1. EKT101 Electric Circuit Theory
Chapter 5 (b)
First-Order and Second Circuits

2. 5.4 Natural and force response in series RLC Circuit
Second order circuit
5.4 Natural and force response in series RLC Circuit

3. Examples of Second Order RLC circuits (1)
What is a 2nd order circuit?
A second-order circuit is characterized by a second-order differential equation. It consists of resistors and the equivalent of two energy storage elements.
RLC Series
RLC Parallel
RL T-config
RC Pi-config

4. Source-Free Series RLC Circuits (1)
The circuit is excited by the energy initially stored in the capacitor and inductor.  natural response of the circuit.
The 2nd order of expression
How to derive and how to solve?

5. Source-Free Series RLC Circuits (2)
At t=0,
So,
Eliminate integral, differentiate to t, rearrange 

6. Source-Free Series RLC Circuits (3)
There are three possible solutions for the following 2nd order differential equation:
General 2nd order Form
where
=>
The types of solutions for i(t) depend on the relative values of a and w.

7. Source-Free Series RLC Circuits (4)
2nd Order differential Equation
Characteristic equation of differential equation
Natural response of series RLC
Where A1 and A2 are determined from initial values i(0) and di(0)/dt.
i(0) = I(0)

8. Source-Free Series RLC Circuits (5)
There are three possible solutions for the following 2nd order differential equation:
1. If a > wo, over-damped case
where
2. If a = wo, critical damped case
where
3. If a < wo, under-damped case
where
B1 = A1+A2 ; B2 = j(A1-A2)

9. Source-Free Series RLC Circuits (5)
Example 1
If R = 10 Ω, L = 5 H, and C = 2 mF in 8.8, find α, ω0, s1 and s2.
What type of natural response will the circuit have?

10. Source-Free Series RLC Circuits (6)
Example 2
The circuit shown below has reached steady state at t = 0-.
If the make-before-break switch moves to position b at t = 0, calculate i(t) for t > 0.

11. For t < 0, the inductor is connected to the voltage source and when the circuit reaches steady state, the inductor acts like a short circuit:
The voltage across the capacitor is:
For t > 0, we have a source-free RLC circuit.
Since ,we have an underdamped case.

12. ω 𝑑 = − =1.6583
i(t) = e -2.5t[B1 cos t + B2 sin t]
Now, we determine B1 and B2 using the initial conditions at t = 0: B1