# 6.6: Exercises

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## Analysis

(All source values are in amps or volts unless specified otherwise)

1. Given the circuit in Figure $$\PageIndex{1}$$, use nodal analysis to determine $$v_c$$. $$I_1 = 3\angle 0^{\circ}$$, $$I_2 = 0.9\angle 0^{\circ}$$.

Figure $$\PageIndex{1}$$

2. Use nodal analysis to find the current through the 120 $$\Omega$$ resistor in the circuit of Figure $$\PageIndex{2}$$. $$I_1 = 0.5\angle 90^{\circ}$$, $$I_2 = 1.6\angle 0^{\circ}$$.

3. Use nodal analysis to find the current through the 43 $$\Omega$$ resistor in the circuit of Figure $$\PageIndex{2}$$. The sources are in phase.

Figure $$\PageIndex{2}$$

4. Given the circuit in Figure $$\PageIndex{2}$$, use nodal analysis to determine $$v_b$$. The sources are in phase.

5. Given the circuit in Figure $$\PageIndex{3}$$, determine $$v_c$$. $$I_1 = 3\angle 0^{\circ}$$, $$I_2 = 2\angle 0^{\circ}$$.

Figure $$\PageIndex{3}$$

6. Use nodal analysis to find the current through the $$j45 \Omega$$ inductor in the circuit of Figure $$\PageIndex{3}$$. $$I_1 = 2\angle 0^{\circ}$$, $$I_2 = 1.5\angle 60^{\circ}$$.

7. Use nodal analysis to find the current through the 4 $$\Omega$$ resistor in the circuit of Figure $$\PageIndex{4}$$. $$I_1 = 1\angle 45^{\circ}$$, $$I_2 = 2\angle 45^{\circ}$$.

Figure $$\PageIndex{4}$$

8. Given the circuit in Figure $$\PageIndex{4}$$, use nodal analysis to determine $$v_c$$. $$I_1 = 6\angle 30^{\circ}$$, $$I_2 = 4\angle 0^{\circ}$$.

9. Given the circuit in Figure $$\PageIndex{5}$$, use nodal analysis to determine $$v_{ac}$$. $$I_1 = 10\angle 0^{\circ}$$, $$I_2 = 6\angle 0^{\circ}$$.

Figure $$\PageIndex{5}$$

10. Use nodal analysis to find the current through the $$j8 \Omega$$ inductor in the circuit of Figure $$\PageIndex{5}$$. $$I_1 = 3\angle 0^{\circ}$$, $$I2 = 5\angle 30^{\circ}$$.

11. Use nodal analysis to find the current through the 22 $$\Omega$$ resistor in the circuit of Figure $$\PageIndex{6}$$. $$I_1 = 800E−3\angle 0^{\circ}$$, $$I_2 = 2.5\angle 0^{\circ}$$, $$I_3 = 2\angle 20^{\circ}$$.

Figure $$\PageIndex{6}$$

12. Given the circuit in Figure $$\PageIndex{6}$$, use nodal analysis to determine $$v_c$$. $$I_1 = 4\angle 90^{\circ}$$, $$I_2 = 10\angle 120^{\circ}$$, $$I_3 = 5\angle 0^{\circ}$$.

13. Given the circuit in Figure $$\PageIndex{7}$$, use nodal analysis to determine $$v_c$$. $$I_1 = 3E−3\angle 0^{\circ}$$, $$I_2 = 10E−3\angle 0^{\circ}$$, $$I_3 = 2E−3\angle 0^{\circ}$$.

Figure $$\PageIndex{7}$$

14. Use nodal analysis to find the current through the $$−j2$$ k$$\Omega$$ capacitor in the circuit of Figure $$\PageIndex{7}$$. $$I_1 = 1E−3\angle 0^{\circ}$$, $$I_2 = 5E−3\angle 0^{\circ}$$, $$I_3 = 6E−3\angle −90^{\circ}$$.

15. Use nodal analysis to find the current through the 3.3 k$$\Omega$$ resistor in the circuit of Figure $$\PageIndex{8}$$. $$E = 36\angle 0^{\circ}$$, $$I = 4E−3\angle −120^{\circ}$$.

Figure $$\PageIndex{8}$$

16. Given the circuit in Figure $$\PageIndex{8}$$, write the node equations and determine $$v_c$$. $$E = 18\angle 0^{\circ}$$, $$I = 7.5E−3\angle −30^{\circ}$$.

