Lab 2: Identification of the
Transfer Function of a DC Motor
Theodore Hintz
Lab 2: Identification of the Transfer Function of a DC Motor
Theodore Hintz
Electrical Engineering Department
University of Texas Arlington
Arlington, Texas USA
Abstract—The purpose of this lab experiment is to extract the
parameters necessary to approximate the behavior of a DC
motor. These parameters will then be used to simulate a DC
motor plant and compare them to the actual DC motor plant they
were extracted from. It was found that the extracted parameters
accurately characterized the DC motor plant.
For the sweep to be accomplished in SIMULINK, the block
diagram in Figure 1 is used. By allowing the step input
amplitude to equal one, the slider gain is equal to the input
voltage applied to the motor.
Index Terms—DC Gain, Rise Time, Dead Zone, Linear,
Nonlinear
INTRODUCTION
In order to precisely control a DC motor in real world
applications, certain parameters must be extracted from a given
motor to approximate its real world behavior. Often times, the
DC gain and rise time of a motor are sufficient parameters to
accurately describe the behavior of a motor.
The DC gain relates the input voltage of a motor to the
output voltage of a motor, thus relating the input voltage of a
motor to the output speed. The rise time is a measurement of
the amount of time it takes for the motor to reach
approximately 63.2% of its steady state response with the
steady state response of a motor being the output voltage as
t → ∞. The transfer function of a DC motor given τ = rise time
and g = DC gain is shown below.
𝐻(𝑠) =
Figure 1: SIMULINK Motor Plant Control
Using the slider gain pictured above, the input voltage is
incremented and the output voltage recorded throughout the
sweep. The resulting plot of Vout vs. Vin is shown below in
Figure 1.
𝑉𝑜𝑢𝑡(𝑠)
𝑔
=
𝑉𝑖𝑛(𝑠)
𝜏𝑠 + 1
During this lab experiment both of these parameters will be
extracted using a SIMULINK computer/plant interface. With
these parameters a DC motor can be sufficiently and accurately
described.
PARAMETER EXTRACTION
DC GAIN
First the DC motor plant is connected to SIMULINK using
an NI-1200 card in a computer. Through this card and
SIMULINK an input voltage can be sent to the plant and the
resulting output data can be viewed. First, to extract the DC
gain parameter, an input voltage is swept from -5V to 5V.
From -2V to 2V the voltage is stepped in 0.2V increments,
between -5V to -2V and 2V to 5V the voltage is stepped in .5V
increments. Each time the voltage is incremented, the steady
state DC value of the output voltage is recorded. Once the
voltage sweep is complete, Vout vs. Vin can be plotted and the
DC gain can be calculated from the slope of the resulting line.
Figure 2: Vout Vs. Vin
When calculating the DC gain, the dead zone cannot be
included in the slope calculation because it has nonlinear
behavior. The dead zone of a DC motor is defined as a certain
range of input voltages that cause no output voltage. This
occurs because this certain range of input voltages is not high
enough to cause the DC motor to overcome its internal
mechanical resistances, thus causing no output voltage. In
Figure 2, the dead zone can clearly be seen between -1V and
1V.
The saturation region must also be excluded when
calculating the DC gain. The saturation region occurs when the
change in output voltage to input voltage approaches zero. This
region can be clearly seen for input voltages less than -4V and
greater than 4V.
In order to calculate the DC gain, values from the linear
region must be used. The linear region for positive input
voltages is approximately between 1V and 4V. The DC gain
can be calculated using the formula below.
𝑔=
Here, a 5V input is used for both the simulation and the real
motor response. The resulting data and the SIMULINK block
diagram are shown below.
𝑉𝑜𝑢𝑡(𝑓𝑖𝑛𝑎𝑙) − 𝑉𝑜𝑢𝑡(𝑖𝑛𝑖𝑡𝑖𝑎𝑙)
𝑉𝑖𝑛(𝑓𝑖𝑛𝑎𝑙) − 𝑉𝑖𝑛(𝑖𝑛𝑖𝑡𝑖𝑎𝑙)
Using experimental data the DC gain g is calculated.
Figure 4: SIMULINK Simulated Motor
2.7𝑉 − .38𝑉
𝑔=
= .77 𝑉/𝑉
4𝑉 − 1𝑉
RISE TIME
The rise time is the approximate time it takes the system to
achieve 63.2% of its steady state response. This number is
approximated by τ = 1-1/e ≈ 63.2%. τ then corresponds to
fc = 1/(2πτ), the cutoff frequency of a first order system. The
cutoff frequency occurs when a given system drops 3db from
its stead state value or equivalently, when it reaches half
power.
In order accurately approximate the rise time of the DC
motor, it is calculated for three different input voltages and
then averaged. For each input voltage, the step response of the
motor is observed. Then, the time at which an output voltage is
63.2% of the steady state output voltage is observed, and this is
recorded as the rise time. The resulting data is shown below in
Table 1.
Vin
5V
3V
-5V
Vout
2.946V
1.9679V
-3.1689V
Avg. Rise Time
τ
.208s
.200s
.200s
Figure 5: Step Response of Real Vs. Simulated Models
In Figure 5 it is observed that the gain of the simulated
motor is significantly higher than that of the actual motor. If
the gain of the simulated motor is tweaked to g = .59 then the
step responses match as shown in Figure 6 below.
