Chap 1c- Dynamic Characteristic of Instrument- Updated

8/9/2019 Chap 1c- Dynamic Characteristic of Instrument- Updated

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DYNAMIC CHARACTERISTICSOF INSTRUMENT


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INTRODUCTION

Instrument rarely respond instantaneously tochanges in the measured variables

Instead, they exhibit a characteristicslowness/sluggishness due to mass, thermalcapacitance, fluid capacitance or electric capacitance

Pure delay in time is often encountered where theinstrument are ‘wait’ for some reaction to take place.

The dynamic behavior of an instrument isdetermined by subjecting its primary element tosome known and predetermined variation inmeasured quantity


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Dynamics characteristic determined by analyzing theresponse of the sensor to a variable input signal.

The reason for dynamics characteristic is the

presence of energy- storing elements:- Inertia (mass)

- Capacitances (electrical, thermal)

INTRODUCTION cont’d


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Type of dynamic input

The dynamic response characteristic - desired input is notconstant but varies rapidly with the time.

The dynamic inputs to an instrument may be of thefollowing types:--

Periodic input-Varying cyclically with time or repeatingitself after a constant interval. The input may be ofharmonic or non harmonic type.


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Transient input-Varying non-cyclically with time. Thesignal is of a definite duration and becomes zero after acertain period of time.

Random input-Varying randomly with time, with nodefinite period and amplitude. This may be continuous, but not cyclic


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Types of the input function

1) Impulse function – if a system is subjected toshock input

2) Step function – if a system is subjected to

sudden disturbances3) Ramp function – if the inputs are graduallychanging function of time

4) Sinusoidal function – oscillation of spring,

ocean waves


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Consider first the ramp function shown in the upper left. It is zero

for tT, and goes linearly from 0 to 1 as time goesfrom 0 to T.

If we let T→0, we get a unit step function, γ(t) (upper right).

If we take the derivative of our ramp function (lower left), we get arectangular pulse with height 1/T (the slope of the line) and widthT. This rectangular pulse has area (height·width) of one.

If we take the limit as T→0, we get a pulse of infinite height, zero width, but still with an area of one; this is the unit impulse and we

represent it by δ(t). Since we can't show the height of the impulse onour graph, we use the vertical axis to show the area. The unitimpulse has area=1, so that is the shown height.


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First Order Instrument

For studying the dynamic, it is necessary to representeach instrument by it mathematical model, from

which the governing relation between its input &output is obtained as below figure,


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input output

Xi (t) Xo (t)

K

1 + D

Block diagram of a first order instrument

Relation between the output and input signals is

(1+ D) Xo = K Xi (t) where, Xo is output signals in function of time

Xi is input signals in function of timeK is constant

is time constantD is the time derivative operator

With the term in the block diagram above as transfer function of theinstrument, K = Xo (t)

(1+ D) Xi (t)


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The governing equation of a first order system is

1 ( )o

o o o i

dx

a a x b x t dt or K = Xo (t)(1+ D) Xi (t)

where, K = bo /ao is static sensitivity

= a1 /ao is time constantD = d/dt is time derivative operator

For the static case, D = 0 , which is a zeroth order system for which

Where K is the static sensitivity

( ) ( )o i

x t Kx t


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(a) Xo/Xi = 0.8 and = 0.2 s

for 1st order instrument with sinusoidal inputXo/Xi = 1/ [1+ ()2] = 0.8

1/ [1+ (0.2)2] = 0.8

=3.75 rad/s

where = 2π f , so f = 0.597 Hz

(b) Xo/Xi = 0.8 and = 0.2 sfor 1st order instrument with step input

Xo/Xi = 1 – e – t/ = 0.8

0.2 = e – t/0.2

t = 0.322 s

Solution :


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First Order Response

The simplest time response is shown in Figure 1 asthe output change in time following a step input as inFigure 2.


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This is called first order - the time response isdetermined by the solution of a first-orderdifferential equation.

A general equation can be written for this responseindependent of the sensor, the variable beingmeasured, or the static transfer function.

The equation gives the sensor output as a function oftime following the step input:


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The sensor output is in error during the transitiontime of the output from bi to bf

The actual variable value was changedinstantaneously to a new value at t = 0.

Equation (I .7) describes transducer output very well except during the initial time period-that is, atthe start of the response near t = 0.

Thermometer, thermocouples & pressure gages(bellow/diaphragm) – may be called first order

type.


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Time Constant: The time constant, τ, is part of thespecification of the sensor. Its significance can beseen by writing Equation (1.7) as:

The quantity on the left is the change in output as afunction of time, whereas (bf - bi) is the total changethat will occur.

The square-bracketed term is the fraction of totalchange as a function of time


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To find the change that has occurred at a timenumerically equal to τ, set t = τ in Equation (1.8) andfind

One time constant represents the time at which the output

value has changed by approximately 63% of the total change. τ is sometimes referred to as the 63% time, the response time,

or the e-folding time.

For a step change, the output response has

approximately reached its final value after five timeconstants, since from Equation (1.8) we find


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Example:

A sensor measures temperature linearly with a statictransfer function of 33 mV/ C and has a 1.5-s timeconstant. Find the output 0.75 s after the inputchanges from 200C to 410C. Find the error intemperature this represents.

Solution :


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This corresponds to an indicated temperature of

So the error is 12.7 0C. because the actualtemperature is 410C


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Real-Time Effects

The concept of the exponential time response andassociated time constant is based on a suddendiscontinuous change of the input value

In the real world, such instantaneous changes occurrarely if ever, and thus we have presented a worstcase situation in the time response.

In general, a sensor should be able to track any

changes in the physical dynamic variable in a timeless than one time constant


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Second-Order Response

A step change in the input causes the output tooscillate for a short period of time before settlingdown to a value that corresponds to the new input

Such oscillation (and the decay of the oscillation

itself) is a function of the sensor This output transient generated by the transducer is

an error and must be accounted for in anymeasurement involving a transducer with this

behavior. Liquid manometers & pressure gage (mass in system,

spring & viscous damping mean) – second order type


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This is called a second-order- the time behavior isdescribed by a second-order differential equation.

A typical output curve that might be expected from atransducer having a second-order response for adiscontinuous change in the input.

Characteristic second-order oscillatory time response of asensor


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It is impossible to describe this behavior by a simpleanalytic expression, as it is with the first-orderresponse.

The general behavior (the general nature of theresponse) can be described in time as:

This equation shows the basic damped oscillationoutput of the device. The damping constant,α, andnatural frequency, f n are characteristics of thetransducer itself and must be considered in manyapplications.


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Solution :


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Solution :

Chap 1c- Dynamic Characteristic of Instrument- Updated

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