Applied Circuit Analysis
Chapter 11
AC Voltage and Current
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Overview
• This chapter will cover alternating current.
• A discussion of complex numbers is
included prior to introducing phasors.
• Applications of phasors and frequency
domain analysis for circuits including
resistors, capacitors, and inductors will be
covered.
• The concept of impedance and admittance is
also introduced.
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Alternating Current
• Alternating Current, or AC, is the dominant
form of electrical power that is delivered to
homes and industry.
• In the late 1800’s there was a battle between
proponents of DC and AC.
• AC won out due to its efficiency for long
distance transmission.
• AC is a sinusoidal current, meaning the
current reverses at regular times and has
alternating positive and negative values.
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Generating AC
• In the DC circuits the power source we used
was a battery.
• But batteries cannot provide the very large
amount of power required by homes and
industry.
• Electrical power can also be created by
rotating a coil of wire in a static magnetic
field.
• This rotation causes the output voltage to
alternate between positive and negative.
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Generating AC II
• The output voltage can be
expressed mathematically:
v  Vm sin 
• Where Vm is the maximum
voltage and  is the angle
(in radians) of rotation of
the coil.
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Alternating Voltage
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SINUSOIDAL ac VOLTAGE
CHARACTERISTICS/DEFINITIONS
FIG. 13.2 Various sources of ac power: (a)
generating plant; (b) portable ac generator; (c) windpower station; (d) solar panel; (e) function generator
Sinusoidal Voltage
• If the angle  is replaced with an expression
indicating time (t), we get:
v  t   Vm sin t
• Where
– Vm is the amplitude of the sinusoid
–  is the angular frequency in radians per second
– t is the time in seconds
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Sinusoids
• Sinusoids are interesting to us because there
are a number of natural phenomenon that are
sinusoidal in nature.
• It is also a very easy signal to generate and
transmit.
• Also, through Fourier analysis, any practical
periodic function can be made by adding
sinusoids.
• Lastly, they are very easy to handle
mathematically.
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Sinusoids II
• A sinusoidal forcing function produces both a
transient and a steady state response.
• When the transient has died out, we say the circuit is
in sinusoidal steady state.
• A sinusoidal voltage may be represented as:
v  t   Vm sin t
• The function repeats itself every T seconds.
• This is called the period
T
2

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Sinusoids III
• The period is inversely related to another
important characteristic, the frequency
f 
1
T
• The units of this is cycles per second, or
Hertz (Hz)
• It is often useful to refer to frequency in
angular terms:
  2 f
• Here the angular frequency is in radians per
second
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Radians
• Although degrees are more commonly used
in engineering, radians are the more natural
unit for sinusoids.
• Thus it is important to establish the
relationship between degrees and radians.
• A 360-degree revolution is equal to 2
radians.
• Or:
 
Radians  
 180
 180 

  Radians
  Degrees Degrees  

  
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EXAMPLE
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Phase
• More generally, we need to account for relative
timing of one wave versus another.
• This can be done by including a phase shift, :
• Consider the two sinusoids:
v1  t   Vm sin t and v2 t   Vm sin t   
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v2 leads v1 by θ or v1 lags v2 by θ
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EXAMPLE
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EXAMPLE
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Phase II
• If two sinusoids are in phase, then this
means that the reach their maximum and
minimum at the same time.
• Sinusoids may be expressed as sine or
cosine.
• The conversion between them is:
sin t  180    sin t
cos t  180    cos t
sin t  90    cos t
cos t  90   sin t
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Phase Relations
• Now let us compare two
sinusoids operating at the
same frequency:
v1  t   Vm sin t
v2  t   Vm sin t   
• The starting point of v2 occurs
first in time.
• We thus say that v2 leads v1 by
, or v1 lags v2.
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Phase Relations II
• We can make this comparison because the
two sinusoids are at the same frequency.
• The difference angle must be less than 180
degrees to be called leading or lagging.
• Sinusoids may also be expressed as
cosines.
• This may require transforming one waveform
to allow proper comparison.
• We can use trigonometric identities to do so:
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Trig Identities
sin(t )   sin(t )
cos(t )  cos(t )
sin(t  180 )   sin(t )
cos(t  180 )   cos(t )
sin(t  90 )   cos(t )
cos(t  90 )  sin(t )
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EXAMPLE
SOLUTIONS
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