Electronics/RCL time domain

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Figure 1: RCL circuit
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

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: