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Sinusoidal Alternating
Waveforms
13.1 INTRODUCTION
The analysis thus far has been limited to dc networks, networks in which the currents or volt- ages are fixed in magnitude except for transient effects. We now turn our attention to the analy- sis of networks in which the magnitude of the source varies in a set manner. Of particular interest is the time-varying voltage that is commercially available in large quantities and is commonly called the ac voltage. (The letters ac are an abbreviation for alternating current. ) To be absolutely rigorous, the terminology ac voltage or ac current is not sufficient to describe the type of signal we will be analyzing. Each waveform in Fig. 13.1 is an alternating wave- form available from commercial supplies. The term alternating indicates only that the wave- form alternates between two prescribed levels in a set time sequence. To be absolutely correct, the term sinusoidal, square-wave, or triangular must also be applied. The pattern of particular interest is the sinusoidal ac voltage in Fig. 13.1. Since this type of signal is encountered in the vast majority of instances, the abbreviated phrases ac voltage and ac current are commonly applied without confusion. For the other patterns in Fig. 13.1, the descriptive term is always present, but frequently the ac abbreviation is dropped, resulting in the designation square-wave or triangular waveforms. One of the important reasons for concentrating on the sinusoidal ac voltage is that it is the voltage generated by utilities throughout the world. Other reasons include its application throughout electrical, electronic, communication, and industrial systems. In addition, the chapters to follow will reveal that the waveform itself has a number of characteristics that re- sult in a unique response when it is applied to basic electrical elements. The wide range of the- orems and methods introduced for dc networks will also be applied to sinusoidal ac systems. Although the application of sinusoidal signals raise the required math level, once the notation
- Become familiar with the characteristics of a sinusoidal waveform including its general format, **_average value, and effective value.
- Be able to determine the phase relationship_** between two sinusoidal waveforms of the same **_frequency.
- Understand how to calculate the average and_** **_effective values of any waveform.
- Become familiar with the use of instruments_** designed to measure ac quantities.
Objectives
Sinusoidal Alternating
Waveforms
0 t
v
Triangular wave
0 t
v
Square wave
0 t
v
Sinusoidal
FIG. 13. Alternating waveforms.
540 ⏐⏐⏐ SINUSOIDAL ALTERNATING WAVEFORMS
(a) (b) (c) (d) (e)
Inverter
FIG. 13. Various sources of ac power: (a) generating plant; (b) portable ac generator; (c) wind-power station; (d) solar panel; (e) function generator.
given in Chapter 14 is understood, most of the concepts introduced in the dc chapters can be applied to ac networks with a minimum of added difficulty.
13.2 SINUSOIDAL ac VOLTAGE
CHARACTERISTICS AND DEFINITIONS
Generation
Sinusoidal ac voltages are available from a variety of sources. The most common source is the typical home outlet, which provides an ac voltage that originates at a power plant. Most power plants are fueled by water power, oil, gas, or nuclear fusion. In each case, an ac generator (also called an alternator ), as shown in Fig. 13.2(a), is the primary component in the energy-conversion process. The power to the shaft developed by one of the energy sources listed turns a rotor (constructed of alternating magnetic poles) inside a set of windings housed in the stator (the sta- tionary part of the dynamo) and induces a voltage across the windings of the stator, as defined by Faraday’s law:
Through proper design of the generator, a sinusoidal ac voltage is devel- oped that can be transformed to higher levels for distribution through the power lines to the consumer. For isolated locations where power lines have not been installed, portable ac generators [Fig. 13.2(b)] are available that run on gasoline. As in the larger power plants, however, an ac gen- erator is an integral part of the design. In an effort to conserve our natural resources and reduce pollution, wind power, solar energy, and fuel cells are receiving increasing interest from various districts of the world that have such energy sources avail- able in level and duration that make the conversion process viable. The turning propellers of the wind-power station [Fig. 13.2(c)] are connected directly to the shaft of an ac generator to provide the ac voltage described above. Through light energy absorbed in the form of photons, solar cells
e N
d f dt
542 ⏐⏐⏐ SINUSOIDAL ALTERNATING WAVEFORMS
1 cycle
T 1
1 cycle
T 2
1 cycle
T 3
FIG. 13. Defining the cycle and period of a sinusoidal waveform.
