Chapter 15
Alternating Current
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Voltages of ac sources alternate in
polarity and vary in magnitude |
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Voltages produce currents that vary in
magnitude and alternate in direction |
Alternating Current
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A sinusoidal ac waveform starts at zero |
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Increases to a positive maximum |
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Decreases to zero |
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Changes polarity |
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Increases to a negative maximum |
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Returns to zero |
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Variation is called a cycle |
Generating AC Voltages
Generating AC Voltages
AC Voltage-Current
Conventions
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Assign a reference polarity for source |
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When voltage has a positive value |
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Its polarity is same as reference
polarity |
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When voltage is negative |
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Its polarity is opposite that of the
reference polarity |
AC Voltage-Current
Conventions
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Assign a reference direction for
current that leaves source at positive reference polarity |
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When current has a positive value |
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Its actual direction is same as current
reference arrow |
AC Voltage-Current
Conventions
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When current is negative |
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Its actual direction is opposite that
of current reference arrow |
Frequency
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Number of cycles per second of a
waveform |
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Frequency |
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Denoted by f |
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Unit of frequency is hertz (Hz) |
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1 Hz = 1 cycle per second |
Period
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Period of a waveform |
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Time it takes to complete one cycle |
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Time is measured in seconds |
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The period is the reciprocal of
frequency |
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T = 1/f |
Amplitude and
Peak-to-Peak Value
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Amplitude of a sine wave |
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Distance from its average to its peak |
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We use Em for amplitude |
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Peak-to-peak voltage |
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Measured between minimum and maximum
peaks |
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We use Epp or Vpp |
Peak Value
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Peak value of an ac voltage or current |
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Maximum value with respect to zero |
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If a sine wave is superimposed on a dc
value |
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Peak value of combined wave is sum of
dc voltage and peak value of ac waveform amplitude |
The Basic Sine Wave
Equation
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Voltage produced by a generator is |
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e = Em sin a |
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Em
is maximum (peak) voltage |
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a is
instantaneous angular position of rotating coil of the generator |
The Basic Sine Wave
Equation
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Voltage at angular position of sine
wave generator |
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May be found by multiplying Em
times the sine of angle at that position |
Angular Velocity
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Rate at which the generator coil
rotates with respect to time, w (Greek letter omega) |
Angular Velocity
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Units for w are revolutions/second, degrees/sec,
or radians/sec. |
Radian Measure
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w is
usually expressed in radians/second |
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2p radians = 360° |
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To convert from degrees to radians,
multiply by p/180 |
Radian Measure
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To convert from radians to degrees,
multiply by 180/p |
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When using a calculator |
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Be sure it is set to radian mode when
working with angles measured in radians |
Relationship between w,T, and f
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One cycle of a sine wave may be
represented by a = 2p
rads or t = T sec |
Voltages and Currents as
Functions of Time
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Since a = wt, the equation e = Em sin a becomes e(t) = Em sin
wt |
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Also, v(t) = Vm sin wt and i(t) = Im sin wt |
Voltages and Currents as
Functions of Time
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Equations used to compute voltages and
currents at any instant of time |
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Referred to as instantaneous voltage or
current |
Voltages and Currents
with Phase Shifts
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If a sine wave does not pass through
zero at t = 0, it has a phase shift |
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For a waveform shifted left |
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v = Vm sin(wt + q) |
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For a waveform shifted right |
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v = Vm sin(wt - q) |
Phasors
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Rotating vectors whose projection onto
a vertical or horizontal axis can be used to represent sinusoidally varying
quantities |
Phasors
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A sinusoidal waveform |
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Produced by plotting vertical
projection of a phasor that rotates in the counterclockwise direction at a
constant angular velocity w |
Phasors
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Phasors apply only to sinusoidally
varying waveforms |
Shifted Sine Waves
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Phasors used to represent shifted
waveforms |
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Angle q is position of phasor at t
= 0 seconds |
Phase Difference
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Phase difference is angular
displacement between waveforms of same frequency |
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If angular displacement is 0° |
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Waveforms are in phase |
Phase Difference
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If angular displacement is not 0o,
they are out of phase by amount of displacement |
Phase Difference
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If v1 = 5 sin(100t) and v2
= 3 sin(100t - 30°), v1 leads v2 by 30° |
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May be determined by drawing two waves
as phasors |
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Look to see which one is ahead of the
other as they rotate in a counterclockwise direction |
Average Value
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To find an average value of a waveform |
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Divide area under waveform by length of
its base |
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Areas above axis are positive, areas
below axis are negative |
Average Value
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Average values also called dc values |
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dc meters indicate average values
rather than instantaneous values |
Sine Wave Averages
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Average value of a sine wave over a
complete cycle is zero |
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Average over a half cycle is not zero |
Sine Wave Averages
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Rectified full-wave average is 0.637
times the maximum value |
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Rectified half-wave average is 0.318
times the maximum value |
Effective Values
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Effective value or RMS value of an ac
waveform is an equivalent dc value |
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It 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 Values
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In North America, house voltage is 120
Vac. |
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Voltage is capable of producing the
same average power as a 120 V battery |
Effective Values
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To determine effective power |
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Set Power(dc) = Power(ac) |
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Pdc = pac |
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I2R = i2R where i = Im sin wt |
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By applying a trigonometric identity |
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Able to solve for I in terms of Im |
Effective Values
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Ieff = .707Im |
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Veff = .707Vm |
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Effective value is also known as the
RMS value |