.202s
Table 1: Rise Time
COMPARING REAL MODEL AND SIMULATION
In order to test the transfer function and the simulations
validity, the step response of the real motor and the step
response of the motor transfer function are observed and
compared.
𝑉𝑜𝑢𝑡(𝑠)
𝑔
𝐻(𝑠) =
=
𝑉𝑖𝑛(𝑠)
𝜏𝑠 + 1
In this model, g = .77 and τ = .202s, these parameters are
then plugged into a SIMULINK simulation and the step
response is compared to the motors at the same input voltage.
Figure 6: Modified Gain Step Response
From Figure 6 it is observed that the gains and rise time of
both the real and simulated motor step responses match. It can
be concluded that the gain parameter extracted was inaccurate
but the rise time was accurate.
TROUBLESHOOTING INACCURACIES
As inaccuracies were observed, the original data used for
the gain parameter was remeasured and rechecked, ultimately
yielding the same inaccurate results. Further troubleshooting
involved the inspection of the plant used. All power supply
voltages were measured but the 5V power supply used for the
plants measuring equipment checked in at .9V, far from
nominal. After this was observed a new power supply was put
in its place but the same inaccurate results were found.
Whenever a power supply was connected to the plant being
used, the measurement equipment power supply measured .9V,
if the power supply was connected to a different plant, it
measured 5V. From the troubleshooting performed it can be
deduced that a component in the plant used for measuring has
likely failed or is malfunctioning.
DATA AND DISCUSSION
DATA
All of the data used for parameter extraction and the
resulting parameter are shown in the table below.
Vin
Vout
τ
5V
3V
-5V
2.946V
1.9679V
-3.1689V
.208s
.200s
.200s
Avg. Rise Time
Vout 2
Vout 1
Vin 2
Vin 1
.202s
2.7V
.38V
4.0V
1.0V
Avg. Gain
.77
Table 2: Parameter Extraction Data
The gain can be sufficiently described with two significant
figures. Since there are many environmental variables when
recording input voltage and output voltage, the second decimal
place would surely change for a given set of data generated
from the same plant. Two significant figures would also be a
valid estimate for the rise time given the amount of
environmental variables, but since an average rise time is the
calculation used, more significant figures were used to produce
a more accurate result.
If the plant was able to provide SIMULINK with the
measured input voltage, a more accurate gain and rise time
could be calculated. Since a power supply is usually a few
decimal places off of its intended value, a small amount of
error will be introduced into the gain calculation.
DISCUSSION
If the motor were to be used in a control system, it would
be most effect if it was operated in the linear region, from
approximately 1V to 4V or -4V to -1V. In these regions, the
first order transfer function model will be most accurate, thus a
more effective control system can be designed.
If one were to design a speed controller for the motor the
dead zone should be accounted for. If the dead zone were not
accounted for, a controller might expect a speed from the
motor at a given input voltage that occurs in the dead zone,
according to the linear relationship of g. If this were the case, a
controller would have a period where no output voltage
occurred. To design around this, a threshold input voltage
could be set for each side of the dead zone so that when the
controller detected it, it could suddenly switch the input voltage
as to avoid the dead zone and continue operation.
If a normal operating speed were to correspond to 2V
output, it would be most effective to approximate the gain
using values just greater than and just less than 2V. For
example, if Vout = 2.1V and Vout = 1.9V were used in the gain
approximation, the final result would be an accurate linear
approximation of the gain at Vout = 2V.
By using a first order transfer function to model the plant,
the parameters for the model can easily be experimentally
extracted. If a second order model were used it would be
difficult to experimentally extract all of the parameters
necessary to describe the behavior of the plant. With this first
order approximation it is an accurate description and easy to
describe.
𝑉𝑜𝑢𝑡(𝑠)
𝑔
. 77
𝐻(𝑠) =
=
=
𝑉𝑖𝑛(𝑠)
𝜏𝑠 + 1
. 202𝑠 + 1
The above transfer function has no zeros but has 1 pole.
This pole occurs at s = - 1/.202 = - 4.95 which is approximately
the minimum input voltage. Since this pole is in the left half
plane (LHP), the system is stable
CONCLUSION
During this lab experiment the methodology used to extract
the DC motors parameters was executed correctly. Through
troubleshooting it was concluded that the inaccurate gain
calculated was attributed to equipment malfunction. Overall it
was an effective tool in the methodology of motor parameter
extraction and testing the validity of the derived models.
REFERENCES
[1]
Gene F. Franklin, J. David Powell and Abbas Emami-Naeini,
Feedback Control of Dynamic Systems, 6th edition, Pearson Higher
Education, 2010.
APPENDICIES
The only code used in this experiment was the code used to plot, this code is shown below.
Figure 2:
ylabel('Vout') %Label y axis
xlabel('Vin') %Label x axis
title('Vout Vs. Vin')
%Title graph
plot(speed(:,1),speed(:,2))
%Plot real motor step response
Figure 5 and 6
plot(speed(:,1),speed(:,2))
%Plot real motor step response
hold on
%Hold graph for second graph
plot(scope(:,1),scope(:,2),'--')
%Plot simulated motor step response
xlabel('Time (s)')
%Label x axis
ylabel('Voltage (V)')
%Label y axis
leg = legend('Real Motor','Simulated Motor')
%Label different lines