Measurement Systems.pdf

Module 2 Measurement Systems Version 2 EE IIT, Kharagpur 1

Module 2 Measurement Systems Version 2 EE IIT, Kharagpur 1

Lesson 3 Measurement Systems Specifications Version 2 EE IIT, Kharagpur 2

Instructional Objectives At the end of this lesson, the student will be able to 1.

Define the different terms used for characterizing the performance of an instrument/ measurement system.

2.

Compare the performances of two similar type of instruments, looking at the specifications

3.

Write down the performance specifications of a measurement system from its test data.

Introduction One of the most frequent tasks that an Engineer involved in the design, commissioning, testing, purchasing, operation or maintenance related to industrial processes, is to interpret manufacturer’s specifications for their own purpose. It is therefore of paramount importance that one understands the basic form of an instrument specification and at least the generic elements in it that appear in almost all instrument specifications. Different blocks of a measurement system have been discussed in lesson-2. The combined performance of all the blocks is described in the specifications. Specifications of an instrument are provided by different manufacturers in different wrap and quoting different terms, which sometimes may cause confusion. Moreover, there are several application specific issues. Still, broadly speaking, these specifications can be classified into three categories: (i) static characteristics, (b) dynamic characteristics and (iii) random characteristics.

1.

Static Characteristics

Static characteristics refer to the characteristics of the system when the input is either held constant or varying very slowly. The items that can be classified under the heading static characteristics are mainly:

Range (or span) It defines the maximum and minimum values of the inputs or the outputs for which the instrument is recommended to use. For example, for a temperature measuring instrument the input range may be 100-500 oC and the output range may be 4-20 mA.

Sensitivity It can be defined as the ratio of the incremental output and the incremental input. While defining the sensitivity, we assume that the input-output characteristic of the instrument is approximately linear in that range. Thus if the sensitivity of a thermocouple is denoted as 10 μ V / 0C , it indicates the sensitivity in the linear range of the thermocouple voltage vs. temperature characteristics. Similarly sensitivity of a spring balance can be expressed as 25 mm/kg (say), indicating additional load of 1 kg will cause additional displacement of the spring by 25mm.

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Again sensitivity of an instrument may also vary with temperature or other external factors. This is known as sensitivity drift. Suppose the sensitivity of the spring balance mentioned above is 25 mm/kg at 20 oC and 27 mm/kg at 30oC. Then the sensitivity drift/oC is 0.2 (mm/kg)/oC. In order to avoid such sensitivity drift, sophisticated instruments are either kept at controlled temperature, or suitable in-built temperature compensation schemes are provided inside the instrument.

Linearity Linearity is actually a measure of nonlinearity of the instrument. When we talk about sensitivity, we assume that the input/output characteristic of the instrument to be approximately linear. But in practice, it is normally nonlinear, as shown in Fig.1. The linearity is defined as the maximum deviation from the linear characteristics as a percentage of the full scale output. Thus, ΔO (1) Linearity = Omax − Omin where, ΔO = max(ΔO1 , ΔO2 ) . Output

Output

OMAX

OMAX

ΔΟ1

H

ΔΟ2 OMIN

OMIN IMIN Fig. 1 Linearity

IMAX

Input

IMIN

IMAX

Input

Fig. 2 Hysteresis

Hysteresis Hysteresis exists not only in magnetic circuits, but in instruments also. For example, the deflection of a diaphragm type pressure gage may be different for the same pressure, but one for increasing and other for decreasing, as shown in Fig.2. The hysteresis is expressed as the maximum hysteresis as a full scale reading, i.e., referring fig.2, H (2) Hysteresis = X 100. Omax − Omin

Resolution In some instruments, the output increases in discrete steps, for continuous increase in the input, as shown in Fig.3. It may be because of the finite graduations in the meter scale; or the Version 2 EE IIT, Kharagpur 4

1 instrument has a digital display, as a result the output indication changes discretely. A 3 -digit 2 voltmeter, operating in 0-2V range, can have maximum reading of 1.999V, and it cannot measure any change in voltage below 0.001V. Resolution indicates the minimum change in input variable that is detectable. For example, an eight-bit A/D converter with +5V input can measure 5 or 19.6 mv. Referring to fig.3, resolution is also defined in terms the minimum voltage of 8 2 −1 of percentage as: ΔI Resolution = (3) X 100 I max − I min The quotient between the measuring range and resolution is often expressed as dynamic range and is defined as: measurement range Dynamic range = (4) resolution And is expressed in terms of dB. The dynamic range of an n-bit ADC, comes out to be approximately 6n dB. Output

ΔΙ Imin

Imax

Input

Fig. 3 Resolution

Accuracy Accuracy indicates the closeness of the measured value with the actual or true value, and is expressed in the form of the maximum error (= measured value – true value) as a percentage of full scale reading. Thus, if the accuracy of a temperature indicator, with a full scale range of 0500 oC is specified as ± 0.5%, it indicates that the measured value will always be within ± 2.5 oC of the true value, if measured through a standard instrument during the process of calibration. But if it indicates a reading of 250 oC, the error will also be ± 2.5 oC, i.e. ± 1% of the reading. Thus it is always better to choose a scale of measurement where the input is near full-scale value. But the true value is always difficult to get. We use standard calibrated instruments in the laboratory for measuring true value if the variable.

Precision Precision indicates the repeatability or reproducibility of an instrument (but does not indicate accuracy). If an instrument is used to measure the same input, but at different instants, spread Version 2 EE IIT, Kharagpur 5

over the whole day, successive measurements may vary randomly. The random fluctuations of readings, (mostly with a Gaussian distribution) is often due to random variations of several other factors which have not been taken into account, while measuring the variable. A precision instrument indicates that the successive reading would be very close, or in other words, the standard deviation σ e of the set of measurements would be very small. Quantitatively, the precision can be expressed as: measured range (5) Precision = σe The difference between precision and accuracy needs to be understood carefully. Precision means repetition of successive readings, but it does not guarantee accuracy; successive readings may be close to each other, but far from the true value. On the other hand, an accurate instrument has to be precise also, since successive readings must be close to the true value (that is unique).

2.

Dynamic Characteristics

Dynamic characteristics refer to the performance of the instrument when the input variable is changing rapidly with time. For example, human eye cannot detect any event whose duration is more than one-tenth of a second; thus the dynamic performance of human eye cannot be said to be very satisfactory. The dynamic performance of an instrument is normally expressed by a differential equation relating the input and output quantities. It is always convenient to express the input-output dynamic characteristics in form of a linear differential equation. So, often a nonlinear mathematical model is linearised and expressed in the form: d n x0 d n −1 x 0 dx 0 d m xi d m −1 xi dx + a + ⋅ ⋅ ⋅ ⋅ + a + a x = b + b + ⋅ ⋅ ⋅ ⋅ +b1 i + b0 xi n −1 1 0 0 m m −1 n n −1 m m −1 dt dt dt dt dt dt (6) where xi and x 0 are the input and the output variables respectively. The above expression can also be expressed in terms of a transfer function, as: x ( s ) bm s m + bm −1 s m −1 ⋅ ⋅ ⋅ +b1 s + b0 (7) = G ( s) = 0 xi ( s ) a n s n + bn −1 s n −1 ⋅ ⋅ ⋅ + a1 s + a 0 Normally m 2000, the flow is turbulent. In the present case we will assume that the flow is turbulent, that is the normal case for practical situations. We consider the fluid flow through a closed channel of variable cross section, as Version 2 EE IIT, Kharagpur 3

shown in fig. 1. The channel is of varying cross section and we consider two cross sections of the channel, 1 and 2. Let the pressure, velocity, cross sectional area and height above the datum be expressed as p1, v1, A1 and z1 for section 1 and the corresponding values for section 2 be p2, v2, A2 and z2 respectively. We also assume that the fluid flowing is incompressible. Now from Bernloulli’s equation: p1 v12 p v2 + + z1 = 2 + 2 + z2 (1) γ 2g γ 2g where γ is the specific weight of the fluid. 2 1

p1 v1 z1

Flow

p2 v2 z2

Fig. 1 Flow through a varying cross section

If z1=z2, then p1

γ

+

v12 p v2 = 2+ 2 2g γ 2g

(2)

If the fluid is incompressible, then v1 A1 = v2 A2 . Therefore, 2g v22 − v12 = ( p1 − p2 ) γ or, A2 2g ( p1 − p2 ) v22 (1 − 22 ) = A1 γ Therefore, 1 2g 1 2g v2 = ( p1 − p2 ) = ( p1 − p2 ) 2 4 γ γ A2 1− β (1 − 2 ) A1 Considering circular cross section, we define β as the ratio of the two diameters, i.e. d A β = 2 , and so, 2 = β 2 . d1 A1 Therefore, the volumetric flow rate through the channel can be expressed as:

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Q = v2 A2 =

2g

A2 1− β

4

γ

( p1 − p2 )

(3)

From the above expression, we can infer that if there is an obstruction in the flow path that causes the variation of the cross sectional area inside the closed flow channel, there would be difference in static pressures at two points and by measuring the pressure difference, one can obtain the flow rate using eqn. (3). However, this expression is valid for incompressible fluids (i.e. liquids) only and the relationship between the volumetric flow rate and pressure difference is nonlinear. A special signal conditioning circuit, called square rooting circuit is to be used for getting a linear relationship.

