AC Waveforms Electronics Performance Objective Upon completion of this lesson, the student will be able to explain what alternating current (AC) electricity is and how it is generated. The student will be able to demonstrate how to perform calculations to convert one value of AC voltage into other values by completing the AC Waveforms Exam. Specific Objectives
Understand how a sine wave of alternating voltage is generated.
Explain the three ways to express the amplitude of a sinusoidal waveform and the relationship between them.
Define the following values for a sine wave: peak, peak-to-peak, root mean square, average, and instantaneous.
Calculate the RMS, average, and peak-to-peak values of a sine wave when the peak value is known.
Calculate the instantaneous value of a sine wave.
Convert peak, peak-to-peak, average, and RMS voltage and current values from one value to another.
Explain the importance of the .707 constant and how it is derived.
Define frequency and period and list the units of each.
Calculate the period when the frequency is known and frequency when the period is known.
Explain the sine, cosine, and tangent trigonometric functions.
Understand the concept of phase angles.
Calculate the value of the sine of any angle between 0and 360. Terms
Alternating current (AC)- a flow of electric charge that periodically changes direction.
Direct current (DC)- the unidirectional flow of electric charge.
Transformer- a device that uses electromagnetic induction to change one value of AC voltage into a different value of AC voltage.
Generator- a device that changes mechanical energy into electrical energy.
Sine wave- a type of repeating curved wave shape created by rotational motion through a magnetic field and described by the mathematical sine function.
Wave- a disturbance traveling through a medium.
Waveform- a graphic representation of a wave.
Period (P)- the time required to complete one cycle of a waveform.
Frequency (f)- the number of cycles of a waveform that occur in one second of time.
Amplitude- height of a wave.
Peak- the maximum positive or negative deviation of a waveform from its zero reference level.
Peak-to-peak- the measurement from the highest amplitude peak to the lowest peak.
Root mean square (RMS)- the value of AC voltage that creates the same heat as the same numerical value of DC voltage.
Average voltage- the DC equivalent voltage over a half cycle of an AC voltage wave.
Instantaneous voltage- the voltage at a single point or instant of time. Time It should take approximately four, 45-minute class periods to teach the lesson and three, 45-minute class periods to practice calculations, complete the lab and worksheets, and take the quiz. TEKS Correlations This lesson, as published, correlates to the following TEKS. Any changes/alterations to the activities may result in the elimination of any or all of the TEKS listed. Electronics
130.368 (c) o (5) The student implements the concepts and skills that form the technical knowledge of
electronics using project-based assessments. The student is expected to: (A) apply Ohm's law, Kirchoff's laws, and power laws; (B) demonstrate an understanding of magnetism and induction as they relate to electronic circuits; and (C) demonstrate knowledge of the fundamentals of electronics theory.
o (6) The student applies the concepts and skills to simulated and actual work situations. The student is expected to:
(A) measure and calculate resistance, current, voltage, and power in series, parallel, and complex circuits; and (B) apply electronic theory to generators, electric motors, and transformers.
o (8) The student learns the function and application of the tools, equipment, and materials used in electronics through project-based assignments. The student is expected to:
(A) safely use tools and laboratory equipment to construct and repair circuits; and (B) use precision measuring instruments to analyze circuits and prototypes.
Interdisciplinary Correlations Precalculus
111.35 (c) o (1) The student defines functions, describes characteristics of functions, and translates among
verbal, numerical, graphical, and symbolic representations of functions, including polynomial, rational, power (including radical), exponential, logarithmic, trigonometric, and piecewise-defined functions. The student is expected to:
(D) recognize and use connections among significant values of a function (zeros, maximum values, minimum values, etc.), points on the graph of a function, and the symbolic representation of a function.
o (3) The student uses functions and their properties, tools and technology, to model and solve meaningful problems. The student is expected to:
(A) investigate properties of trigonometric and polynomial functions; (B) use functions such as logarithmic, exponential, trigonometric, polynomial, etc. to model real-life data; and (E) solve problems from physical situations using trigonometry, including the use of Law of Sines, Law of Cosines, and area formulas and incorporate radian measure where needed.
Plan layout and installation of electrical wiring, equipment, or fixtures, based on job specifications and local codes.
Connect wires to circuit breakers, transformers, or other components.
Test electrical systems or continuity of circuits in electrical wiring, equipment, or fixtures, using testing devices, such as ohmmeters, voltmeters, or oscilloscopes, to ensure compatibility and safety of system.
Use a variety of tools or equipment, such as power construction equipment, measuring devices, power tools, and testing equipment, such as oscilloscopes, ammeters, or test lamps.
Inspect electrical systems, equipment, or components to identify hazards, defects, or the need for adjustment or repair, and to ensure compliance with codes.
Prepare sketches or follow blueprints to determine the location of wiring or equipment and to ensure conformance to building and safety codes.
Diagnose malfunctioning systems, apparatus, or components, using test equipment and hand tools to locate the cause of a breakdown and correct the problem.
Work from ladders, scaffolds, or roofs to install, maintain, or repair electrical wiring, equipment, or fixtures.
Advise management on whether continued operation of equipment could be hazardous.
Maintain current electrician's license or identification card to meet governmental regulations. Soft Skills
Instructing Accommodations for Learning Differences These lessons accommodate the needs of every learner. Modify the lessons to accommodate your students with learning differences by referring to the files found on the Special Populations page of this website. Preparation
Mitchel E. Schultz. (2007). Grob’s basic electronics fundamentals of DC and AC circuits. Columbus, Ohio: McGraw Hill.
Instructional Aids
AC Waveforms (with Exercise Key) slide presentation and notes
Study Guide 1
Study Guide 1 Answer Key
Study Guide 2
Study Guide 2 Answer Key
AC Waveforms Exam
AC Waveforms Exam Answer Key
This lesson discusses Alternating Current and various ways to describe the amplitude of a sinusoidal waveform. The time and frequency measurement of a waveform are reviewed, and an introduction to the trigonometric function are presented.
Show o An electrical outlet.
Ask o Most of us take for granted the huge infrastructure necessary to bring electricity into our homes.
Does anyone really know the characteristics of this common energy source?
Say o Today we are going to start learning about this important energy source. Let’s get started.
I. Introduction and objectives A. Overview the lesson objectives. B. Introduce important terms. C. Discuss how AC is an important technology that
people tend to take for granted. D. Discuss AC and the use of common trigonometric
functions in real world examples.
II. What is AC? A. Contrast/compare alternating current (AC) and
direct current (DC). B. Explain that movement of electrons can perform
work whether the movement is always in the same direction or back and forth.
C. Discuss motors that are used in common household items (fans, air conditioners, washing machines, dryers, refrigerators, and pumps).
D. Electronic devices can create AC, but electronics does not work on AC.
E. Electronic devices typically use a power supply to convert AC into DC for use.
III. Why change voltage?
A. Discuss the importance of transformers in power distribution and increased efficiency over long distances with AC voltage.
B. A transformer can step voltage up (or down) but power in always equals power out.
C. As voltage is stepped up, current goes down and current is what creates heat as it travels through wires in a power distribution system.
D. AC is the only type of voltage that works in a transformer.
E. Until very recently there was no good method to step up or down DC voltages.
NOTES TO TEACHER Slides 1-4 Talk about things you are going to be covering in this lesson and be careful to not make it sound more difficult than it is. Slides 5-7 Slides 8-9 In the late 1800s there was a huge battle between the advocates of DC (Thomas Edison) and the advocates of AC (Charles Westinghouse). This battle was called “the war of currents” and makes for a good research paper.
A. A waveform is a plot of voltage vs time. B. A waveform is the shape of a signal. C. There are several ways to characterize the amount
of voltage given by a changing waveform. D. Peak-to-peak is the value seen by an oscilloscope. E. The peak value is the absolute value of the
maximum amplitude. F. RMS is a heating value. G. Average is the DC equivalent voltage. H. Angles are given in either degrees or radians.
