EE221 - Linear Circuits

	



    

    

        

		    Topics: Applications of the Laplace Transform

			Reading: Chapter 14.
			
Lecture Notes: 
			
    
			   
  • Fing the Laplace transform of  pulse inputs.
			   
  • The inverse Laplace transform using partial fraction expansion and using 
			        MATLAB.
			   
  • We can distinguish at least three ways to use the Laplace transform to find the complete response of a circuit:
			   
      
			      
    • Represent the circuit by a single differential equation. (In general, the order of that equation will be 
			      equal to the sum of the number of capacitors plus inductors.) Take the 
				  Laplace transform of that differential equation ...
			      
    • Apply Kirchhoff's laws in the time domain to get several equations that involve derivatives of capacitor voltages 
			      and/or inductor currents. Take the Laplace transforms of these equations ...
			      
    • Represent the circuit in the s-domain. Write mesh or node equations...
			      
    • Here's an example that illustrates these three methods. 
			   
    
			   
  • Linear circuits can be represented, characterized or specified by their
			   
      
			      
    • transfer function
			      
    • step response 
			      
    • impulse response
			   
    
			   These 3 representations are equivalent: given any one of the three, 
			   we can determine the other two.
			   
  • We can use convolution to determine the output of a linear circuit.  MATLAB 
			       will help.
			   
  • Consider a design problem in which we are given
			   
       
			      
    • A circuit having some unspecified pararmeters, e.g. resistances, capacitances or gains of dependent sources.
			      
    • A specified impulse or step response.
			   
    			
               We are asked to determine 
			   
       
			      
    • Is given circuit is able to have the specified response?
			      
    • If so, what are the required values of the unspecified circuit parameters?
			   
    
			   In such problems, we
			   
       
			      
    • Determine the required transfer function from the specified step or impulse responses.
			      
    • Determime possibles transfer function from the circuit.
			      
    • Compare the required and possible transfer functions to see if they are compatible and, if so, 
			      to determine values of the unspecified circuit parameters. 
			   
    
			   
  • A circuit is stable if all of the poles of its transfer function lie 
			   in the open left half of the s-plane. Is a circuit is stable, then we can obtain its 
			   network function from its transfer function.
			
 
				
    

	




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