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					Experiment 3: DC Motor Voltage–to–Speed Transfer
Function Estimation by Step–Response and Frequency–
Response
Introduction

This lab introduces new methods for estimating the transfer function of a plant (in our
case, the DC motor). Recall that in Experiment 2 the transfer function of the motor was
obtained by measuring the various physical parameters of the motor and applying them to
the known mathematical model. The methods to be used in this lab, called step– and
frequency– response methods, are indirect and can be used even if a good mathematical
model of the plant is unavailable.


Experiment 3 Prelab
   1. For a first order system, how can the transfer function be estimated from the step
      response, i.e. how are the DC-gain and time constant found?
   2. Consider A  sin( t ) as the input to a linear system with Transfer Function G(s).
      Write down the expression for the output of the system.
   3. Write down expressions for the 2 sine waves depicted in Figure 1. The plot is
      showing magnitude vs. time with time being 0 - 2π s. The larger amplitude signal
      is the input and the smaller amplitude signal is the output.
                                                                              Km
   4. Sketch the Bode diagram, both magnitude and phase, for the system
                                                                            s m  1
   5. Using the Bode plot from the previous question (#4), determine the DC gain and
      time constant for the system. Mark them clearly on your graph.




                                        Figure 1
Simulation of Motor Transfer Functions in Simulink.

First your TA will give you a short lecture on how to use Simulink. You will then build a
Simulink simulation to compare to the actual output data from the motor. First simulate
the transfer function you found in Lab #2. In order to compare this transfer function to
the transfer function to be identified in this lab you will have to add the gain of the
amplifier, Kamp = 2.4, and the gain of the tachometer, Ktach=0.03. The transfer function
                                           VTach ( s)          K K K
that will be identified in this lab is                         Amp m Tach . First start by
                                       Vinput _ to _ Amp ( s)    s m  1
simulating the motor transfer function found in Lab #2. Use Figure 2 and the steps below
to build a Simulink simulation of the motor.


                                           K_m
                               2.4                     0.03       simout
                                         tau_m.s+1
                      Step              Transfer Fcn           To Workspace
                               Kamp                    Ktach




                                         Figure 2
Creating a SIMULINK simulation

   1) Load MATLAB
   2) Load Simulink by typing “simulink” on MATLAB’s prompt
   3) Create a new model by going to File->New->Model (This will be done in the
       Simulink Library Browser window)
   4) To set the simulation time go to Simulation->Configuration Parameters. Under
       “Time Simulation” change “Stop Time” to 1 sec.
   5) To create the model in Figure 2 we’ll use some simulation boxes.
   6) From the Simulink Library Browser window select “Commonly Used Blocks”
   7) Drag a “Gain” block to the Model created in step 3.
   8) Go back to the Simulink Library Browser window and look for “Sink”
   9) Drag a “simout - To Workspace” block to the Model (This box will output your
       measurements as variables in MATLAB: simout = Voltage and tout = time)
   10) Go back to the Simulink Library Browser window and look for “Sources”
   11) Drag a “Step” block source to the Model
   12) Go back to the Simulink Library Browser window and look for “Continuous”
   13) Drag a “Transfer Fcn” block to the Model
   14) Now connect the boxes so they look like Figure 2 in your lab. To connect them
       use the small triangles on the edges of each block. Click on them and drag the
       connection to the next block.
   15) Double click on each box and change the values so they match the system you are
       simulating.
   16) Make sure the “Step” block source is set as following:
          a. Step Time = 0 (the source is turned on at 0 seconds)
           b. Initial Value = 1.25 (That is the initial value you input on VEE)
           c. Final Value = 2.5. (In the next section of this lab you will change this
               final value from 2.5 to 4.5 to match the current identification run.)
           d. Sample time = 0
   17) Make sure your “simout” block is set as following:
           a. Under Parameters go to “Save Format” and set it to “Array”.
   18) After setting up every block, hit the play symbol (“Start simulation”) on the bar,
       or go to Simulation->Start.
   19) Graph your responses using the following MATLAB command prompt
       command:
           a. plot(tout,simout)
   20) Show you’re your work to your TA.
   21) Print this step response and your simulation block diagram to be included in the
       post lab.




