9.8: Exercises

Analysis

1. Determine the AC peak and RMS voltages, DC offset, frequency, period and phase shift for the following expression: \(v(t) = 10 \sin 2\pi 1000 t\)

2. Determine the AC peak and RMS voltages, DC offset, frequency, period and phase shift for the following expression: \(v(t) = 0.4 \sin 2\pi 5000 t\)

3. Determine the peak AC portion voltage, DC offset, frequency, period and phase shift for the following expression: \(v(t) = −3 + 20 \sin 2\pi 50 t\)

4. Determine the peak AC portion voltage, DC offset, frequency, period and phase shift for the following expression: \(v(t) = 12 + 2 \sin 2\pi 20000 t\)

5. Determine the AC peak and RMS voltages, DC offset, frequency, period and phase shift for the following expression: \(v(t) = 10 \sin (2\pi 100 t + 45^{\circ} )\)

6. Determine the AC peak and RMS voltages, DC offset, frequency, period and phase shift for the following expression: \(v(t) = 5 \sin (2\pi 1000 t − 90^{\circ} )\)

7. Determine the peak AC portion voltage, DC offset, frequency, period and phase shift for the following expression: \(v(t) = 10 + 1 \sin (2\pi 400 t − 45^{\circ} )\)

8. Determine the peak AC portion voltage, DC offset, frequency, period and phase shift for the following expression: \(v(t) = 10 + 10 \sin (2\pi 5000 t + 30^{\circ} )\)

9. A 1 kHz sine wave has a phase of 72\(^{\circ}\). Determine the time delay. Repeat for a 20 kHz sine wave.

10. A 2 kHz sine wave has a phase of 18\(^{\circ}\). Determine the time delay. Repeat for a 100 kHz sine wave.

11. An oscilloscope measures a time delay of 0.2 milliseconds between a pair of 500 Hz sine waves. Determine the phase shift.

12. An oscilloscope measures a time delay of −10 microseconds between a pair of 20 kHz sine waves. Determine the phase shift.

13. Convert the following from rectangular to polar form:

a) \(10 + j10\)

b) \(5 − j10\)

c) \(−100 + j20\)

d) \(3k + j4k\)

14. Convert the following from rectangular to polar form:

a) \(2k + j1.5k\)

b) \(8 − j8\)

c) \(−300 + j300\)

d) \(−1k − j1k\)

15. Convert these from polar to rectangular form:

a) \(10\angle 45^{\circ} \)

b) \( 0.4\angle 90^{\circ} \)

c) \(−9\angle 60^{\circ} \)

d) \(100\angle −45^{\circ} \)

16. Convert these from polar to rectangular form:

a) \(−4\angle 60^{\circ} \)

b) \(−0.9\angle 30^{\circ} \)

c) \(5\angle 120^{\circ} \)

d) \(6\angle −135^{\circ} \)

17. Perform the following computations:

a) \((10 + j10) + (5 + j20)\)

b) \((5 + j2) + (−5 + j2)\)

c) \((80 − j2) − (100 + j2)\)

d) \((−65 + j50) − (5 − j200)\)

18. Perform the following computations:

a) \((100 + j200) + (75 + j210)\)

b) \((−35 + j25) + (15 + j8)\)

c) \((500 − j70) − (200 + j30)\)

d) \((−105 + j540) − (5− j200)\)

19. Perform the following computations:

a) \((100 + j200) \cdot (75 + j210)\)

b) \((−35 + j25) \cdot (15 + j8)\)

c) \((500 − j70) / (200 + j30)\)

d) \((−105 + j540) / (5− j200)\)

20. Perform the following computations:

a) \((10 + j10) \cdot (5 + j20)\)

b) \((5 + j2) \cdot (−5 + j2)\)

c) \((80 − j2) / (100 + j2)\)

d) \((−65 + j50) / (5− j200)\)

21. Perform the following computations:

a) \((10\angle 0^{\circ} ) \cdot (10\angle 0^{\circ} )\)

b) \((5\angle 45^{\circ} ) \cdot (−2\angle 20^{\circ} )\)

c) \((20\angle 135^{\circ} ) / (40\angle −10^{\circ} )\)

d) \((8\angle 0^{\circ} ) / (32\angle 45^{\circ} )\)

22. Perform the following computations:

a) \((0.3\angle 0^{\circ} ) \cdot (3\angle 180^{\circ} )\)

b) \((5\angle −45^{\circ} ) \cdot (−4\angle 20^{\circ} )\)

c) \((0.05\angle 95^{\circ} ) / (0.04\angle −20^{\circ} )\)

d) \((500\angle 0^{\circ} ) / (60\angle 225^{\circ} )\)

23. Perform the following computations:

a) \((0.3\angle 0^{\circ} ) + (3\angle 180^{\circ} )\)

b) \((5\angle −45^{\circ} ) + (−4\angle 20^{\circ} )\)

c) \((0.05\angle 95^{\circ} ) − (0.04\angle −20^{\circ} )\)

d) \((500\angle 0^{\circ} ) − (60\angle 225^{\circ} )\)

24. Perform the following computations:

a) \((10\angle 0^{\circ} ) + (10\angle 0^{\circ} )\)

b) \((5\angle 45^{\circ} ) + (−2\angle 20^{\circ} )\)

c) \((20\angle 135^{\circ} ) − (40\angle −10^{\circ} )\)

d) \((8\angle 0^{\circ} ) − (32\angle 45^{\circ} ) \)

