Chapter 4. Overview of Measurement Systems
4.1. Transducers
Of most interest are the physical properties and performance characteristics of a transducer. Some examples are given in Tables 4.1 and 4.2.
Table 4.1. Physical properties
| Property | Method of measurement |
|---|---|
| Strain | Strain gauge, a resistive transducer whose resistance changes with length |
| Temperature | Resistance thermometer, thermocouple, thermistor, thermopile |
| Humidity | Resistance change of hygroscopic material |
| Pressure | Movement of the end of a coiled tube under pressure |
| Voltage | Moving coil in a magnetic field |
| Radioactivity | Electrical pulses resulting from ionization of gas at low pressure |
| Magnetic field | Deflection of a current carrying wire |
Table 4.2. Performance characteristics
| Static | Dynamic | Environmental |
|---|---|---|
| Sensitivity | Response time | Operating temperature range |
| Zero offset | Damping | Orientation |
| Linearity | Natural frequency | Vibration/shock |
| Range | Frequency response | |
| Span | ||
| Resolution | ||
| Threshold | ||
| Hysteresis | ||
| Repeatability |
A consideration of these characteristics influences the choice of transducer for a particular application. Further characteristics that are often important are the operating life, storage life, power requirements, and safety aspects of the device as well as cost and availability of service.
In industrial situations, the property being measured or controlled is called the controlled variable. Process control is the procedure used to measure the controlled variable and control it to within a tolerance level of a set point. The controlled variable is one of several process variables and is measured using a transducer and controlled using an actuator.
4.2. Methods of Measurement
All measurements involve a comparison between a measured quantity and a reference standard. There are two fundamental methods of measurement:
Null method (Figure 4.1)
- Direct comparison
- No loading
- Can be relatively slow
Figure 4.1. Null method: Bridge circuit. An unknown component is inserted into the bridge and the values of the others are altered to achieve balance condition. At balance, no current flows through the galvanometer G: R1R4 = Lu/C3; R1/C4 = Ru/C3.
Deflection method (Figure 4.2)
- Indirect comparison
- Deflection from zero until some balance condition achieved
- Limited in precision and accuracy
- Loading (transducer itself takes some energy from the system being measured)
- Relatively fast
Figure 4.2. Deflection method: Moving coil voltmeter. Although such a meter is designed to have a very high internal impedance, it has to draw some current from the circuit being measured in order to cause a deflection of the pointer. This may affect the operation of the circuit itself and lead to inaccurate readings—especially if the output resistance of the voltage source being measured is large.
4.3. Sensitivity
An important parameter associated with every transducer is its sensitivity. This is a measure of the magnitude of the output divided by the magnitude of the input.
For example, the sensitivity of a thermocouple may be specified as 10 μV/°C indicating that for each degree change in temperature between the sensor and the “reference” temperature, the output signal changes by 10 μV. The sensitivity may not be a constant across the working range.
The output voltage of most transducers is in the millivolt range for interfacing in a laboratory or light industrial applications. For heavy industrial applications, the output is usually given as a current rather than a voltage. Such devices are usually referred to as transmitters rather than transducers.
In most applications, the chances are that the signal produced by the transducer contains noise, or unwanted information. The proportion of wanted to unwanted signal is called the signal-to-noise ratio or SNR (usually expressed in decibels), see Figure 4.3.
Figure 4.3. Signal-to-noise ratio.
The higher the SNR the better. In electronic apparatus, noise signals often arise due to thermal random motion of electrons and is called white noise. White noise appears at all frequencies.
The first stage of any amplification of signal is the most critical when dealing with noise. In most sensitive equipment, a preamplifier is connected very close to the transducer to minimize noise and the resulting amplified signal passed to a main, or power, amplifier.
The noise produced by a transducer limits its ability to detect very small signals. A measure of performance is the detectivity given by
For example, if d = 106V−1 for a voltmeter, it means that the device can measure a voltage as low as 10−6V.
The least detectable input is often referred to as the noise floor of the instrument. The magnitude of the noise floor may be limited by the transducer itself or the effect of the operating environment.
4.4. Zero, Range, Linearity, and Span
The range of a transducer is specified by the maximum and minimum input and output signals. As an example consider a thermocouple that has an input range of −100 to +300° C and an output range of −1 to +10 mV.
The span or full-scale deflection (fsd) is the maximum variation in the input or output, see Figure 4.4. From this, we see that the thermocouple in the preceding example has an input span of 400°C and an output span S of 11 mV.
Figure 4.4. Span.
The zero and span calibration controls are shown in Figure 4.5.
Figure 4.5. The zero and span calibration controls.
The percentage of nonlinearity describes the deviation of a linear relationship between the input and the output (Figure 4.6):
Figure 4.6. Nonlinearity.
A linear output can be obtained by using a lookup table or altering the output signal electronically.
Zero offset errors can occur because of calibration errors, changes or aging of the sensor, a change in environmental conditions, and the like. The error is a constant over the range of the instrument.
A change in sensitivity, or a span error, results in the output being different from the correct value by a constant percentage. That is, the error is proportional to the magnitude of the output signal (change in slope).
4.5. Resolution, Hysteresis, and Error
A continuous increase in the input signal sometimes results in a series of discrete steps in the output signal due to the nature of the transducer (Figure 4.7).
