Transcript Chapter 8-Part2

If 02 <  2 s1 and s2 are real and distincts  overdamped
where
A1 and A 2 are determined by solving the following equations:
If  2 < 02 s1 and s2 are complex conjugate and distincts  underdamped
s1 =    j 02   2 =    j d
s 2 =    j 02   2 =    j d
where
B 1 and B 2 are determined by solving the following equations:
If  2  02 s1 and s2 are real and identical  critically damped
s1  s 2     1
2RC
D1 and D 2 are determined by solving the following equations:
02 <  2
s1 and s 2 are real and distincts
 2 < 02
s1 and s 2 are complex conjugate
and distincts
 overdamped
 underdamped
s1 =    j 02   2 =    j d
 2  02
s1 and s 2 are real and identical
 critically damped
s1  s 2     1
2RC
s 2 =    j 02   2 =    j d
A1 and A 2 are determined by
B 1 and B 2 are determined by
D1 and D 2 are determined by
follows
We will show first the indirect approach solution , then the direct approach
t
 i(t)=

1
v dτ + i(0)
L
0
KCL
Differentiating once we obtain
(Natural Response Solutions)
the equations into
and solve for i L as i L =  v  C dv
R
dt
where A1' , A 2' , B 1' , B 2' , D1' and D 2'
are arbitray constants
The primed constants
A1'
interms of A1
D 2' can be found
D2
(cumbersome or time-consuming)
The primed constants A1'
D 2' can be found indirectly interms of A1
D 2 using v(0) and
dv(0)
dt
di L (0)
dt
The solution i L (t) for the 2ed order differential equation with constant forcing function (step response)
or we can find A1'
D 2' directly using i L (0) and
i (t )  I f + function of the same form as the natural response
v (t )  V f + function of the same form as the natural response
where I f ,V f represent the the final value of the response function which can be zero as v
in the example above
24 mA
400 W
The initial energy stored in the RLC circuit above is zero
At t = 0 , a dc current source of 24 mA is applied to the circuit.
The value of R is 400 W
24 mA
400 W
24 mA
400 W
Because 02 <  2 s1 and s2 are real and distincts  overdamped