Transfer Function Estimation by Step Response

The transfer function of an unknown plant can be obtained by analyzing its step response.
As seen in Lab 2, the motor’s Voltage to Angular Velocity transfer function can be
                                                 ( s )      Km
approximated by a first order transfer function,                   . The goal of this section
                                                 E a ( s ) s m  1
will be to estimate the values for Km and τm by analyzing the response of the motor to a
step input voltage. Figure 2 above is the block diagram of the setup that will be needed
for this experiment. A step function will be input to the motor, and the signal from the
tachometer will be observed. Remember that the voltage of the tachometer is proportional
to the angular velocity of the motor. Note that the motor parameter values in Figure 2
(Km and τm) are the ones to be estimated.

Testing Procedure:
   Connect the HP 0-20V power supply to the motor’s amplifier input.
   Scope the tachometer on channel 1 of the scope.
   Turn on all the equipment and then open the Agilent VEE file:
      “n:/labs/ge320/exp3/step.vxe”
   Initialize communication between the computer and the instruments. To do so, hit
      “Reset” on the window for the power supply and Send New Scope Parameters
      on the window for the scope. Turn the power supply output enable OFF.
   Set the current to 0.2 amps, and the voltage to 1.25 volts. Turn output enable ON.
   Change the voltage to 2.5 volts.
   The step response should appear on the scope. Turn off the motor by switching
      OFF the output enable. Show this observation to the TA.
      Press Collect Data from Scope to capture the rising exponential displayed on the
       scope. Use this data to calculate Km and τm. Remember that KAmp=2.4 and
       KTach=0.03. Input these new values for Km and τm into your Simulink simulation
       and compare visually the actual data plot to the simulation plot.
      Repeat the experiment 4 more times always starting with the voltage at 1.25V and
       incrementing your final voltages from 3.0 to 4.5 volts in 0.5 volt increments.
       Also compare to simulation (don’t forget to change the step size). Initially set the
       scope’s y scale to 1 V per division. When the step response goes beyond the top
       of the scope screen change the sensitivity to 2 V per division so the full response
       is displayed. Always remember to make the changes for the scope from the
       Agilent VEE program instead of using the knobs on the actual scope. Ask your TA
       if you have any questions

Transfer Function Estimation by Frequency Response

A very powerful tool for identifying systems is the frequency response method. The key
idea behind frequency response techniques is to input sinusoids into the linear system and
study the response. A well-known result of linear theory establishes that the steady-state
response of a linear system to a sinusoid is a sinusoid, with the same frequency as the
input sinusoid, but, possibly, with a different amplitude and phase. The change in
amplitude is equal to G( j) and the phase shift is Angle(G( j)) , where G(s) is the
transfer function of the system, evaluated at the input frequency. Figure 3 shows a
graphical representation of this idea.


              sin ( ωt )                        | G( j ) | sin( t  Angle(G( j ))) |
                                 G(s)



                 Figure 2 Steady-State response of a linear system to a sinusoid


The goal of this section will be to input sine waves of varying frequencies and measure
the ratio of amplitudes of output to input sine waves and the phase shift between the two
waveforms. That will provide enough information to identify the linear system G(s). The
input wave needs to have a DC offset of sufficient magnitude (Why?). When calculating
the ratio between output and input amplitudes, you will have to eliminate the DC offsets
both for the input and output signals. An easy way to do this is to compare the peak to
peak voltages instead of the amplitudes. From your prelab you should have a good idea
on how to use this information to obtain the transfer function of your system. If you still
have doubts ask your TA.

