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Alternating Voltage and Current Topics Covered in Chapter 15 15-1: Alternating Current Applications 15-2: Alternating-Voltage Generator 15-3: The Sine Wave 15-4: Alternating Current 15-5: Voltage and Current Values for a Sine Wave 15-6: Frequency Chapter 15 © 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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Topics Covered in Chapter 15 15-7: Period 15-8: Wavelength 15-9: Phase Angle 15-10: The Time Factor in Frequency and Phase 15-11: Alternating Current Circuits with Resistance 15-12: Nonsinusoidal AC Waveforms 15-13: Harmonic Frequencies 15-14: The 60-Hz AC Power Line 15-15: Motors and Generators 15-16: Three-Phase AC Power McGraw-Hill© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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15-1: Alternating Current Applications A transformer can only operate with alternating current to step up or step down an ac voltage. A transformer is an example of inductance in ac circuits where the changing magnetic flux of a varying current produces an induced voltage. Capacitance is important with the changing electric field of a varying voltage. The effects of inductance and capacitance depend on having an ac source. An important application is a resonant circuit with L and C that is tuned to a particular frequency.

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15-2: Alternating-Voltage Generator Characteristics of Alternating Current Alternating voltage and alternating current vary continuously in magnitude and reverse in polarity. One cycle includes the variations between two successive points having the same value and varying in the same direction. Frequency is measured in hertz (Hz).

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15-2: Alternating-Voltage Generator The conductor loop rotates through the magnetic field to generate induced ac voltage across open terminals. At the horizontal position, the loop does not induce a voltage because the conductors do not cut across the flux. At the vertical position, conductors cut across the flux and produce maximum v. Each of the longer conductors has opposite polarity of induced voltage. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 15-2: Loop rotating in magnetic field to produce induced voltage v with alternating polarities. (a) Loop conductors moving parallel to magnetic field results in zero voltage. (b) Loop conductors cutting across magnetic field produce maximum induced voltage.

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15-2: Alternating-Voltage Generator The Cycle One complete revolution of the loop around the circle is a cycle. The half-cycle of revolution is called an alternation.

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15-2: Alternating-Voltage Generator Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 15-3: One cycle of alternating voltage generated by rotating loop. Magnetic field, not shown here, is directed from top to bottom, as in Fig. 15-2. The voltage waveform shown in Fig. 15-3 is called a sine wave, sinusoidal wave, or sinusoid because the amount of induced voltage is proportional to the sine of the angle of rotation in the circular motion producing the voltage.

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15-2: Alternating-Voltage Generator Angular Measure and Radian Measure The cycle of voltage corresponds to rotation of the loop around a circle, so parts of the cycle are described in angles. The radian (rad) is an angle equivalent to 57.3°. A radian is the angular part of the circle that includes an arc equal to the radius r of the circle. A circle’s circumference equals 2 π r, so one cycle equals 2 π rad. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 15-3(a).

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15-2: Alternating-Voltage Generator Angular Measure and Radian Measure Angular MeasurementRadian Equivalent Zero degreesZero radians 360°2 π rad 180°½ × 2 π rad, or π rad 90°½ × π rad, or π/2 rad 270° (180°+ 90°)π rad + π/2 rad = 3π/2 rad

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15-2: Alternating-Voltage Generator Amplitude 0 0 °90 °180 °270 °360 ° 2 rad rad /2 rad 0 rad 3 /2 rad Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Angular Measure and Radian Measure

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15-3: The Sine Wave The voltage waveform pictured here is called a sine wave, sinusoidal wave, or sinusoid. The induced voltage is proportional to the sine of the angle of rotation in the circular motion producing the voltage. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 15-1(a): Waveform of ac power-line voltage with frequency of 60 Hz. Two cycles are shown. Oscilloscope readout.

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15-3: The Sine Wave With a sine wave, the induced voltage increases to a maximum at 90°, when the loop is vertical, just as the sine of the angle of rotation increases to a maximum at 90°. The instantaneous value of a sine-wave voltage for any angle of rotation is expressed in the formula: v = V M sin Θ Θ (theta) is the angle sin = the abbreviation for sine V M = the maximum voltage value v = the instantaneous value of voltage at angle Θ.

