Electrical Engineering ⇒ Topic : Step Response of R-L-C Circuit

STEP RESPONSE OF R-L-C CIRCUIT

Figure (a) shows a series R-L-C circuit having zero initial conditions, i.e., initially the circuit was relaxed. Let the switch be closed at t = O.

The circuit equation is given by

                    Figure (a) Series R-L-C circuit

                   ........ (1)

The Laplace transform of Eq. (1) can be written as

            ........... (2)

                             ............ (2a)

The three possible solutions for I(s) are as follows:
Case (i): Both a and b are real and a ≠ b ,i.e.,

              ............. (3)

     ................ (4)

     ................... (5)

Using Eqs. (4) and (5), Eq. (3) can be written as

                .............. (6)
The inverse Laplace transform of Eq. (6) is given by

                                                      .............(7)

Taking the inverse Laplace transform of Eq. (7), we have

                                                         ................ (8)

Case (iii): Roots are imaginary, i.e.and a and b are complex conjugates

                                  .......... (9)

Taking the inverse Laplace transform of Eq. (9), we obtain

                                                      ............... (10)