Solved: Four branches terminate at a common node. The reference direction of each branch

Find the phasor transform of each trigonometric function: a) v = 170cos(377r - 40) V. b) i = 10 sin (1000/1 + 20) A. c) i = [5 cos (art + 36.87) + 10cos(art -53.13)] A. d) v = [300cos(20,00077t + 45) - 100 sin(20,0007rr + 30)] mV.

Find the time-domain expression corresponding to each phasor: a) V = 18.6/-54 V. b) I = (20/45^ - 50/-30) mA. c) V = (20 + /80 - 30/15) V.

The current in the 20 mH inductor is 10 cos (10,000? + 30) mA. Calculate (a) the inductive reactance; (b) the impedance of the inductor; (c) the phasor voltage V; and (d) the steady-state expression for v(t). 20 mH

The current in the 20 mH inductor is 10 cos (10,000? + 30) mA. Calculate (a) the inductive reactance; (b) the impedance of the inductor; (c) the phasor voltage V; and (d) the steady-state expression for v(t). 20 mH

Four branches terminate at a common node. The reference direction of each branch current (*j, i2, /3, and /4) is toward the node. If NOTE: Also try Chapter Problem 9.15. ix = 100 cos (G>* + 25) A, i2 = 100cos(o)f + 145) A, and 13 = 100 cos (eat - 95) A, find/4.

Using the values of resistance and inductance in the circuit of Fig. 9.15, let Ys = 125 /-60 V and a) = 5000 rad/s. Find a) the value of capacitance that yields a steady-state output current i with a phase angle of -105. b) the magnitude of the steady-state output current f

A 20 fl resistor is connected in parallel with a 5 mH inductor. This parallel combination is connected in series with a 5 ft resistor and a 25 ^iF capacitor. a) Calculate the impedance of this interconnection if the frequency is 2 krad/s. b) Repeat (a) for a frequency of 8 krad/s. c) At what finite frequency does the impedance of the interconnection become purely resistive? d) What is the impedance at the frequency found in (c)?

The interconnection described in Assessment Problem 9.7 is connected across the terminals of a voltage source that is generating v = 150 cos 4000f V. What is the maximum amplitude of the current in the 5 mH inductor?

Use a A -to-Y transformation to find the current I in the circuit shown.

Find the Thevenin equivalent with respect to terminals a,b in the circuit shown.

Use the node-voltage method to find the steadystate expression for v(t) in the circuit shown. The sinusoidal sources are is = 10 cos cot A and vs = 100 sin cot V, where to - 50 krad/s.

Use the mesh-current method to find the phasor current I in the circuit shown.

A linear transformer couples a load consisting of a 360 O resistor in series with a 0.25 H inductor to a sinusoidal voltage source, as shown. The voltage source has an internal impedance of 184 + /0 Cl and a maximum voltage of 245.20 V, and it is operating at 800 rad/s. The transformer parameters are Ri = 100 fl, Lt = 0.5 H, R2 = 40 O, L2 = 0.125 H, and k = 0.4. Calculate (a) the reflected impedance; (b) the primary current; and (c) the secondary current. NOTE: Also try Chapter Problems 9.76 and 9.77. Source Transformer d Load

The source voltage in the phasor domain circuit in the accompanying figure is 25 /0 kV. Find the amplitude and phase angle of V2 an d ^2-

A 10 O resistor and a 5 /xF capacitor are connected PSPICE i n parallel. This parallel combination is also in par1 allel with the series combination of an 8 O resistor and a 300 /xH inductor. These three parallel branches are driven by a sinusoidal current source whose current is 922 cos(20,000r + 30) 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 v{t).

a) Show that, at a given frequency w, the circuits in Fig. P9.17(a) and (b) will have the same impedance between the terminals a,b if 2 J 2 /e, - a>z LjR2 R\ + co 2 L 2 2 U Rl^i R\ + (o2 L\ 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 5 kH resistor connected in parallel with a 1.25 H inductor.

a) Show that at a given frequency eo, the circuits in Fig. P9.17(a) and (b) will have the same impedance between the terminals a,b if Ro = R] + a?L\ u = R\ + o)2 L\ 2f (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 kft resistor connected in series with a 4 H inductor.

a) Show that at a given frequency to, the circuits in Fig. P9.19(a) and (b) will have the same impedance between the terminals a,b if /?i = Q = R2 1 + o?R\c\ 1 + arRJCJ w 2RJC2 b) Find the values of resistance and capacitance that when connected in series will have the same impedance at 40 krad/s as that of a 1000 ft resistor connected in parallel with a 50 nF capacitor.

a) Using component values from Appendix H, combine at least one resistor, inductor, and capacitor in series to create an impedance of 300 - /400 ft 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 of 40 + /20 ft at a frequency of 5000 rad/s. (HintUse the results of Problem 9.18.)b) Using component values from Appendix H, combine at least one resistor and one capacitor in parallel to create an impedance of 40 - /20 ft at a frequency of 5000 rad/s. (Hint: Use the result of Problem 9.20.)

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 9.22(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 9.22(b), gives an equivalent impedance that is purely resistive at a frequency of 5000 rad/s.

Three branches having impedances of 3 + /4 O, 16 - /12 ft, and -/ 4 ft, 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 w/ A, what is the maximum amplitude of the current in the purely capacitive branch?

a) For the circuit shown in Fig. P9.25, find the frePSPICE quency (in radians per second) at which the impedance Zab is purely resistive, b) Find the value of Za b at the frequency of (a).

Find the admittance K,b in the circuit seen in Fig. P9.26. Express K(lb in both polar and rectangular form. Give the value of Yab in millisiemens.

Find the impedance Zab in the circuit seen in Fig. P9.27. Express Za b in both polar and rectangular form.

The circuit shown in Fig. P9.28 is operating in the sinusoidal steady state. Find the value of co if i(} = 40 sin (a)/ + 21.87) mA. vg = 40cos(o>/ - 15) V.

The circuit in Fig. P9.29 is operating in the sinuPSPICE soidal steady state. Find the steady-state expression M"LTISIM f orVw (f )i f u = 40 cos 50,000/ V.

Find the steady-state expression for /(/) in the circuit in Fig. P9.31 if vs = 100 sin 50/ mV.

Find the steady-state expression for v(> in the circuit of Fig. P9.32 if ig = 500 cos 2000/ m A.

The phasor current Ia in the circuit shown in PSPICE Fig. P9.33 is 2/0 A. lumsiM a) Find Ib, Ic, and Vg. b) If a) = 800 rad/s, write the expressions for ib(t), /c(/), and vM).

The circuit in Fig. P9.34 is operating in the sinusoidal PSPICE steady state. Find v0(t) if /,(/) = 3 cos 200/ mA.

Find the value of Z in the circuit seen in Fig. P9.35 if Vg = 100 - /50 V, I g = 30 + /20 A, and \i = 140 + /30 V.

Find Ib and Z in the circuit shown in Fig. P9.36 if VJJ = 25/(T V and I a = 5 /90 A.

Find Zab for the circuit shown in Fig P9.37.

a) The frequency of the source voltage in the circuit in Fig. P9.38 is adjusted until ig is in phase with Vr What is the value of co in radians per second? b) If vg = 20 cos a)t V (where a> is the frequency found in [a]), what is the steady-state expression for vnl

The frequency of the sinusoidal voltage source in PSPICE the circuit in Fig. P9.39 is adjusted until the current "1ULTISIM i ;._ ia is in phase with vg. a) Find the frequency in hertz. b) Find the steady-state expression for ig (at the frequency found in [a]) if vg = 30 cos wt V.

a) The source voltage in the circuit in Fig. P9.41 is Vg - 50 cos 50,000f V. Find the values of L such that ig is in phase with vg when the circuit is operating in the steady state. b) For the values of L found in (a), find the steadystate expressions for ig.

The frequency of the sinusoidal current source in PSPICE the circuit in Fig. P9.42 is adjusted until va is in mTISIM phase with ir a) What is the value of a) in radians per second? b) If ig = 2.5 cos oit mA (where to is the frequency found in [a]), what is the steady-state expression for u,?

The device in Fig. P9.43 is represented in the frequency domain by a Norton equivalent. When a resistor having an impedance of 5 kft is connected across the device, the value of V0 is 5 /15 V. When a capacitor having an impedance of -/ 3 kft is connected across the device, the value of I() is 4.5 - /6 mA. Find the Norton current I N and the Norton impedance Z N .

