter: https://www.bctengineer.com/md/semester_4/instrumentation_i/chapter-2

## Question

Which bridge is used for the measurement of inductance of coil having high quality factor? Explain how the measurement is done and why the bridge is preferred.

## Short answer

The Hay bridge measures high-Q inductors (Q>10) using a series R4–C4 arm. At balance, L1 ≈ R2R3C4, Q = 1/(ωC4R4). It is preferred because high Q requires only a small R4, while a Maxwell bridge would need an impractically large R4.

## Answer

The **Hay bridge** is used to measure the inductance of coils with a high quality factor (Q > 10).

#### Circuit
```text
          A
        /   \
  L1,R1(Z1)  R3 (Z4)
      /         \
     B---[D]----D
      \         /
    R2 (Z2)   R4 + C4 (series) (Z3)
        \     /
          C
   source across A-C, detector across B-D
```
Arm AB: unknown coil (L1 in series with R1). Arm BC: R2. Arm CD: R4 in series with a standard capacitor C4. Arm DA: R3. The detector is between B and D.

#### Measurement
Adjust R3 (or R2) and R4 until the detector shows zero. At balance, Z1·Z3 = Z2·Z4, with Z3 = R4 − j/(ωC4):

(R1 + jωL1)(R4 − j/ωC4) = R2R3

Equate the real and imaginary parts:
```text
 R1R4 + L1/C4 = R2R3
 ωL1R4 − R1/(ωC4) = 0   ->  R1 = ω²L1R4C4
```
Solving:
```text
 L1 = R2R3C4 / (1 + ω²C4²R4²)
 R1 = ω²C4²R2R3R4 / (1 + ω²C4²R4²)
 Q  = ωL1/R1 = 1/(ωC4R4)
```
For high Q (ωC4R4 ≪ 1), the denominator is nearly 1, so L1 ≈ R2R3C4.

#### Why it is preferred for high Q
- Q = 1/(ωC4R4), so a high Q needs only a **small R4**, which is cheap and practical.
- In the Maxwell bridge the same Q would need a very large parallel R4 (Q = ωC4R4), which is impractical.
- For high Q the term ω²C4²R4² is negligible, so L1 ≈ R2R3C4 is almost independent of frequency.
- The standard capacitor is nearly lossless and easy to obtain, which gives good accuracy.
- Drawback: the exact balance equations contain ω, so the source frequency must be known and stable, and the bridge is not suitable for low-Q coils.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which bridge measures inductance of a coil with Q > 10?
   A. Maxwell bridge
   B. Schering bridge
   C. Wien bridge
   D. Hay bridge
   Answer: D. The Hay bridge, with a series R4-C4 arm, suits high-Q coils. (Key point: Bridge for high Q)

2. In the Hay bridge, how are the standard capacitor C4 and resistor R4 connected?
   A. In series
   B. In parallel
   C. C4 in the source branch only
   D. R4 across the detector
   Answer: A. Arm CD has R4 in series with C4, unlike Maxwell where they are in parallel. (Key point: Hay bridge arm)

3. What is the Hay bridge expression for the quality factor?
   A. Q = omega C4 R4
   B. Q = R2R3C4
   C. Q = 1/(omega C4 R4)
   D. Q = R2R3/R4
   Answer: C. Equating real and imaginary parts gives Q = omega L1/R1 = 1/(omega C4 R4). (Key point: Q expression)

4. Why is the Hay bridge preferred for high-Q coils over the Maxwell bridge?
   A. It is frequency independent
   B. High Q needs only a small R4, while Maxwell would need a very large R4
   C. It needs no standard capacitor
   D. It works only for low Q
   Answer: B. In Hay Q = 1/(omega C4 R4), so a high Q needs small R4; Maxwell (Q = omega C4 R4) would need an impractically large R4. Hay's drawback is that balance depends on frequency. (Key point: Why preferred)

## Related in Instrumentation I

- [Maxwell bridge balance equation and suitability](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_i%3Amaxwell-bridge-inductance) — Instrumentation I, chapter 2
- [AC bridge balance and Schering bridge](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_i%3Aschering-bridge-capacitance) — Instrumentation I, chapter 2

## Related in other subjects

- [Self induction and inductance of a coil](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aself-induction-inductance-permeability) — Engineering Physics, chapter 8
- [Coil on 240 V, power, current, inductance](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Acoil-1-3kva-1-2kvar-240v) — Basic Electrical Engineering, chapter 6
- [Coil inductance and resistance from two frequencies](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Acoil-resistance-inductance-127hz-456hz) — Basic Electrical Engineering, chapter 6
- [Inductance and average induced emf of iron ring coil](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Airon-rod-ring-inductance-average-emf) — Electrical Machines I, chapter 1