How does an LVDT measure displacement?

AC drives the central primary coil, and the voltages induced in two secondaries on either side are subtracted. With the iron core centred the two voltages are equal and the difference is zero; as the core moves toward one side, coupling to that secondary strengthens and a difference voltage proportional to core position appears. Core and coils never touch, so there is no wear and no fundamental resolution limit.

What sets the linear range?

First, core length and coil layout. Here (20 mm secondaries, 30 mm core) the geometric range is ±10 mm, until the core end reaches the secondary's end. In practice flux spreads at the core ends, so output departs from proportionality near the range ends: with 1.5 mm spreading, error is 0.46% at ±8 mm and 3.89% at ±10 mm.

Why use the ratio (difference/sum)?

Changes in excitation voltage, frequency or temperature that scale all coil voltages cancel when the difference is divided by the sum. A 2% excitation change shifts the difference output by 2% but leaves the ratio almost unchanged. In this simple model the sum of the two secondaries is constant with core position, so ratio and difference linearity match.

LVDT (Linear Variable Differential Transformer Displacement Sensor)

Category: Electromagnetic Analysis – Sensors | Revised 2026-10-01

CAE visualization for lvdt sensor theory - technical simulation diagram

LVDT (Differential Transformer Displacement Sensor)

Theory: two couplings that change with core position

Overview

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Can an LVDT's catalogue measuring range be used in full?

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Linearity often degrades at the ends. An LVDT works by an iron core sharing the primary's flux between two secondaries: as the core moves to one side, coupling to that side strengthens and the other weakens. While the core is well inside the coils, coupling changes in proportion to core travel and output is linear. As the core end approaches a coil end, proportionality breaks down — and because flux spreads out at the core ends, the breakdown begins before the geometric end. That spreading depends on core–coil gap and materials, so detailed work uses electromagnetic field analysis. Here the idea is shown with a simple overlap model.

Overlap model

$$ c(z) = \tfrac12\left[\tanh\frac{z + L_c/2}{w} - \tanh\frac{z - L_c/2}{w}\right],\qquad M_i(x) \propto \int_{\text{coil } i} c(z - x)\,dz $$

Secondaries are 20 mm long at [5, 25] mm and [−25, −5] mm with the primary between. Core length L_c = 30 mm. w is the length over which flux spreads at the core ends; as w → 0, coupling equals the overlap length. This is an illustrative stand-in for field analysis.

Output and linearity error

$$ D(x) = M_1 - M_2,\qquad Q(x) = \frac{M_1 - M_2}{M_1 + M_2},\qquad \varepsilon_{lin} = \frac{\max|D - kx|}{|kR|} $$

k is the slope of the best line through the origin over ±R; error is expressed relative to output at ±R.

Coffee Break Trivia

Aircraft control surfaces and LVDTs

In fly-by-wire aircraft, where control surfaces move on electrical commands, LVDTs commonly measure the position of hydraulic actuators for feedback. With no contact they don't wear, tolerate vibration and temperature changes, and rarely fail; multiple coil sets are often built into one actuator so position remains known if one fails. Simple construction and high reliability are why LVDTs keep being chosen decade after decade.

Worked examples

Example 1: linearity error by range and fringing

Fringing w	±4 mm	±6 mm	±8 mm	±10 mm
0.5 mm	0.00%	0.00%	0.00%	1.54%
1.5 mm	0.00%	0.04%	0.46%	3.89%
3.0 mm	0.19%	0.71%	2.28%	5.93%

Central sensitivity is nearly the same in all cases (2.0 per mm, normalised by coil length). The geometric range is ±10 mm, but within 0.5% error usable range is about ±8 mm at w = 1.5 mm and about ±5 mm at w = 3 mm.

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If larger fringing narrows the range, why not put the core right against the coil?

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A smaller gap does reduce fringing, but creates other problems. The core moves inside a tube; too small a gap and mounting misalignment or thermal expansion cause rubbing, losing the no-contact advantage. If the core runs off-centre, gap changes alter coupling and produce output changes unrelated to displacement. In practice designers lengthen the core so the working range sits nearer the coil centre, or wind coils more densely at the ends to compensate end coupling. Which measure works depends on core permeability, coil shape and excitation frequency (eddy currents push flux toward the surface), so field analysis is used to compute and verify the output curve. The overlap model here is for grasping beforehand that range ends can't be used, and roughly how much.

Modelling workflow

Estimate geometric range from core length and coil layout.
Use axisymmetric field analysis to get secondary voltages versus core position.
Include eddy currents at excitation frequency and core permeability.
Compute linearity of difference and ratio outputs and set usable range.
Vary core eccentricity and temperature to check output changes.

Coffee Break Trivia

“A sensor whose zero shifted after warm-up”

On one test machine, the LVDT zero drifted gradually as the equipment warmed. Investigation found the two secondaries had slightly different turn counts; as copper resistance changed with temperature, the phases of the two voltages diverged and the difference never fully reached zero. Processing output as a ratio (difference/sum) with phase-matched demodulation reduced temperature-induced zero drift.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Using the full geometric range	Linearity drops at ends	Set range from error
Difference output only	Output varies with excitation	Use ratio output
Ignoring eddy currents	Sensitivity varies with frequency	Solve at excitation frequency
Ignoring core eccentricity	Output change unrelated to motion	Sweep eccentricity
Ignoring phase	Residual or drifting null	Phase-matched demodulation

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I'd like to learn related topics.

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Related pages include magnetic encoder, eddy-current sensor, Hall-effect sensor, Rogowski coil and search coil.

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