# 1.8: Exercises

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## 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.

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