Engr 16 Today’s Topic 2nd Order Transient Engr 16 Lec 5-2 2nd Order Transient Resp.                Engr 16 Today’s Topic 2nd Order Transient Characteristic Equation Discriminant & Damping Form of Natural Response Intro to Impedance Method

Engr 16 Lec 5-2 2nd Order Transient Resp. DC Shortcut If all inputs are DC sources, capacitors will eventually become an open and inductors will eventually become a short. Use to find DC Steady State as you come to t = 0 and as you go to t = ∞.

Engr 16 Lec 5-2 2nd Order Transient Resp. Charging Capacitor Choking Inductor Simple series connection

Differential Equation 𝑉 𝑆 =𝐿𝐶 𝑑 2 𝑣 𝐶 (𝑡) 𝑑 𝑡 2 + RC 𝑑𝑣 𝐶 (𝑡) 𝑑𝑡 + 𝑣 𝐶 (𝑡) Characteristic Equation 0=𝐿𝐶 𝑠 2 +𝑅𝐶𝑠+1 Roots of Char. Equa. Discriminant determines natural response shape 𝑠 1,2 = −𝑅𝐶± 𝑅 2 𝐶 2 −4𝐿𝐶 2𝐿𝐶 Over Damped: dicsr>0, two real roots 𝑣 𝐶−𝑛𝑎𝑡 𝑡 = 𝐾 1 𝑒 𝑠 1 𝑡 + 𝐾 2 𝑒 𝑠 2 𝑡 Under Damped: dicsr<0, complex conjugate roots 𝑣 𝐶−𝑛𝑎𝑡 𝑡 =𝐾 𝑒 −𝛼𝑡 cos⁡( 𝜔 𝑛 𝑡+𝜑) Critically Damped: dicsr=0, one real repeated root 𝑣 𝐶−𝑛𝑎𝑡 𝑡 = 𝐾 1 𝑒 −𝛼𝑡 + 𝐾 2 𝑡 𝑒 −𝛼𝑡

Impedance Method Derive transfer function using 1/Cs & Ls & R Find the forced response (t>0) Find the format of the natural response (t>0) Find the state at 0-: Vc(0-) and IL(0-) Find the initial conditions Evaluate general solution for constants Total = (Forced+Natural)|init condits