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# Series RLC circuit

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How to find the equation of current of the RLC circuit after the switch is moved?

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It all depends on the values of the components:If

the system will very slowly decay until the energy of the system reaches 0.

If

the system undergoes something which will look like a part of a oscillation and loses its energy very quickly

If

it oscillates with decreasing amplitude until its energy reaches 0.

In our case:

it will very slowly decay without any oscillation.In order to find the equation of current of this RLC circuit we must be introduced to 2 things:

Neper angular frequency -> a feature of damped systems

In the case of the series RLC circuit it is:

The natural angular frequency -> the frequency response of the system under no damping

In a RLC circuit it is equal to:

The equation of current of this overdamped system takes this form:

where :

(1)

and:

Now lets go calculate s1 and s2:

After the switch is closed:

and due to L1:

so the voltage between L1 is:

By substituting the values of s1,s2 VL1(0+),L1 and IL1(0+) to (1) and solving the system we get:

This is the equation of the current of the loop consisting of C1,L1,R1 after the switch is closed.

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