Appendix G: (Not required information but I put it here incase it might help)

b. Build a Simulink program based on the transfer function in Eq. (2-8) with R= 10k12 and C = 10uF. to conduct simulations. Then write a MATLAB plot program to plot the Simulink output response, the v.(t) in Eq. (2-6). (See Appendix G for details) 1. Build the Simulink program. II. Plot the Simulink simulation result, v.(t), the voltage across the capacitor. Observe how the changes of R and C will affect the response. RC dv.(t) + v.(t) = vi(t). (2-6) dt Throughout this section, the resistor R and the capacitor C of the RC circuit in Figure 2.2 are assumed R = 10k and C = 10uF. Note that the tolerance of the resistor is +5%, and the tolerance of the capacitor can be as high as +20%. Appendix G: Simulating RC Response with Simulink Introduction It is often advantageous to simulate a system's behavior for theoretical analysis and comparison with experimental trials. Many tools exist for simulating electrical and dynamic systems, one of them being Simulink. Simulink is an extension of MATLAB that allows programs and models to be built using a graphical user interface. This is different from the more familiar m-code approach where commands are entered in the MATLAB Command Window or are written in an m-file function or script. In many cases, Simulink proves to be more intuitive and practical for simulating complicated systems. Simulink models are designed graphically by connecting blocks in a flow-chart manner. The blocks can be dragged and dropped into place and interconnected. Many blocks have tunable parameters. After creating a model, the model can be run to generate an output. This output can then be plotted or used for further analysis in Matlab or using other software packages. Initial Setup and Creating a New Model Begin by setting a variable, RC, in the MATLAB Command Window to the RC value of the circuit. After defining RC=l, you will see the RC icon appears in the Workspace window showing its Value is 1. If we double click the RC icon, it will open the window named Variables - RC allowing us to see the content inside. In general, the variable can be more than a single number. It can be an array or a matrix, and hence the ability to see the whole content of the variables is meaningful. The value of RC (the time constant of the RC circuit) will be used later in the Simulink simulation. MATLAB R2021b-academic use . PL APPS VARL VIEW @ Search Documentation Bor-Chin New from Print Transpose Sort X Workspace Value Open Rows Columns 1 Insert Delete 1 Selection VARIABLE SELECTION EDIT ada: Users bcc88 Documents MATLAB Current Folder O Variables - RC Name RC X Untitled_1.m A 1x1 double Untitled.m 1 2 3 runfgWindowsXP_777-20. 1 RC_StepResponsesxc plotRCresponse.m Details Command Window >> RC = Name - RC 1 4 5 RC Select a file to view details 1 >> simulink fx >> Then, to start using Simulink, we type simulink' as the next command on the same Command Window, and hit the Enter key to pop up the following Simulink Start Page window: Simulink Start Page X SIMULINK New Examples Open Search All Recent > My Templates Learn More Projects #From Source Control Simulink Learn 0 to Simulink Onramp Lo Statelow Onramp to Control Design Onramp with Simulink Simscape Onramp to Circuit Simulation Onramp Blank Model Blank Subsystem Blank Library Blank Project Double click the 'Blank Model" icon to open a Simulink model window named 'untitled'. You can rename the file anytime by clicking 'Save' and then type the new name. To build a Simulink simulation model, we first open the Simulink Library Browser window by clicking the Library Browser icon on the Simulink model window. Untitled Simulink Library Browser Entouch SIMULATION DEBUG MODELING FORMAT APPS - O Stop Time Simulink Open Save Print TILE Ster Library Browser UEDA Inspector VIEW STS HENNE SALATE 6 untitled Simulink Commonly used Continuous Dashboard Additional Math Commonly Discontinuities & Discrete Used Blocks Discrete Logic and Bit Of Lookup Tables Math Operation Continuous Dashboard Matrix Operator Messages & Eve Ma Model Verificatic Model Wide Ut Discontinuities Discrete Ports & Subsyste Signal Attributes Signal Routing Sinks Logic and it Lookup Ready 100% Variabletup Auto On the Library Browser window, we will select and open the "Continuous' folder to drag a Transfer Fen block, which represent a continuous-time system, onto the Simulink model window as shown in the following figure. tid Simulink academic use SIMULATION DEBUG MCOEUNG FORMAT APPS BLOCK Etichete Open - Save Stop Time 100 Noma Library Howar BRARY Restart - Forward Back RE Dets sector BEWATE PD Control (200R untitled BON State Space Simulink/Continuous Simulink Commonly used Continuous Dashboard Discontinues Discrete Logic and Bit Opera Lookup Tables Math Operations Matti Operations Messages & Events Model Verification Model Wide Utilitie Ports & Subsystems Signal Aibutes Signal Routing Sinks *+1 1 Transfer For Transport Ready 100% Variabletto The Transfer Fen block now shows the transfer function with 1 as numerator and stl as the denominator, but both of them can be replaced by any polynomial