T =0.4 s
1 s
(b)
T =1 s
(a)
T =0.5 s
1 s
(c)
FIG. 13. Demonstrating the effect of a changing frequency on the period of a sinusoidal waveform.
FIG. 13. Heinrich Rudolph Hertz. Courtesy of the Smithsonian Institution, Photo No. 66,606. German (Hamburg, Berlin, Karlsruhe) (1857–94) Physicist Professor of Physics, Karlsruhe Polytechnic and University of Bonn Spurred on by the earlier predictions of the English physicist James Clerk Maxwell, Heinrich Hertz pro- duced electromagnetic waves in his laboratory at the Karlsruhe Polytechnic while in his early 30s. The rudimentary transmitter and receiver were in essence the first to broadcast and receive radio waves. He was able to measure the wavelength of the electromag- netic waves and confirmed that the velocity of propa- gation is in the same order of magnitude as light. In addition, he demonstrated that the reflective and refractive properties of electromagnetic waves are the same as those for heat and light waves. It was in- deed unfortunate that such an ingenious, industrious individual should pass away at the very early age of 37 due to a bone disease.
Periodic waveform: A waveform that continually repeats itself af- ter the same time interval. The waveform in Fig. 13.3 is a periodic waveform. Period ( T ): The time of a periodic waveform. Cycle: The portion of a waveform contained in one period of time. The cycles within T 1 , T 2 , and T 3 in Fig. 13.3 may appear different in Fig. 13.4, but they are all bounded by one period of time and therefore satisfy the definition of a cycle.
Frequency ( f ): The number of cycles that occur in 1 s. The frequency of the waveform in Fig. 13.5(a) is 1 cycle per second, and for Fig. 13.5(b), 2^1 ⁄ 2 cycles per second. If a waveform of similar shape had a pe- riod of 0.5 s [Fig. 13.5(c)], the frequency would be 2 cycles per second.
The unit of measure for frequency is the hertz (Hz), where
The unit hertz is derived from the surname of Heinrich Rudolph Hertz (Fig. 13.6), who did original research in the area of alternating currents and voltages and their effect on the basic R, L, and C elements. The fre- quency standard for North America is 60 Hz, whereas for Europe it is pre- dominantly 50 Hz. As with all standards, any variation from the norm will cause difficul- ties. In 1993, Berlin, Germany, received all its power from eastern plants, whose output frequency was varying between 50.03 Hz and 51 Hz. The result was that clocks were gaining as much as 4 minutes a day. Alarms went off too soon, VCRs clicked off before the end of the program, and so on, requiring that clocks be continually reset. In 1994, however, when power was linked with the rest of Europe, the precise standard of 50 Hz was reestablished and everyone was on time again.
1 hertz 1 Hz 2 1 cycle per second 1 cps 2
FREQUENCY SPECTRUM ⏐⏐⏐ 543
0 0.2 0.4 0.6 0.8 1.0 1.2 1.
8 V
–8 V
v
t (s)
FIG. 13. Example 13.1.
EXAMPLE 13.1 For the sinusoidal waveform in Fig. 13.7.
a. What is the peak value? b. What is the instantaneous value at 0.3 s and 0.6 s? 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? f. What is the frequency of the waveform?
Solutions:
a. 8 V. b. At 0.3 s, 8 V; at 0.6 s, 0 V. c. 16 V. d. 0.4 s. e. 3.5 cycles. f. 2.5 cps, or 2.5 Hz.
