Introduction to Frequency Response

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Introduction to Frequency Response

• Roundup before the confusion

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Frequency

• Sine wave

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Frequency Response Analysis

• Any signal can be broken into sine and cosine
  waves of different frequencies and amplitudes
  (Fourier)
• A dynamic systems response can be described in
  terms of the frequency content of the input and
  output signals
• The transfer function describes the effect of the
  system on changing an input into an output

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Frequency response

• The process is described in terms of changes in
  amplitude (size) and phase (shift in time) of the
  signal from input to output.

Process
Output amplitude
Input amplitude
Output amplitude / Input amplitude amplitude
ratio (AR)
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Shift
Period
Phase shift ? Shift/Period 2p or Phase
shift ? Shift/Period 360 A delay is a
negative phase shift (as we will see later)
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• For a linear system, AR and phase vary with
  frequency, but not amplitude, of the input
  signal.
• Frequency is usually measured in radians per
  second (or per minute) and is denoted by ?

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Response of First Order System to Sine WaveK1,
?1 sec, ?0, ?1 rad/s
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Response of First Order System to Sine WaveK1,
?1 sec, ?0, ?3 rad/s
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Response of First Order System to Sine WaveK1,
?1 sec, ?0, ?10 rad/s
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Response of First Order System to Sine WavesK1,
?1 sec, ?0
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Whats going on (Physically)?

• Faster sine waves have the same amplitude, but
  smaller integral before returning to zero.
• First order systems have a capacitance,
  integrating the difference between the input and
  output flows. Smaller integral means smaller
  physical changes.
• There is an initial transient before the output
  sine wave asserts itself. We will ignore this
  transient. We are only interested in the
  sustained behaviour.

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Amplitude Ratio and Phase Shift using Transfer
Functions

1. Replace S with j? in the transfer function G(s)
   ?G(j?)
2. Rationalize G make it equal to a jb, where a
   and b may be functions of ? (G is now a complex
   number that is a function of ?)
3. AR G sqrt(a2 b2)
4. ? tan-1(b/a)