25. Determine the capacitive reactance of a 1 \(\mu\)F capacitor at the following frequencies:

a) 10 Hz

b) 500 Hz

c) 10 kHz

d) 400 kHz

e) 10 MHz

26. Determine the capacitive reactance of a 220 pF capacitor at the following frequencies:

a) 10 Hz

b) 500 Hz

c) 10 kHz

d) 400 kHz

e) 10 MHz

27. Determine the capacitive reactance at 50 Hz for the following capacitors:

a) 10 pF

b) 470 pF

c) 22 nF

d) 33 \(\mu\)F

28. Determine the capacitive reactance at 1 MHz for the following capacitors:

a) 22 pF

b) 560 pF

c) 33 nF

d) 4.7 \(\mu\)F

29. Determine the inductive reactance of a 100 mH inductor at the following frequencies:

a) 10 Hz

b) 500 Hz

c) 10 kHz

d) 400 kHz

e) 10 MHz

30. Determine the inductive reactance of a 100 mH inductor at the following frequencies:

a) 10 Hz

b) 500 Hz

c) 10 kHz

d) 400 kHz

e) 10 MHz

31. Determine the inductive reactance at 1 kHz for the following inductors:

a) 10 mH

b) 500 mH

c) 10 \(\mu\)H

d) 400 \(\mu\)H

32. Determine the inductive reactance at 500 kHz for the following inductors:

a) 1 mH

b) 40 mH

c) 2 \(\mu\)H

d) 50 \(\mu\)H

33. Draw phasor diagrams for the following:

a) \(5 + j2\)

b) \(−10 −j20\)

c) \(8\angle 45^{\circ} \)

d) \(2\angle −35^{\circ} \)

34. Draw phasor diagrams for the following:

a) \(60j−20\)

b) \(−40 + j500\)

c) \(0.05\angle −45^{\circ} \)

d) \(−15\angle 60^{\circ} \)

35. The fundamental of a certain square wave is a 5 volt peak, 1 kHz sine. Determine the amplitude and frequency of each of the next five harmonics.

36. The fundamental of a certain triangle wave is a 10 volt peak, 100 Hz sine. Determine the amplitude and frequency of each of the next five harmonics.

Design

37. Determine the capacitance required for the following reactance values at 1 kHz:

a) 560 \(\Omega\)

b) 330 k\(\Omega\)

c) 470 k\(\Omega\)

d) 1.2 k\(\Omega\)

e) 750 \(\Omega\)

38. Determine the capacitance required for the following reactance values at 20 Hz:

a) 56 k\(\Omega\)

b) 330 k\(\Omega\)

c) 470 k\(\Omega\)

d) 1.2 k\(\Omega\)

e) 750 \(\Omega\)

39. Determine the inductance required for the following reactance values at 100 MHz:

a) 560 \(\Omega\)

b) 330 k\(\Omega\)

c) 470 k\(\Omega\)

d) 1.2 k\(\Omega\)

e) 750 \(\Omega\)

40. Determine the inductance required for the following reactance values at 25 kHz:

a) 56 \(\Omega\)

b) 33 k\(\Omega\)

c) 470 k\(\Omega\)

d) 1.2 k\(\Omega\)

e) 750 \(\Omega\)

41. Which of the following have a reactance of less than 100 \(\Omega\) for all frequencies below 1 kHz?

a) 2 mH

b) 99 mH

c) 470 pF

d) 10000 \(\mu\)F

42. Which of the following have a reactance of less than 8 \(\Omega\) for all frequencies above 10 kHz?

a) 10 nH

b) 5 mH

c) 56 pF

d) 470 \(\mu\)F

43. Which of the following have a reactance of at least 1k \(\Omega\) for all frequencies above 20 kHz?

a) 2 mH

b) 200 mH

c) 680 pF

d) 33 \(\mu\)F

44. Which of the following have a reactance of at least 75 \(\Omega\) for all frequencies below 5 kHz?

a) 680 \(\mu\)H

b) 10 mH

c) 82 pF

d) 33 nF

Challenge

45. Determine the negative and positive peak voltages, RMS voltage, DC offset, frequency, period and phase shift for the following expression: \(v(t) = −10 \sin (2\pi 250 t + 180^{\circ} )\)

46. Determine the negative and positive peak voltages, DC offset, frequency, period and phase shift for the following expression: \(v(t) = 1 − 100 \sin 2\pi 50000 t\)

47. Assume you have a DC coupled oscilloscope set as follows: time base = 100 microseconds/division, vertical sensitivity = 1 volt/division. Sketch the display of this waveform: \(v(t) = 2 + 3 \sin 2\pi 2000 t\)

48. Assume you have a DC coupled oscilloscope set to the following: time base = 20 microseconds/division, vertical sensitivity = 200 millivolts/division. Sketch the display of this waveform: \(v(t) = −0.2 + 0.4 \sin 2\pi 10000 t\)

49. A 200 \(\Omega\) resistor is in series with a 1 mH inductor. Determine the impedance of this combination at 200 Hz and at 20 kHz.

50. A 1 k\(\Omega\) resistor is in series with an inductor. If the combined impedance at 10 kHz is \(1.41 k\angle 45^{\circ} \), determine the inductance in mH.