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2. Sinusoidal Alternating Waveforms A.C.

3. Alternating waveforms
Introduction
Alternating waveforms
The term alternating indicates only that the waveform alternates between two prescribed levels in a set time sequence. 2

4. Sinusoidal ac Voltage Characteristics and Definitions
Generation
An ac generator (or alternator) powered by water power, gas, or nuclear fusion is the primary component in the energy-conversion process.
The energy source turns a rotor (constructed of alternating magnetic poles) inside a set of windings housed in the stator (the stationary part of the dynamo) and will induce voltage across the windings of the stator. 3

5. Various sources of ac power: (a) generating plant; (b) portable ac generator; (c) wind-power station; (d) solar panel; (e) function generator.

6. Sinusoidal ac Voltage Characteristics and Definitions
Generation:
Wind power and solar power energy are receiving increased interest from various districts of the world.
The turning propellers of the wind-power station are connected directly to the shaft of an ac generator.
Light energy in the form of photons can be absorbed by solar cells. Solar cells produce dc, which can be electronically converted to ac with an inverter.
A function generator, as used in the lab, can generate and control alternating waveforms. 4

7. Symbol for a sinusoidal voltage source

8. The electromechanical ac generator, cutaway view (courtesy of General Electric).

9. AC Generator
The ac generator has slip rings that pick up the induced voltage through a complete rotation cycle
The induced voltage is related to the number of lines of flux being cut. When the loop is moving parallel with the lines of flux, no voltage is induced. When the loop is moving perpendicular to the lines of flux, the maximum voltage is induced

10. Basic ac generator operation.

11. One revolution of the wire loop generates one cycle of the sinusoidal voltage.

12. Frequency is directly proportional to the rate of rotation of the wire loop in an ac generator.

13. Multi-pole ac Generator
By increasing the number of poles, the number of cycles per revolution can be increased

14. Four poles achieve a higher frequency than two poles for the same r p s.

15. Electronic Signal Generators
In the lab, we usually use a signal generator to produce a variety of waveforms at a wide range of frequencies
An oscillator in the signal generator produces the repetitive wave
We are able to set the frequency and amplitude of the signal from the signal generator

16. Typical signal generators. (Copyright Tektronix, Inc
Typical signal generators. (Copyright Tektronix, Inc. Reproduced by permission.)

17. Function generator.

18. Sinusoidal ac Voltage Characteristics and Definitions
Waveform: The path traced by a quantity, such as voltage, plotted as a function of some variable such as time, position, degree, radius, temperature and so on.
Instantaneous value: The magnitude of a waveform at any instant of time; denoted by the lowercase letters (e1, e2).
Peak amplitude: The maximum value of the waveform as measured from its average (or mean) value, denoted by the uppercase letters Em (source of voltage) and Vm (voltage drop across a load). 5

19. Sinusoidal ac Voltage Characteristics and Definitions
Peak value: The maximum instantaneous value of a function as measured from zero-volt level.
Peak-to-peak value: Denoted by Ep-p or Vp-p, the full voltage between positive and negative peaks of the waveform, that is, the sum of the magnitude of the positive and negative peaks.
Periodic waveform: A waveform that continually repeats itself after the same time interval. 6

20. Important parameters for a sinusoidal voltage.

21. Sine Wave
The sine wave is a common type of alternating current (ac) and alternating voltage

22. Alternating current & voltage.

23. Sinusoidal ac Voltage Characteristics and Definitions
Period (T): The time interval between successive repetitions of a periodic waveform (the period T1 = T2 = T3), as long as successive similar points of the periodic waveform are used in determining T
Cycle: The portion of a waveform contained in one period of time
Frequency: (Hertz) the number of cycles that occur in 1 s 7

24. Period of a Sine Wave
The time required for a sine wave to complete one full cycle is called the period (T)
A cycle consists of one complete positive, and one complete negative alternation
The period of a sine wave can be measured between any two corresponding points on the waveform

25. Graph of one cycle of a sine wave.

26. Defining the cycle and period of a sinusoidal waveform.

27. Measurement of the period of a sine wave.
Thomas L. Floyd Electronics Fundamentals, 6e Electric Circuit Fundamentals, 6e
Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

28. Frequency of a Sine Wave
Frequency ( f ) is the number of cycles that a sine wave completes in one second
The more cycles completed in one second, the higher the frequency
Frequency is measured in hertz (Hz)
Relationship between frequency ( f ) and period (T) is:
f = 1/T

29. Illustration of frequency.

30. Demonstrating the effect of a changing frequency on the period of a sinusoidal waveform.

31. The period of a given sine wave is the same for each cycle.

32. Cycles in one second of time

33. Cycles in one second of time

34. Areas of application for specific frequency bands.

35. Seismogram from station BNY (Binghamton University) in New York due to magnitude 6.7 earthquake in Central Alaska that occurred at 63.62°N, °W, with a depth of 10 km, on Wednesday, October 23, 2002.

