TRANFER FUNCTION ANALYSIS 1.1

ABSTRACT:

        HP 48 G(X) Program for use in control systems.
        It includes:
                        Simulation
                        Predeterminated Inputs
                        Bode Plot (Magn. and Phase)
                        Root Locus
                        Nyquist Plot
                        Nichols Plot

        All in easy to use and graphical Menus, like built-in apps.



INDEX

        1  Introduction.
        2  Installation.
        3  Getting started.
        4  Main Menu.
        5  Transfer function: H(s).
        6  Predeterminated input.
        7  User's defined input.
        8  Graph Input.
        9  Output, Simulation.
        10 Bode Plot.
        11 Root Locus.
        12 Nyquist Plot.
        13 Nichols Plot.
        14 Final Words





1 INTRODUCTION

        This package was developed for use in analysis and design of
control systems using menus and graphic capabilities of HP48G(X).
        The purpose of this programs is to help students in control
courses.
        Written in UserRPL.
        The main features are:
                Predeterminated Inputs:
                        Square, triangle, saw, half rectified sine,
                        full rectified sine, sine and step waves.
                        And user's inputs
                Simulation :
                        Graph of the response of the system.
                Bode Plots, Root Locus, Nyquist Plot and Nichols plot.

        I 'll hope you enjoy my program. Thank you for using it.

        Excuse for my poor english. My native language is spanish,
        I'm from Chile -SouthAmerica-




2 INSTALLATION

        Just download the archive: tfa
        The directory tfa will be created.


3 GETTING STARTED

        Enter to TFA directory, then press CST, the soft keys are:

        EXECUTE: Starts the program. Go the Main Menu.

        PICTURE: Shows the current graphic.

        ABOUT  : Info about me.

        OFF    : Powers off the HP.
						 



4 MAIN MENU

        The Main Menu shows all capabilities through a graphic menu.
        These are:

        - H(s): Input the transfer function.
        - Input: Choice one of the predeterminated inputs.
        - Output: Simulates and graphics the output.
        - Graph Input: Graphics the input.
        - User Input: Choice your own input function.
        - Bode Plot: graphs Bode Plot.
        - Root Locus: graphs Root Locus.
        - Nyquist Plot: polar graphic.
        - Nichols Plot: magnitude and phase plane.
        - About: Info about me.



5 H(s)

        Here you enter the transfer function. It has the next form:

                       numerator
                H(s) = ---------    *  exp( -delay*s)
                      denominator


        Numerator and denominator are both polynomials and must be
        entering in the HP form:
                a[n]*s^n  + a[n-1]*s^n-1 + ... +  a[0] is equals to
                [  a[n]   a[n-1] ...  a[0]  ]

        The delay is a positive number.

        For example:

        Num.   [ 2.65 ]
        Den.   [ 1 1]

        Delay 0.8

        OK



                                   2.65   exp(-0.8*s)
        actually is:     H(s)= -------------
                                  s + 1




6 INPUT :

        Here you can choice one of the predeterminated inputs:

        Square:  AMP1, TIME1, AMP2 and TIME2 are the next:

        AMP1 | _______         _______
             |        |       |       |
             |        |       |       |
             |        |       |       |
        AMP2 |        |_______|       |________
             |
             |___________________________________
             <-TIME1-><-TIME2->


        Triangle: AMP1, TIME1, AMP2 and TIME2 are the next:

              |
          AMP1|      / \                     / \
              |    /     \                 /     \
              |  /         \             /         \
              |              \         /             \
              |                \     /                 \
          AMP2|                  \ /
              |________________________________________

               <- TIME1  -><-   TIME2  ->

        Saw : AMP1, TIME1, AMP2 and TIME2 are the next:

       AMP2 |                   /                    /
            |                 /  |                 /  |
            |               /    |               /    |
            |             /      |             /      |
            |           /        |           /        |
            |         /          |         /          |
            |       /            |       /            |
            |     /              |     /              |
            |   /                |   /                |
       AMP1 | /                  | /                  |
            |___________________________________________
              <-TIME1-><-TIME2->

        In the tree shapes AMP1 and AMP2 can be positive or negative.




        Sine, Half rectified Sine and Full rectified Sine:

                Here you must specific: AMPLITUDE, FRECUENCY & PHASE
        where

                Sine(T) = AMPLITUDE*SIN(2PI*FRECUENCY+PHASE)

                The rectified waves are exactly that.


        Step: AMPLITUDE and DELAY are:

                 |             ___________________________
       AMPLITUDE |            |
                 |            |
                 |            |
                 |            |
                 |            |
                 |            |
                 |            |
               0 |____________|____________________________
                  <-DELAY---->


        Pulse: AMPLITUDE, WIDTH & DELAY are:

                 |             ____________
       AMPLITUDE |            |            |
                 |            |            |
                 |            |            |
                 |            |            |
                 |            |            |
                 |            |            |
                 |            |            |
               0 |____________|____________|______________
                  <-DELAY----> <--WIDTH--->

       In the last two shapes AMPLITUDE can be both positive or negative.





