Chapter 15:AC Fundamentals

AC FundamentalsChapter 151Introduction2

Alternating Current3Voltages of ac sources alternate in polarity and vary in magnitudeVoltages produce currents that vary in magnitude and alternate in direction34

Alternating Current5A sinusoidal ac waveform starts at zeroIncreases to a positive maximumDecreases to zeroChanges polarityIncreases to a negative maximumReturns to zeroVariation is called a cycle 5Generating AC Voltages6

6Generating AC Voltages7

7AC Voltage-Current Conventions 8Assign a reference polarity for sourceWhen voltage has a positive valueIts polarity is same as reference polarityWhen voltage is negativeIts polarity is opposite that of the reference polarity89

AC Voltage-Current Conventions 10Assign a reference direction for current that leaves source at positive reference polarityWhen current has a positive valueIts actual direction is same as current reference arrow10AC Voltage-Current Conventions 11When current is negativeIts actual direction is opposite that of current reference arrow11Instantaneous Values12

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Frequency14Number of cycles per second of a waveform FrequencyDenoted by fUnit of frequency is hertz (Hz)1 Hz = 1 cycle per second1415

Period16Period of a waveform Time it takes to complete one cycleTime is measured in secondsThe period is the reciprocal of frequencyT = 1/f16Amplitude and Peak-to-Peak Value17Amplitude of a sine wave Distance from its average to its peakWe use Em for amplitudePeak-to-peak voltageMeasured between minimum and maximum peaksWe use Epp or Vpp1718

Peak Value19Peak value of an ac voltage or current Maximum value with respect to zeroIf a sine wave is superimposed on a dc valuePeak value of combined wave is sum of dc voltage and peak value of ac waveform amplitude19The Basic Sine Wave Equation20Voltage produced by a generator ise = Em sin Em is maximum (peak) voltage is instantaneous angular position of rotating coil of the generator20The Basic Sine Wave Equation21Voltage at angular position of sine wave generatorMay be found by multiplying Em times the sine of angle at that position

21Angular Velocity22Rate at which the generator coil rotates with respect to time, (Greek letter omega)22Angular Velocity23Units for are revolutions/second, degrees/sec, or radians/sec.

23Radian Measure24 is usually expressed in radians/second2 radians = 360To convert from degrees to radians, multiply by /180

24Radian Measure25To convert from radians to degrees, multiply by 180/When using a calculatorBe sure it is set to radian mode when working with angles measured in radians

25Relationship between ,T, and f26One cycle of a sine wave may be represented by = 2 rads or t = T sec

26Voltages and Currents as Functions of Time27Since = t, the equation e = Em sin becomes e(t) = Em sin tAlso, v(t) = Vm sin t and i(t) = Im sin t27Voltages and Currents with Phase Shifts28If a sine wave does not pass through zero at t = 0, it has a phase shiftFor a waveform shifted leftv = Vm sin(t + )For a waveform shifted rightv = Vm sin(t - )

28Phasors29Rotating vectors whose projection onto a vertical or horizontal axis can be used to represent sinusoidally varying quantities

29Phasors30A sinusoidal waveformProduced by plotting vertical projection of a phasor that rotates in the counterclockwise direction at a constant angular velocity 30Phasors31Phasors apply only to sinusoidally varying waveforms

31Example32

Shifted Sine WavesPhasors used to represent shifted waveformsAngle is position of phasor at t = 0 seconds

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Phase Difference35Phase difference is angular displacement between waveforms of same frequencyIf angular displacement is 0Waveforms are in phase35Phase Difference36If angular displacement is not 0o, they are out of phase by amount of displacement

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Phase Difference40If v1 = 5 sin(100t) and v2 = 3 sin(100t - 30), v1 leads v2 by 30May be determined by drawing two waves as phasors Look to see which one is ahead of the other as they rotate in a counterclockwise direction40Average Value41To find an average value of a waveformDivide area under waveform by length of its baseAreas above axis are positive, areas below axis are negative41Average Value42Average values also called dc valuesdc meters indicate average values rather than instantaneous values4243

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Sine Wave Averages45Average value of a sine wave over a complete cycle is zeroAverage over a half cycle is not zero45Sine Wave Averages46Rectified full-wave average is 0.637 times the maximum valueRectified half-wave average is 0.318 times the maximum value46Effective Values47Effective value or RMS value of an ac waveform is an equivalent dc valueIt tells how many volts or amps of dc that an ac waveform supplies in terms of its ability to produce the same average power

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Effective Values51In North America, house voltage is 120 Vac. Voltage is capable of producing the same average power as a 120 V battery51Effective Values52To determine effective powerSet Power(dc) = Power(ac)Pdc = pacI2R = i2R where i = Im sin tBy applying a trigonometric identityAble to solve for I in terms of Im52Effective Values53Ieff = .707ImVeff = .707VmEffective value is also known as the RMS value53