Orifice meter Depending on the type of obstruction, we can have different types of flow meters. Most common among them is the orifice type flowmeter, where an orifice plate is placed in the pipe line, as shown in fig.2. If d1 and d2 are the diameters of the pipe line and the orifice opening, then the flow rate can be obtained using eqn. (3) by measuring the pressure difference (p1-p2). Flow profile

Orifice Plate Vena Contacta

Flow d1

p1

d2

p2 Fig. 2 Orifice type flowmeter

Corrections The flow expression obtained from eqn.(3) is not an accurate expression in the actual case, and some correction factor, named as discharge co-efficient (Cd) has to be incorporated in (3), as C A 2g ( p1 − p2 ) Q = v2 A2 = d 2 (4) 4 γ 1− β Cd is defined as the ratio of the actual flow and the ideal flow and is always less than one. There are in fact two main reasons due to which the actual flow rate is less than the ideal one (obtained from eqn. (3)). The first is that the assumption of frictionless flow is not always valid. The amount of friction depends on the Reynold’s number (RD). The more important point is that, the minimum flow area is not the orifice area A2, but is somewhat less and it occurs at a distance from the orifice plate, known as the Vena Contracta, and we are taking a pressure tapping around Version 2 EE IIT, Kharagpur 5

that point in order to obtain the maximum pressure drop. As a result, the correction factor Cd 104, the flow is totally turbulent and Cd is independent on RD. In this range, the typical value of Cd for orifice plate varies between 0.6 and 0.7.

Orifice Plate, Venturimeter and Flow nozzle The major advantages of orifice plate are that it is low cost device, simple in construction and easy to install in the pipeline as shown in fig.3. The orifice plate is a circular plate with a hole in the center. Pressure tappings are normally taken distances D and 0.5D upstream and downstream the orifice respectively (D is the internal diameter of the pipe). But there are many more types of pressure tappings those are in use.

Permanent Pressure drop

Fig. 3 Orifice plate and permanent pressure drop

The major disadvantage of using orifice plate is the permanent pressure drop that is normally experienced in the orifice plate as shown in fig.3. The pressure drops significantly after the orifice and can be recovered only partially. The magnitude of the permanent pressure drop is around 40%, which is sometimes objectionable. It requires more pressure to pump the liquid. This problem can be overcome by improving the design of the restrictions. Venturimeters and flow nozzles are two such devices. The construction of a venturimeter is shown in fig.4. Here it is so designed that the change in the flow path is gradual. As a result, there is no permanent pressure drop in the flow path. The discharge coefficient Cd varies between 0.95 and 0.98. The construction also provides high mechanical strength for the meter. However, the major disadvantage is the high cost of the meter.

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Flow nozzle is a compromise between orifice plate and venturimeter. The typical construction is shown in fig. 5.

p2

p1

Fig. 4 Venturimeter

p1

p2 Fig. 5 Flow nozzle

In general, few guidelines are to be followed for installation of obstruction type flowmeters. Most important among them is that, no other obstruction or bending of the pipe line is not allowed near the meter. Though this type of flowmeters are most popular in industries, their accuracy is low for low flow rates. As a result, they are not recommended for low flow rate measurement.

Flow measurement of compressible fluids So far we have discussed about the flow measurement of incompressible fluids (liquids). For of compressible fluids, i.e. gases, the flow rates are normally expressed in terms of mass flow rates. The same obstruction type flowmeters can be used, but an additional correction factor needs to be introduced to take in to account the compressibility of the gas used. The mass flow rate gases can be expressed as :

⎡ C A W =Y ⎢ d 2 ⎢⎣ 1 − β 4 where,

2 g ( p1 − p2 ) ⎤ ⎥ v1 ⎥⎦

(5)

v1= specific volume of the gas in m3/kgf

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p1 − p2 1 p1 K C K= Specific heat ratio p of the gas at state 1 Cv And all other terms are as defined in (1). Y = 1 − (0.41 + 0.35β 4 )

Pitot Tube Pitot tube is widely used for velocity measurement in aircraft. Its basic principle can be understood from fig. 6(a). If a blunt object is placed in the flow channel, the velocity of fluid at the point just before it, will be zero. Then considering the fluid to be incompressible, from eqn. (2), we have, p1 v12 p v2 + = 2+ 2 γ 2g γ 2g Now v 2 = 0 . Therefore, v12 p − p1 = 2 2g γ

or,

v1 =

2g

γ

(6)

( p 2 − p1 )

However, as mentioned earlier corrections are to be incorporated for compressible fluids. The typical construction of a Pitot tube is shown in fig. 6(b). Blunt object

PL V1

V2 = 0 Fig. 6(a) Pitot Tube: Basic Principle

p1

p2

Fig. 6(b) Pitot Tube: Construction

Rotameter The orificemeter, Venturimeter and flow nozzle work on the principle of constant area variable pressure drop. Here the area of obstruction is constant, and the pressure drop changes with flow rate. On the other hand Rotameter works as a constant pressure drop variable area meter. It can be only be used in a vertical pipeline. Its accuracy is also less (2%) compared to other types of flow meters. But the major advantages of rotameter are, it is simple in construction, ready to install and the flow rate can be directly seen on a calibrated scale, without the help of any other device, e.g. differential pressure sensor etc. Moreover, it is useful for a wide range of variation of flow rates (10:1). Version 2 EE IIT, Kharagpur 8

The basic construction of a rotameter is shown in fig. 7. It consists of a vertical pipe, tapered downward. The flow passes from the bottom to the top. There is cylindrical type metallic float inside the tube. The fluid flows upward through the gap between the tube and the float. As the float moves up or down there is a change in the gap, as a result changing the area of the orifice. In fact, the float settles down at a position, where the pressure drop across the orifice will create an upward thrust that will balance the downward force due to the gravity. The position of the float is calibrated with the flow rate. p2

Float

Tapered pipe

p1 Orifice area

Flow Fig. 7 Basic construction of a rotameter.

Let us consider,

γ 1 = Specific weight of the float γ 2 = specific weight of the fluid v f = volume of the float Af = Area of the float.

At = Area of the tube at equilibrium (corresponding to the dotted line) From equation (4), for incompressible fluid, we have, for the orifice, Cd A2 2g ( p1 − p2 ) Q= (7) A2 2 γ 2 1− ( ) A1 Now consider the free body diagram of the float, shown in fig. 8. Let, W

Fd

Fu Fig. 8 Forces acting on the float

Fd = Downward thrust on the float Fu = Upward thrust on the float Version 2 EE IIT, Kharagpur 9

W = Apparent weight of the float At balance, W = Fu − Fd

V f (γ 1 − γ 2 ) = p1 Af − p2 Af or, Therefore, vf p1 − p2 = (γ 1 − γ 2 ) Af Substituting the above expression in (7), we obtain: Q=

⎡ 2g v f ⎤ (γ 1 − γ 2 ) ⎥ ⎢ 2 ⎧ At − Af ⎫ ⎣⎢ γ 2 Af ⎦⎥ 1− ⎨ ⎬ ⎩ At ⎭ Cd ( At − Af )

(8)

2

⎧ At − Af ⎫ The term within the third bracket in the above expression is constant. If ⎨ ⎬ > γ 2 . For measurement of mass flow rate (W), we can write, W = γ 2Q = K K1 (γ 1 − γ 2 )γ 2 (11) dW = 0 , can be satisfied, if we select γ 1 = γ 2 . This can be achieved by using a The condition, dγ 2 hollow float, or a plastic float.

Electromagnetic Flowmeter Electromagnetic flowmeter is different from all other flowmeters due to its uniqueness on several accounts. The advantages of this type of flowmeter can be summarized as: 1. It causes no obstruction to flow path. 2. It gives complete linear output in form of voltage. 3. The output is unaffected by changes in pressure, temperature and viscosity of the fluid. 4. Reverse flow can also be measured. 5. Flow velocity as low as 10-6m/sec can be measured.

Electrodes v

e0

B

B e0

v Fig. 10 Electromagnetic Flowmeter

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Electromagnetic flowmeters are suitable for measurement of velocity of conducting (Mercury) and weakly conducting (water) liquid. The basic principle of operation can be understood from fig. 10. It works on the principle of basic electromagnetic induction; i.e. when a conductor moves along a magnetic field perpendicular to the direction of flow, a voltage would be induced perpendicular to the direction of movement as also to the magnetic filed. The flowing liquid acts like a conductor. External magnetic field is applied perpendicular to the direction of the flow and two electrodes are flushed on the wall of the pipeline as shown. The expression for the voltage induced is given by: eo = B l v (12) where l is the length of the conductor (diameter d in this case) and v is the velocity of the liquid. The above expression shows the complete relationship between the voltage induced and the velocity. However, the magnetic field applied is not d.c. if the liquid medium is water or any other polarizable liquid. This is because, if the magnetic field is d.d. the voltage induced will also be d.c. and a small amount of d.c. current will flow if a measuring circuit is connected to the terminals. This small d.c. current will cause electrolysis; oxygen and hydrogen bubbles will be formed and they will stick to the electrodes surfaces for some time. This will provide an insulating layer on the electrodes surfaces that will disrupt the voltage generation process. As a result, the magnetic field applied for these cases is a.c., or pulsed d.c. excitation. The meter can only be used for liquids having moderate conductivities (more than 10 μ mho / cm ). As a result, it is not suitable for gases or liquid hydrocarbons. The accuracy is around ±1% .

Turbine type Flowmeter Turbine type flowmeter is a simple way for measuring flow velocity. A rotating shaft with turbine type angular blades is placed inside the flow pipe. The fluid flowing through the pipeline will cause rotation of the turbine whose speed of rotation can be a measure of the flowrate. Referring fig.11, let blades make an angle α with the body. Then, ωr R = tan α − v where, − Q v = Average velocity of the fluid = A Q = Volumetric flowrate A = Effective flow area of the pipe R = Radius of the blade ω r = Angular speed of the blade. From the above expression, the volumetric flow rate can be related with the angular speed, as: (13) ωr = k Q where, tan α k= RA

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ωr

ωrR

α v

Fig. 11 Turbine type flowmeter

The speed of rotation of the turbine can be measured using several ways, such as, optical method, inductive pick up etc.

Vortex type Flowmeter Formation of vortex on a flowing stream by an obstruction like straw or stone is a common observation. But what is probably not commonly known is the fact that, the frequency of vortex formation is proportional to flow velocity. Transmitter

d Flow

Blunt object

Receiver Karman Votex

Fig. 12 Vortex type flowmeter

Fig.12 shows the basic principle of vortex type flowmeter. It is based on the principle of vertex shading. When a blunt object is placed on the passage of a flowing stream, vortices are formed. A vortes of this sort is called Karman Vortex. If the flow is turbulent and the Reynold’s number is RD > 104 , then the frequency of vortex formation is given by: N f = st v (14) d where, d= width of the blunt object Version 2 EE IIT, Kharagpur 13

v = velocity of the fluid Nst = A constant, called Strouhal Number. The fig. 12 shows a typical arrangement of measurement of frequency of vorticex formation using ultrasonic technique. Formation of a vortex will modulate the intensity of ultrasound received by the receiver, and the frequency of modulation can be measured easily.

Conclusion In this lesson, we have learnt about various techniques of flow measurement in industrial processes. It has been seen that most of the flow measurement techniques are based on the principle of obstruction type flowmeter. Orifice meters and venture meters are the two most popular types of transducers for flow measurement. However, they require, additional differential pressure transducers for converting the differential pressure generated into appropriate electrical signals and also square rooting devices in order to obtain a linear output proportional to flowrate. Comparatively, electromagnetic flowmeter provides a direct method for measurement of flowrate and gives a proportional voltage output with respect to flow. It also does not provide any obstruction to the flow path; as a result, there is no pressure drop. But this technique is suitable for conducting fluids only and cannot be used for gases. Moreover, often the polarization property of water creates problems and calls for an involved signal conditioning circuit. There are few other types of flowmeters, whose principles of operations could not be discussed here due to paucity of space. One of them is the ultrasonic flowmeter. This type of flowmeter is also non-intrusive type, i.e., it does not provide any obstruction to the flow passage. But it is quite costly, compared to other flowmeters. Positive displacement flowmeter is an integral type of flowmeter, in the sense, that it measures total flow in a given amount of time, and finds wide use in water meters, petrol pumps etc. Its construction is normally different from other types of flowmeters, though turbine type flowmeter with a counter to count the number of revolutions can also be used for this purpose.

Review Exercise 1. 2. 3. 4. 5. 6. 7.

What is meant by discharge coefficient in an orifice type flowmeter? Compare the advantages and disadvantages of an orifice meter and a venturimeter. Can a rotameter be used in a horizontal pipe line? If not, explain why? The magnetic field applied to an electromagnetic flowmeter is not constant, but time varying. Why? What are the flowmeters where the output is frequency varying with flow velocity? What is the difference between a constant area variable pressure drop flowmeter and a constant pressure drop variable area flowmeter? The pressure drop across an orifice is measured for a particular flow rate. If the flow rate is doubled, keeping all other parameters constant, what would happen to the pressure drop? a) It will remain the same. b) It will also be doubled. c) It will be halved. d) It will increase four times. Version 2 EE IIT, Kharagpur 14

8.

A rotameter designed to measure the flow rate of water is used to measure the flow rate of brine (specific gravity 1.15), without altering the scale. Would it more, or less? Justify.

Answer Q5. Q7. Q8.

Turbine type flowmeter and Vortex type flowmeter. (d) Less (refer eqn.(8)).

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Module 2 Measurement Systems Version 2 EE IIT, Kharagpur 1

Lesson 8 Measurement of Level, Humidity and pH Version 2 EE IIT, Kharagpur 2

Instructional Objectives At the end of this lesson, the student will be able to: 1. Name different methods for level and moisture measurements 2. Explain the basic techniques of level and humidity measurement 3. Explain the principle of pH measurement 4. Explain the necessity of using special measuring circuit for pH measurement

1.

Introduction

Level, humidity and pH are three important process parameters and their measurement find wide application in chemical and manufacturing industries. In this chapter we would provide a brief overview of the different techniques adopted for measurement of liquid level and humidity. The basic principle of pH measurement and the construction of pH electrodes are explained in section 4.

2.

Level Measurement

There are several instances where we need to monitor the liquid level in vessels. In some cases the problem is simple, we need to monitor the water level of a tank; a simple float type mechanism will suffice. But in some cases, the vessel may be sealed and the liquid a combustible one; as a result, the monitoring process becomes more complex. Depending upon the complexity of the situation, there are different methods for measuring the liquid level, as can be summarized as follows: (a) Float type (b) Hydrostatic differential pressure gage type (c) Capacitance type (d) Ultrasonic type (e) Radiation technique. Some of the techniques are elaborated in this section.

Hydrostatic Differential Pressure type The hydrostatic pressure developed at the bottom of a tank is given by: p=hρ g where h is the height of the liquid level and ρ is the density of the liquid. So by putting two pressure tapings, one at the bottom and the other at the top of the tank, we can measure the differential pressure, which can be calibrated in terms of the liquid level. Such a schematic arrangement is shown in Fig. 1 . The drum level of a boiler is normally measured using this basic principle. However proper care should be taken in the measurement compensate for variation of density of water with temperature and pressure.

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Capacitance type This type of sensors are widely used for chemical and petrochemical industries; and can be used for a wide range of temperature (-40 to 200 oC) and pressure variation (25 to 60 kg/cm2). It uses a coaxial type cylinder, and the capacitance is measured between the inner rod and the outer cylinder, as shown in Fig. 2. The total capacitance between the two terminals is the sum of (i) capacitance of the insulating bushing, (ii) capacitance due to air and liquid vapour and (iii) capacitance due to the liquid. If the total capacitance measured when the tank is empty is expressed as C1, then the capacitance or the liquid level of h can be expressed as: 2πε 0 (ε 1 − ε 2 )h Ct = C1 + ln (r2 / r1 ) where, ε 1 is the relative permittivity of the liquid and ε 2 is the relative permittivity of the air and liquid vapour (≈ 1) . Hence a linear relationship can be obtained with the liquid level. The advantage of capacitance type sensor is that permittivity of the liquid is less sensitive to variation of temperature and can be easily compensated.

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Ultrasonic type Ultrasonic method can be effectively used for measurement of liquid level in a sealed tank. An ultrasonic transmitter/receiver pair is mounted at the bottom of the tank. Ultrasonic wave can pass through the liquid, but gets reflected at the liquid-air interface, as shown in Fig.3. The time taken to receive the pulse is measured, that can be related with the liquid level. For accurate measurement, variation of speed of sound with the liquid density (and temperature) should be properly compensated.

Radiation technique Radioactive technique also finds applications in measurement of level in sealed containers. Radioactive ray gets attenuated as it passes through a medium. The intensity of the radiation as it passes a distance x through a medium is given by: I ( x) = I 0 e−α x where I0 is the incidental intensity and α is the absorption co-efficient of the medium. Thus if we measure the intensity of the radiation, knowing I0 , and α, x can be determined. There are several techniques which are in use. In one method, a float with a radioactive source inside is allowed to move along a vertical path with the liquid level. A Geiger Muller Counter is placed at the bottom of the tank along the vertical path and the intensity is measured. The basic scheme is shown in Fig. 4. The method used in a batch filling process of bottles, uses a source-detector assembly that can slide along the two sides of the bottle, as shown in Fig. 5 . As soon as the source-detector assembly passes through the liquid-air interface, there would be a large change in the signal received by the detector. Radioactive methods, though simple in principle, find limited applications, because of possible radiation hazards. However radioactive methods are routinely used for level measurement of grains and granular solids.

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3.

Humidity Measurement

Humidity measurement finds wide applications in different process industries. Moisture in the atmosphere must be controlled below a certain level in many manufacturing processes, e.g., semiconductor devices, optical fibres etc. Humidity inside an incubator must be controlled at a very precision level. Textiles, papers and cereals must be dried to a standard storage condition in order to prevent the quality deterioration. The humidity can be expressed in different ways: (a) absolute humidity, (b) relative humidity and (c) dew point. Humidity can be measured in different ways. Some of the techniques are explained below.

Hygrometer Many hygroscopic materials, such as wood, hair, paper, etc. are sensitive to humidity. Their dimensions change with humidity. The change in dimension can be measured and calibrated in terms of humidity.

Psychrometer Psychrometric method for measurement of relative humidity is a popular method. Two bulbs are used- dry bulb and wet bulb. The wet bulb is soaked in saturated water vapour and the dry bulb is kept in the ambient condition. The temperature difference between the dry bulb and wet bulb is used to obtain the relative humidity through a psychrometric chart. The whole process can also be automated.

Dew point measurement If a gas is cooled at constant pressure to the dew point, condensation of vapour will start. The dew point can be measured by placing a clean glass mirror in the atmosphere. The temperature of the mirror surface is controlled and reduced slowly; vapour starts condensation over the mirror. Optical method is used to detect the condensation phenomena, and the temperature of the mirror surface is measured.

Conductance/Capacitance method of measurement Many solids absorb moisture and their values of the conductance or capacitance change with the degree of moisture absorption. Moisture content in granules changes the capacitance between two electrodes placed inside. By measuring the capacitance variation, the moisture content in the Version 2 EE IIT, Kharagpur 6

granules can be measured. Similarly, moisture content in paper and textiles change their resistance. A schematic arrangement for measurement of moisture content in paper or textiles using Resistance Bridge is shown in Fig. 6.

Infrared Technique Water molecule present in any material absorb infrared wave at wavelengths 1.94µm, 2.95 µm and 6.2µm. The degree of absorption of infrared light at any of these wavelengths may provide a measure of moisture content in the material.

4.

Measurement of pH

pH is a measure of hydrogen ion concentration in aqueous solution. It is an important parameter to determine the quality of water. The pH value is expressed as: 1 pH = log10 C Where C is the concentration of H+ ions in a solution. In pure water, the concentration of H+ ions is 10-7 gm/ltr at 25o C. So the pH value is 1 pH = =7. log10 10−7 The advantage of using pH scale is that the activities of all strong acids and bases can be brought down to the scale of 0-14. The pH value of acidic solutions is in the range 0-7 and alkaline solutions in the range 7-14. The pH value of a solution is measured by using pH electrode. It essentially consists of a pair of electrodes: measuring and reference electrode, both dipped in the solution of unknown pH. These two electrodes essentially form two half-cells; the total potential developed is the difference between the individual electric potential developed in each half cell. While the potential developed in the reference cell is constant, the measuring cell potential is dependent on the hydrogen ion concentration of the solution and is governed by Nernst’s equation: RT E = E0 + ln(a C ) nF Where: Version 2 EE IIT, Kharagpur 7

E= e.m.f of the half cell E0= e.m.f of the half cell under saturated condition R= Gas constant (8.314 J/ 0C) T= Absolute temperature (K) N= valance of the ion F= Faraday Constant = 96493 C a= Activity co-efficient (0 ≤ a ≤ 1) ; for a very dilute solution, a → 1 C= molar concentration of ions.

Measuring Electrode The measuring electrode is made of thin sodium ion selective glass. A potential is developed across the two surfaces of this glass bulb, when dipped in aqueous solution. This potential is sensitive to the H+ ion concentration, having a sensitivity of 59.2 mv/pH at 250C. Fig. 7 shows the basic schematic of a measuring probe. The buffer solution inside the glass bulb has a constant H+ ion concentration and provides electrical connection to the lead wire.

Reference Electrode The basic purpose of a reference electrode is to provide continuity to the electrical circuit, since the potential across a single half cell cannot be measured. With both the measuring and reference cells dipped in the same solution, the potential is measured across the two lead wires. A reference electrode should satisfy the following basic requirements: The potential developed should be independent of H+ ion concentration. (i) The potential developed should be independent of temperature (ii) (iii) The potential developed should not change with time. Considering all these requirements, two types of reference electrodes are commonly used: (i) Calomel (Mercury-Mercurous Chloride) and (ii) Silver-Silver Chloride. The construction of a Calomel reference electrode is shown in Fig. 8. The electrical connection is maintained through the salt bridge.

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Sometimes the reference and measuring electrodes are housed together, as shown in Fig. 9. This type of electrode is known as Combination Electrode. The reference electrode used in this case is Silver-Silver Chloride. The combination is dipped in the solution whose pH is to be measured and the output voltage is the difference between the e.m.f.s generated by the measuring glass electrode and the reference electrode.

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Measuring scheme The sensitivity of pH probe is around 59.2mv/pH at 250C. This sensitivity should be sufficient for measurement of voltage using ordinary electronic voltmeters. But, that is not the case; special measuring circuits are required for measurement of pH voltage. This is because of the fact that the internal resistance of the pH probe as a voltage source is very high, in the order of 108-109 Ω. This is because of the fact; the electrical path between the two lead wires is completed through the glass membrane. As a result, the input resistance for of the measuring device must be at least ten times electrode resistance of the electrode. FET-input amplifier circuits are normally used for amplifying the voltage from the pH probe. Not only that, the insulation resistance between the leads must also be very high. They are normally provided with moisture resistance insulation coating. The voltage in the pH probe is temperature dependent, as evident from Nernst equation. As a result suitable temperature compensation scheme should also be provided in the measuring scheme.

Review Questions 1. How would you measure level of a liquid inside a sealed tank? Explain with a schematic arrangement any one of the methods. 2. Name different techniques used for level measurement of a liquid. Explain the principle of operation of hydrostatic differential pressure level gage. 3. Name few instances where measurement of humidity/ moisture finds important applications in industry. 4. How the moisture content in solids can be measured? Give an example and show the schematic arrangement. Version 2 EE IIT, Kharagpur 10

5. Define pH of a solution. What is the hydrogen ion concentration of a solution if the pH of the solution is 5.0? 6. Explain with simple sketches the construction of measuring electrode and reference electrode. Why two electrodes are required for pH measurement? 7. Why temperature compensation scheme should be provided in pH measurement? 8. What special arrangements are to be provided for amplifying the voltage generated in a pH electrode? Justify.

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Module 2 Measurement Systems Version 2 EE IIT, Kharagpur

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Lesson 9 Signal Conditioning Circuits Version 2 EE IIT, Kharagpur

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Instructional Objective The reader, after going through the lesson would be able to: 1. Identify the different building blocks of a measuring system and explain the function of each block. 2. Design an unbalanced wheatstone bridge and determine its sensitivity and other parameters. 3. Able to explain the advantage of using push-pull configuration in unbalanced a.c. and d.c. bridges. 4. Define CMRR of an amplifier and explain its importance for amplifying differential signal. 5. Compare the performances of single input amplifiers (inverting and non-inverting) in terms of gain and input impedance. 6. Draw and derive the gain expression of a three-op.amp. instrumentation amplifier.

1.

Introduction

It has been mentioned in Lesson-2 that a basic measurement system consists mainly of the three blocks: sensing element, signal conditioning element and signal processing element, as shown in fig.1. The sensing element converts the non-electrical signal (e.g. temperature) into electrical signals (e.g. voltage, current, resistance, capacitance etc.). The job of the signal conditioning element is to convert the variation of electrical signal into a voltage level suitable for further processing. The next stage is the signal processing element. It takes the output of the signal conditioning element and converts into a form more suitable for presentation and other uses (display, recording, feedback control etc.). Analog-to-digital converters, linearization circuits etc. fall under the category of signal processing circuits. The success of the design of any measurement system depends heavily on the design and performance of the signal conditioning circuits. Even a costly and accurate transducer may fail to deliver good performance if the signal conditioning circuit is not designed properly. The schematic arrangement and the selection of the passive and active elements in the circuit heavily influence the overall performance of the system. Often these are decided by the electrical output characteristics of the sensing element. Nowadays, many commercial sensors often have in-built signal conditioning circuit. This arrangement can overcome the problem of incompatibility between the sensing element and the signal conditioning circuit.

Input Measurand

Sensing element

Signal conditioning element

Signal processing element

Output Electrical output

Fig. 1 Elements of a measuring system.

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If one looks at the different cross section of sensing elements and their signal conditioning circuits, it can be observed that the majority of them use standard blocks like bridges (A.C. and D.C.), amplifiers, filters and phase sensitive detectors for signal conditioning. In this lesson, we would concentrate mostly on bridges and amplifiers and ponder about issues on the design issues.

2.

Unbalanced D.C. Bridge

We are more familiar with balanced wheatstone bridge, compared to the unbalanced one; but the later one finds wider applications in the area of Instrumentation. To illustrate the properties of unbalanced d.c. bridge, let us consider the circuit shown in fig.2 .Here the variable resistance can be considered to be a sensor, whose resistance varies with the process parameter. The output voltage is e0 , which varies with the change of the resistance x (= ΔR / R ) . The arm ratio of the bridge is p and E is the excitation voltage.

R2 = R(1+x)

R1 = pR e0

R3 = R

R4 = pR

E Fig. 2 Unbalanced D.C. bridge. Then, ⎡ R (1 + x) R ⎤ − e0 = ⎢ ⎥E ⎣ pR + R (1 + x ) pR + R ⎦ px = E ( p + 1 + x )( p + 1)

(1)

From the above expression, several conclusions can be drawn. These are: A. e0 vs. x Characteristics is nonlinear (since x is present in the denominator as well as in the numerator). B.

Maximum sensitivity of the bridge can be achieved for the arm ratio p=1. Version 2 EE IIT, Kharagpur

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The above fact can easily be verified by differentiating e0 with respect to p and equating to zero; i.e. de0 = 0 gives, dp x( p + 1 + x )( p + 1) − px ( 2 p + 2 + x) = 0 or, p 2 = 1 + x,

i.e. p = 1 + x ≈ 1 , for small x.

(2)

C.

Nonlinearity of the bridge decreases with increase in the arm ratio p, but the sensitivity is also reduced. e This fact can be verified by plotting 0 vs. x for different p, as shown in fig. 3. E D. For unity arm ratio (p=1), and for small x, we can obtain an approximate linear relationship as, x (3) e0 = E . 4

Unbalanced voltage (eo/E)

0.25 0.2

0.15

p=1

0.1

p = 10

0.05 p = 100

0 0

0.2 0.4 0.6 0.8

1

1.2 1.4 1.6 1.8

2

x

Fig. 3 Bridge characteristics for different arms ratio.

E.

We have seen that the maximum sensitivity of the bridge is attained at the arm ratio p=1. Instead of making all the values of R1, R2, R3, R4 equal under balanced condition, it could also be achieved by selecting different values with R1= R2, R3 = R4 for x = 0. But this is not advisable, since the output impedance of the bridge will be higher in the later case. So, from the requirement of low output impedance of a signal-conditioning element, it is better to construct the basic bridge with all equal resistances.

F.

It may appear from the above discussions, that, there is no restriction on selection of the bridge excitation voltage E. Moreover, since, more the excitation voltage, more is the output voltage sensitivity, higher excitation voltage is preferred. But the Version 2 EE IIT, Kharagpur

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restriction comes from the allowable power dissipation of resistors. If we increase E, there will be more power loss in a resistance element and if it exceeds the allowable power dissipation limit, self heating will play an important role. In this case, the temperature of the resistance element will increase, which again will change the resistance and the power loss. Sometimes, this may lead to the permanent damage of the sensor (as in case of a thermistor).

Push-pull Configuration The characteristics of an unbalanced wheatstone bridge with single resistive element as one of the arms can greatly be improved with a push-pull arrangement of the bridge, comprising of two identical resistive elements in two adjacent arms: while the resistance of one sensor decreasing, the resistance of the other sensor is increasing by the same amount, as shown in fig.4. The unbalanced voltage can be obtained as: ⎡ R(1 + x) R⎤ − e0 = ⎢ ⎥E ⎣ R(1 + x) + R(1 − x) 2 R ⎦ ⎡1 + x 1 ⎤ =⎢ − ⎥E 2⎦ ⎣ 2 x = E 2

(4)

R2 = R(1+x)

R1 = R(1-x)

e0 R3 = R

R4 = R

E Fig. 4 Unbalanced D.C. bridge with push pull configuration of resistance sensors.