7 USER'S INPUT

        Just enter the input of your systems i(T), where T=TIME.

        EXAMPLE: From the Main Menu:
        User Input
                   I(T) : '2*COS(10*T)*(1-EXP(-T))'
                   OK

        Watch the use of ' ' and the independent time variable is "T"
        ( not "t").




8 GRAPH INPUT

        Just shows the input.
        See 9 for explanation of graphic parameters.

        EXAMPLE1  From the Main Menu:

        Input
                Triangle
                          AMP.1: -1
                          TIME1:  1
                          AMP.2:  1
                          TIME2:  1
                          OK
                { An inverted triangle wave}



        Graph Input
                FINAL TIME:    5
                MAX. AMP  :  1.2
                MIN. AMP  : -1.2
                H. SIZE   :    1
                V. SIZE   :    2

        Press OK and you will get the 2 vertical screens graph of the
        input.


        EXAMPLE2  From the Main Menu:

        User Input

                I(T) = '2*COS(10*T)*(1-EXP(-T))'

        Graph Input
                FINAL TIME:    2
                MAX. AMP  :    2
                MIN. AMP  :   -2
                H. SIZE   :    1
                V. SIZE   :    1

        Press OK for the graph.



9 OUTPUT

        Here you can obtain the time response to your system:
                          _______
                         |       |
         input(t)------> | H(s)  |------> output(t)
                         |_______|

        The parameters are:

        FINAL TIME: the time interval is 0 to FINAL TIME.

        RESOLUTION: number of points that will be plot.

        MAX. AMP : maximun amplitude of the output.

        MIN. AMP : minimun amplitude of the output.

        H. SIZE : horizontal size of graph in screen units.

        V. SIZE : vertical size of graph in screen units.

        The two last parameters could create a graph larger than the
        display, for example a 4 screens graphic.

        EXAMPLE 1:
                           ____________________
                          |                    |
        R(S)----> + ----> | 1.42/(S*(1.09S+1)) | ---------> C(S)
                  -       |____________________|    |
                  ^                                 |
                  |_________________________________|

        It's clear that

                C(S)       1.42
                ---- = --------------
                R(S)   1.09S^2+S+1.42

        And the input is

                r(t) = 2*sin(2*pi*1*t)

        So, from the main menu :

        H(s)
                NUM: [1.42]
                DEN: [1.09 1 1.42]

                DELAY: 0

                OK

        Input
                Sine
                        AMPLITUDE: 2
                        FRECUENCY: 1
                        PHASE:     0

                        OK

        Output
                FINAL TIME:   15
                RESOLUTION:  200
                MAX. AMP  :  0.3
                MIN. AMP  : -0.3
                H. SIZE   :    2
                V. SIZE   :    1

                OK


                The graph will have two screens in horizontal scale.
                Now press OK an you'll get te graph. Move with cursors.
                The resolution is directly related to numerical solution,
                but 200 is good enough.

        This part of program may take a LONG TIME. Be patience.

        EXAMPLE 2:
        
        H(S)
                NUM: [10 10]
                DEN: [1 7 10]

                DELAY: 0.3

                OK

        Input
                Saw
                        AMP.1: -1
                        TIME1:  1
                        AMP.2:  1
                        TIME2:  2

                        OK
                { Ascending saw wave}

       Graph Input

                FINAL TIME:  8
                MAX. AMP  :  2
                MIN. AMP  : -2

                OK
                { See the ascending saw }

        Output
                FINAL TIME:  8
                RESOLUTION:  100
                MAX. AMP  :  2
                MIN. AMP  : -2
                H. SIZE   :  1
                V. SIZE   :  1

                { Press OK and wait two minutes, please }

        See the time delay, and the output curve.




10 BODE PLOT

        In this routine you can get the Bode Plot of H(s).

        When you graph the MAGNITUDE and PHASE you must enter the min and
        max decades, i.e. :

                initial_w =  10^(min. decade)
                final_w   =  10^(max. decade)

        In the Magnitude plot you must enter also the max. and min. dB's.

        In the Angle plot you could get a discontinuity, it is because of
        limitated range of ARG function, the truth graph is just the
        continue one.


        EXAMPLE 1 From the Main Menu:

        H(S)
                NUM: [10 10]
                DEN: [1 7 10]

                DELAY: 0


        Bode Plot
                MAGNITUDE
                         MAX DB :  10
                         MIN DB : -30
                         MAX DEC:   2
                         MIN DEC:  -1
                         H SIZE :   1
                         V SIZE :   2

                { Here you get the Bode Plot from 0.1 to 100 [rad/sec] }

        Bode Plot
                ANGLE
                        MAX DEC :   1
                        MIN DEC :  -1
                        H SIZE  :   1
                        V SIZE  :   1



11 ROOT LOCUS

        This part graphics the root locus of the next equation:

        f(s) = B(s) + K*A(s) = 0

        where B(s) and A(s) are polynomials of s, for different K's.