13.3 FREQUENCY SPECTRUM
Using a log scale (described in detail in Chapter 20), a frequency spec- trum from 1 Hz to 1000 GHz can be scaled off on the same axis, as shown in Fig. 13.8. A number of terms in the various spectrums are probably fa- miliar to you from everyday experiences. Note that the audio range (hu- man ear) extends from only 15 Hz to 20 kHz, but the transmission of radio signals can occur between 3 kHz and 300 GHz. The uniform process of defining the intervals of the radio-frequency spectrum from VLF to EHF is quite evident from the length of the bars in the figure (al- though keep in mind that it is a log scale, so the frequencies encompassed within each segment are quite different). Other frequencies of particular interest (TV, CB, microwave, and so on) are also included for reference purposes. Although it is numerically easy to talk about frequencies in the megahertz and gigahertz range, keep in mind that a frequency of 100 MHz, for instance, represents a sinusoidal waveform that passes through 100,000,000 cycles in only 1 s—an incredible number when we compare it to the 60 Hz of our conventional power sources. The Intel ® Pentium®^ 4 chip manufactured by Intel can run at speeds over 2 GHz. Imagine a product able to handle 2 billion instructions per second—an incredible achievement.
FREQUENCY SPECTRUM ⏐⏐⏐ 545
Since the frequency is inversely related to the period—that is, as one increases, the other decreases by an equal amount—the two can be re- lated by the following equation:
f Hz T seconds (s)
or (13.3)
EXAMPLE 13.2 Find the period of periodic waveform with a fre- quency of
a. 60 Hz. b. 1000 Hz.
Solutions:
a.
(a recurring value since 60 Hz is so prevalent)
b.
EXAMPLE 13.3 Determine the frequency of the waveform in Fig. 13.9.
Solution: From the figure, T (25 ms 5 ms) or (35 ms 15 ms) 20 ms, and
In Fig. 13.10, the seismogram resulting from a seismometer near an earthquake is displayed. Prior to the disturbance, the waveform has a rel- atively steady level, but as the event is about to occur, the frequency begins
f
T
20 10 ^3 s
50 Hz
T
f
1000 Hz
10 ^3 s 1 ms
T
f
60 Hz
0.01667 s or 16.67 ms
T
f
f
T
0 t (ms)
10 V
e
5 15 25 35
FIG. 13. Example 13.3.
Relatively low frequency, low amplitude
Relatively high frequency, high amplitude Relatively high frequency, low amplitude
East–West BNY OCT23(296), 10:41 GMT
Time (minutes) from 10:41:00.000 GMT
0
1
2
X 10+ 50 55 60 65 70 75 80 85 90
FIG. 13. Seismogram from station BNY (Binghamton University) in New York due to magnitude 6.7 earthquake in Central Alaska that occurred at 63.62°N, 148.04°W, with a depth of 10 km, on Wednesday, October 23, 2002.
546 ⏐⏐⏐ SINUSOIDAL ALTERNATING WAVEFORMS
(a)
e
e
t
i
(b)
i t
FIG. 13. (a) Sinusoidal ac voltage sources; (b) sinusoidal current sources.
i
R, L, or C v t
t
FIG. 13. The sine wave is the only alternating waveform whose shape is not altered by the response characteristics of a pure resistor, inductor, or capacitor.
r
r
57.296°
1 radian
FIG. 13. Defining the radian.
to increase along with the amplitude. Finally, the earthquake occurs, and the frequency and the amplitude increase dramatically. In other words, the relative frequencies can be determined simply by looking at the tightness of the waveform and the associated period. The change in amplitude is im- mediately obvious from the resulting waveform. The fact that the earth- quake lasts for only a few minutes is clear from the horizontal scale.
Defined Polarities and Direction
You may be wondering how a polarity for a voltage or a direction for a current can be established if the waveform moves back and forth from the positive to the negative region. For a period of time, a voltage has one po- larity, while for the next equal period it reverses. To take care of this problem, a positive sign is applied if the voltage is above the axis, as shown in Fig. 13.11(a). For a current source, the direction in the sym- bol corresponds with the positive region of the waveform, as shown in Fig. 13.11(b). For any quantity that will not change with time, an uppercase letter such as V or I is used. For expressions that are time dependent or that rep- resent a particular instant of time, a lowercase letter such as e or i is used. The need for defining polarities and current direction becomes quite obvious when we consider multisource ac networks. Note in the last sen- tence the absence of the term sinusoidal before the phrase ac networks. This phrase will be used to an increasing degree as we progress; sinusoidal is to be understood unless otherwise indicated.
