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Electric circuits

Sinusoidal ac Voltage Characteristics

2.

3.

For the square-wave signal in Fig. 13.81:

a. What is the peak value?

b. What is the instantaneous value at 5 ms and at 11 ms?

c. What is the peak-to-peak value of the waveform?

d. What is the period of the waveform?

e. How many cycles are shown?

For the periodic waveform in Fig. 13.82:

a. What is the peak value?

b. What is the instantaneous value at 3 ms and at 9 ms?

c. What is the peak-to-peak value of the waveform?

d. What is the period of the waveform?

e. How many cycles are shown?

  1. For the sinusoidal waveform in Fig. 13.80:

a. What is the peak value?

b. What is the instantaneous value at 15 ms and at 20 ms?

c. What is the peak-to-peak value of the waveform?

d. What is the period of the waveform?

e. How many cycles are shown?

i (mA) 20

PROBLEMS ⏐⏐⏐ 581

10

0 –10

–20

v (V) 40

5 10 15 20 25 30 35 40 45 50 t(ms)

FIG. 13.80

Problem 1.

0 1 23 45 67 89 –40

1011 12t(􏰄s)

FIG. 13.81

Problem 2.

v (mV) 8

0 1 2 3 4 5 6 7 8 9 10t(􏰄s) –8

FIG. 13.82

Problem 3.

582 ⏐⏐⏐ SINUSOIDAL ALTERNATING WAVEFORMS SECTION 13.3 Frequency Spectrum

  1. Find the period of a periodic waveform whose frequency is a. 25 Hz. b. 40 MHz.

c. 25 kHz. d. 1 Hz.

  1. Find the frequency of a repeating waveform whose period is
  2. Find the angular velocity of a waveform with a frequency of a. 50 Hz. b. 600 Hz.

c. 2 kHz. d. 0.004 MHz.

  1. Find the frequency and period of sine waves having an an- gular velocity of

a. 754 rad/s.

c. 6000 rad/s.

*15. Given f 􏰕 60 Hz, determine how long it will take the sinu-

soidal waveform to pass through an angle of 45°.

*16. If a sinusoidal waveform passes through an angle of 30° in 5 ms, determine the angular velocity of the waveform.

SECTION 13.5 General Format for the Sinusoidal Voltage or Current

  1. Find the amplitude and frequency of the following waves: a. 20 sin 377t b. 5 sin 754t

c. 106 sin 10,000t d. 􏰈6.4 sin 942t

  1. Sketch 5 sin 754t with the abscissa

a. angle in degrees. b. angle in radians. c. time in seconds.

*19. Sketch 􏰈7.6 sin 43.6t with the abscissa

a. angle in degrees. b. angle in radians. c. time in seconds.

  1. If e 􏰕 300 sin 157t, how long (in seconds) does it take this waveform to complete 1/2 cycle?
  2. Given i 􏰕 0.5 sin 􏰁, determine i at 􏰁 􏰕 72°.
  3. Given y 􏰕 20 sin 􏰁, determine y at 􏰁 􏰕 1.2p.

*23. Given y 􏰕 30 􏰖 10􏰈3 sin 􏰁, determine the angles at which

y will be 6 mV.

*24. If y 􏰕 40 V at 􏰁 􏰕 30 and t 􏰕 1 ms, determine the mathe-

matical expression for the sinusoidal voltage.

SECTION 13.6 Phase Relations

  1. Sketch sin(377t 􏰇 60°) with the abscissa

a. angle in degrees. b. angle in radians. c. time in seconds.

  1. Sketch the following waveforms:

a. 1/60 s. c. 40 ms.

b. 0.01 s. d. 25 ms.

b. 8.4 rad/s. d. 1 rad>s.

  1. If a periodic waveform has a frequency of 20 Hz, how long (in seconds) will it take to complete five cycles?
  2. Find the period of a sinusoidal waveform that completes 80 cycles in 24 ms.
  3. What is the frequency of a periodic waveform that com- pletes 42 cycles in 6 s?
  4. For the oscilloscope pattern of Fig. 13.83:

a. Determine the peak amplitude.

b. Find the period.

c. Calculate the frequency.

Redraw the oscilloscope pattern if a 􏰇25 mV dc level were added to the input waveform.

SECTION 13.4

Vertical sensitivity = 50 mV/div. Horizontal sensitivity = 10 􏰄s/div.

FIG. 13.83

Problem 9.