as long as the degree of the numerator polynomial is less than or equal to that of the denominator polynomial. For the RC circuit system, the denominator polynomial is RCs+l while the numerator still remains as 1. To make changes, we double click the Transfer Fcn block to pop up the 'Block Parameters: Transfer Fen' window, where we can change the coefficients of the numerator and the denominator. Now we revise the denominator coefficients from [1 l) to [RC 1], and click 'Apply'. Then the denominator polynomial of the Transfer Fen block on the Simulink model is changed accordingly from st! to RCs +1 as shown in the following figure. untitled - Simulink academic use SIMULATION DEBUG MODELING FORMAT APPS Block Parameters: Transfer Fon Parameter tunability controls the runtime tunability level for numerator and denominator coefficients. 'Auto': Allow Simulink to choose the most appropriate tunability Open - Stop Time 10.0 level New Normal Print Library Browser LIBRARY 109 Signals PREPARE Fast Restart Step Run Back- SIMULATE FILE Optimized': Tunability is optimized for performance. "Unconstrained': Tunability is unconstrained across the simulation targets. Parameters Numerator coefficients: untitled Mode Browser untitled Denominator coefficients: [RC 1] RC -5+1 Transfer For Parameter tunability: Optimized Absolute tolerance: auto State Name: (e.g., "position') OK Cancel Help Ready 10096 Apply To simulate the step response of the system, a step input Source must be placed into the model. It can be found in the Source folder of the Library Browser as illustrated. MODELING FORMA 100 Stop Time Normal Fast Restart 0000 CO PRERARE titled-Simulinkacademic use Block Parameters: Step Step Output a step Main Signal Attributes Step time: 0 Initial value: 0 Final value: 5 Sample time: 0 Interpret vector parameters as 1-0 Enable zero-crossing detection Signal Generator Simulink Library Browser APPS Enter search terms Simulink/Sources Step Back - Matrix Operations SIMULATI Messages & Events Model Verification Model-Wide Utilitie Ports & Subsystems Signal Attributes Signal Routing Sinks Sources String User-Defined functi Additional Math & > Quick Insert Aerospace Blockset Audio Toolbox Automated Driving Toc RC se RC StepResponse 1 Simulink Library Browser Enter search term Simulink/Sinks Logic and Bit Operations Lookup Tables Math Operations Matrix Operations Messages & Events Model Verification Model-Wide Utilities Ports & Subsystems Signal Attributes Signal Routing Sinks Sources String User-Defined Functions Additional Math & Discrete Terminator To File Save format: Array Save 2-D signals as: 3-0 array (concatenate along the Log fixed-point data as a fi object Sample time (-1 for inherited): -1 out. Vout RC -5+1 sout TO Workspace Transfer Font To Workspace XY Grape OK Cancel Help Ready 125% autoode 45 Note that the name of the To Workspace block has been renamed as Vout, and the Save format is Array. In order to record the simulated data as functions of time, we need to add a Clock block from the library Sources and another To Workspace block to the Simulink model. The To Workspace block parameters should be tuned similarly to those illustrated earlier. Run the simulation by clicking the black triangle) at the top of the model window. This resembles a play button from a DVD player or similar device. Simulink Library Browser TODEUNG FORMAT APPS Enter search term Stop Time 10.0 Blod Parameters: To Workspace BLOC To log a bus signal, use "Timeseries" save format. Parameters Variable name: time Run Normal Fast Restart Step Back - REPARE SIMULATE Band-Limited White Noise Limit data points to last: Inf Chirp Signal Decimation: 1 Simulink/Sources Matrix Operat Messages & E Model Verifica Model-Wide Ports & Subsy Signal Attribu Signal Routing Sinks Sources String User-Defined Additional Ma Quick Insert Aerospace Blocks Audio Toolbox Automated Drivit VITOCAD Rarke. out time Clock 1 RC 5+1 out. Vout Save format: Array Save 2-D signals as: 3-D array (concatenate along thir Log fixed-point data as a fi object Sample time (-1 for inherited): -1 Constant M Counter OK Cancel 100% Help Apply The simulation results are stored in the array Vout (the output voltage of the RC circuit) and another array time (the timing information corresponding to Vout). MATLAR R021b-academicus RC StepResponse - Simulink academic use SIMULATION DEBUG MODELING FORMAT Search Documentation Bor-Chin APPS Transpose Insert Delete Sort Stop Time 100 LE LIBRARY PREPARE Normal Fast Restart Step Back- SIMULATE Step Forward RC Stepesponte RC StepResponse Workspace Name out E RC podsur de Open BOW Columns New from Print Selection VARIABLE SELECTION EDIT UDC Users becs Documents MATLAB Current Folder Variables-out Name out Untitled 1.m txt Simulation Output Untitled.m Value Property runt Windows XP RC_StepRespons- Vout 5.81 double Initial conditions E time Sex double Gyro sensor.docx tout 58x1 double agtm_simulate 6D. e Simulation. 