The document presents information on frequency response systems and Bode plots. It defines frequency response as a measure of the output spectrum of a system in response to a stimulus. A Bode plot is a graphical representation of a system's frequency response in terms of gain and phase shift. It shows the logarithm of the magnitude and phase angle as functions of frequency. The document discusses different system types (0, 1, 2) and how to identify them based on the slope of the log magnitude curve at different frequencies. It also explains the impact of different transfer function components like constants, poles, and zeros on the shape of Bode plots.
2. Content
Introduction
Bode plot
Some definitions of Bode plot
System Type
Log magnitude and angle diagram
curve
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3. Introduction
- Frequency response is the quantitative
measure of the output spectrum of a system or device in
response to a stimulus, and is used to characterize the
dynamics of the system. It is a measure of magnitude
and phase of the output as a function of frequency, in
comparison to the input.
- The frequency response is characterized
by the magnitude of the system's response, typically
measured in decibels (dB) or as a decimal, and
the phase, measured in radians or degrees, versus
frequency in radians/sec or Hertz (Hz).
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4. Bode Plot
What is Bode Plot?
-Bode plot is a graphical representation system of
a signal frequency response in terms of gain and
phase shift.
-The log magnitude and frequency response
curve as function of log w are called bode plots or
bode diagrams.
-A Bode Plot is a useful tool that shows the
gain and phase response of a given LTI system for
different frequencies.
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6. Decibel :
In feedback-system work the unit commonly used for
the logarithm of the magnitude is the decibel (dB).
Log magnitude :
The logarithm of the magnitude of a transfer function
G(jω) express in decibel is 20 log |G(jω)| dB .
this quantity is called log magnitude .
Octave & Decade :
-An octave is a frequency band from f1 to f2,
where f2/f1=2 .
-An decade is a frequency band from f1 to f2 ,
where f1/f2=10
Bode Plot (Contd.)
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7. Bode Plot
(Contd.)
Properties of bode plot :
1) As a number double , the decibel
value increased by 6 dB .
2) As a number increase by a factor 10,
the decibel value increase by 20 dB .
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8. Significance :
(1) the mathematical operations of multiplication
and division are transformed to addition and subtraction
.
(2) the work of obtaining the transfer function is
largely graphical instead of analytical.
(3) It gives us log magnitude and angle at a time .
Bode Plot (Contd.)
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9. Bode Plot
(Contd.)
Components of bode plot-
1.Constant:
The constant is a frequency invariant function.
And log magnitude is -
The plot of constant in a bode plot is horizontal
line. The constant raises or lowers the Lm curve of the
complete transfer function by a fixed amount.
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10. Bode Plot
(Contd.)
2. јω factor:
For , јω factor appearing in the denominator
has a log magnitude,
And the angle is constant to –90 degree.
For , јω factor appearing in the numerator has a
log magnitude,
And the angle is constant to 90 degree .
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11. Bode Plot
(Contd.)
3. 1+ јωT factor:
The factor 1+ јωT appearing in the
denominator has a log magnitude is,
For, very small value of ω, that is ωT <<1 ,
so, the plot of the Lm at small frequencies is “0”
dB .
For every large value of ω ,that is ωT >>1 ,
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12. In here, 1/T is known as corner frequency .
so, before corner frequency the slope of the factor is= 0 dB
after corner frequency the slope of the factor is = -20 dB
with angle varies from 0 to -90 degree .
Now, The factor 1 +јωT appearing in the numerator has a log magnitude is ,
Similarly , slope before corner frequency is = 0 dB
& slope after corner frequency is = 20 dB
with the angle varies from 0 to +90 degree .
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13. Bode Plot
(Contd.)
4. Quadratic factor:
Quadratic factors in the denominator of the transfer
function have the form –
So, the slope of the function before corner frequency = 0 dB
& the slope of the function after corner frequency = -40 dB
With angle varies from 0 to -180 degree .
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14. System Types
System type and gain as related to
magnitude curve-
1.Type “0” system
2.Type “1” system
3.Type “2” system
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15. System Types
#Type “0” System :
-Transfer Function of type “0” system is,
At low frequency , log-magnitude of transfer function – Lm (G
(jω) ) = 20log k0
At high frequency , log magnitude of transfer function Lm
(1/(1+jω)) = -20 dB/dec
-Characteristics:
1) The slope at low frequency is zero .
2)The magnitude at low frequencies is 20log k0 .
3) The gain k0 is the steady-state step error coefficient .
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17. #Type “1” system :
A second-order Type 1 system has a transfer function of the form –
At low frequency slope of the transfer function is = -20 dB
& At high frequency slope of the transfer function is = -40 dB
With angle varies from o to -180 degree .
System Types(contd.)
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18. -Charactaristics:
1. The slope at low frequencies is -20 dB/dec .
2.The intercept of the low frequency slope of -20 dB/dec with 0 dB
axis occurs at the frequency ωx .
3.The value of the low frequency slopeof -20 d/dec at frequency ω=1
is equal to 20log k1 .
4. The gain k1 is the steady-state ramp error coefficient .
Fig: Log magnitude curve
System Types(contd.)
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19. #Type ‘2’ system:
A third order type ‘2’ system has a transfer function of the form-
Fig: Log magnitude plot for type ‘2’ system .
System Types(contd.)
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