The document discusses the static and dynamic performance characteristics of measuring instruments. It describes how instruments can be modeled as zero-order, first-order, or second-order systems depending on how their output responds to changes in input over time. Zero-order instruments have an immediate output response, while first-order instruments exhibit a lag due to a time constant. Second-order instruments may also oscillate before reaching steady-state. Examples are given like thermometers and potentiometers to illustrate different order responses. Dynamic inputs like step, ramp and periodic signals are also discussed to analyze instrument behavior under transient and steady-state conditions.

2. Static Performance of Instrument
The static characteristics of instruments are
related with steady state response.
•The relationship between the output and the
input when the input does not change, or the
input is changing at a slow rate.
•Range & Span
•Linearity & Sensitivity
•Environmental effects
•Hysteresis
•Resolution
•Repeatability
•Death space

3. The quantity to be measured by an instrument could be either constant or rapidly
varying with time.
The quantities to be measured by an instrument which are either constant or vary
very slowely with time , the quantities are termed as static could be either
constant or rapidly varying with time.

4. Performance Characteristics of Measuring Instruments
The response of an instrument to a particular input is the guiding factor to decide
its choice out of the available options. The input to the instrument can be
constant or varying with time. Therefore performance characteristics can be
headed into two types as:
• Static Performance Characteristics
• Dynamic Performance Characteristics
Static Calibration: - Static Calibration refers to a procedure where an input, either
constant or slowly time varying, is applied to an instrument and corresponding output is
measured with all other input kept constant at some value. The change in input quantity
will create a corresponding change in the output. The functional relationship between the
output and input is referred to as static calibration. The static calibration can be expressed
analytically as Q=f(x) (i.e. output as function of input), or graphically or in tabular format.
A graphical representation between Q and x is known as calibration curve.

5. Aspect of dynamic response are :
(i) Fidelity/Faithfulness (ii) Speed of response and delay caused
Fidelity : It is system’s ability to present faithfully the information in the measurand
Fidelity is defined in terms of :
1. Amplitude response
2. Frequency response
3. Phase response
1. Amplitude response
 It is the ability of the system to treat all input amplitudes
equally and uniformly . Practically, it may not be possible. So, it
is desired that for over a specified range of input amplitudes
the ratio of output amplitude to input amplitude should
remain constant.

6. 2. Frequency response
 It is the ability of the system to treat inputs of all frequencies
equally and uniformly . Again, Practically it is desired that the
ratio of output to input amplitudes should remain constant
over some desired frequency range.
Frequency response
3. Phase response
 It is the ability of the system to treat inputs of all
frequencies uniformly in terms of causing time delay or
causing a shift timewise.
output
Rise time Time
Output of the
System response

7. Speed of response and delay time
Are defined or judged by the followings :
(i) Dead time : It is time taken by the system to begin to respond after a change in the
input .
(ii) Delay time or rise time : It is time required for the output to attain a certain
magnitude (usually 90%) of the step input .
Output of the
System response
Rise time
output
Time
(iii) Response time : It is time required
for the output to attain a specified value
(usually± 90%) of the steady state or
final value.
(vi) Slew time : It is the maximum rate of
change that can be handled by the
system.
(v) Time constant : It is the time
required for the output of a first order
measuring system to attain 63.2 % of its
final value when subjected to a step
input.

8. Modeling for dynamic performance
 For studying the dynamic response of an instrument it is to be represented by a
mathematical model. That a relationship between input and output of the instrument
is to be determined taking into account the physical parameters of its elements.
 A generalized relationship for a second order system can be expressed as:
𝑨 𝟐
𝒅 𝟐
𝜽 𝟎
𝒅𝒕 𝟐
+ 𝑨 𝟏
𝒅𝜽 𝟎
𝒅𝒕
+ 𝑨 𝟎 𝜽 𝟎 = 𝑩 𝟎 𝜽𝒊
Where ɵi = input signal /information
ɵ0 = output signal /information
A2, A1, A0, B0 are constants and represent system parameters ,
the significant of which shall be explained later

9.  The order of the system (instrument) is the order of the differential equation
representing it. Above , differential equation contains second order derivative, so
called a second order system.
Systems of various orders
Zeros order systems
 Zeros order systems is represented by a differential equation of zero order as :
𝑨 𝟎 𝜽 𝟎 = 𝑩 𝟎 𝜽𝒊
𝑶𝑹 𝜽 𝟎=
𝑩 𝟎
𝑨 𝟎
𝜽𝒊
𝑶𝑹 𝜽 𝟎= 𝑲 𝜽𝒊
Where, K is the static sensitivity of the system and is the only parameters
which characterized a zero order system.

