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**Unformatted text preview: **Chapter 21
Alternating Current Circuits
and Electromagnetic Waves AC Circuit
■ An AC circuit consists of a combination
of circuit elements and an AC generator
or source
■ The output of an AC generator is
sinusoidal and varies with time
according to the following equation
■ Δv = ΔVmax sin 2pƒt Δv is the instantaneous voltage
■ ΔVmax is the maximum voltage of the generator
■ ƒ is the frequency at which the voltage changes, in
Hz
■ Resistor in an AC Circuit
Consider a circuit
consisting of an AC source
and a resistor
■ The graph shows the
current through and the
voltage across the resistor
■ The current and the
voltage reach their
maximum values at the
same time
■ The current and the
voltage are said to be in
phase
■ More About Resistors in an AC
Circuit
■ The direction of the current has no effect
on the behavior of the resistor
■ The rate at which electrical energy is
dissipated in the circuit is given by
■ P = i2 R where i is the instantaneous current
■ the heating effect produced by an AC current with
a maximum value of Imax is not the same as that of
a DC current of the same value
■ The maximum current occurs for a small amount
of time
■ rms Current and Voltage
■ The rms current is the direct
current that would dissipate the
same amount of energy in a
resistor as is actually dissipated by
the AC current ■ Alternating voltages can also be
discussed in terms of rms values Ohm’s Law in an AC Circuit
■ rms values will be used when
discussing AC currents and voltages
■ AC ammeters and voltmeters are
designed to read rms values
■ Many of the equations will be in the
same form as in DC circuits ■ Ohm’s circuit ■ ΔVrms Law for a resistor, R, in an AC
= Irms R ■ Also applies to the maximum values of v
and i Capacitors in an AC Circuit
■ Consider a circuit containing a capacitor and
an AC source
■ The current starts out at a large value and
charges the plates of the capacitor
■ There is initially no resistance to hinder the flow
of the current while the plates are not charged ■ As the charge on the plates increases, the
voltage across the plates increases and the
current flowing in the circuit decreases More About Capacitors in an AC
Circuit
■ The current
reverses direction
■ The voltage across
the plates
decreases as the
plates lose the
charge they had
accumulated
■ The voltage across
the capacitor lags
behind the current
by 90° Capacitive Reactance and Ohm’s
Law
■ The impeding effect of a capacitor on the
current in an AC circuit is called the
capacitive reactance and is given by ■ When ƒ is in Hz and C is in F, XC will be in ohms ■ Ohm’s
■ Law for a capacitor in an AC circuit ΔVrms = Irms XC Inductors in an AC Circuit
■ ■ ■ Consider an AC circuit
with a source and an
inductor
The current in the
circuit is impeded by
the back emf of the
inductor
The voltage across the
inductor always leads
the current by 90° Inductive Reactance and Ohm’s
Law
■ The effective resistance of a coil in
an AC circuit is called its inductive
reactance and is given by
■ XL = 2pƒL ■ When ƒ is in Hz and L is in H, XL will be in ohms ■ Ohm’s
■ ΔVrms Law for the inductor = Irms XL The RLC Series Circuit
■ The resistor,
inductor, and
capacitor can be
combined in a
circuit
■ The current in the
circuit is the same
at any time and
varies sinusoidally
with time Current and Voltage Relationships
in an RLC Circuit
■ ■ ■ The instantaneous
voltage across the
resistor is in phase
with the current
The instantaneous
voltage across the
inductor leads the
current by 90°
The instantaneous
voltage across the
capacitor lags the
current by 90° Phasor Diagrams
■ ■ ■ To account for the
different phases of the
voltage drops, vector
techniques are used
Represent the voltage
across each element
as a rotating vector,
called a phasor
The diagram is called
a phasor diagram Phasor Diagram for RLC Series
Circuit
■ ■ ■ The voltage across the
resistor is on the +x
axis since it is in phase
with the current
The voltage across the
inductor is on the +y
since it leads the
current by 90°
