University of Tennessee at Chattanooga Step Response Engineering 329 By Gold Team: Jason Price Jered Swartz Simon Ionashku 2-3-
2 INTRODUCTION: The purpose of the experiments was to investigate and understand the steady-state behavior and step response parameters of a single-input, single-output system. The system consists of a generator that is supplied with power from an electrical motor, and then in return generates a voltage. The input, or manipulated variable, will be set by the user and the output, or controlled variable, will be measured using data acquisition software. The input and output are both functions of time. The Gold team s primary focus will be on the output range falling between 75-95 volts as measured by the data acquisition software. The results will be used to analyze the steady-state and step response performance of the system. The main objective of the experiments is to develop the steady-state operating curve and to establish the First-Order Plus Dead Time (FOPDT) parameters for the system. The report begins with a background and theory of the laboratory, followed by the procedure of the experiment. The results obtained from the experiment are shown next, then the discussion of results. The conclusions and recommendations will be the next section and the appendix will be the last section of the report. The background and theory section will display and explain the mathematical formulas and physical models that are used to analyze the data obtained in the experiment. The order in which the experiment was done and how the experiment was conducted are described in the procedure section of the report. The results section will include the data obtained from the experiment in tabular and graphical format. The significance of the results will be explained in the discussion part of the report. The conclusions and recommendations section will detail the principles that were
3 demonstrated by the results. All of the remaining data collected from the experiment will be placed in the appendix.
4 BACKGROUND AND THEORY: Modern electronics and software packages give individuals the opportunity to collect, analyze and interpret large amounts of data obtained in modern control systems. These systems are expected to operate within desired ranges over its operating cycle. Analysis of a systems performance prior to their implementation is crucial for continued use and reliability in service. The system that will be used in the experiment consists of an electrical motor, generator, controllers, transmitters, and light bulbs. A schematic of the system can be seen in Figure 1. Figure 1. Schematic of Voltage Control System
5 The user selects an input power percentage and enters it into the computer. The computer sends a signal to the voltage control actuator, VCZ 272. The voltage control actuator relays the voltage input to the 5 HP, 3-phase motor. The mechanical energy of the motor is transmitted through a connected shaft to an electrical generator which creates voltage potential through magnetic induction and supplies electric current to a bank of light bulbs. In the process, the generator also sends a voltage output signal to the voltage transmitter, VT 272, which relays the signal to the voltage recording control, VRC 272. LabVIEW software is used to display the data on the computer. LabVIEW will graph the input and output functions of the system and the analysis will be done by using EXCEL. The block diagram in Figure 2 shows the control system with the input and output functions. The input is the percentage of power delivered to the motor, M(t), which is the manipulated variable. The output is the generated voltage from the generator, C(t), which is the controlled variable. The system responds by delivering a voltage output for different percentages of power input that is dependent on time. Input M(t) Motor Power (%) Motor- Generator System Output C(t) Generated Voltage (V) Figure 2. Block Diagram of the Control System
6 Steady-State One performance characteristic of great interest is the steady state operating curve (SSOC). The goal of this experiment is to study SSOC characteristic for a voltage system by examining its response to a series of constant power inputs. This experiment is intended to display the steady-state conditions of a control system by the input of a manipulated variable and the output of a controlled variable. A steady-state operating curve will be developed using the experimental data obtained from the sensors in the control system. Step Response A step response is the time behavior of the system's output to a sudden change of input power percentage. The objective of this experiment is to determine the First-Order Plus Dead Time (FOPDT) parameters for the system. One of these parameters is steadystate gain (K), which is the steady-state change in output divided by the sustained change in input. Another parameter is the dead time (t o ), which is the amount of time it takes the system's output to respond to the change in input. The time constant of the system will be calculated, which is when the output response reaches 63.2% of its final change. The last parameter that will be determined is the response time (τ) of the system, which is the amount of time it takes for a system in steady-state operation to reach steady-state operation again after a change in input. The response time is usually five time constants, because the output reaches over 99% of its change in five time constants.
