Displacement and position sensors: LVDT, potentiometer, encoders

Why it matters

Almost every mechatronic axis — a CNC slide, a robot joint, a valve stem — closes its loop on a position measurement. The choice between a potentiometer, an LVDT and an encoder sets the achievable resolution, life, robustness and cost of the whole machine.

Key ideas

Resistive potentiometer

• A resistive track (wire-wound, cermet or conductive plastic) with a sliding wiper; linear and rotary (single or multi-turn) forms.
• Supplied with V_s across the track, the wiper voltage is a fraction of V_s equal to the fractional travel x — an absolute, analogue position signal that is cheap and simple.
• Limits: contact wear and electrical noise, finite resolution in wire-wound types (one turn of wire), and loading error when the meter or ADC input resistance R_L is not much larger than the pot resistance R_p. The loading error is zero at both ends and largest near x ≈ 2/3 of travel.

Linear variable differential transformer (LVDT)

• One primary coil and two identical secondaries on a former, with a movable ferromagnetic core. The primary is excited with AC (typically 1–10 kHz).
• The secondaries are connected in series opposition, so the output is the difference of the two induced voltages. At the null (core centred) the two are equal and the output is ideally zero; in practice a small residual null voltage remains due to harmonics and stray capacitance.
• Moving the core increases coupling to one secondary and reduces it to the other, so the output amplitude rises linearly with displacement over the rated range, and the phase flips by 180° as the core crosses null. A phase-sensitive (synchronous) demodulator converts this to a DC voltage whose sign gives direction.
• Non-contact, so no friction or wear, essentially infinite resolution (limited by electronics), rugged; but needs AC excitation and demodulation, and the core adds mass. The RVDT is the rotary version.

Encoders

• Incremental: a disc (rotary) or scale (linear) with N lines read optically or magnetically gives pulse trains A and B 90° out of phase (quadrature) plus an index pulse Z once per revolution. Counting edges gives relative position; the lead/lag of A and B gives direction. Counting both edges of both channels (×4 decoding) gives 4N counts per revolution. Position is lost at power-off, so a homing routine is needed.
• Absolute: each position has a unique n-bit code (usually Gray code so only one bit changes between neighbours, avoiding large read errors at transitions). Resolution is 2ⁿ positions per revolution, and position is known immediately at power-up. Multi-turn versions also count revolutions.
• Speed is obtained by counting pulses in a fixed time (good at high speed) or timing between pulses (good at low speed).

Others to know by name: capacitive displacement sensors (sub-micron gaps), eddy-current probes (shaft runout), magnetostrictive linear transducers (long hydraulic cylinders), resolvers (robust rotary feedback for servo motors).

Formulas

V_o = x·V_s (unloaded potentiometer)

• V_o: output (V), V_s: supply (V), x: fractional wiper travel (0–1). Valid when R_L ≫ R_p.

V_o = x·V_s / [1 + (R_p/R_L)·x·(1 − x)] (loaded potentiometer)

• R_p: total pot resistance (Ω), R_L: load resistance (Ω).

V_o = S·d (LVDT, within linear range)

• S: sensitivity (V/mm, often given per volt of excitation as mV/V/mm), d: core displacement from null (mm). Magnitude only; sign comes from phase.

Counts per revolution = 4·N (quadrature ×4 decoding)

• N: lines (pulses) per revolution, PPR.

Angular resolution = 360°/(4N) (incremental, ×4) ; = 360°/2ⁿ (n-bit absolute)

Linear resolution = p / (counts per revolution)

• p: lead of the screw (mm/rev) for a rotary encoder on a lead-screw drive.

n_rpm = 60·f / (counts per revolution)

• f: count frequency (counts/s). Shaft speed from encoder pulse rate.

Worked examples

Example 1 (standard) — potentiometer loading. A 10 kΩ linear pot is supplied with 10 V. The wiper is at 40 % of travel and drives a meter of 20 kΩ input resistance. Find the reading and the loading error.

1. Ideal output: V_o = x·V_s = 0.4 × 10 = 4.00 V.
2. Loaded output: V_o = x·V_s / [1 + (R_p/R_L)·x(1 − x)].
3. R_p/R_L = 10/20 = 0.5; x(1 − x) = 0.4 × 0.6 = 0.24; denominator = 1 + 0.5 × 0.24 = 1.12.
4. V_o = 4.00/1.12 = 3.571 V.
5. Error = 4.00 − 3.571 = 0.429 V = 10.7 % of reading = 4.3 % of full scale.
6. Answer: reading ≈ 3.57 V, loading error ≈ 0.43 V. A buffer amplifier or R_L ≥ 10·R_p cures it.

Example 2 (GATE level) — encoder on a lead screw. A 1000-PPR incremental encoder on a ball screw of 5 mm lead is decoded ×4. Find (a) the linear resolution, (b) the table speed when the counter sees 200 000 counts/s, and (c) the bits needed for an absolute encoder to give at least the same angular resolution.

1. Counts/rev = 4 × 1000 = 4000.
2. (a) Resolution = 5 mm / 4000 = 0.00125 mm = 1.25 µm.
3. (b) Shaft speed = 200 000 / 4000 = 50 rev/s = 3000 rpm; table speed = 50 × 5 = 250 mm/s.
4. (c) Need 2ⁿ ≥ 4000 → n = 12 (4096 positions, 0.088° per step).
5. Answers: (a) 1.25 µm, (b) 250 mm/s (3000 rpm), (c) 12 bits.

Example 3 (quick). An LVDT with sensitivity 40 mV/mm gives 0.6 V (in phase with the excitation reference). Displacement = 0.6/0.040 = 15 mm on the in-phase side of null.

Common mistakes

• Treating LVDT output as DC or ignoring phase: amplitude alone cannot tell which side of null the core is on.
• Writing the LVDT output in terms of turns ratio — the turns are fixed; it is the core position that changes the mutual inductances.
• Forgetting ×4 decoding (or applying it when the question says ×1 counting).
• Assuming an incremental encoder knows position at power-up.
• Ignoring potentiometer loading when the load resistance is comparable to the pot resistance.
• Mixing pitch and lead on multi-start screws: linear travel per revolution is the lead.

For GATE ME

Typical items: encoder resolution and speed from pulse counts, bits needed for an absolute encoder, loaded potentiometer output, LVDT displacement from output and sensitivity, and MCQs on LVDT null/phase behaviour and incremental vs absolute encoders. Practise the loading formula and unit conversions between counts, degrees, mm and rpm.

Quick check

1. Why are LVDT secondaries connected in series opposition?
2. A 500-PPR encoder is read with ×4 decoding. How many counts per revolution?
3. Why is Gray code used in absolute encoders?
4. At what positions is potentiometer loading error zero?
5. What information do channels A and B together give that one channel cannot?

Answers: 1. So the output is the difference of the induced voltages: zero at null, rising with displacement, with phase showing direction. 2. 2000. 3. Only one bit changes between adjacent positions, avoiding large misreads at transitions. 4. At both ends of travel (x = 0 and x = 1). 5. Direction of motion (and ×4 resolution).