Download Chapter 36. AC Circuits

Chapter 36. AC Circuits
Today, a “grid” of AC
electrical distribution systems
spans the United States and
other countries. Any device
that plugs into an electric
outlet uses an AC circuit. In
this chapter, you will learn
some of the basic techniques
for analyzing AC circuits.
Chapter Goal: To understand
and apply basic techniques of
AC circuit analysis.
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AC Circuits
Alternating current refers to currents and voltages which
are harmonic functions.
“AC power” refers to the 60 hz alternating voltage
available from a common outlet.
AC circuit analysis has a more general motivation and
application. Any voltage or current may be represented
as a sum of harmonic voltages or currents and the
response of a linear circuit understood in terms of the
response for different frequencies.
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Fourier analysis
Any function may be represented by a sum of sin and cos
functions.
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AC Sources and Phasors
An AC signal is described by an amplitude, a frequency, and a phase.
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AC Current in a resistor
If an ideal
resistor is subject
to an alternating
voltage, an
alternating
current flows.
Ohm’s law V=IR
applies to the
instantaneous
currents and
voltages.
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AC Current in a resistor
The resistor voltage vR is
given by
where VR is the peak or
maximum voltage. The
current through the resistor is
where IR = VR/R is the peak current.
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EXAMPLE 36.1 Finding resistor voltages
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EXAMPLE 36.1 Finding resistor voltages
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EXAMPLE 36.1 Finding resistor voltages
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EXAMPLE 36.1 Finding resistor voltages
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EXAMPLE 36.1 Finding resistor voltages
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AC current in a capacitor
A capacitor differentiates a
voltage signal.
An AC voltage applied to a
capacitor corresponds to an AC
current at the same frequency but
with a different phase.
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Capacitor Circuits
Suppose the instantaneous voltage across a capacitor is:
The instantaneous current in the circuit is
The AC current to and from a capacitor leads the
capacitor voltage by π/2 rad, or 90°.
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Capacitive Reactance
The capacitive reactance XC is defined as
The units of reactance, like those of resistance, are ohms.
Reactance relates the peak voltage VC and current IC:
NOTE: Reactance differs from resistance in that it does not
relate the instantaneous capacitor voltage and current
because they are out of phase. That is, vC ≠ iCXC.
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EXAMPLE 36.3 Capacitor current
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Power filtering with a capacitor current
A capacitor has a high impedance at low frequency (behaves
like a huge resistor) and low impedance at high frequency
(behaves like a short).
The low pass filter, a DC input voltage charges the capacitor
in time RC and there after the output voltage equals the input.
An additional high frequency voltage component is shorted
out.
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AC inductor circuit
An ideal inductor is circuit
active only under AC
conditions.
An ideal inductor integrates a
signal.
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Inductor Circuits
If the instantaneous voltage across a single inductor is
the instantaneous inductor current is
The AC current through an inductor lags the inductor
voltage by π/2 rad, or 90°.
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Inductive Reactance
The inductive reactance XL is defined as
Reactance relates the peak voltage VL and current IL:
NOTE: Reactance differs from resistance in that it does not
relate the instantaneous inductor voltage and current
because they are out of phase. That is, vL ≠ iLXL.
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Inductive Reactance
In contrast to a capacitor,
an inductor offers a
vanishing impedance at
low frequency and a high
impedance at high
frequency.
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Inductive filters
An inductor lets low
frequencies slip through
while choking off high
frequencies.
Low pass
filter
In the low pass filter a DC
input voltage transfers to
the output – the inductor is
just a conducting wire –
while a high frequency is
choked. In the high pass
filter, the inductor shorts
out low frequency voltages.
High pass
filter
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EXAMPLE 36.5 Current and voltage of an
inductor
QUESTION:
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EXAMPLE 36.5 Current and voltage of an
inductor
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EXAMPLE 36.5 Current and voltage of an
inductor
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EXAMPLE 36.5 Current and voltage of an
inductor
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AC circuit combinations
The response of a
combination of circuit
elements can be
understood by series and
parallel addition of the
response of the elements
by accounting
simultaneously for
amplitude and phase
effects.
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Applications
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The Series RLC Circuit
The impedance Z of a series RLC circuit is defined as
Impedance, like resistance and reactance, is measured in
ohms. The circuit’s peak current is related to the source emf
and the circuit impedance by
Z is at a minimum, making I a maximum, when XL = XC, at
the circuit’s resonance frequency:
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The resonance of a RLC circuit
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Applications
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Power in AC Circuits
The root-mean-square current Irms is related to the peak
current IR by
Similarly, the root-mean-square voltage and emf are
The average power supplied by the emf is
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EXAMPLE 36.7 Lighting a bulb
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EXAMPLE 36.7 Lighting a bulb
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Impedance
This chapter introduces analysis of time dependent linear
circuits. A passive circuit element is described by an
internal resistance and a frequency dependent reactance.
The combination is the impedance Z= R+iX.
Not demonstrated here but for your amusement: series and
parallel combinations of passive elements may be
understood by reduction using familiar rules for the
parallel (add inverses) and series (addition) addition of
resistance but with complex impedance.
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