1

Given the initial condition, , can be determined using transient

analysis to be . Hence,

.

Both and can also be determined using Laplace transform. Using KVL, we can write

2

Taking the Laplace transform yields

Taking the inverse Laplace transform yields .

Since

giving rise to , we can rewrite

to be . Taking the Laplace

transform yields

Taking the inverse Laplace transform yields . Note that partial fraction is required and used in order for the inverse Laplace transform of to be performed to obtain .

3

Partial Fraction

In general, we need to find the inverse Laplace transform of a function which is of the form

where and are real constants, and and are positive integers. is called a proper rational function if , an improper rational function if . Note that only proper rational function can be expanded as a sum of partial fractions.

4

Distinct Real Roots

Consider the proper rational function,

, where the roots

of the denominator are all real and distinct, we can expand the function using partial fraction as follows:

Using the cover-up method, we can find and to be

Hence,

and taking the inverse transform yields

5

Distinct Complex Roots

Given

, we can expand the function using partial

fraction as follows:

Using the cover-up method, we can find and to be

Note that the coefficients associated with complex conjugate roots, and , are themselves complex conjugates. Thus, we only need to find one of the two coefficients, either or .

6

After finding the values for and , we can proceed in two ways. One way is to use the polar form, along with the general transform formula

to obtain

The other way is to proceed using the rectangular form as follows:

7

We can also determine the inverse transform without using complex numbers as follows:

After finding using the cover-up method, we multiply both side of the equation by the denominator and obtain

Hence,

Finally,

8

Repeated Real Roots

Given

, we can expand the function as follows:

Using the cover-up method, we can find and to be

To find , we need to multiply both sides of by and then

determine to be

9

To find , we multiply both sides of by and then find to be

Generally, given that

, we can determine

After finding the values for , and , we determine the inverse transform to be

10

Repeated Complex Roots

Given

, we can expand the function as follows:

Using the previously presented methods, we can find and to be

Hence,

The general formula for repeated complex roots is

11

Improper Rational Functions

For an improper rational function,

, we must

first rewrite it as a proper rational function using long division and then perform partial fraction as follows:

Finally, taking the inverse transform yields

12

Initial and Final Value Theorems

Without having to find the inverse Laplace transform of , we can evaluate and to see if they conform with known circuit behavior using the initial and final value theorems as follows:

(initial value theorem)

(final value theorem)

For instance, in the example presented in the first slide, the known circuit response are and . We can verify them as follows:

The initial and final value theorems are useful for verifying if the known circuit response at and are described by .