Page 32 - Circuit Analysis II with MATLAB Applications

Chapter 1 Second Order Circuits

e 12t = 348
---------
97
or

12t = ln § 348 · 1.2775
--------- =
© 97 ¹
and

t = t max = 1.2775 0.106 s
---------------- =
12
By substitution into (1.55)
1
v max = -- 291e – 4x0.106 – 261e – 16x0.106 = 23.76 V (1.57)
-
6
A useful quantity, especially in electronic circuit analysis, is the settling time, denoted as , and it is
t
S
t
defined as the time required for the voltage to drop to 1% of its maximum value. Therefore, is an
S
indication of the time it takes for vt to damp-out, meaning to decrease the amplitude of vt to
t
approximately zero. For this example, 0.01 u 23.76 = 0.2376 V , and we can find by substitution
S
into (1.55). Then,
1 – 4t – 16t
0.01v max = 0.2376 = --- 291e – 261e (1.58)

6
and we need to solve for the time . To simplify the computation, we neglect the second term inside
t
the parentheses of (1.58) since this component of the voltage damps out much faster than the other
component. This expression then simplifies to

1 – 4t
0.2376 = -- 291e s
-
6
or
– 4t = ln 0.005 = – 5.32

S
or
t = 1.33 s (1.59)
S

Example 1.4

For the circuit of Figure 1.13, i 0 = 2A and v 0 = 5V , and the resistor is to be adjusted so
C
L
that the natural response will be critically damped. Compute and sketch vt for t ! 0 .

1-20 Circuit Analysis II with MATLAB Applications
Orchard Publications