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.
Since is a polynomial of degree 0, the solution must be a constant such that:
Substituting into equation 5:
Solve for :
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