1x1 Simulation gtm design stun Errorlessage gtr_design FGS Explore SimulationMetadata 18trimA antiwin Command Window Details >> RC=1 out me WORCE RC-5+1 out. Vout RC = Select a file to view details >> simulink >> Ready 100% autolode45) The Vout and time arrays can be employed to plot the value of Vout as function of time. On the MATLAB Command window, type the command plot (out.time, out. Vout) and press the Enter key will plot in a MATLAB figure as shown below. The step response rises up from Vout O at time = 0 exponentially with time constant = 1 second, and reaches the steady state value Vout= 5 volt shortly after 6 seconds. This figure is not suitable for use in a report or presentation, though, because it lacks titles and labels. MATLAB R2021bcademic use Search Documentation Figure 1 P4 Bor-Chin File Edit View Insert Tools Desktop Window Help H ET 4.5 Open Rows Colom Transpose Newtom Print Insert Delete Sort Selection VARIABLE SELECTOR Users bec88 Documents MATLAB Current Folder Variables-Out Name out Untitled_1.m 1x1 Simulation Output Untitled.mn rung Windows XP Property Value SRC StepRespons Vout 581 double Initial conditions time 58x1 double Gyro sensor dock tout 5&xt double 9gtm_simulate 60. SimulationM. 11 Simulation gtm_design_sfun Error Message gtm_design_FG_S Explore Simulation Metadata 18trimA antiwin Command Window Details Workspace Name out RC 3.5 3 2.5 2 1.5 >> simulink >> plot time. Vout) Unrecognized function of variable time! Select a file to view details 05 >> plot (out. time, out. Vout) fx >> 0 2 10 An m-file MATLAB script can be created to plot the step response with titles, labels, and grid as shown below. Run the script to generate a new plot. Plot RC step response of the simulated system plot (out.time, out. Vout) xlabel ('Time [sec]') ylabel ('Vout [volts]') grid title('Step Response of RC Circuit') The newly plotted figure will contain appropriate titles, labels, and grid lines. Analyze SECTION MATLAB R2021-academic use Figure 1 - Search Documentation File Edit View Insert Tools Desktop Window Help % ad DE DE Compare Profile Find New Oper Save Go To Refactor N Bookmark - FA Step Response of RC Circuit FILE MAGATE COO ANALVIE Users bcc88 - Documents MATLAB 4.5 Current Folder Variables-out Editor plot Cresponse.mx Works Name plotRCresponse.mx + Name- Untitled_1.m Plot RC response of the simulated system out Untitled. 2 plot(out.time, out. Vout) HRC 3.5 3 rung WindowsXP-7- > simulink >> plot (time, Vout) Unrecognized function of variable time 0.5 >> plot (out.time, out. Vout) >> plotrCresponse >> 0 0 2 8 10 4 6 Time [sec! To import the figure into Microsoft Word or PowerPoint, the figure should be copied and pasted as shown below. DO NOT COPY & PASTE A SCREEN CAPTURE. In the figure window, select Edit Copy Figure. Figure File Edit View Insert Tools Desktop Window Help Undo Redo C. Cut Inse of RC Circuit Copy Chi Paste EV Car Clipboard Delete Select All Copy Figure Copy Options Figure Properties. Aus Properties Current Object Properties. Colormap Find Files Clear Figure Clear Command Window Clear Command History Clear Workspace 6 Time feat 8 TO The copied figure can be pasted directly into Microsoft Word or PowerPoint. It appears much more clearly than a screen capture, with minimal pixilation and distortion. Step Response of RC Circuit 4.5 3.5 Vout (volts 2 1.5 1 0.5 0 2 4 6 8 10 Time [sec] In many occations, we may want to show the waveforms of the output and the input on the same graph. To achieve this, we slightly modify the Simulink model and the m-file to include the input signal, Vin, as shown in the following Simulink window and the MATLAB Editor window. AMATLAB R2021b-academic use RC Step Response - Simulink academic Search Documentation PBor-Chin SIMULATI DEBUG MODELING FORMAT APPS Stop Time 1010 Compare Print NAVIGATE CODE ANALYZE SECTION Run Step Stop FLE LIBRARY PREPARE Normal Fast Restart Step Run Back - SIMULATE Work RC StepResponse RC StepResponse Name out HRC out time 4 1 RC 3+1 out. Vout FILE RUN CUsers bacB8 Documents MATLAB Current Fold Editor - plotRCresponse.m X Variables-out Na.. plotRCresponse.mx + Untitle Plot RC response of the simulated system Untitle 2 plot(out. time, out. Vin, out.time, out. Vout) Brunfgw. 3 xlabel("Time [sec]"> RC Ste. 4 ylabel("Vout (volts) plotRC. 5 legend("Vin", 'Vout"> Initial c. 6 grid Gyro se. title("Step Response of RC Circuit) 8 gtm_si. gtm_de. gtm de Command Window f18trim. >> plot RCresponse f18trim. fx >> f18trim. BE 18stab... plotRCresp. A 6. UTF-8 CRLF In 5 out. Vin script Col 20 Ready 100% autoode45) Run the modified Simulink model and then the modified plot m-file, we will obtain the following MATLAB figure that includes both the step input Vin (in blue) with magnitude 5 volt and the step response Vout (in red). Step Response of RC Circuit 5 4.5 Vin Vout 4 3.5 3 Vout (volts) -2.5 1.5 1 0.5 0 0 2 4 8 10 6 Time (sec! Remember to SAVE YOUR WORK, including the Simulink model file (*slx or * mdl) and the MATLAB m- file (*.m). Saving these files allows them to be reused and referenced later. When used as part of a lab report, they are required in the lab report submission. Sinusoidal Response Model The step response model can be modified by changing the Source block to create a sinusoidal response model. Replace the Step block with a Sine Wave block. Remember to adjust the block parameters as