Second Order Transient Analysis

Second Order Transient Analysis

How do recognize it?

A transient problem is one that asks you to find voltage (or current) vs. time. It also includes some instant event: a switch opens or closes, the power supply turns on, a part is suddenly pulled out the circuit, a fuse blows. Your answer can either be a formula for v(t) or i(t) or it might be a graph of v or i on the vertical axis and time on the horizontal axis.
Second-order problems have two independent energy storage elements (capacitors and/or inductors). Note that two capacitors in parallel or series only count as one capacitor because you can find the equivalent capacitance (same with inductors).

How do you do it?

1. For first-order, the only form possible was an exponential response. For second-order there are three possible forms. You can compute which one based on the following three constants that are coefficients of the differential equation that describes these types of circuits:

   Constant	Series RLC	Parallel RLC
   a	1	1
   b	R/L	1/(RC)
   c	1/(LC)	1/(LC)

   Note that in each case, R is the Thevenin resistance seen by the LC pair. If the LC pair is in parallel but the Thevenin R is in series, you may need to convert to a Norton so that all the elements are in parallel.
2. Now compare three constants to determine which of the three response forms you have:
   • if b^2 > 4*a*c then you have overdamped.
   • if b^2 = 4*a*c then you have critically damped.
   • if b^2 < 4*a*c then you have underdamped.