Looking at the above expression, one can immediately appreciate the advantage of using pushpull configuration. First of all, the nonlinearity in the bridge output can be eliminated completely. Secondly, the sensitivity is doubled compared to a single sensor element bridge. The same concept can also be applied to A.C. bridges with inductive or capacitive sensors. These applications are elaborated below. Version 2 EE IIT, Kharagpur 6

3.

Unbalanced A.C. Bridge with Push-pull Configuration

Figures 5(a) and (b) shows the schematic arrangements of unbalanced A.C. bridge with inductive and capacitive sensors respectively with push-pull configuration. Here, the D.C. excitation is replaced by an A.C. source and two fixed resistances of same value are kept in the two adjacent arms and the inductive (or the capacitive) sensors are so designed that if the inductance (capacitance) increases by a particular amount, that of the other one would decrease by the same amount. For fig. 5(a), ⎡ jwL(1 + x) R⎤ e0 = ⎢ − ⎥E , ⎣ jwL (1 + x) + jwL(1 − x) 2 R ⎦

where w is the angular frequency of excitation, L is the nominal value of the inductance and x = ΔL . Simplifying, we obtain, L

L(1-x)

L(1+x)

e0

e0 R3 = R

R4 = R

C(1+x)

C(1-x)

R3 = R

R4 = R

~

~

E, ω

E, ω

Fig. 5 Unbalanced A.C. bridge with push-pull configuration: (a) for inductive sensor, and (b) for capacitive sensor.

x E, 2 which again shows the linear characteristics of the bridge. e0 =

(5)

For the capacitance sensor with the arrangement shown in fig. 5(b), we have:

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1 ⎤ ⎡ R⎥ jwC (1 + x) ⎢ E e0 = − ⎢1 2R ⎥ + 1 jwC (1 − x) ⎦⎥ ⎣⎢ jwC (1 + x) ⎡ jwC (1 − x) R⎤ =⎢ − ⎥E ⎣ jwC (1 + x) + jwC (1 − x) 2 R ⎦ x (6) =− E 2 where x = ΔC . As expected, we would also obtain here a complete linear characteristic, C irrespective of whatever is the value of x. But here is a small difference between the performance of an inductive sensor bridge and that of a capacitance sensor bridge (equation (5) and (6)): a negative sign. This negative sign in an A.C. bridge indicates that the output voltage in fig. 4(b) will be 1800 out of phase with the input voltage E. But this cannot be detected, if we use a simple A.C. voltmeter to measure the output voltage. In fact, if the value of x were negative, there would also be a phase reversal in the output voltage, which cannot be detected, unless a special measuring device for sensing the phase is used. This type of circuit is called a Phase Sensitive Device (PSD) and is often used in conjunction with inductive and capacitive sensors. The circuit of a PSD rectifies the small A.C. voltage into a D.C. one; the polarity of the D.C. output voltage is reversed, if there is a phase reversal.

Capacitance Amplifier Here we would present another type of circuit configuration, suitable for push-pull type capacitance sensor. The circuit can also be termed as a half bridge and a typical configuration has been shown in fig.6. Here two identical voltage sources are connected in series, with their common point grounded. This can be also achieved by using a center-tapped transformer. Two sensing capacitors C1 and C2 are connected as shown in the fig. 5 and the unbalanced current flows through an amplifier circuit with a feedback capacitor Cf . Now the current through the capacitors are: I 1 = V . jwC1 and I 2 = −V . jwC 2 Hence the unbalanced current: I = I 1 + I 2 = V . jw(C1 − C 2 ) And the voltage output of the amplifier: C − C2 I =− 1 V0 = − V (7) jwC f Cf As expected, a linear response can also be obtained by connecting a push-pull configuration of capacitance in fig.6. The gain can be adjusted by varying Cf. However, this is an ideal circuit, for a practical circuit, a high resistance has to be placed in parallel with Cf.

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C1

Cf

+ V

~ -

+ V

~

+

V0

C2 Fig. 6 A capacitance amplifier.

4.

Amplifiers

An Amplifier is an integral part of any signal conditioning circuit. However, there are different configurations of amplifiers, and depending of the type of the requirement, one should select the proper configuration.

Inverting and Non-inverting Amplifiers These two types are single ended amplifiers, with one terminal of the input is grounded. From the schematics of these two popular amplifiers, shown in fig.7, the voltage gain for the inverting amplifier is: e0 R =− 2 ei R1 while the voltage gain for the noninverting amplifier is: e0 R = 1+ 2 ei R1 Apparently, both the two amplifiers are capable of delivering any desired voltage gain, provided the phase inversion in the first case is not a problem. But looking carefully into the circuits, one can easily understand, that, the input impedance of the inverting amplifier is finite and is approximately R1 , while a noninverting amplifier has an infinite input impedance. Definitely, the second amplifier will perform better, if we want that, the amplifier should not load the sensor (or a bridge circuit).

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R2 R2 R1 -

R1

+

-

e0

e0 ei

+

Fig. 7 (a) Inverting amplifier, (b) noninverting amplifier.

Differential Amplifier Differential amplifiers are useful for the cases, where both the input terminals are floating. These amplifiers find wide applications in instrumentation. A typical differential amplifier with single op.amp. configuration is shown in fig.8. Here, by applying superposition theorem, one can easily obtain the contribution of each input and add them algebraically to obtain the output voltage as: R4 R R e0 = (1 + 2 ) e2 − 2 e1 (8) R1 R3 + R4 R1 If we select R4 R2 = , (9) R3 R1 then, the output voltage becomes: R e0 = 2 (e2 − e1 ) (10) R1 R2 R1 e1

e0

R3 e2

+

R4

Fig. 8 Differential amplifier.

However, this type of differential amplifier with single op. amp. configuration also suffers from the limitation of finite input impedance. In fact, several criteria are used for judging the Version 2 EE IIT, Kharagpur 10

performance of an amplifier. These are mainly: (i) offset and drift, (ii) input impedance, (iii) gain and bandwidth, and (iv) common mode rejection ratio (CMRR). The performance of an operational amplifier is judged by the gain- bandwidth product, which is fixed by the manufacturer’s specification. In the open loop, the gain is very high (around 105) but the bandwidth is very low. In the closed loop operation, the gain is low, but the achievable bandwidth is high. Normally, the gain of a single stage operational amplifier circuit is kept limited around 10, thus large bandwidth is achievable. For larger gains, several stages of amplifiers are connected in cascade. CMRR is a very important parameter for instrumentation circuit applications and it is desirable to use amplifiers of high CMRR when connected to instrumentation circuits. The CMRR is defined as: A CMRR = 20 log10 d (11) Ac where, Ad is the differential mode gain and Ac is the common mode gain of the amplifier. The importance of using a high CMRR amplifier can be explained with the following example:

Example -1 The unbalanced voltage of a resistance bridge is to be amplified 200 times using a differential amplifier. The configuration is shown in fig. 9 with R= 1000Ω and x=2 x 10-3. Two amplifiers are available: one with Ad =200 and CMRR= 80 dB and the other with Ad =200 and CMRR= 60dB. Find the values of V0 for both the cases and compute errors.

R

R2 = R(1+x) v0

+10V

Amplifier R

R

Fig. 9 Solution

Here x=2 x 10-3. Using (3), x ei = x10 = 5 mv = v d 4 The common mode voltage to the amplifier is vc = 5V , half the supply voltage. Version 2 EE IIT, Kharagpur 11

For amp.-1, Ad = 200 , 20 log Therefore,

Ad = 80dB Ac

Ad 200 = 10 4 , or, Ac = 4 = 0.02 . Ac 10

v0 = Ad v d + Ac vc = 200 × 5 × 10 −3 + 0.02 × 5 = 1.1 V So, Ideally, the voltage should have been 1.0 V, 200 times the bridge unbalanced voltage, but due to the presence of common voltage, 10% error is introduced. In the second case, CMRR is 60 dB, all other values remaining same. For this case, 200 Ac = 3 = 0.2 . Therefore, 10 v0 = Ad v d + Ac vc = 200 × 5 × 10 −3 + 0.2 × 5 = 2.0 V an error of magnitude 100% is introduced due to the common mode voltage! Referring to fig. 8, if we consider, the op. amp. to be an ideal one, then by selecting the resistances, such that, R4 R2 = , R3 R1 the effect of the common mode voltage can be eliminated completely, as is evident from eqn. (10). But if the resistance values differ, due to the tolerance of the resistors, the common mode voltage will cause error in the output voltage. The other alternative in the above example is to apply +5 and –5V at the bridge supply terminals, instead of +10V and 0V.

Instrumentation Amplifier Often we need to amplify a small differential voltage few hundred times in instrumentation applications. A single stage differential amplifier, shown in fig.8 is not capable of performing this job efficiently, because of several reasons. First of all, the input impedance is finite; moreover, the achievable gain in this single stage amplifier is also limited due to gain bandwidth product limitation as well as limitations due to offset current of the op. amp. Naturally, we need to seek for an improved version of this amplifier. A three op. amp. Instrumentation amplifier, shown in fig.10 is an ideal choice for achieving the objective. The major properties are (i) high differential gain (adjustable up to 1000) (ii) infinite input impedance, (iii) large CMRR (80 dB or more), and (iv) moderate bandwidth. From fig. 10, it is apparent that, no current will be drown by the input stage of the op. amps. (since inputs are fed to the non inverting input terminals). Thus the second property mentioned above is achieved. Looking at the input stage, the same current I will flow through the resistances R1 and R2. Using the properties of ideal op. amp., we can have: e −e e −e e − e2 I = 1 i1 = i1 i 2 = i 2 (12) R1 R2 R1 from which, we obtain,

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e1 = ei1 +

R1 (ei1 − ei 2 ) R2

e2 = ei 2 −

R1 (ei1 − ei 2 ) R2

Therefore,

2 R1 )(ei1 − ei 2 ) R2 The second stage of the instrumentation amplifier is a simple differential amplifier, and hence, using (10), the over all gain: R R 2R e0 = 4 (e2 − e1 ) = 4 (1 + 1 )(ei 2 − ei1 ) (13) R3 R3 R2 e1 − e2 = (1 +

ei1

+

e1

-

I

R4

R1 ei1

I

R2

R3 R3

e0 +

ei2 I

R1

ei2

+

R4

e2

Fig. 10 Three op. amp. Instrumentation Amplifier.

Thus by varying R2 very large gain can be achieved, but the relationship is inverse. Since three op. amps. are responsible for achieving this gain, the bandwidth does not suffer. There are many commercially available single chip instrumentation amplifiers in the market. Their gains can be adjusted by connecting an external resistance, or by selecting the gains (50, 100 or 500) through jumper connections.

5.

Concluding Remarks

Several issues have to be taken into consideration for the design of a signal conditioning circuit. Linearity, sensitivity, loading effect, bandwidth, common mode rejection are the important issues that affect the performance of the signal conditioning circuits. In this lesson, we have learnt about different configurations of unbalanced D.C. and A.C bridges, those are suitable for resistive, capacitive and inductive type transducers. Besides the characteristics of different types of amplifiers using common operational amplifiers have also been discussed in details. However, the actual design is dependent on the particular sensing element to be used and its characteristics. Version 2 EE IIT, Kharagpur 13

Several other types of signal conditioning circuits (e.g. phase sensitive detector, filters and many others) have been left out in the discussion.

Problems 1. A resistance temperature detector using copper as the detecting element has a resistance of 100Ω at 0oC. The resistance temperature co-efficient of copper is 0.00427/oC at 0oC. The sensing element is put in an unbalanced wheatstone bridge as in fig.2, the other arms are fixed resistances of 100Ω each. Plot the unbalanced voltage vs. temperature for temperature variation from 0oC to 100oC, if the excitation voltage is E = 2V. Are the characteristics linear or nonlinear? Justify your answer. 2. Explain the advantage of using push-pull arrangement in a bridge circuit. 3. For what arm ratio the sensitivity of an unbalanced wheatstone bridge is maximum? 4. A noninverting amplifier provides higher input impedance to the measuring circuit compared to an inverting amplifier- justify. 5. Define CMRR of an op. amp. Why is it important for designing a measurement system? 6. Design a differential amplifier of gain 10. 7. Discuss the main features of an instrumentation amplifier. 8. A differential amplifier circuit shown in fig. 8 has the resistances: R1 = 10K, R2 = 100K, R3 = 11K and R4 = 100K. Assuming the op. amp. To be an ideal one, find the CMRR of the amplifier. 9. A simple capacitance amplifier circuit is shown in fig. P1. C1 represents a capacitive sensor whose nominal value is 50 pF. C2 is a fixed capacitor of 25 pf. Find the output voltage if the sinusoidal excitation voltage 1V peak-to peak at frequency 1kHz. Assume the op.amp. to be an ideal one.

C1 ei

e0 +

Fig. P1.

Answers 1. For 100oC change in temperature is change in resistance for the RTD is 42.7Ω. So the condition ΔR R 30), one can safely approximate the variance as, _ 1 n Variance V = ∑ ( xi − x) 2 = σ 2 (6) n i =1 The term standard deviation is often used as a measure of uncertainty in a set of measurements. Standard deviation is also used as a measure of quality of an instrument. It has been discussed in Lesson-3 that precision, a measure of reproducibility is expressed in terms of standard deviation.

Propagation of Error Quite often, a variable is estimated from the measurement of two parameters. A typical example may be the estimation of power of a d.c circuit from the measurement of voltage and current in the circuit. The question is that how to estimate the uncertainty in the estimated variable, if the uncertainties in the measured parameters are known. The problem can be stated mathematically as, y = f ( x1 , x2 ,....., xn ) Let (7) If the uncertainty (or deviation) in xi is known and is equal to Δxi , (i = 1,2,..n) , what is the overall uncertainty in the term y? Differentiating the above expression, and applying Taylor series expansion, we obtain, ∂f ∂f ∂f Δy = Δx1 + Δx2 + ...... + Δxn (8) ∂x1 ∂x2 ∂xn Since Δxi can be either +ve or –ve in sign, the maximum possible error is when all the errors are positive and occurring simultaneously. The term absolute error is defined as, ∂f ∂f ∂f Absolute error : Δy = Δx1 + Δx2 + ...... + Δxn (9) ∂x1 ∂x2 ∂xn But this is a very unlikely phenomenon. In practice, x1 , x2 ,....., xn are independent and all errors do not occur simultaneously. As a result, the above error estimation is very conservative. To alleviate this problem, the cumulative error in y is defined in terms of the standard deviation. Squaring equation (8), we obtain, 2

2

⎛ ∂f ⎞ ⎛ ∂f ⎞ ∂f ∂f ⎟⎟ (Δx2 ) 2 + ...... + 2 ⎟⎟ (Δx1 ) 2 + ⎜⎜ (Δy ) = ⎜⎜ .(Δx1Δx2 ) + ....... (10) ∂x1 ∂x2 ⎝ ∂x2 ⎠ ⎝ ∂x1 ⎠ If the variations of x1 , x2 ,..... are independent, positive value of one increment is equally likely to be associated with the negative value of another increment, so that the some of all the cross 2

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product terms can be taken as zero, in repeated observations. We have already defined variance V as the mean squared error. So, the mean of (Δy ) 2 for a set of repeated observations, becomes the variance of y, or 2

2

⎛ ∂f ⎞ ⎛ ∂f ⎞ ⎟⎟ V ( x2 ) + ...... ⎟⎟ V ( x1 ) + ⎜⎜ V ( y ) = ⎜⎜ ∂ ∂ x x ⎝ 2⎠ ⎝ 1⎠ So the standard deviation of the variable y can be expressed as: 2 ⎡⎛ ∂f ⎞ 2 ⎤ ⎛ ∂f ⎞ 2 2 ⎟⎟ σ ( x2 ) + ......⎥ σ ( y ) = ⎢⎜⎜ ⎟⎟ σ ( x1 ) + ⎜⎜ ⎢⎣⎝ ∂x1 ⎠ ⎥⎦ ⎝ ∂x2 ⎠

1

(11)

2

(12)

Limiting Error Limiting error is an important parameter used for specifying the accuracy of an instrument. The limiting error (or guarantee error) is specified by the manufacturer to define the maximum limit of the error that may occur in the instrument. Suppose the accuracy of a 0-100V voltmeter is specified as 2% of the full scale range. This implies that the error is guaranteed to be within ± 2V for any reading. If the voltmeter reads 50V, then also the error is also within ± 2V. As a 2 result, the accuracy for this reading will be ×100 = 4% . If the overall performance of a 5 measuring system is dependent on the accuracy of several independent parameters, then the limiting or guarantee error is decided by the absolute error as given in the expression in (9). For example, if we are measuring the value of an unknown resistance element using a wheatstone bridge whose known resistors have specified accuracies of 1%, 2% and 3% respectively, then,

Since Rx =

R1 R2 , we have, R3

ΔRx = or,

R2 R RR ΔR1 + 1 ΔR2 − 1 2 2 ΔR3 R3 R3 R3

ΔRx ΔR1 ΔR2 ΔR3 = + − Rx R1 R2 R3

Then following the logic given to establish (9), the absolute error is computed by taking the positive values only and the errors will add up; as a result the limiting error for the unknown resistor will be 6%.

Importance of the Arithmetic Mean It has been a common practice to take a number of measurements and take the arithmetic mean to estimate the average value. But the question may be raised: why mean? The answer is: The most probable value of a set of dispersed data is the arithmetic mean. The statement can be substantiated from the following proof. Let x1 , x2 , x3 ,...., xn be a set of n observed data. Let X be the central value (not yet specified). So the deviations from the central value are ( x1 − X ), ( x2 − X ),....(xn − X ). Version 2 EE IIT, Kharagpur 6

The sum of the square of the deviations is: S sq = ( x1 − X ) 2 + ( x2 − X ) 2 + ... + ( xn − X ) 2

= x12 + x22 + .... + xn2 − 2 X ( x1 + x2 + ... + xn ) + nX 2 So the problem is to find X so that Ssq is minimum. So, dS sq = −2( x1 + x2 + ... + xn ) + 2nX = 0 dX

or, X =

_ 1 ( x1 + x2 + ... + xn ) = x n

So the arithmetic mean is the central value in the least square sense. If we take another set of readings, we shall reach at a different mean value. But if we take a large number of readings, definitely we shall come very close to the actual value (or universal mean). So the question is, how to determine the deviations of the different set of mean values obtained from the actual value?