needed. For this simulation, the amplitude and frequency are chosen to be 1 and 1 rad/s, respectively. Notice that the simulation duration time, located at the top of the model window, near the run button, has been modified to 40 sec. This will allow the simulation to run long enough for the steady-state sinusoidal response to be observed. MATLAB R2021b-cademic PRC Sinusoidal Response - Simulink acade Bor-Chin SIMUL DEBUG MODE FORMAT APPS P. Search Documentation Big Stop Time 400 Compare New Open Save Print NAVIGATE CODE ANALYZE SECTION Run Step Stop Normal BLE LERARY PREPARE Fast Restart Step Run Back - SIMULATE FILE Works RC_Sinusoidal Response RC Sinusoidal Response 1 Users bcc88 Documents MATLAB Editor - plotrCresponse sinusoidal.mx Variables out. Vout plotrCresponse_sinusoidal.m + Plot RC response of the simulated system 2 plot(out.time, out. Vin, out.time, out. Vout) xlabel('Time [sec]') ylabel('Vout (volts]') legend('Vin', 'Vout) 6 grid title("Sinusoidal Response of RC Circuit) Name out HERC 3 4 5 out. Vout ta Current FOLL Na. Untitle. Untitle runfg. RC_Ste RC Sin plotRC Initial Gyro se gtm_si. gtm de. atm de f1 trim f1 trim f18trim plotrCresp. RC - +1 7 8 out Vin Command Window Documentation for legend >> plotrCresponse_sinusoidal fx >> UTF-8 CRLF script Col1 Ready 100% auto ode45) Run the simulation model and the m-file plotRCresponse_sinusoidal.m to obtain the following MATLAB figure showing the waveforms of the input and output of the RC circuit system. Sinusoidal Response of RC Circuit 0.8 Vin Vout 0.6 0.4 0.2 Voutvolts) NOT 0 -0.2 -0.4 Acceptable !!! -0.6 -0.8 -1 0 5 10 15 20 25 30 35 40 Time [sec] Notice that the plotted waveforms of the input and the output appear jagged and unsmooth. This is a result of Simulink using a simulation-step size that is too large. To ensure higher-fidelity simulation results, the Simulink model parameters need to be configured to limit the maximum simulation-step size used during simulation. These parameters can be adjusted by selecting SIMULATION PREPARE CONFIGURATION & SIMULATION Model Setting... in the model window. Configuration Parameters: RC_Sinusoidal Response/Configuration (Active) Q Search Simulation time Start time 00 Stop time 400 Solver Data Import Export Math and Data Types Diagnostics Hardware Implementation Model Referencing Simulation Target Code Generation Coverage HDL Code Generation Solver selection Type: Variable-step - Solver auto (Automatic solver selection) Solver details RG/103 Max step size Min step size auto Relative tolerance Te 3 Absolute tolerance auto Auto scale absolute tolerance Disable A Initial step auto Shape preservation Number of consecutive min steps 1 Zero-crossing options Zoro-crossing control Use local settings Time tolerance 10"128"eps Number of consecutive zero crossings: 1000 - Algorithm Nonadaptive Signal threshold. Tasking and sample time options Automatically handle rate transition for data transfer Allow multiple tasks to access inputs and outputs Higher priority value indicates higher task priority OK Cancel Help Many model-wide settings can be configured in the configuration window shown above. In the upper-middle section of the window the Max step size parameter should be changed to RC/100. This ensures that the maximum simulation-step size will be no more than 1/100th of the RC value. These settings are sufficient for the range of RC values expected for this lab. Click APPLY to save the altered parameters and return to the model window. Rerun the simulation with the new parameters and plot the results. Notice that the plots now appear as smooth curves. This is a result of the simulation running with smaller simulation-step sizes, yielding higher resolution and greater precision on the logged data. Sinusoidal Response of RC Circuit 1 0.8 Vin Vout 0.6 0.4 0.2 Vout [volts] 0 -0.2 -0.4 -0.6 -0.8 -1 0 5 10 15 25 30 35 40 20 Time [sec] This sinusoidal response plot is better than the previous one; however, the waveforms are too crowded for the measurement of the oscillation period and the phase difference between the input and the output. Since for the sinusoidal response only the steady state response are of interest, we will enlarge the portion of the graph between 25 second and 35 second marks by revising the m-file plotRCresponse_sinusoidal.m as follows: %Plot RC response of the simulated system plot(out.time ( 2500:3500), out. Vin(2500:3500), out.time(2500:3500), out. Vout(2500:3500)) xlabel('Time [sec]') ylabel('Vout [volts]') legend('Vin', 'Vout') grid grid minor title('Sinusoidal Response of RC Circuit') By running the m-file program, we have the following portion of the steady-state sinusoidal response between 25 second and 35 second. Sinusoidal Response of RC Circuit 0.8 - Vin Vout 0.6 0.4 0.2 Vout (volts) 0 A -0.2 -0.4 -0.6 -0.8 -1 24 26 28 30 32 34 36 Time [sec] The period (T) of the output waveform is exactly the same as that of the input: T = 32.15-25.87 = 6.28 sec 21 O= T = 1.0005 rad/sec =1 rad/sec 1 The amplitude of the output waveform is approximately 0.71=0.707=- 2 The output waveform lags behind the input waveform by 32.15-31.37 = 0.78sec . Hence, the phase shift will -0.78 x360 = -44.7 =-45. 