Lecture 7 ac waves
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- Sine waves are fundamental alternating current (AC) waveforms characterized by amplitude and period. The period is the time for one complete cycle, while amplitude is the maximum voltage or current value. - Frequency is the number of cycles per second, measured in Hertz (Hz). Period and frequency are reciprocals, so if one value is known, the other can be calculated. - Sinusoidal voltages can be produced by rotating a conductor in a magnetic field, such as in an AC generator, where the output is a sine wave.
Lecture 7 ac waves
- 2. The sinusoidal waveform (sine wave) is the fundamental alternating current (ac) and alternating voltage waveform. Sine waves Electrical sine waves are named from the mathematical function with the same shape.
- 3. Summary Sine waves are characterized by the amplitude and period. The amplitude is the maximum value of a voltage or current; the period is the time interval for one complete cycle. Sine waves 0 V 10 V -10 V 15 V -15 V -20 V t ( s) 0 25 37.5 50.0 20 V The amplitude (A) of this sine wave is 20 V The period is 50.0 s A T
- 4. Summary The period of a sine wave can be measured between any two corresponding points on the waveform. Sine waves T T T T T T By contrast, the amplitude of a sine wave is only measured from the center to the maximum point. A
- 5. 3.0 Hz SummarySummary Frequency Frequency ( f ) is the number of cycles that a sine wave completes in one second. Frequency is measured in hertz (Hz). If 3 cycles of a wave occur in one second, the frequency is 1.0 s
- 6. Summary The period and frequency are reciprocals of each other. Summary Period and frequency T f 1 and f T 1 Thus, if you know one, you can easily find the other. If the period is 50 s, the frequency is 0.02 MHz = 20 kHz. (The 1/x key on your calculator is handy for converting between f and T.)
- 7. Summary Sinusoidal voltages are produced by ac generators and electronic oscillators. Summary Sinusoidal voltage sourcesGeneration of a sine wave N S Motion of conductor Conductor B C D A A B C D A B B C D A C B C D A D When a conductor rotates in a constant magnetic field, a sinusoidal wave is generated. When the conductor 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. B C D A
- 8. Generators convert rotational energy to electrical energy. A stationary field alternator with a rotating armature is shown. The armature has an induced voltage, which is connected through slip rings and brushes to a load. The armature loops are wound on a magnetic core (not shown for simplicity). AC generator (alternator) N S slip rings armature brushes Small alternators may use a permanent magnet as shown here; other use field coils to produce the magnetic flux.
- 9. AC generator (alternator) By increasing the number of poles, the number of cycles per revolution is increased. A four-pole generator will produce two complete cycles in each revolution.
- 10. Function generators Function selection Frequency Output level (amplitude) DC offset CMOS output Range Adjust Duty cycle Typical controls: Outputs Readout Sine Square Triangle
- 11. Sine wave voltage and current values There are several ways to specify the voltage of a sinusoidal voltage waveform. The amplitude of a sine wave is also called the peak value, abbreviated as VP for a voltage waveform. 0 V 10 V -10 V 15 V -15 V -20 V t ( s) 0 25 37.5 50.0 20 V The peak voltage of this waveform is 20 V. VP
- 12. 0 V 10 V -10 V 15 V -15 V -20 V t ( s) 0 25 37.5 50.0 20 V The voltage of a sine wave can also be specified as either the peak-to-peak or the rms value. The peak-to- peak is twice the peak value. The rms value is 0.707 times the peak value. Sine wave voltage and current values The peak-to-peak voltage is 40 V. The rms voltage is 14.1 V. VPP Vrms
- 13. 0 V 10 V -10 V 15 V -15 V -20 V t ( s) 0 25 37.5 50.0 20 V For some purposes, the average value (actually the half- wave average) is used to specify the voltage or current. By definition, the average value is as 0.637 times the peak value. Sine wave voltage and current values The average value for the sinusoidal voltage is 12.7 V. Vavg
- 14. Angular measurements can be made in degrees (o) or radians. The radian (rad) is the angle that is formed when the arc is equal to the radius of a circle. There are 360o or 2p radians in one complete revolution. Angular measurement R R 1.0 -1.0 0.8 -0.8 0.6 -0.6 0.4 -0.4 0.2 -0.2 0 0 2ppp 2 p 4 p 4 3 p 2 3p 4 5 p 4 7
- 15. Because there are 2p radians in one complete revolution and 360o in a revolution, the conversion between radians and degrees is easy to write. To find the number of radians, given the number of degrees: degrees 360 rad2 rad p rad rad2 360 deg p To find the number of degrees, given the radians: Angular measurement
- 16. Instantaneous values of a wave are shown as v or i. The equation for the instantaneous voltage (v) of a sine wave is Sine wave equation where If the peak voltage is 25 V, the instantaneous voltage at 50 degrees is sinpVv Vp = = Peak voltage Angle in rad or degrees 19.2 V
- 17. Sine wave equation v = = 19.2 VVp sin Vp 90 500 = 50 Vp Vp = 25 V A plot of the example in the previous slide (peak at 25 V) is shown. The instantaneous voltage at 50o is 19.2 V as previously calculated.