17. Given the circuit in Figure $$\PageIndex{9}$$, use nodal analysis to determine $$v_c$$. $$E = 40\angle 180^{\circ}$$, $$I = 20E−3\angle 0^{\circ}$$.

Figure $$\PageIndex{9}$$

18. Use nodal analysis to find the current through the 2.2 k$$\Omega$$ resistor in Figure $$\PageIndex{9}$$. $$E = 240\angle 0^{\circ}$$, $$I = 100E−3\angle 0^{\circ}$$.

19. Use nodal analysis to find $$v_{bc}$$ in the circuit of Figure $$\PageIndex{10}$$.

Figure $$\PageIndex{10}$$

20. Use nodal analysis to find the current through the 2.7 k$$\Omega$$ resistor in the circuit of Figure $$\PageIndex{11}$$.

Figure $$\PageIndex{11}$$

21. Given the circuit in Figure $$\PageIndex{12}$$, use nodal analysis to determine $$v_{ba}$$. $$E_1 = 1\angle 0^{\circ}$$, $$E_2 = 2\angle 0^{\circ}$$.

Figure $$\PageIndex{12}$$

22. Given the circuit in Figure $$\PageIndex{13}$$, use nodal analysis to determine $$v_{ad}$$. $$E_1 = 9\angle 0^{\circ}$$, $$E_2 = 5\angle 40^{\circ}$$.

Figure $$\PageIndex{13}$$

23. Use nodal analysis to find $$v_{cb}$$ in the circuit of Figure $$\PageIndex{14}$$. $$E_1 = 10\angle −180^{\circ}$$, $$E_2 = 25\angle 0^{\circ}$$.

Figure $$\PageIndex{14}$$

24. Given the circuit in Figure $$\PageIndex{15}$$, use nodal analysis to determine $$v_{bc}$$. $$E = 20\angle 0^{\circ}$$, $$R_1$$ = 10 k$$\Omega$$, $$R_2$$ = 30 k$$\Omega$$, $$R_3$$ = 1 k$$\Omega$$, $$X_C = −j15$$ k$$\Omega$$, $$X_L = j20$$ k$$\Omega$$.

Figure $$\PageIndex{15}$$

25. Given the circuit in Figure $$\PageIndex{16}$$, write the mesh loop equations and determine $$v_b$$.

Figure $$\PageIndex{16}$$

26. Use mesh analysis to find the current through the 2.7 k$$\Omega$$ resistor in the circuit of Figure $$\PageIndex{16}$$.

27. Use mesh analysis to find the current through the 75 $$\Omega$$ resistor in the circuit of Figure $$\PageIndex{10}$$.

28. Given the circuit in Figure $$\PageIndex{10}$$, write the mesh loop equations and determine $$v_c$$.

29. Given the circuit in Figure $$\PageIndex{11}$$, write the mesh loop equations and determine $$v_b$$.

30. Use mesh analysis to find the current through the 1.8 k$$\Omega$$ resistor in the circuit of Figure $$\PageIndex{11}$$.

31. Use mesh analysis to find the current through the $$j200 \Omega$$ inductor in Figure $$\PageIndex{12}$$. $$E_1 = 1\angle 0^{\circ}$$, $$E_2 = 2\angle 0^{\circ}$$.

32. Given the circuit in Figure $$\PageIndex{12}$$, write the mesh loop equations and determine $$v_b$$. Consider using parallel simplification first. $$E_1 = 36\angle −90^{\circ}$$, $$E_2 = 24\angle −90^{\circ}$$.

33. Given the circuit in Figure $$\PageIndex{13}$$, use mesh analysis to determine $$v_{cd}$$. $$E_1 = 0.1\angle 0^{\circ}$$, $$E_2 = 0.5\angle 0^{\circ}$$.

34. Use mesh analysis to find the current through the 600 $$\Omega$$ resistor in the circuit of Figure $$\PageIndex{13}$$. $$E_1 = 9\angle 0^{\circ}$$, $$E_2 = 5\angle 40^{\circ}$$.

35. Use mesh analysis to find the current through the $$−j200 \Omega$$ capacitor in the circuit of Figure $$\PageIndex{17}$$. $$E_1 = 18\angle 0^{\circ}$$, $$E_2 = 12\angle 90^{\circ}$$.

