Capacitive Transducers

Capacitive sensors are based on changes in capacitance in response to physical movement. These sensors find their applications mainly in humidity, moisture and displacement sensing.

Reactance of a capacitance C is given by 1/(jωC), since i = C (dv/dt). These sensors have high impedance at low frequencies, as clear from the reactance expression for a capacitor. Also, capacitive sensors are non-contacting devices in the common usage. They require specific signal-conditioning hardware. In addition to analog capacitive sensors, digital (pulse-generating) capacitive transducers such as digital tachometers are also available.

A capacitor is formed by two plates, which can store an electric charge. The stored charge generates a potential difference between the plates. The capacitance C of a two-plate capacitor is given by

where

A is the common /overlapping area of the two plates; m²

d is the gap width between the two plates; m
ε is the dielectric constant or permittivity,  ε = ε_rε_o; F/m
ε_r is the relative permittivity,
ε_o is the permittivity of vacuum; 8.85x10^-12 F/m.

The capacitive transducer work on the principle of change of capacitance which may be caused by:

(i)                 Change in overlapping area A,

(ii)               Change in the distance d between the plates, and

(iii)             Change in dielectric constant

The cause of these changes can be displacement, force and pressure. We will discuss the various types below:

A.    Transducers based on Change in overlapping area

From the characteristic equation of capacitance it is clear that capacitance is directly proportional to the overlapping plate area A. For a parallel plate capacitor, the capacitance is given by-

x = length of overlapping plates; m

w = width of overlapping part of plates; m

Sensitivity

This type of a capacitive transducer is suitable for measurement of linear displacements ranging from 1 mm to 10 mm.

For a cylindrical capacitor, the capacitance is given by-

x = length of overlapping part of cylinders; m

D₂ = inner diameter of outer cylindrical electrode; m

D₁ = outer diameter of inner cylindrical electrode; m

Capacitive transducers can be employed to measure angular displacement also. If we have two plates- one fixed and one rotating, then their overlapping are is a function of angle between the overlapping edges.

The maximum capacitance is when the two plates completely overlap each other.

If the angle of overlap area is θ 

A.    Transducers based on Change in distance between plates

The capacitance between plates is inversely proportional to the distance between them.

The relationship between capacitance C and distance between the plates d is hyperbolic.

The sensitivity increases as x decreases.

The percent change in C is proportional to the percent change in x.

A.    Transducers based on change in Dielectric constant

Measurement of displacement

Normal capacitance when dielectric medium is partially overlapped with metal plates-

If the dielectric material is moved a distance ‘x’ in direction as shown, the capacitance changes by ‘∆C’.

Measurement of Liquid Level

This type of transducer is predominantly used in the form of two concentric cylinders for measuring the level of fluids in tanks. A non-conducting fluid forms the dielectric material. The method is generally based on the difference between the dielectric constant of the liquid and that of the gas or air above it. Two concentric metal cylinders are used for capacitance as shown in Figure below.

Capacitive Differential Transducer

A normal parallel plate capacitive transducer exhibits non-linear response. By using differential arrangement, we can get linear response for capacitive differential transducers. The arrangement is shown below:

It consists of two fixed plates and one moving plate whose displacement is to be measured. It acts like two capacitors in series.

Let C₁ and C₂ be the capacitance of individual parallel plate combination when the movable plate is at middle position. Thus,

For a voltage ‘V’ applied across the fixed plates, the voltage appearing across individual plate combination is equal when the movable plat is at middle position.

 

Differential voltage ∆V= 0.

Let the movable plate is moved by a ‘x’. Therefore the new values of C₁ and C₂ are given as-

This method can have accuracy upto 0.1% and measurement range can be from tens of nm to 10 mm.

Charge Amplifier Circuit

An op-amp circuit with a feedback capacitor Cf, which is similar to a charge amplifier, may be used with a variable-capacitance transducer. A circuit of this type is shown in Figure below. The transducer capacitance is denoted by Cs. The charge balance at node A gives VrefCs + VoCf = 0. The circuit output is given by

Position measurement- Encoder

Shaft Encoders

Any transducer that generates a coded digital signal of a measured quantity is known as an encoder. Shaft encoders are used to measure angular displacements and angular velocities. High resolution, high accuracy, and digital output are some of the relative advantages of shaft encoders.

Resolution: Depends on the word size of the encoder output and the number of pulses generated per revolution of the encoder.

Accuracy:  Due to noise immunity and reliability of digital signals.