The sinusoidal voltage source in the circuit in Fig. P9.44 is developing a voltage equal to 247.49 cos (lOOOf+ 45) V. a) Find the Thevenin voltage with respect to the terminals a,b. b) Find the Thevenin impedance with respect to the terminals a,b. c) Draw the Thevenin equivalent.

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

Use source transformations to find the Norton equivalent circuit with respect to the terminals a,b for the circuit shown in Fie. P9.46.

Find the Thevenin equivalent circuit with respect to the terminals a,b for the circuit shown in Fig. P9.47

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

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

Find the Thevenin impedance seen looking into the terminals a,b of the circuit in Fig. P9.51 if the frequency of operation is (25/TT) kHz.

Find the Norton equivalent circuit with respect to the terminals a,b for the circuit shown in Fig. P9.52 whenVy = 5/0 V.

The circuit shown in Fig. P9.53 is operating at a frequency of 10 rad/s. Assume a is real and lies between -10 and +10, that is, -10 < a < 10. a) Find the value of a so that the Thevenin impedance looking into the terminals a,b is purely resistive. b) What is the value of the Thevenin impedance for the a found in (a)? c) Can a be adjusted so that the Thevenin impedance equals 500 /500 O? If so, what is the value of a? d) For what values of a will the Thevenin impedance be inductive?

Use the node-voltage method to find the steadyPSPICE state expression for v()(t) in the circuit in Fig. P9.54 if MULTISIM % = 20cos(2000r - 36.87) V,vg2 = 50sin(2000r - 16.26) V.

Use the node-voltage method to find \(> in the circuit in Fig. P9.55.

Use the node-voltage method to find the phasor voltage V in the circuit shown in Fig. P9.56.

Use the node voltage method to find the steady-state PSPICE expressions for the branch currents /a, ib, and /c in the MULTISIM circuit seen in Fig. P9.57 if v& = 50sinl0fVV and Vb = 25 cos (106 / + 90) V.

Us e the node-voltage method to find Vf) and I in the circuit seen in Fig. P9.58.

Us e the node-voltage method to find the phasor voltage V in the circuit shown in Fig. P9.59. Express the voltage in both polar and rectangular form.

Us e the mesh-current method to find the steadystate expression for i(){t) in the circuit in Fig. P9.61 if va = 60 cos 40,000/ V, vh = 90 sin (40,000* + 180) V.

Us e the mesh-current method to find the phasor current lg in the circuit in Fig. P9.56.

Us e the mesh-current method to find the branch currents I. I h , I c , and I d in the circuit shown in Fig. P9.63.

Use the mesh-current method to find the steadyPSPICE s tat e expression for va in the circuit seen in " " Fig< p9. 6 4 if v equa | s \ 3 0 CO s 10,000/ V.

Use the concept of current division to find the PSPICE steady-state expression for i(> in the circuit in Mum pig. P9.65 if/^ = 125 cos 12,500* mA.

Use the concept of voltage division to find the PSPICE steady-state expression for v()(t) in the circuit in LTISIM Fi g p 9 6 6 i f v = 7 5 co s 20,000/ V.

The op amp in the circuit seen in Fig. P9.67 is ideal. PSPICE Find the steady-state expression for v(>(t) when - =2cosl(/Y V

The op amp in the circuit in Fig. P9.68 is ideal. MULTISIM a ) ^ n

The sinusoidal voltage source in the circuit shown in PSPICE pig P9.69 is generating the voltage v = 4 cos 200r V. MULTISIM ..11,- 1 i If the op amp is ideal, what is the steady-state expression for v0(t)1

The operational amplifier in the circuit shown in PSPICE Fig. P9.71 is ideal. The voltage of the ideal sinuMULTISIM , 1 . ir i r\f\* t / soida 1 source is vg = 30 cos 10t V. a) How small can Ca be before the steady-state output voltage no longer has a pure sinusoidal waveform? b) For the value of C0 found in (a), write the steady-state expression for va.

a) Find the input impedance Zab for the circuit in Fig. P9.72. Express Zab as a function of Z and K where K = (R2/R\). b) If Z is a pure capacitive element, what is the capacitance seen looking into the terminals a,b?

