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LESSON 15: BODE PLOTS
OF TRANSFER FUNCTIONS
ET 438a Automatic Control Systems Technology
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Learning Objectives
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After this presentation you will be able to:
Compute the magnitude of a transfer function for a given
radian frequency.
Compute the phase shift of a transfer function for a given
radian frequency.
Construct a Bode plot that shows both magnitude and
phase shift as functions of transfer function input frequency
Use MatLAB instructions it produce Bode plots of transfer
functions.
Construction of Bode Plots
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Bode plots consist of two individual graphs: a) a semilog plot of gain vs frequency b) a semilog plot of phase shift vs frequency. Frequency is the logarithmic axis on both plots.
Bode plots of transfer functions give the frequency response of a control system
To compute the points for a Bode Plot:
- Replace Laplace variable, s, in transfer function with jw
- Select frequencies of interest in rad/sec (w=2pf)
- Compute magnitude and phase angle of the resulting complex expression.
Construction of Bode Plots
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Bode plot calculations- magnitude/phase
Gain: dB 20 log(G(j ))
X(j) G(j ) Y(j) w
w w w
Transfer Function
X(jw) Y(jw)
i
o x
y
X
Y
Where for a given frequency Phase: o i
To find the magnitude and phase shift of a complex number in rectangular form
given: z a jb
z a 2 b^2
a
tan 1 b
Example 15-1 Solution (2)
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Complex G (j 0. 001 ) 1 15902000 j 0. 001 1 ^2000 j 1. 590 566. 9 j 901 gain
w a G(j ) a^2 b^2 tan-^1 b
dB 20 log( 1065 ) 60. 5 dB
G(j 0 .001) 1065
G(j 0 .001) 566. 9 901
a 566. 9 b - 2 2
Magnitude Calculation (^) Phase shift
58
- 9 tan^901 a tan -^1 b -^1
Ans
Ans
At 0.001 rad/sec, the system has a gain of 60.5 dB and the output changes in height lag the flow changes by 58 degrees
Constructing Bode Plots Using MatLAB
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MatLAB has control system toolbox functions for defining Linear Time-invariant systems (LTI) and constructing the Bode plots.
Use tf and bode functions to create LTI and plot. Introducing zpk function
sys = zpk(z,p,k) Turns arrays of zeros, poles and gains into LTI called sys
Where z = array of transfer function zeros p = array of transfer function poles k = array of transfer function gains
Constructing Bode Plots Using MatLAB
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Example 15-2: Construct the Bode plot for the given transfer function shown in factored form using MatLAB control toolbox functions.
0. 001 s 1 0. 001 s 1
- 005 s V(s)
V(s) i
o
Solution: Transfer function has one zero at s=0 and two poles at s=-1/0.001=- 1000 Dividing the transfer function denominator and numerator by 0.001 places it into standard form
^ s^1000 s^1000
5 s
001 s^1
001
001
001 s^1
001
001
001 s^0.^005
001
001 s 10. 001 s 1 1
001
005 s^1 V(s)
V(s) i
o ^ ^
Constructing Bode Plots Using MatLAB
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MatLAB code to produce Bode plot of given transfer function Enter at command prompt of into m-file k= [5] p=[1000 1000] z=[0] sys=zpk(z,p,k) bode(sys)
Magnitude (dB)
-270 101 102 103 104 105
Phase (deg)
Bode Diagram
Frequency (rad/s)
Magnitude plot
Phase plot
w=
End Lesson 15: Bode Plots of
Transfer Functions
ET 438a Automatic Control Systems Technology
lesson15et438a.pptx