VIII. Time-based waveform terms (frequency and period)
A. Frequency and period have nothing to do with amplitude.
B. Frequency and period are time-based terms. C. Wavelength is another term that often gets used in
the same context as frequency and period, but wavelength is not a time-based term.
D. Wavelength is a distance that relates to the size of the wave.
E. Inverse means one over something (there is a button on a calculator that performs this function:
).
IX. Waveform terminology
A. Each of these terms has a different value associated with it.
B. The ability to convert from one value to any other value is important to know.
C. We give the descriptions first then the formulas needed to perform conversions.
D. There are three terms in each of the formulas, and in most cases, the third term is a constant.
E. For instantaneous voltage, all three terms are variables, where any two terms are necessary to calculate the third.
F. Regular trigonometric functions take the sine of an angle to get a value, and inverse functions give the angle from the ratio value.
MI
NOTES TO TEACHER Slides 34-38 Slides 39-41 Slides 42-51 Have students practice calculating each of these values, particularly those using the inverse function.
A. Use the formula relationship circles as a memory aide.
B. Have students work example problems in slide presentation.
C. Create more problems for students to practice working.
D. Students will complete the table on the last page of Study Guide 1.
I. Problems 1 and 2 as guided practice II. Problems 3-5 as independent practice
XI. Trigonometric functions
A. The motion of a conductor through the magnetic field can be resolved into a right triangle.
B. The sine function is used because motion perpendicular to the magnetic field is the opposite side of the triangle.
C. Right-triangle side and angle relationships (hypotenuse, opposite, adjacent).
D. Basic trigonometric functions (sine, cosine, and tangent).
E. Motion parallel to the magnetic field would be the adjacent side of the triangle.
F. Trigonometric exercise - Study Guide 2 i. Triangles 1 and 2 are teacher-guided
practice ii. Triangles 3 and 4 are students’
independent practice
XII. AC Waveforms Exam
NOTES TO TEACHER Slides 52-61 Pass out Study Guide 1 and have students complete the table for guided and independent practice. Slides 62-69 Upon completion of slides, pass out student Study Guide 2 for guided and independent practice. Slides 70-76 Review answer keys with students. Administer the AC Waveforms Exam and grade with answer key.
Study Guide 2 - triangles 1 and 2 Independent Practice
Study Guide 1 - problems 3 and 4
Study Guide 2 - triangles 3 and 4 Review Have students restate lesson objectives.
Informal Assessment The teacher monitors students during guided and independent practice activities and may assign a grade for the independent activities. Formal Assessment Students will complete the AC Waveforms Exam. Extension To add an interdisciplinary component, students will write a short research paper on “the war of the currents.” Students will create their own problems for independent practice. See lesson outline (page 5/Notes to Teacher).
Waveform A wave is a disturbance traveling through a medium. A waveform is a graphic representation of a wave. Like a wave, a waveform depends on movement and on time. The ripple on the surface of a pond is a movement of water in time. Wave shapes tell you a great deal about the signal. Any time you see a change in the vertical dimension of a signal, you know that this amplitude change represents a change in voltage. Wave shapes alone are not the whole story. To completely describe a waveform, you will need to find its particular parameters. Depending on the signal, these parameters might be frequency, period, amplitude, width, rise time, or phase. Frequency The frequency of a waveform is the number of cycles of the waveform that occur in one second of time. The common unit of measurement is hertz (Hz). Period The period of a waveform, which sometimes is called its time, is the time required to complete one cycle of a waveform. It is measured in units of seconds, such as seconds, tenths of seconds, milliseconds, or microseconds.
