Shaft Torque Sensor: Working Principle, 3 Coupling Architectures, Spec Sheet, Diagnostics

Updated 2026-05-11 by the Sino-Inst engineering team.

A shaft torque sensor sits between a motor and the load and reads the twisting force on the rotating shaft. The output is a 4-20 mA, frequency, CAN, or RS485 signal that a test bench, dynamometer, or process controller logs in real time. Most field failures on these instruments are not the sensor itself — they are misalignment, vibration coupling from the pump or motor, or a coupling that was the wrong choice for the speed and torque envelope.

This guide covers what a shaft torque sensor measures, the three working principles (strain gauge, SAW, magnetoelastic), the three mechanical coupling architectures you have to choose between, how to read a spec sheet, the three dominant failure modes, a 5-step diagnostic checklist, maintenance intervals by signal-coupling type, and the decision tree for re-zero / re-cal / replace.

Contents

Shaft torque sensor in drivetrain test bench

Shaft Torque Sensor Definition and Role in the Drivetrain

A shaft torque sensor — also called a rotary torque transducer or in-line torque sensor — measures the torsional moment T (in N·m or lbf·ft) that one side of the drivetrain delivers to the other while the shaft is turning at speed n. Inserted between the prime mover and the load, it sees real-time torque without disturbing the rotational coupling. The output reports both static torque (when the bench is locked) and dynamic torque (during transient acceleration, gear-shift, or stall).

The sensor’s job is to convert a mechanical twist into a clean electrical signal that survives slip rings, brush wear, or non-contact telemetry over the life of the test stand. For the broader concept of how torque transducers fit into instrumentation, see our torque transducer overview.

Working Principle: Strain Gauge, SAW, and Magnetoelastic

Three sensing technologies dominate shaft torque measurement. They differ by 10× in price, by signal coupling, and by what they tolerate at speed.

  • Strain gauge. Four resistive gauges bonded to the shaft in a Wheatstone bridge sense the surface strain caused by the torsional shear. Accuracy ±0.1 % FS, range 0.1 N·m to 100 kN·m. The signal must leave the rotating shaft via slip rings, brushless rotary transformer, or digital telemetry — that signal-coupling choice drives maintenance.
  • Surface Acoustic Wave (SAW). Piezoelectric resonators on the shaft surface shift frequency under stress. The signal is read non-contact via an RF antenna. No moving wires, no brushes, no slip-ring wear; accuracy ±0.5 % FS, range 1 N·m to 5 kN·m. Wider temperature window than strain gauge but more sensitive to electromagnetic interference.
  • Magnetoelastic. A ferromagnetic shaft region is magnetised; torsional stress changes its permeability, which a pickup coil reads. Non-contact, rugged, but accuracy is ±1 % FS at best. Used in automotive electric power steering (EPS) and high-throughput production lines where ±1 % is enough.

Three Mechanical Coupling Architectures: Inline, Flange, and Rotor-Stator

Once the sensing technology is fixed, the mechanical architecture decides how the sensor mounts in the drivetrain.

ArchitectureSpeed limitBest forTrade-off
Inline shaft-to-shaft15,000-30,000 rpmDyno test cells, gearbox R&DNeeds precise alignment; flex couplings essential
Flange-to-flange5,000-10,000 rpmMarine, wind turbine, large industrial drivesHeavy; pricey for low torque
Rotor-stator (split, non-contact)20,000+ rpmRetrofit on existing shafts, EPS, surface telemetryAir gap and runout matter; calibration is more complex

Pick inline for new builds where the test bench is being designed around the sensor. Pick flange-to-flange for big drives where the shaft is too large to interrupt. Pick rotor-stator when you cannot break the shaft and just need to clamp a stator around a magnetised section. For the broader test-bench torque measurement context, see our stationary vs rotary torque sensors guide.

