In the circuit in Fig. 8.1, R = 5 k, L = 8 H, C = 125 nF,

The circuit elements in the circuit in Fig. 8.1 are R = 125 , L = 200 mH and C = C = 5 F The initial inductor current is and the initial capacitor voltage is 25 V. a) Calculate the initial current in each branch of the circuit. (b) Find v (t) for t 0. (c) Find il (t) for t 0.

The resistance in Problem 8.1 is decreased to 100 . Find the expression for for .

The resistance in Problem 8.1 is decreased to 80 Find the expression for for v(t) t 0

The resistance, inductance, and capacitance in a parallel RLC circuit are 2000 250 mH, and 10 nF respectively. a) Calculate the roots of the characteristic equation that describe the voltage response of the circuit. b) Will the response be over-, under-, or critically damped? c) What value of R will yield a damped frequency of ? d) What are the roots of the characteristic equation for the value of R found in (c)? e) What value of R will result in a critically damped response

Suppose the inductor in the circuit shown in Fig. 8.1 has a value of . The voltage response for is a) Determine the numerical values of wo a, C, and R. b) Calculate iR(t), iL(t), iC(t) and for t 0+

The natural voltage response of the circuit in Fig. 8.1 is when the capacitor is 250 F. Find (a) L; (b) R; (c) Vo; (d) Io and (e) iL(t).

The voltage response for the circuit in Fig. 8.1 is known to be The initial current in the inductor ( ) is mA, and the initial voltage on the capacitor ( ) is 5 V. The resistor has a value of . a) Find the values of C,L D1 and D2. b) Find ict for t 0+.

In the circuit shown in Fig. 8.1, a 20 mH inductor is shunted by a 500 nF capacitor, the resistor R is adjusted for critical damping, and I0 = 120 mA.a) Calculate the numerical value of R. b) Calculate v (t) for t 0 c) Find v (t) when iC(t) = 0. d) What percentage of the initially stored energy remains stored in the circuit at the instant iC(t) is 0?

The natural response for the circuit shown in Fig. 8.1 is known to be v(t) = -11e-100t + 20e-400t V, t 0. If C = 2 mF and L = 12.5 H find iL(0+) in milliamperes.

The resistor in the circuit in Example 8.4 is changed to 3200 . a) Find the numerical expression for v(t) when t 0 b) Plot v(t) versus t for the time interval 0 t 7 ms. Compare this response with the one in Example 8.4 (R = 20 k ) and Example 8.5 ( R = 4 k). In particular, compare peak values of v(t) and the times when these peak values

The two switches in the circuit seen in Fig. P8.11 operate synchronously. When switch 1 is in position a, switch 2 is in position d.When switch 1 moves to position b, switch 2 moves to position c. Switch 1 has been in position a for a long time. At the switches move to their alternate positions. Find for v t 0 .

The resistor in the circuit of Fig. P8.11 is decreased from 50 to 40 Find v(t) for t 0.

The resistor in the circuit of Fig. P8.11 is decreased from 50 32 to . Find v(t) for t 0.

The switch in the circuit of Fig. P8.14 has been in position a for a long time.At t = 0 the switch moves instantaneously to position b. Find for v(t) for t 0.

The inductor in the circuit of Fig. P8.14 is increased to 80 mH. Find v (t) for t 0

The inductor in the circuit of Fig. P8.14 is increased to 125 mH. Find v (t) for t 0

a) Design a parallel RLC circuit (see Fig. 8.1) using component values from Appendix H, with a resonant radian frequency of 5000 . Choose a resistor or create a resistor network so that the response is critically damped. Draw your circuit. b) Calculate the roots of the characteristic equation for the resistance in part (

a) Change the resistance for the circuit you designed in Problem 8.5(a) so that the response is underdamped. Continue to use components from Appendix H. Calculate the roots of the characteristic equation for this new resistance. b) Change the resistance for the circuit you designed in Problem 8.5(a) so that the response is overdamped. Continue to use components from Appendix H. Calculate the roots of the characteristic equation for this new resistance.

