# Electronics/RCL time domain1

*Figure 1: RCL circuit*

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**. The system has two distinct real solutions:

2. If the system is said to be **critically damped**. The system has one real solution:

- Let :

3. If the system is said to be **underdamped**. The system has two complex solutions:

- By Euler's formula ():

- Let and