Standard deviation of the mean Here we shall try to find out the standard deviation of the mean value obtained from the universal mean or actual value. Consider a set of n number of readings, x1 , x2 , x3, ...., xn . The mean value of this set expressed as: 1 ( x1 + x2 + ... + xn ) = f ( x1 + x2 + ... + xn ) n Using (11) for the above expression, we can write: _

x=

2

2

2

⎛ ∂f ⎞ ⎛ ∂f ⎞ ⎛ ∂f ⎞ ⎟⎟ V ( xn ) ⎟⎟ V ( x1 ) + ⎜⎜ ⎟⎟ V ( x2 ) + ...... + ⎜⎜ V ( x) = ⎜⎜ ⎝ ∂x1 ⎠ ⎝ ∂x2 ⎠ ⎝ ∂xn ⎠ 1 = 2 [V ( x1 ) + V ( x2 )..... + V ( xn )] n _

Now the standard deviation for the readings x1 , x2 ,..., xn is defined as: 1

⎡1 ⎤ 2 σ = ⎢ [V ( x1 ) + V ( x2 )..... + V ( xn )]⎥ , where n is large. ⎣n ⎦ Therefore, _ 1 σ2 V ( x ) = 2 ( n.σ 2 ) = n n Hence, the standard deviation of the mean, _

σ ( x) =

σ

(13)

n _

which indicates that the precision can be increased, (i.e. σ ( x) reduced) by taking more number of

observations. But the improvement is slow due to the

n factor. Version 2 EE IIT, Kharagpur 7

Example: Suppose, a measuring instrument produces a random error whose standard deviation is 1%. How many measurements should be taken and averaged, in order to reduce the standard deviation of the mean to = 10 ; or , n > 100. 0 .1 n

Least square Curve Fitting

y

Often while performing experiments, we obtain a set of data relating the input and output variables (e.g. resistance vs. temperature characteristics of a resistive element) and we want to fit a smooth curve joining different experimental points. Mathematically, we want to fit a polynomial over the experimental data, such that the sum of the square of the deviations between the experimental points and the corresponding points of the polynomial is minimum. The technique is known as least square curve fitting. We shall explain the method for a straight line curve fitting. A typical case of least square straight line fitting for a set of dispersed data is shown in Fig. 1. We want to obtain the best fit straight line out of the dispersed data shown.

x x x x

x

x

x

0

x

Fig. 1 Least square straight line fitting.

Suppose, we have a set of n observed data ( x1 , y1 ), ( x2 , y2 ),..., ( xn , yn ) . We want to estimate a straight line

y∗ = a0 + a1 x (14) such that the integral square error is minimum. The unknowns in the estimated straight line are the constants a0 and a1 . Now the error in the estimation corresponding to the i-th reading: ei = yi − y ∗ = yi − a0 − a1 xi The integral square error is given by, n

Se = ∑ ei2 i =1

n

= ∑ ( yi − a0 − a1 xi ) 2 i =1

For minimum integral square error,

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∂Se ∂Se =0, =0 ∂a0 ∂a1 or, n ∂Se = − 2∑ ( yi − a0 − a1 xi ) = 0 ∂a0 i =1 n ∂Se = − 2∑ xi ( yi − a0 − a1 xi ) = 0 ∂a1 i =1 From (15) and (16), we obtain,

and

n

∑y i =1

− a 0 .n − a 1 ∑ x i = 0 ,

i

i =1

n

n

∑x y −a ∑x −a ∑x i =1

(16)

n

n

i

(15)

0

i

i =1

1

i

i =1

2 i

=0.

Solving, we obtain, n

a1 =

n

n

n∑ xi yi − ∑ xi ∑ yi i =1

i =1

n

i =1

n

n∑ xi − (∑ xi ) 2 2

i =1

i =1

or, _ _ 1 n x y x − ∑ i i .y n i =1 a1 = 2 1 n 2 _ x − x ∑i n i =1

(17)

_

_

where x and y are the mean values of the experimental readings xi and yi respectively. Using (14), we can have, _

_

a0 = y − a1 x

(18)

Calibration and error reduction It has already been mentioned that the random errors cannot be eliminated. But by taking a number of readings under the same condition and taking the mean, we can considerably reduce the random errors. In fact, if the number of readings is very large, we can say that the mean value will approach the true value, and thus the error can be made almost zero. For finite number of readings, by using the statistical method of analysis, we can also estimate the range of the measurement error. On the other hand, the systematic errors are well defined, the source of error can be identified easily and once identified, it is possible to eliminate the systematic error. But even for a simple instrument, the systematic errors arise due to a number of causes and it is a tedious process to identify and eliminate all the sources of errors. An attractive alternative is to calibrate the instrument for different known inputs.

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Calibration is a process where a known input signal or a series of input signals are applied to the measuring system. By comparing the actual input value with the output indication of the system, the overall effect of the systematic errors can be observed. The errors at those calibrating points are then made zero by trimming few adjustable components, by using calibration charts or by using software corrections. Strictly speaking, calibration involves comparing the measured value with the standard instruments derived from comparison with the primary standards kept at Standard Laboratories. In an actual calibrating system for a pressure sensor (say), we not only require a standard pressure measuring device, but also a test-bench, where the desired pressure can be generated at different values. The calibration process of an acceleration measuring device is more difficult, since, the desired acceleration should be generated on a body, the measuring device has to be mounted on it and the actual value of the generated acceleration is measured in some indirect way. The calibration can be done for all the points, and then for actual measurement, the true value can be obtained from a look-up table prepared and stored before hand. This type of calibration, is often referred as software calibration. Alternatively, a more popular way is to calibrate the instrument at one, two or three points of measurement and trim the instrument through independent adjustments, so that, the error at those points would be zero. It is then expected that error for the whole range of measurement would remain within a small range. These types of calibration are known as single-point, two-point and three-point calibration. Typical input-output characteristics of a measuring device under these three calibrations are shown in fig.2. The single-point calibration is often referred as offset adjustment, where the output of the system is forced to be zero under zero input condition. For electronic instruments, often it is done automatically and is the process is known as auto-zero calibration. For most of the field instruments calibration is done at two points, one at zero input and the other at full scale input. Two independent adjustments, normally provided, are known as zero and span adjustments.

Actual Input variable (a)

Ideal

Actual Input variable (b)

Output variable

Ideal

Output variable

Output variable

One important point needs to be mentioned at this juncture. The characteristics of an instrument change with time. So even it is calibrated once, the output may deviate from the calibrated points with time, temperature and other environmental conditions. So the calibration process has to be repeated at regular intervals if one wants that it should give accurate value of the measurand through out.

Ideal

Actual Input variable (c)

Fig. 2 (a) single point calibration, (b) two point calibration, (c) three point calibration

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Conclusion Errors and calibration are two major issues in measurement. In fact, knowledge on measurement remains incomplete without any comprehensive idea on these two issues. In this chapter we have tried to give a brief overview about errors and calibration. The terms error and limiting error have been defined and explained. The different types of error are also classified. The methods for reducing random errors through repetitive measurements are explained. We have also discussed the least square straight line fitting technique. The propagation of error is also discussed. However, though the importance of mean and standard deviation has been elaborated, for the sake of brevity, the normal distribution, that random errors normally follows, has been left out. The performance of an instrument changes with time and many other physical parameters. In order to ensure that the instrument reading will follow the actual value within reason accuracy, calibration is required at frequent intervals. In this process we compare and adjust the instrument readings to give true values at few selected readings. Different methods of calibration, e.g., single point calibration, two point calibration and three point calibration have been explained.

References 1. 2. 3. 4.

M.B.Stout: Basic Electrical Measurements, 2/e, Prentice Hall of India, New Delhi, 1981. R.Pallas-Areny and J.G.Webster: Analog Signal Processing, John Wiley, NY, 1999. R.B. Northrup: Introduction to Instrumentation and Measurements (2/e), CRC Press, Boca Raton, 2005. J.W. Dally, W.F. Riley and K.G. McConnell: Instrumentation for Engineering Measurements (2/e), John Wiley & Sons, NY, 2003.

Review Questions 1. 2. 3. 4. 5.

Define error. A temperature indicator reads 189.80C when the actual temperature is 195.50C. Find the percentage error in the reading. Distinguish between gross error and systematic error. Write down two possible sources of systematic error. Explain the term limiting error. In a multiple range instrument it is always advisable to take a reading where the indication is near the full scale: justify. The most probable value of a set of dispersed data is the arithmetic mean: justify. The resistance value at a temperature t of a metal wire, Rt is given by the expression, R t =R 0 (1+αt) where, R 0 is the resistance at 0oC, and α is the resistance temperature coefficient. The resistance values of the metal wire at different temperatures have been tabulated as given below. Obtain the values of R 0 and α using least square straight line fitting. Temperature (oC)

20

40

60

80

100

Resistance (ohm)

107.5

117.0

117.0

128.0

142.5

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6. 7.

Most of the instruments have zero and span adjustments. What type of calibration is it? Explain three point calibration and its advantage over the other types of calibration.

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Module 3 Process Control Version 2 EE IIT, Kharagpur 1

Lesson 11 Introduction to Process Control Version 2 EE IIT, Kharagpur 2

Instructional Objectives At the end of this lesson, the student should be able to

1.

•

Distinguish with examples the difference between sequential control and continuous process control.

•

Identify three special features of a process.

•

Differentiate between manipulating variable and disturbance.

•

Distinguish between a SISO system and MIMO system and give at least one example in each case.

•

Develop linearised mathematical models of simple systems.

•

Give an example of a time delay system.

•

Identify the parameters on which the time delay is dependent.

•

Sketch the step response of a first order system with time delay.

•

State and explain the significance of transfer function matrix.

Introduction

We often come across the term process indicating a set up or a plant that we want to control. Thus by a process we may mean a unit of chemical plant (say, a distillation column), or a manufacturing system (say, an assembly shop), or a food processing industry and so on. We may want to automate the process; we may also like to control certain parameters of the system output (say, level of a tank, pressure of steam etc.). Broadly speaking, there could be two types of control; we might want to carry out. The first one is called sequential control, where the control action is carried out in a sequence. A good example for this type of operation could be in an automated car manufacturing system, where the assembly of parts is carried out in a sequence (on a conveyor line). Here the control action is sequential in nature and works in a preprogrammed open loop fashion (implying that there is no feedback of the output signal to the controller). Programmable Logic Controller (PLC) is often used to carry out these operations. But there are cases, where the control action needed is continuous in nature and precise control of the output variable is required. Take for example, the drum level control of a boiler. Here, the water level of the drum has to be maintained within a small band, in spite of variations of steam flow rate, steam pressure etc. This type of control is sometimes called modulating control, as the control variable is modulated to keep the process variable at a constant value. Feedback principle is used for these types of control. Now onwards, we would concentrate on the control of these types of processes. But in order to design a controller effectively, we must have a thorough knowledge about the dynamics of the process. A mathematical model of the process dynamics often helps us to understand the process behaviour under different operational conditions. In this lesson, we would discuss the basic characteristics of this type of processes where continuous control is used for controlling certain variables at the outputs.

