Unit Step Function - University of Tennessee at...

Lanze Berry

9/18/2012

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Unit Step Function

By Lanze Berry

University of Tennessee at Chattanooga

Engineering 3280L

Blue Team (Khanh Nguyen, Justin Cartwright)

Course: ENGR 3280L

Section: 001

Date: September 18, 2012

Instructor: Dr. Jim Henry

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Introduction

There were two objectives to attain in this laboratory. The first objective was to continue

to familiarize ourselves with our system, both the apparatus and the fundamentals of controlling

and guiding it. The focus in this laboratory was to reach an understanding of the unit step

function evaluated at specified inputs. The second objective was to collect experimental data on

the current output of our system for each percent power that was input into the system.

This report will go over in detail the background and theory behind the experiment which

will then be followed by the procedure. The results of the experiment will then be presented and

then there will be a discussion of the results. The final two sections will include the conclusions

and recommendations. Lastly, there will be an appendix containing raw data.

The background and theory section will contain information relevant to the set up of the

lab. This will include all the necessary technical data and schematics. The procedure will

provide an analysis of the experimental methodology. The results will be given in the form of

graphs, tables, and charts of the data obtained during the experiment. The conclusion segment

will reiterate the goals of the experiment and also explain the principles behind it. The

recommendations section will follow and highlight potential criteria of the experiment that can

be changed or improved. The appendix will contain any additional graphs or data not included

in the body of the report.

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Background and Theory

There are four main types of unit input functions. There are unit steps functions, pulses,

unit impulse functions, and sine waves. This laboratory will focus on the unit step function only

and show the dynamics of it. A unit step function will be developed using the data acquired by

the computer. This will help show the significance of the unit step function and help provide an

understanding of the experiment.

The materials used in this lab are a computer, motor, generator, controller, transmitters

and light bulbs. A computer program called LabVIEW is used to graph the readings of the

experiment. The computer is used to send a constant input power signal to the motor. The motor

will then cause the generator to produce amps. A signal will then be sent back to the computer

as an output. Once the signal is received by the computer, LabVIEW will then graph the input

and output readings. Once this has been done, the data can be read by EXCEL, which will help

in the analysis of the data. Figure 1 shows the schematic of the motor-generated system.

Figure 1: Schematic of the Motor- Generated System

Computer

Motor Generator

Light Bulbs

JT 101

JRC 101

ERT 101

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Figure 1 demonstrates in schematic form the procedure of sending an input and receiving the

output. The computer sends a signal to the power transmitter (JT 101) which in turn relays it to

the power-recording controller (JRC 101). Once the signal is received, it is sent to the motor

where the percentage power is the input. The motor begins to turn and the mechanical energy is

converted to electrical energy. The generator uses electromagnetic induction to convert the

energy. The new electrical energy is then used to create current to the light bulbs. The generator

then sends an amps output signal back to the (ERT) and relays the signal to the computer where

it is interpreted in LabVIEW as a graph. The block diagram is shown in Figure 2.

Figure 2: Block Diagram of the Motor-Generated System

Figure 2 shows the input to the system on the left hand side of the block. The input is the time

allotted (in seconds) for the experiment and the motor power (in a percentage from 0 to 100).

The output response from the system is shown on the right side of the block. The output C(t)

demonstrates that it is a function of time. The output also contains the generated amps. The

system responds by giving a different amp output to each different input containing the time and

motor power percentage used.

In the previous experiment, an input percentage was entered to attain a steady state curve

within a desired output range (in amps). This data was collected using LabVIEW and then the

Motor-

Generator

System

Input M(t)

Motor Power (%)

Output C(t)

Generated Voltage (V) Generated Amps (A)

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data was plotted in excel. Figure 3 shows the SSOC for the input percentage of 72.5 and the

output within the 3 to 4 amp range.

Figure 3: SSOC for 72.4% Input

Each group member had a specific output target of 1 – 2 amps, 2 – 3 amps and 3 – 4 amps. Once

each individual collected data, it was compiled into one table. This data is shown in Table 1.

Motor Input Power (%) Average Current (Amps) Mean 2x (Std. Dev.)

