Foundations of Analog and Digital Circuits Mas - Wordpress ...

Energy
0.5 L i L 2 (0)
0 | | | | | | |
π/2ωd π/ωd 3π/2ωd 2π/ωd 5π/2ωd 3π/ωd t
|
w M
w T α e -2αt
w E
12.3 Stored Energy in Transient, Series RLC Circuit CHAPTER TWELVE 653
factor term,
1
2 Cv 2 1
C (0) +
2 Li 2 L (0)
,
is the initial stored energy (wT(0)) in the system. The second factor represents
the decay of energy with time. Finally, by rewriting the third factor as
cos 2
⎛
⎝ωdt + tan −1
⎛
⎞⎞
⎝
L iL(0)
⎠⎠
=
C vC(0)
v C
|
|
1 + cos 2 ωdt + tan −1 L
2
e -αt
| | | | | |
π/2ωd π/ωd 3π/2ωd 2π/ωd 5π/2ωd 3π/ωd t
iL(0)
C vC(0)
we can see that the energy is sloshing back and forth between the capacitor and
inductor, twice per cycle of the transient ring.
Through a comparison with the results of Section 12.1, we also see that
for large Q, that is, for a relatively short-circuited R and hence light damping,
the introduction of a resistor into the circuit causes an exponential decay of the
states and the stored energy. A sketch of the energy versus time is shown in
Figure 12.23 for the case
vC(0) = 0.
The length of time for the stored energy to dissipate can now be readily calculated.
Obviously the controlling function in Equation 12.77 is the exponential
term e−2αt . This can be rewritten using Equation 12.65 as
Since ω d ωo for large Q, then in Q cycles
decay = e −ωot/Q . (12.78)
ωot ω dt = 2πQ.
FIGURE 12.23 Energy in RLC
transient.