10.  It is a system where output is directly proportional to the input, no matter how
input varies. Output , thus , is a true and faithful reproduction of input .
 A mechanical lever (force is proportional to displacement ) and a linear electrical
potentiometer (voltage proportional to displacement of the wiper) represent a zero
order system.
Kɵi ɵ0

11. First order system
 Such a system is represented by a first order differential equation
.
𝑨 𝟏
𝒅𝜽 𝟎
𝒅𝒕
+ 𝑨 𝟎 𝜽 𝟎 = 𝑩 𝟎 𝜽𝒊
𝑨𝒊
𝑨 𝟎
𝒅𝜽 𝟎
𝒅𝒕
+ 𝜽 𝟎 =
𝑩 𝟎
𝑨 𝟎
𝜽𝒊
𝝉
𝒅𝜽 𝟎
𝒅𝒕
+ 𝜽 𝟎 = 𝑲𝜽𝒊
Is called time constant Where, 𝝉 =
𝑨 𝒊
𝑨 𝒐
Is called static sensitivity Where, K =
𝑩 𝒊
𝑨 𝒐

12.  In terms of D- operator,
𝐷 ≡
𝑑
𝑑𝑡
𝜏𝐷𝜃0 + 𝜃0 = 𝐾𝜃𝑖
𝜃0
𝜃𝑖
=
𝐾
𝜏𝐷 + 1
 Above equation represents the transfer – function operator for the first
order system.
 Temperature measuring devices like thermocouple, mercury in glass
thermometer and electrical network of resistance and capacitance can be
represented by a first order.
 A first order mechanical system is represented by a spring and a dashpot
assuming the mass to be zero . In this case time constant is
𝜏 =
𝑐
𝑘
=
𝐷𝑎𝑚𝑝𝑖𝑛𝑔 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑆𝑡𝑖𝑓𝑓𝑛𝑒𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑝𝑟𝑖𝑛𝑔
=
𝑁
𝑚 𝑠−1
𝑁
𝑚
= 𝑠 (𝑠𝑒𝑐𝑜𝑛𝑑)

13. Dynamic Inputs
Periodic Input :- Repeating cyclically with time or repeating itself after a constant

18. The dynamic characteristics of a measuring instrument describe its behaviour
between the time a measured quantity changes value and the time when the
instrument output attains a steady value in response.
In any linear, time-invariant measuring system, the following general relation can
be written between input and output for time t > 0:
where qi is the measured quantity, q0 is the output reading and a0 . . . an, b0 . . . bm
are constants.
If we limit consideration to that of step changes in the measured quantity only, then
equation (2.1) reduces to:

19. Zero order instrument
If all the coefficients a1 . . . an other than a0 in equation (2.2) are assumed zero, then:
where K is a constant known as the instrument sensitivity as defined earlier.
Any instrument that behaves according to equation (2.3) is said to be of zero order
type. Following a step change in the measured quantity at time t, the instrument
output moves immediately to a new value at the same time instant t, as shown in
Figure 2.10.
A potentiometer, which measures motion, is a good example of such an instrument,
where the output voltage changes instantaneously as the slider is displaced along the
potentiometer track.

21. First order instrument
If all the coefficients a2 . . . an except for a0 and a1 are assumed zero in
equation (2.2) then:
Any instrument that behaves according to equation (2.4) is known as a first order
instrument. If d/dt is replaced by the D operator in equation (2.4), we get:
Defining K = b0/a0 as the static sensitivity and τ = a1/a0 as the time constant of
the system, equation (2.5) becomes:
If equation (2.6) is solved analytically, the output quantity q0 in response to a step
change in qi at time t varies with time in the manner shown in Figure 2.11. The time
constant of the step response is the time taken for the output quantity q0 to reach
63% of its final value.

22. The liquid-in-glass thermometer is a good example of a first order instrument. It is
well known that, if a thermometer at room temperature is plunged into boiling
water, the output e.m.f. does not rise instantaneously to a level indicating 100°C,
but instead approaches a reading indicating 100°C in a manner similar to that
shown in Figure 2.11.
A large number of other instruments also belong to this first order class: this is of
particular importance in control systems where it is necessary to take account of the
time lag that occurs between a measured quantity changing in value and the
measuring instrument indicating the change. Fortunately, the time constant of many
first order instruments is small relative to the dynamics of the process being
measured, and so no serious problems are created.

23. Example 2.3
A balloon is equipped with temperature and altitude measuring instruments and has
radio equipment that can transmit the output readings of these instruments back to
ground. The balloon is initially anchored to the ground with the instrument output
readings in steady state. The altitude-measuring instrument is approximately zero order
and the temperature transducer first order with a time constant of 15 seconds.