50 1.13 0.021

60 1.43 0.015

65 1.73 0.012

67 1.77 0.012

69 1.95 0.009

69.5 2 0.51

70 2.01 0.4

70.5 2.1 0.5

71 2.16 0.881

72 2.2 0.373

74 3.6 0.006

76 3.7 0.005

79 3.8 0.005

82 3.9 0.006

Table 1: Steady State Curve Data Points

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Using Table 1, a plot was created using the points. Figure 4 shows the relationship of the input

percentages versus the output amperage.

Figure 4: Steady State Operating Curve

Figure 3 shows that that the input percentage between about 68% - 73% had a very clumped

output of about 1.8 amps to 2.2 amps. This section of the data prevents the line from showing an

ideal linear curve. There were difficulties staying in the 2 – 3 amps output range. If the input

percentage was too high, the output exceeded 3 amps. If the input percentage was too low, the

output did not reach 2 amps. When observing the graph, it can be seen that the first four data

points and the last four data points share about the same linear curve. It can be seen that this data

was somewhat consistent.

current (amps) = 0.0582(Input%) - 1.9671

0

1

2

3

4

5

6

45 50 55 60 65 70 75 80 85

Ou

tpu

t (A

mp

s)

Input (%)

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Procedure

The experiment begins with using a computer. A program called LabVIEW is used as

system control and it collects data throughout the duration of the experiment. This program was

accessible through the internet and the Control Systems class website. Once the Current

Information Systems group was selected, a choice appeared to begin collected data for the Unit

Step Function. Once this was selected, a screen appeared with credentials to fill out (name,

email, etc…). Also, a space was provided to enter the how many seconds the experiment was to

last and there was also a space to enter at what percentage the power should be. Once the

criterion was complete, the program would conduct the experiment according to the time and

percentage entered. The program then records and plots the data into a data base. This process

was done three times by each group member for a unit step function. The data was then

translated into excel to plot a steady state operating curve and to statistically analyze the data.

Figure 5 shows the flow of information in LabVIEW.

Figure 5: Flow of Information to and from LabVIEW

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Results

Each experiment conducted collected data to create plots of the current output vs. time vs.

power input percentages in excel. Figure 6 shows a sample of a graph obtained from the

experimental data.

Figure 6: Graph of data collected for Unit Step Up Function

The output current is on the left side of the graph and measured on the secondary y-axis. The

time in seconds is located on the x-axis and the input percentage is located on the primary axis.

The blue line represents the input percentage. Figure 6 has an input percentage of 39% and has a

20% step height. Each graph had criteria that needed to be determined on it. Figure 5 shows

how the time constant was to be determined for the same input percentage as well as the gain and

the dead time. Each was calculated and shown on the graph.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0

10

20

30

40

50

60

70

3 4 5 6 7

Ou

tpu

t (A

MP

S)

Inp

ut(

%)

Time (sec)

Unit Step Up Graph

Motor Input Power (%)

Output Current (Amps)

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Figure 7: Input Step Percentage of 39.5% to 59.5%

In Figure 7, the analysis of the unit step is shown. Fit 2 is used for the analysis. The gain (K) is

calculated to be 0.43 amps. The time constant (τ) is calculated to be 0.43 seconds. The dead time

(t0) was found to be 0.11 seconds. The results gathered by analyzing all of the experimental runs

were then used to create a unit step function. The data that created this function is shown in

Table 1. Once the output started to get higher, in the range of 3 – 4 amps, the measurements

started to become very inconsistent (which can be seen in the appendix for input % of 65 and

higher). Table 1 also identifies this. The time constant is about double the other measurents for

this range.

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Input (%) Gain (cm-amps/%) Dead Time (sec) Time constant (sec)

38 0.0420 0.1070 0.420

39 0.0430 0.1100 0.430

40 0.0450 0.1080 0.450

41 0.0450 0.1070 0.460

42 0.0460 0.1080 0.470

43 0.0460 0.1070 0.480

50 0.0500 0.1080 0.323

51 0.0490 0.1070 0.323

52 0.0499 0.1080 0.161

53 0.0516 0.1080 0.161

53.5 0.0502 0.1080 0.323

54.5 0.0499 0.1080 0.308

65 0.0652 0.1080 0.716

66 0.0688 0.1080 0.761

67 0.0733 0.1080 0.819

68 0.0780 0.1080 0.878

69 0.0820 0.1070 0.936

69.5 0.0830 0.1080 0.935

Table 2: Summary of Results for the Unit Step Function

Bar graphs were created to show the comparison of the gain, dead time, and time constant in

each section of amps output for the unit step up. A unit step down was not conducted due to the

motor not working correctly and data not being collected. The bar graphs are shown in Figures 8

