---
title: "Linearising capacitive displacement transducer"
subject: "Instrumentation I"
chapter: 3
key: "semester_4/instrumentation_i:capacitive-transducer-linearisation"
source: https://www.bctengineer.com/q/semester_4/instrumentation_i/capacitive-transducer-linearisation
site: BCT Engineering
---

# Linearising capacitive displacement transducer

Instrumentation I · Chapter 3: Transducer

- Asked in: year not given, 8 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/instrumentation_i/capacitive-transducer-linearisation
- Whole chapter: https://www.bctengineer.com/md/semester_4/instrumentation_i/chapter-3

## Question

Explain how non-linear characteristics of a capacitive transducer, used for the measurement of displacement using the principle of change in capacitance due to change in distance between plates, can be made linear.

## Short answer

Nonlinearity from C = εA/d is linearised by an op-amp with the transducer in feedback, giving Vo = −Vi·C1·d/(εA), linear in gap. Alternatively, a differential arrangement gives (C1−C2)/(C1+C2) = x/d, linear with doubled sensitivity and reduced errors.

## Answer

For parallel plates of area A at distance d with permittivity ε: **C = εA/d**. So C is inversely proportional to d, which makes it **non-linear** for a plate displacement x: C = εA/(d ± x).

The linearisation methods are as follows.

#### 1. Op-amp circuit with the transducer in the feedback path
A fixed capacitor C1 is placed at the input, and the displacement transducer Cx is placed in the feedback path.
```text
                 Cx = εA/d (transducer)
           +--------||--------+
           |                  |
 Vi o--||--+----(-)\          |
       C1           >---------+----o Vo
       gnd------(+)/
```
For an ideal op-amp: Vo = −Vi · (Z_f/Z_in) = −Vi · (1/jωCx)/(1/jωC1) = −Vi · C1/Cx.

Substituting Cx = εA/d:

**Vo = −Vi · C1 · d /(εA)**

Since C1, ε, A and Vi are constant, the output is **directly proportional to the gap d**, i.e. to the displacement. The output is therefore linear.

#### 2. Differential (push-pull) arrangement
Two fixed plates with a moving centre plate form C1 = εA/(d − x) and C2 = εA/(d + x). Using them in adjacent bridge arms, or taking the ratio:

(C1 − C2)/(C1 + C2) = x/d

This is linear in x. It also doubles the sensitivity, cancels even-order non-linearity, and reduces temperature and stray effects.

#### 3. Other ways
- Use variation of the overlapping area (sliding plates) instead, where C ∝ x is linear.
- Restrict the displacement to a small fraction of d (x ≪ d); the error is then small.

Method 1 or 2 is the usual textbook answer.

## Check yourself

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

1. Why is a capacitive transducer with varying plate distance non-linear?
   A. C is proportional to d
   B. C is inversely proportional to d (C = eps A/d)
   C. C is independent of area
   D. C depends on frequency only
   Answer: B. C = eps A/d, so capacitance varies inversely with the gap, not linearly with displacement. (Key point: Source of non-linearity)

2. In the op-amp linearisation method with C1 at the input and the transducer Cx in the feedback, Vo equals:
   A. -Vi Cx / C1
   B. -Vi C1 d/(eps A), proportional to the gap d
   C. Vi C1 eps A/d
   D. -Vi d/C1
   Answer: B. Vo = -Vi C1/Cx and Cx = eps A/d, so Vo is proportional to d. (Key point: Op-amp method)

3. For the differential capacitor C1 = eps A/(d - x), C2 = eps A/(d + x), the ratio (C1 - C2)/(C1 + C2) equals:
   A. d/x
   B. x/d
   C. 2x/d
   D. x^2/d^2
   Answer: B. The ratio simplifies to x/d, which is linear in x. (Key point: Differential arrangement)

4. Which alternative also gives a linear capacitance change?
   A. Increasing the gap greatly
   B. Using x much larger than d
   C. Varying the plate area (sliding plates), where C is proportional to x
   D. Changing frequency
   Answer: C. With area variation C is proportional to overlap, hence linear; small x relative to d also keeps error small. (Key point: Other methods)

## Related in Instrumentation I

- [Capacitive transducer for liquid level](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_i%3Acapacitive-liquid-level) — Instrumentation I, chapter 3

## Related in other subjects

- [Principle applied in two-way linear measurement](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atwo-way-linear-measurement-principle) — Surveying I, chapter 1
- [Principle of optical distance measurement](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aoptical-distance-measurement-principle) — Surveying II, chapter 2
- [Second-order non-linear kinematic wave; FDM principle](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Asecond-order-nonlinear-kinematic-principle) — Computational Techniques in Civil Engineering, chapter 5
- [Load characteristics and capacitive load voltage rise](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Aload-characteristics-capacitive-load) — Electric Machines II, chapter 1, asked 3×