Measurement Range, Accuracy, and the Spec Sheet Decoded

Five spec-sheet fields decide whether a shaft torque sensor fits the loop:

  1. Nominal torque (T_nom) and overload limit. Spec the sensor for 1.3-1.5 × your worst-case dynamic peak, not your steady-state. An overload limit of 200 % of T_nom is the industry standard but check the catalog — some economy units stop at 130 %.
  2. Accuracy class. ±0.05 % FS is custody-grade. ±0.2 % FS is standard for dyno work. ±1 % FS is production-line acceptable. Read whether the figure is “linearity” alone or “linearity + hysteresis + repeatability combined” — the latter is the honest one.
  3. Speed rating and bandwidth. Maximum continuous rpm and signal bandwidth (Hz). A bandwidth lower than 2× your shaft frequency aliases — at 3000 rpm = 50 Hz fundamental, you need at least 100 Hz bandwidth to catch the twist signature.
  4. Signal coupling and output. Slip ring, rotary transformer, RF telemetry, or SAW interrogation. Output: ±10 V analogue, 4-20 mA, frequency (PWM), or digital (CAN/Modbus/Profinet).
  5. Environment. Operating temperature, IP rating, vibration class, and EMC compliance. A test cell next to a 200 kW VFD will spray broadband noise; pick a sensor with explicit EMC class B compliance and shielded cabling.

The calibration certificate that ships with the sensor is your traceability proof. For the calibration practices that keep the certificate meaningful between annual recals, see our note on how to calibrate field instruments — the principles transfer directly.

Three Dominant Failure Modes and What They Look Like on the Trace

Failure modeSymptom on the traceRoot causeRecoverable?
Zero driftStatic reading creeps away from 0 over hoursStrain gauge adhesive aging, slip-ring contact resistance, temperature gradient on the bodyYes — re-zero, then re-cal if it returns
Span lossSensor reads only 80-90 % of the dyno’s calibrated torqueCracked gauge bond, partial slip-ring brush wear, SAW antenna detuning from cable shiftSometimes — recal often masks the underlying damage
Noise / oscillation30-200 Hz pickup on the trace, no actual torque changeVibration coupling from upstream pump or motor; ground-loop noise in the signal cableYes — isolate or move the cable
Catastrophic breakReading flatlines or saturates to ±FSOverload event, fatigue crack through gauge, shaft seizureNo — sensor is end-of-life

5-Step Diagnostic Checklist When Readings Look Wrong

  1. Re-zero at no-load. Disengage the load, let the shaft turn at idle for 60 seconds, command a zero. If the trace returns to 0.000 N·m and stays there for 5 minutes, the offset is just thermal drift — not a fault.
  2. Shunt-cal check. Push the shunt-cal button or send the shunt-cal command. The sensor should output its rated shunt value within ±0.5 %. If it does not, the span path (bridge resistance, signal-conditioner gain) is degraded.
  3. Two consecutive run-ups. Run the dyno to nominal torque twice in succession. If the second run reads more than 0.2 % FS different from the first, you have hysteresis from a sticky coupling or a strain-gauge crack.
  4. Alignment and runout check. Dial-indicate the input and output shafts at the coupling. More than 0.05 mm TIR loads the sensor body with bending moments that read as false torque.
  5. Cross-check with motor electrical torque. A modern VFD reports motor torque from current and rotor angle to ±5 %. If the dyno torque sensor diverges from the VFD-derived torque by more than 5 % across the speed range, suspect the sensor, not the motor.

Maintenance Intervals by Signal-Coupling Type

  • Slip-ring + brush coupling: Inspect brushes every 2000 operating hours; replace at 5000 hours or 25 % brush remaining. Clean slip ring with isopropyl alcohol monthly. Recal annually.
  • Rotary-transformer (brushless) coupling: Inspect every 8000 hours; no consumable parts. Recal every 18-24 months unless mounted in harsh vibration.
  • RF telemetry coupling: Battery or inductively powered. Battery-powered units need pack replacement at 2-3 years. Antenna alignment check yearly. Recal every 24 months.
  • SAW (non-contact, no power on rotor): No rotor maintenance. Stator antenna realign every 12 months or after any motor swap. Recal every 24-36 months — drift is mostly in the interrogator electronics, not on the rotor.

For the broader installation hygiene that applies to any rotating-shaft instrument — vibration isolation, cable routing, EMC — see our pressure transmitter installation guide. The signal-conditioner mounting principles transfer directly.