13.4 THE SINUSOIDAL WAVEFORM
The terms defined in the previous section can be applied to any type of periodic waveform, whether smooth or discontinuous. The sinusoidal waveform is of particular importance, however, since it lends itself read- ily to the mathematics and the physical phenomena associated with elec- tric circuits. Consider the power of the following statement: The sinusoidal waveform is the only alternating waveform whose shape is unaffected by the response characteristics of R, L, and C elements. In other words, if the voltage across (or current through) a resistor, in- ductor, or capacitor is sinusoidal in nature, the resulting current (or volt- age, respectively) for each will also have sinusoidal characteristics, as shown in Fig. 13.12. If any other alternating waveform such as a square wave or a triangular wave were applied, such would not be the case. The unit of measurement for the horizontal axis can be time (as ap- pearing in the figures thus far), degrees , or radians. The term radian can be defined as follows: If we mark off a portion of the circumference of a circle by a length equal to the radius of the circle, as shown in Fig. 13.13, the angle resulting is called 1 radian. The result is
where 57.3° is the usual approximation applied. One full circle has 2p radians, as shown in Fig. 13.14. That is,
2 p rad 360° (13.5)
1 rad 57.296° 57.3°
548 ⏐⏐⏐ SINUSOIDAL ALTERNATING WAVEFORMS
(a) (b)
v , i , etc.
0 4
2 4
(^3)
4
(^5) 2
(^3) 4
(^7) 2 (radians)
v , i , etc.
0 45 ° 90 ° 135 ° ^ (degrees)
225 ° 270 ° 315 ° 360 ° 180 °^
FIG. 13. Plotting a sine wave versus (a) degrees and (b) radians.
For comparison purposes, two sinusoidal voltages are plotted in Fig. 13.15 using degrees and radians as the units of measurement for the hor- izontal axis. It is of particular interest that the sinusoidal waveform can be derived from the length of the vertical projection of a radius vector rotating in a uniform circular motion about a fixed point. Starting as shown in Fig. 13.16(a) and plotting the amplitude (above and below zero) on the coor- dinates drawn to the right [Figs. 13.16(b) through (i)], we will trace a complete sinusoidal waveform after the radius vector has completed a 360° rotation about the center. The velocity with which the radius vector rotates about the center, called the angular velocity, can be determined from the following equation:
Substituting into Eq. (13.8) and assigning the lowercase Greek letter omega (v) to the angular velocity, we have
and (13.10)
Since v is typically provided in radians per second, the angle a ob- tained using Eq. (13.10) is usually in radians. If a is required in degrees, Eq. (13.7) must be applied. The importance of remembering the above will become obvious in the examples to follow. In Fig. 13.16, the time required to complete one revolution is equal to the period ( T ) of the sinusoidal waveform in Fig. 13.16(i). The radians subtended in this time interval are 2p. Substituting, we have
(rad/s) (13.11)
In words, this equation states that the smaller the period of the sinu- soidal waveform of Fig. 13.16(i), or the smaller the time interval before one complete cycle is generated, the greater must be the angular velocity of the rotating radius vector. Certainly this statement agrees with what we have learned thus far. We can now go one step further and apply the fact
v
2 p T
a v t
v
a t
Angular velocity
distance 1 degrees or radians 2 time 1 seconds 2
THE SINUSOIDAL WAVEFORM ⏐⏐⏐ 549
0 ° 45 °^90 °^135 ° 180 °
225 ° 270 ° 315 ° 360 °
T (period)
Sine wave
(i) α
α = 360°
0 °
315 ° (h) α
α = 315°
0 °
(g) α
α = 270° 270 °
0 °
(f) α
α = 225° 225 °
0 °
(e) α
α = 180°
180 °
0 °
(d) α
α = 135° 45 ° 90 ° 135 °
0 ° (c) α
α = 90° 90 °
0 °
(b) α
α = 45° 45 °
Note equality
0 ° (a) α = 0° α
FIG. 13. Generating a sinusoidal waveform through the vertical projection of a rotating vector.