Encoder Types

Shaft encoders can be classified into two categories depending on the nature and the method of interpretation of the transducer output: (1) incremental encoders and (2) absolute encoders.

Incremental Encoder

The output of an incremental encoder is a pulse train signal, which is generated when the transducer disk rotates. The number of pulses and the number of pulse per unit time gives the measurement of angular displacement and angular velocity of the device on which the encoder disk is mounted. With an incremental encoder, displacement is obtained with respect to some reference point or marker. That is, incremental encoder giver relative position of a body wrt its initial position. The reference point can be the home position of the moving component (say, determined by a limit switch) or a reference point on the encoder disk, as indicated by a reference pulse (index pulse) generated at that location on the disk. The index pulse count determines the number of full revolutions.

Absolute Encoder

An absolute encoder has many pulse tracks on its transducer disk. When the disk of an absolute encoder rotates, several pulse trains are generated simultaneously. The number of pulse train is equal to the number of tracks on the disk. At a given instant, the magnitude of each pulse signal will be either ‘1’ (HIGH) or ‘0’ (LOW) depending on opaque and transparent segment of disk.. Hence, the set of pulse trains gives an encoded binary number at any instant. This encoded binary data gives the absolute position of the body on which the encoder is mounted. The pulse voltage can be made compatible with some digital interface logic (e.g TTL). Consequently, the direct digital readout of an angular position is possible with an absolute encoder. Absolute encoders are commonly used to measure fractions of a revolution. However, complete revolutions can be measured using an additional track, which generates an index pulse, as in the case of an incremental encoder. The same signal generation (and pick-off) mechanism may be used in both types (incremental and absolute) of transducers.

Encoder Technologies

Four techniques of transducer signal generation may be identified for shaft encoders:

1.      Optical method- we will discuss only this method in the post.

2.      Sliding contact (electrical conducting) method

3.      Magnetic saturation (reluctance) method

4.      Proximity sensor method

By far, the optical encoder is most common and cost-effective. The other three methods may be used in special circumstances, where the optical method may not be suitable (e.g., under extreme tem peratures or in the presence of dust, smoke, etc.). For a given type of encoder (incremental or absolute), the method of signal interpretation is identical for all four types of signal generation listed previously. Now we briefly describe the principle of signal generation for all four techniques and consider only the optical encoder in the context of signal interpretation and processing.

Optical Encoder

The optical encoder uses an opaque disk (coded disk) that has one or more circular tracks, with some arrangement of identical transparent windows (slits) in each track. A parallel beam of light (e.g., from a set of light-emitting diodes or LEDs) is projected to all tracks from one side of the disk. The transmitted light is picked off using a series of photosensors on the other side of the disk, which typically has one sensor for each track. This arrangement is shown in Figure a, which indicates just one track and one pick-off sensor. The light sensor could be a silicon photodiode or a phototransistor. Since the light from the source is interrupted by the opaque regions of the track, the output signal from the photosensor is a series of voltage pulses. This signal can be interpreted through edge detection or level detection to obtain the increments in the angular position and also the angular velocity of the disk.

The sensor element of such a measuring device is the encoder disk, which is coupled to the rotating object directly or through a gear mechanism. The transducer stage is the conversion of disk motion (analog) into the pulse signals, which can be coded into a digital word.

If the direction of rotation is not important, an incremental encoder disk requires only one primary track that has equally spaced and identical pick-off regions. A reference track that has just one window may be used to generate the index pulse, to initiate pulse counting for angular position measurement and to detect complete revolutions.

Note: A transparent disk with a track of opaque spots will work equally well as the encoder disk of an optical encoder. In either form, the track has a 50% duty cycle (i.e., the length of the transparent region is equal to the length of the opaque region).

Direction of Rotation

An incremental encoder generally have a second track placed at quarter-pitch separation from the first track pattern (pitch = center-to-center distance between adjacent windows) to generate a quadrature signal, which will indicate the direction of rotation.