The voltage across the
capacitor is on the –y
axis since it lags behind
the current by 90° Phasor Diagram, cont
■ The phasors are
added as vectors to
account for the
phase differences in
the voltages
■ ΔVL and ΔVC are on
the same line and so
the net y component
is ΔVL - ΔVC ΔVmax From the Phasor
Diagram
■ The voltages are not in phase, so they
cannot simply be added to get the
voltage across the combination of the
elements or the voltage source ■ f is the phase angle between the
current and the maximum voltage Impedance of a Circuit
■ The impedance, Z,
can also be
represented in a
phasor diagram Impedance and Ohm’s Law
■ Ohm’s Law can be applied to the
impedance
■ ΔVmax = Imax Z Summary of Circuit Elements,
Impedance and Phase Angles Problem Solving for AC
Circuits
■ Calculate as many unknown
quantities as possible
■ For example, find XL and XC
■ Be careful of units -- use F, H, Ω ■ Apply Ohm’s Law to the portion of
the circuit that is of interest
■ Determine all the unknowns asked
for in the problem Power in an AC Circuit
■ No power losses are associated with
capacitors and pure inductors in an AC
circuit In a capacitor, during one-half of a cycle energy
is stored and during the other half the energy is
returned to the circuit
■ In an inductor, the source does work against
the back emf of the inductor and energy is
stored in the inductor, but when the current
begins to decrease in the circuit, the energy is
returned to the circuit
■ Power in an AC Circuit,
cont
■ The average power delivered by
the generator is converted to
internal energy in the resistor
■ Pav = IrmsΔVR = IrmsΔVrms cos f
■ cos f is called the power factor of the
circuit ■ Phase shifts can be used to
maximize power outputs Resonance in an AC Circuit
■ Resonance occurs at
the frequency, ƒo,
where the current has
its maximum value
■ ■ To achieve maximum
current, the
impedance must have
a minimum value
This occurs when XL =
XC Resonance, cont
■ Theoretically, if R = 0 the current would be
infinite at resonance
■ ■ Tuning a radio
■ ■ Real circuits always have some resistance A varying capacitor changes the resonance frequency
of the tuning circuit in your radio to match the station
to be received Metal Detector
■
■ The portal is an inductor, and the frequency is set to a
condition with no metal present
When metal is present, it changes the effective
inductance, which changes the current which is
detected and an alarm sounds Transformers
■ An AC transformer
consists of two
coils of wire wound
around a core of
soft iron
■ The side connected
to the input AC
voltage source is
called the primary
and has N1 turns Transformers, 2
■ The other side, called the secondary,
is connected to a resistor and has N2
turns
■ The core is used to increase the
magnetic flux and to provide a
medium for the flux to pass from one
coil to the other
■ The rate of change of the flux is the
same for both coils Transformers, 3
■ The voltages are related by ■ When N2 > N1, the transformer is
referred to as a step up transformer
■ When N2 < N1, the transformer is
referred to as a step down transformer Transformer, final
■ The power input into the primary
equals the power output at the
secondary
■ I1ΔV1 ■ You ■ This
■ In = I2ΔV2 don’t get something for nothing assumes an ideal transformer real transformers, power efficiencies
typically range from 90% to 99% Electrical Power
Transmission
■ When transmitting electric power over
long distances, it is most economical to
use high voltage and low current
■ ■ In Minimizes I2R power losses practice, voltage is stepped up to
about 230 000 V at the generating
station and stepped down to 20 000 V at
the distribution station and finally to 120
V at the customer’s utility pole James Clerk Maxwell
■ ■ Electricity and
magnetism were
originally thought to
be unrelated
in 1865, James Clerk
Maxwell provided a
mathematical theory
that showed a close
relationship between
all electric and
magnetic phenomena Maxwell’s Starting Points
■ Electric field lines originate on positive
charges and terminate on negative charges
■ Magnetic field lines always form closed loops