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15-3: The Sine Wave Characteristics of the Sine-Wave AC Waveform: The cycle includes 360° or 2π rad. The polarity reverses each half-cycle. The maximum values are at 90° and 270°. The zero values are at 0° and 180°. The waveform changes its values the fastest when it crosses the zero axis. The waveform changes its values the slowest when it is at its maximum value.

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15-4: Alternating Current When a sine wave of alternating voltage is connected across a load resistance, the current that flows in the circuit is also a sine wave. The sine wave frequency of an alternating voltage is the same as the alternating current through a series connected load resistance.

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15-4: Alternating Current Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 15-5: A sine wave of alternating voltage applied across R produces a sine wave of alternating current in the circuit. (a) Waveform of applied voltage. (b) AC circuit. Note the symbol for sine-wave generator V. (c) Waveform of current in the circuit.

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15-4: Alternating Current After the first half-cycle, polarity reverses and current flows in the opposite direction. The negative half-cycle of applied voltage is as useful as the positive half-cycle in producing current. The direction does not matter in the application. The motion of electrons against resistance produces power dissipation. Only v and i waveforms can be compared.

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15-5: Voltage and Current Values for a Sine Wave The following specific magnitudes are used to compare one wave to another: Peak value: maximum value V M or I M. This applies to the positive or negative peak. Peak-to-peak: usually, but not always, double the peak value, as it measures distance between two amplitudes. Average value: Arithmetic average of all values in one half-cycle (the full cycle average = 0). Root-Mean-Square (RMS) or Effective Value: Relates the amount of a sine wave of voltage or current to the DC values that will produce the same heating effect.

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15-5: Voltage and Current Values for a Sine Wave The average value is 0.637 × peak value. The rms value is 0.707 × peak value. The peak value is 1.414 × rms value. The peak-to-peak value is 2.828 × rms value.

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15-5: Voltage and Current Values for a Sine Wave Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 15-6: Definitions of important amplitude values for a sine wave of voltage or current.

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15-5: Voltage and Current Values for a Sine Wave 120 V + 100 V rms is the effective value. The heating effect of these two sources is identical. Same power dissipation The default sine wave ac measurement is V rms.

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15-6: Frequency Frequency ( f ) is the number of cycles per second. Cycle is measured between two successive points having the same value and direction. One cycle per second is 1 Hz.

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15-6: Frequency Amplitude 0 Time 1 sec f = 2 Hz 0.5 sec Sine Wave Frequency (two cycles shown) Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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15-6: Frequency Audio and Radio Frequencies Audio range (frequencies of electrical variations heard as sound waves by the human ear) is 16 to 16,000 Hz Radio-frequency range is up to 300,000 MHz (300 GHz) The higher the frequency, the higher the pitch of a sound. Amplitude has no relation to frequency. Amplitude influences the loudness of a sound.

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15-6: Frequency Sonic and Ultrasonic Frequencies Sound waves are variations in pressure generated by mechanical vibrations rather than electrical variations. Sound waves in the audible frequency range (16– 16,000 Hz) are sonic frequencies. Ultrasonic waves are above the audible range of frequencies (16,000–several MHz).

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15-7: Period Period (T) is the time per cycle. T = 1/f f = 1/T The higher the frequency, the shorter the period.

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15-7: Period Amplitude 0 Time 0.0167 s f = 1/T = 1/.0167 = 60 Hz T Period (T) Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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15-8: Wavelength Wavelength (λ) is the distance a wave travels in one cycle. λ = v/f, where: λ = wavelength λ = v = velocity of wave (such as sound or light) in applications, velocity can be influenced by electromagnetic fields, air pressure, etc. f = frequency 1130 ft/s f Hz

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15-8: Wavelength The higher the frequency, the lower the wavelength. The velocity of a radio wave is 3 × 10 10 cms/s (3 × 10 8 meters/s).

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15-8: Wavelength Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 15-9: Wavelength λ is the distance traveled by the wave in one cycle.

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15-9: Phase Angle Phase angle (Θ) is the angular difference between the same points on two different waveforms of the same frequency. Two waveforms that have peaks and zeros at the same time are in phase and have a phase angle of 0°. When one sine wave is at its peak while another is at zero, the two are 90° out of phase. When one sine wave has just the opposite phase of another, they are 180° out of phase.

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15-9: Phase Angle Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 15-10: Two sine-wave voltages 90° out of phase. (a) Wave B leads wave A by 90°. (b) Corresponding phasors V B and V A for the two sine-wave voltages with phase angle Θ = 90°. The right angle shows quadrature phase.

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15-9: Phase Angle Phase-Angle Diagrams Similar to vectors, phasors indicate the amplitude and phase angle of ac voltage or current. A vector quantity has direction in space, but a phasor angle represents a difference in time. The length of the phasor represents the amplitude of the waveform. The angle represents the phase angle of the waveform.

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15-9: Phase Angle Phase-Angle Diagrams The phasor corresponds to the entire cycle of voltage. The phase angle of one wave can be specified only with respect to another as a reference. Usually the reference phasor is horizontal. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 15-11: Leading and lagging phase angles for 90°. (a) When phasor V A is the horizontal reference, phasor V B leads by 90°. (b) When phasor V B is the horizontal reference, phasor V A lags by −90°.

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15-10: The Time Factor in Frequency and Phase The physical factors represented on waveforms are variations in amplitude, usually on the vertical scale, with respect to equal intervals on the horizontal scale, which can represent either distance or time. The angle of 360° represents the time for one cycle, or the period T. The phase angle between two waves of the same frequency indicates a specific difference in time. The time for a phase angle can be calculated as t = Θ 360 × 1 f

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15-11: Alternating Current Circuits with Resistance Series AC Circuit with R. The 4-A current is the same in all parts of the series circuit. (Note: This principle applies for either an ac or dc source.) The series voltage drops are equal to V = I x R The sum of the individual IR drops equals the applied voltage (120V). Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 15-16: Series ac circuit with resistance only.

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15-11: Alternating Current Circuits with Resistance Parallel AC Circuit with R. The voltage across the parallel branches is the same as the applied voltage. Each branch current is equal to the applied voltage (120V) divided by the branch resistance. Total line current is the sum of the branch currents (18A). Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 15-17: Parallel ac circuit with resistance only.

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15-11: Alternating Current Circuits with Resistance Series-Parallel AC Circuit with R. The main line current I T produced by the 120V source is equal to V/R T. Since the branch resistances are equal, the 4-A I T divides equally. Parallel branch currents add to equal the 4-A current in the main line. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 15-18: Series-parallel ac circuit with resistance only.

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15-12: Nonsinusoidal AC Waveforms In many electronic applications, other waveforms besides sine and cosine are important. Some of those forms are shown below. Square wave Sawtooth wave Pulse wave Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Common in digital electronic circuitry Used in timing and control circuitry Used in digital and control circuitry

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15-12: Nonsinusoidal AC Waveforms Key Similarities and Differences between Sinusoidal and Nonsinusoidal Waveforms For all waveforms, the cycle is measured between two points having the same amplitude and varying in the same direction. Peak amplitude is measured from the zero axis to the maximum positive or negative value. Peak-to-peak amplitude is better for measuring nonsinusoidal waveshapes because they can have unsymmetrical peaks.

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15-12: Nonsinusoidal AC Waveforms Key Similarities and Differences between Sinusoidal and Nonsinusoidal Waveforms The rms value 0.707 applies only to sine waves. Phase angles apply only to sine waves. All the waveforms represent ac voltages. Positive values are shown above the zero axis, and negative values are shown below the axis.

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15-13: Harmonic Frequencies Harmonic frequencies are exact multiples of the fundamental frequency. Harmonics are useful in analyzing distorted sine waves or nonsinusoidal waveforms. A common unit for frequency multiples is the octave, which is a range of 2:1. Doubling the frequency range (e.g., from 200 to 400 Hz) raises the frequency by one octave. Another unit for representing frequency multiples is the decade. A decade corresponds to a 10:1 range in frequencies (e.g., 30 kHz to 300 kHz).

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15-13: Harmonic Frequencies Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 15-20: Fundamental and harmonic frequencies for an example of a 100-Hz square wave.