7 PROCEDURE: A computer and LabVIEW software will be used to control the experiment and acquire experimental data. The experiment can be accessed and performed remotely from any computer with internet access. The user enters the desired input power percentage and duration of the experiment into LabVIEW. The program records and plots the data acquired from the experimental run into a database that can be opened in EXCEL. A schematic of the LabVIEW operating steps can be seen in Figure 3. Figure 3. LabVIEW Schematic Steady-State The generator output range for the experiment was set from 7 to 95 volts, therefore by trial and error the input power percentage was found to be from 83% to 95%. The experiment was conducted for seven different input power percentages in 2% intervals for seconds. The experimental data was compiled and averaged to develop a steady-state operating curve for the system.
8 Step Response Since the team's voltage output range is from 7 to 95 volts, the input range from 83 to 94% was used. This range was divided into four segments; 83-85%, 86-88%, 89-91%, and 92-94%. In these four segments a step response of 2% was used to step up and step down using a time of seconds for each. Three trials were completed for each step up and step down. The experimental data was used to develop graphs in order to determine the steady-state gain, time response, dead time, and time constant.
9 RESULTS: Steady-State A plot was made for each of the seven experimental data sets detailing voltage output versus time versus input power percentage. An example is provided below in Figure 4 for 89% input power. JRA 1// Figure 4. Sample Data at 89% Input Time in seconds is on the x-axis, input power in percent is on the primary y-axis, and output voltage in volts is on the secondary y-axis. The output voltage is represented by the red line and the input power is represented by the blue line. The output of the system was considered to be in steady-state operation between and seconds for each data set. The average and twice the standard deviation was analyzed in this time range and the results can be seen in Table 1. This graph also shows a time delay in the output before steady-state operation is reached.
Table 1. Summary of the Steady-State Operating Curve Results INPUT, M(t) (%) OUTPUT VOLTAGE, C(t) (V) STANDARD DEVIATION (2x) (V) 83 72. ±.29 85 75.4 ±.25 87 8.1 ±. 89 84. ±.27 91 87.5 ±.24 93 9.1 ±.29 95 93.6 ±.31 The data in Table 1 was used to create a plot to display the steady-state operating curve for the system in Figure 5. Output Operating Range Input Operating Range JRA 1// Figure 5. Steady-State Operating Curve The power input percentage in percent is shown on the x-axis and the voltage output in volts is shown on the y-axis. The graph displays the motor input power
11 percentage as a function of the generator's output voltage. The steady-state operating curve between 83% and 95% input power has very close linear relationship with a slope or gain of approximately 1.8 V/%. Uncertainties were assigned to the data points with a confidence level of 95%. The uncertainties can be seen by vertical lines at each data point. These were found by using the standard deviation of the mean for the output voltage and multiplying by two. The uncertainties for the outputs are shown in Table 1. Step Response A plot was made for each experimental data trial detailing voltage output versus time versus input power percentage and showing the step response of the system. An example of the gain calculation is provided below in Figure 6 for 83% input power with a 2% step response. JRA 1/31/ Figure 6. Step Response, Gain
12 Time in seconds is on the x-axis, input power in percent is on the primary y-axis, and output voltage in volts is on the secondary y-axis. The output voltage is represented by the lilac line and the input power is represented by the green line. The change in input power percentage (Δm) is 2% and the change in output voltage (Δc) is 3.5 volts. Therefore, the gain was calculated by dividing the change in output by the change in input, which results in a gain of 1.95 volts/% in this example. An example of the dead time calculation is provided below in Figure 7 for 83% input power with a 2% step response. JRA 1/31/ Figure 7. Step Response, Dead Time Time in seconds is on the x-axis, input power in percent is on the primary y-axis, and output voltage in volts is on the secondary y-axis. The output voltage is represented by the lilac line and the input power is represented by the green line. The dead time can be determined by drawing a parallel line at the steepest point of the output step response
13 and a horizontal line to approximate the average steady-state line of the output. Then a vertical line is drawn from the intersection of these two lines. The time difference between this vertical line and the time that the input line begins the 2% step is the dead time. The dead time in this example was determined to be.7 seconds. An example of the time constant calculation is provided below in Figure 8 for 83% input power with a 2% step response. JRA 1/31/ Figure 8. Step Response, Time Constant Time in seconds is on the x-axis, input power in percent is on the primary y-axis, and output voltage in volts is on the secondary y-axis. The output voltage is represented by the lilac line and the input power is represented by the green line. The time constant is found by calculating 63.2% of the change in output (Δc), which is 2.21 volts in the example. A horizontal line is drawn to represent the mean steady-state output before the step response and another horizontal line is drawn at 63.2% of the change in output from