GENERAL FORMAT FOR THE SINUSOIDAL VOLTAGE OR CURRENT ⏐⏐⏐ 551
0
ππ, 180 °^2 ππ, 360° α α (° or rad) Am
Am
FIG. 13. Basic sinusoidal function.
13.5 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 and a is the unit of measure for the horizontal axis, as shown in Fig. 13.18. The equation a v t states that the angle a through which the rotat- ing vector in Fig. 13.16 will pass is determined by the angular velocity of the rotating vector and the length of time the vector rotates. For example, for a particular angular velocity (fixed v), the longer the radius vector is permitted to rotate (that is, the greater the value of t ), the greater the num- ber of degrees or radians through which the vector will pass. Relating this statement to the sinusoidal waveform, for a particular angular velocity, the longer the time, the greater the number of cycles shown. For a fixed time interval, the greater the angular velocity, the greater the number of cycles generated. Due to Eq. (13.10), the general format of a sine wave can also be written
with v t as the horizontal unit of measure. For electrical quantities such as current and voltage, the general for- mat is
i Im sin v t Im sin a e Em sin v t Em sin a
where the capital letters with the subscript m represent the amplitude, and the lowercase letters i and e represent the instantaneous value of current and voltage, respectively, at any time t. This format is particularly important be- cause it presents the sinusoidal voltage or current as a function of time, which is the horizontal scale for the oscilloscope. Recall that the horizon- tal sensitivity of a scope is in time per division, not degrees per centimeter.
EXAMPLE 13.8 Given e 5 sin a, determine e at a 40° and a 0.8p.
Solution: For a 40°,
e 5 sin 40° 5(0.6428) 3.21 V
For a 0.8p,
and e 5 sin 144° 5(0.5878) 2.94 V
The angle at which a particular voltage level is attained can be deter- mined by rearranging the equation
e Em sin a
a 1 ° 2
p 1 0.8p 2 144°
Am sin v t
Am sin a
552 ⏐⏐⏐ SINUSOIDAL ALTERNATING WAVEFORMS
v (V)
4
1 90 °
10
0 t 1
2 t 2
180 °
FIG. 13. Example 13.9.
in the following manner:
which can be written
Similarly, for a particular current level,
EXAMPLE 13.
a. Determine the angle at which the magnitude of the sinusoidal func- tion y 10 sin 377 t is 4 V. b. Determine the time at which the magnitude is attained.
Solutions: a. Eq. (13.15):
However, Fig.13.19 reveals that the magnitude of 4 V (positive) will be attained at two points between 0° and 180°. The second in- tersection is determined by a 2 180° 23.578° 156.42° In general, therefore, keep in mind that Eqs. (13.15) and (13.16) will provide an angle with a magnitude between 0° and 90°. b. Eq. (13.10): a v t, and so t a/v. However, a must be in radi- ans. Thus,
and
For the second intersection,
Calculator Operations
Both sin and sin^1 are available on all scientific calculators. You can also use them to work with the angle in degrees or radians without hav- ing to convert from one form to the other. That is, if the angle is in ra-
t 2
a v
2.73 rad 377 rad>s
7.24 ms
a 1 rad 2
p 180°
1 156.422° 2 2.73 rad
t 1
a v
0.412 rad 377 rad>s
1.09 ms
a 1 rad 2
p 180°
1 23.578° 2 0.412 rad
a 1 sin^1
y Em
sin^1
4 V
10 V
sin^1 0.4 23.58°
a sin^1
i Im
a sin^1
e Em
sin a
e Em
554 ⏐⏐⏐ SINUSOIDAL ALTERNATING WAVEFORMS
10
0 ° 30 ° 90 °
180 ° 270 ° 360 ° αα (°)
e
10
FIG. 13. Example 13.10, horizontal axis in degrees.
