DE4401 - APTE 5601
INTRODUCTION TO
AC CIRCUIT THEORY
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DC and AC
• DC stands for “Direct Current,”
– voltage maintains constant polarity over time
– current maintains constant direction over time.
• AC stands for “Alternating Current”:
– voltage changes polarity over time
– current changes direction over time
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Why AC?
• With AC, it is possible to build electric generators,
motors and power distribution systems that are far more
efficient than DC
War of Currents
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AC is necessary because...
• AC generators and AC motors tend to be simpler than
DC generators and DC motors.
– This relative simplicity translates into greater reliability and lower
cost of manufacture.
• The transformer's ability to step AC voltage up or down
with ease gives AC an advantage unmatched by DC in
the realm of power distribution.
– When transmitting electrical power over long distances, it is far
more efficient to do so with stepped-up voltages and steppeddown currents (smaller-diameter wire with less resistive power
losses), then step the voltage back down and the current back
up at the receiving end for consumer use.
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Main AC advantage: power distribution
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Can’t be done with DC
• Transformer technology has made long-range electric
power distribution practical. Without the ability to
efficiently step voltage up and down, it would be costprohibitive to construct power systems for anything but
close-range (within a few miles at most) use.
• Transformers only work with AC, not DC. It works based
on the phenomenon of mutual inductance, which relies
on changing magnetic fields. Direct current (DC) can
only produce steady magnetic fields, thus transformers
simply will not work with direct current.
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Waveform of DC
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AC waveform: sine wave
• When graphed, the voltage produced by AC generator
(alternator) takes on a distinct shape known as a sine
wave:
One cycle of a wave is one complete
evolution of its shape until the point that it
is ready to repeat itself.
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Sinewave Generation
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Sinewave Details
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Other waveforms
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Sinusoidal waveform
• Sinusoidal waveform is given by I = Iosin (ωt+ϕ)
– Io is amplitude
– ω is angular frequency in radian per second: ω = 2p / T
– t is time and ϕ is the phase angle.
• The period T of a wave is the amount of time it takes to
complete one cycle (unit is second s).
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Frequency
• Frequency is the number of complete cycles that a wave
completes in a given amount of time.
• Standard unit for frequency is Hertz (abbreviated Hz).
Frequency of 1 Hz represents the number of periods
completed during one second of time.
f=1/T
w=2pf
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Radians
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Conversions
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Some well known frequencies...
• In New Zealand , power system frequency is 50 Hz. This
corresponds to the period T = 20ms.
• In US, power system frequency is 60 Hz.
• Audio frequencies: from 20 Hz to 20kHz
• A radio station broadcast at a frequency of 100 MHz.
• Microwave oven operates at 2.54 GHz
• k (kilo) = 1000 Hz; M (mega) = 1 000 000 Hz;
G(giga)=1000000000 Hz.
• In music, frequency is the same as pitch, which is the
essential property distinguishing one note from another
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AC generator in the laboratory
• In our lab , we will use
an Audio Generator .
This means that the
frequency of an AC
wave taken from the
generator Output is
within an audio range
(frequencies we can
hear).
• We can adjust both
frequency (period) and
magnitude of the wave
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Oscilloscope
• The main purpose
of an oscilloscope
is to give an
accurate visual
representation of
electrical signals.
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Your instrument
• Every large instrument manufacturer has additional
resources online: pdf files, videos, forums, FAQ
– Get the model number of your instrument and find the manual
online
• Tektronix: http://www.tek.com/learning
• Agilent: http://www.home.agilent.com/agilent/home.jspx
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Oscilloscope
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Measurement of AC amplitude
• With DC, where quantities of voltage and current are
constant, we have little trouble expressing how much
voltage or current we have in any part of a circuit. But
how do you measure magnitude of something that is
constantly changing?
• The amplitude of an AC waveform is its height as
depicted on a graph over time. An amplitude
measurement can take the form of peak, peak-to-peak,
average, or RMS quantity.
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Peak amplitude
• Peak amplitude is the height of an AC waveform as
measured from the zero mark to the highest positive or
lowest negative point on a graph. Also known as the
crest amplitude of a wave.
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Peak-to-peak amplitude
• Peak-to-peak amplitude is the total height of an AC
waveform as measured from maximum positive to
maximum negative peaks on a graph. Often abbreviated
as “P-P”.
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Average value of all points ?
Average amplitude is the mathematical “mean” of
all a waveform's points over the period of one cycle.
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Average of absolute values
• Technically, the average amplitude of any waveform with
equal-area portions above and below the “zero” line on a
graph is zero. However, as a practical measure of
amplitude, a waveform's average value is often
calculated as the mathematical mean of all the points'
absolute values (taking all the negative values and
considering them as positive)
– For a sine wave, the average value so calculated is
approximately 0.637 of its peak value.
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Ability to do useful work
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DC equivalent to AC
• assign a “DC equivalent” measurement to any AC
voltage or current:
– whatever magnitude of DC voltage or current would produce the
same amount of heat energy dissipation through an equal
resistance
An RMS
voltage
produces
the same
heating
effect as a
the same DC
voltage
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Root Mean Square
• For example, 10 volts AC RMS is the amount of voltage
that would produce the same amount of heat dissipation
across a resistor of given value as a 10 volt DC power
supply.
• The qualifier “RMS” stands for Root Mean Square, and is
a way of expressing an AC quantity of voltage or current
in terms functionally equivalent to DC.
– To calculate the Root Mean Square of a set of numbers, square
all the numbers in the set and then find the arithmetic mean of
the squares. Take the square root of the result.
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RMS for sine wave
• For a sine wave, the RMS value is
approximately 0.707 of its peak value
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RMS Peak Average Conversions
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Non Sinusoidal Average is Different
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RMS is most often stated
• When talking of AC, RMS values are so commonly used
that, unless otherwise stated, you may assume that RMS
values are intended*. For instance, normal domestic AC
in New Zealand is 230 Volts AC with frequency 50 Hz.
• The RMS voltage is 230 volts, so:
– the peak value Vm= V.√2 = 325 volts.
– So the active wire goes from +325 volts to -325 volts and back
again 50 times per second.
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It depends on the use, though...
• For instance, when determining the proper size of wire to
conduct electric power from a source to a load, RMS
current measurement is the best to use, because the
principal concern with current is overheating of the wire,
which is a function of power dissipation caused by
current through the resistance of the wire.
• However, when rating insulators for service in highvoltage AC applications, peak voltage measurements
are the most appropriate, because the principal concern
here is insulator “flashover” caused by brief spikes of
voltage, irrespective of time.
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For other wave shapes..
• Formulas for RMS and average value given above are
for sine waves only.
• If you have other wave shapes, you cannot use these
formulas!
• Find appropriate formula in literature or online OR
calculate the value from definition
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Comparing two sine waveforms
I = Iosin (ωt+ϕ)
• Looking at :
– Amplitude
– Frequency
– Phase
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In Phase
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Pos Neg Phase
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Phase Difference
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Phase shift
• Phase shift occurs when two waveforms are “out of step”
with each other.
• Calculations for AC circuit analysis must take into
consideration both amplitude and phase shift of voltage and
current waveforms to be completely accurate. This requires
the use of complex numbers.
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Phase: always relative
• phase is
always a
relative
measurement
between two
waveforms
rather than an
absolute
property
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Revision
• Period
• Frequency
• For Sine wave: must know how to measure / read / calculate
–
–
–
–
Peak value
Peak-to-Peak value
Average value
RMS value
• Phase shift
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