An incremental encoder typically has the following five pinouts:

1. Ground

2. Index Channel

3. A Channel

4. +5V dc power

5. B Channel

Pins for Channel A and Channel B give the quadrature signals shown in Figure a and b, and the Index pin gives the reference pulse signal shown in Figure c. Figure 2a shows the sensor outputs (v1 and v2) when the disk rotates in the clockwise (cw) direction; and Figure 2b shows the outputs when the disk rotates in the counterclockwise (ccw) direction. Several methods can be used to determine the direction of rotation using these two quadrature signals. For example,

1.      By phase angle between the two signals

2.      By clock counts to two adjacent rising edges of the two signals

3.      By checking for rising or falling edge of one signal when the other is at high

4.       For a high-to-low transition of one signal check the next transition of the other signal

Pulsed signal output of incremental encoder. (a) CW rotation; (b) CCW rotation; (c) Index pulse

Method 1: It is clear from Figure 2a and b that in the CW rotation, v1 lags v2 by a quarter of a cycle (i.e., a phase lag of 90°) and in the CCW rotation, v1 leads v2 by a quarter of a cycle. Hence, the direction of rotation may be obtained by determining the phase difference of the two output signals, using phase detecting circuitry.

Method 2: A rising edge of a pulse can be determined by comparing successive signal levels at fixed time periods (can be done in both hardware and software). Rising-edge time can be measured using pulse counts of a high-frequency clock. Suppose that the counting (timing) begins when the v1 signal begins to rise (i.e., when a rising edge is detected). Let n1 = number of clock cycles (time) up to the time when v2 begins to rise; and n2 = number of clock cycles up to the time when v1 begins to rise again. Ten, the following logic applies:

If n₁ > n₂ – n₁àCW rotation

If n₁ < n₂ – n₁àCCW rotation

Method 3: In this case, firstly high level logic of v₂ is detected and then check if v₁ is at rising or falling edge.

If v₁ is at rising edge when v₂ is at logic high à CW rotation

If v₁ is at falling edge when v₂ is at logic high à CCW rotation

Method 4: Detect a high to low transition in signal v₁.

If the next transition in signal v₂ is Low to High à CW rotation

If the next transition in signal v₂ is High to Low à CCW rotation

Strain Gauge

Introduction

A strain gauge is a resistive sensor whose resistance is a function of applied strain. Under the effect of external force, the length and cross-section area of a conductor changes. Due to this resistance change takes place. The resistivity of the conductor also changes and this property is known as the piezoresistive effect and is expressed through the gauge factor G_f of the conductor. The value of Guage factor is ≈ 2 for most conductors. For Platinum its value is 6. For silicon its value is upto 150.

Early strain gauges were fine metal filaments. Modern strain gauges are manufactured primarily as metallic foil using the constantan or Semiconductor elements (e.g., silicon with trace impurity boron). They are manufactured by first forming a thin film (foil) of metal or a single crystal of Semiconductor material and then cutting it into a suitable grid pattern. This process is much more economical and is more precise than making strain gauges with metal filaments.

The strain-gauge element is formed on a backing film of electrically insulated material of polymide plastic. This element is cemented or bonded using epoxy, onto the member whose strain is to be measured. strain from the object must be reliably coupled to the gauge wire, whereas the wire must be electrically isolated from the object. The coefficient of thermal expansion of the backing should be matched to that of the wire. Alternatively, a thin film of insulating ceramic substrate is melted onto the measurement surface, on which the strain gauge is mounted directly. The direction of sensitivity is the major direction of elongation of the strain-gauge element. To measure strains in more than one direction, multiple strain gauges (e.g., various rosette configurations) are available as single units. These units have more than one direction of sensitivity.

Gage resistance values range from 30 to 3000 ohms, with 350 ohms being the most
commonly used.

Theory

The change of electrical resistance of a material when mechanically deformed is the property used in resistance-type strain gauges. The resistance R of a conductor of length ℓ and area of cross-section A is given by

Under the influence of tensile force F, the length of the conductor increases by dL and diameter decreases by dD and the resistance changes by dR.

 

Differentiating both sides,

where

If the change in the value of resistivity of a material under strain is neglected, the Gauge factor can be written as

The Poisson’s ration for all metals is between 0 to 0.5. This gives a gauge factor of about 2.

Strain gauge Applications

Many variables—including displacement, acceleration, pressure, temperature, liquid level, stress, force, and torque—can be determined using strain measurements. Some variables (e.g., stress, force, and torque) can be determined by measuring the strain of the dynamic object itself at suitable locations. In other situations, an auxiliary front-end device may be required to convert the measurand into a proportional strain. For instance, pressure or displacement may be measured by converting them to a measurable strain using a diaphragm, bellows, or a bending element. Acceleration may be measured by first converting it into an inertia force of a suitable mass (seismic mass) element, then subjecting a cantilever (strain member) to that inertia force, and finally, measuring the strain at a high-sensitivity location of the cantilever element. Temperature may be measured by measuring the thermal expansion or deformation in a bimetallic element.