In the circuit in Fig. 8.1, R = 5 k, L = 8 H, C = 125 nF, Vo = 30V and Io = 6mA a) Find v (t) for t 0 b) Find the first three values of t for which dv/dt is zero. Let these values of t be denoted t1,t2 and t3 c) Show that t3 - t1 = Td d) Show that e) Calculate t2 - t1 = Td/2 and f) Sketch v(t) versus t for 0 t t.

a) Find v(t) for t 0 in the circuit in Problem 8.19 if the 5 k resistor is removed from the circuit. b) Calculate the frequency of v(t) in hertz. c) Calculate the maximum amplitude of v(t) in volts.

Assume the underdamped voltage response of the circuit in Fig. 8.1 is written as v(t) = (A1 + A2)e-at cos vdt + j(A1 - A2)e-at sin vdtThe initial value of the inductor current is I0 and the initial value of the capacitor voltage v0 is Show that A2 is the conjugate of (Hint: Use the same process as outlined in the text to find A1 and A2)

Show that the results obtained from Problem 8.21 that is, the expressions for A1 A2 are consistent with Eqs. 8.30 and 8.31 in the text

The initial value of the voltage in the circuit in Fig. 8.1 is zero, and the initial value of the capacitor current, ic(0+) is 45 mA. The expression for the capacitor current is known to be ic(t) = A1e-200t + A2e-800t , t 0+ when R is 250 Find a) the values of L, C, and A1 A2 nt: diC(0+) dt = - diL(0+) dt - diR(0+) dt = -v(0) L - 1 R iC(0+) C b) the expression for v(t), t 0 c) the expression for iR(t) 0 d) the expression for iL(t) 0.

For the circuit in Example 8.6, find, for t 0 (a) v(t); (b) iR(t); and (c) iC(t)

For the circuit in Example 8.7, find for t 0 (a) v(t) and (b) ic(t

For the circuit in Example 8.8, find v(t) for t 0.

Assume that at the instant the 2A dc current source is applied to the circuit in Fig. P8.27, the initial current in the 25 mH inductor is and the initial voltage on the capacitor is 50 V (positive at the upper terminal). Find the expression for iL(t) for t 0 if R equals 125 Figure P8.27

The resistance in the circuit in Fig. P8.27 is changed to Find for t 0 .

The resistance in the circuit in Fig. P8.27 is changed to Find iL(t) for t 0 .

The switch in the circuit in Fig. P8.30 has been open a long time before closing at At the time the switch closes, the capacitor has no stored energy. Find for v (t) for t 0 .

The switch in the circuit in Fig. P8.31 has been open for a long time before closing at Find io (t) for t 0.

a) For the circuit in Fig. P8.31, find v for t 0 b) Show that your solution for v is consistent with the solution for i in Problem 8.31.

There is no energy stored in the circuit in Fig. P8.33 when the switch is closed at t = 0. Find i(t) for t 0.

a) For the circuit in Fig. P8.33, find v for t 0 b) Show that your solution for io is consistent with the solution for in Problem 8.33.

The switch in the circuit in Fig. P8.35 has been in the left position for a long time before moving to the right position at . Find a) for , b) for .

Use the circuit in Fig. P8.35 a) Find the total energy delivered to the inductor. b) Find the total energy delivered to the 40 resistor. c) Find the total energy delivered to the capacitor. d) Find the total energy delivered by the current source. e) Check the results of parts (a) through (d) against the conservation of energy principle.

The switch in the circuit in Fig. P8.37 has been open a long time before closing at Find iL(t) for t 0.

Switches 1 and 2 in the circuit in Fig. P8.38 are synchronized. When switch 1 is opened, switch 2 closes and vice versa. Switch 1 has been open a long time before closing at Find for Section 8.4 8.39 The current in the circuit in Fig. 8.3 is known to be The capacitor has a value of 80 nF; the initial value of the current is 7.5 mA; and the initial voltage on the capacitor is 30 V. Find the values of R, L, and 8.40 Find the voltage across the 80 nF capacitor for the circuit described in Problem 8.39.Assume the reference polarity for the capacitor voltage is positive at the upper terminal. 8.41 The initial energy stored in the 31.25 nF capacitor in the circuit in Fig. P8.41 is . The initial energy stored in the inductor is zero. The roots of the characteristic equation that describes the natural 9 mJ B2. B1, i = B1e-2000t cos 1500t + B2e-2000tsin 1500t, t 0. t = 0. iL(t) for t 0.

The current in the circuit in Fig. 8.3 is known to be i = B1e-2000t cos 1500t + B2e-2000tsin 1500t, t 0. The capacitor has a value of 80 nF; the initial value of the current is 7.5 mA; and the initial voltage on the capacitor is 30 V. Find the values of R, L B1 and B2.

Find the voltage across the 80 nF capacitor for the circuit described in Problem 8.39.Assume the reference polarity for the capacitor voltage is positive at the upper terminal.

The initial energy stored in the 31.25 nF capacitor in the circuit in Fig. P8.41 is 9 J The initial energy stored in the inductor is zero. The roots of the characteristic equation that describes the natural behavior of the current i are -4000 s-1 and 16,000 s-1 a) Find the numerical values of R and L. b) Find the numerical values of i(0) and di(0)/dt immediately after the switch has been closed. c) Find i(t) for t 0, d) How many microseconds after the switch closes does the current reach its maximum value? e) What is the maximum value of i in milliamperes? f) Find vL(t) for t 0.

In the circuit in Fig. P8.42, the resistor is adjusted for critical damping. The initial capacitor voltage is 15 V, and the initial inductor current is 6 mA. a) Find the numerical value of R. b) Find the numerical values of i and di/dt immediately after the switch is closed. c) Find vC(t) for t 0.

a) Design a series RLC circuit (see Fig. 8.3) using component values from Appendix H, with a resonant radian frequency of 20 . Choose a resistor or create a resistor network so that the response is critically damped. Draw your circuit. b) Calculate the roots of the characteristic equation for the resistance in part (a).

a) Change the resistance for the circuit you designed in Problem 8.43(a) so that the response is underdamped. Continue to use components from Appendix H. Calculate the roots of the characteristic equation for this new resistance. b) Change the resistance for the circuit you designed in Problem 8.43(a) so that the response is overdamped. Continue to use components from Appendix H. Calculate the roots of the characteristic equation for this new resistance.

The circuit shown in Fig. P8.45 has been in operation for a long time. At t = 0 the two switches move to the new positions shown in the figure. Find a) i (t) for for t 0 b) (t) for t 0

The switch in the circuit shown in Fig. P8.46 has been in position a for a long time. At the switch is moved instantaneously to position b. Find i(t) for t 0.

The switch in the circuit shown in Fig. P8.47 has been closed for a long time. The switch opens at Find for .

he switch in the circuit in Fig. P8.48 has been in position a for a long time.At the switch moves instantaneously to position b. a) What is the initial value of va? b) What is the initial value of d va/dt ? c) What is the numerical expression for for va (t) for t 0?

The initial energy stored in the circuit in Fig. P8.49 is zero. Find for v (t) for t 0

The resistor in the circuit shown in Fig. P8.49 is changed to 250 . The initial energy stored is still zero. Find v (t) for t 0

The resistor in the circuit shown in Fig. P8.49 is changed to 312.5 . The initial energy stored is still zero. Find v (t) for t 0

The switch in the circuit of Fig. P8.52 has been in position a for a long time. At the switch moves instantaneously to position b. Find for .

The circuit shown in Fig. P8.53 has been in operation for a long time.At t = 0 the source voltage suddenly drops to 150 V. Find v (t) for t 0.

The two switches in the circuit seen in Fig. P8.55 operate synchronously. When switch 1 is in position a, switch 2 is closed. When switch 1 is in position b, switch 2 is open. Switch 1 has been in position a for a long time.At it moves instantaneously to position b. Find vc (t) for t 0

The switch in the circuit shown in Fig. P8.55 has been closed for a long time before it is opened at t = 0. Assume that the circuit parameters are such that the response is underdamped. a) Derive the expression for as a function of V a, vd,C, and R for t 0 b) Derive the expression for the value of t when the magnitude of v is maximum

The circuit parameters in the circuit of Fig. P8.55 are R = 480 , L = 8 mH, C = 50 nF and vg = -24 V a) Express v (t) numerically for t 0 b) How many microseconds after the switch opens is the inductor voltage maximum? c) What is the maximum value of the inductor voltage? d) Repeat (a)(c) with R reduced to 96 .

