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The system we will be employing in this activity is a simple electrical circuit consisting of a resistor R and a capacitor C in series. The Arduino board will be used for generating the input to Billing Boats Rc Model Matlab this RC circuit and for measuring the output of the circuit. Specifically, the input to the circuit will be a 5-Volt step, generated from one of the board's Digital Outputsbilling boats rc model matlab across the resistor and capacitor.Model Matlab Boats Billing Rc Billing Boats Rc Model Matlab

The output of the circuit will be the voltage across the capacitor which will be read via one of the board's Analog Inputs. This data is then fed to Simulink for visualization and for comparison to our resulting simulation model output. The purpose of this activity is to demonstrate how to model a simple electrical.

Specifically, a first principles approach based on the underlying physics of the circuit and a blackbox approach based on recorded data will be employed. The associated experiment is employed to demonstrate the blackbox approach, as well as to demonstrate the accuracy of the resulting models.

This activity also provides a physical example of the common class of first-order systems. First we will employ Billing Boats Rc Model Matlab our understanding of the underlying physics of the RC circuit to derive the structure of the system model. We will term this process "modeling from first principles. Current flows from a higher potential to a lower potential, therefore, the direction Billing Boats Rc Model Matlab of the current will be clockwise in this case billing boats rc model matlab.

The loop law states that the sum of voltages around a closed loop must equal zero. Thus, the loop law produces the following governing equation for Billing Boats Rc Model Matlab the circuit. An alternative to the intregro-differential equation model of a dynamic system is the transfer function. Next we must perform some algebra to re-arrange the above into the form of its output divided by its input. In this case, Billing Boats Rc Model Matlab our input is and our output is. Therefore, we must eliminate the current from the above since it is neither an input nor an output, and we must introduce the output into the above equation.

Solving the above equation Billing Boats Rc Model Matlab for we arrive at the following. Then we can recognize that the output voltage billing boats rc model matlab the capacitor is. Taking the Laplace transform and again solving forwe arrive at the following. Setting the two previous equations equal Billing Boats Rc Model Matlab to one another, we can eliminate. Then re-arranging into the desired form of output divided by input, we produce the resulting transfer function model.

Recognizing the above as a first-order system, we can manipulate the transfer function so that it Billing Boats Rc Model Matlab has the standard form shown. In this form billing boats rc model matlab can see by inspection how the parameters of the circuit affect its response.

Specifically, the values of the resistor and capacitor affect the circuit's speed of Billing Boats Rc Model Matlab response as indicated by their influence on the circuit's time constant. The circuit components, however, cannot influence the circuit's steady-state performance as indicated by the fact that the DC gain always billing boats rc model matlab 1.

In this experiment we will record the output voltage of the RC circuit for a step in input voltage. Based on the resulting time response of billing boats rc model matlab output voltage, we will fit a model to the Billing Boats Rc Model Matlab data. This approach is sometimes referred to as blackbox modeling or data-driven modeling.

After we have generated such a model, we will compare it to the first-principles derived model we created previously. Our simple RC circuit can be implemented on Model Boats Billing Matlab Rc a breadboard and connected to the Arduino board as shown.

One thing to note is that if you employ an electrolytic capacitor, its orientation matters. Specifically, if your capacitor has legs of different lengths and one leg is marked by a negative sign, then you have an electrolytic capacitor.

Orient an electrolytic capacitor so that the leg marked by the negative sign connects to the lower potential part of the circuit ground in this case.

The Arduino board is employed Billing Boats Rc Model Matlab to receive the input command from Simulink and to apply the input voltage to the circuit via a Digital Output. The board also acquires the output voltage data from the circuit via an Analog Input and communicates the data Billing Boats Rc Model Matlab Billing Boats Rc Model Matlab to Simulink.

If the sampling rate is at least 10 times faster than the circuit's time constant, we will be able to get a clear picture of the response. In this experiment, we chose and to produce a circuit Billing Boats Rc Model Matlab with time constant second. This is a low power application so most billing boats rc model matlab and capacitors available will work fine, just make sure that the capacitor is rated for at least 5 Volts.

In this experiment, we Billing Boats Rc Model Matlab Billing Boats Rc Model Matlab will employ Simulink to read the data from the board and to plot the data in real time. In particular, we will employ the IO package from the MathWorks.

For details on how to use the IO package, refer to Billing Boats Rc Model Matlab the following link. The Simulink model we will use is shown below and can be downloaded herewhere you may need to change the port to which the Arduino board is connected the port is COM5 billing boats rc model Billing Boats Rc Model Matlab matlab this case.

As shown below, the input voltage command is generated by a Pulse Generator block. This block is chosen because of its generality, though for this experiment we could billing boats rc model matlab used billing boats rc Billing Boats Rc Model Matlab Billing Boats Rc Model Matlab model matlab Step block. The Pulse Generator block generates values of 0 billing boats rc model matlab 1 which billing boats rc model matlab then fed to billing boats rc model matlab Arduino Digital Write block.

Since we are using Billing Boats Rc Model Matlab channel 8 for the digital output, we double-click on the Arduino Digital Write block to set the Pin to 8 from the drop-down menu.

An input of 0 to the Digital Write block causes an output of 0 Volts to be generated at the corresponding pin, while an input of 1 to the Digital Write block generates an output of 5 Volts. This scaling is captured by the Gain block that is included prior to the Scope block. In this Billing Boats Rc Model Matlab model the Pulse Generator block billing boats rc model matlab set to output 0 for the first 5 seconds of the run in order to discharge the billing boats rc model matlab completely in case it had built up Billing Boats Rc Model Matlab charge before stepping to 1 for the last 5 seconds of the run.