Strain Gauge measurement

The preferred circuit for use in strain-gauge measurements is the Wheatstone bridge. One or more of the four resistors R1, R2, R3, and R4 in the bridge may represent strain gauges. The output relationship for the Wheatstone bridge circuit is given by

When this output voltage is zero, the bridge is balanced. It follows from Equation above that for a balanced bridge,

Equation above is valid for any value of the load resistance R_L, because when the bridge is balanced, current i through the load become zero.

Bridge Sensitivity

Strain-gauge measurements are calibrated with respect to a balanced bridge. When a strain gauge in
the bridge deforms, the balance is upset. If one of the arms of the bridge has a variable resistor, it can be
adjusted to restore the balance. The amount of this adjustment measures the amount by which the resistance of the strain gauge has changed, thereby measuring the applied strain. This is known as the null-balance method of strain measurement. This method is inherently slow because of the time required to balance the bridge each time a reading is taken.

A more common method, which is particularly suitable for making dynamic readings from a strain-gauge bridge, is to measure the output voltage resulting from the imbalance caused by the deformation of an active strain gauge in the bridge. To determine the calibration constant of a strain-gauge bridge, the sensitivity of the bridge output to changes in the four resistors in the bridge should be known. For small changes in resistance, this may be determined as dVo/Vin.

Tee Rosette

Rectangular Rosette

Figure below shows a rectangular rosette which is used when orientation of principal axes is unknown. The gauge elements are mounted in 0°, 45° and 90° orientation. Typical setup is for each gauge element to become an active arm in separate Wheatstone bridge circuits (i.e. a Wheatstone bridge for each gauge element of the rosette).

Strain gauge installation

Proper strain gauge installation is necessary to obtain accurate
experimental stress results. Skilled technicians usually install strain gauges after attending a
strain gauge school or training course. The decision on locations of the desired stress
measurements is usually made by the test sponsor or mechanical design team of the test item.
The instrumentation engineer must then ensure the following items are addressed. Failure to
proceed in this fashion will result in poor gauge adhesion, faulty data, and engineer frustration.

a. The supplies should be on hand for the technicians to apply the gauges when the time comes.

b. The type of gauge, the material to which the gauge is to be mounted, and the environment in which the gauge will be exposed.

c. Are the principal axes known (many times they are not)?

d. How long do these gauges need to be used?

e. Will the gauges be submersed in a liquid?

f. Will the gauges be exposed to temperature extremes?

g. How long will the lead wires need to be?

h. Can the bridge completion resistors be co-located at the site of the strain gauge?

Once all of the supplies are on hand and the desired locations of the gauges known, the installation process can begin. The following steps should be followed.

a. Surface preparation. The surfaces to which the gauges are to be applied must be free of paint, oil, grease, etc. The surface preparation usually involves paint stripping, sanding, and degreasing.

b. Install gauge or rosette. The gauge (or rosette) is applied to the surface with an adhesive. Adhesives exist for a variety of environments and desired lifespan. Cure times vary from a few hours to several days. Cure temperatures can vary from room temperature to several hundred degrees Centigrade. During the curing process, evenly distributed pressure is usually required on the applied gauge(s). 

c. Install bridge. The bridge completion resistor assemblies and small wire terminals or solder tabs may be bonded near the gauge. Small gauge wire is typically used to connect the gauge to the bridge completion packs and to the wire solder tabs. The larger gauge lead wires are then soldered to these tabs and completion networks (not directly to the gauge itself).

d. Epoxies and sealants. Finally, depending on the test environment, several additional coatings of specialized epoxies and sealants may be needed to protect the gauge(s), bridge completion packs, and/or wire tabs.

e. Patience and skill. Skipping steps is not advisable.

GATE QUESTIONS

GATE-2007

GATE-2008

GATE-2009

GATE-2010

GATE-2012

GATE-2014

Semiconductor strain gages typically have much higher gage factors than those of metallic strain gages, primarily due to :

(A) higher temperature sensitivity

(B) higher Poisson’s ratio

(C) higher piezoresitive coefficient

(D) higher magnetostrictive coefficient

GATE-2015

A p-type semiconductor strain gauge has a nominal resistance of 1000 Ω and a gauge factor of +200 at 25 °C. The resistance of the strain gauge in ohms when subjected to a strain of +10−4 m/m at the same temperature is __________ Ω.

GATE-2018

A 1000 Ohm strain gage (Rg) has a gage factor of 2.5. It is connected in the bridge as shown in
figure. The strain gage is subjected with a positive strain of 400 um/m. The output Vo (in

mV) of the bridge is (up to two decimal places) _____.