-10. The bar graphs show that gain greatly increases once the output reaches the 3 to 4 amp

range. Figure 9 displays the dead time and shows that the dead time stays very consistent

throughout the different output ranges. Figure 10 is a graph of the time constant. This is a very

inconsistent measurement. Each output range has a different average.

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Figure 8: Gain of the 20% Unit Step Up

Figure 9: Dead Time of the 20% Unit Step Up

0.103

0.104

0.105

0.106

0.107

0.108

0.109

0.11

0.111

1-2 amps 2-3 amps 3-4 amps

Dead Time : to, (s)

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Figure 10: Time Constant of the 20% Unit Step Up

The unit step function for the motor-generator system (using Table 1) is shown in Figure 11.

Figure 11: Unit Step Function for the System

0

0.2

0.4

0.6

0.8

1

1.2

1-2 amps 2-3 amps 3-4 amps

Time Constant: τ, (s)

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Discussion

This experiment dealt with analyzing the unit step function using signal processing. The

results explore the gain, dead time, time constant, and height of the step. The graphical

representations provide the best understanding of the unit step function. Through the graph, it is

visually seen where the input percentage jump is located and how it affects the output. The

current increases almost simultaneously with the input jump. Using just the graph, all the details

necessary for the experiment are provided. The dead time can be found using slope of the

current and the start of the input percentage jump. The time constant and the gain can also be

calculated with help of the graph. Input percentages ranged from 38% to 69.5%. These

percentages produced an output between 1 to 4 amps. The gain, dead time and time constant

were found for a 20% unit step up. The step down was not calculated because it could not be

conducted. Bar graphs were created to demonstrate the relationship of each output range (low

being 1 to 2, medium being 2 to 3, and upper being 3 to 4 amps).

It is determined that the motor input level greatly affects the output. When the jump of

the input percentage occurs, the amps greatly increase to follow that jump which in turn

represents the unit step. The initial part of the graph is disregarded because the data is constant

until the step begins. Figure 11 has a time range of four to seven seconds because that is the part

of the experiment to be focused on. Everything prior and after that time zone range is constant

and doesn’t provide any crucial data for evaluating the unit step function. Inconsistencies were

found when the output reached the range of 3 – 4 amps. The graphs (located in the appendix)

show that the output could not reach a constant state after the unit step in the desired output

range.

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Conclusions and Recommendations

The experiments that were conducted were done to collect data and to analyze the data to

further understand the Unit Step Function. The results that were obtained show clear unit steps

in each of the graphs and present the principles of the unit step function. The experiment

validates that when there was an input percentage increase; the output also increased and then

became constant.

The primary objectives of this experiment were to collect experimental data and to

understand and know how to analyze the unit step function. Each of these objectives was

reached. The data was collected using LabVIEW and put into an excel spreadsheet where it was

then created into a graph. Using the graph, the unit step function was analyzed and the dead

time, time constant, and gain were found. The results clearly present the data in concise and

detailed graphs.

The values for the dead time proved that it was a very consistent measurement throughout

all the outputs. The gain of each input percentage steadily increased as the output range

increased from 1 to 4 amps. The time constant proved to fluctuate tremendously with each

output range. The time constant ranged from 0.161 - 0.935 seconds.

A recommendation would be to have a way to evaluate why an output range could not

obtain a constant state. In the output range of 3-4 amps the data got somewhat inconsistent with

a much higher time constant.

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Appendix

Figure 12: 38% Input with 20% Step

Figure 13: 39.5% Input with 20% Step

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References

1. http://chem.engr.utc.edu/3280L

2. Carlos A. Smith and Armanso Corripio, Principles and Practices of Automatic Process

Control, John Wiley & Sons Inc, 3rd

Ed., 2006

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