The Sinusoidal Waveform

  1. Convert the following degrees to radians: a. 45° b. 60° c. 270° d. 170°
  2. Convert the following radians to degrees: 1

a. 50 sin(vt 􏰇 0°) c. 2 cos(vt 􏰇 10°)

b. 5 sin(vt 􏰇 60°) d. 􏰈20 sin(vt 􏰇 2°)

a. p/4 b. p/6 c. 10p d. 0.6p

  1. Find the angular velocity of a waveform with a period of
  2. Write the analytical expression for the waveforms in Fig. 13.84 with the phase angle in degrees.

1

a. 2 s. b. 0.3 ms. c. 4 ms. d. 26 s.

FIG. 13.84

Problem 27.

16

v (V)

i (mA)

25

30°

f = 60 Hz

2􏰀 3

f = 1000 Hz

0 􏰃t 0 􏰃t

(a)

(b)

–3

v (V) 0.01

0 100°

(a)

f = 40 Hz

i (mA) 2

  1. Write the analytical expression for the waveforms in Fig. 13.85 with the phase angle in degrees.
  2. Findthephaserelationshipbetweenthefollowingwaveforms: y 􏰕 4 sin(vt 􏰇 50°)

i 􏰕 6 sin(vt 􏰇 40°)

  1. Findthephaserelationshipbetweenthefollowingwaveforms:

y 􏰕 25 sin(vt 􏰈 80°) i􏰕5􏰖10􏰈3 sin(vt􏰈10°)

  1. Findthephaserelationshipbetweenthefollowingwaveforms: y 􏰕 0.2 sin(vt 􏰈 60°)

i 􏰕 0.1 sin(vt 􏰇 20°)

*32. Findthephaserelationshipbetweenthefollowingwaveforms:

y 􏰕 2 cos(vt 􏰈 30°) i 􏰕 5 sin(vt 􏰇 60°)

*33. Findthephaserelationshipbetweenthefollowingwaveforms: y 􏰕 􏰈4 cos(vt 􏰇 90°)

i 􏰕 􏰈2 sin(vt 􏰇 10°)

*34. Thesinusoidalvoltagey􏰕200sin(2p1000t􏰇60°)isplot- ted in Fig. 13.86. Determine the time t1 when the waveform crosses the axis.

FIG. 13.86

Problem 34.

*35. The sinusoidal current i 􏰕 4 sin(50,000t 􏰈 40°) is plotted in Fig. 13.87. Determine the time tI when the waveform crosses the axis.

FIG. 13.87

  1. For the oscilloscope display in Fig. 13.88:

a. Determine the period of the waveform.

b. Determine the frequency of each waveform.

c. Find the rms value of each waveform.

d. Determine the phase shift between the two waveforms

and determine which leads and which lags.

ei

Vertical sensitivity = 0.5 V/div. Horizontal sensitivity = 1 ms/div.

FIG. 13.88

Problem 36.

Average Value

v

200

t1

–􏰀 0 t1 􏰀 2􏰀 t

60°

Problem 35.

Problem 37.

􏰃t

0

􏰃t

FIG. 13.85

Problem 28.

3􏰀 4

SECTION 13.7

37.

Find the average value of the periodic waveform in Fig. 13.89.

(b)

f = 10 kHz

PROBLEMS ⏐⏐⏐ 583

v (V)

6 3

–3

1 3 t(s)

1 cycle

i

–􏰀 0 t1

4A

40°

􏰀 2􏰀 t(􏰄s)

0

2

FIG. 13.89

584 ⏐⏐⏐ SINUSOIDAL ALTERNATING WAVEFORMS

38.

Find the average value of the periodic waveform in Fig. 13.90.

i (mA) 20

0 468t(ms) –5

FIG. 13.90

Problem 38.

Find the average value of the periodic waveform in Fig. 13.91.

39.

  1. Find the rms values of the following sinusoidal waveforms:

a. y 􏰕 140 sin(377t 􏰇 60°)

b. i 􏰕 6 􏰖 10􏰈3 sin(2p 1000t)

c. y 􏰕 40 􏰖 10􏰈6 (sin(2p 5000t 􏰇 30°)

  1. Write the sinusoidal expressions for voltages and currents having the following rms values at a frequency of 60 Hz with zero phase shift:

a. 10 V b. 50 mA c. 2 kV

  1. Find the rms value of the periodic waveform in Fig. 13.94 over one full cycle.

i (mA) 20

Circular

SECTION 13.8

Vertical sensitivity = 10 mV/div. Horizontal sensitivity = 10 􏰄s/div.

FIG. 13.93

Problem 41.

Effective (rms) Values

v (V)

3 2 1

1 cycle

40.

0 􏰀 2􏰀􏰂 –5

FIG. 13.91

Problem 39.

