ece131l exp1

Interpretation of Results Module 1 is about Linear Constant-Coefficient Differential Equations. The part 1 of the module is about regeneration and representation of homogenous solution of the given differential equations. To represent a differential equation to homogenous solution, first we need to convert it to its auxiliary equation to find the roots which is to be used as the parameter of the homogenous solution. The form of the homogenous solution is dependent on the roots that is derived on the auxiliary equation in which it has three cases and these are, for case 1 the roots are real and distinct, for case 2 the roots are real and repeated, and lastly for case 3 the roots are complex and conjugate. The part 2 of the module is about determining the particular solution of the given differential equation with varying r(t). The Methods of Undetermined Coefficients (MUC) was used to determine the coefficients and form of the particular solution. For part 3 the homogenous and particular solution was solve to determine the total solution for the given differential equation in part 2 (a). The part 4 of the module is about determining the total solution by the used of Laplace transform. By taking the Laplace transform of the particular solution and converting the characteristic solution into S domain from T domain. By taking the Inverse Laplace of the particular solution over the

characteristic solution both in S domain thereby resulting to the total solution back in T domain. The part 5 of the module is about determining the poles and zeroes of the transfer function. With the used of RESIDUE and by using the real and imaginary numbers, we are able to determine the location of the poles and zeroes. In the command, R represents the residue, P represents the poles and k is the direct term. The same method was done for part 6. The part 7 of the module is about determining the partial fraction expansion and the Inverse Laplace as well. The EXPAND command was used to find the coefficients of the numerator and denominator making it easier to compute. The command ILAPLACE was used for part 8 to directly get the Inverse Laplace equivalent for the given in part 7 and to verify the derived Inverse Laplace on that part. For the last part of the module or the seatwork, the transfer function was determined by using KVL which is in T domain the derived equation was then converted to S domain to obtain the Laplace Transform to solve for the value of Vc(s).

Conclusion: Based on what I’ve learned from this module the characteristic or homogenous solution, the right side of the equation is being neglected and can be taken by taking the roots of the auxiliary equations derived from the differential equation or MATLAB where the command is ROOTS([coefficients of the auxiliary equation]). I’ve learned that the total solution can be derived immediately from the transfer function by using the DSOLVE command in which it will display both the characteristic and particular solution since their sum is equal to total solution. The particular solution can be obtained using the same equation but identifying the characteristic solution first is better in order to know the particular solution. The total solution (sum of characteristic solution and particular solution) can be derived using the Laplace transform. It is done by converting the time domain function to s domain function and then by dividing the particular solution (in s domain) to characteristic solution (in s domain). The poles and zeroes can be determine from a transfer function by the use of RESIDUE command which represented by [R,P,K]=residue(B,A) wherein R represents residue, P represents poles, k represents the direct

term then A serves as the denominator’s coefficients while B serves as the numerator’s coefficients. The series RLC circuit can be solve by using a Laplace Transform to the equation obtained from KVL and it can determine the voltage across any element on the circuit.

Mapúa Institute of Technology School of Electrical, Electronics and Computer Engineering Feedback and Control Systems Laboratory ECE131L/B12

CONTROL SYSTEMS Module No.: 1 Linear Constant-Coefficient Differential Equations

Submitted by: Sapalaran, Ma. Carmela P.

Submitted to: Engr. Ernesto Vergara

Submitted on: July 30, 2015