In the circuit shown in Fig. 8.1, a 20 mH inductor is shunted by a 500 nF capacitor, the

The resistance and inductance of the circuit in Fig. 8.5 are and 20 mH, respectively. a) Find the value of C that makes the voltage response critically damped. b) If C is adjusted to give a neper frequency of , find the value of C and the roots of the characteristic equation. c) If C is adjusted to give a resonant frequency of , find the value of C and the roots of the characteristic equation. 20 krad>s 5 krad>s 100 Answer: (a) (b) (c) s2 = -74,641 rad>s. s1 = -5359 rad>s, C = 125 nF, s2 = -5000 - j5000 rad>s; s1 = -5000 + j5000 rad>s, C = 1 mF, 500 nF;

The element values in the circuit shown are and The initial current in the inductor is and the initial voltage on the capacitor is 0 V. The output signal is the voltage . Find (a) (b) (c) (d) (e) and (f) when v(t) t 0.

A 10 mH inductor, a capacitor, and a variable resistor are connected in parallel in the circuit shown. The resistor is adjusted so that the roots of the characteristic equation are The initial voltage on the capacitor is 10 V, and the initial current in the inductor is 80 mA. Find a) R; b) c) and in the solution for ; and d) iL(t). B1 B2 v dv(0+)>dt; -8000 ; j6000 rad>s. 1 mF Answer: (a) (b) (c) (d) + (82>3)sin 6000t] mA when t 0. iL(t) = 10e-8000t [8 cos 6000t B B2 = -80>3 V; 1 = 10 V, -240,000 V>s; 62.5 ;

The resistor in the circuit in Assessment Problem 8.4 is adjusted for critical damping. The inductance and capacitance values are 0.4 H and respectively. The initial energy stored in the circuit is 25 mJ and is distributed equally between the inductor and capacitor. Find (a) R; (b) (c) (d) and in the solution for ; and (e) t 0+ v iR, . I0 D1 D2 V ; 0; 10 mF, Answer: (a) (b) 50 V; (c) 250 mA; (d) 50 V; (e) t 0+. iR(t) = (-500te-500t + 0.50e-500t ) A,

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 C, and R. b) Calculate and for t 0+ iR(t), iL(t), iC(t) .

The natural voltage response of the circuit in Fig. 8.1 is when the capacitor is 250 F. Find (a) L; (b) R; (c) (d) and (e)L(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 , L, and b) Find 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 for c) Find when d) What percentage of the initially stored energy remains stored in the circuit at the instant is 0?

The natural response for the circuit shown in Fig. 8.1 is known to be If in milliamperes.

The resistor in the circuit in Example 8.4 is changed to a) Find the numerical expression for when b) Plot versus t for the time interval Compare this response with the one in Example 8.4 ( ) and Example 8.5 ( ). In particular, compare peak values of and the times when these peak values occur.

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 to . Find for t 0.

The resistor in the circuit of Fig. P8.11 is decreased from to . Find for t 0.

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

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

The inductor in the circuit of Fig. P8.14 is increased to 125 mH. Find 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).

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, and a) Find for b) Find the first three values of t for which is zero. Let these values of t be denoted and c) Show that d) Show that e) Calculate and f) Sketch versus t for 0 t t2 v(t) .

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

Assume the underdamped voltage response of the circuit in Fig. 8.1 is written as The initial value of the inductor current is and the initial value of the capacitor voltage is Show that is the conjugate of (Hint: Use the same process as outlined in the text to find and )

Show that the results obtained from Problem 8.21 that is, the expressions for and 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, is 45 mA. The expression for the capacitor current is known to be when R is Find a) the values of L, C, and A1 A2

For the circuit in Example 8.6, find, for (a) (b) and (c) iC(t).

For the circuit in Example 8.7, find, for (a) and (b) C(t).

For the circuit in Example 8.8, find 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 for if R equals 12.5 .

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 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 0.

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

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

There is no energy stored in the circuit in Fig. P8.33 when the switch is closed at Find t 0.

a) For the circuit in Fig. P8.33, find for b) Show that your solution for 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 vC (t) t 0.

Use the circuit in Fig. P8.35 a) Find the total energy delivered to the inductor. b) Find the total energy delivered to the 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 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 t 0.