24. temperature on the ground, T0, is 10°C and the temperature Tx at an altitude of x
metres is given by the relation: Tx = T0 - 0.01x
(a) If the balloon is released at time zero, and thereafter rises upwards at a velocity of
5 metres/second, draw a table showing the temperature and altitude measurements
reported at intervals of 10 seconds over the first 50 seconds of travel. Show also
in the table the error in each temperature reading.
(b) What temperature does the balloon report at an altitude of 5000 metres?
Solution
In order to answer this question, it is assumed that the solution of a first order
differential equation has been presented to the reader in a mathematics course. If the
reader is not so equipped, the following solution will be difficult to follow.
Let the temperature reported by the balloon at some general time t be Tr. Then Tx
is related to Tr by the relation:

26. This result might have been inferred from the table above where it can be seen that
the error is converging towards a value of 0.75. For large values of t, the transducer
reading lags the true temperature value by a period of time equal to the time constant of15
seconds. In this time, the balloon travels a distance of 75 metres and the temperature
falls by 0.75°. Thus for large values of t, the output reading is always 0.75° less than
it should be.

27. Second order instrument
If all coefficients a3 . . . an other than a0, a1 and a2 in equation (2.2) are assumed zero,
then we get:

28. This is the standard equation for a second order system and any instrument whose
response can be described by it is known as a second order instrument.
If equation (2.9) is solved analytically, the shape of the step response obtained depends
on the value of the damping ratio parameter .
For case (A) where ξ = 0, there is no damping and the instrument output exhibits
constant amplitude oscillations when disturbed by any change in the physical quantity
measured.
For light damping of ξ = 0.2, represented by case (B), the response to a step change in
input is still oscillatory but the oscillations gradually die down.
Further increase in the value of reduces oscillations and overshoot still more, as
shown by curves (C) and (D), and finally the response becomes very overdamped as
shown by curve (E) where the output reading creeps up slowly towards the correct
reading.

30. 2. Transient Input : varying non- cyclically with time. As shown in figure, the signal is
of a definite duration and becomes zero after certain period of time

31. FLOW MEASUREMENTS
 There are different types of flow measuring techniques that are used in
industries.
 The common types of flowmeters that find industrial applications can be
listed as below:
(a) Obstruction type (differential pressure or variable area)
(b) Inferential (turbine type)
(h) Mass flowmeter (Coriolis).
(c) Electromagnetic
(d) Positive displacement (integrating (e) fluid dynamic (vortex shedding),
(f) Anemometer (g) ultrasonic

32. Obstruction type flowmeter
 Obstruction or head type flowmeters are of two types: differential pressure
type and variable area type.
OBSTRUCTION
FLOW TYPE
differential
pressure type
- Orifice meter
- Venturimeter
- Pitot tube
variable area type

33.  In all the cases, an obstruction is created in the flow passage and the pressure drop
across the obstruction is related with the flow rate.
Basic Principle
 It is well know that flow can be of two types: viscous and turbulent. Whether a flow
is viscous or turbulent can be decided by the Reynold’s number RD. If RD > 2000, the
flow is turbulent.
 We consider the fluid flow through a closed channel of variable cross section, as
shown in fig. 1.

34.  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.
From Bernloulli’s equation:

35.  This expression is valid for incompressible fluids (i.e. liquids) only and the relationship
between the volumetric flow rate and pressure difference is nonlinear.
Orifice meter
 An orifice plate is placed in the pipe line, as shown in fig.2. The orifice plate is a
circular plate with a hole in the center.

36.  Pressure tappings are normally taken distances D and 0.5D upstream and
downstream the orifice respectively (D is the internal diameter of the pipe).
 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.

37. venturimeter
 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 major disadvantage is the high cost of the meter.

38.  Flow nozzle is a compromise between orifice plate and venturimeter. The typical
construction is shown in fig. 5.
Flow nozzle
 They are not recommended for low flow rate measurement.

39. Flow measurement of compressible fluids
 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 :

40. 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,

41.  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).

42. Rotameter
 The orificemeter, Venturimeter and flow nozzle work on the principle of constant
area variable pressure drop.
 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).
 The basic construction of a rotameter is shown in fig. 7. It consists of a vertical
pipe, tapered downward.

45. Construction of the float
The construction of the float decides heavily, the performance of the rotameter. In
general, a float should be designed such that:
(a) it must be held vertical
(b) it should create uniform turbulence so as to make it insensitive to viscosity
(c) it should make the rotameter least sensitive to the variation of the fluid density.
 A typical construction of the float is shown in fig. 9. The top section of the float has a
sharp edge and several angular grooves. The fluid passing through these grooves,
causes the rotation of the float. The turbulence created in this process reduces the
viscous force considerably.

47. 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).
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
 Cd is defined as the ratio of the actual flow and the ideal flow and is always less
than one.

48. MASS FLOWMETERS
 The measurement of mass flow can be obtained as the product of volumetric flow and
density or as a direct measurement of the mass flow of the flowing process gas, liquid,
or solids.
 The mass flow of homogeneous gases is most frequently measured by thermal
flowmeters.
 The main advantage of these detectors is their good accuracy and very high
rangeability.
 The main disadvantage is their sensitivity to specific heat variations in the process
fluid due to composition or temperature changes.
 The mass flow of liquids and gases can be directly detected by angular-momentum
devices or indirectly through the measurement of volumetric flow and density.