When to Re-Zero, When to Re-Cal, When to Replace

  1. Re-zero when the offset is less than 0.5 % FS and stable after a 60-second idle. Operators can do this themselves. Trend the offset week-on-week; growing drift is the early signal of a slip-ring or adhesive problem.
  2. Re-cal when the offset is between 0.5 % and 2 % FS, or when shunt-cal disagrees with the certificate by more than 0.5 %. Requires a calibration arm and traceable weights, typically a service-house job.
  3. Replace when the sensor saturates, when two run-ups disagree by more than 1 % FS at nominal torque, when shunt-cal fails completely, or when the unit has seen an overload above 150 % T_nom. Once the strain bond or SAW resonator is cracked, recal cannot restore traceability.

Featured Shaft Torque Sensors

807 Rotary Torque Sensor (15,000 rpm)

Inline strain-gauge, ±0.1 % FS, rotary-transformer signal coupling, ±10 V / 4-20 mA / frequency output.

120 Reaction Torque Sensor

Static torque wrench / motor / engine test, ±0.5 % FS, 0-1000 N·m, 4-20 mA + Modbus.

56 Micro Reaction Torque Sensor

Small-torque bench, ±0.2 % FS, 0-10 N·m, low-noise strain-gauge bridge.

FAQ

How does a shaft torque sensor work?

Three principles dominate: strain gauges in a Wheatstone bridge sense torsional surface strain; SAW resonators shift frequency under torque; magnetoelastic sensors detect permeability changes in a magnetised shaft region. The signal leaves the rotating element via slip rings, rotary transformer, RF telemetry, or non-contact SAW interrogation.

What is a shaft torque sensor?

A rotary torque transducer inserted in a drivetrain that measures the torque transmitted through the shaft while it is turning. Output is ±10 V, 4-20 mA, frequency, or digital (CAN/Modbus).

What is shaft torque?

The torsional moment T = F·r transmitted around the axis of a rotating shaft, measured in N·m (SI) or lbf·ft (US). Power transmitted is P = T × ω, where ω is angular velocity in rad/s.

What is the difference between an industrial shaft torque sensor and an automotive steering shaft torque sensor?

Industrial sensors are precision test instruments — ±0.1 to 0.5 % FS, recalibrated annually, used on dynos and R&D benches. Automotive EPS sensors are embedded in the steering column to feed the electric power steering ECU. They are typically magnetoelastic, ±1-2 % FS, and not designed for traceable measurement.

How is a shaft torque sensor installed?

Insert between input and output shafts using flexible couplings on both sides to absorb alignment error. Keep the input-output runout under 0.05 mm TIR. Route signal cable away from motor power leads, and ground at the conditioner end only to avoid ground loops.

What measurement range does a shaft torque sensor cover?

Strain-gauge sensors span 0.01 N·m (micro reaction) to 100 kN·m (marine drive). SAW typically 1 N·m to 5 kN·m. Spec the nominal torque at 1.3-1.5 × your worst-case peak, not your steady-state.

How often should a shaft torque sensor be calibrated?

Annually for slip-ring units, every 18-24 months for rotary-transformer and RF telemetry, every 24-36 months for SAW. Always recal after any overload event above 150 % nominal torque.

Need help picking a shaft torque sensor for your test bench or drive? Our engineers can quote and ship within 24 hours — message us with peak torque, shaft speed, accuracy band, and signal-coupling preference.

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Torque Transducer Selection: Reaction vs Rotary, Slip Ring vs Telemetry, and Accuracy Class

Updated: May 10, 2026 — by Sino-Inst Engineering Team

For shaft-mounted sensing on rotating drivetrains — strain gauge, SAW, magnetoelastic, and the maintenance/diagnostic playbook — see our shaft torque sensor guide.

A torque transducer turns mechanical torque on a shaft or flange into a calibrated electrical signal — usually 4-20 mA, 0–10 V, or a digital frame. The wrong choice is expensive: a reaction sensor on a rotating dyno cannot read the torque you actually need, and a high-bandwidth rotary sensor on a static lab fixture is paying for capability you will never use. This page is the decision tree we hand new application engineers: reaction vs rotary first, then signal-transmission type within rotary, then accuracy class, then four install pitfalls that crash commissioning. Wheel-torque, aerospace and automotive applications get their own callouts because the duty cycle changes everything.

Contents

Reaction vs Rotary: Decision Matrix for Test Rigs

The first question is not about the brand. It is whether the torque measurement has to happen on a rotating shaft (rotary) or can be picked off a stationary flange (reaction). The two architectures are different machines with different failure modes.