For the waveform in Fig. 13.92:

a. Determine the period.

b. Find the frequency.

c. Determine the average value.

d. Sketch the resulting oscilloscope display if the vertical

channel is switched from DC to AC.

0

–1 –2

v (V)

3 2 1

1 cycle

1 2 3 4 5

7 8 9 10 11 12 t(s)

6

FIG. 13.94

Problem 44.

  1. Find the rms value of the periodic waveform in Fig. 13.95 over one full cycle.

Vertical sensitivity = 10 mV/div. Horizontal sensitivity = 0.2 ms/div.

FIG. 13.92

Problem 40.

*41. For the waveform in Fig. 13.93:

a. Determine the period.

b. Find the frequency.

c. Determine the average value.

d. Sketch the resulting oscilloscope display if the vertical

FIG. 13.95

channel is switched from DC to AC.

Problem 45.

0

–1 –2 –3

1 2 3 4 5

12 t(s)

6 7 8 9 10 11

  1. What are the average and rms values of the square wave in Fig. 13.96?

SECTION 13.9

v (V)

1 cycle

10

0

–10

8 t (ms)

4

  1. Determine the reading of the meter for each situation in Fig. 13.98.

FIG. 13.97

Problem 47.

Horizontal sensitivity (b)

= 50 􏰄􏰄s/div.

PROBLEMS ⏐⏐⏐ 585

FIG. 13.96

Problem 46.

*47. For each waveform in Fig. 13.97, determine the period, fre- quency, average value, and rms value.

Vertical sensitivity = 20 mV/div. Horizontal sensitivity = 10 􏰄􏰄􏰄􏰄s/div.

(a)

ac Meters and Instruments

Vertical sensitivity = 0.2 V/div.

d’Arsonval movement

Voltmeter

Idc = 4 mA ac

++

2 k􏰅 v = 16 sin(377t + 20°)

rms scale (half-wave rectifier)

(a)

(b)

––

FIG. 13.98

Problem 48.

586 ⏐⏐⏐ SINUSOIDAL ALTERNATING WAVEFORMS GLOSSARY

Alternatingwaveform Awaveformthatoscillatesaboveandbe- low a defined reference level.

Angular velocity The velocity with which a radius vector pro- jecting a sinusoidal function rotates about its center.

Average value The level of a waveform defined by the condition that the area enclosed by the curve above this level is exactly equal to the area enclosed by the curve below this level.

Calibration factor A multiplying factor used to convert from one meter indication to another.

Clamp Meter® A clamp-type instrument that will permit non- invasive current measurements and that can be used as a con- ventional voltmeter or ohmmeter.

Cycle A portion of a waveform contained in one period of time. Effective value The equivalent dc value of any alternating volt-

age or current.

Electrodynamometer meters Instruments that can measure

both ac and dc quantities without a change in internal circuitry. Frequency(f) Thenumberofcyclesofaperiodicwaveformthat

occur in 1 second.

Frequency counter An instrument that will provide a digital

display of the frequency or period of a periodic time-varying

signal.

Instantaneous value The magnitude of a waveform at any in-

stant of time, denoted by lowercase letters.

Lagging waveform A waveform that crosses the time axis at a

point in time later than another waveform of the same frequency. Leading waveform A waveform that crosses the time axis at a point in time ahead of another waveform of the same frequency.

Oscilloscope An instrument that will display, through the use of a cathode-ray tube, the characteristics of a time-varying signal.

Peak amplitude The maximum value of a waveform as mea- sured from its average, or mean, value, denoted by uppercase letters.

Peak-to-peakvalue Themagnitudeofthetotalswingofasignal from positive to negative peaks. The sum of the absolute val- ues of the positive and negative peak values.

Peak value The maximum value of a waveform, denoted by up- percase letters.

Period(T) Thetimeintervalnecessaryforonecycleofaperiodic waveform.

Periodic waveform A waveform that continually repeats itself after a defined time interval.

Phase relationship An indication of which of two waveforms leads or lags the other, and by how many degrees or radians.

Radian (rad) A unit of measure used to define a particular seg- ment of a circle. One radian is approximately equal to 57.3°; 2p rad are equal to 360°.

Root-mean-square (rms) value The root-mean-square or effec- tive value of a waveform.

Sinusoidal ac waveform An alternating waveform of unique characteristics that oscillates with equal amplitude above and below a given axis.

VOM Amultimeterwiththecapabilitytomeasureresistanceand both ac and dc levels of current and voltage.

Waveform Thepathtracedbyaquantity,plottedasafunctionof some variable such as position, time, degrees, temperature, and so on.

Sample Solution

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