What is Second Order?

1. 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 • 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 • Applying KVL around the loop • After differentiation with respect to t, the roots:

4. The Source-Free Series RLC

5. 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 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 ) • Both roots s1 and s2 are negative and real • The response is

8. Critically Damped case (α = ω0 ) • Roots s1 and s2 : • The response is

9. Underdamped case (α < ω0 ) • Roots s1 and s2 : • The response is

10. Step Response of Series RLC • Applying KVL around the loop for t>0,

11. Step Response of Series RLC • : Transient response • : Steady-state response

12. Step Response of Series RLC The transient response for the overdamped, critically damped and underdamped cases : • Overdamped • Critically damped • Underdamped