Lab Report 4: Root Locus and Proportional Controller

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1 Lab Report 4: Root Locus and Proportional Controller University of Tennessee at Chattanooga Engineering 32 Blue Team Kevin Schrumpf Justin Anchanattu Justin Rehagen April 1, 212

2 Introduction The first three experiments were learning about the voltage system in the steady-state, step response, and step function. The assignment for the laboratory now is to maintain the root locus and run a proportional controller experiment. The team s laboratory station is the Voltage System like we see in everyday life. The mission in this experiment was to observe the data obtained by the root locus and the proportional controller experiment. The Root Locus was found by plugging in our own K (gain), τ, and. Once the correct values were put in we could then find all our damping points which are the following: Critical damping at, 1/5 th decay 45,1/1 th decay 7, Quarter decay 78, and Ultimate 9. Then the team ran an experiment to find the proportional controller for the voltage output within the guidelines set by the customer. The output range was set to be 7 to 85 volts. In this document the Background & Theory section will cover the technical aspects and arrangement of the Voltage System. The Procedure section will provide an in-depth look at the steps taken and the methods used to collect the experimental data. The Results section will present selected data through arranged tables and graphs. The Discussion section will explain the importance of the collected data and discuss its relation to this laboratories objectives and whether or not the requirements were sufficiently met. The Conclusions section will briefly recap this document with the Appendix following containing the remaining data and graphs. 1

3 Background & Theory An electric motor is connected to a voltage generator in the laboratory. The shaft of the electric motor is connected to the shaft of the voltage generator via a coupler. The experiment is run from a webpage that allows the user to run the motor at different inputs (Henry, 212). The webpage then runs the electric motor remotely via the internet. The software records the various data points, plots them, and makes them available in an excel data file. See Figure 1 below. Input, m(t) Motor Input, (%) Voltage System Output, c(t) Voltage, (Volts) Figure 1: Block diagram of the System The value m(t) was input as a percentage. The possible percentage range was from % to 1%. The goal was to run the experiment at different values of m(t) to get desired output values in the range of 7 to 85 volts. The output, c(t), was expressed as volts. Each individual experiment was run until the output stabilized. This constant output is called the steady-state operating value for experiment. Excel was used to calculate the average value over this steady state range. In addition, the standard deviation for each run was calculated. Two times the standard deviation will incorporate 95% confidence 2

4 with the data points of the raw data, etc. This lab will use two times the standard deviation as the uncertainty. Figure 2, below, shows the motor that is used to produce the voltage that makes the light bulbs energize. Figure 2a Figure 2b 3

5 Figure 3 shows a schematic of the entire control and data-collection system described above. In the diagram, the computer is connected to the power recording controller (JRC). This is the m(t) controller. It is connected to the variable frequency drive (VFD) which actually controls the percentage of power input to the motor. The generator is connected to both the lights and the voltage recording transmitter (ERT), which is the measuring device that reports to the computer the amount of voltage output by the system. ERT VFD Figure 3: Schematic of voltage control and data collection system 4

6 PROCEDURE As mentioned above, the equipment is controlled remotely via computer interface. The software controlling this is called LabVIEW. The flow of information is explained below in Figure 4. Figure 4: Schematic of remote controller for motor and generator. To process a report, simply log onto the laboratory web site (Henry, 212), select the desired experiment (steady-state, step response, or sine), and enter the necessary information, including the location (EMCS 44). In this experiment, sine experiment, we had to enter a desired percentage input, how long the experiment was to run for the root locus and the proportional experiment. 5

7 Microsoft Excel was used to assimilate the data and create the necessary plots. This was done by various members of the team, and the work was divided equally into three categories based on output: 7-75 Watts; 75- Watts; and -85 Watts. One of the purposes of these experiments was to obtain the root locus for the voltage system. The root locus consists of the following values: critically damping,, under damping,, under damping, under damping, ultimate ROOT LOCUS PLOT 6 IMAGINARY AXIS REAL AXIS - Figure 5: Measurements taken to obtain the Root Locus. 6

