# ge320_Lab3

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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

3) Create a new model by going to File->New->Model (This will be done in the
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)
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
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

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
   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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