14 the mean steady-state output line. A vertical line is drawn at the intersection of the 63.2% change in output line and the step response curve. Another vertical line is drawn at beginning of the output step response. The time difference between these two lines is the time constant, which is.25 seconds in the example. The response time is calculated by multiplying the time constant by five. Therefore, the time response for the step response in the examples is 1.25 seconds. The experimental data was used to create graphs to display the average FOPDT parameters for the step up and step down responses of the system. The data can be seen in Table 3 in the appendix. The values for the gain, dead time, and time constant are the mean values for the three experimental trials. Table 2 shows the average parameter values and their uncertainties for step up and step down responses. The uncertainties were calculated using the student t's statistical method with three trials. This was then multiplied by two. Table 2. Summary of Step Responses K, V/% ±, V/% T o, s ±, s τ, s ±, s T r, s ±, s STEP UP 1.72.35.5.3.26. 1.31 1.51 STEP DOWN 1.44.24.3.1.8.11..57
15 JRA 1/31/ Figure 9. Average Gain The type of step response is shown on the x-axis and the average gain in volts per percent is shown on the y-axis. The graph shows that the gain in the step up responses stays nearly the same, regardless of input power percentage. The gain in the step down responses increases as the input power percentage increases. The gain is also higher for the step up responses than the step down responses. The uncertainty bars indicate the level of confidence of the mean values.
16 JRA 1/31/ Figure. Average Dead Time The type of step response is shown on the x-axis and the average dead time in seconds is shown on the y-axis. The graph shows that the dead time in the step up responses is greater than its' corresponding step down response. The uncertainty bars indicate the level of confidence of the mean values. The step up from 86% input power shows a high level of uncertainty compared to the mean dead time.
17 JRA 1/31/ Figure 11. Average Time Constant The type of step response is shown on the x-axis and the average time constant in seconds is shown on the y-axis. The graph shows that the time constant in the step up responses decreases as the input power percentage increases. The time constant in the step down responses stays nearly the same across the range of input power percentages. The uncertainty bars indicate the level of confidence of the mean values. Several of the time constants have a high level of uncertainty for the step responses.
18 Figure 12. Average Time Response The type of step response is shown on the x-axis and the average time response in seconds is shown on the y-axis. The graph shows that the time response in the step up responses decreases as the input power percentage increases. The time response in the step down responses stays nearly the same across the range of input power percentages. The uncertainty bars indicate the level of confidence of the mean values. Several of the time responses have a high level of uncertainty for the step responses.
19 DISCUSSION: Steady-State The steady-state operating curve in Figure 5 details that the control system is in its' operating range within the input power limits of 83% and 95% outlined in the experiment, because of the linear relationship of the curve. This is important because this region could be represented by a first-order linear differential equation. This means that one could predict the output voltage in this range with this type of equation. The gain, or slope of the steady-state operating curve, was estimated to be 1.8 V/%. The uncertainties of the voltage output were between.% and.31%, which would indicate that the mean voltage output calculated for each input power percentage is a very close representation of the data acquired. It was determined that within the specified power input range of 83% and 95% that the system is in steady-state operation. Step Response The average gains of the system for several step up and step down responses are shown in Figure 9 and indicate that the gain is nearly constant for step up responses. The gain in the step down responses increases as the input power percentage increases. The gain is also higher for the step up responses than the step down responses. The average dead times of the system for the same step up and step down responses are shown in Figure and indicate that the dead time in the step up responses is greater than its' corresponding step down response. There were high levels of uncertainty associated with the average dead time values. The average time constants and time responses of the system for the same step up and step down responses are shown in Figure 11 and indicate that the time constant and time response is greater with a step up response than with a
step down response. There were high levels of uncertainty associated with the average time constant and time response values. The four FOPDT parameter values were higher for the step up responses than the step down responses.