TypeHow it worksSpeed limitTypical useWhere it fails
Reaction (static)Strain-gauged element fixed between motor casing and frame; measures the reaction torque the casing tries to twist0 rpm — sensor body does not turnHand-tool calibration, screwdriver QC, motor bench tests where dynamic response below 10 Hz is enoughMisses transient peaks faster than 10 Hz; reaction frame compliance leaks signal
Rotary (in-line)Strain-gauged shaft section runs in line with the load; signal exits via slip ring, telemetry, or rotary transformerUp to ~50,000 rpm with telemetry, ~5,000 rpm with slip ringEngine dyno, transmission test, anything with peak torque events > 100 HzBearing wear, alignment-induced bending error, signal coupling drop-out
Reaction with rotation isolationReaction body with bearings on both sides; lets the shaft turn through but reads only torsionUp to ~10,000 rpmCompromise option for benches that change between static and rotary testsBearing drag adds zero offset — must re-zero at each rpm

Rule of thumb: if the application is an engine, gearbox or driveshaft test, you need rotary. If it is a torque-wrench check, a hand-tool calibration jig, or a motor-bench locked-rotor test, reaction is cheaper, simpler, and more accurate at zero rpm. The middle case (a bench that needs to do both) is where engineers usually overspend — most teams are better off with two dedicated sensors than one compromise unit. The same “match-the-tool-to-the-duty” logic appears in our note on selecting and installing pressure transmitters.

Signal Transmission in Rotary Types: Slip Ring, Telemetry, SAW, Rotary Transformer

Once the rotary box is checked, the second decision is how the signal leaves the spinning shaft. Each architecture trades RPM, accuracy and maintenance differently.

CouplingRPM ceilingAccuracy classService intervalBest for
Slip ring (brush)~5,000 rpm0.1–0.25 %500–1,000 h brush replacementLab benches, low-cost dynos, intermittent duty
Rotary transformer~15,000 rpm0.1–0.2 %Bearings only; no contact wearProduction dyno cells, automotive end-of-line
Digital telemetry (radio / inductive)~50,000 rpm0.05–0.1 %Battery / inductive supply; bearings onlyHigh-speed turbo, e-motor dyno, wind-turbine drivetrain
SAW (Surface-Acoustic-Wave)~30,000 rpm0.5–1.0 % (improving)None — no electronics on shaftOEM-built drivetrains, harsh ambient (heat, dust, vibration)

Slip-ring sensors are still common on low-RPM workbenches because they are cheap, but anyone running 24/7 production should look hard at telemetry — the bandwidth is higher and the brush-change downtime disappears. SAW transducers are the rising option for OEM-integrated sensing where you cannot bond strain gauges to the customer’s shaft; today’s accuracy is lower but trending toward 0.2 % class.

Accuracy Class and the Spec-Sheet Lines That Mislead

Datasheets bundle several error sources under the headline “accuracy class”. Pull them apart before you compare quotes.

  1. Combined error vs linearity-only. A 0.05 % combined-error sensor includes linearity, hysteresis, repeatability and temperature effect. A 0.05 % linearity-only number is meaningless without the rest. Insist on the combined-error figure.
  2. Of-reading vs of-full-scale. A 200 Nm sensor with 0.1 % FS is ±0.2 Nm everywhere. A 0.1 % of-reading sensor is ±0.02 Nm at 20 Nm — ten times tighter at the bottom of the range. Most low-torque QC checks need of-reading.
  3. Temperature coefficient. Look for two numbers: TC of zero and TC of span, both in % FS / °C. A 0.005 %/°C span TC over a 30 °C swing is 0.15 % — already worse than a 0.1 % accuracy claim if the bench is not climate-controlled.
  4. Overload survival. Static overload (limit of zero shift) is usually 150 % FS; mechanical overload (catastrophic failure) is 200–300 % FS. Test fixtures with sudden engagement need at least 200 % static.
  5. Calibration traceability. A DKD or NIST-traceable cert with at least 5 calibration points and uncertainty < 1/3 of the sensor accuracy is the minimum for any custody-grade or warranty-grade work.

The same way a pressure transmitter gets oversized when teams confuse FS error with reading error, torque sensors get oversized when teams pick a 1 kNm range to “have headroom” and then fight 0.5 % FS error at the 100 Nm working point. Right-size the range first; the accuracy class follows. Our walkthrough on DP transmitter installation uses the same “size to the working point, not the headroom” principle.