Figure $$\PageIndex{17}$$

36. Given the circuit in Figure $$\PageIndex{17}$$, use mesh analysis to determine $$v_{ac}$$. $$E_1 = 1\angle 0^{\circ}$$, $$E_2 = 500E−3\angle 0^{\circ}$$.

37. Given the circuit in Figure $$\PageIndex{14}$$, use mesh analysis to determine $$v_c$$. $$E_1 = 10\angle −180^{\circ}$$, $$E_2 = 25\angle 0^{\circ}$$.

38. Use mesh analysis to find the current through the 22 k$$\Omega$$ resistor in the circuit of Figure $$\PageIndex{14}$$. $$E_1 = 24\angle 0^{\circ}$$, $$E_2 = 36\angle 0^{\circ}$$.

39. Use mesh analysis to find the current through the $$j300 \Omega$$ inductor in Figure $$\PageIndex{18}$$. $$E_1 = 1\angle 0^{\circ}$$, $$E_2 = 10\angle 90^{\circ}$$.

Figure $$\PageIndex{18}$$

40. Given the circuit in Figure $$\PageIndex{18}$$, use mesh analysis to determine $$v_a$$. $$E_1 = 100\angle 0^{\circ}$$, $$E_2 = 90\angle 0^{\circ}$$.

41. Given the circuit in Figure $$\PageIndex{15}$$, use mesh analysis to determine $$v_{bc}$$. $$E = 10\angle 0^{\circ}$$, $$R_1$$ = 1 k$$\Omega$$, $$R_2$$ = 2 k$$\Omega$$, $$R_3$$ = 3 k$$\Omega$$, $$X_C = −j4$$ k$$\Omega$$, $$X_L = j8$$ k$$\Omega$$.

42. Use mesh analysis to find the current through resistor $$R_3$$ in the circuit of Figure $$\PageIndex{15}$$. $$E = 20\angle 0^{\circ}$$, $$R_1$$ = 10 k$$\Omega$$, $$R_2$$ = 30 k$$\Omega$$, $$R_3$$ = 1 k$$\Omega$$, $$X_C = −j15$$ k$$\Omega$$, $$X_L = j20$$ k$$\Omega$$.

43. Use mesh analysis to find the current through resistor $$R_3$$ in Figure $$\PageIndex{19}$$. $$E = 60\angle 0^{\circ}$$, $$R_1$$ = 1 k$$\Omega$$, $$R_2$$ = 2 k$$\Omega$$, $$R_3$$ = 3 k$$\Omega$$, $$X_C = −j10$$ k$$\Omega$$, $$X_L = j20$$ k$$\Omega$$.

Figure $$\PageIndex{19}$$

44. Given the circuit in Figure $$\PageIndex{19}$$, use mesh analysis to determine $$v_{bc}$$. $$E = 120\angle 90^{\circ}$$, $$R_1$$ = 100 k$$\Omega$$, $$R_2$$ = 20 k$$\Omega$$, $$R_3$$ = 10 k$$\Omega$$, $$X_C = −j5$$ k$$\Omega$$, $$X_L = j20$$ k$$\Omega$$.

45. Given the circuit in Figure $$\PageIndex{20}$$, use mesh analysis to determine $$v_b$$. Consider using source conversion. $$E = 12\angle 0^{\circ}$$, $$I = 10E−3\angle 0^{\circ}$$.

Figure $$\PageIndex{20}$$

46. Use mesh analysis to find the current through the 3 $$\Omega$$ resistor in the circuit in Figure $$\PageIndex{20}$$. Consider using source conversion. $$E = 15\angle 90^{\circ}$$, $$I = 10E−3\angle 0^{\circ}$$.

47. Use mesh analysis to find the current through the 2.2 k$$\Omega$$ resistor in the circuit in Figure $$\PageIndex{21}$$. $$E = 3.3\angle 0^{\circ}$$, $$I = 2.1E−3\angle 0^{\circ}$$.

Figure $$\PageIndex{21}$$

48. Given the circuit in Figure $$\PageIndex{21}$$, use mesh analysis to determine $$v_b$$. $$E = 10\angle 0^{\circ}$$, $$I = 30E−3\angle 90^{\circ}$$.

49. Given the circuit in Figure $$\PageIndex{22}$$, use nodal analysis to determine $$v_{ab}$$.

Figure $$\PageIndex{22}$$

50. Use nodal analysis to find the current through the 100 mH inductor in the circuit of Figure $$\PageIndex{22}$$.

51. Use nodal analysis to find the current through the 330 $$\Omega$$ resistor in the circuit of Figure $$\PageIndex{23}$$.

Figure $$\PageIndex{23}$$

52. Given the circuit in Figure $$\PageIndex{23}$$, write the node equations and determine $$v_b$$.

53. Given the circuit in Figure $$\PageIndex{19}$$, use nodal analysis to determine $$v_{bc}$$. $$E = 120\angle 0^{\circ}$$, $$R_1$$ = 1 k$$\Omega$$, $$R_2$$ = 2 k$$\Omega$$, $$R_3$$ = 3 k$$\Omega$$, $$X_C = −j10$$ k$$\Omega$$, $$X_L = j20$$ k$$\Omega$$.

54. Determine the current through the 10 k$$\Omega$$ resistor in the circuit of Figure $$\PageIndex{24}$$ if $$I_1 = 10E−3\angle −90^{\circ}$$.

Figure $$\PageIndex{24}$$

55. Determine $$v_b$$ in the circuit of Figure $$\PageIndex{24}$$ if the source $$I_1 = 20E−3\angle 0^{\circ}$$.

56. Determine $$v_c$$ in the circuit of Figure $$\PageIndex{25}$$ if the source $$E = 3\angle 120^{\circ}$$.

Figure $$\PageIndex{25}$$

57. Determine the current through the 5 k$$\Omega$$ resistor in the circuit of Figure $$\PageIndex{25}$$ if $$E = 10\angle 0^{\circ}$$.

58. In the circuit of Figure $$\PageIndex{26}$$, determine the capacitor current if the source $$E = 12\angle 0^{\circ}$$.

Figure $$\PageIndex{26}$$

59. In the circuit of Figure $$\PageIndex{26}$$, determine $$v_c$$ if the source $$E = 8\angle 90^{\circ}$$.

60. In the circuit of Figure $$\PageIndex{27}$$, determine $$v_b$$ if the source $$E = 12\angle −90^{\circ}$$.

Figure $$\PageIndex{27}$$

61. In the circuit of Figure $$\PageIndex{27}$$, determine the current flowing into the 1 k$$\Omega$$ resistor if the source $$E = 6\angle 0^{\circ}$$.

62. In the circuit of Figure $$\PageIndex{28}$$, determine the current flowing into the 600 $$\Omega$$ resistor if $$I_1 = 1E−3\angle 180^{\circ}$$.

Figure $$\PageIndex{28}$$

63. Determine $$v_a$$ and $$v_b$$ in the circuit of Figure $$\PageIndex{28}$$ if the source $$I_1 = 2E−3\angle 0^{\circ}$$.

64. Determine $$v_a$$ in the circuit of Figure $$\PageIndex{29}$$ if the source $$E = 2\angle 0^{\circ}$$.

Figure $$\PageIndex{29}$$

65. Given the circuit in Figure $$\PageIndex{29}$$, determine the current flowing through the 1 k$$\Omega$$ resistor. Assume that $$E = 15\angle 45^{\circ}$$.

66. Given the circuit in Figure $$\PageIndex{30}$$, determine the current flowing through the 3 k$$\Omega$$ resistor if the source $$E = 25\angle 33^{\circ}$$.

Figure $$\PageIndex{30}$$

67. Given the circuit in Figure $$\PageIndex{30}$$, determine $$v_{ab}$$. Assume the source $$E = 15\angle −112^{\circ}$$.

68. In the circuit of Figure $$\PageIndex{31}$$, determine $$v_d$$.

Figure $$\PageIndex{31}$$

69. Given the circuit in Figure $$\PageIndex{31}$$, determine the current flowing through the 1 k$$\Omega$$ resistor.

70. Given the circuit in Figure $$\PageIndex{32}$$, determine the current flowing through the 100 $$\Omega$$ resistor.

Figure $$\PageIndex{32}$$

71. Determine $$v_d$$ in the circuit of Figure $$\PageIndex{32}$$.

72. Determine $$v_{ab}$$ in the circuit of Figure $$\PageIndex{33}$$. $$E = 10\angle 0^{\circ}$$

Figure $$\PageIndex{33}$$

## Challenge

73. Given the circuit in Figure $$\PageIndex{34}$$, write the node equations. $$E_1 = 50\angle 0^{\circ}$$, $$E_2 = 35\angle 120^{\circ}$$, $$I = 500E−3\angle 90^{\circ}$$.

