2. Prelab_2

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Oct 30, 2023

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ENSC3015 Lab 2 Linear Time Invariant Systems Analysis using SIMULINK Pre-Lab 1 Consider the following RLC circuit: By applying KVL to the circuit, one can derive the following equation: L di ( t ) dt + Ri ( t ) + C v = x ( t ) (1) LC d 2 y ( t ) dt 2 + RC dy ( t ) dt + y ( t ) = x ( t ) (2) As per Figure 1, C = 1 6 F , L=1 H , R=5Ω, hence we arrive at: d 2 y ( t ) dt 2 + 5 dy ( t ) dt + 6 y ( t ) = 6 x ( t ) (3) Now, we’ve been asked to determine the transfer function, impulse response function, and zero-state response of this system to a unit step input as well as the frequency response to a sinusoidal input. This will be detailed in following subsections. Transfer Function Deriving the transfer function: 1
Impulse Response Function And subsequently the impulse response function by applying inverse Laplace to the transfer function H(s): Zero State Response The zero state response can be derived as follows: Frequency Response Given x(t) = sin (2 πt ), the frequency response of H(s) is as follows: Note: after substituting 2 π as ω into H(j ω ), we can see that it sits in the 3rd quadrant. Thus the resultant phase response must reflect this with a 180 shift. 2
Pre-Lab 2 The following SIMULINK circuit connection was created with a step time of 0. Note the Scope output graph. Scope output graph: 3
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Replacing the Scope sink with the To Workspace sink, our circuit connection changes as follows. Note (1) The code in the MATLAB window which enables us to generate an output plot of the SIMULINK and (2) Figure 1 being the output plot. The step response was previously derived as: s ( t ) = 3 e 2 t + 2 e 3 t + 1 (4) Plotting the derived theoretical response, superimposed with the SIMULINK output produces the following graph (see window entitled ”Figure 1”). Again noting the MATLAB code used to output said graph. Theoretical response (as shown in blue) is displayed within a time range of 0 to 5s, just to ensure that the scope output (simulated response as shown in pink), is visible for comparison purposes. The graphs are essentially identical, underscoring the fact that SIMULINK purely serves as an alternate means of executing the same mathematical procedure. 4

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