Difference between revisions of "Transfer Functions"
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−  {{chheaderOutput FeedbackTransfer Functions  +  {{chheaderOutput FeedbackTransfer FunctionsFrequency Domain Analysis}} 
This chapter introduces the concept of the ''transfer function'', which is a compact description of the inputoutput relation for a linear system. Combining transfer functions with block diagrams gives a powerful method of dealing with complex systems. The relationship between transfer functions and other system descriptions of dynamics is also discussed.  This chapter introduces the concept of the ''transfer function'', which is a compact description of the inputoutput relation for a linear system. Combining transfer functions with block diagrams gives a powerful method of dealing with complex systems. The relationship between transfer functions and other system descriptions of dynamics is also discussed.  
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{{chaptertable left}}  {{chaptertable left}}  
== Textbook Contents ==  == Textbook Contents ==  
−  {{am05pdf  +  {{am05pdfam08xferfcns28Sep12Transfer Functions}} 
* 1. Frequency Domain Analysis  * 1. Frequency Domain Analysis  
* 2. Derivation of the Transfer Function  * 2. Derivation of the Transfer Function  
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== Supplemental Information ==  == Supplemental Information ==  
* [[#Frequently Asked QuestionsFrequently Asked Questions]]  * [[#Frequently Asked QuestionsFrequently Asked Questions]]  
+  * [[#ErrataErrata]]  
* Wikipedia entries: {{wikipediaTransfer_functionTransfer Function}}  * Wikipedia entries: {{wikipediaTransfer_functionTransfer Function}}  
* [[#Additional InformationAdditional Information]]  * [[#Additional InformationAdditional Information]]  
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<li><p>Block diagrams that consist of transfer functions can be manipulated using ''block diagram algebra''. The following table gives the transfer functions for some common interconnections of linear systems  <li><p>Block diagrams that consist of transfer functions can be manipulated using ''block diagram algebra''. The following table gives the transfer functions for some common interconnections of linear systems  
+  <center>  
{  {  
    
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 [[Image:xferfcns_feedback.png]]   [[Image:xferfcns_feedback.png]]  
    
