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Unformatted text preview: is: d 2 x(t )
2 2 0 x(t ) f (t ) 2
. with , 0 2L
75 EE1002 Introduction to Circuits and Systems The general solution to above equation is x(t ) x p (t ) xc (t ) The particular equation will be dependent on the forcing function as in case of first order differential equation. However, the complementary solution will be a solution to the second order homogenous differential equation: d 2 xc (t )
dx (t )
2 2 c 0 xc (t ) 0 . 2
We start by substituting xc (t ) Ke st , and get ( s 2 2 s 0 ) Ke st 0 . 2
As we are interested in Ke st 0 , s 2 2 s 0 0 , which is called the characteristics equation. The two roots of the characteristics equation are 2
s1 2 0
s2 2 0 The solution will then be xc (t ) K1e s1t K 2 e s2t . The constants K 1 , K 2 can be solved from the initial conditions of xc (0 ), xc (0 ). , its value decides the shape of the response. 0
Overdamped: 1 . Critically damped: 1 and under damped: 1 . The under damped The damping ratio is defined as 2
case will have oscilla...
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This document was uploaded on 01/20/2014.
- Winter '14