If a waveform is to be properly described in terms of its period or frequency, it must be a repetitious waveform. A repetitious waveform is one in which each following cycle is identical to the previous cycle. Waveform Amplitude Specifications In addition to frequency and period values, a third major specification of a waveform is the amplitude or height of the wave. There are three possible ways to express the amplitude of a sinusoidal waveform: peak, peak-to-peak, and root-mean-square (RMS).
1. Peak The peak amplitude of a sinusoidal waveform is the maximum positive or negative deviation of a waveform from its zero reference level. The sinusoidal waveform is a symmetrical waveform, so the positive peak value is the same as the negative peak value as shown in Figure 2.2. If the positive peak has a value of 10 volts, then the negative peak will also have a value of 10 volts. When measuring the peak value of a waveform, either positive or negative peaks can be used.
2. Peak–to-Peak The peak-to-peak amplitude is simply a measurement of the amplitude of a waveform taken from its positive peak to its negative peak as shown in Figure 2.3.
For sinusoidal waveform, if the positive peak value is 10 volts in magnitude, then the negative peak value of the same waveform is also 10 volts. Measuring from peak-to-peak, there is a total of 20 volts. Therefore, the value of the sinusoidal waveform in Figure 2.2 can be specified as either 10 volts peak or 20 volts peak-to-peak.
For the non-sinusoidal waveform shown in Figure 2.4, the peak-to-peak value of the voltage can be determined by adding the magnitude of the positive and the negative peak. In this example, the peak-to-peak amplitude is 18 volts plus 2 volts for a total of 20 volts, peak-to-peak. 3. Root-Mean-Square The third specification for AC waveform is called root-mean-square, abbreviated RMS. This term allows the comparison of AC and DC circuit values. Root-mean-square values are the most common methods of specifying sinusoidal waveforms. In fact, almost all AC voltmeter and ammeters are calibrated so that they measure AC values in terms of RMS amplitude. RMS Relations to DC Heating Effect The RMS value is also known as the effective value and is defined in terms of the equivalent heating effect of direct current. The RMS value of a sinusoidal voltage is equivalent to the value of a DC voltage, which causes an equal amount of heat due to the circuit current flowing through a resistance. The RMS value of a sinusoidal voltage or current waveform is 70.7 percent or 0.707 of its peak amplitude value. VRMS = 0.707 Vpeak
IRMS = 0.707 Ipeak A sinusoidal voltage with peak amplitude of 1 volt has the same effect as a DC voltage of 0.707 volts as far as its ability to reproduce the same amount of heat in a resistance. Because the AC voltage of 1 volt peak or 0.707 volts RMS is as effective as a DC voltage of 0.707 volts, the RMS value of voltage is also referred to as the effective value.
Determining the 0.707 Constant How is the 70.7 percent of peak-value constant derived? Essentially, the words root-mean-square define the mathematical procedure used to determine the constant.
The Sine Wave and Sine Trigonometric Function The term sinusoidal has been used to describe a waveform produced by an AC generator. The term sinusoidal comes from a trigonometric function called the sine function. Right-Triangle Side and Angle Relationships Trigonometry is the study of triangles and their relationships involving lengths and angles of triangles. The
basic triangle studied in trigonometry is a right triangle, which is a triangle that has a 90◦ angle as one of its
three angles. A 90◦ triangle has a unique set of relationships from which the rules for trigonometry are derived. To help distinguish the sides of a right triangle from one another, a name is given to each side. The sides of the triangle are named with respect to the angle theta. The side of the triangle across from or opposite to the angle theta is called the opposite side. The longest side of a right triangle is called the hypotenuse. The remaining side is called the adjacent side because it lies beside or adjacent to the angle. These three names are commonly abbreviated to their first initials, O, H, and A. Basic Trigonometric Functions In trigonometry, these ratios have specific names. The three most commonly-used ratios in the study of right triangles are called sine, cosine, and tangent. The sine of the angle theta is equal to the ratio formed by the length of the opposite side divided by the length of the hypotenuse:
Sine ө = opposite
hypotenuse
The cosine of the angle theta is equal to the ratio formed by length of the adjacent side divided by the length
of the hypotenuse:
Cosine ө = adjacent
hypotenuse The tangent of the angle theta is equal to the ratio formed by length of the opposite side divided by the length of the adjacent side:
Waveform A wave is a disturbance traveling through a medium. A waveform is a graphic representation of a wave. Like a wave, a waveform depends on movement and on time. The ripple on the surface of a pond is a movement of water in time. Wave shapes tell you a great deal about the signal. Any time you see a change in the vertical dimension of a signal, you know that this amplitude change represents a change in voltage. Wave shapes alone are not the whole story. To completely describe a waveform, you will need to find its particular parameters. Depending on the signal, these parameters might be frequency, period, amplitude, width, rise time, or phase. Frequency The frequency of a waveform is the number of cycles of the waveform that occur in one second of time. The common unit of measurement is hertz (Hz). Period The period of a waveform, which sometimes is called its time, is the time required to complete one cycle of a waveform. It is measured in units of seconds, such as seconds, tenths of seconds, milliseconds, or microseconds.