Assume that the capacitor voltage in the circuit of Fig. 8.15 is underdamped. Also assume that no energy is stored in the circuit elements when the switch is closed. a) Show that b) Show that dvC>dt = (v2 0>vd)Ve-atsin vdt. when where n = 0, 1, 2, . . . .c) Let and show that d) Show that where = 1 Td ln vC(t1) - vC(t3) - V

The voltage across a capacitor in the circuit of Fig. 8.15 is described as follows: After the switch has been closed for several seconds, the voltage is constant at 100 V. The first time the voltage exceeds 100 V, it reaches a peak of 163.84 V. This occurs /7 ms after the switch has been closed.The second time the voltage exceeds 100 V, it reaches a peak of 126.02 V. This second peak occurs 3/7 after the switch has been closed. At the time when the switch is closed, there is no energy stored in either the capacitor or the inductor. Find the numerical values of R and L. (Hint: Work Problem 8.57 first.)

Show that, if no energy is stored in the circuit shown in Fig. 8.19 at the instant vg jumps in value dv/dt, then equals zero at t = 0.

a) Find the equation for vo (t) for 0 t tin sat the circuit shown in Fig. 8.19 if vo1(0) and vo(0) = 8 V. b) How long does the circuit take to reach saturation

a) Rework Example 8.14 with feedback resistors R1 and R2 removed. b) Rework Example 8.14 with vo1(0) = -2 V and vo(0) = 4 V.

a) Derive the differential equation that relates the output voltage to the input voltage for the circuit shown in Fig. P8.62. b) Compare the result with Eq. 8.75 when R1C1 = R2C2 = RC in Fig. 8.18. c) What is the advantage of the circuit shown in Fig. P8.62?

The voltage signal of Fig. P8.63(a) is applied to the cascaded integrating amplifiers shown in Fig. P8.63(b). There is no energy stored in the capacitors at the instant the signal is applied. a) Derive the numerical expressions for and for the time intervals and 0.5 s t tsat. b) Compute the value of tsat.

The circuit in Fig. P8.63(b) is modified by adding a resistor in parallel with the capacitor and a 5 M resistor in parallel with the capacitor. As in Problem 8.63, there is no energy stored in the capacitors at the time the signal is applied. Derive the numerical expressions for and for the time intervals 0 t 0.5 s and t 0.5 s.

We now wish to illustrate how several op amp circuits can be interconnected to solve a differential equation. a) Derive the differential equation for the springmass system shown in Fig. P8.65(a). Assume that the force exerted by the spring is directly proportional to the spring displacement, that the mass is constant, and that the frictional force is directly proportional to the velocity of the moving mass. b) Rewrite the differential equation derived in (a) so that the highest order derivative is expressed as a function of all the other terms in the equation. Now assume that a voltag equal to d2 x>dt2 is available and by successive integrations generates dx/dt and x. We can synthesize the coefficients in the equations by scaling amplifiers, and we can combine the terms required to generate d2 x>dt2t by using a summing amplifier. With these ideas in mind, analyze the interconnection shown in Fig. P8.65(b). In particular, describe the purpose of each shaded area in the circuit and describe the signal at the points labeled B, C, D, E, and F, assuming the signal at A represents d2 x/dt2 Also discuss the parameters R; R1, C1; C2;R3 R4 ; R5 , R6 and R7;R8, in terms of the coefficients in the differential equation.

a) Suppose the circuit in Fig. 8.21 has a 5 nH inductor and a 2 pF capacitor. Calculate the frequency, in GHz, of the sinusoidal output for t 0 b) The dc voltage source and series-connected resistor in Fig. 8.21 are used to establish the initial energy in the inductor. If V = 10 V and R = 25 , calculate the initial energy stored in the inductor. c) What is the total energy stored in the LC circuit for any time t 0 ?

Consider the LC oscillator circuit in Fig. 8.21. Assume that , and . a) Calculate the value of capacitance , that will produce a sinusoidal output with a frequency of for . b) Write the expression for the output voltage for .

Suppose the inductor and capacitor in the LC oscillator circuit in Fig. 8.21 are not ideal, but instead have some small resistance that can be lumped together. Assume that , , , and , just as in Problem 8.66. Suppose the resistance associated with the inductor and capacitor is . 1. Calculate the values of the neper frequency , and the resonant radian frequency, . 2. Is the response of this circuit over-, under-, or critically damped? 3. What is the actual frequency of oscillation, in ? 4. Approximately how long will the circuit oscillate?