Figure $$\PageIndex{34}$$

74. Given the circuit in Figure $$\PageIndex{34}$$, use either mesh or nodal analysis to determine $$v_{ed}$$. $$E_1 = 9\angle 0^{\circ}$$, $$E_2 = 12\angle 0^{\circ}$$, $$I = 50E−3\angle 0^{\circ}$$.

75. Given the circuit in Figure $$\PageIndex{35}$$, use mesh analysis to determine $$v_{fc}$$. $$E_1 = 12\angle 0^{\circ}$$, $$E_2 = 48\angle 0^{\circ}$$, $$E_3 = 36\angle 70^{\circ}$$.

Figure $$\PageIndex{35}$$

76. Find voltage $$v_{bc}$$ in the circuit of Figure $$\PageIndex{36}$$ using either mesh or nodal analysis. $$E = 100\angle 0^{\circ}$$, $$R_1 = R_2 = 2$$ k$$\Omega$$, $$R_3$$ = 3 k$$\Omega$$, $$R_4$$ = 10 k$$\Omega$$, $$R_5$$ = 5 k$$\Omega$$, $$X_{C1} = X_{C2} = −j2$$ k$$\Omega$$.

Figure $$\PageIndex{36}$$

77. Given the circuit in Figure $$\PageIndex{37}$$, use nodal analysis to find $$v_{ac}$$. $$I_1 = 8E−3\angle 0^{\circ}$$, $$I_2 = 12E−3\angle 0^{\circ}$$, $$E = 50\angle 0^{\circ}$$.

Figure $$\PageIndex{37}$$

78. Given the circuit in Figure $$\PageIndex{38}$$, use nodal analysis to determine $$v_{ad}$$. $$I_1 = 0.1\angle 0^{\circ}$$, $$I_2 = 0.2\angle 0^{\circ}$$, $$I_3 = 0.3\angle 0^{\circ}$$.

Figure $$\PageIndex{38}$$

79. Given the circuit in Figure $$\PageIndex{39}$$, determine $$v_{ad}$$. $$E_1 = 15\angle 0^{\circ}$$, $$E_2 = 6\angle 0^{\circ}$$, $$I = 100E−3\angle 0^{\circ}$$.

Figure $$\PageIndex{39}$$

80. Given the circuit in Figure $$\PageIndex{40}$$, determine $$v_{ad}$$. $$E_1 = 22\angle 0^{\circ}$$, $$E_2 = −10\angle 0^{\circ}$$, $$I = 2E−3\angle 0^{\circ}$$.

Figure $$\PageIndex{40}$$

81. Given the circuit in Figure $$\PageIndex{41}$$, determine $$v_{ab}$$. $$I_1 = 1.2\angle 0^{\circ}$$, $$I_2 = 2\angle 120^{\circ}$$, $$E = 75\angle 0^{\circ}$$.

Figure $$\PageIndex{41}$$

82. Given the circuit in Figure $$\PageIndex{42}$$, determine $$v_{ad}$$. $$I_1 = 0.8\angle 0^{\circ}$$, $$I_2 = 0.2\angle 180^{\circ}$$, $$I_3 = 0.1\angle 0^{\circ}$$, $$E = 15\angle 0^{\circ}$$.

Figure $$\PageIndex{42}$$

## Simulation

83. Perform a transient analysis simulation on the circuit of problem 25 (Figure $$\PageIndex{16}$$) to verify the results for $$v_b$$.

84. Investigate the variation of $$v_b$$ due to frequency in problem 25 (Figure $$\PageIndex{16}$$) by performing an AC simulation. Run the simulation from 10 Hz up to 100 kHz.

85. Investigate the variation of $$v_b$$ due to component tolerance in problem 25 (Figure $$\PageIndex{16}$$) by performing a Monte Carlo simulation. Apply a 10% tolerance to the resistors and capacitor.

86. Perform a transient analysis simulation on the circuit of problem 28 (Figure $$\PageIndex{10}$$) to verify the results for $$v_c$$.

87. Investigate the variation of $$v_b$$ due to frequency in problem 28 (Figure $$\PageIndex{10}$$) by performing an AC simulation. Run the simulation from 1 Hz up to 10 kHz.

88. Investigate the variation of $$v_b$$ due to component tolerance in problem 28 (Figure $$\PageIndex{10}$$) by performing a Monte Carlo simulation. Apply a 10% tolerance to the resistors and capacitors.

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