        Here B(s) is called IND. COEFS and A(s) is K COEFS.
        Both B and A MUST HAVE the same dimension.

        The graph is plotted in the complex plane, so you must enter
        the dimension of this plane, and how much screens you want to
        use.

        EXAMPLE
                Plot the Root Locus of:

                     s(s+1)(s^2+4s+16)
                K= - -----------------
                           s+1

                or equivalently:

                s^4 + 3*s^3 + 12*s^2 -16*s + K*(s+1) = 0

        From the Main Menu:

        Root Locus
                    K COEFS :   [ 0 0  0   1 1 ]
                    IND. COEFS: [ 1 3 12 -16 0 ]

                    OK

                    MAX RE:  10
                    MIN RE: -10
                    MAX IM:  10
                    MIN IM: -10
                    H. SIZE:  1
                    V. SIZE:  1

                    OK

                    MIN K: 0
                    MAX K: 50
                    RESOLUTION : 100
                    { The resolution gives the number of points to be plotted}

                    OK  { wait please}

                    Now press CANCEL an you'll back in the
                    "INPUT K RANGE" menu.

                    MIN K: 50
                    MAX K: 10000
                    RESOLUTION : 50
                    { K->INFINITE}

                    OK { wait please }


        As you have seen there's no place for time delay in this section.

        If you have a system with transport lag or dead time and it's very
        small then you can aproximate:

                EXP(-delay*s) = 1 - delay*s
        or
                EXP(-delay*s) = 1 / ( delay*s + 1)




12 NYQUIST PLOT

        Here you can get a polar plot of the transfer function H(s),
        i.e., the plot of its real and imaginary parts when s = jw.

        You must specific the real and imaginary ranges together with
        the frecuency interval.

        When the tranfer function have a pole in zero you can't evaluate
        it in w = 0, instead use w=.001 (or a small value).

        EXAMPLE1  Get the polar plot of:

                        2.65*EXP(-.8*S) /(1+S)
        From the Main Menu:

        H(s)
                NUM : [2.65]
                DEN : [1 1]

                DELAY: 0.8

                OK

        NYQUIST PLOT
                MAX RE:  3
                MIN RE: -3
                MAX IM:  2
                MIN IM: -2
                H SIZE:  2
                V SIZE:  2

                OK

                MIN W : 0
                MAX W : 10
                RESOLUTION : 50

                OK {GRAPHING..}
                CANCEL

                MIN W : 10
                MAX W : 40
                RESOLUTION : 25

                OK {GRAPHING..}

        EXAMPLE2 :  Let's add a pole in zero.

        H(s)
                NUM : [2.65]
                DEN : [1 1 0]

                DELAY : 0.8

        Nyquist Plot
                Same as before { just press OK}

                MIN W : 0.01 { it cannot be zero}
                MAX W : 10
                RESOLUTION : 50
                OK




13 NICHOLS PLOT

        Here you can get a log-magnitude versus phase plot for a frecuency
        range of interest of the transfer function.
        You must specific a decibel range of magnitude and the frecuency
        range.
        Like Bode Plot you could get a discontinuity, just meke te graph
        continue.

        EXAMPLE:

        H(S)
                NUM: [ 1]
                DEN: [1 0.8 1 0]
                DELAY: 0

        Nichols Plot:
                MAX DB:  10
                MIN DB: -10
                H SIZE:   1
                V SIZE:   1
                OK
                MIN W :  .1  { A pole in zero}
                MAX W :  2
                RESOLUTION : 100

                OK { Graph. Look the discontinuity, the graphic could be
                     in the negative angle halfplane }




14 FINAL WORDS

        I hope that this software will be useful for you.
        I would like to have it when I did "Modern Control" course in
        EE Department of University of Chile.
        I'm not so interesting in Control Area, I like Digital systems
        and Telecomunications.

        Now I'm 22 years old (8/1995).

        Thanks to Joseph K. Horn and Patrick J. Yagle for help me about
        INFORM comand.
        Thanks to Sylvana Arenas for her supporting.

        Future Versions? . Sure I'm just thinking in new routines and
        improving the current ones (stability and accuracy), in the
        next versions i'll add state space analysis ( H(s)<->ABCD )
        and maybe a graphical environment.

        Others Programs? . Sure I have a lot programas , most of them
        are about numerical computations ( roots, integrals, differential
        equations (ordinary and parcial), linear systems ( iterative)...);
        electric and electronic circuits and electromagnetics equations.
        None of them have been distribuited, because they are not so good
        for me.

        If you are reading this doc  please E-mail me, becuase I want to
        know how many people will use or hate my program .
        If you find a bug, have a critic or just use this program feel
        free to contact me (PLEASE, All my friends said: nobody will use
        your program).

        Thanks you and Good Luck.


        Juan Francisco Roco Arriagada.

        E-mail:  sroco@cec.uchile.cl