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Department of Chemical Engineering I.I.T. Bombay, India
Department of Chemical Engineering I.I.T. Bombay, India
Frequency Response Response of the process to signals of varying - - PowerPoint PPT Presentation
Frequency Response Response of the process to signals of varying - - PowerPoint PPT Presentation
Department of Chemical Engineering I.I.T. Bombay, India Frequency Response Response of the process to signals of varying frequencies. A process can pass certain frequencies and block others. Implicit Assumption: Any signal can be
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Department of Chemical Engineering I.I.T. Bombay, India
Analysis of Dynamics via frequency response W=0.1 rad/sec W=1 rad/sec W=5 rad/sec
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Department of Chemical Engineering I.I.T. Bombay, India
) ( ) ( ) ( s G s u s y
u(t) = sin (w t)
2 2
) ( s w s u
2 2
) ( ) ( w s w s G s y
Frequency Response Generation Process Transfer function Input signal And then invert this in the time domain to get the frequency response.
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Department of Chemical Engineering I.I.T. Bombay, India
Basic Complex Algebra Given a complex number z=a+ j b, it can be represented in the complex plane as shown below.
Real axis
- Im. axis
b a
Magnitude of z denoted by |z|= (a2+b2)0.5 and angle is tan-1 (b/a) and z can be more compactly represented as z=|z| ej
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Department of Chemical Engineering I.I.T. Bombay, India
Short cut Method for Frequency Response Generation Step 1: s=jw in G(s) to obtain G(jw) Step 2: Rationalize G(jw) to obtain Real and imaginary parts Step 3: The output sine wave will have an amplitude Aout=A(Real2 + Imag2) and the phase angle is =tan-1( Imag/ Real). Illustrative example
1 1 ) ( s s G 1 1 ) ( jw jw G
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Department of Chemical Engineering I.I.T. Bombay, India
Short cut Method for Frequency Response Generation
1 1 1 1 1 ) (
2 2 2 2 2 2
w jw w w jw jw G 1 ) Im( and 1 1 ) Re(
2 2 2 2
w w jw w jw
Output Amplitude is therefore
2 2 2 2 2 2
1 1 1 w w w 1 1
2 2
w
Phase is therefore
) ( tan 1 w
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Department of Chemical Engineering I.I.T. Bombay, India
Representation of Frequency Response Information Bode Plots: Plot the amplitude ratio (AR) and the phase lag on two separate plots called the Magnitude and the phase plots. For the illustrative example, Magnitude plots are typically logarithmic but phase plots are semi-logarithmic.
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Department of Chemical Engineering I.I.T. Bombay, India
Representation of Frequency Response Information Nyquist Plots: Plot the amplitude ratio (AR) and the phase lag
- n the same plot in the imaginary plane.Treat G(jw) as a
complex number and plot the Real(G) and Imag(G) on the diagram. Note: Unlike the bode plot, the phase angle is implicit in the plot here.
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Department of Chemical Engineering I.I.T. Bombay, India
Frequency Response of several systems in series If g(s) = g1(s)g2(s)g3(s) … gN(s), then they can all be expressed in the frequency domain as g(jw)=g1(jw)g2(jw)g3(jw) … gN(jw) Using the polar form of representation g(jw)=|g1||g2||g3||g4| …. |gN| ej, where , is the summation of the individual phase angles. Thus, the bode plots of higher order systems can be constructed from the local building blocks.
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Department of Chemical Engineering I.I.T. Bombay, India
General behaviour of the frequency response plots The amplitude ratio plots generally decay at higher frequencies, more so for chemical processes. The rate of decay is a function
- f the order of the process.
The phase plot asymptotically approaches (p-q) times –90o at high frequencies, where p is the number of poles and q is the number of zeros. Can you guess why ? Also, what is the frequency response of a pure delay process ?
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Department of Chemical Engineering I.I.T. Bombay, India
Interpretations from the frequency response First Order Processes
1 ) ( jw K jw G
1
2 2
w K
AR and Phase are:
) ( tan 1 w
As 0, AR/K 1 and log (AR/K) log(1); gives the low frequency asymptote As , and log (AR/K) -log() ; gives the high frequency asymptote It is convenient to work in terms of a magnitude ratio:
) 1 log( 2 1 ) log(
2 2
w K AR
(The two asymptotes meet at =1), I.e. at =1/. Thus the natural time constant has a role in the frequency response.)
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Department of Chemical Engineering I.I.T. Bombay, India
Pure Capacity System: Frequency Response
) log( ) log( 90 ) ( tan , ) ( , ) (
1
w K AR w K AR jw K jw G s K s G
-
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Department of Chemical Engineering I.I.T. Bombay, India
Pure Lead System: Frequency Response
) ( tan , 1 ) 1 ( ) ( , ) 1 ( ) (
1 2 2
w w K AR jw K jw G s K s G
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Department of Chemical Engineering I.I.T. Bombay, India
Lead - Lag System: Frequency Response
) ( tan ) ( tan , 1 1 1 ) 1 ( ) ( , 1 ) 1 ( ) (
1 1 2 2 2 2
w w w w K AR jw jw K jw G s s K s G
Effective Lag Effective Lead
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Department of Chemical Engineering I.I.T. Bombay, India
Second-order Systems: Frequency Response
s frequencie higher at ) 2log(
- to
and s frequencie low at ), 1 log( ) log( ) 1 ( 2 1 ) 1 ( 2 1 ) log( , 1 1 1 ) 1 )( 1 ( ) 1 ( ) ( , ) 1 )( 1 ( ) 1 ( ) (
2 2 2 2 1 2 2 2 2 2 1 2 2 1 2 1
w K AR w w K AR w w K AR jw jw K jw G s s K s G
Amplitude
s frequencie lower at s frequencie higher at 180
-
Phase
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Department of Chemical Engineering I.I.T. Bombay, India
General Second-order Systems: Frequency Response
2 2 1 2 2 2 2 2 2
1 2 tan , ) 2 ( ) 1 ( 1 , ) 1 2 ( ) ( ) ( w w w w K AR s s K s G
Effect of damping factor
It can be shown that the AR curve shows a hump for all values of less than 0.707 and the maxima occurs at
Transfer Function
2
2 1 1
r
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Department of Chemical Engineering I.I.T. Bombay, India
Other higher order systems: Frequency Response Lead time constant > lag time constants Lead time constant < lag time constants
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Department of Chemical Engineering I.I.T. Bombay, India
Time Delay Systems
w K AR e jw G e s G
jw s