Testing Procedure:
    Open the file “n:/labs/ge320/exp3/freq.vxe” and initiate communication with the
       instruments by sending the scope and function generator parameters and settings.
   A sine wave will be used to drive the motor. The current of the function generator
    is not enough to drive the motor. Therefore, you need to use the amplifier in the
    PATCH PANEL. Connect the amplifier output to the motor. Connect the thumb
    switch to the audio jack on the amplifier.
   Use channel 2 of the oscilloscope to scope the signal from the function generator,
    BEFORE THE AMPLIFIER, and channel 1 to scope the signal from the
    tachometer. DO NOT TURN ANYTHING ON BEFORE YOUR TA CHECKS
    YOUR CONNECTIONS
   From the computer, in the function generator window, set the frequency of the
    input. You will have to repeat the experiment for frequencies 0.1Hz, 0.5Hz, 1Hz,
    2Hz, 3Hz, 4Hz and 5Hz. You need to identify the interval at which a 45˚ phase –
    shift occurs (Why?). To find the 45˚ point more accurately pick three more
    frequencies within that interval and find the gain and phase shift.
   To create the magnitude plot, find the ratio of peak to peak output voltage verses
    peak to peak input voltage. This is call “gain.” When creating your Bode plot it
    is common to convert this “gain” to units of decibels (dB). A dB =
    20*log10(gain). To measure the phase shift use two adjacent peaks of the input
    and output waves and measure the time difference between them. The output
    wave is said to be “behind” the input wave. You need to convert this time
    measurement to degrees. To do that, you know that 1 period of the wave is 360 ˚
    (period = 1/frequency).
   Press the thumb switch and keep it pressed. Wait until the system reaches steady
    state. For slow frequencies you will have to wait a little longer. Collect the data
    displayed in the scope, repeat until you get a clean waveform. Once you have a
    clean waveform, release the thumb switch. Using the collected data in Agilent
    VEE, measure the frequency’s gain and phase shift.
   Repeat for the frequencies indicated. As you increase the frequency, you need to
    change the Timebase of the scope. You don’t need to change it for each repetition
    of the experiment but make sure that you don’t have more than 3 full periods of
    the wave on the scope’s screen.
   Create a bode plot with the data found. One semilog plot of magnitude (in dB)
    verses input frequency and one semilog plot of phase (in degrees) verses input
    frequency.
   From these two plots find values for Km and τm. (Post Lab Question #4)
Lab 3: Post Lab
Include the answers to the following questions in you lab report.
   1. Calculate the transfer function for the motor with the data obtained from the step–
      response experiment.
   2. Comment on the similarity/differences between the first–order transfer function
      estimated in the step–response experiment with the similar transfer function
      obtained in experiment 2. Remember to account for the gain of the amplifier and
      the gain of the tachometer.
   3. Using the first–order transfer function estimated in the step–response experiment,
      design an open–loop control system that will spin the motor at 100 rad/sec and
      200 rad/sec. That is, you will have to identify the voltage that needs to be applied
      to the motor to spin the motor at the required speeds. Roughly, how long should
      we have to wait to achieve the required speed to within 1%?
   4. Calculate the transfer function for the motor with the data obtained from the
      frequency–response experiment.
   5. In the frequency–response experiment, why did we need a DC offset?
   6. How can we infer the order of the system from the Bode plot of the frequency-
      response experiment?
   7. Include the block diagram from your Simulink simulations (step responses). Just
      include two step responses that compare the transfer function found in Lab 2 to
      one of the transfer functions found with the step response method.
Data Sheet

STEP RESPONSE OF A DC MOTOR

Step size     Vout (V) Steady Time       constant DC gain
From - To     State           (sec)

1.25 – 2.5

1.25 – 3.0

1.25 – 3.5

1.25 - 4

1.25 – 4.5


FREQUENCY RESPONSE OF A DC MOTOR
Frequency (Hz) Vin p-p (V) Vout p-p (V) Vout      Gain      t (sec)   Phase shift
                                             gain in dB                (degrees)
                                        V in

0.1

0.5

1

2

3

4

5

Extra measurements for 45 phase shift

				
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