6.28 be = b. Build a Simulink program based on the transfer function in Eq. (2-8) with R= 10k12 and C = 10uF. to conduct simulations. Then write a MATLAB plot program to plot the Simulink output response, the v.(t) in Eq. (2-6). (See Appendix G for details) 1. Build the Simulink program. II. Plot the Simulink simulation result, v.(t), the voltage across the capacitor. Observe how the changes of R and C will affect the response. RC dv.(t) + v.(t) = vi(t). (2-6) dt Throughout this section, the resistor R and the capacitor C of the RC circuit in Figure 2.2 are assumed R = 10k and C = 10uF. Note that the tolerance of the resistor is +5%, and the tolerance of the capacitor can be as high as +20%. Appendix G: Simulating RC Response with Simulink Introduction It is often advantageous to simulate a system's behavior for theoretical analysis and comparison with experimental trials. Many tools exist for simulating electrical and dynamic systems, one of them being Simulink. Simulink is an extension of MATLAB that allows programs and models to be built using a graphical user interface. This is different from the more familiar m-code approach where commands are entered in the MATLAB Command Window or are written in an m-file function or script. In many cases, Simulink proves to be more intuitive and practical for simulating complicated systems. Simulink models are designed graphically by connecting blocks in a flow-chart manner. The blocks can be dragged and dropped into place and interconnected. Many blocks have tunable parameters. After creating a model, the model can be run to generate an output. This output can then be plotted or used for further analysis in Matlab or using other software packages. Initial Setup and Creating a New Model Begin by setting a variable, RC, in the MATLAB Command Window to the RC value of the circuit. After defining RC=l, you will see the RC icon appears in the Workspace window showing its Value is 1. If we double click the RC icon, it will open the window named Variables - RC allowing us to see the content inside. In general, the variable can be more than a single number. It can be an array or a matrix, and hence the ability to see the whole content of the variables is meaningful. The value of RC (the time constant of the RC circuit) will be used later in the Simulink simulation. MATLAB R2021b-academic use . PL APPS VARL VIEW @ Search Documentation Bor-Chin New from Print Transpose Sort X Workspace Value Open Rows Columns 1 Insert Delete 1 Selection VARIABLE SELECTION EDIT ada: Users bcc88 Documents MATLAB Current Folder O Variables - RC Name RC X Untitled_1.m A 1x1 double Untitled.m 1 2 3 runfgWindowsXP_777-20. 1 RC_StepResponsesxc plotRCresponse.m Details Command Window >> RC = Name - RC 1 4 5 RC Select a file to view details 1 >> simulink fx >> Then, to start using Simulink, we type simulink' as the next command on the same Command Window, and hit the Enter key to pop up the following Simulink Start Page window: Simulink Start Page X SIMULINK New Examples Open Search All Recent > My Templates Learn More Projects #From Source Control Simulink Learn 0 to Simulink Onramp Lo Statelow Onramp to Control Design Onramp with Simulink Simscape Onramp to Circuit Simulation Onramp Blank Model Blank Subsystem Blank Library Blank Project Double click the 'Blank Model" icon to open a Simulink model window named 'untitled'. You can rename the file anytime by clicking 'Save' and then type the new name. To build a Simulink simulation model, we first open the Simulink Library Browser window by clicking the Library Browser icon on the Simulink model window. Untitled Simulink Library Browser Entouch SIMULATION DEBUG MODELING FORMAT APPS - O Stop Time Simulink Open Save Print TILE Ster Library Browser UEDA Inspector VIEW STS HENNE SALATE 6 untitled Simulink Commonly used Continuous Dashboard Additional Math Commonly Discontinuities & Discrete Used Blocks Discrete Logic and Bit Of Lookup Tables Math Operation Continuous Dashboard Matrix Operator Messages & Eve Ma Model Verificatic Model Wide Ut Discontinuities Discrete Ports & Subsyste Signal Attributes Signal Routing Sinks Logic and it Lookup Ready 100% Variabletup Auto On the Library Browser window, we will select and open the "Continuous' folder to drag a Transfer Fen block, which represent a continuous-time system, onto the Simulink model window as shown in the following figure. tid Simulink academic use SIMULATION DEBUG MCOEUNG FORMAT APPS BLOCK Etichete Open - Save Stop Time 100 Noma Library Howar BRARY Restart - Forward Back RE Dets sector BEWATE PD Control (200R untitled BON State Space Simulink/Continuous Simulink Commonly used Continuous Dashboard Discontinues Discrete Logic and Bit Opera Lookup Tables Math Operations Matti Operations Messages & Events Model Verification Model Wide Utilitie Ports & Subsystems Signal Aibutes Signal Routing Sinks *+1 1 Transfer For Transport Ready 100% Variabletto The Transfer Fen block now shows the transfer function with 1 as numerator and stl as the denominator, but both of them can be replaced by any polynomial as long as the degree of the numerator