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15 1 05 104 volts 0 05 1 15 0 0002 0006 001 0014 Time (s) 0018 002
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Figure 554 Secondary ignition voltage response Focus on Computer-Aided Tools: The Matlab m- le containing the numerical analysis and plotting commands for this example may be found in the CD that accompanies this book An EWB simulation is also enclosed
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59 Derive the differential equation for the series circuit of Figure 539(b) Show that one can write the equation either as RT dvC 1 1 d 2 vC + + vC = vT (t) dt 2 L dt LC LC or as 1 RT diL 1 dvT (t) d 2 iL + iL = + 2 dt L dt LC L dt 510 Determine the roots of the characteristic equation of the series RLC circuit of Figure 539(b) with R = 100 , C = 10 F, and L = 1 H 511 For the series RLC circuit of Figure 539(b), with L = 1 H and C = 10 F, nd the ranges of values of R for which the circuit response is overdamped and underdamped, respectively
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CYU 51 CYU 52 CYU 53 CYU 54 CYU 55 CYU 56 CYU 57 CYU 58 CYU 59 CYU 510 1 dvC 1 + vC = vS dt RC RC 1 1 dv + v = iS dt RC C R diL R + iL = iS dt L L vC (0 ) = 8 V and iR (0+ ) = 2 A vC = 75 75e t/0025 V, 0 t < 005 s; vC = 6485e (t 005)/00375 V, t 005 s t90% = 125 s 545 s The output pulse has a higher peak 50 j 31225 Overdamped: R > 63246 ; underdamped: R < 63246
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HOMEWORK PROBLEMS
Section 1: First-Order Transients 51 Just before the switch is opened at t = 0, the current
through the inductor is 170 mA in the direction shown in Figure P51 Did steady-state conditions exist just before the switch was opened L = 09 mH R1 = 6 k R3 = 3 k VS = 12 V R2 = 6 k
Figure P53
C R2 t=0 R1 + + _ V1
54 Determine the current through the capacitor just
before and just after the switch is closed in Figure P53 Assume steady-state conditions for t < 0
t=0 +
+ V _ S _
R2 iL R1 L R3 + VR3 _
V1 = 12 V R1 = 400 m
C = 150 F R2 = 22 k
55 Just before the switch is opened at t = 0 in Figure
P51, the current through the inductor is 170 mA in the direction shown Determine the voltage across R3 just after the switch is opened VS = 12 V R1 = 6 k R3 = 3 k L = 09 mH R2 = 6 k
Figure P51
52 At t < 0, the circuit shown in Figure P52 is at
VS1 = 35 V C = 11 F R2 = 7 k VS2 = 130 V R1 = 17 k R3 = 23 k
steady state The switch is changed as shown at t = 0
56 Determine the voltage across the inductor just before
and just after the switch is changed in Figure P56 Assume steady-state conditions exist for t < 0 VS = 12 V R1 = 22 k Rs = 07 L = 100 mH
Determine at t = 0+ the initial current through R3 just after the switch is changed
C + + VS 1 _
t=0 R3 R2 + + V _ S2
t=0 Rs + + _ Vs R1 L
Figure P56
Figure P52
57 Steady-state conditions exist in the circuit shown in
Figure P57 at t < 0 The switch is closed at t = 0 V1 = 12 V R2 = 22 k C = 047 F R1 = 068 k R3 = 18 k
53 Determine the current through the capacitor just
before and just after the switch is closed in Figure P53 Assume steady-state conditions for t < 0 V1 = 12 V R1 = 068 k C = 05 F R2 = 18 k
Determine the current through the capacitor at t = 0+ , just after the switch is closed
5
Transient Analysis
512 Determine vC (t) for t > 0 The voltage across the
ic(t) R2 C R3 t=0 R1 + + V 1 _
capacitor in Figure P512 just before the switch is changed is given below vC (0 ) = 7 V R1 = 7 k
Io = 17 mA R2 = 33 k
C = 055 F
Figure P57
R2 Io R1 C + VC(t)
58 At t > 0, the circuit shown in Figure P52 is at
VS1 = 35 V C = 11 F R2 = 7 k VS2 = 130 V R1 = 17 k R3 = 23 k
steady state The switch is changed as shown at t = 0
Figure P512
Determine the time constant of the circuit for t > 0
513 Determine iR3 (t) for t > 0 in Figure P59
VS1 23 V L = 23 mH R2 = 13 VS2 = 2o V R1 = 07 R3 = 330 k
59 At t < 0, the circuit shown in Figure P59 is at
VS1 = 13 V L = 170 mH R2 = 43 k VS2 = 13 V R1 = 27 R3 = 29 k
steady state The switch is changed as shown at t = 0
514 Assume DC steady-state conditions exist in the
circuit shown in Figure P514 for t < 0 The switch is changed at t = 0 as shown VS1 = 17 V R1 = 14 k R3 = 14 k VS2 = 11 V R2 = 13 k C = 70 nF
Determine the time constant of the circuit for t > 0
t=0 R1 + + V _ S1 L R2
+ _ VS 2 +
Determine: a v(t) for t > 0 b The time required, after the switch is operated, for V (t) to change by 98 percent of its total change in voltage
Figure P59
t=0 + + _ VS 1 R1 C + V _ S2 + R2 + V(t)
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