Lecture 2: Measurement and Instrumentation

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Lecture 2 Measurement and Instrumentation
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Time vs. Frequency Domain

• Different ways of looking at a problem
• Interchangeable no information is lost in
  changing from one domain to another
• Benefits from changing perspective the solution
  to difficult problems can often become quite
  clear in the other domain

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Time Domain Analysis

• The traditional way of observing signals is to
  view them in the time domain. Record in the time
  domain typically describes the variation of
  system output or system parameter over time.

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Record from Seismograph

• This record (of displacement, also called
  seismogram) displays 24 hours of data, beginning
  approximately 8 hours before the mainshock,
  recorded by the BDSN station BKS at a distance of
  525 km.

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Frequency Domain Analysis

• It was shown over one hundred years ago by Jean
  B. Fourier that any waveform that exists in the
  real world can be generated by adding up sine
  waves.
• where the Fourier coefficients of f(x) can be
  calculated using the so-called Euler formulas

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Frequency Component of a Signal

• This frequency domain representation of our
  signal is called the spectrum of the signal, in
  which every sine wave from the signal appears as
  a vertical line. Its height represents its
  amplitude and its position represents its
  frequency.
• Each sine wave line of the spectrum is called a
  component of the total signal.

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Spectrum of Various Signal Types

• Figures on the right show a few common signals in
  both the time and frequency domains.

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Laplace Transformation
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Measurement System

• Components
• Transducer convert a physical quantity into a
  time-varying electrical signal, i.e. analog
  signal
• Signal conditioner modify/enhance the analog
  signal (profiltering, amplification, etc)
• Analog-to-digital converter convert analog
  signal into digital format
• Digital signal processing
• Recorder display or data storage

Actual Input
Measurement System
Recorder output
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Transfer Function

• Block diagram and transfer function provide an
  efficient way to describe a dynamic system
• Transfer Function
• Used to describe linear time-invariant systems
• Can be expressed using the Laplace transform of
  the ratio of the output and input variables of
  the system
• Can be experimentally determined by curve-fitting
• Frequency response (sinusoidal input)
• Impulse response (short-duration pulse input)

Input, r(t)
Output, y(t)
System
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Frequency Response of a System
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Characteristics of a Good Measurement System

• Amplitude Linearity
• Adequate Bandwidth
• Phase Linearity

A Amplitude f frequency f phase
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Amplitude Linearity
a proportion constant

• Factors impact linearity
• Limited range of input amplitude
• Bandwidth

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Frequency Bandwidth

• Requirement a measurement system should replica
  all frequency components
• Unit-Decibel scale
• Bode plot frequency response curve of a system,
  i.e. a plot of amplitude ratio Aout/Ain, vs. the
  input frequency

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Bode Plot
Amplitude (dB)
fL
fc
fH
Frequency
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Bandwidth

• Bandwidth the range of frequencies where the
  input of the system is not attenuated by morn
  than -3dB
• Low high cutoff frequency
• Bandwidth
• Questions
• what happen if a square wave is measured with a
  limited BW system
• How to determine the BW of a measurement system