When the switch is closed, a voltage step is applied to the RCL circuit. Take the time the switch was closed to be 0s such that the voltage before the switch was closed was 0 volts and the voltage after the switch was closed is a voltage V. This is a step function given by where V is the magnitude of the step and for and zero otherwise.

To analyse the circuit response using transient analysis, a differential equation which describes the system is formulated. The voltage around the loop is given by:

where is the voltage across the capacitor, is the voltage across the inductor and the voltage across the resistor.

Substituting into equation 1:

The voltage has two components, a natural response and a forced response such that:

substituting equation 3 into equation 2.

when then :

The natural response and forced solution are solved separately.

**Solve for **

Since is a polynomial of degree 0, the solution must be a constant such that:

Substituting into equation 5:

**Solve for :**

Let:

Substituting into equation 4 gives:

Therefore has two solutions and

where and are given by:

The general solution is then given by:

Depending on the values of the Resistor, inductor or capacitor the solution has three posibilies.

1. If the system is said to be **overdamped**

2. If the system is said to be **critically damped**

3. If the system is said to be **underdamped**

## Example:Edit

Given the general solution

R | L | C | V |

0.5H | 1kΩ | 100nF | 1V |

Thus by Euler's formula ( ):

Let and

**Solve for and :**

From equation \ref{eq:vf}, for a unit step of magnitude 1V. Therefore substitution of and into equation \ref{eq:nonhomogeneous} gives:

for the voltage across the capacitor is zero,

for , the current in the inductor must be zero,

substituting from equation \ref{eq:B1} gives

For , is given by:

is given by:

For , is given by: