# Laplace transforms: Simple Electrical Network

From Class Wiki

### Problem Statement

Using the formulas

Solve the system when V0 = 50 V, L = 4 h, R = 20 Î©, C = 10^{-4} f, and the currents are initially zero.

## Solution

Solve the system when V0 = 50 V, L = 4 h, R = 20 Î©, C = 10^{-4} f, and the currents are initially zero.

Applying the Laplace transform to each equation gives

Solving for

We find the partial decomposition

Let

Comparing the coefficients we get

Thus

Now we do the same for where we solve the function in terms of and decomposing the partial fraction resulting in

In order to make it nicer on us we need to complete the square as follows

Thus

Taking the Inverse Laplace transform gives

## Initial Value Theorem

## Final Value Theorem

## Bode Plots

The following are bode plots for the transfer functions