(^0) π α α (rad) — 2 π — 6
π —^32 π 2 π
10
e
10
FIG. 13. Example 13.10, horizontal axis in radians.
0 1.
10 15 20
10
T = 20 ms
5 t (ms)
10
FIG. 13. Example 13.10, horizontal axis in milliseconds.
( – )
Am
(2 – )
Am sin ^
FIG. 13. Defining the phase shift for a sinusoidal function that crosses the horizontal axis with a positive slope before 0°.
Solutions: a. See Fig. 13.24. (Note that no calculations are required.) b. See Fig. 13.25. (Once the relationship between degrees and radians is understood, no calculations are required.) c. See Fig 13.26.
T
20 ms 12
1.67 ms
T
20 ms 4
5 ms
T
20 ms 2
10 ms
360°: T
2 p v
2 p 314
20 ms
EXAMPLE 13.11 Given i 6 10 ^3 sin 1000 t, determine i at t 2 ms.
Solution:
13.6 PHASE RELATIONS
Thus far, we have considered only sine waves that have maxima at p/ and 3p/2, with a zero value at 0, p, and 2p, as shown in Fig. 13.25. If the waveform is shifted to the right or left of 0°, the expression becomes
where u is the angle in degrees or radians that the waveform has been shifted. If the waveform passes through the horizontal axis with a positive- going (increasing with time) slope before 0°, as shown in Fig. 13.27, the expression is
Am sin 1 v t u 2 (13.18)
Am sin 1 v t u 2
i 16 10 ^3 2 1sin 114.59° 2 1 6 mA2 10.9093 2 5.46 mA
a 1 ° 2
p rad
1 2 rad 2 114.59°
a v t 1000 t 1 1000 rad>s2 1 2 10 ^3 s 2 2 rad
PHASE RELATIONS ⏐⏐⏐ 555
v (p^ +^ v)
A (^) m (2p + v)
FIG. 13. Defining the phase shift for a sinusoidal function that crosses the horizontal axis with a positive slope after 0°.
0
A (^) m
90 °
cos sin
p 2 p
p
(^3) p 2
FIG. 13. Phase relationship between a sine wave and a cosine wave.
At v t a 0°, the magnitude is determined by Am sin u. If the wave- form passes through the horizontal axis with a positive-going slope after 0°, as shown in Fig. 13.28, the expression is
Finally, at v t a 0°, the magnitude is Am sin (u), which, by a trigonometric identity, is Am sin u. If the waveform crosses the horizontal axis with a positive-going slope 90° (p/2) sooner, as shown in Fig. 13.29, it is called a cosine wave; that is,
or (13.21)
The terms leading and lagging are used to indicate the relationship between two sinusoidal waveforms of the same frequency plotted on the same set of axes. In Fig. 13.29, the cosine curve is said to lead the sine curve by 90°, and the sine curve is said to lag the cosine curve by 90°. The 90° is referred to as the phase angle between the two waveforms. In language commonly applied, the waveforms are out of phase by 90°. Note that the phase angle between the two waveforms is measured be- tween those two points on the horizontal axis through which each passes with the same slope. If both waveforms cross the axis at the same point with the same slope, they are in phase. The geometric relationship between various forms of the sine and co- sine functions can be derived from Fig. 13.30. For instance, starting at the sin a position, we find that cos a is an additional 90° in the counter- clockwise direction. Therefore, cos a sin(a 90°). For sin a we must travel 180° in the counterclockwise (or clockwise) direction so that sin a sin (a 180°), and so on, as listed below:
In addition, note that
If a sinusoidal expression appears as e Em sin v t
the negative sign is associated with the sine portion of the expression, not the peak value Em. In other words, the expression, if not for convenience, would be written
e Em (sin v t )
sin 1 a 2 sin a cos 1 a 2 cos a
cos a sin 1 a 90° 2 sin a cos 1 a 90° 2 sin a sin 1 a 180° 2 cos a sin 1 a 270° 2 sin 1 a 90° 2 etc.
sin v t cos 1 vt 90° 2 cos a v t
p 2
b
sin 1 v t 90° 2 sin a v t
p 2
b cos v t
Am sin 1 v t u 2
α
α α
α
sin( α +90°)
cos( α –90°)
FIG. 13. Graphic tool for finding the relationship between specific sine and cosine functions.