Four Install Pitfalls That Crash Commissioning

  • Coupling misalignment. A rotary torque sensor is intolerant of bending and side load. Use a double-flexure or bellows coupling on each side, with maximum 0.05 mm parallel and 0.05° angular misalignment. Rigid couplings inject side load that shows up as zero drift correlated with rpm — the symptom that wastes weeks of debugging.
  • Overload during start-up transients. A 200 Nm sensor on a 50 kW motor will see 5–8× nominal torque at locked-rotor start. Add a soft-start, a slip clutch, or oversize the sensor for the transient — not the steady-state.
  • EMC pickup. VFD-driven dynos radiate broadband noise on the strain-gauge cable. Use double-shielded cable, ground the shield at the amplifier end only, and keep the cable away from the motor power leads — separation by at least 30 cm or in a separate cable tray.
  • Forgetting the temperature-rise during burn-in. A torque sensor that runs for 8 hours will heat 5–10 °C from bearing friction even with no external load. Re-zero after the bench reaches thermal equilibrium, not at cold start.

Application Matrix: Dyno, Gearbox QA, Motor Test, Wind Turbine, Wheel, Aerospace

ApplicationArchitectureSignal couplingAccuracy classRange note
Engine dyno (cars / trucks)Rotary in-lineTelemetry or rotary transformer0.1 %Size for transient peak (~3× rated)
Gearbox end-of-line QARotary in-lineTelemetry0.05–0.1 %Need 1 kHz bandwidth for backlash inspection
Motor / e-motor benchReaction or rotary, depending on rpmTelemetry above 5 krpm0.05 % combinedWatch overload during locked-rotor
Wind-turbine drivetrainRotary in-line, hollow shaftTelemetry, IP670.1–0.2 %Size for hub-side peak gust torque
Wheel-torque (vehicle)Wheel-rim mounted, hollow flangeTelemetry (battery or inductive)0.1–0.25 %Size for braking peak (5–10× steady torque); IP69K for road testing
Aerospace benchRotary in-line, often non-contactDigital telemetry, redundant0.05 % combinedOften dual-channel sensors for safety case; AS9100 calibration
Hand-tool calibrationReactionHardwired0.1 % of readingNeed traceable 5-point cert
Screwdriver / fastener QCReaction (small, micro range)Hardwired0.25 % of readingWatch axial thrust pickup

Wheel-torque transducers deserve a separate note. They mount between the wheel hub and the rim, take road shock plus brake-temperature swings, and need IP69K sealing because they live below the splash line. Battery telemetry is normal because the rim cannot be slip-ringed. For an adjacent flow-side analogue of the “match the sensor to the working point, not the headroom” rule, see how we size straight-pipe runs ahead of flow meters.

Hollow-Type Reaction Torque Transducer

Flange-to-flange reaction body. 1–2,000 Nm, 0.1 % combined error, ±200 % overload. For motor benches and screwdriver QC stations.

901 Inline Contactless Rotary Torque Sensor

Rotary transformer coupling. 0.5 Nm to 5 kNm range, up to 15 krpm, 0.1 % accuracy. Built for production dyno cells and automotive end-of-line.

SI-T40B Digital Telemetry Torque Transducer

No bearings, no slip rings. Up to 50 krpm, 0.05 % combined error, digital frame output. Ideal for high-speed e-motor and turbo testing.

FAQ

What is a torque transducer?

A torque transducer is a sensor that converts mechanical torque on a shaft or flange into a calibrated electrical output (4-20 mA, 0–10 V, or digital). It is the primary instrument for engine dynos, hand-tool calibration, gearbox QA, and any work where torque is the controlled or measured variable.

What is the difference between a torque transducer and a torque sensor?

The two terms are used interchangeably in industry. Strictly, a “transducer” implies signal conditioning and a calibrated electrical output, while “sensor” can mean only the strain-gauge element. In practice both refer to the same instrument.

Which is better, slip ring or telemetry torque sensor?

Telemetry is better above ~5,000 rpm or for 24/7 production duty — no brush wear, no service interval, higher bandwidth. Slip ring is better for low-cost lab benches under 5,000 rpm with intermittent use.

How accurate are torque transducers?

