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Lecture 4

BIOE3270 Lecture 4: BIOE3270-Notes4.pdf

by OneClass440147 , Winter 2011
5 Pages
68 Views
Winter 2011

Department
Biosystems Engineering
Course Code
BIOE 3270
Professor
Paliwal
Lecture
4

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Dynamic Response of Measuring Instruments
Transducer:
A device that converts a signal from one physical form to a corresponding signal having a different physical form
Physical form: mechanical, thermal, magnetic, optical, electrical
Transducers are energy modifiers or converters
Sensor: receives a stimulus and responds
Dynamic Characteristics: The properties of the system transient response to an input.
y=f(x)
The dynamic response is typically modeled by a constant coefficient linear differential equation

ax
dky(t)
dtkKa2
d2y(t)
d2ta1
dy(t)
dt aoy(t)x(t)
I practice, these models are limited to zero, first, and second order
Zero Order Instruments: have an output that is proportional to the input at all times
Static gain of the instrument (measure of its sensitivity)
o Example would be a strain gage where

R
L
A
(output is proportional to the length of the wire.
No delays
The sensor only changes the amplitude of the signal
Unit Step Function:
x(t)=0,

t0
x(t)=1, t>0
o For a zero order instrument, y(t)=Kx(t) y(t)=0, for

t0
; and y=K for t>0 therefore response of a zero
order instrument to a unit step function with height k, would be K
First Order Instruments: output is given by a non-homogeneous first-order differential equation

d
dt y(t)y(t)Kx(t)
o

is called the time constant of the instrument
o The response of a first order instrument to the unit step function

d
dt y(t)y(t)K
x(t)=1
Initial condition y(0)=0

y(t)K(1et
)
When t=

y(
)K(1e)0.632K
Second Order Instrument: second order non-homogenous differential equation

d2
dt2y(t)2
d
dt y(t)
2y(t)K
2x(t)

- constant, damping factor if the instrument

- angular frequency of the instrument
Temperature Measurement
Two necessities for temperature measurement:
o Reference temperature
o Rule for measuring the difference between a certain temperature from the reference point
Temperature Scales:
Fahrenheit: Devised by Gabriel Fahrenheit
o Earliest standard scale
o Wanted to fix human body temperature at 100 degrees + he wanted whole numbers for the
freezing and boiling points
32o F freezing pt. of water
212o F boiling pt. of water
98.6o F human body temperature
Celsius: devised by Andres Celsius
o Set out to fix the boiling point of H2O @ 0o an freezing pt. @ 100(this was later reversed)
o It became popular because if the 100 division
o The unit called ‘centigrade’
o In 1947 the general committee on Weight & Measures ruled that the unit be called ‘Celsius
Kelvin: William Kelvin
o Discovered the principle of energy as it related to the heat of the matter
o This led to the idea of ‘absolute zero’, a temperature below which it is impossible to go since the
matter would have zero energy at this point.
o 1oC difference in temperature = 1 K temperature difference
273.16 K = 0.01oC & 611 Pa (Triple point of water)
o A unique combination of temperature and pressure at which the solid, liquid and vapor phases co-
exist at equilibrium
Four Basic temperature measurement methods:
1. Thermal expansion: e.g. Hg thermometer
2. Thermoelectricity: e.g. thermocouple
3. Resistance: e.g. thermistor
4. Radiation: e.g. pyrometers
Choice of a particular thermometer depends on:
o Accuracy
o Recording requirements
o Control
o Requirements

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Description
Dynamic Response of Measuring Instruments Transducer:  A device that converts a signal from one physical form to a corresponding signal having a different physical form Physical form: mechanical, thermal, magnetic, optical, electrical  Transducers are energy modifiers or converters Sensor: receives a stimulus and responds Dynamic Characteristics: The properties of the system transient response to an input. y=f(x)  The dynamic response is typically modeled by a constant – coefficient linear differential equation k 2 a d y(t) K a d y(t) a dy(t)a y(t)  x(t) x dtk 2 d t 1 dt o  I practice, these models are limited to zero, first, and second order Zero Order Instruments: have an output that is proportional to the input at all times  Static gain of the instrument (measure of its sensitivity) L o Example would be a strain gage where R   (output is proportional to the length of the wire. A  No delays  The sensor only changes the amplitude of the signal Unit Step Function:  x(t)=0, t 0 x(t)=1, t>0 o For a zero order instrument, y(t)=Kx(t) y(t)=0, fort 0; and y=K for t>0 therefore response of a zero  order instrument to a unit step function with height k, would be K First Order Instruments: output is given by a non-homogeneous first-order differential equation d   y(t) y(t)  Kx(t) dt o  is called the time constant of the instrument o The response of a first order instrument to the unit step function  d   y(t) y(t) K x(t)=1 dt Initial condition y(0)=0 t y(t) K(1e )  When t=  y()  K(1e)  0.632K  Second Order Instrument: second order non-homogenous differential equation   2 d d 2 2 2y(t)2 y(t) y(t) K x(t) dt dt  - constant, damping factor if the instrument  - angular frequency of the instrument  Temperature Measurement  Two necessities for temperature measurement:  o Reference temperature o Rule for measuring the difference between a certain temperature from the reference point Temperature Scales: Fahrenheit: Devised by Gabriel Fahrenheit o Earliest standard scale o Wanted to fix human body temperature at 100 degrees + he wanted whole numbers for the freezing and boiling points 32 F – freezing pt. of water 212 F – boiling pt. of water 98.6 F – human body temperature Celsius: devised by Andres Celsius o Set out to fix the boiling point o2 H O @ 0 an freezing pt. @ 100(this was later reversed) o It became popular because if the 100 division o The unit called ‘centigrade’ o In 1947 the general committee on Weight & Measures ruled that the unit be called ‘Celsius’ Kelvin: William Kelvin o Discovered the principle of energy as it related to the heat of the matter o This led to the idea of ‘absolute zero’, a temperature below which it is impossible to go since the matter would have zero energy at this point. o 1 C difference in temperature = 1 K temperature difference 273.16 K = 0.01 C & 611 Pa (Triple point of water) o A unique combination of temperature and pressure at which the solid, liquid and vapor phases co- exist at equilibrium Four Basic temperature measurement methods: 1. Thermal expansion: e.g. Hg thermometer 2. Thermoelectricity: e.g. thermocouple 3. Resistance: e.g. thermistor 4. Radiation: e.g. pyrometers Choice of a particular thermometer depends on: o Accuracy o Recording requirements o Control o Requirements o Temperature range o Location (ambient temperature) Thermal Expansion: Liquid-in-glass thermometer: o Based on expansion/contraction of liquid in a glass tube o Hg  freezes @ -39 C  Range: -35 C to 500 C o Alcohol  Range: -70 to 120 C Partial immersion: immersed to an indicated level Complete/Full immersion: entire thermometer is
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