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2.

Characteristics of a Process

Different processes have different characteristics. But, broadly speaking, there are certain characteristics features those are more or less common to most of the processes. They are: (i) (ii) (iii)

The mathematical model of the process is nonlinear in nature. The process model contains the disturbance input The process model contains the time delay term.

In general a process may have several input variables and several output variables. But only one or two (at most few) of the input variables are used to control the process. These inputs, used for manipulating the process are called manipulating variables. The other inputs those are left uncontrolled are called disturbances. Few outputs are measured and fed back for comparison with the desired set values. The controller operates based on the error values and gives the command for controlling the manipulating variables. The block diagram of such a closed loop process can be drawn as shown in Fig. 1.

In order to understand the behavour of a process, let us take up a simple open loop process as shown in Fig. 2. It is a tank containing certain liquid with an inflow line fitted with a valve V1 and an outflow line fitted with another valve V2. We want to maintain the level of the liquid in the tank; so the measured output variable is the liquid level h. It is evident from Fig.2 that there are two variables, which affect the measured output (henceforth we will call it only output) - the liquid level. These are the throttling of the valves V1 and V2. The valve V1 is in the inlet line, and it is used to vary the inflow rate, depending on the level of the tank. So we can call the inflow rate as the manipulating variable. The outflow rate (or the throttling of the valve V2 ) also affect the level of the tank, but that is decided by the demand, so not in our hand. We call it a disturbance (or sometimes as load). The major feature of this process is that it has a single input (manipulating variable) and a single output (liquid level). So we call it a Single-Input-Single-Output (SISO) process. We would see afterwards that there are Multiple-Input-Multiple-Output (MIMO) processes also.

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3.

Mathematical Modeling

In order to understand the behaviour of a process, a mathematical description of the dynamic behaviour of the process has to be developed. But unfortunately, the mathematical model of most of the physical processes is nonlinear in nature. On the other hand, most of the tools for analysis, simulation and design of the controllers, assumes, the process dynamics is linear in nature. In order to bridge this gap, the linearization of the nonlinear model is often needed. This linearization is with respect to a particular operating point of the system. In this section we will illustrate the nonlinear mathematical behaviour of a process and the linearization of the model. We will take up the specific example of a simple process described in Fig.2. Let Qi and Qo are the inflow rate and outflow rate (in m3/sec) of the tank, and H is the height of the liquid level at any time instant. We assume that the cross sectional area of the tank be A. In a steady state, both Qi and Qo are same, and the height H of the tank will be constant. But when they are unequal, we can write, dH (1) Qi − Qo = A dt But the outflow rate Qo is dependent on the height of the tank. Considering the Valve V2 as an orifice, we can write, (please refer eqn.(4) in Lesson 7 for details)

Qo =

Cd A2 1− β 4

2g

γ

( P1 − P2 )

(2)

We can also assume that the outlet pressure P2=0 (atmospheric pressure) and P1 = ρ gH

(3)

Considering that the opening of the orifice (valve V2 position) remains same throughout the operation, equation (2) can be simplified as: Qo = C H (4) Where, C is a constant. So from equation (1) we can write that, dH (5) Qi − C H = A dt The nonlinear nature of the process dynamics is evident from eqn.(5), due to the presence of the term H . Version 2 EE IIT, Kharagpur 5

In order to linearise the model and obtain a transfer function between the input and output, let us assume that initially Qi =Qo =Qs; and the liquid level has attained a steady state value Hs. Now suppose the inflow rate has slightly changed, then how the height will change? Now expanding Qo in Taylor’s series, we can have: •

Qo = Qo ( H s ) + Qo ( H s ) ( H − H s ) + ..... Taking the first order approximation, from eqn.(4), Qo ( H s ) = C H s = Qs

(6)

•

C 2 Hs Then from (1) and (6), we can write, d (H − H s ) C dH =A (7) Qi − Qs − (H − H s ) = A dt dt 2 Hs Now, we define the variables q and h, as the deviations from the steady state values, q = Qi − Qs (8) h = H − Hs We can write from (7), dh 1 (9) q=A + h dt R 2 Hs (10) Where, R = C It can be easily seen, that eqn.(9) is a linear differential equation. So the transfer function of the process can easily be obtained as: h( s ) R (11) = q(s) τ s + 1 Where, τ = RA . Qo ( H s ) =

It is to be noted that all the input and output variables in the transfer function model represent, the deviations from the steady state values. If the operating point (the steady state level Hs in the present case) changes, the parameters of the process (R and τ ) will also change. The importance of linearisation needs to be emphasized at this juncture. The mathematical models of most of the physical processes are nonlinear in nature; but most of the tools for design and analysis are for linear systems only. As a result, it is easier to design and evaluate the performance of a system if its mathematical model is available in linear form. Linearised model is an approximation of the actual model of the system, but it is preferred in order to have a physical insight of the system behaviour. It is to be kept in mind that this model is valid as long as the variation of the variables around the operating point is small. There are few systems whose dynamic behaviour is highly nonlinear and it is almost impossible to have a linear model of a system. For example, it is possible to develop the linearised transfer function model of an a.c. servomotor, but it is not possible for a step motor. Referring to Fig. 2, if the valve V1 is motorized and operated by electrical signal, we can also develop the model relating the electrical input signal and the output. Again, we have so far assumed that the opening of the valve V2 to be constant, during the operation. But if we also Version 2 EE IIT, Kharagpur 6

consider its variation, that would also affect the dynamics of the tank model. So, the effect of disturbance can be incorporated in the overall plant model, as shown in Fig.3, by introducing a disturbance transfer function D(s). D(s) can be easily by using the same methodology as described earlier in this section.

4.

Higher Order System Model

We have considered a single tank and developed the linearised model of it. So it has a single time constant τ . But there are more complex processes. If there are two tanks coupled together, as shown in Fig.4, then we would have two time constants τ 1 and τ 2 . But it is evident that the dynamics of two tanks are coupled. Considering the change in the inflow q(t) as the input and the change in the level of the second tank h2 (t ) as the output variable, with a little bit of calculation, it can be shown that the transfer function of this coupled tank system is, h2 ( s ) R2 = (12) 2 q( s) τ 1τ 1s + (τ 1 + τ 2 + A1R2 ) s + 1 The constants are similar to the earlier section with added suffixes corresponding to tank 1 and tank 2 respectively. In this case we have neglected the effects of the disturbances.

5.

Time delay

It has been mentioned earlier that one of the major characteristics of a process is the presence of time delay. This time delay term is often referred as “transportation lag”, since it is generated due to the delay in transportation of the output to the measuring point. The presence and effect of time delay can be easily explained with an example of a simple heat exchanger, as shown in Fig. 5.

Version 2 EE IIT, Kharagpur 7

In this case the transfer of heat takes place between the steam in the jacket and water in the tank. The measured output is the water temperature at the outlet T(t). For controlling this temperature, we may vary the steam flow rate at its inlet. So the manipulating variable is the steam flow rate. We can also identify a number of input variables those act as the disturbance, thus affecting the temperature at the water outlet; for example, inlet steam temperature, inlet water temperature and the water flow rate. The temperature transducer should be placed at a location in the water outlet line just after the tank (location A in Fig. 5). But suppose, due to the space constraint, the transducer was placed at location B, at a distance L from the tank. In that case, there would be a delay sensing this temperature. If T(t) is the temperature measured at location A, then the temperature measured at location B would be T (t − τ d ) . The time delay term τ d can be expressed in terms of the physical parameters as: τd = L v (13) where L is the distance of the pipeline between locations A and B; and v is the velocity of water through the pipeline. Noting from the Laplace Transformation table,

L f (t − τ d ) = e − sτ F ( s ) d

(14)

we can conclude that an additional term of e − sτ d would be introduced in the transfer function of the system due to the time delay factor. Thus the transfer function of an ordinary first order plant with time delay is Ke − sτ d G (s) = 1 + sτ and its step response to a unit step input is as shown in Fig. 6.

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It can be seen that though the input has been at t = 0, the output remains zero till t = τ d . This time delay present in the system may often be the main cause for instability of a closed loop system operation.

6.

Multiple Input Multiple Output Systems

So far we have considered the behviour of single input single output (SISO) systems only. In these cases, we had a single manipulating variable to control a single output variable. But in many cases, we have a number of inputs to control a number of outputs simultaneously, and the input-outputs are not decoupled. This will be evident if we consider a system, slightly modified system from that one shown in Fig. 4. In the modified system, we have added another inlet flow line in tank 2, as shown in Fig. 7.

If we consider the changes in inflow rates q1 and q2 are in inputs and the changes in the liquid levels of the two tanks h1 and h2 as the outputs, then the complete input-output behaviour can be modeled using the transfer function matrix, as shown below: ⎡ h1 ( s ) ⎤ ⎡G11 ( s ) ⎢ h ( s ) ⎥ = ⎢G ( s ) ⎣ 2 ⎦ ⎣ 21

G12 ( s ) ⎤ ⎡ q1 ( s ) ⎤ G22 ( s ) ⎥⎦ ⎢⎣ q2 ( s ) ⎥⎦

(15)

We define G ( s ) as the transfer function matrix and G12 ( s ) ⎤ ⎡G11 ( s ) G ( s) = ⎢ G22 ( s ) ⎥⎦ ⎣G21 ( s )

Version 2 EE IIT, Kharagpur 9

In general, if there are m inputs and p outputs, then the order of the transfer function matrix is p X m. The MIMO system can also be further classified depending on the number of inputs and outputs. If the number of inputs is more than the number of outputs (m>p), then the system is called an overactuated system. If the number of inputs is less than the number of outputs (m