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Applications of the Laplace Transform

physic

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Applications of the Laplace Transform

Part One: Analysis of Network (6-2, 6-3)

1.     Review of Resistive Network



1)     Elements

       

2) Superposition

                   

 


3) KVL and KCL

      

4) Equivalent Circuits

                            


5) Nodal Analysis and Mesh Analysis

Mesh analysis

                 Solve for I1 and I2.

2.     Characteristics of Dynamic Network

              Dynamic Elements ® Ohm’s Law: ineffective

1) Inductor   

2)     Capacitor

3) Example (Problem 5.9):

          Why so simple? Algebraic operation!

Dynamic Relationships (not Ohm’s Law) Complicate the analysis

          Using Laplace Transform

                

          Define ‘Generalized Resistors’ (Impedances)


                   
  

                         As simple as resistive network!

           Solution proposed for dynamic network:

                         All the dynamic elements Þ Laplace Trans. Models.

                          Þ       As Resistive Network

           Key: Laplace transform models of (dynamic) elements.

3.     Laplace transform models of circuit elements.

1)     Capacitor

                  

                Important: We can handle these two ‘resistive network elements’!

2) Inductor

3) Resistor   V(s) = RI(s)

4)Sources

               

5)  Mutual Inductance (Transformers)

 (make sure both i1 and i2 either away

or toward the polarity marks to make

the mutual inductance M positive.)

      Circuit (not transformer) form:

                  

  


Benefits of transform

    ß

Let’s write the equations from this circuit form:


                 The Same

ß    Laplace transform model: Obtain it by using inductance model

       

Just ‘sources’ and ‘generalized resistors’  (impedances)!

4.     Circuit Analysis: Examples

Key: Remember very little, capable of doing a lot

How: follow your intuition, resistive network

‘Little’ to remember: models for inductor, capacitor and mutual inductance.

Example 6-4: Find Norton Equivalent circuit

                   

Assumption:

*Review of Resistive Network

   1) short-circuit current through the load:

   2) Equivalent Impedance or Resistance or :

      A: Remove all sources

      B: Replace  by an external source

      C: Calculate the current generated by the external source ‘point a’

      D: Voltage / Current Þ

*Solution

    1) Find

                    

                       

 2) Find

 

Vtest(s)

 

(Will I(s) be zero?  We don’t know yet!)

 


                     condition: 1 ohm = 3/s

                 or I(s) = 0

              =>I(s) = 0 =>Zs = ¥

3)   

    Example 6-5: Loop Analysis  (including initial condition)

          Question: What are i0 and v0?

                          What is ?

Why this direction?

 
   Solution

1)     Laplace Transformed Circuit


            

Why this direction?

 
 

2)     KVL Equations

              

                 Important: Signs of the sources!

3)     Simplified (Standard form)

6-4     Transfer Functions

1.  Definition of a Transfer Function

(1)   Definition



                               

 System analysis: How the system processes the input to form the output, or

                                                     Input : variable used and to be adjusted

                                                                 to change or influence the output.

                                                            Can you give some examples for input and output?

 

       Quantitative Description of  ‘ how the system processes

                           the input to form the output’: Transfer Function H(s)

(2)   d input

The resultant output y(t) to  d (t) input: unit impulse response

In this case: X(s) = L [d (t)] = 1

                   Y(s) = Laplace Transform of the unit impulse response

=>  H(s) = Y(s)/X(s) = Y(s)

                    Therefore: What is the transfer function of a system?

                           Answer : It is the Laplace transform of the unit impulse response

                                           of the system.

(3)   Facts on Transfer Functions

    * Independent of input, a property of the system structure and parameters.

    * Obtained with zero initial conditions.

        (Can we obtain the complete response of  a system based on its transfer 

         function and the input?)

    * Rational Function of s (Linear, lumped, fixed)

    * H(s): Transfer function

                 H( j2pf ) or H( jw ): frequency response function of the system

                     (Replace s in H(s) by j2pf or jw)

                  |H( j2pf )|  or  |H( jw )|: amplitude response function

                  ÐH(j2pf)  or  ÐH( jw ): Phase response function

2.  Properties of Transfer Function for Linear, Lumped stable systems

(1)   Rational Function of s

Lumped, fixed, linear system =>

                             

Corresponding differential equations:

                           

 (2)     all real!   Why?  Results from real system components.

    Roots of N(s), D(s): real or complex conjugate pairs.

Poles of the transfer function: roots of D(s)

Zeros of the transfer function: roots of N(s)

    Example:

  (3) H(s) = N(s)/D(s) of bounded-input bounded-output (BIBO) stable

  system

 

  * Degree of N(s) £ Degree of D(s)

        Why? If degree N(s) > Degree D(s)

           where degree N(s) <degree D(s)

       Under a bounded-input  x(t) = u(t) => X(s) = 1/s

                    ( not bounded!)

  * Poles: must lie in the left half of  the s-plan (l. h. p)

                                  i.e.,   

              Why?

                            

                           (Can we also include k=1 into this form?   Yes!)

                       

  * Any restriction on zeros?  No  (for BIBO stable system)

3.  Components of System Response

                    

     Because x(t) is input, we can assume

                             

     Laplace transform of the differentional equation

                          

                            D(s): System parameters

                            C(s): Determined by the initial conditions (initial states)

                Initial-State Response (ISR) or Zero-Input Response (ZIR):

                                           

                Zero-State Response (ZSR)  (due to input)

                                          

        From another point of view:

                          Transient Response: Approaches zero as tà

                          Forced Response: Steady-State response if the forced

                                                       response is a constant

        How to find (1) zero-input response or initial-state response? No problem!