For the circuit in Fig. P9.73 suppose vt = 20 cos(2000f - 36.87) V v2 = 10cos(5000/ + 16.26) V a) What circuit analysis technique must be used to find the steady-state expression for vv(t)1 b) Find the steady-state expression for vC)(t).

For the circuit in Fig. P9.61, suppose v.d = 5 cos 80,000/ V vb = -2.5 cos 320,000/ V. b) Find the coefficient of coupling. c) Find the energy stored in the magnetically coupled coils at t = 1007T /xs and t = 200-7T ^s. Figure P9.77 30 a a) What circuit analysis technique must be used to find the steady-state expression for j(r)? b) Find the steady-state expression for / 0 (/)?

A series combination of a 300 O resistor and a 100 m H inductor is connected to a sinusoidal voltage source by a linear transformer. The source is operating at a frequency of 1 krad/s. A t this frequency, the internal impedance of the source is 100 + /13.74 CI. The rms voltage at the terminals of the source is 50 V when it is not loaded. The parameters of the linear transformer are R\ = 41.68 O, L{ = 180 mH , R2 = 500 ft, L2 = 500 mH , and M = 270 mH . a) What is the value of the impedance reflected into the primary? b) Wha t is the value of the impedanc e seen from the terminals of the practical source?

The sinusoidal voltage source in the circuit seen in PSPICE Fig. P9.76 is operating at a frequency of 200 krad/s. LTISIM ^ e coe ff}c j etl t 0 f coupling is adjusted until the peak amplitude of ix is maximum. a) What is the value of kl b) What is the peak amplitude of /j if vg = 560 cos(2 X 10 V ?

a) Find the steady-state expressions for the currents ig and iL in the circuit in Fig. P9.77 when PSPICE MU LTISIM vg = 70 cos 5000/ V.b) Find the coefficient of coupling. c) Find the energy stored in the magnetically coupled coils at t = 1007T /xs and t = 200-7T ^s.

For the circuit in Fig. P9.78, find the Thevenin equivalent with respect to the terminals c,d.

The value of k in the circuit in Fig. P9.79 is adjusted so that Za b is purely resistive when a> = 4 krad/s. Find Zab.

a) Show that the impedanc e seen looking into the terminals a,b in the circuit in Fig. P9.81 on the next page 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. P9.82 is given by the expression Zab - ZL 1 + N-, b) Show that if the polarity terminal of either one of the coils is reversed that

Find the impedance Za b in the circuit in Fig. P9.83 if ZL = 80/60'H .

Show by using a phasor diagram what happens to PSPICE the magnitude and phase angle of the voltage v in MULTISIM th e circui t i n Fi g p 9 8 4 a s R^ j s yaric d fro m zer o t Q infinity. The amplitude and phase angle of the source voltage are held constant as Rx varies

The parameters in the circuit shown in Fig. 9.53 are /?, = 0.1 il,o)Lx = 0.8 ft,fl2 = 24 il,(oL2 = 32 ft, and VL = 240 + / 0 V. 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 Vv? 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 lYvl = |V/J = 240 V

a) For the circuit shown in Fig. P9.86, compute Vv and V/. b) Construct a phasor diagram showing the relationship between Vs , V/, and the load voltage of 240/0 V. c) Repeat parts (a) and (b), given that the load voltage remains constant at 240 /0 V, when a capacitive reactance of - 5 Cl 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 Problems 359 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 farmer's 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 didn't 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. P9.87. The impedances of the line conductors and the neutral conductor are assumed negligible. a) Calculate the branch currents It , I2, I3, I4, I5, and I 6 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 I]-I

Suppose the 40 ft resistance in the distribution cirpRAcncAL cuit in Fie. 9.58 is replaced bv a 20 ft resistance. PERSPECTIVE r a) Recalculate the branch current in the 2 (1 resistor, I2. b) Recalculate the primary current, Ip. c) On the basis of your answers, is it desirable to have the resistance of the two 120 V loads be equal?

a) Find the primary current I p for (c) and (d) in P nwr PRACTICAL Problem 9.90. K m moa n PERSPECTIVE Fuse B( 100 A) b) Do your answers make sense in terms of known circuit behavior?