Industrial-grade combined error runs 0.05 % to 0.5 % FS. Lab-grade reference transducers reach 0.02 %. Always confirm whether the spec is combined error or linearity-only, and whether it is of-reading or of-full-scale.

What is a wheel torque transducer?

A wheel torque transducer mounts between the vehicle wheel hub and the rim to measure driveshaft torque under real road conditions. It uses telemetry (no slip rings — the rim cannot be brushed), is sized for braking peak loads (5–10× steady torque), and needs IP69K sealing because it lives below the splash line. Common in vehicle dynamics, ABS calibration and tire-traction studies.

What is the maximum RPM for a rotary torque sensor?

Around 5,000 rpm for slip ring, 15,000 rpm for rotary transformer, and up to 50,000 rpm for digital telemetry. SAW sensors reach about 30,000 rpm. Above these limits, balancing and bearing life become the limit.

How do I calibrate a torque transducer in the field?

Use a calibration arm with a traceable mass, or a master torque sensor in series. Apply at least 5 ascending and 5 descending points to capture hysteresis. Lab calibrations use dead-weight benches per ASTM E2428 or ISO 6789.

What does “torque transducer accuracy class” mean?

It is the combined error (linearity + hysteresis + repeatability + temperature effect) expressed as a percentage of full scale. A 0.1 % class sensor has worst-case combined error of ±0.1 % FS over the rated temperature range.

Send the rated torque, the operating rpm, the application (dyno, hand-tool, gearbox, e-motor, wheel-torque), and the bench supply voltage. Sino-Inst engineers reply with a model number, a coupling recommendation, and a calibration certificate plan.

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Stationary Torque Transducers

Stationary Torque Transducers are also called Static Torque Transducers, or Stationary Transducers.

For the diagnostic playbook on rotating shaft sensors — 3 failure modes, 5-step checklist, and re-zero / re-cal / replace decision — see our shaft torque sensor diagnostic guide.

Static torque transducers are compact, robust and maintenance-free, offering many advantages. Sino-Inst’s Stationary Torque Transducers are based on strain gauge technology and provide high-precision signal transmission. Designed to perform static torque measurements and dynamic rotation (limited angles) with high precision in both clockwise and counterclockwise directions.

Featured Stationary Torque Transducer for Sale

Features

  1. It is suitable for detecting the torque value of non-rotating body. The maximum deflection angle is ≯360 degrees.
  2. It can enter the working state after turning on the machine, no preheating process is required.
  3. High detection accuracy, good stability and strong anti-interference.
  4. It can continuously measure positive and negative torque without repeated zero adjustment.
  5. Small size, light weight, easy to install.
  6. The sensor can be used independently from the secondary instrument, as long as the +24V power supply is provided according to the pin number of the socket, the current, voltage or frequency signal whose impedance is proportional to the torque can be output.

Stationary Torque Transducer Working principle

The static torque sensor has round shafts at both ends. It is praised by users for its excellent stability, high measurement accuracy, high cost performance and simpler monitoring method.

The measuring elastic body of the sensor does not participate in relative motion. The testing elastic body is subject to relative reaction force. When installing, one end is fixed and the other end is stressed.

Measurement of torque:
Using strain gauge electrical measurement technology, a strain bridge is formed on the elastic shaft, and the electric signal of the elastic shaft torsion can be measured by supplying power to the strain bridge.
After the strain signal is amplified, it becomes a frequency signal proportional to the torsional strain through voltage/frequency conversion.
Or directly output standard 4-20mA current signal and 1-5V voltage signal.

Read more about: Types Of Torque Transducers

More Torque Measurement Solutions

Sino-Inst is a manufacturer of Stationary Torque Transducers. We offer more than 20 types of torque transducers, 60% are dynamic torque sensors, 40% are stationary torque sensors.

For example: electric motor, engine, internal combustion engine, etc. Can be used to make viscometers.

The application range of Stationary Torque Transducers is mainly used in test systems such as experimental machines and static torque detection.

Sino-Inst’s Stationary Torque Transducers, made in China, have good quality, with better prices. Our Torque Sensors are widely used in China, India, Pakistan, USA and other countries.

Sino-Inst’s entire team is well trained, so we can ensure that each customer’s needs are met. If you need any help with your product requirements, whether it is a Stationary Torque Transducers, level sensors, or other equipment, please give us a call.

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