fardinabbasi / Laplace_Transform

Signals and Systems [ECE 538] - CA#4 - Spring 2022 - University of Tehran - Dr. Akhavan

Laplace Transform

Modeling mechanical and electrical systems using Laplace transform in MATLAB.

Part 1: Electrical System

Simulation of the following RLC Circuit Using the Laplace Transform.

The relationship between the inductor's current $(I_L)$ and the input current $(I_s)$ is obtained using the Laplace transform through the following procedure.

Finally, here is the final equation: $$I_L = \frac{{128I_s - 128I_L}}{{s^2}} - \frac{{32I_L}}{s}$$

The block diagram below illustrates the system response when a step input is applied as the input current.

Furthermore, the simulation result obtained from Simulink is provided below.

Part 2: Mechanical System

Simulation of the following Spring-Mass-Damper System Using the Laplace Transform.

The relationship between the input force $(F)$ and the output distance from the wall $(X)$ can be derived using the Laplace transform by following the procedure outlined below.

The minimum value for parameter $B$, which ensures that the transfer function's poles are real, is 2.

$\Delta = B^2 - 4 \rightarrow \beta = B_{\text{min}} = 2$

The block diagram below illustrates the system response when a step input is applied as the force.

Furthermore, the simulation result with different values for $B$ obtained from Simulink is provided below.

$B = \beta =2$	$B = 0 (Complex Poles)$
$B = 1 < \beta (Complex Poles)$	$B = 4 > \beta$

1. $B = \beta =2$: After applying the force, the system stabilizes over time.
2. $B = 0$: The poles being purely imaginary result in a non-oscillatory response.
3. $B = 1 < \beta$: The poles being complex result in a damped oscillation.
4. $B = 4 > \beta$: With real poles, the system oscillates over time but at a slower rate compared to the case when $B = \beta =2$.