Bode Diagrams

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Bode Diagrams
Plot of magnitude vs. frequency. Plot of
phase vs. frequency.
Frequency (radians/s)
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Bode Diagrams
Tells us how the output varies for given input
frequencies, e.g. are high frequencies reduced?
Input Sin (?t )
Input Sin (?t )
Time s
Time s
Frequency (radians/s) now 1/10 of previous
Output k Sin (?t ?)
Output k Sin (?t ?)
Time s
Time s
Time s
At high frequencies the output is reduced and a
phase lag of 180 is introduced. At low
frequencies the output closely matches the input.
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Bode Diagrams
Plot of magnitude vs. frequency. Plot of
phase vs. frequency. BACKGROUND
Bel log POUT P Power
PIN ?B 10 log POUT 20 log
VOUT PIN VIN
PROPERTIES OF LOGARITHMS logna
logxa logxb logn(ab) (1ogna lognb) -
(lognc 1ognd) cd
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Bode Approximations
Frequency response of simple functions
1 3 5 G(s) K sn2 (lsT2)
snl (lsTl) 2 4
1 Constant Gain 2 Poles _at_ origin 3
Zeroes _at_ origin 4 Single real poles 5
Single real zeroes 6 Complex poles
Prior to the widespread availability
of computers, systems still needed to be
controlled! How can we plot Bode diagrams by
hand?
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Bode Diagrams
CONSTANT GAIN G(j?) k dB
20log(k) Mag 20log(k)
. ? ? 0 if K ve
-180 if K -ve
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Bode Diagrams
POLES _at_ ORIGIN G(j?) 1 dB
20log 1 (j?)L (j?)L
Mag Slope-20db/decade per order
? 0 ? -90 per
order
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Bode Diagrams
ZEROS _at_ ORIGIN G(j?) (j?)L dB 20log
(j?)L As with poles but reversed.
Magnitude 20dB/decade slope per order.
Phase 90 per order. Mag Slope
20db/decade per order ? ?
0 90 per order
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Bode Diagrams
SINGLE REAL POLES G(j?) 1 dB 20log
1 1 j ? 1 j? ?i ?i
?i is called the BREAK PREQUENCY, at this
frequency the curve is approx. 3dB down from the
asymptotes. Asymptotes can be seen at the
zero slope for ? ltlt ?i and at the -20dB/decade
slope for ? gtgt ?i . At the break frequency
?i, the phase passes through -45. Mag ?i
10?i 3dB -20
? ? -45 -90
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Bode Diagrams

• SINGLE REAL ZEROS
• G(j?) 1 j? dB 20log 1 j?
• ?i ?i
• As single real poles but reversed.
• Zero slope asymptote for ? ltlt ?i.
• 20dB/decade asymptote for ? gtgt ?i.
• 3dB up at break frequency.
• Phase passes through 45 at ?i.
• Mag
• 20
• 3dB
• ?i 10?
• ?
• 90

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Bode Diagrams

• EXAMPLE
• Consider the system
• G(j?) 1 j?
• 1 j?
• 10
• It can be seen that this system has a zero at ?i
  1 and a pole at ?i 10.
• The following asymptote plot can be constructed
  using
• this information
• Mag 20
• 1 10 100
• 90

Zero ? Pole Zero ? Pole
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Bode Diagrams
EXAMPLE Consider the system G(j?)
1 j? 1 j? 10 Using
the asymptotes constructed from the transfer
function the following Bode plot can be produced
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Bode Diagrams
QUESTION Given the following transfer function,
construct the Bode plot. Hint Matlab
command to check results s tf('s') G
tf(1.6666 5,0.001666 0.1033 1 0) bode(G)