1. *
07/16/96
What is Second Order?
second-order circuit : characterized by second-order differential equation
consists of resistors and the equivalent of two energy storage elements *

2. Finding Initial and Final Values*
07/16/96
Finding Initial and Final Values
Combine R, L & C
Find v(0), i(0), dv(0)/dt, di(0)/dt, i(∞) & v(∞).
t(0-)  time before switching event
t(0+)  time after switching event
Capacitor voltage always continuous 
v(0+)=v(0-)
Inductor current always continuous 
i(0+) = i(0-) *

3. The Source-Free Series RLC*
07/16/96
The Source-Free Series RLC
Applying KVL around the loop
After differentiation with respect to t, the roots: *

4. The Source-Free Series RLC*
07/16/96
The Source-Free Series RLC *

5. The Source-Free Series RLC*
07/16/96
The Source-Free Series RLC
Roots equation or natural frequencies (Np/s)
Where
 neper freq/damping factor (Np/s)
ω0  resonant freq./undamped natural freq (rad/s) *

6. The Source-Free Series RLC*
07/16/96
The Source-Free Series RLC
From natural frequencies, there are three type of solutions:
If α > ω0  overdamped case
If α = ω0  critically damped case
If α < ω0  underdamped case *

7. Overdamped case (α > ω0 )*
07/16/96
Overdamped case (α > ω0 )
Both roots s1 and s2 are negative and real
The response is *

8. Critically Damped case (α = ω0 )*
07/16/96
Critically Damped case (α = ω0 )
Roots s1 and s2 :
The response is *

9. Underdamped case (α < ω0 )*
07/16/96
Underdamped case (α < ω0 )
Roots s1 and s2 :
The response is *

10. Step Response of Series RLC*
07/16/96
Step Response of Series RLC
Applying KVL around the loop for t>0, *

11. Step Response of Series RLC*
07/16/96
Step Response of Series RLC
: Transient response
: Steady-state response *

12. Step Response of Series RLC*
07/16/96
Step Response of Series RLC
The transient response for the overdamped, critically damped and underdamped cases :
Overdamped
Critically damped
Underdamped *