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Figure 1.13 Commercially available LVDT sensor

From (1.3), if we apply a constant amplitude sinusoidal signal as the input, the amplitude of the output voltage is proportional to the distance between the plates.

LVDT sensors (see Figure 1.13) consist of one primary and two secondary windings on a hollow cylinder. The primary winding is in the middle, and the secondary windings have equal number of turns, series coupled, and they are at the ends of the cylinder (see Figure 1.14). A sinusoidal signal with a voltage of 0.5–5 V and frequency 1–20 kHz is applied to the primary winding. A magnetic core which measures the position moves inside the cylinder, and the movement of this core varies the magnetic field linking the primary winding to the secondary windings. Because the secondary windings are in opposition, the movement of the core to one position increases the induced voltage in one secondary coil and decreases the induced voltage in the other secondary coil. The net voltage difference is proportional to the position of the core inside the cylinder. Thus, by measuring the induced voltage we know the position of the core. The strong relationship between the core position and the induced voltage yields a design that exhibits excellent resolution. Most commercially available LVDTs come with built-in signal-conditioning circuitry that provides an easy interface to a computer. The device operates from a d.c. supply and the signal conditioner provides the a.c. signal required for the operation of the circuit, as well as the demodulation of the output signal to give a useful d.c.

CORE

Secondary

V1

Vo = V1 − V2

a.c.

Primary

V2

Secondary

Figure 1.14 LVDT sensor circuit diagram

20 INTRODUCTION

voltage output. The range of an LVDT is from ±125 µm to ±75 mm and the sensitivity ranges from 0.6 to 30 mV per 25 µm under normal excitation of 3–6 V.

The advantages of LVDT are:

low cost;

robust design;

no hysteresis effect;

fast response time;

no friction resistance;

long life.

The main disadvantage of the LVDT is that the core must have direct contact with the measured surface, which may not always be possible.

1.7.3 Velocity and acceleration sensors

Velocity is the differentiation of position, and in general position sensors can be used to measure velocity. The required differentiation can be done either in hardware (e.g. using operational amplifiers) or by the computer. For more accurate measurements velocity sensors should be used. There are two types of velocity sensors: linear sensors, and rotary sensors.

Linear velocity sensors can be constructed using a pair of coils and a moving magnet. When the coils are connected in series, the movement of the magnet produces additive voltage which is proportional to the movement of the magnet.

One of the most widely used rotary velocity sensors is the tachometer (or tachogenerator). A tachometer (see Figure 1.15) is connected to the shaft of a rotating device (e.g. a motor) and produces an analog d.c. voltage which is proportional to the speed of the shaft. If ω is the angular velocity of the shaft, the output voltage of the tachometer is given by

Vo = kω,

where k is the gain constant of the tachometer.

Another popular velocity sensor is the optical encoder. This basically consists of a light source and a disk with opaque and transparent sections where the disk is attached to the rotating shaft. A light sensor at the other side of the wheel detects light and a pulse is produced when the transparent section of the disk comes round. The encoder’s controller counts the pulses in a given time, and this is proportional to the speed of the shaft. Figure 1.16 shows a typical commercial encoder.

Figure 1.15 Commercially available tachometer

Figure 1.16 Commercially available encoder

Acceleration is the differentiation of velocity, or the double differentiation of position. Thus, in general, position sensors can be used to measure acceleration. The differentiation can be done either by using operational amplifiers or by a computer program. For accurate measurement of the acceleration, semiconductor accelerometers can be used. For example, the ADXL202 is an accelerometer chip manufactured by Analog Devices Inc. This is a low-cost 8-pin chip with two outputs to measure the acceleration in two dimensions. The outputs are digital signals whose duty cycles are proportional to the acceleration in each of the two axes. These outputs can be connected directly to a microcontroller and the acceleration can be measured very easily, requiring no A/D converter. The measurement range of the ADXL202 is ±2 g, where g is acceleration due to gravity, and the device can measure both dynamic acceleration (e.g. vibration), and static acceleration (e.g. gravity).

1.7.4 Force sensors

Force sensors can be constructed using position sensors. Alternatively, a strain gauge can be used to measure force accurately. There are many different types of strain gauges. A strain gauge can be made from capacitors and inductors, but the most widely used types are made from resistors. A wire strain gauge is made from a resistor, in the form of a metal foil. The principle of operation is that the resistance of a wire increases with increasing strain and decreases with decreasing strain.

In order to measure strain with a strain gauge, it must be connected to an electrical circuit, and a Wheatstone bridge is commonly used to detect the small changes in the resistance of the strain gauge.

Strain gauges can be used to measure force, load, weight pressure, torque or displacement. Force can also be measured using the principle of piezoelectricity. A piezoelectric sensor produces voltage when a force is applied to its surface. The disadvantage of this method is that the voltage decays after the application of the force and thus piezoelectric sensors are only

useful for measuring dynamic force.

1.7.5 Pressure sensors

Early pressure measurement was based on using a flexible device (e.g. a diaphragm) as a sensor; the pressure changed as the device moved and caused a dial connected to the device to move

Figure 1.17 Commercially available magnetic flow rate sensor (Sparling Instruments Inc.)

Magnetic flow meters are based on Faraday’s law of magnetic induction. Here, the liquid acts as a conductor as it flows through a pipe. This induces a voltage which is proportional to the flow rate. The faster the flow rate, the higher is the voltage. This voltage is picked up by the sensors mounted in the meter tube and electronic means are used to calculate the flow rate based on the cross-sectional area of the tube. Advantages of magnetic flow rates are as follows:

Corrosive liquids can be used.

The measurement does not change the flow stream.

Figure 1.17 shows a typical magnetic flow meter.

1.7.7 Air flow sensors

Air flow is usually measured using anemometers. A classical anemometer (see Figure 1.18) has a rotating vane, and the speed of rotation is proportional to the air flow. Hot wire anemometers

Figure 1.18 Classical anemometer

2

System Modelling

The task of mathematical modelling is an important step in the analysis and design of control systems. In this chapter, we will develop mathematical models for the mechanical, electrical, hydraulic and thermal systems which are used commonly in everyday life. The mathematical models of systems are obtained by applying the fundamental physical laws governing the nature of the components making these systems. For example, Newton’s laws are used in the mathematical modelling of mechanical systems. Similarly, Kirchhoff’s laws are used in the modelling and analysis of electrical systems.

Our mathematical treatment will be limited to linear, time-invariant ordinary differential equations whose coefficients do not change in time. In real life many systems are nonlinear, but they can be linearized around certain operating ranges about their equilibrium conditions. Real systems are usually quite complex and exact analysis is often impossible. We shall make approximations and reduce the system components to idealized versions whose behaviours are similar to the real components.

In this chapter we shall look only at the passive components. These components are of two types: those storing energy (e.g. the capacitor in an electrical system), and those dissipating energy (e.g. the resistor in an electrical system).

The mathematical model of a system is one or more differential equations describing the dynamic behaviour of the system. The Laplace transformation is applied to the mathematical model and then the model is converted into an algebraic equation. The properties and behaviour of the system can then be represented as a block diagram, with the transfer function of each component describing the relationship between its input and output behaviour.

2.1 MECHANICAL SYSTEMS

Models of mechanical systems are important in control engineering because a mechanical system may be a vehicle, a robot arm, a missile, or any other system which incorporates a mechanical component. Mechanical systems can be divided into two categories: translational systems and rotational systems. Some systems may be purely translational or rotational, whereas others may be hybrid, incorporating both translational and rotational components.

Microcontroller Based Applied Digital Control D. Ibrahim

C 2006 John Wiley & Sons, Ltd. ISBN: 0-470-86335-8

28 SYSTEM MODELLING

2.1.1 Translational Mechanical Systems

The basic building blocks of translational mechanical systems are masses, springs, and dashpots (Figure 2.1). The input to a translational mechanical system may be a force, F , and the output the displacement, y.

Springs store energy and are used in most mechanical systems. As shown in Figure 2.2, some springs are hard, some are soft, and some are linear. A hard or a soft spring can be linearized for small deviations from its equilibrium condition. In the analysis in this section, a spring is assumed massless, or of negligible mass, i.e. the forces at both ends of the spring are assumed to be equal in magnitude but opposite in direction.

where k is known as the stiffness constant. The spring when stretched stores energy given by

This energy is released when the spring contracts back to its original length.

In some applications springs can be in parallel or in series. When n springs are in parallel, then the equivalent stiffness constant keq is equal to the sum of all the individual spring stiffnesses ki :

Figure 2.2 (a) Hard spring, (b) soft spring, (c) linear spring