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 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 . 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 and a) Find the numerical values of R and L. b) Find the numerical values of and immediately after the switch has been closed. c) Find for 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 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 immediately after the switch is closed. c) Find 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 the two switches move to the new positions shown in the figure. Find

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 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 t 0+ v . o

The 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 ? b) What is the initial value of ? c) What is the numerical expression for for ?

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

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

The resistor in the circuit shown in Fig. P8.49 is changed to . The initial energy stored is still zero. Find for

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.

The circuit shown in Fig. P8.53 has been in operation for a long time.At the source voltage suddenly drops to 150 V. Find for Figure P8.53 25  250 mH vo 200 V 2.

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 for t 0. c

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

The circuit parameters in the circuit of Fig. P8.55 are and a) Express numerically for 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 when where n = 0, 1, 2,

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 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 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 jumps in value, then equals zero at t = 0.

a) Find the equation for for in the circuit shown in Fig. 8.19 if and b) How long does the circuit take to reach saturation?

a) Rework Example 8.14 with feedback resistors and removed. b) Rework Example 8.14 with 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 in Fig. 8.18. c) What is the advantage of the circuit shown in Fig. P8.62? Figure P8.62 VCC VCC

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 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 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 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 voltage equal to is available and by successive integrations generates and x. We can synthesize the coefficients in the equations by scaling amplifiers, and we can combine the terms required to generate 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 Also discuss the parameters R; and 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 . b) The dc voltage source and series-connected resistor in Fig. 8.21 are used to establish the initial energy in the inductor. If and , calculate the initial energy stored in the inductor. c) What is the total energy stored in the LC circuit for any time

Consider the LC oscillator circuit in Fig. 8.21. Assume that , , and L = 1 nH. a) Calculate the value of capacitance, C, that will produce a sinusoidal output with a frequency of 2 GHz for . b) Write the expression for the output voltage, vo(t), 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 V 10 V, R 25 , L 5 nH, and C 2 pF, just as in Problem 8.66. Suppose the resistance associated with the inductor and capacitor is 10 m . a) Calculate the values of the neper frequency, and the resonant radian frequency, . v0 a, = = = = b) Is the response of this circuit over-, under-, or critically damped? c) What is the actual frequency of oscillation, in GHz? d) Approximately how long will the circuit oscillate?

The sinusoidal voltage source in the circuit shown in Fig. P9.69 is generating the voltage If the op amp is ideal, what is the steady-state expression for ? vo(t)

The capacitor in the circuit seen in Fig. P9.69 is replaced with a variable capacitor. The capacitor is adjusted until the output voltage leads the input voltage by a) Find the value of C in microfarads. b) Write the steady-state expression for when C has the value found in (a).

The op amp in the circuit in Fig. P9.71 is ideal. a) Find the steady-state expression for b) How large can the amplitude of be before the amplifier saturates?

The op amp in the circuit seen in Fig. P9.72 is ideal. Find the steady-state expression for when vg = 2 cos 106 t V.

The operational amplifier in the circuit shown in Fig. P9.73 is ideal. The voltage of the ideal sinusoidal source is a) How small can be before the steady-state output voltage no longer has a pure sinusoidal waveform? b) For the value of found in (a), write the steady-state expression for vo.

The value of k in the circuit in Fig. P9.74 is adjusted so that is purely resistive when Find Zab.

For the circuit in Fig. P9.75, find the Thvenin equivalent with respect to the terminals c,d.

a) Find the steady-state expressions for the currents and in the circuit in Fig. P9.76 when b) Find the coefficient of coupling. c) Find the energy stored in the magnetically coupled coils at and t = 1250p ms

The sinusoidal voltage source in the circuit seen in Fig. P9.77 is operating at a frequency of 200 krad/s. The coefficient of coupling is adjusted until the peak amplitude of is maximum. a) What is the value of k? b) What is the peak amplitude of if g = 560 cos(2 * 105 t) V ?

A series combination of a resistor and a inductor is connected to a sinusoidal voltage source by a linear transformer. The source is operating at a frequency of 400 . At this frequency, the internal impedance of the source is The rms voltage at the terminals of the source is 75 V when it is not loaded. The parameters of the linear transformer are and a) What is the value of the impedance reflected into the primary? b) What is the value of the impedance seen from the terminals of the practical source?

