At first glance, it may appear from Eq. 9.69 that an

a) Verify that Eq. 9.9 is the solution of Eq. 9.8. This can be done by substituting Eq. 9.9 into the lefthand side of Eq. 9.8 and then noting that it equals the right-hand side for all values of t > 0 At Eq. 9.9 should reduce to the initial value of the current

Use the concept of the phasor to combine the following sinusoidal functions into a single trigonometric expression: a) y = 30 cos(200t - 160) + 15 cos(200t + 70), b) y = 90 sin(50t - 20) + 60 cos(200t - 70),c)y = 50 cos(5000t - 60) + 25 sin(5000t + 110) - 75 cos(5000t - 30) d) y = 10 cos (vt + 30) + 10 sin vt + 10 cos(vt + 150).

A 400 Hz sinusoidal voltage with a maximum amplitude of 100 V at is applied across the terminals of an inductor. The maximum amplitude of the steady-state current in the inductor is 20 A. a) What is the frequency of the inductor current? b) If the phase angle of the voltage is zero, what is the phase angle of the current? c) What is the inductive reactance of the inductor? d) What is the inductance of the inductor in millihenrys? e) What is the impedance of the inductor?

A 80 kHz sinusoidal voltage has zero phase angle and a maximum amplitude of 25 mV. When this voltage is applied across the terminals of a capacitor, the resulting steady-state current has a maximum amplitude of 628.32 A.a) What is the frequency of the current in radians per second? b) What is the phase angle of the current? c) What is the capacitive reactance of the capacitor? d) What is the capacitance of the capacitor in microfarads? e) What is the impedance of the capacitor?

The expressions for the steady-state voltage and current at the terminals of the circuit seen in Fig. P9.14 are vg = 300 cos(5000pt + 78) V ig = 6 sin (5000pt + 123) A, a) What is the impedance seen by the source? b) By how many microseconds is the current out of phase with the voltage?

A 20 resistor, a 50 mH inductor, and a 32 F capacitor are connected in series.The series-connected elements are energized by a sinusoidal voltage source whose voltage is 25 cos(500t - 60)V. a) Draw the frequency-domain equivalent circuit. b) Reference the current in the direction of the voltage rise across the source, and find the phasor current. c) Find the steady-state expression for i(t).

A resistor and a inductor are connected in parallel. This parallel combination is also in parallel with the series combination of a resistor and a capacitor. These three parallel branches are driven by a sinusoidal current source whose current is 125 sin(2500t + 60) A. a) Draw the frequency-domain equivalent circuit. b) Reference the voltage across the current source as a rise in the direction of the source current, and find the phasor voltage. c) Find the steady-state expression for

Three branches having impedances of 3 + j 4 ,16 - j12 and -j respectively, are connected in parallel. What are the equivalent (a) admittance, (b) conductance, and (c) susceptance of the parallel connection in millisiemens? (d) If the parallel branches are excited from a sinusoidal current source where i = 8 cos vt what is the maximum amplitude of the current in the purely capacitive branch?

a) Show that, at a given frequency the circuits in Fig. P9.18(a) and (b) will have the same impedance between the terminals a,b if R1 = v2 L2 2R2 R2 2 + v2 L2 2, L1 = R2 2L2 R2 2 + v2 L2 2 .b) Find the values of resistance and inductance that when connected in series will have the same impedance at 4 krad/s as that of a resistor connected in parallel with a 1.25 H inductor.

a) Show that at a given frequency the circuits in Fig. P9.19(a) and (b) will have the same impedance between the terminals a,b if R2 = R2 1 + v2 L2 1 R1 , L2 = R2 1 + v2 L2 1 v2 L1 .(Hint: The two circuits will have the same impedance if they have the same admittance.) b) Find the values of resistance and inductance that when connected in parallel will have the same impedance at 1 krad/s as an 8 resistor connected in series with a 4 H inductor

a) Show that at a given frequency the circuits in Fig. P9.20(a) and (b) will have the same impedance between the terminals a,b if R1 = R2 1 + v2 R2 2C2 2,C1 = 1 + v2 R2 2C2 2 v2 R2 2C2 .b) Find the values of resistance and capacitance that when connected in series will have the same impedance at as that of a resistor connected in parallel with a 50 nF capacitor.

a) Show that at a given frequency the circuits in Fig 9.20(a) and (b) will have the same impedance between the terminals a,b if (Hint: The two circuits will have the same impedance if they have the same admittance.) b) Find the values of resistance and capacitance that when connected in parallel will give the same impedance at 50 krad/s as that of a resistor connected in series with a capacitance of 40 nF.

Find the impedance Zab in the circuit seen in Fig. P9.22. Express Zab in both polar and rectangular form.

Find the admittance Yab in the circuit seen in Fig. P9.23. Express Yab in both polar and rectangular form. Give the value of Yab in millisiemens.

a) For the circuit shown in Fig. P9.24, find the frequency (in radians per second) at which the impedance Zab is purely resistive. b) Find the value of Zab at the frequency of (a).

a) Using component values from Appendix H, combine at least one resistor, inductor, and capacitor in series to create an impedance of 300 - j400 at a frequency of 10,000 rad s. b) At what frequency does the circuit from part (a) have an impedance that is purely resistive?

a) Using component values from Appendix H, combine at least one resistor and one inductor in parallel to create an impedance 40 + j20 of at a frequency of 5000 rad s. (Hint: Use the results of Problem 9.19.) b) Using component values from Appendix H, combine at least one resistor and one capacitor in parallel to create an impedance of at a frequency of 40 - j20 5000 rad s. (Hint: Use the result of Problem 9.21.)

a) Using component values from Appendix H, find a single capacitor or a network of capacitors that, when combined in parallel with the RL circuit from Problem 9.26(a), gives an equivalent impedance that is purely resistive at a frequency of 5000 rad/s. b) Using component values from Appendix H, find a single inductor or a network of inductors that, when combined in parallel with the RC circuit from Problem 9.26(b), gives an equivalent impedance that is purely resistive at a frequency of 5000 rad/

Find the steady-state expression for io(t) in the circuit in Fig. P9.28 if Figure P9.28 if vs vs = 80 cos 2000t V.

The circuit in Fig. P9.29 is operating in the sinusoidal steady state. Find the steady-state expression for vo(t) if vg 60 sin 8000t V. Figure P9.29

The circuit in Fig. P9.30 is operating in the sinusoidal steady state Find if Figure P9.30 5  10  2.5 mH 20  12.5 mF io vs v = 25 sin 4000t V

a) For the circuit shown in Fig. P9.31, find the steadystate expression for i v g = 25 cos 50,000t mA. if b) By how many microseconds does lead ?

Find and Z in the circuit shown in Fig. P9.32 if j2  j3  j5  4  Z Ia j3  1  Ia = 5 l90 V A

Find the value of Z in the circuit seen in Fig. P9.33 if and V1 = 140 + j 30 V

Find the steady-state expression for in the circuit of Fig. P9.34 if Figure P9.34 50 10 mH 100 2mF ig vo ig = 60 cos 10,000t mA.

The circuit shown in Fig. P9.35 is operating in the sinusoidal steady state. Find the value of if io = 40 sin (vt + 21.87) mA, vg = 40 cos (vt - 15) V.

The phasor current in the circuit shown in Fig. P9.36 is a) Find , and b) If write expressions for and ig i (t). c(t), ia v = 1500 rad>

The frequency of the sinusoidal voltage source in the circuit in Fig. P9.37 is adjusted until the current io is in phase with vg a) Find the frequency in hertz. b) Find the steady-state expression for (at the frequency found in [a]) if vg = 90 cos vt V Figure P9.37

a) The frequency of the source voltage in the circuit in Fig. P9.38 is adjusted until is in phase with What is the value of in radians per second? b) If (where is the frequency found in [a]), what is the steady-state expression for ?

The frequency of the sinusoidal voltage source in the circuit in Fig. P9.39 is adjusted until ig is in phase with vg a) What is the value of in radians per second? b) If vg = 15 cos vt V (where is the frequency found in [a]), what is the steady-state expression for ig ? Figure P9.39

a) The source voltage in the circuit in Fig. P9.40 is vg = 40 cos 1000t V. Find the values of L such that ig is in phase with when the circuit is operating in the steady state. b) For the values of L found in (a), find the steadystate expressions for ig Figure P9.40

The circuit shown in Fig. P9.41 is operating in the sinusoidal steady state. The capacitor is adjusted until the current ig is in phase with the sinusoidal voltage vg. a) Specify the capacitance in microfarads if vg = 80 cos 5000t V. b) Give the steady-state expression for ig when C has the value found in (a).

Find Zab for the circuit shown in Fig P9.42

The sinusoidal voltage source in the circuit in Fig. P9.43 is developing a voltage equal to 50 sin 400t V a) Find the Thvenin voltage with respect to the terminals a,b. b) Find the Thvenin impedance with respect to the terminals a,b. c) Draw the Thvenin equivalent.

Use source transformations to find the Norton equivalent circuit with respect to the terminals a,b for the circuit shown in Fig. P9.44.

Use source transformations to find the Thvenin equivalent circuit with respect to the terminals a,b for the circuit shown in Fig. P9.45.

Find the Norton equivalent circuit with respect to the terminals a,b for the circuit shown in Fig. P9.46.

The device in Fig.P9.47 is represented in the frequency domain by a Thvenin equivalent.When a resistor having an impedance of is connected across the device, the value of is When an inductor having an impedance of is connected across the device, the value of is Find the Thvenin volatge and the Thvenin impedance .

Find the Norton equivalent with respect to terminals a,b in the circuit of Fig. P9.48. Figure P9.48

Find the Thvenin equivalent circuit with respect to the terminals a,b of the circuit shown in Fig. P9.49. Figure P9.49

Find the Norton equivalent circuit with respect to the terminals a,b for the circuit shown in Fig. P9.50 when Vs = 5l0 V. Figure P9.50

The circuit shown in Fig. P9.51 is operating at a frequency of 10 rad/s. Assume is real and lies between -10 and + 10, that is -10 a10., a) Find the value of so that the Thvenin impedance looking into the terminals a,b is purely resistive. b) What is the value of the Thvenin impedance for the found in (a)? c) Can a be adjusted so that the Thvenin impedance equals 500 - j500 ? If so, what is the value of a? d) For what values of will the Thvenin impedance be inductive?