36. Peak Values of Sine Waves
The peak value of a sine wave is the value of voltage or current at the positive or negative maximum with respect to zero
Peak values are represented as:
Vp and Ip

37. Peak values.

38. where: Vpp = 2Vp and Ipp = 2Ip
Peak-to-Peak Values
The peak-to-peak value of a sine wave is the voltage or current from the positive peak to the negative peak
The peak-to-peak values are represented as:
Vpp and Ipp
where: Vpp = 2Vp and Ipp = 2Ip

39. Peak-to-peak value.

40. Instantaneous Values of Sine Waves
The instantaneous values of a sine wave voltage (or current) are different at different points along the curve, having negative and positive values
Instantaneous values are represented as:
v and i

41. General Format for the Sinusoidal Voltage or Current
The basic mathematical format for the sinusoidal waveform is:
where:
Am is the peak value of the waveform
 is the unit of measure for the horizontal axis 14

42. Sine Wave Formula The general expression for a sine wave is:
y = A sin 
Where: y = an instantaneous value (v or i)
A = amplitude (maximum value)
 = angle along the horizontal axis

43. Example of instantaneous values of a sinusoidal voltage.

44. Sine wave angles.

45. One cycle of a generic sine wave showing amplitude and phase.

46. Illustration of the instantaneous value of a voltage sine wave at  = 60º.

47. Right triangle derivation of sine wave formula, v = Vp sin .

48. Sine wave represented by a rotating phasor.

49. Average Value
The algebraic sum of the areas must be determined, since some area contributions will be from below the horizontal axis.
Area above the axis is assigned a positive sign and area below the axis is assigned a negative sign.
The average value of any current or voltage is the value indicated on a dc meter – over a complete cycle the average value is the equivalent dc value. 21

50. Average Value Understanding the average value using a sand analogy:
The average height of the sand is that height obtained if the distance form one end to the other is maintained while the sand is leveled off. 20

51. Average Value of a Sine Wave
The average value is the total area under the half-cycle curve divided by the distance in radians of the curve along the horizontal axis
Vavg = 0.637Vp
Iavg = 0.637Ip

52. Half-cycle average value.

53. Effective (rms) Values
How is it possible for a sinusoidal ac quantity to deliver a net power if, over a full cycle the net current in any one direction is zero (average value = 0).
Irrespective of direction, current of any magnitude through a resistor will deliver power to that resistor – during the positive and negative portions of a sinusoidal ac current, power is being delivered at each instant of time to the resistor.
The net power flow will equal twice that delivered by either the positive or the negative regions of sinusoidal quantity. 22

54. RMS Value of a Sine Wave
The rms (root mean square) value, or effective value, of a sinusoidal voltage is equal to the dc voltage that produces the same amount of heat in a resistance as does the sinusoidal voltage
Vrms = 0.707Vp
Irms = 0.707Ip

55. An experimental setup to establish a relationship between dc and ac quantities.

56. When the same amount of heat is being produced by the resistor in both setups, the sinusoidal voltage has an rms value equal to the dc voltage.

57. Effective (rms) Values
The formula for power delivered by the ac supply at any time is:
The average power delivered by the ac source is just the first term, since the average value of a cosine wave is zero even though the wave may have twice the frequency of the original input current waveform. 23

58. Effective (rms) Values
The equivalent dc value is called the effective value of the sinusoidal quantity or
and
Where: Im and Em are max (peak) values 24

59. Effective (rms) Values
Instrumentation
A true rms meter will read the effective value of any waveform and is not limited to only sinusoidal waveforms.
You should make sure that your meter is a true rms meter, by checking the manual, if waveforms other than purely sinusoidal are to be encountered. 25

60. Angular Measurement of a Sine Wave
A degree is an angular measurement corresponding to 1/360 of a circle or a complete revolution
A radian (rad) is the angular measure along the circumference of a circle that is equal to the radius of the circle
There are 2 radians or 360 in one complete cycle of a sine wave

61. Phase Relations If a sinusoidal expression should appear as
e = - Emsin t
the negative sign is associated with the sine portion of the expression, not the peak value Em .
Phase Measurements
When determining the phase measurement we first note that each sinusoidal function has the same frequency, permitting the use of either waveform to determine the period.
Since the full period represents a cycle of 360°, the following ratio can be formed: 19

62. The Sine Wave
If we define x as the number of intervals of r (the radius) around the circumference of a circle, then
C = 2r = x • r and we find x = 2
Therefore, there are 2 rad around a 360° circle, as shown in the figure. 10

