Capacitive & Inductive Sensors for GATE RA 2027: Principles, LVDT & Signal Conditioning

Capacitive and inductive sensors are two of the most important non-contact sensing technologies in the GATE Robotics and Automation (RA) 2027 syllabus. They appear under the “Sensors” line of the syllabus alongside signal-conditioning circuits, and they are favourites for both conceptual and numerical questions. This guide explains how capacitive and inductive sensors work, derives their governing equations, walks through solved examples, and shows the signal-conditioning circuits you must know for the exam.

Capacitive sensors: working principle

A capacitive sensor is essentially a parallel-plate capacitor whose capacitance changes with the measurand. The governing relationship is:

where C is capacitance (F), ε₀ = 8.854 × 10^-12 F/m is the permittivity of free space, ε_r is the relative permittivity of the dielectric, A is the overlapping plate area (m²), and d is the gap between plates (m). Any measurand that changes A, d, or ε_r changes the capacitance, and this is the basis of all capacitive sensing.

Three modes of capacitive sensing

• Varying distance (d): Moving one plate changes the gap. This gives a non-linear (hyperbolic) response C ∝ 1/d, most sensitive at small gaps. Used for small displacement and pressure diaphragms.
• Varying area (A): Sliding or rotating plates change the overlap. This gives a linear response C ∝ A. Used for linear and angular displacement.
• Varying dielectric (ε_r): Moving a dielectric slab between plates, or changing the medium, changes ε_r. Used for level sensing, humidity, and material analysis.

Worked example: capacitive displacement

Inductive sensors: working principle

Inductive sensors convert displacement into a change of inductance or mutual inductance. The self-inductance of a coil is:

where N is the number of turns, μ the permeability of the magnetic path, A the cross-sectional area, l the magnetic path length, and ℝ = l/(μA) the reluctance. Changing the reluctance — usually by moving a ferromagnetic core — changes L. This is the principle of variable-reluctance and LVDT sensors.

LVDT and variable-reluctance sensors

The Linear Variable Differential Transformer (LVDT) is the most examined inductive sensor. It has one primary winding and two identical secondary windings connected in series opposition, with a movable ferromagnetic core.

At the central (null) position the two secondary EMFs cancel and the output is zero. As the core moves, one secondary couples more flux than the other, so the differential output |V_s1 − V_s2| rises linearly with displacement, while the phase (0° or 180°) tells you the direction. Key LVDT advantages: frictionless, infinite resolution, high linearity (±0.25%), and robustness — which is exactly why examiners love it.

Eddy-current proximity sensors

An eddy-current sensor drives a coil with a high-frequency AC current, creating an alternating magnetic field. When a conductive target approaches, circulating eddy currents are induced in it; these oppose the coil field, reducing the coil’s effective inductance and increasing its apparent resistance. The change is detected as a shift in oscillator amplitude or frequency. Eddy-current sensors are non-contact, work only on conductive targets, and are widely used as industrial proximity switches and for vibration/shaft-position measurement.

Signal conditioning circuits

Because capacitance and inductance are reactive, they cannot be read with a simple DC bridge. The standard signal-conditioning approaches you should know are:

• AC bridge (Wheatstone-type with reactive arms): the sensor forms one arm of an AC-excited bridge; the bridge is balanced at null and the off-null voltage is proportional to the change in C or L.
• Phase-sensitive (synchronous) detector: demodulates the AC bridge output using a reference locked to the excitation. This recovers both magnitude and sign (direction) — essential for the LVDT.
• Oscillator / tank circuit: the sensor sets the frequency of an LC or RC oscillator (f = 1/2π√(LC)); the measurand is read as a frequency shift. Common for eddy-current and capacitive proximity switches.
• Charge amplifier: converts the very small capacitance/charge changes into a usable voltage with a high-input-impedance op-amp.

Capacitive vs inductive: comparison

Key formulas

Common mistakes to avoid

• Assuming the gap-type capacitive sensor is linear — it is non-linear (C ∝ 1/d). Only the area-type is linear.
• Forgetting that an LVDT needs a phase-sensitive detector to resolve direction; a plain rectifier loses the sign.
• Using a DC bridge for capacitive/inductive sensors — they require AC excitation.
• Believing eddy-current sensors work on any material — they need a conductive target.

Frequently asked questions

Why are capacitive gap sensors non-linear?

Because capacitance varies inversely with the gap (C = εA/d), so a fixed change in d produces a larger change in C at small gaps than at large gaps. The response is hyperbolic, not straight-line. Linearising circuits or area-varying geometry are used when a linear output is required.

What is the main advantage of an LVDT over a potentiometer?

The LVDT has no sliding electrical contact, so there is no friction and no wear. This gives it effectively infinite resolution, long life, and excellent repeatability — whereas a resistive potentiometer wears out and adds mechanical load.

Do inductive proximity sensors detect plastic or wood?

No. Inductive and eddy-current sensors rely on inducing currents in a conductive/metallic target. For non-conductive materials such as plastic, wood, liquid or grain, a capacitive proximity sensor is used instead.

How is direction of motion recovered from an LVDT?

By phase-sensitive (synchronous) detection. The output magnitude gives displacement; the phase relative to the excitation (0° vs 180°) tells whether the core moved above or below the null point.

This guide is part of the complete GATE RA 2027 Syllabus overview. Continue with the previous sensor guide on Resistive Sensors & Signal Conditioning, and review the fundamentals in Basics of Mechatronics.