Introduction to Frequency Response

1
Introduction to Frequency Response

• Roundup before the confusion

2
Frequency

• Sine wave

3
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

4
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)
5
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)
6

• 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
10
Response of First Order System to Sine WavesK1,
?1 sec, ?0
11
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.

12
Amplitude Ratio and Phase Shift using Transfer
Functions

• Replace S with j? in the transfer function G(s)
  ?G(j?)
• 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 ?)
• AR G sqrt(a2 b2)
• ? tan-1(b/a)