# Laplace transforms: Simple Electrical Network

### From Class Wiki

## Contents |

### Problem Statement

Using the formulas

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

## Solution

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

Applying the Laplace transform to each equation gives

Solving for *I*_{2}(*s*)

We find the partial decomposition

Let

Comparing the coefficients we get

Thus

Now we do the same for *I*_{1} where we solve the function in terms of *I*_{1} and decomposing the partial fraction resulting in

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

*s*^{2} + 500*s* + 2500 = 0

Thus

Taking the Inverse Laplace transform gives

## Initial Value Theorem

## Final Value Theorem

## Bode Plots

The following are bode plots for the transfer functions