At first glance, it may appear from Eq. 9.69 that an inductive load could make the reactance seen looking into the primary terminals (i.e., ) look capacitive. Intuitively, we know this is impossible. Show that can never be negative if is an inductive reactance.

a) Show that the impedance seen looking into the terminals a,b in the circuit in Fig. P9.80 is given by the expression b) Show that if the polarity terminals of either one of the coils is reversed,

a) Show that the impedance seen looking into the terminals a,b in the circuit in Fig. P9.81 is given by the expression b) Show that if the polarity terminal of either one of the coils is reversed that

Find the impedance in the circuit in Fig. P9.82 if ZL = 200 l -45 .

Show by using a phasor diagram what happens to the magnitude and phase angle of the voltage in the circuit in Fig. P9.83 as is varied from zero to infinity. The amplitude and phase angle of the source voltage are held constant as varies.

The parameters in the circuit shown in Fig. 9.53 are and a) Calculate the phasor voltage b) Connect a capacitor in parallel with the inductor, hold constant, and adjust the capacitor until the magnitude of I is a minimum. What is the capacitive reactance? What is the value of ? c) Find the value of the capacitive reactance that keeps the magnitude of I as small as possible and that at the same time makes

a) For the circuit shown in Fig. P9.85, compute and b) Construct a phasor diagram showing the relationship between and the load voltage of c) Repeat parts (a) and (b), given that the load voltage remains constant at when a capacitive reactance of is connected across the load terminals.

You may have the opportunity as an engineering graduate to serve as an expert witness in lawsuits involving either personal injury or property damage. As an example of the type of problem on which you may be asked to give an opinion, consider the following event. At the end of a day of fieldwork, a farmer returns to his farmstead, checks his hog confinement building, and finds to his dismay that the hogs are dead. The problem is traced to a blown fuse that caused a 240 V fan motor to stop. The loss of ventilation led to the suffocation of the livestock. The interrupted fuse is located in the main switch that connects the farmstead to the electrical service. Before the insurance company settles the claim, it wants to know if the electric circuit supplying the farmstead functioned properly. The lawyers for the insurance company are puzzled because the farmers wife, who was in the house on the day of the accident convalescing from minor surgery, was able to watch TV during the afternoon. Furthermore, when she went to the kitchen to start preparing the evening meal, the electric clock indicated the correct time.The lawyers have hired you to explain (1) why the electric clock in the kitchen and the television set in the living room continued to operate after the fuse in the main switch blew and (2) why the second fuse in the main switch didnt blow after the fan motor stalled.After ascertaining the loads on the three-wire distribution circuit prior to the interruption of fuse A, you are able to construct the circuit model shown in Fig. P9.86. The impedances of the line conductors and the neutral conductor are assumed negligible. a) Calculate the branch currents and prior to the interruption of fuse A. b) Calculate the branch currents after the interruption of fuse A. Assume the stalled fan motor behaves as a short circuit. c) Explain why the clock and television set were not affected by the momentary short circuit that interrupted fuse A. d) Assume the fan motor is equipped with a thermal cutout designed to interrupt the motor circuit if the motor current becomes excessive. Would you expect the thermal cutout to operate? Explain. e) Explain why fuse B is not interrupted when the fan motor stalls.

a) Calculate the branch currents in the circuit in Fig. 9.58. b) Find the primary current Ip.

Suppose the resistance in the distribution circuit in Fig. 9.58 is replaced by a resistance. a) Recalculate the branch current in the resistor, b) Recalculate the primary current, c) On the basis of your answers, is it desirable to have the resistance of the two 120 V loads be equal?

A residential wiring circuit is shown in Fig. P9.89. In this model, the resistor is used to model a 250 V appliance (such as an electric range), and the resistors and are used to model 125 V appliances (such as a lamp, toaster, and iron). The branches carrying and are modeling what electricians refer to as the hot conductors in the circuit, and the branch carrying is modeling the neutral conductor. Our purpose in analyzing the circuit is to show the importance of the neutral conductor in the satisfactory operation of the circuit. You are to choose the method for analyzing the circuit. a) Show that is zero if b) Show that if c) Open the neutral branch and calculate and if and d) Close the neutral branch and repeat (c). e) On the basis of your calculations, explain why the neutral conductor is never fused in such a manner that it could open while the hot conductors are energized.

a) Find the primary current for (c) and (d) in Problem 9.89. b) Do your answers make sense in terms of known circuit behavior?