If a waveform is to be properly described in terms of its period or frequency, it must be a repetitious waveform. A repetitious waveform is one in which each following cycle is identical to the previous cycle. Waveform Amplitude Specifications In addition to frequency and period values, a third major specification of a waveform is the amplitude or height of the wave. There are three possible ways to express the amplitude of a sinusoidal waveform: peak, peak-to-peak, and root-mean-square (RMS).
1. Peak The peak amplitude of a sinusoidal waveform is the maximum positive or negative deviation of a waveform from its zero reference level. The sinusoidal waveform is a symmetrical waveform, so the positive peak value is the same as the negative peak value as shown in figure 2.2. If the positive peak has a value of 10 volts, then the negative peak will also have a value of 10 volts. When measuring the peak value of a waveform, either positive or negative peaks can be used.
2. Peak–to-Peak The peak-to-peak amplitude is simply a measurement of the amplitude of a waveform taken from its positive peak to its negative peak as shown in figure 2.3.
For sinusoidal waveform, if the positive peak value is 10 volts in magnitude, then the negative peak value of the same waveform is also 10 volts. Measuring from peak-to-peak, there is a total of 20 volts. Therefore, the value of the sinusoidal waveform in Figure 2.2 can be specified as either 10 volts peak or 20 volts peak-to-peak.
For the non-sinusoidal waveform shown in Figure 2.4, the peak-to-peak value of the voltage can be determined by adding the magnitude of the positive and the negative peak. In this example, the peak-to-peak amplitude is 18 volts plus 2 volts for a total of 20 volts, peak-to-peak. 3. Root-Mean-Square The third specification for AC waveform is called root-mean-square, abbreviated RMS. This term allows the comparison of AC and DC circuit values. Root-mean-square values are the most common methods of specifying sinusoidal waveforms. In fact, almost all AC voltmeter and ammeters are calibrated so that they measure AC values in terms of RMS amplitude. RMS Relations to DC Heating Effect The RMS value is also known as the effective value and is defined in terms of the equivalent heating effect of direct current. The RMS value of a sinusoidal voltage is equivalent to the value of a DC voltage, which causes an equal amount of heat due to the circuit current flowing through a resistance. The RMS value of a sinusoidal voltage or current waveform is 70.7 percent or 0.707 of its peak amplitude value. VRMS = 0.707 Vpeak
IRMS = 0.707 Ipeak
A sinusoidal voltage with peak amplitude of 1 volt has the same effect as a DC voltage of 0.707 volts as far as its ability to reproduce the same amount of heat in a resistance. Because the AC voltage of 1 volt peak or 0.707 volts RMS is as effective as a DC voltage of 0.707 volts, the RMS value of voltage is also referred to as the effective value.
Determining the 0.707 Constant How is the 70.7 percent of peak-value constant derived? Essentially, the words root-mean-square define the mathematical procedure used to determine the constant.
