10.7: 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.