polynomial is less than or equal to that of the denominator polynomial. For the RC circuit system, the denominator polynomial is RCs+l while the numerator still remains as 1. To make changes, we double click the Transfer Fcn block to pop up the 'Block Parameters: Transfer Fen' window, where we can change the coefficients of the numerator and the denominator. Now we revise the denominator coefficients from [1 l) to [RC 1], and click 'Apply'. Then the denominator polynomial of the Transfer Fen block on the Simulink model is changed accordingly from st! to RCs +1 as shown in the following figure. untitled - Simulink academic use SIMULATION DEBUG MODELING FORMAT APPS Block Parameters: Transfer Fon Parameter tunability controls the runtime tunability level for numerator and denominator coefficients. 'Auto': Allow Simulink to choose the most appropriate tunability Open - Stop Time 10.0 level New Normal Print Library Browser LIBRARY 109 Signals PREPARE Fast Restart Step Run Back- SIMULATE FILE Optimized': Tunability is optimized for performance. "Unconstrained': Tunability is unconstrained across the simulation targets. Parameters Numerator coefficients: untitled Mode Browser untitled Denominator coefficients: [RC 1] RC -5+1 Transfer For Parameter tunability: Optimized Absolute tolerance: auto State Name: (e.g., "position') OK Cancel Help Ready 10096 Apply To simulate the step response of the system, a step input Source must be placed into the model. It can be found in the Source folder of the Library Browser as illustrated. MODELING FORMA 100 Stop Time Normal Fast Restart 0000 CO PRERARE titled-Simulinkacademic use Block Parameters: Step Step Output a step Main Signal Attributes Step time: 0 Initial value: 0 Final value: 5 Sample time: 0 Interpret vector parameters as 1-0 Enable zero-crossing detection Signal Generator Simulink Library Browser APPS Enter search terms Simulink/Sources Step Back - Matrix Operations SIMULATI Messages & Events Model Verification Model-Wide Utilitie Ports & Subsystems Signal Attributes Signal Routing Sinks Sources String User-Defined functi Additional Math & > Quick Insert Aerospace Blockset Audio Toolbox Automated Driving Toc RC se RC StepResponse 1 Simulink Library Browser Enter search term Simulink/Sinks Logic and Bit Operations Lookup Tables Math Operations Matrix Operations Messages & Events Model Verification Model-Wide Utilities Ports & Subsystems Signal Attributes Signal Routing Sinks Sources String User-Defined Functions Additional Math & Discrete Terminator To File Save format: Array Save 2-D signals as: 3-0 array (concatenate along the Log fixed-point data as a fi object Sample time (-1 for inherited): -1 out. Vout RC -5+1 sout TO Workspace Transfer Font To Workspace XY Grape OK Cancel Help Ready 125% autoode 45 Note that the name of the To Workspace block has been renamed as Vout, and the Save format is Array. In order to record the simulated data as functions of time, we need to add a Clock block from the library Sources and another To Workspace block to the Simulink model. The To Workspace block parameters should be tuned similarly to those illustrated earlier. Run the simulation by clicking the black triangle) at the top of the model window. This resembles a play button from a DVD player or similar device. Simulink Library Browser TODEUNG FORMAT APPS Enter search term Stop Time 10.0 Blod Parameters: To Workspace BLOC To log a bus signal, use "Timeseries" save format. Parameters Variable name: time Run Normal Fast Restart Step Back - REPARE SIMULATE Band-Limited White Noise Limit data points to last: Inf Chirp Signal Decimation: 1 Simulink/Sources Matrix Operat Messages & E Model Verifica Model-Wide Ports & Subsy Signal Attribu Signal Routing Sinks Sources String User-Defined Additional Ma Quick Insert Aerospace Blocks Audio Toolbox Automated Drivit VITOCAD Rarke. out time Clock 1 RC 5+1 out. Vout Save format: Array Save 2-D signals as: 3-D array (concatenate along thir Log fixed-point data as a fi object Sample time (-1 for inherited): -1 Constant M Counter OK Cancel 100% Help Apply The simulation results are stored in the array Vout (the output voltage of the RC circuit) and another array time (the timing information corresponding to Vout). MATLAR R021b-academicus RC StepResponse - Simulink academic use SIMULATION DEBUG MODELING FORMAT Search Documentation Bor-Chin APPS Transpose Insert Delete Sort Stop Time 100 LE LIBRARY PREPARE Normal Fast Restart Step Back- SIMULATE Step Forward RC Stepesponte RC StepResponse Workspace Name out E RC podsur de Open BOW Columns New from Print Selection VARIABLE SELECTION EDIT UDC Users becs Documents MATLAB Current Folder Variables-out Name out Untitled 1.m txt Simulation Output Untitled.m Value Property runt Windows XP RC_StepRespons- Vout 5.81 double Initial conditions E time Sex double Gyro sensor.docx tout 58x1 double agtm_simulate 6D. e Simulation. 