Find in the circuit shown in Fig. P9.52 when the circuit is operating at a frequency of Figure P9.52.

Find the Thvenin impedance seen looking into the terminals a,b of the circuit in Fig. P9.53 if the frequency of operation is (25/ p) kHz. Figure P9.53

Use the node-voltage method to find Vo in the circuit in Fig. P9.54.

Use the node-voltage method to find the phasor voltage in the circuit shown in Fig. P9.55.

Use the node voltage method to find the steady-state expression for , in the circuit seen in Fig. P9.56 if i vg = 20 cos (2500t + 90) V. g = 5 cos 2500t Aand Figure P9.56

Use the node-voltage method to find the steadystate expression for in the circuit in Fig. P9.57 if vg1 = 25 sin (400t + 143.13) V,vg2 = 18.03 cos (400t + 33.69) V.

Use the node-voltage method to find the phasor voltage in the circuit shown in Fig. P9.58. Express the voltage in both polar and rectangular form. Figure P9.58

Use the node-voltage method to find and in the circuit seen in Fig. P9.59.

Use the mesh-current method to find the phasor current Ig in the circuit in Fig. P9.55.

Use the mesh-current method to find the steadystate expression for vo(t) in the circuit in Fig. P9.57.

Use the mesh-current method to find the branch currents Ia,Ib,Ic and Id in the circuit shown in Fig. P9.62.

Use the mesh-current method to find the steady-state expression for in the circuit in Fig. P9.63 if va = 18 sin 4000t V, vb = 12 cos 4000t V

Use the mesh-current method to find the steadystate expression for in the circuit seen in Fig. P9.64 if equals Figure P9.64 110 mH 4 mH i 4 mF 100 i vg 10  vo v 75 cos 5000t V. g

Use the concept of voltage division to find the steady-state expression for in the circuit in Fig. P9.65 if Figure P9.65 vg = 120 cos 100,000t V.

Use the concept of current division to find the steady-state expression for io in the circuit in Fig. P9.66 if Figure P9.66 ig = 60 cos 250t mA.

For the circuit in Fig. P9.67. Suppose v1 = 20 cos(2000t - 36.87) V, v2 = 10 cos(5000t + 16.26) V a) What circuit analysis technique must be used to find the steady-state expression for vo(t)? b) Find the steady-state expression for vo(t) .

For the circuit in Fig. P9.63, suppose va = 10 cos 16,000t V vb = 20 cos 4000t V. a) What circuit analysis technique must be used to find the steady-state expression for ? b) Find the steady-state expression for io(t)

The sinusoidal voltage source in the circuit shown in Fig. P9.69 is generating the voltage vg = 20 cos 5000t V. If the op amp is ideal, what is the steady-state expression for ?

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 135 a) Find the value of C in microfarads. b) Write the steady-state expression for vo(t) 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 vo(t) 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 vg = 30 cos 106 t V. a) How small can be before the steady-state output voltage no longer has a pure sinusoidal waveform? b) For the value of Co 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 = 4 krad/s. Find Figure P9.74

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 vg = 168 cos 800t V. b) Find the coefficient of coupling. c) Find the energy stored in the magnetically coupled coils at t = 625p ms and t = 1250 s Figure P9.76

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 vg = 560 cos(2 * 105 t) V

A series combination of a 60 resistor and a 50 mH 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 (10 + j12.75) . The rms voltage at the terminals of the source is 75 V when it is not loaded. The parameters of the linear transformer are R1 = 8.34 ,L 2 = 100 , L2 = 250 mH,M = 135 mH 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 Xab can never be negative if XL 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 Zab = a1 + N1 N2 b 2 ZL 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 Zab = ZL a1 + N1 N2 b 2, Show that if the polarity terminal of either one of the coils is reversed that Zab = ZL a1 - N1 N2 b 2 .

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 Rx is varied from zero to infinity. The amplitude and phase angle of the source voltage are held constant Rx as varies.

The parameters in the circuit shown in Fig. 9.53 are R = 0.1,L1 = 0.8 , R = 24 ,wL = 32 and vL = 240 + j0v. a) Calculate the phasor voltage Vs. b) Connect a capacitor in parallel with the inductor, hold VL 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 |Vs| = |VL| = 240 V..

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. Figure P9.85 Vs 240 0 V Vl 0.1  j0.8  8  j6  j5  -5 240

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 I1-I6 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 2 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 R1 = R2 I . b) Show V . 1 = V that if R1 = R2 c) Open the neutral branch and calculate V1 and V2 if R1 = 40 and R2 = 400 and R3 = 8 . 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 Ip for (c) and (d) in Problem 9.89. b) Do your answers make sense in terms of known circuit behavior?