Resistive, inductive and capacitive displacement sensors

Resistive, inductive and capacitive displacement sensors

FIGURE 6.1

Representative cutaways of linear-motion (a) and rotary (b) potentiometers.

FIGURE 6.2 (a) Schematic diagrams depict a potentiometer as a resistor with an arrow representing the wiper. This schematic shows a pot used as a variable voltage divider — the preferred configuration for precision measurement. RP is the total resistance of the pot, RL is the load resistance, vr is the reference or supply voltage, and vo is the output voltage. (b) shows an ideal linear output function where x represents the wiper position, and xP is its maximum position. TABLE 6.1 Fundamental Potentiometer Characteristics Advantages

Disadvantages

Easy to use Low cost Nonelectronic High-amplitude output signal Proven technology

Limited bandwidth Frictional loading Inertial loading Wear

FIGURE 6.3 Independent linearity is the maximum amount by which the actual output function deviates from a line of best fit.

FIGURE 6.4 Linearity can be greatly influenced by the ratio of load resistance, RL, to potentiometer resistance, RP .

(

)(

)

x x P RL RP vo = vr RL RP + x x P − x x P

(

) (

) (

)

2

(6.1)

FIGURE 6.7 A basic inductive sensor consists of a magnetic circuit made from a ferromagnetic core with a coil wound on it. The coil acts as a source of magnetomotive force (mmf) that drives the flux through the magnetic circuit and the air gap. The presence of the air gap causes a large increase in circuit reluctance and a corresponding decrease in the flux. Hence, a small variation in the air gap results in a measurable change in inductance.

py g For the explanation of the basic principles of inductive sensors, a simple magnetic circuit is shown in Figure 6.7. The magnetic circuit consists of a core, made from a ferromagnetic materia,l with a coil of n number of turns wound on it. The coil acts as a source of magnetomotive force (mmf) which drives the flux Φ through the magnetic circuit. If one assumes that the air gap is zero, the equation for the magnetic circuit can be expressed as:

mmf = Flux × Reluctance = Φ × ℜ

A - turns

(6.2)

such that the reluctance ℜ limits the flux in a magnetic circuit just as resistance limits the current in an electric circuit. By writing the mmf in terms of current, the magnetic flux may be expressed as:

Φ = ni ℜ

weber

(6.3)

In Figure 6.7, the flux linking a single turn is by Equation 6.3; but the total flux linking by the entire n number of the turns of the coil is

Ψ = nΦ = n2 i ℜ

weber

(6.4)

Equation 6.4 leads to self inductance L of the coil, which is described as the total flux (Ψ weber) per unit current for that particular coil; that is

L = Ψ I = n2 ℜ

(6.5)

This indicates that the self inductance of an inductive element can be calculated by magnetic circuit properties. Expressing ℜ in terms of dimensions as:

ℜ = l µµ 0 A where l = the total length of the flux path µ = the relative permeability of the magnetic circuit material µ0 = the permeability of free space (= 4π × 10–7 H/m) A = the cross-sectional area of the flux path

(6.6)

FIGURE 6.9 A variable-differential reluctance sensor consists of an armature moving between two identical cores separated by a fixed distance. The armature moves in the air gap in response to a mechanical input. This movement alters the reluctance of coils 1 and 2, thus altering their inductive properties. This arrangement overcomes the problem of nonlinearity inherent in single coil sensors.

FIGURE 6.10 A typical commercial variable differential sensor. The iron core is located half-way between the two E frames. Motion of the core increases the air gap for one of the E frames while decreasing the other side. This causes reluctances to change, thus inducing more voltage on one side than the other. Motion in the other direction reverses the action, with a 180° phase shift occurring at null. The output voltage can be processed, depending on the requirements, by means of rectification, demodulation, or filtering. The full-scale motion may be extremely small, on the order of few thousandths of a centimeter.

FIGURE 6.11 A variable-reluctance tachogenerator is a sensor which is based on Faraday’s law of electromagnetic induction. It consists of a ferromagnetic toothed wheel attached to the rotating shaft and a coil wound onto a permanent magnet extended by a soft iron pole piece. The wheel rotates in close proximity to the pole piece, thus causing the flux linked by the coil to change. The change in flux causes an output in the coil similar to a square waveform whose frequency depends on the speed of the rotation of the wheel and the number of teeth.

When the wheel rotates with a velocity ω, the flux may mathematically be expressed as:

()

Ψ θ = A + B cos mθ

(6.12)

where A = the mean flux B = the amplitude of the flux variation m = the number of teeth The induced emf is given by:

()

( () ) (

E = − d Ψ θ dt = − d Ψ θ d θ × d θ d t

)

(6.13)

or

E = bmω sin mωt

(6.14)

FIGURE 6.23 A variable distance capacitive displacement sensor. One of the plates of the capacitor moves to vary the distance between plates in response to changes in a physical variable. The outputs of these transducers are nonlinear with respect to distance x having a hyperbolic transfer function characteristic. Appropriate signal processing must be employed for linearization.

