Sensors

A transducer converts one form of energy or physical quantity into another, usually into an electrical signal. A sensor is the sensing element of a transducer.

Classification

1. By energy source: active (self-generating) and passive.
2. By function: primary (sensing) and secondary.
3. By output signal: analog and digital.
4. By principle: resistive, capacitive, inductive, piezoelectric, thermoelectric, photoelectric, Hall effect.
5. By quantity measured: displacement, force, pressure, temperature, flow and so on.
6. By operating mode: transducers (input-to-output converters) and inverse transducers.

Active and passive

Primary and secondary

• Primary transducer: first element in contact with the measured quantity; it converts it into a mechanical or other signal. Example: a bourdon tube turns pressure into displacement.
• Secondary transducer: converts the output of the primary into an electrical signal. Example: a potentiometer or LVDT attached to the bourdon tube gives voltage.

A load cell is a combination: an elastic member (primary) with strain gauges (secondary).

(a) Potentiometer transducer

A resistance potentiometer (pot) has a resistive element (wire-wound, carbon or conductive plastic) with a sliding contact (wiper) mechanically linked to the moving member. A supply $E_s$ is applied across the full length $L$ and the output is taken between the wiper and one end. With no load, the output is linear:

$$e_o = E_s\frac{x}{L}$$

   Es o----+----------+
           |   Rp     |
           |  ____    |
           +-|_||_|---+
              wiper ->|--- o e_o
   Displacement x  ---|

Advantages: simple, cheap, high output, no amplifier needed, works with dc, linear or rotary.

Limitations:

• Wear and friction of the sliding contact; limited life.
• Finite resolution for wire-wound types (one turn step).
• Loading error with finite load resistance.
• Limited frequency response because of wiper inertia; noise at contact.

(b) Loading error

With a load $R_m$ across the wiper output, and $x$ = fractional position:

$$\frac{e_o}{E_s} = \frac{x}{1 + x(1-x)\dfrac{R_p}{R_m}}$$

$R_p/R_m = 10/50 = 0.2$, $E_s = 10$ V.

Answer: $e_o = 4.76$ V at 50% (error 0.238 V) and $7.23$ V at 75% (error 0.271 V).

The loading error is largest at about 2/3 travel. To reduce it:

• Make $R_m \gg R_p$ (use a high-input-impedance voltmeter or buffer amplifier).
• Use a smaller $R_p$ (limited by power dissipation).
• Use a tapped (non-linear) pot to compensate.

A thermocouple produces an emf when two dissimilar metals joined at two junctions are at different temperatures (Seebeck effect).

Laws of thermocouples

1. Law of homogeneous circuit: a thermoelectric current cannot be sustained in a circuit of a single homogeneous metal by heat alone, however the section varies. A temperature gradient in a uniform wire adds no emf.
2. Law of intermediate metals: a third metal inserted in the circuit (e.g. the voltmeter leads) adds no net emf if its two junctions are at the same temperature.
3. Law of intermediate temperatures: if the emf between junction temperatures $T_1$ and $T_2$ is $E_{1,2}$ and between $T_2$ and $T_3$ is $E_{2,3}$, then $E_{1,3} = E_{1,2} + E_{2,3}$. This lets tables with a 0 $^\circ$C reference be used for any cold junction.

(a) Junctions at 300 $^\circ$C and 100 $^\circ$C

$$E_{100,300} = E_{0,300} - E_{0,100} = 12.209 - 4.096 = 8.113\ \text{mV}$$

(b) Hot junction temperature

The instrument reads 7.5 mV with cold junction at 30 $^\circ$C. Referred to 0 $^\circ$C:

$$E_{0,T} = E_{30,T} + E_{0,30} = 7.5 + 1.203 = 8.703\ \text{mV}$$

This lies between 200 $^\circ$C (8.138 mV) and 300 $^\circ$C (12.209 mV).

$$T = 200 + \frac{8.703 - 8.138}{12.209 - 8.138}\times 100 = 200 + \frac{0.565}{4.071}\times 100 = 213.9\ ^\circ\text{C}$$

Answer: (a) 8.113 mV; (b) $T \approx 213.9\ ^\circ$C.

(a) Thermistors

A thermistor (thermal resistor) is a semiconductor (sintered metal oxides of Mn, Ni, Co, Cu) whose resistance changes strongly with temperature. Its sensing element is a bead, disc or rod.

For an NTC thermistor, the resistance falls as temperature rises:

$$R = R_0\exp\left[\beta\left(\frac{1}{T} - \frac{1}{T_0}\right)\right]$$

where $T$ and $T_0$ are in kelvin. The temperature coefficient is $\alpha = \dfrac{1}{R}\dfrac{dR}{dT} = -\dfrac{\beta}{T^2}$.

Advantages: high sensitivity (about 4%/K), small size and fast response, high resistance so lead resistance is negligible.

Limitations: strongly nonlinear, small range (up to about 300 $^\circ$C), interchangeability poor, self-heating error.