8 In order to determine the damping points in the root locus there was an Excel sheet provided Dr. Henry in the Lab website. All the student had to do is plug in their values for their own K (gain), τ, and. Once these numbers are plugged in you can get all the damping points in the root locus. In order to determine the parameters (K, τ, and ) In order to determine what the FOPDT parameters were for this experiment the amplitude ratio and phase angle had to be observed for each frequency. To determine the amplitude C AR = M ratio the following formula was used: Where C is the amplitude of the output signal in Volts and M is the amplitude of the input signal as a percentage. This is demonstrated in the figure below. M, however, can be determined without inspecting the graph since it was one of the parameters used to run the experiment. Figure 6: Illustrations of input and output signals( Not out actual experiment) To calculate the phase angle the following formula was used: t PA = T 36 7

9 Where t is the period of the output signal in seconds and T is the period of the input signal in seconds. This is demonstrated in the figure below. Figure 7: Illustrations of period for the input and output signal( Not out actual experiment) The first FOPDT parameter that can be observed is the gain K. The gain can be found by looking at the bode plot for Amplitude Ratio versus Frequency. The gain K is the asymptotic value of AR at low frequencies. Please refer to figure 8 for an illustration. The second FOPDT parameter is t the first order time constant. τ can be found by rearranging the following equation: K AR =, ω = 2πf ω τ By solving for τ, the formula can be rearranged into the following equation: ( K ) 2 1 AR τ = 2πf Where K is the gain, AR is the ultimate amplitude ratio (ARu), and f is the ultimate frequency (Fu). 8

10 The third and final FOPDT parameter is the dead time t. t can be found by rearranging the following equation: φ = ω t + tan 1 ( ω ), ω = 2πf τ By solving for t, the following formula was derived: t φ + tan 1 ( 2πf τ ) = 2πf Since τ has already been found, it can be further simplified into the following equation: ( K ) 1 φ + tan ( 1) AR t = 2πf Where Ø is the phase angle (-1 degrees or π), K is the gain, AR is the ultimate amplitude ratio (ARu), and f is the ultimate frequency (Fu). The data were plotted graphically, shown in the APPENDIX section of this report, and also a model of the results was performed using modeling guidelines presented in class by Dr. Henry 2 Figure 5 is an example of a Root Locus of an 85% input with a K (gain) value of 1.5 Volts/%, a τ value of.14 seconds, a value of.4 seconds, and a value of.1 %/Volts. critically damping was taken at zero degrees above the x-axis which gave a value of.5 %/Volts., under damping was taken 45 degrees above the x- axis which yielded a value of 1.18 %/Volts., under damping was taken at 7 degrees which presented a value of 2.64 %/Volts. under damping was recorded at 78 degrees which generated a value of 3.49 %/Volts. ultimate was recorded at 9 degrees which produced a value of 5.33 %/Volts. 9

11 Three different Root Locus were obtained from an input range of 7%-85% with different K, τ,, and values in order to obtain the necessary data to calculate the average and the standard deviation. 1

12 RESULTS ROOT LOCUS PLOT 6 IMAGINARY AXIS REAL AXIS - Figure 9: Example of a Root Locus Diagram with the parameters of an % input. Above, in Figure 9, is the actual data collected from the parameter values from an % input. K had a value of 1.5 Volts/%, τhad a value of.14 seconds, and had a value of.4 seconds. All inputs from 7 to 85% had parameters very close to each other, and the graphs for the other inputs will be posted in the APPENDIX section. 11

13 Input ( % ) Input Value(%) Output(Volts) SET-P(Volts) Output ( Volts ) Time ( sec ) 55 Figure 1: Actual data collected for a % Input. Kc Figure 11: Model vs Experiment Data. 12

14 Figure 1 is the data collected from a % input. The green line represents the set point of the graph, the blue line represents the input of the graph, and the purple line represents the output of the graph. The input value started at % and ended at approximately 82%. These inputs gave us an output value which started at about 6 volts and ended at about 65 volts. Figure 11 is the data for the average Model vs. Experiment. The data shows all the damping values for the experiment. 13

15 DISCUSSION The root locus was able to show that the data we got from the bode plots were valid results. The FOPDT parameters were the values used to find the damping values for the root locus experiment. The parameters were similar to each other. The graphs for the root locus ended up being similar. The FOPDT parameters could be calculated by mathematical equations. The proportional controller experiments were done successfully. The values were changed constantly to obtain the correct data needed for the experiment. 14

16 CONCLUSIONS AND RECOMMENDATIONS This lab demonstrated the user s ability to derive the FOPDT parameters for a root locus, and run an experiment for the proportional controller. The overall results seem to indicate that the bode plot data was correct from observing the data obtain the root locus. This lab also demonstrated the user s ability to run an experiment for the proportional controller. 15

17 APPENDIX Below are the plots from all experiments including steady state operating curve, step response, frequency tests, bode plots, sine response, root locus, and proportional controller. Input (Percent) Voltage Output for Step Input of 85-9% Input Value(%) Time (Sec) Voltage Output (Volts) 16

18 Voltage Output (Volts) Voltage Output for Step Input of 9-95% Time (sec) Output(Volt s) 95 Input (Percentage of Maximum, %) 9 85 Voltage Output (Volts) Voltage Output for Step Input of 95-1% Time (sec) Output (V) Input (Percentage of Maximum, %) 17

19 Voltage Output for Constant Input of 85% Output (Volts) Output(Volts) Time (seconds) Voltage Output for Constant Input of 86% Output (Volts) Output (Volts) Time (seconds) 18

20 Voltage Output for Constant Input of 87% Output (Volts) Time (seconds) Voltage Output Output (Volts) Voltage Output for Constant Input of 88% Time (seconds) Output(Volts) 19

21 Output (Volts) Voltage Output for Constant Input of 89% Time (seconds) Voltage Output Output (Volts) Voltage Output for Constant Input of 9% Voltage Output Time (seconds) 2

22 9 Voltage Output for Constant Input of 91% Output (Volts) Time (Seconds) Voltage Output Output (Volts) Voltage Output for Constant Input of 92% Time (Seconds) Voltage Output 21

Output (Volts) 9 7 6 5 4 3 2 1 Voltage Output for Constant Input of 93% 5 1 15 2 25 3 Time (Seconds) Voltage Output Voltage Output for Step Input of 85-9% Input

23 Output (Volts) Voltage Output for Constant Input of 93% Time (Seconds) Voltage Output Voltage Output for Step Input of 85-9% Input (Percent) Input Value(%) Δ Δm t τ Time (Sec) Δ ΔC Voltage Output (Volts) 22

24 Model Amplitude Ratio v. Frequency (Hz) 1 Amplitude Ratio Amplitude Ratio ARu K 1.1 Model AR Frequency (Hz).1 Model Phase Angle (Degrees) v. Frequency (Hz) Fu (Hz) Phase Angle (Degrees) 1 line Phase Angle Model PA Frequency (Hz)

25 Amplitude Ratio v. Frequency 1. Amplitude Ratio Frequency Phase Angle v. Frequency Phase Angle Frequency 24

26 Model Phase Angle (Degrees) v. Frequency (Hz) Fu (Hz) Phase Angle (Degrees) Phase Angle 1 line Model PA Frequency (Hz) -24 Model Amplitude Ratio v. Frequency (Hz) 1 Amplitude Ratio Amplitude Ratio ARu K 1.1 Model AR Frequency (Hz).1 25

27 ROOT LOCUS PLOT 6 4 IMAGINARY AXIS REAL AXIS -6 26

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University of Tennessee at. Chattanooga

University of Tennessee at. Chattanooga 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