()

C x = εA x = ε r ε 0 A x where ε εr ε0 x A

(6.23)

= the dielectric constant or permittivity = the relative dielectric constant (in air and vacuum εr ≈ 1) = 8.854188 × 10–12 F/m–1, the dielectric constant of vacuum = the distance of the plates in m = the effective area of the plates in m2

The capacitance of this transducer is nonlinear with respect to distance x, having a hyperbolic transfer function characteristic. The sensitivity of capacitance to changes in plate separation is

dC dx = −ε r ε 0 A x 2

(6.24)

Equation 6.24 indicates that the sensitivity increases as x decreases. Nevertheless, from Equations 6.23 and 6.24, it follows that the percent change in C is proportional to the percent change in x. This can be expressed as:

dC C = − dx x

(6.25)

This type of sensor is often used for measuring small incremental displacements without making contact with the object.

FIGURE 6.24 A variable area capacitive displacement sensor. The sensor operates on the variation in the effective area between plates of a flat-plate capacitor. The transducer output is linear with respect to displacement x. This type of sensor is normally implemented as a rotating capacitor for measuring angular displacement.

Variable Area Displacement Sensors Alternatively, the displacements may be sensed by varying the surface area of the electrodes of a flat plate capacitor, as illustrated in Figure 6.24. In this case, the capacitance would be:

(

)

C = ε r ε 0 A − wx d

(6.26)

where w = the width wx = the reduction in the area due to movement of the plate Then, the transducer output is linear with displacement x. This type of sensor is normally implemented as a rotating capacitor for measuring angular displacement. The rotating capacitor structures are also used as an output transducer for measuring electric voltages as capacitive voltmeters.

FIGURE 6.25 A variable dielectric capacitive displacement sensor. The dielectric material between the two parallel plate capacitors moves, varying the effective dielectric constant. The output of the sensor is linear.

Variable Dielectric Displacement Sensors In some cases, the displacement may be sensed by the relative movement of the dielectric material between the plates, as shown in Figure 6.25. The corresponding equations would be:

[ (

)]

C = ε 0 w ε 2l − ε 2 − ε1 x

(6.27)

where ε1 = the relative permittivity of the dielectric material ε2 = the permittivity of the displacing material (e.g., liquid) In this case, the output of the transducer is also linear. This type of transducer is predominantly used in the form of two concentric cylinders for measuring the level of fluids in tanks. A nonconducting fluid forms the dielectric material. Further discussion will be included in the level measurements section.

FIGURE 6.26 A differential capacitive sensor. They are essentially three terminal capacitors with one fixed center plate and two outer plates. The response to physical variables is linear. In some versions, the central plate moves in response to physical variable with respect to two outer plates, and in the others, the central plate is fixed and outer plates are allowed to move.

Differential Capacitive Sensors Some of the nonlinearity in capacitive sensors can be eliminated using differential capacitive arrangements. These sensors are basically three-terminal capacitors, as shown in Figure 6.26. Slight variations in the construction of these sensors find many different applications, including differential pressure measurements. In some versions, the central plate moves in response to physical variables with respect to the fixed plates. In others, the central plate is fixed and outer plates are allowed to move. The output from the center plate is zero at the central position and increases as it moves left or right. The range is equal to twice the separation d. For a displacement d, one obtains:

(

)

(

)

(

)

(

)

(

2δC = C1 − C2 = ε r ε 0lw d − δd − ε r ε 0lw d + δd = 2ε r ε 0lwδ d d 2 + δd 2

)

(6.28)

and

(

C1 + C2 = 2C = ε r ε 0lw d − δd + ε r ε 0lw d + δd = 2ε r ε 0 lwd d 2 + δd 2

)

(6.29)

Giving approximately:

δ C C = δd d

(6.30)

FIGURE 6.28 A capacitive pressure sensor. These pressure sensors are made from a fixed metal plate and a flexible diaphragm. The flat flexible diaphragm is clamped around its circumference. The bending of the flexible plate is proportional to the applied pressure P. The deformation of the diaphragm results in changes in capacitance.

FIGURE 6.31 A capacitive liquid level sensor. Two concentric metal cylinders are used as electrodes of a capacitor. The value of the capacitance depends on the permittivity of the liquid and that of the gas or air above it. The total permittivity changes depending on the liquid level. These devices are usually applied in nonconducting liquid applications.

Deflection bridges

ì Z1 ì ü Z2 ü 1 1 ETh = VS í − = − V ý ý Sí + + + + Z Z Z Z 1 Z / Z 1 Z / Z 4 2 3 4 1 3 2 î 1 î

ETh , MIN ↔ Z1, MIN , when ETh , MIN = 0 ETh , MAX ↔ Z1, MAX

Z 4 / Z1 = Z 3 / Z 2 (Wheatstone)

Capacitive and inductive bridges

Capacitive:

Inductive:

Z1 = 1 /( jωC0 ) , Z 2 = R2 , Z 3 = R3 , Z 4 = 1 /( jωCh )

Z1 = jωL1 , Z 2 = R , Z 3 = R , Z 4 = jωL2

ü ü ì ì 1 1 1 1 ETh = VS í − − ý ý = VS í î1 + C0 / Ch 1 + R3 / R2 î1 + Z 4 / Z1 1 + Z 3 / Z 2

ì ü ì L1 1 1 1ü − − ý ETh = VS í ý = VS í î L1 + L2 2 î1 + Z 4 / Z1 1 + Z 3 / Z 2

V1 = 15V − R D I D ∆Vout = −4∆VGS

V+ > V−

Vout > 0

Op-amp rules:

1.

The inputs draw no current. This is in the ideal case; the 741 actually draws about 0.08 µA. We ignore this in future discussions.

2.

The output does whatever is necessary so that the feedback keeps the ( + ) and ( - ) inputs at the same potential.