Study Guide 2 Answer Key The Sine Wave and Sine Trigonometric Function
The term sinusoidal has been used to describe a waveform produced by an AC generator. The term sinusoidal comes from a trigonometric function called the sine function. Right-Triangle: Side and Angle Relationships Trigonometry is the study of triangles and their relationship involving lengths and angles of triangles. The basic
triangle studied in trigonometry is a right triangle, which is a triangle that has a 90◦ angle as one of its three
angles. A 90◦ triangle has a unique set of relationships from which the rules for trigonometry are derived. To help distinguish the sides of a right triangle from one another, a name is given to each side. The sides of the triangle are named with respect to the angle theta. The side of the triangle across from or opposite to the angle theta is called the opposite side. The longest side of a right triangle is called the hypotenuse. The remaining side is called the adjacent side because it lies beside or adjacent to the angle. These three names are commonly abbreviated to their first initials, O, H, and A. Basic Trigonometric Functions In trigonometry, these ratios have specific names. The three most commonly-used ratios in the study of right triangles are called sine, cosine, and tangent.
1. The sine of the angle theta is equal to the ratio formed by the length of the opposite side divided by the length of the hypotenuse.
Sine ө = opposite
hypotenuse
2. The cosine of the angle theta is equal to the ratio formed by length of the adjacent side divided by the
length of the hypotenuse.
Cosine ө = adjacent
hypotenuse 3. The tangent of the angle theta is equal to the ratio formed by length of the opposite side divided by
Triangle #1 Given the adjacent side = 10’ and the opposite side = 8’, what is the length of the hypotenuse side? Step 1 - Find the degree angle Tangent = Opposite Adjacent = 8_ = 0.8
10
= .8 enter 2nd
tan on your calculator = 38.65980825 round up to 38.66 = 38.66° Step 2 - Change the degree angle to cosine Hypotenuse = Adjacent Cosine 38.66 = 10’ _______ (take 38.66° enter cosine on your calculator your answer is .780866719 0.780866719) = 12.8’
Triangle #2 Given the adjacent side = 6.8 rods and the opposite side = 5.3 rods, what is the length of the hypotenuse side? Step 1 - Find the degree angle Tangent = Opposite Adjacent = 5.3 rods = 0.779411765 6.8 rods = enter 2nd tangent 0.779411765on your calculator = 37.93° Step 2 - Change the degree angle to sine Hypotenuse = Opposite Sine = 5.3 rods Sine 37.93 (enter 37.93, enter sine on the calculator) = 5.3 rods 0.6146982793 = 8.62 rods
Triangle #3 Given the hypotenuse side = 125 miles and the adjacent side = 85 miles, what is the length of the opposite side? Step 1 - Find the angle in degrees Cosine = Adjacent Hypotenuse = 85 miles_ = 0.68 125 miles = enter 2nd function button on calculator, then enter cosine 0.68 = 47.15635696 (round off to 47.16 degrees) Step 2 - Change the degree angle to sine Opposite = Sine x Hypotenuse enter sine 47.16 on calculator = 0.7332553462 = 0.7332553462 x 125 miles = 91.65691828 (round off to 91.7) = 91.7 miles
Triangle #4 Given the hypotenuse = 56’ and the opposite side = 23.2’, what is the length of the adjacent side? Step 1 - Find the angle in degrees Sine = Opposite__ Hypotenuse = 23.2’ = 0.4142857143 56’ = enter 2nd sine 0.414285 (or inv sine) = 24.47 degrees Step 2 - Change the degree angle to cosine Adjacent = Cosine x Hypotenuse = enter cosine 24.47 on the calculator = 0.910178279 x 56’ = 50.96998362 (round off to 51’) = 51’
Complete the table by calculating the missing or incomplete values for the RMS, peak amplitudes, the peak-to-peak values, the average values and/or the instantaneous values.
Complete the table by calculating the missing or incomplete values for the RMS, peak amplitudes, the peak-to-peak values, the average values and the instantaneous values.
