* 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 *

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-) *

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: *

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

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) *

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 *

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

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

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

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

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

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 *