- 18. Phase shift where f = Phase shift The phase of a sine wave is an angular measurement that specifies the position of a sine wave relative to a reference. To show that a sine wave is shifted to the left or right of this reference, a term is added to the equation given previously. f sinPVv
- 19. Phase shift Voltage(V) 270 3600 90 180 40 45 135 225 315 0 Angle () 30 20 10 -20 -30 - 40 405 Peak voltage Reference Notice that a lagging sine wave is below the axis at 0o Example of a wave that lags the reference v = 30 V sin ( - 45o) …and the equation has a negative phase shift
- 20. Phase shift Voltage(V) 270 3600 90 180 40 45 135 225 3150 Angle () 30 20 10 -20 -30 -40 Peak voltage Reference -45 -10 Notice that a leading sine wave is above the axis at 0o Example of a wave that leads the reference v = 30 V sin ( + 45o) …and the equation has a positive phase shift
- 21. 00 90 90 180180 360 The sine wave can be represented as the projection of a vector rotating at a constant rate. This rotating vector is called a phasor. Phasors
- 22. Phasors allow ac calculations to use basic trigonometry. The sine function in trigonometry is the ratio of the opposite side of a right triangle to the adjacent side. hypotenuse right angle opposite side adjacent side hypotenuse oppositeside sin= Phasors
- 23. The position of a phasor at any instant can be expressed as a positive angle, measured counterclockwise from 0 or as a negative angle equal to - 360. Phasors positive angle of negative angle of - 360 phasor
- 24. Angular velocity of a phasor When a phasor rotates through 360 or 2p radians, one complete cycle is traced out. The velocity of rotation is called the angular velocity (). = 2pf The instantaneous voltage at any point in time is given by v = Vpsin 2pf (Note that this angular velocity is expressed in radians per second.)
- 25. Example
- 26. Sine wave Alternating current Period (T) Frequency (f) Hertz Current that reverses direction in response to a change in source voltage polarity. The time interval for one complete cycle of a periodic waveform. A type of waveform that follows a cyclic sinusoidal pattern defined by the formula y = A sin . Selected Key Terms A measure of the rate of change of a periodic function; the number of cycles completed in 1 s. The unit of frequency. One hertz equals one cycle per second.
- 27. Instantaneous value Peak value Peak-to-peak value rms value The voltage or current value of a waveform at its maximum positive or negative points. The voltage or current value of a waveform measured from its minimum to its maximum points. The voltage or current value of a waveform at a given instant in time. Selected Key Terms The value of a sinusoidal voltage that indicates its heating effect, also known as effective value. It is equal to 0.707 times the peak value. rms stands for root mean square.
- 28. Radian Phasor Amplitude Pulse Harmonics The maximum value of a voltage or current. A type of waveform that consists of two equal and opposite steps in voltage or current separated by a time interval. A unit of angular measurement. There are 2p radians in one complete 360o revolution. Selected Key Terms The frequencies contained in a composite waveform, which are integer multiples of the pulse repetition frequency. A representation of a sine wave in terms of its magnitude (amplitude) and direction (phase angle).
- 29. Quiz 1. In North America, the frequency of ac utility voltage is 60 Hz. The period is a. 8.3 ms b. 16.7 ms c. 60 ms d. 60 s
- 30. Quiz 2. The amplitude of a sine wave is measured a. at the maximum point b. between the minimum and maximum points c. at the midpoint d. anywhere on the wave
- 31. Quiz 3. An example of an equation for a waveform that lags the reference is a. v = -40 V sin () b. v = 100 V sin ( + 35o) c. v = 5.0 V sin ( - 27o) d. v = 27 V
- 32. 4. In the equation v = Vp sin , the letter v stands for the a. peak value b. average value c. rms value d. instantaneous value Quiz
- 33. 7. The number of radians in 90o are a. p/2 b. p c. 2p/3 d. 2p Quiz
- 34. Quiz Answers: 1. b 2. a 3. c 4. d 5. b 6. b 7. a 8. c 9. a 10. b