1x1 Simulation gtm design stun Errorlessage gtr_design FGS Explore SimulationMetadata 18trimA antiwin Command Window Details >> RC=1 out me WORCE RC-5+1 out. Vout RC = Select a file to view details >> simulink >> Ready 100% autolode45) The Vout and time arrays can be employed to plot the value of Vout as function of time. On the MATLAB Command window, type the command plot (out.time, out. Vout) and press the Enter key will plot in a MATLAB figure as shown below. The step response rises up from Vout O at time = 0 exponentially with time constant = 1 second, and reaches the steady state value Vout= 5 volt shortly after 6 seconds. This figure is not suitable for use in a report or presentation, though, because it lacks titles and labels. MATLAB R2021bcademic use Search Documentation Figure 1 P4 Bor-Chin File Edit View Insert Tools Desktop Window Help H ET 4.5 Open Rows Colom Transpose Newtom Print Insert Delete Sort Selection VARIABLE SELECTOR Users bec88 Documents MATLAB Current Folder Variables-Out Name out Untitled_1.m 1x1 Simulation Output Untitled.mn rung Windows XP Property Value SRC StepRespons Vout 581 double Initial conditions time 58x1 double Gyro sensor dock tout 5&xt double 9gtm_simulate 60. SimulationM. 11 Simulation gtm_design_sfun Error Message gtm_design_FG_S Explore Simulation Metadata 18trimA antiwin Command Window Details Workspace Name out RC 3.5 3 2.5 2 1.5 >> simulink >> plot time. Vout) Unrecognized function of variable time! Select a file to view details 05 >> plot (out. time, out. Vout) fx >> 0 2 10 An m-file MATLAB script can be created to plot the step response with titles, labels, and grid as shown below. Run the script to generate a new plot. Plot RC step response of the simulated system plot (out.time, out. Vout) xlabel ('Time [sec]') ylabel ('Vout [volts]') grid title('Step Response of RC Circuit') The newly plotted figure will contain appropriate titles, labels, and grid lines. Analyze SECTION MATLAB R2021-academic use Figure 1 - Search Documentation File Edit View Insert Tools Desktop Window Help % ad DE DE Compare Profile Find New Oper Save Go To Refactor N Bookmark - FA Step Response of RC Circuit FILE MAGATE COO ANALVIE Users bcc88 - Documents MATLAB 4.5 Current Folder Variables-out Editor plot Cresponse.mx Works Name plotRCresponse.mx + Name- Untitled_1.m Plot RC response of the simulated system out Untitled. 2 plot(out.time, out. Vout) HRC 3.5 3 rung WindowsXP-7- > simulink >> plot (time, Vout) Unrecognized function of variable time 0.5 >> plot (out.time, out. Vout) >> plotrCresponse >> 0 0 2 8 10 4 6 Time [sec! To import the figure into Microsoft Word or PowerPoint, the figure should be copied and pasted as shown below. DO NOT COPY & PASTE A SCREEN CAPTURE. In the figure window, select Edit Copy Figure. Figure File Edit View Insert Tools Desktop Window Help Undo Redo C. Cut Inse of RC Circuit Copy Chi Paste EV Car Clipboard Delete Select All Copy Figure Copy Options Figure Properties. Aus Properties Current Object Properties. Colormap Find Files Clear Figure Clear Command Window Clear Command History Clear Workspace 6 Time feat 8 TO The copied figure can be pasted directly into Microsoft Word or PowerPoint. It appears much more clearly than a screen capture, with minimal pixilation and distortion. Step Response of RC Circuit 4.5 3.5 Vout (volts 2 1.5 1 0.5 0 2 4 6 8 10 Time [sec] In many occations, we may want to show the waveforms of the output and the input on the same graph. To achieve this, we slightly modify the Simulink model and the m-file to include the input signal, Vin, as shown in the following Simulink window and the MATLAB Editor window. AMATLAB R2021b-academic use RC Step Response - Simulink academic Search Documentation PBor-Chin SIMULATI DEBUG MODELING FORMAT APPS Stop Time 1010 Compare Print NAVIGATE CODE ANALYZE SECTION Run Step Stop FLE LIBRARY PREPARE Normal Fast Restart Step Run Back - SIMULATE Work RC StepResponse RC StepResponse Name out HRC out time 4 1 RC 3+1 out. Vout FILE RUN CUsers bacB8 Documents MATLAB Current Fold Editor - plotRCresponse.m X Variables-out Na.. plotRCresponse.mx + Untitle Plot RC response of the simulated system Untitle 2 plot(out. time, out. Vin, out.time, out. Vout) Brunfgw. 3 xlabel("Time [sec]"> RC Ste. 4 ylabel("Vout (volts) plotRC. 5 legend("Vin", 'Vout"> Initial c. 6 grid Gyro se. title("Step Response of RC Circuit) 8 gtm_si. gtm_de. gtm de Command Window f18trim. >> plot RCresponse f18trim. fx >> f18trim. BE 18stab... plotRCresp. A 6. UTF-8 CRLF In 5 out. Vin script Col 20 Ready 100% autoode45) Run the modified Simulink model and then the modified plot m-file, we will obtain the following MATLAB figure that includes both the step input Vin (in blue) with magnitude 5 volt and the step response Vout (in red). Step Response of RC Circuit 5 4.5 Vin Vout 4 3.5 3 Vout (volts) -2.5 1.5 1 0.5 0 0 2 4 8 10 6 Time (sec! Remember to SAVE YOUR WORK, including the Simulink model file (*slx or * mdl) and the MATLAB m- file (*.m). Saving these files allows them to be reused and referenced later. When used as part of a lab report, they are required in the lab report submission. Sinusoidal Response Model The step response model can be modified by changing the Source block to create a sinusoidal response model. Replace the Step block with a Sine Wave block. Remember to adjust the block parameters as needed. For this simulation, the amplitude