63. Angular measurement showing relationship of the radian to degrees.

64. Angular measurements starting at 0º and going counterclockwise.

65. Phase reference.

66. The Sine Wave
The quantity  is the ratio of the circumference of a circle to its diameter.
For 180° and 360°, the two units of measurement are related as follows: 11

67. The Sine Wave
The sinusoidal wave form can be derived from the length of the vertical projection of a radius vector rotating in a uniform circular motion about a fixed point.
The velocity with which the radius vector rotates about the center, called the angular velocity, can be determined from the following equation: 12

68. horizontal axis in degrees

69. horizontal axis in radians

70. horizontal axis in milliseconds

71. The Sine Wave The angular velocity () is: or
Since () is typically provided in radians per second, the angle  obtained using  = t is usually in radians.
The time required to complete one revolution is equal to the period (T) of the sinusoidal waveform. The radians subtended in this time interval are 2. or 13

72. General Format for the Sinusoidal Voltage or Current
The equation  = t states that the angle  through which the rotating vector will pass is determined by the angular velocity of the rotating vector and the length of time the vector rotates.
For a particular angular velocity (fixed ), the longer the radius vector is permitted to rotate (that is, the greater the value of t ), the greater will be the number of degrees or radians through which the vector will pass.
The general format of a sine wave can also be as: 15

73. Demonstrating the effect of  on the frequency and period.

74. Phase of a Sine Wave
The phase of a sine wave is an angular measurement that specifies the position of a sine wave relative to a reference
When a sine wave is shifted left or right with respect to this reference, there is a phase shift

75. Expressions for Shifted Sine Waves
When a sine wave is shifted to the right of the reference by an angle , it is termed lagging
When a sine wave is shifted to the left of the reference by an angle , it is termed leading

76. Ohms’s Law and Kirchhoff’s Laws in AC Circuits
When time-varying ac voltages such as a sinusoidal voltage are applied to a circuit, the circuit laws that were studied earlier still apply
Ohm’s law and Kirchhoff’s laws apply to ac circuits in the same way that they apply to dc circuits

77. sinusoidal voltage produces a sinusoidal current

78. Illustration of Kirchhoff’s voltage law in an ac circuit

79. Superimposed dc and ac Voltages
DC and ac voltages will add algebraically, to produce an ac voltage “riding” on a dc level

80. Superimposed dc and ac voltages.

81. Sine waves with dc levels.

82. Pulse Waveforms
A pulse has a rapid transition (leading or rising edge) from a baseline to an amplitude level, then, after a period of time, a rapid transition (trailing or falling edge) back to the baseline level
Pulses can be positive-going, or negative-going, depending upon where the baseline is
The distance between rising and falling edge is termed the pulse width

83. Non-ideal Pulse
A non-ideal pulse has a rising and falling time interval, measured between 10% and 90% of its Amplitude
Pulse width is taken at the half-way point

84. Pulse Waveforms

85. Repetitive Pulses
Any waveform that repeats itself at fixed intervals is periodic
The time from one pulse to the corresponding point on the next pulse is the period, T ( f =1/T )
The duty cycle is the ratio of the pulse width (tw) to the period (T), and is usually expressed as %
Duty cycle = (tw/T)100%
Square waves have a 50% duty cycle

86. Repetitive Pulses

87. Square wave

88. Triangular and Sawtooth Waveforms
Triangular and sawtooth waveforms are formed by voltage or current ramps (linear increase/decrease)
Triangular waveforms have positive-going and negative-going ramps of equal slope
The sawtooth waveform is a special case of the triangular wave consisting of two ramps, one of much longer duration than the other. A sawtooth voltage is sometimes called a sweep voltage

89. Alternating triangular waveform.

90. Alternating sawtooth waveform.

91. Harmonics
A repetitive non-sinusoidal waveform is composed of a fundamental frequency (repetition rate of the waveform) and harmonic frequencies
Odd harmonics are frequencies that are odd multiples of the fundamental frequency
Even harmonics are frequencies that are even multiples of the fundamental frequency
Composite waveforms vary from a pure sine wave, they may contain only even harmonics, only odd harmonics or both

92. Odd Harmonics Produce a Square Wave

93. Summary
Conversions of sine wave values are:

94. Four-channel digital phosphor oscilloscope
Four-channel digital phosphor oscilloscope. Tektronix TDS3000B series oscilloscope.

95. AC-GND-DC switch for the vertical channel of an oscilloscope.

96. A typical dual-channel digital oscilloscope..

97. Connection for voltage probe compensation.

98. Compensation waveforms.

99. Proper Display

101. A typical dual-channel analog oscilloscope.

102. Proper triggering stabilizes a repeating waveform.