1. 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

2. 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 = ∞.

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

4. 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

5. 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