and frequency are chosen to be 1 and 1 rad/s, respectively. Notice that the simulation duration time, located at the top of the model window, near the run button, has been modified to 40 sec. This will allow the simulation to run long enough for the steady-state sinusoidal response to be observed. MATLAB R2021b-cademic PRC Sinusoidal Response - Simulink acade Bor-Chin SIMUL DEBUG MODE FORMAT APPS P. Search Documentation Big Stop Time 400 Compare New Open Save Print NAVIGATE CODE ANALYZE SECTION Run Step Stop Normal BLE LERARY PREPARE Fast Restart Step Run Back - SIMULATE FILE Works RC_Sinusoidal Response RC Sinusoidal Response 1 Users bcc88 Documents MATLAB Editor - plotrCresponse sinusoidal.mx Variables out. Vout plotrCresponse_sinusoidal.m + Plot RC response of the simulated system 2 plot(out.time, out. Vin, out.time, out. Vout) xlabel('Time [sec]') ylabel('Vout (volts]') legend('Vin', 'Vout) 6 grid title("Sinusoidal Response of RC Circuit) Name out HERC 3 4 5 out. Vout ta Current FOLL Na. Untitle. Untitle runfg. RC_Ste RC Sin plotRC Initial Gyro se gtm_si. gtm de. atm de f1 trim f1 trim f18trim plotrCresp. RC - +1 7 8 out Vin Command Window Documentation for legend >> plotrCresponse_sinusoidal fx >> UTF-8 CRLF script Col1 Ready 100% auto ode45) Run the simulation model and the m-file plotRCresponse_sinusoidal.m to obtain the following MATLAB figure showing the waveforms of the input and output of the RC circuit system. Sinusoidal Response of RC Circuit 0.8 Vin Vout 0.6 0.4 0.2 Voutvolts) NOT 0 -0.2 -0.4 Acceptable !!! -0.6 -0.8 -1 0 5 10 15 20 25 30 35 40 Time [sec] Notice that the plotted waveforms of the input and the output appear jagged and unsmooth. This is a result of Simulink using a simulation-step size that is too large. To ensure higher-fidelity simulation results, the Simulink model parameters need to be configured to limit the maximum simulation-step size used during simulation. These parameters can be adjusted by selecting SIMULATION PREPARE CONFIGURATION & SIMULATION Model Setting... in the model window. Configuration Parameters: RC_Sinusoidal Response/Configuration (Active) Q Search Simulation time Start time 00 Stop time 400 Solver Data Import Export Math and Data Types Diagnostics Hardware Implementation Model Referencing Simulation Target Code Generation Coverage HDL Code Generation Solver selection Type: Variable-step - Solver auto (Automatic solver selection) Solver details RG/103 Max step size Min step size auto Relative tolerance Te 3 Absolute tolerance auto Auto scale absolute tolerance Disable A Initial step auto Shape preservation Number of consecutive min steps 1 Zero-crossing options Zoro-crossing control Use local settings Time tolerance 10"128"eps Number of consecutive zero crossings: 1000 - Algorithm Nonadaptive Signal threshold. Tasking and sample time options Automatically handle rate transition for data transfer Allow multiple tasks to access inputs and outputs Higher priority value indicates higher task priority OK Cancel Help Many model-wide settings can be configured in the configuration window shown above. In the upper-middle section of the window the Max step size parameter should be changed to RC/100. This ensures that the maximum simulation-step size will be no more than 1/100th of the RC value. These settings are sufficient for the range of RC values expected for this lab. Click APPLY to save the altered parameters and return to the model window. Rerun the simulation with the new parameters and plot the results. Notice that the plots now appear as smooth curves. This is a result of the simulation running with smaller simulation-step sizes, yielding higher resolution and greater precision on the logged data. Sinusoidal Response of RC Circuit 1 0.8 Vin Vout 0.6 0.4 0.2 Vout [volts] 0 -0.2 -0.4 -0.6 -0.8 -1 0 5 10 15 25 30 35 40 20 Time [sec] This sinusoidal response plot is better than the previous one; however, the waveforms are too crowded for the measurement of the oscillation period and the phase difference between the input and the output. Since for the sinusoidal response only the steady state response are of interest, we will enlarge the portion of the graph between 25 second and 35 second marks by revising the m-file plotRCresponse_sinusoidal.m as follows: %Plot RC response of the simulated system plot(out.time ( 2500:3500), out. Vin(2500:3500), out.time(2500:3500), out. Vout(2500:3500)) xlabel('Time [sec]') ylabel('Vout [volts]') legend('Vin', 'Vout') grid grid minor title('Sinusoidal Response of RC Circuit') By running the m-file program, we have the following portion of the steady-state sinusoidal response between 25 second and 35 second. Sinusoidal Response of RC Circuit 0.8 - Vin Vout 0.6 0.4 0.2 Vout (volts) 0 A -0.2 -0.4 -0.6 -0.8 -1 24 26 28 30 32 34 36 Time [sec] The period (T) of the output waveform is exactly the same as that of the input: T = 32.15-25.87 = 6.28 sec 21 O= T = 1.0005 rad/sec =1 rad/sec 1 The amplitude of the output waveform is approximately 0.71=0.707=- 2 The output waveform lags behind the input waveform by 32.15-31.37 = 0.78sec . Hence, the phase shift will -0.78 x360 = -44.7 =-45. 6.28 be =