  

       (2) zero-state response?  No prolbem!

         How to find (1) transient response? All terms which go to 0 as tà¥

                             (2) forced response?  All terms other than transient terms.

Example 6-7

    Input

    Output

    Initial capacitor voltage:

    RC = 1 second

Solution

(1)   Find total response                                                                                                   

                                                                           

(2)   Find zero-input response and zero-state response

              Zero-input response: 

              Zero-state response:

                                                 

(3)   Find transient and forced response

                    

      Which terms go to zero as tà¥?

                                 

      What are the other terms:

                                     

4.  Asymptotic and Marginal Stability

System:  (1)  Asymptotically stable if   as  tॠ (no input) for all




possible initial conditions, y(0), y’(0), … y(n-1)(0)

                            è Internal stability, has nothing to do with external input/output

               (2)   Marginally stable

                               all t>0 and all initial conditions

               (3) Unstable

             *grows without bound for at least some values

                                     of the initial condition.

               (4) Asymptotically stable (internally stable)

                              =>must be BIBO stable.  (external stability)

6-5 Routh Array

1. Introduction

  

    System H(s) = N(s)/D(s)   asymptotically stable ó all poles in l.h.p (not

include jw axis.

     How to determine the stability?

Factorize D(s):

   

     Other method to determine (just) stability without factorization? 

                             Routh Array

(1) Necessary condition

 All        (when  is used)

ð     any   =>  system unstable!

         Why?

   Denote    to esnure stability

          

         When all Re(pj) > 0 , all coefficients must be greater than zero.  If some coefficient is not greater than zero, there must 

          be at least a Re(pj) <= 0  (i.e., )

                                                =>  system unstable

(2) Routh Array

 Question:  All  implies system stable?

Not necessary

          Judge the stability: Use Routh Array (necessary and sufficient)

2. Routh Array Criterion

Find how many poles in the right half of the s-plane

(1)    Basic Method


                   Formation of Routh Array

Number of sign changes in the first column of the array

=> number of poles in the r. h. p.

Example 6-8

         


sign: Changed once =>one pole in the r.h.p

                    verification:

                               

Example 6-9

                                      

             Sign: changed twice => two poles in r.h.p.

(2)   Modifications for zero entries in the array

Case 1: First element of a row is zero

ð     replace 0 by  ε (a small positive number)

Example 6-10

Case 2: whole row is zero (must occur at odd power row)

  construct an auxiliary polynomial and the perform differentiation

            Example: best way.


Example 6-11


S

 

 




(3)   Application: Can not be replaced by MatLab

                          Range of some system parameters.

Example:

             

    Stable system

             to ensure system stable!

6-6          Frequency Response and Bode Plot

      Transfer Function

      Frequency Response

Why?

 


     Amplitude Response: 

                     Real positive number: function of

               Phase Response:


      

      Interest of this section

                In particular, obtain

                          

                                                     What are these?


Important Question: What is a Bode Plot?

                     How to obtain them without much computations?

                                            Asymptotes only!

1.   Bode plots of factors

(1)   Constant factor k:

 (2)  s

                         Can we plot it?

:                 Can we plot  for them?

Phase    s:

       *:

(3)



step 1: Coordinate systems

step 2: corner frequency   

          

step 3: Label 0.1wc, wc , 10wc

Line

 

Point 1

 
step 4: left of wc :


Point 2

 
step 5: right of wc :

        Why?

 

        If

                  

        If

                  

Example:

                           What is T : T = 0.2

                           What is wc : wc = 1/T = 5

Example :  0.2s + 1

Example : (0.2s + 1)2,     (0.2s + 1)-2


Example : (Ts + 1)±N


(4)     (Complex  --- Conjugate poles)


Step 3 : Before     :  

   Right of   :  

Line

 
   point 1: ( , )

   point 2: ( , )

Example:

    Actual  and z (show Fig 6-20)

          What’s resonant frequency: reach maximum: 

          Under what condition we have a resonant frequency:

   

      :  see fig 6-21

What about : ?

2.   Bode plots: More than one factors

Can we sum two  plots into one?

Can we sum two  plots into one?

                     Yes!


3.   MatLab

                         

                                    Show result in fig 6-24

6.7 Block Diagrams

1.  What is a block diagram?


 Concepts: Block, block transfer function,

                  Interconnection, signal flow, direction

                  Summer

                  System input, system output

                  Simplification, system transfer function

2.  Block


                                                                                 


                                                                               Assumption: Y(s) is determined

                                                                               by input (X(s)) and block transfer

                                                                              function (G(s)).  Not affected by

                                                                               the load.

              Should be vary careful in

            analysis of practical systems about the accuracy of this assumption.

3.  Cascade connection

4.  Summer

5. Single-loop system

             

 

                                      

          Let’s find           Closed-loop transfer function

                  Equation (1)           

                  Equation (2)


                                  

6. More Rules and Summary: Table 6-1


Example 6-14: Find Y(s)/X(s)

                    

Example 6-15: Armature- Controlled dc servomotor

            Input : Ea (armature voltage)

            Output :  (angular shift)


             Can we obtain ?

Example 6-16  Design of control system

 

              Design of K such that closed loop system stable.

        

        Routh Array:

                                                  

System stable if k>0.  If certain performance is required in addition to the stability, k must be further designed.








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