Double-clicking on the block allows us to set the Pin to 0 from the drop-down menu. We also will set the Sample Time to "0. This is Billing Boats Rc Model Matlab 10 times faster than the circuit's time constant and hence is sufficiently fast.

The other blocks in the model can also be set to have a Sample Time of "0. This conversion billing boats rc model matlab be understood by recognizing that the Arduino Board employs a bit analog-to-digital converter, which means for the default that an Analog Input channel billing boats rc model matlab a voltage between 0 and 5 V and slices that range into pieces.

Therefore, 0 corresponds to 0 V and corresponds to 5 V. The given Simulink model then plots the commanded input voltage and recorded output voltage on a scope and also writes the output data, as an array, to the MATLAB Billing Boats Rc Model Matlab Billing Boats Rc Model Matlab Billing Boats Rc Model Matlab Billing Boats Rc Model Matlab workspace for further analysis.

The remaining blocks are part of the standard Simulink library, specifically, they can be found under the Math, Sources, and Sinks libraries. Once the Simulink model has been created, it can then be run to collect Billing Boats Rc Model Matlab the input voltage and output voltage data. Note that we plot only the last five seconds of the run, thereby excluding any discharging of the capacitor that may have taken place. Based on the on the above figure, we can Billing Boats Rc Model Matlab Billing Boats Rc Model Matlab Billing Boats Rc Model Matlab fit a model to the recorded data.

Recognizing that the observed data has billing boats rc model matlab shape of a first-order step response, we will assume the following standard first-order model for the circuit billing boats rc model matlab Billing Boats Rc Model Matlab to match our first-principles model. We can then identify the system parameters and from the recorded response data.

Specifically, the steady-state value of the response indicates that the DC gain is. Note, we verified with a Voltmeter that the output voltage generated via the Digital Output was very close to billing boats rc model matlab Volts.

Recalling that by definition the time constant represents the time it takes the system's response to reach of its total change, can Billing Boats Rc Model Matlab Billing Boats Rc Model Matlab Billing Boats Rc Model Matlab be calculated from the following where Since the step input occurs at approximately 0. In reality, we only have data at intervals of 0. Using a smaller sample period would allow us to improve the accuracy of our identification of Billing Boats Rc Model Matlab the system parameters.

In conclusion, our resulting billing boats rc model matlab model is the following. In the this section we will validate the two circuit models we have derived, the first-principles model and the blackbox model, by comparing the Billing Rc Model Matlab Boats Billing Boats Rc Model Matlab step responses predicted by the models to actual data recorded billing boats rc model matlab the circuit. We will generate a Simulink model of this governing equation by first solving the equation for its highest-order "derivative.

This equation is then represented in Simulink as shown below by the highlighted blocks where R and C are variables. We have inserted these blocks into the existing Simulink model so that we may directly compare the simulated output voltage to the actual Billing Boats Rc Model Matlab Billing Boats Rc Model Matlab Billing Boats Rc Model Matlab Billing Boats Rc Model Matlab Billing Boats Rc Model Matlab recorded output voltage. The Unit Delay block is added so that the timing of the simulated input voltage better matches the timing of the input voltage generated by the Arduino board.

The To Workspace blocks are set to have Array Rc Boats Billing Matlab Model outputs. We could have also represented the first-principles model as a transfer function, but then we wouldn't be able to directly include initial conditions an initial charge on the capacitor. The charge on the capacitor is the output of Billing Boats Rc Model Matlab the Integrator block and its initial condition is set by double-clicking on the Integrator block.

In order to make our Simulink model cleaner, we can create billing boats rc model matlab subsystem for our first-principles model. Specifically, we can select the blocks highlighted in the above figure and then right-click. From the resulting menu we then choose Create Subsystem from Selection. For the first parameter, enter "Resistance" under the Prompt column and "R" under the Name column. Repeat this Billing Boats Rc Model Matlab process two more times for the "Capacitance" and the "Initial charge on capacitor" as shown.

This window is then closed and the settings saved by clicking the OK button. If you double-click on RC Circuit Model again, billing boats rc Billing Boats Rc Model Matlab model matlab will find that the mask with desired parameters has been generated. For this experiment, recall that we are employing a resistor, a capacitor, and we are observing the step response when when the capacitor is initially discharged. Recall Billing Boats Rc Model Matlab that the Pulse Generator block is set to output 0 for the first 5 seconds of the run in order to discharge the capacitor completely before stepping to 1.

The above figure shows pretty good, though not exact, agreement between the output voltage predicted by our first-principles model and the physical circuit. We will discuss the accuracy of this model momentarily, but let's first investigate the performance of our blackbox model. Recall that in the System identification experiment Billing Boats Rc Model Matlab section we estimated the system parameters as and leading to the following blackbox model.

We can generate the blackbox model in Simulink using the Transfer Function block. Double-clicking on the Transfer Function block we can set Numerator coefficients equal to Billing Boats Rc Model Matlab the variable "K" and the Denominator coefficients equal to "[tau 1]" where tau is also a variable, the time constant. We can then create a subsystem with a mask for this blackbox model in the same manner we did for Billing Boats Rc Model Matlab Boats Matlab Model Billing Rc Billing Boats Rc Model Matlab the first-principles model.

The resulting Simulink model is shown below and can be downloaded. Remember to set the parameters for the blackbox subsystem parameters K and tau. Recall in the above that our first principles model did not exactly predict Billing Boats Rc Model Matlab the output generated by the actual physical circuit. One factor contributing to this difference could be that the components the resistor and capacitor have actual values that differ from their nominal labels.

We can improve our model by measuring the Model Rc Boats Billing Matlab actual resistance and capacitance using an Ohmmeter and a capacitance meter, respectively.

11:02:

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