LVDT- Linear Variable Differential transformer

LVDT- Linear Variable Differential transformer

Position and displacement may be sensed by methods of electromagnetic induction. The most commonly used inductive transducer to convert linear motion into electrical signals is the Linear Variable Differential Transformer or LVDT as they are commonly known.

An LVDT consists of one primary coil and two secondary coils. The primary coil carries ac excitation (Vref) that induces a steady ac voltage in the secondary coils. The induced amplitude depends on flux coupling between the coils.

There are two techniques for changing the coupling.

• One is the movement of a core made of ferromagnetic material within the flux path. This changes the reluctance of the path, which, in turn, alters the coupling between the coils. This is the basis for the operation of a LVDT (linear variable differential transformer), a RVDT (rotary variable differential transformer).
• The other method is to physically move one coil with respect to another.

Generally, the core is made of high permeability, nickel iron which is hydrogen annealed. It gives low harmonics, low null voltage.

The ac excitation in the primary coil produces an alternating magnetic field which induces an alternating current in the secondary coils. The secondary coils are connected in series opposition. Hence the polarity of voltage induces is opposite in the two secondary coils. When the core is positioned in the magnetic center of the transformer, the secondary output signals cancel and there is no output voltage. This is known as “null position”.

Moving the core away from the central position unbalances the induced magnetic flux ratio between the secondaries, developing an output. As the core moves, reluctance of the flux path changes. Hence, the degree of flux coupling depends on the axial position of the core. At a steady state, the amplitude of the induced voltage is proportional to the core displacement. Consequently, voltage may be used as a measure of a displacement. The LVDT provides the direction as well as magnitude of the displacement. The direction is determined by the phase angle between the primary (reference) voltage and the secondary voltage. Excitation voltage is generated by a stable oscillator.

The LVDT is considered a passive transducer because the displacement of the core, which is being measured, itself provides energy for changing the induced voltage in the secondary coil. Even though an external power supply is used to energize the primary coil, which in turn induces a steady voltage at the carrier frequency in the secondary coil, which is not relevant in the definition of a passive transducer.

Note: Because of opposed secondary windings, the LVDT provides the direction as well as the magnitude of displacement. When the output signal is demodulated, its sign gives the direction. If the output signal is not demodulated, the direction is determined by the phase angle between the primary (reference) voltage and the secondary (output) voltage, which includes the carrier signal.

For an LVDT to measure transient motions accurately, the frequency of the reference voltage (the carrier frequency) has to be at least 10 times larger than the largest significant (useful) frequency component in the measured motion, and typically can be as high as 20 kHz. For quasi-dynamic displacements and slow transients of the order of a few hertz, a standard ac supply (at 50/60 Hz line frequency) is adequate. The performance (particularly sensitivity and accuracy) is known to improve with the excitation frequency, however. Since the amplitude of the output signal is proportional to the amplitude of the primary signal, the reference voltage should be regulated to get accurate results. In particular, the power source should have a low output impedance.

Ideally the output at the null position should be equal to zero. However in practice there exists a small voltage at the null position. This may be due to presence of harmonics in the input supply  voltage, stray magnetic fields, temperature effect among other reasons.

Calibration and Compensation

An LVDT may be calibrated in millimeter per volt (mm/V), in its linear range. In addition, and a displacement offset (mm) may be provided. This typically represents the least squares fit of a set of calibration data. Since ambient temperature and other environmental conditions will affect the LVDT output, in addition to the primary and secondary coils, a reference coil may be available for compensation of the LVDT output.

Advantages of the LVDT include the following:

1.       It is essentially a non-contacting device with no frictional resistance. Lightweight core will result in very small resistive forces.

2.       Hysteresis (both magnetic hysteresis and mechanical backlash) is negligible.

3.       It has low output impedance, typically in the order of 100 Ω.

4.       It provides directional measurements (positive/negative).

5.       It is available in miniature sizes as well (e.g., length of 1 or 2 mm, displacement measurements of a fraction of a millimeter, and maximum travel or stroke of 1 mm).

6.       It has a simple and robust construction (inexpensive and durable)

7.       It has fine resolutions (theoretically, infinitesimal resolution; practically, much better than a coil potentiometer)

Disadvantages

1.      They are sensitive to stray magnetic field.

2.      Output signal need amplification due to its small magnitude.

3.      Measurement is affected by vibrations.

4.      Affected by temperature variation.