A Series RLC Circuit Consider the series RLC circuit shown in Figure 6.57, where R = 1 Ω, L = 1 H, and C = 0.5 F. When the switch is closed at 1 second, the circuit is driven by a 24 V DC voltage source. Assume that all initial conditions are zero. a. Build a Simscape model of the physical system

Question

A Series RLC Circuit

Consider the series RLC circuit shown in Figure 6.57, where [latex]R=1 \Omega, L=1 \mathrm{H}[/latex], and [latex]C=0.5 \mathrm{~F}[/latex]. When the switch is closed at 1 second, the circuit is driven by a [latex]24 \mathrm{~V} D C[/latex] voltage source. Assume that all initial conditions are zero.

a. Build a Simscape model of the physical system and find the loop current [latex]i(t)[/latex] and the voltage across the capacitor [latex]v_{\mathrm{C}}(t)[/latex].

b. Refer to the results obtained in Example 6.1. Build a Simulink model of the system based on the transfer function [latex]I(s) / V_{\mathrm{a}}(s)[/latex] and find the loop current [latex]i(t)[/latex].

c. Refer to the results obtained in Example 6.1. Build a Simulink model of the system based on the transfer function [latex]V_{\mathrm{C}}(s) / V_{\mathrm{a}}(s)[/latex] and find the voltage across the capacitor [latex]v_{\mathrm{C}}(t)[/latex].

Accepted Answer

a. The Simscape block diagram corresponding to the physical system is shown in Figure 6.58, which can be created by following these steps.

1. Type ssc_new at the MATLAB Command window to open the main Simscape library and create a new model.

2. Open the library of Simscape/Foundation Library/Electrical/Electrical Elements and drag the Resistor, Inductor, and Capacitor blocks into the model window. Double-click on these blocks to define the parameters Resistance, Inductance, and Capacitance as [latex]1 \mathrm{Ohm}, 1 \mathrm{H}[/latex], and [latex]0.5 \mathrm{~F}[/latex]. Also, drag the Switch and Electrical Reference blocks into the model window.

3. To add a [latex]24 \mathrm{~V}[/latex] DC voltage source, open the library of simscape/Foundation Library/Electrical/Electrical Sources and drag the DC Voltage Source block into the model window. Double-click on the block and define the parameter Constant voltage as [latex]24 \mathrm{~V}[/latex].

4. To measure the loop current and the voltage across the capacitor, open the library of Simscape/Foundation Library/Electrical/Electrical Sensors and drag both the Current Sensor and Voltage Sensor blocks into the model window. Each sensor has ports + and -, through which the sensor is connected to the circuit. The third port, I for the Current Sensor or V for the Voltage Sensor, is a physical signal port that outputs either current or voltage value.

5. Note that the Switch is closed at 1 second. To add the time source, open the library of Simulink/Sources and drag the Clock block, which is connected to the Simulink-PS Converter block and then the Switch block. Double-click on the Switch block and type 1 for the Threshold. This implies that the switch is closed if the time is greater than 1 second, otherwise the switch is open.

6. To display the loop current and the voltage across the capacitor, open the library of Simulink/Sinks and drag two Scope blocks, which are connected to the sensor blocks through the PS-Simulink Converter blocks. Note that the Current Sensor is connected in series with the circuit and the Voltage Sensor is connected in parallel with the Capacitor block.

7. Orient the blocks and connect them as shown in Figure 6.58.

Set the simulation time to 15 seconds and run the model. The plots of the resulting loop current [latex]i(t)[/latex] and the voltage across the capacitor [latex]v_{\mathrm{C}}(t)[/latex] are shown in Figures 6.59 and 6.60, respectively.

b. Refer to the results obtained in Example 6.1. The transfer function relating the input [latex]v_{a}(t)[/latex] to the output [latex]i(t)[/latex] is

[latex]\frac{I(s)}{V_{\mathrm{a}}(s)}=\frac{C s}{L C s^{2}+R C s+1}.[/latex]

When the switch is closed at 1 second, the circuit is driven by a [latex]24 \mathrm{~V}[/latex] DC voltage source. Mathematically, this can be modeled using a Step block with the step time set as 1 and the Final value set as 24. The corresponding Simulink block diagram is shown in Figure 6.61, where a Transfer Fcn block is used to represent the series RLC circuit. Double-click on the block and type [latex]\left[\begin{array}{ll}1 & 0\end{array}\right][/latex] for the Numerator coefficient and [ [latex]L{ }^{*} C R^{*} C[/latex] 1] for the Denominator coefficient to define the transfer function [latex]I(s) / V_{\mathrm{a}}(s)[/latex].

c. Similarly, the transfer function relating the input [latex]v_{\mathrm{a}}(t)[/latex] to the output [latex]v_{\mathrm{C}}(t)[/latex] is obtained as

[latex]\frac{V_{C}(s)}{V_{\mathrm{a}}(s)}=\frac{1}{L C s^{2}+R C s+1} .[/latex]

The corresponding Simulink block diagram is similar to the one shown in Figure 6.61, except that the Numerator coefficient of the Transfer Fen block is [1], instead of [latex][\begin{array}{ll}1 & 0]\end{array}[/latex]. Running the Simulink models, we will obtain the same current output as shown in Figure 6.59 and the voltage output as shown in Figure 6.60.

Question 6.14: A Series RLC Circuit Consider the series RLC circuit shown i...

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