PHASE RELATIONS ⏐⏐⏐ 557
10 15
i v
2
(^3) 2 2 t
20 ° 80 °
60 °
0
FIG. 13. Example 13.12(b): i leads y by 80°.
d. See Fig. 13.34. Note sin(v t 30°) sin(v t 30° 180°) sin(v t 150°) Y leads i by 160º, or i lags Y by 160º. Or using Note sin(v t 30°) sin(v t 30° 180°) sin(v t 210°) i leads Y by 200º, or Y lags i by 200º.
i
2^ v^3
10 °
110 °
2
0 3 2 ^
(^2) t
100 °
FIG. 13. Example 13.12(c): i leads y by 110°.
2 1
2
2 2
5 2 3 t
10 ° 160 ° 200 ° 360 °
0
i
v
150 °
FIG. 13. Example 13.12(d): y leads i by 160°.
2
2
2 5 2 ^
3
t 0
i
v
150 °
2 3
FIG. 13. Example 13.12(e): y and i are in phase.
e. See Fig. 13.35. By choice i 2 cos(v t 60°) 2 cos(v t 60° 180°) 2 cos(v t 240°)
→
→
→
558 ⏐⏐⏐ SINUSOIDAL ALTERNATING WAVEFORMS
Vertical sensitivity=0.1 V/div. Horizontal sensitivity=50 s/div.
FIG. 13. Example 13.13.
However, cos a sin(a 90°) so that 2 cos(v t 240°) 2 sin(v t 240° 90°) 2 sin(v t 150°) Y and i are in phase.
The Oscilloscope
The oscilloscope is an instrument that will display the sinusoidal alter- nating waveform in a way that will permit the reviewing of all of the waveform’s characteristics. In some ways, the screen and the dials give an oscilloscope the appearance of a small TV, but remember that it can display only what you feed into it. You can’t turn it on and ask for a sine wave, a square wave, and so on; it must be connected to a source or an active circuit to pick up the desired waveform. The screen has a standard appearance, with 10 horizontal divisions and 8 vertical divisions. The distance between divisions is 1 cm on the vertical and horizontal scales, providing you with an excellent opportunity to be- come aware of the length of 1 cm. The vertical scale is set to display volt- age levels, whereas the horizontal scale is always in units of time. The vertical sensitivity control sets the voltage level for each division, whereas the horizontal sensitivity control sets the time associated with each division. In other words, if the vertical sensitivity is set at 1 V/div., each division dis- plays a 1 V swing, so that a total vertical swing of 8 divisions represents 8 V peak-to-peak. If the horizontal control is set on 10 ms/div., 4 divisions equal a time period of 40 ms. Remember, the oscilloscope display presents a si- nusoidal voltage versus time, not degrees or radians. Further, the vertical scale is always a voltage sensitivity, never units of amperes.
EXAMPLE 13.13 Find the period, frequency, and peak value of the si- nusoidal waveform appearing on the screen of the oscilloscope in Fig. 13.36. Note the sensitivities provided in the figure. Solution: One cycle spans 4 divisions. Therefore, the period is
and the frequency is
The vertical height above the horizontal axis encompasses 2 divisions. Therefore,
An oscilloscope can also be used to make phase measurements be- tween two sinusoidal waveforms. Virtually all laboratory oscilloscopes today have the dual-trace option, that is, the ability to show two wave- forms at the same time. It is important to remember, however, that both waveforms will and must have the same frequency. The hookup proce- dure for using an oscilloscope to measure phase angles is covered in de-
Vm 2 div. a
0.1 V
div.
b 0.2 V
f
T
200 10 ^6 s
5 kHz
T 4 div. a
50 ms div.
b 200 M s