Figure 4.7. As an example consider a wire wound potentiometer being used as a distance transducer. The wiper moves over the windings bringing a step change in resistance (R of one turn) with a change in distance.
The resolution of a transducer is defined as the size of the step divided by the fsd or span and is given in percent.
For example, the resolution of a 100-turn potentiometer is 1/100 = 1%.
For a particular input signal, the magnitude of the output signal may depend on whether the input is increasing or decreasing—this is called hysteresis (Figure 4.8).
Figure 4.8. Hysteresis.
In mechanical systems, hysteresis usually occurs due to backlash in moving parts (e.g., gear teeth).
The general response of a transducer is usually given as a percent error (Figure 4.9).
Figure 4.9. Error.
4.6. Fourier Analysis
Analog input signals that require sampling by a digital-to-analog converter system do not usually consist of just a single sinusoidal waveform. Real signals usually have a variety of amplitudes and frequencies that vary with time.
Such signals can be broken down into component frequencies and amplitudes using a method called Fourier analysis. Fourier analysis relies on the fact that any periodic waveform, no matter how complicated, can be constructed by the superposition of sine waves of the appropriate frequency and amplitude.
For example, a square wave can be represented using the sum of individual component sine waves (Figure 4.10):
Figure 4.10. A square wave represented by sine waves.
Fourier analysis, or the breaking up of a signal into its component frequencies, is important when we consider the process
of filtering and the conversion of an analog signal into a digital form.
4.7. Dynamic Response
The dynamic response of a transducer is concerned with the ability of the output to respond to changes at the input (Figure 4.11). The most severe test of dynamic response is to introduce a step signal at the input and measure the time response of the output.
Figure 4.11. Dynamic response.
Of particular interest are the following quantities:
- Rise time
- Response time
- Time constant τ
A step signal at the input causes the transducer to respond to an infinite number of component frequencies. When the input varies in a sinusoidal manner, the amplitude of the output signal may vary depending upon the frequency of the input if the frequency of the input is close to the resonant frequency of the system. If the input frequency is higher than the resonant frequency, then the transducer cannot keep up with the rapidly changing input signal and the output response decreases as a result.
4.8. PID Control
In many systems, a servo feedback loop is used to control a desired quantity. For example, a thermostat can be used in conjunction with an electric heater element to control the temperature in an oven. Such a servo loop consists of a sensor whose output controls the input signal to an actuator.
The difference between the target or set point and the current value of the controlled variable is the error signal Δe. If the error is larger than some preset tolerance or error band, then a correction signal, positive or negative, is sent to the actuator to cause the error to be reduced. In sophisticated systems, the error signal is processed by a PID controller before a correction signal is sent to the actuator. The PID controller determines the magnitude and type of the correction signal to be sent to the actuator to reduce the error signal.
The characteristics of a PID controller are expressed in terms of gains. The correction signal O from the PID controller to the actuator is given by the sum of the error Δe term multiplied by the proportional gain Kp, the integral gain Ki, and the derivative gain Kd.
- The proportional term causes the controller to generate a signal to the actuator whose amplitude is proportional to the magnitude of the error. That is, a large correction is made to correct a large error.
- The integral term is used to ramp the actuator to the final state to overcome friction or hysteresis in the system. It is a long-term correction and allows the system to servo to the target value.
- The derivative signal offers a damping response that reduces oscillation. The magnitude of the derivative correction depends upon the rate of change of the magnitude of the error signal. If the signal changes rapidly, a large correction is made.
The PID correction acts upon the error signal which is itself a function of time. The PID correction is thus also a function of time. For example, in servo motion control, a PID controller is able to cause the moving body (e.g., a robot arm) to accelerate, maintain a constant velocity, and decelerate to the target position (Figure 4.12).
Figure 4.12. Servo motion control.
4.9. Accuracy and Repeatability
Accuracy is a quantitative statement about the closeness of a measured value with the true value. The true value of a quantity is that which is specified by international agreement (Figure 4.13).
Figure 4.13. The kilogram is the unit of mass and is equal to the mass of an international standard kilogram held in Paris.
There is a difference between the accuracy and the precision of physical measurements (Figure 4.14).
Figure 4.14. Accuracy and precision.
High precision need not be accompanied by high accuracy. Precision is measured by the standard deviation of several measurements.
High accuracy may also be accompanied by a wide scatter in the measurement readings leading to low precision.
4.10. Mechanical Models
The response of materials and systems can often be modeled by springs and dashpots. This allows both static and dynamic processes to be modeled mathematically with some convenience (Figure 4.15). Most materials have a mechanical character that falls somewhere in between the two extremes of a solid and a fluid. Springs represent the solidlike characteristics of a system. Dashpots represent the fluidlike aspects of a system.
Figure 4.15. Modeling the response of materials and systems.
If two or more springs are connected in parallel, then they experience a common displacement. In this case, the overall stiffness is given in Figure 4.16.
Figure 4.16. Deflection of springs connected in parallel.
If two (or more) springs are connected in series, then loaded with a common force, then the total overall stiffness is given by (Figure 4.17):
Figure 4.17. Deflection of springs connected in series.
