---
title: "Capacitive transducer for liquid level"
subject: "Instrumentation I"
chapter: 3
key: "semester_4/instrumentation_i:capacitive-liquid-level"
source: https://www.bctengineer.com/q/semester_4/instrumentation_i/capacitive-liquid-level
site: BCT Engineering
---

# Capacitive transducer for liquid level

Instrumentation I · Chapter 3: Transducer

- Asked in: year not given, 6 marks
- Video: [Liquid Level Measurement Using Capacitive Transducer | Sensors And Transducers](https://www.youtube.com/watch?v=pfsHEJWoJpc) — ENGINEERING TUTORIAL
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/instrumentation_i/capacitive-liquid-level
- Whole chapter: https://www.bctengineer.com/md/semester_4/instrumentation_i/chapter-3

## Question

Show how the capacitive transducer can be used for the measurement of liquid level.

## Short answer

A cylindrical capacitor (inner rod, outer tube) has capacitance C = [2πε0/ln(b/a)]·[H + (εr−1)h], a linear function of liquid height h. The change is measured via AC bridge or oscillator and calibrated for level; suitable for corrosive liquids with no moving parts.

## Answer

A capacitive transducer can measure liquid level because the capacitance of a probe depends on how much of it is surrounded by liquid, whose dielectric constant differs from that of air.

#### Arrangement
```text
          |  <- insulated rod (inner electrode, radius a)
   +------|------+  <- metal tank / outer tube (radius b)
   |      |      |
   |  air |      |  height (H - h), εr = 1
   |~~~~~~|~~~~~~|  <- liquid surface
   |liquid|      |  height h, εr of the liquid
   +------|------+
```
A cylindrical capacitor is formed by an inner rod (electrode) and an outer concentric tube (or the metal tank wall). The liquid fills the space between them to height h; the total height is H. For a non-conducting liquid, the liquid itself is the dielectric.

#### Capacitance
The capacitance is that of the part in the liquid plus the part in air (two capacitors in parallel):

C = 2πε0 εr h / ln(b/a) + 2πε0 (H − h) / ln(b/a)

**C = [2πε0 / ln(b/a)] · [H + (εr − 1) h]**

Here εr is the relative permittivity of the liquid. C is a **linear function of h**: C = C0 + k h, where C0 = 2πε0H/ln(b/a) is the empty-tank capacitance and k = 2πε0 (εr − 1)/ln(b/a).

#### Measurement
- The change in capacitance is measured with an AC bridge, an oscillator (whose frequency changes) or a capacitance-to-voltage converter, and the display is calibrated in level.
- For a conducting liquid, the rod is insulated (e.g. Teflon coated) and the liquid acts as the second plate; the capacitance again increases linearly with h.

**Advantages:** no moving parts, suitable for corrosive liquids and for remote indication. **Limitation:** εr changes with temperature and with the type of liquid, so calibration is needed.

## Check yourself

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

1. Why does a capacitive probe measure liquid level?
   A. Liquid changes the plate area only
   B. Capacitance depends on how much of the probe is in liquid, whose dielectric constant differs from air
   C. Liquid reduces the supply voltage
   D. Liquid conducts current to the rod always
   Answer: B. The liquid-filled and air-filled parts act as capacitors in parallel with different permittivities. (Key point: Principle)

2. The capacitance of the probe for liquid height h is:
   A. C = [2 pi e0 / ln(b/a)]·[H + (er - 1) h], linear in h
   B. C proportional to 1/h
   C. C proportional to h^2
   D. C independent of h
   Answer: A. Adding the liquid part and the air part gives C = C0 + k h with k = 2 pi e0 (er - 1)/ln(b/a). (Key point: Capacitance expression)

3. What is a limitation of this method?
   A. It has moving parts
   B. It cannot be used remotely
   C. It cannot be used for corrosive liquids
   D. The relative permittivity changes with temperature and liquid type, so calibration is needed
   Answer: D. No moving parts and suitability for corrosive liquids are advantages; er variation requires calibration. (Key point: Limitation)

## Related in Instrumentation I

- [Linearising capacitive displacement transducer](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_i%3Acapacitive-transducer-linearisation) — Instrumentation I, chapter 3

## Related in other subjects

- [Calibration line and nonlinearity of transducer](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Acalibration-least-squares-nonlinearity) — Instrumentation and Measurement, chapter 3
- [Second-order transducer frequency and step response](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Asecond-order-transducer-numerical) — Instrumentation and Measurement, chapter 4
- [Potentiometer transducer and loading error](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Apotentiometer-transducer-loading-error) — Instrumentation and Measurement, chapter 5
- [Liquid rise in rotating U-tube](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Au-tube-rotating-liquid-rise) — Fluid Mechanics, chapter 2