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15-14: The 60-Hz AC Power Line Almost all homes in the US are supplied alternating voltage between 115 and 125 V rms, at a frequency of 60 Hz. The incoming voltage is wired to all the wall outlets and electrical equipment in parallel. The 120-V source of commercial electricity is the 60-Hz power line or the mains, indicating that it is the main line for all the parallel branches.

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15-14: The 60-Hz AC Power Line Applications in Residential Wiring: Residential wiring uses ac power instead of dc, because ac is more efficient in distribution from the generating station. House wiring uses 3-wire, single-phase power. The voltages for house wiring are 120 V to ground, and 240 V across the two high sides. A value higher than 120 V would create more danger of fatal electric shock, but lower voltages would be less efficient in supplying power.

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15-14: The 60-Hz AC Power Line Applications in Residential Wiring: Higher voltage can supply electric power with less I 2 R loss, since the same power is produced with less I. Although the frequency of house wiring in North America is 60 Hz, many places outside N. America use a 50 Hz standard for house wiring.

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15-14: The 60-Hz AC Power Line The 60-Hz Frequency Reference All power companies in the US, except for those in Texas, are interconnected in a grid that maintains the ac power-line frequency between 59.98 and 60.02 Hz. The frequency is compared with the time standard provided by the Bureau of Standards radio station WWV at Fort Collins, Colorado. This accuracy makes the power-line voltage a good secondary standard for checking frequencies based on 60 Hz.

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15-14: The 60-Hz AC Power Line Grounding Grounding is the practice of connecting one side of the power line to earth or ground. The purpose is safety: Grounding provides protection against dangerous electric shock. The power distribution lines are protected against excessively high voltage, particularly from lightning.

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15-14: The 60-Hz AC Power Line Grounding Plug connectors for the ac power line are configured to provide protection because they are polarized with respect to the ground connections. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 15-22: Plug connectors polarized for ground connection to an ac power line. (a) Wider blade connects to neutral. (b) Rounded pin connects to ground.

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15-14: The 60-Hz AC Power Line Grounding The ground-fault circuit interrupted (GFCI) is a device that can sense excessive leakage current and open the circuit as a protection against shock. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 15-23: Ground-fault circuit interrupter (GFCI).

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15-15: Motors and Generators A generator converts mechanical energy into electric energy. A motor converts electricity into rotary motion. GeneratorMotor ArmatureConnects to the external circuit to provide the generator output voltage. Connects to the electrical source that drives the motor. Field WindingThe field current may be obtained from a separate exciter source, or from its own armature output. Current for the field is produced by the same source that supplies the armature. Slip RingsUsed to transfer induced current in the coil to the brushes. Connect the rotating loop to the stationary wire leads for the external circuit. BrushesComplete the transfer of current to the load. Are spring-mounted to brush against the spinning rings on the rotor.

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15-15: Motors and Generators GeneratorMotor CommutatorSegments reverse the loop connections to the brushes every half-cycle to maintain a constant polarity of output voltage. Segments allow the dc source to produce torque in one direction. AC Induction MotorAre economical and rugged without any troublesome brush arcing. Universal MotorOperates on either ac or dc and is commonly used for small machines such as portable drills and food mixers. AlternatorAC generators are alternators.

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15-16: Three-Phase AC Power Three-phase ac power has three legs that are 120° out of phase. The advantage of three-phase ac voltage is more efficient distribution of power. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 15-25: Three-phase alternating voltage or current with 120° between each phase. (a) Sine waves. (b) Phasor diagram.

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15-16: Three-Phase AC Power The windings may be in the form of a Y (wye) or a delta (Δ). In the Y connection, any pair of terminals is across two coils in series and each coil has 120V. The output voltage across any two terminals is 120 x 1.73 or 208V. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 15-26: Types of connections for three-phase power (a) Wye or Y. (b) Delta or Δ.

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15-16: Three-Phase AC Power In the delta connection, any pair of terminals is across one generator winding and the output is 120V. The other coils are in a parallel branch. The current capacity of the line is increased by the factor 1.73. Fig. 15-26: Types of connections for three-phase power (a) Wye or Y. (b) Delta or Δ. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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