−   Series: <  +   Series: <amsmath>H_{yu} = G_2 G_1</amsmath> 
−   Parallel: <  +   Parallel: <amsmath>H_{yu} = G_1+ G_2 </amsmath> 
−   Feedback: <  +   Feedback: <amsmath>H_{yu} = \frac{G_1}{1 + G_1 G_2}</amsmath> 
}  }  
−  </p></li>  +  </center></p></li> 
<li><p>A ''Bode plot'' is a plot of the magnitude and phase of the frequency response:  <li><p>A ''Bode plot'' is a plot of the magnitude and phase of the frequency response:  
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</ol>  </ol>  
−  == Exercises ==  +  {{chaptertable begin}} 
+  {{chaptertable left}}  
+  
+  <!  
+  == Additional Exercises ==  
+  The following exercises cover some of the topics introduced in this chapter. Exercises marked with a * appear in the printed text.  
<ncl>Transfer Functions Exercises</ncl>  <ncl>Transfer Functions Exercises</ncl>  
+  >  
== Frequently Asked Questions ==  == Frequently Asked Questions ==  
<ncl>Transfer Functions FAQ</ncl>  <ncl>Transfer Functions FAQ</ncl>  
+  == Errata ==  
+  <ncl>Transfer Functions errata v2.11b</ncl>  
+  * [[:Category:Transfer Functions errataFull list of errata starting from first printing]]  
+  * {{submitbug}}  
+  {{chaptertable right}}  
+  
+  == MATLAB code ==  
+  The following MATLAB scripts are available for producing figures that appear in this chapter.  
+  * Figure 8.3: {{matlabfilexferfcnsopamp_wfb.m}}  
+  * Figure 8.5: {{matlabfilexferfcnsbalance_polezero.m}}, {{matlabfile.balance_params.m}}  
+  * Figure 8.10: {{matlabfilexferfcnscruise_pzcancel.m}}, {{matlabfile.cruise_carpar.m}}, {{matlabfile.cruise_conpar.m}}, {{matlabfile.cruise_opcon.m}}, {{matlabfile.cruise_lin.m}}  
+  * Figure 8.14: {{matlabfilexferfcnsbodesketch.m}}  
+  * Figure 8.15: {{matlabfilexferfcnslowpass.m}}, {{matlabfilexferfcnsbandpass.m}}, {{matlabfilexferfcnshighpass.m}}  
+  * Figure 8.16: {{matlabfilexferfcnsgenereg.m}}, {{matlabfile.genereg_params.m}}  
+  * Figure 8.17: {{matlabfilexferfcnsafm_piezobode.m}}  
+  * Figure 8.19: {{matlabfilexferfcnspupil_bode.m}}  
+  See the [[softwaresoftware page]] for more information on how to run these scripts.  
+  
== Additional Information ==  == Additional Information ==  
* {{Lew03Lewis}}, {{Lew03urlChapter 3}}  provides a very complete mathematical description of transfer functions, written in the language of Laplace transforms  * {{Lew03Lewis}}, {{Lew03urlChapter 3}}  provides a very complete mathematical description of transfer functions, written in the language of Laplace transforms  
+  
+  {{chaptertable end}} 
Latest revision as of 00:49, 5 November 2012
Prev: Output Feedback  Chapter 8  Transfer Functions  Next: Frequency Domain Analysis 
This chapter introduces the concept of the transfer function, which is a compact description of the inputoutput relation for a linear system. Combining transfer functions with block diagrams gives a powerful method of dealing with complex systems. The relationship between transfer functions and other system descriptions of dynamics is also discussed.
Textbook ContentsTransfer Functions (pdf, 28Sep12)

Lecture MaterialsSupplemental Information

Chapter Summary
This chapter introduces the concept of a transfer functon for a linear input/output system.
The frequency response of a linear system
is the response of the system to a sinusoidal input at a given frequency. Due to linearity, the response of a system to a more complicated input can be constructed by decomposing the input into the sum of sines and cosines
(The frequency response is described in Chapter 5  Linear Systems).

More, generally an exponential signal is given by
where gives the decay rate of the signal and is the oscillation frequency of the signal. The response to an exponential signal is given by
The transfer function for a linear system is given by
The transfer function represents the steady state response of the system to an exponential input. The transfer function is independent of the choice of coordinates for the state space.
The transfer function for a linear differential equation of the form
is given by
where
The zero frequency gain of a system is given by the magnitude of the transfer function at . It represents the ratio of the steady state value of the output with respect to a step input. For a transfer function of the form , the roots of the polynomial are called the poles of the system and the roots of the polynomial are called the zeros of the system. A pole is also called a mode of the system. The poles correspond to the eigenvalues of the dynamics matrix and determine the stability of the system. The zeros of a transfer function correspond to exponential signals whose transmission is blocked by the system.
Block diagrams that consist of transfer functions can be manipulated using block diagram algebra. The following table gives the transfer functions for some common interconnections of linear systems
Series: Parallel: Feedback: A Bode plot is a plot of the magnitude and phase of the frequency response:
The top plot is the gain curve; the frequency and magnitude are both plotted using a logarithmic scale. The bottom plot is the phase curve and uses a loglinear scale. The dashed lines show straight line approximations of the gain curve and the corresponding phase curve.
The transfer function for a system can be determined from experiments by measuring the frequency response and fitting a transfer function to the data. Formally, the transfer function corresponds to the ratio of the Laplace transforms of the output to the input.
Frequently Asked Questions
Errata

MATLAB codeThe following MATLAB scripts are available for producing figures that appear in this chapter.
See the software page for more information on how to run these scripts. Additional Information 