– they do not begin or end anywhere
■ A varying magnetic field induces an emf and
hence an electric field (Faraday’s Law)
■ Magnetic fields are generated by moving
charges or currents (Ampère’s Law) Maxwell’s Predictions
■ ■
■
■ Maxwell used these starting points and a
corresponding mathematical framework to prove
that electric and magnetic fields play symmetric
roles in nature
He hypothesized that a changing electric field
would produce a magnetic field
Maxwell calculated the speed of light to be 3x10 8
m/s
He concluded that visible light and all other
electromagnetic waves consist of fluctuating
electric and magnetic fields, with each varying
field inducing the other Hertz’s Confirmation of Maxwell’s
Predictions
■ Heinrich Hertz was
the first to
generate and
detect
electromagnetic
waves in a
laboratory setting Hertz’s Basic LC Circuit
■ ■ When the switch is
closed, oscillations
occur in the current
and in the charge on
the capacitor
When the capacitor is
fully charged, the total
energy of the circuit is
stored in the electric
field of the capacitor
■ At this time, the current
is zero and no energy is
stored in the inductor LC Circuit, cont
■
■
■ As the capacitor discharges, the energy stored
in the electric field decreases
At the same time, the current increases and the
energy stored in the magnetic field increases
When the capacitor is fully discharged, there is
no energy stored in its electric field
■ ■
■ The current is at a maximum and all the energy
is stored in the magnetic field in the inductor The process repeats in the opposite direction
There is a continuous transfer of energy
between the inductor and the capacitor Hertz’s Experimental
Apparatus
■ An induction coil is
connected to two
large spheres
forming a capacitor
■ Oscillations are
initiated by short
voltage pulses
■ The inductor and
capacitor form the
transmitter Hertz’s Experiment
■ Several meters away from the
transmitter is the receiver
■ This consisted of a single loop of wire
connected to two spheres
■ It had its own inductance and capacitance ■ When the resonance frequencies of
the transmitter and receiver matched,
energy transfer occurred between
them Hertz’s Conclusions
■ Hertz hypothesized the energy
transfer was in the form of waves
■ These are now known to be
electromagnetic waves ■ Hertz confirmed Maxwell’s theory
by showing the waves existed and
had all the properties of light waves
■ They had different frequencies and
wavelengths Hertz’s Measure of the Speed of
the Waves
■ Hertz measured the speed of the waves
from the transmitter
He used the waves to form an interference
pattern and calculated the wavelength
■ From v = f λ, v was found
■ v was very close to 3 x 108 m/s, the known
speed of light
■ ■ This provided evidence in support of
Maxwell’s theory Electromagnetic Waves Produced
by an Antenna
■ When a charged particle undergoes an
acceleration, it must radiate energy If currents in an ac circuit change rapidly,
some energy is lost in the form of em waves
■ EM waves are radiated by any circuit
carrying alternating current
■ ■ An alternating voltage applied to the
wires of an antenna forces the electric
charge in the antenna to oscillate EM Waves by an Antenna,
cont ■
■ ■
■ Two rods are connected to an ac source, charges oscillate between
the rods (a)
As oscillations continue, the rods become less charged, the field
near the charges decreases and the field produced at t = 0 moves
away from the rod (b)
The charges and field reverse (c)
The oscillations continue (d) EM Waves by an Antenna,
final
■ Because the
oscillating charges
in the rod produce
a current, there is
also a magnetic
field generated
■ As the current
changes, the
magnetic field
spreads out from
the antenna Charges and Fields,
Summary
■ Stationary charges produce only
electric fields
■ Charges in uniform motion
(constant velocity) produce electric
and magnetic fields
■ Charges that are accelerated
produce electric and magnetic
fields and electromagnetic waves Electromagnetic Waves,
Summary
■A changing magnetic field produces
an electric field
■ A changing electric field produces a
magnetic field
■ These fields are in phase
■ At any point, both fields reach their
maximum value at the same time Electromagnetic Waves are
Transverse Waves
■ The E and B fields
are perpendicular
to each other
■ Both fields are
perpendicular to
the direction of
motion
■ Therefore, em
waves are
transverse waves Properties of EM Waves
■ Electromagnetic waves are transverse waves
■ Electromagnetic waves travel at the
speed of light ■ Because em waves travel at a speed that is
precisely the speed of light, light is an
electromagnetic wave Properties of EM Waves, 2
■ The ratio of the electric field to the
magnetic field is equal to the speed of light ■ Electromagnetic waves carry energy as
they travel through space, and this energy
can be transferred to objects placed in
their path Properties of EM Waves, 3
■ Energy carried by em waves is
shared equally by the electric and
magnetic fields Properties of EM Waves,
final
■ Electromagnetic waves transport
linear momentum as well as energy
■ For complete absorption of energy U,
p=U/c
■ For complete reflection of energy U,
p=(2U)/c ■ Radiation pressures can be
determined experimentally Determining Radiation
Pressure
■ This is an apparatus
for measuring
radiation pressure
■ In practice, the
system is contained
in a vacuum
■ The pressure is
determined by the
angle at which
equilibrium occurs The Spectrum of EM Waves
■ Forms of electromagnetic waves
exist that are distinguished by their
frequencies and wavelengths
■c = ƒλ ■ Wavelengths for visible light range
from 400 nm to 700 nm
■ There is no sharp division between
one kind of em wave and the next The EM
Spectrum
■ Note the overlap
between types
of waves
■ Visible light is a
small portion of
the spectrum
■ Types are
distinguished by
frequency or
wavelength Notes on The EM Spectrum
■ Radio Waves ■ Used in radio and television
communication systems ■ Microwaves ■ Wavelengths from about 1 mm to 30 cm
■ Well suited for radar systems
■ Microwave ovens are an application Notes on the EM Spectrum,
2
■ Infrared waves ■ Incorrectly called “heat waves”
■ Produced by hot objects and
molecules
■ Readily absorbed by most materials ■ Visible
■ Part light of the spectrum detected by the
human eye
■ Most sensitive at about 560 nm
(yellow-green) Notes on the EM Spectrum,
3
■ Ultraviolet light Covers about 400 nm to 0.6 nm
■ Sun is an important source of uv light
■ Most uv light from the sun is absorbed in the
stratosphere by ozone
■ ■ X-rays Most common source is acceleration of highenergy electrons striking a metal target
■ Used as a diagnostic tool in medicine
■ Notes on the EM Spectrum,
final
■ Gamma rays ■ Emitted by radioactive nuclei
■ Highly penetrating and cause serious
damage when absorbed by living
tissue ■ Looking at objects in different
portions of the spectrum can
produce different information Doppler Effect and EM
Waves
■A Doppler Effect occurs for em waves,
but differs from that of sound waves
■ For sound waves, motion relative to a
medium is most important
■ ■ For light waves, the medium plays no role since
the light waves do not require a medium for
propagation The speed of sound depends on its frame of
reference
■ The speed of em waves is the same in all
coordinate systems that are at rest or moving with
a constant velocity with respect to each other Doppler Equation for EM
Waves
■ The Doppler effect for em waves f’ is the observed frequency
■ f is the frequency emitted by the source
■ u is the relative speed between the source
and the observer
■ The equation is valid only when u is much
smaller than c
■ Doppler Equation, cont
■ The positive sign is used when the object
and source are moving toward each other
■ The negative sign is used when the
object and source are moving away from
each other
■ Astronomers refer to a red shift when
objects are moving away from the earth
since the wavelengths are shifted toward
the red end of the spectrum ...

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- Winter '18
- ghars
- Alternating Current, electromagnetic waves, EM waves