(b) Numerical

Given $R_0 = 10$ k$\Omega$ at $T_0 = 298.15$ K, $\beta = 3950$ K.

At 50 $^\circ$C ($T = 323.15$ K):

$$R_{50} = 10{,}000\exp\left[3950\left(\frac{1}{323.15} - \frac{1}{298.15}\right)\right] = 10{,}000\,e^{-1.0249} = 3588\ \Omega$$

At 100 $^\circ$C ($T = 373.15$ K):

$$R_{100} = 10{,}000\exp\left[3950\left(\frac{1}{373.15} - \frac{1}{298.15}\right)\right] = 10{,}000\,e^{-2.6628} = 697.5\ \Omega$$

Temperature coefficient at 50 $^\circ$C:

$$\alpha = -\frac{3950}{323.15^2} = -0.0378\ \text{K}^{-1} = -3.78\%/\text{K}$$

Answer: $R_{50} = 3.59$ k$\Omega$, $R_{100} = 697.5\ \Omega$, $\alpha_{50} = -3.78\%/^\circ$C.

The LVDT is a passive inductive transducer that converts linear displacement into an ac voltage whose amplitude is proportional to the displacement and whose phase gives the direction.

Construction

• One primary winding at the centre and two identical secondary windings $S_1$, $S_2$ wound symmetrically on either side, on a hollow non-magnetic former.
• A movable ferromagnetic core (soft iron or nickel-iron) slides inside the former and is attached to the object whose displacement is measured.
• The primary is excited by a 1-10 V, 50 Hz-20 kHz ac supply. The two secondaries are connected in series opposition.

      +---------+--------+---------+
 ~E --|  S1     | Primary|   S2    |-- e_o = e1 - e2
      |  (((    |  (((   |   (((   |
      +----[=== core ===]----------+
                 <-- x -->

Working

The primary field induces emf $e_1$ and $e_2$ in the secondaries, and the output is $e_o = e_1 - e_2$.

• Core at centre (null position): equal flux links both secondaries, $e_1 = e_2$, so $e_o = 0$.
• Core moved toward $S_1$: $S_1$ gets more flux, so $e_1 > e_2$ and $e_o$ is in phase with the primary voltage.
• Core moved toward $S_2$: $e_2 > e_1$ and $e_o$ is $180^\circ$ out of phase with the primary.

The magnitude of $e_o$ is proportional to the displacement over the linear range. A phase-sensitive demodulator is used to obtain a dc signal with sign.

 e_o |      /
     |    /   <- linear range
 ----+--/---------- x
     |/ 
   (phase reversal at null)

Advantages

• Frictionless, infinite resolution, no contact wear, long life.
• High sensitivity (up to about 2 mV/V/mm); good linearity (about 0.25%).
• Rugged, tolerant to shock, vibration, dirt and environment.
• Small null error, and the core is free to move without touching the coils.

Limitations

• Needs ac excitation and a demodulator.
• Stray magnetic fields and temperature affect it; sensitivity drops at high frequencies (the carrier must be at least about 10 times the signal frequency).
• Limited range, and a residual voltage at null.

Applications

Displacement and position, thickness, force and pressure (with an elastic element such as a diaphragm or bourdon tube), acceleration, liquid level and load cells.

A variable-reluctance (inductance) transducer senses the measurand through a change in the inductance of a coil due to a change in the reluctance of its magnetic path.

Principle

The inductance of a coil of $N$ turns is

$$L = \frac{N^2}{\mathcal{R}}, \qquad \mathcal{R} = \frac{l_g}{\mu_0 A}\ \text{(air gap dominates)}$$

so $L = \dfrac{N^2\mu_0 A}{l_g}$. Changing the air gap $l_g$, the area $A$, or the permeability $\mu$ changes $L$.

Single-coil, change of air-gap type

A U- or E-shaped iron core carries the coil. A ferromagnetic armature is separated from it by a gap $l_g$. When the armature moves with the measurand, $l_g$ changes, so $L$ changes.

     +---[ coil ]---+
     |              |
   __|__          __|__   <- iron core
   |___|          |___|
      gap l_g (variable)
   ===================    <- movable armature
         ^ x

The relation $L \propto 1/l_g$ is nonlinear; it is approximately linear only for small gap changes. The iron armature also exerts a magnetic pull on the moving member.

Differential (push-pull) type

Two identical coils are placed on opposite sides of a common armature. When the armature moves up, the gap of one coil decreases ($L_1$ increases) and the other increases ($L_2$ decreases). The coils are placed in adjacent arms of an ac bridge, so the output is proportional to $L_1 - L_2$.

   [ coil 1 ]    gap 1
   ============  <- armature (moves)
   [ coil 2 ]    gap 2

Why differential is preferred

• Even-order nonlinearity cancels, giving better linearity and about twice the sensitivity.
• The magnetic pull forces on the armature cancel.
• Effects of temperature, supply variation and stray fields on both coils cancel in the bridge.

Typical uses: small displacement, pressure (diaphragm) and proximity sensing.