21 CONCLUSIONS AND RECOMMENDATIONS: Steady-State The objective of the experiment was to acquire data from a system and develop a steady-state operating curve for that system. This curve was created by inputting different power percentages to an electric motor which powers a generator and receiving output voltages from the generator. LabVIEW was used for data acquisition and EXCEL was used in creating graphs for data analysis. The steady-state portion of each graph was used to form a steady-state operating curve. The graphs in the Appendix indicate a slight time delay before the system reaches steady-state operation. The results show that between 83% and 95% input power that the system is operating under steady-state conditions. Step Response The objective of the experiment was to acquire data from the voltage system and determine the First-Order Plus Dead Time (FOPDT) parameters for the system. The FOPDT parameters are the gain, dead time, time constant, and time response. These parameters were found by graphing the experimental data of the step responses.. LabVIEW was used for data acquisition and EXCEL was used in creating graphs for data analysis. The results for average gain, dead times, time constants, and time responses can be seen in Figures 9,, 11,12, respectively. The results show that the four FOPDT parameter values were higher for the step up responses than the step down responses. There is a high amount of uncertainty associated with the dead times, time constants, and time responses.
22 APPENDIX: 9 OUTPUT VS. TIME (83% INPUT) 8 8 7 7 6 INPUT, M(t) (%) 6 5 5 15 25 35 TIME (s) Figure 13. Voltage vs. Time for 83% Input Power JRA 1// 5 OUTPUT, C(t) (V) 9 OUTPUT VS. TIME (85% INPUT) 8 8 7 7 6 INPUT, M(t) (%) 6 5 5 15 25 35 TIME (s) Figure 14. Voltage vs. Time for 85% Input Power JRA 1// 5 OUTPUT, C(t) (V)
23 9 8 OUPUT VS. TIME (87% INPUT) 9 8 7 INPUT, M(t) (%) 7 6 5 5 15 25 35 TIME (s) Figure 15. Voltage vs. Time for 87% Input Power JRA 1// 6 5 OUTPUT, C(t) (V) 9 8 OUTPUT VS. TIME (89% INPUT) 9 8 7 INPUT, M(t) (%) 7 6 5 5 15 25 35 TIME (s) Figure 16. Voltage vs. Time for 89% Input Power JRA 1// 6 5 OUTPUT, C(t) (V)
24 9 8 OUTPUT VS. TIME (91% INPUT) 9 8 INPUT, M(t) (%) 7 6 5 5 15 25 35 TIME (s) Figure 17. Voltage vs. Time for 91% Input Power JRA 1// 7 6 5 OUTPUT, C(t) (V) 9 8 OUTPUT VS. TIME (93% INPUT) 9 8 INPUT, M(t) (%) 7 6 5 5 15 25 35 TIME (s) Figure 18. Voltage vs. Time for 93% Input Power JRA 1// 7 6 5 OUTPUT, C(t) (V)
25 9 8 OUTPUT VS. TIME (95% INPUT) 9 8 INPUT, M(t) (%) 7 6 5 5 15 25 35 TIME (s) Figure 19. Voltage vs. Time for 95% Input Power JRA 1// 7 6 5 OUTPUT, C(t) (V)
26 Table 3. Summary of Step Response Data K, V/% ±, V/% T o, s ±, s τ, s ±, s T r, s ±, s 83% UP 1.85.54.7.2.33.32 1.65 1.58 85% DOWN 1.33.34.2.1.7.2.37.12 86% UP 1.71.21.4.4.37.52 1.83 2.58 88% DOWN 1..29.4.2.7..37 1.1 89% UP 1.8.45.6.2.24.17 1.22.87 91% DOWN 1.51.14.3..6.2.32.12 92% UP 1.79.24.3..22.3 1.12.17 94% DOWN 1.52..3.2.11..53 1.2 NOTE: 72 graphs were used to acquire the data in Table 3.