Appendix to 21'4 Parallel RLC circuit

1
Appendix to 21.4 Parallel RLC circuit
IC-IL
?
(a) Parallel RLC circuit and (b) the phase
diagram of the circuit.
2
Parallel RLC, cont.

• The magnitude of the total current is

3
Example
I
Calculate the current I if R10 ?, C20 mF, and
L25 mH. The applied voltage is Vrms20 V and
f60 Hz. IR10 V/10 ?1.00 A IC(10 V)(2?60
Hz)(20?10-6 F)0.08 A IL10V/(2?60 Hz0.025
H)1.06 A I(1.00 A20.08 A-1.06
A2)1/2 I1.40 A
IC
V
I
IL
4
21.7 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 and the
  coil on the right, which is connected to a
  resistor, has N2 turns and is called the secondary

5
Transformers and power transmission
Power line
B
Problem Losses I2R  Solution Increasing the
voltage   PI(?)(V?)constant   A transformer
does the voltage increase
A
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Secondary voltage

• In the secondary coil the induced voltage V2
  arises from mutual inductions and is given by
  Faradays law
•  V2-N2(DFB/Dt)

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Ratio of the voltages

• V2-N2(DFB/Dt)
• V1-N1(DFB/Dt)
•  

V2/V1N2/N1
Secondary voltage
Turns of the secondary coil
Primary voltage
Turns of the primary coil
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Magnetic flux and power

• FB2FB1FB (100 efficient)
• P1P2 (no power losses)
• V1I1V2I2
• N2gtN1, step-up N2ltN1, step-down

Primary current
Secondary current
9
Example
A doorbell requires 0.4 A at 6.0 V. It is
connected via a transformer whose primary coil
contains 2000 turns to a 120.0 V AC line.
Calculate (a) N2 and (b) I1. (a) N2/N1V2/V1 ?
N2N1V2/V1 N2(2000?6.0 V)/(120.0 V) 100 (b)
I1V1I2V2 ? I1I2V2/V1 I1(0.4 A?6.0 V)/(120
V)0.02 A  Step-down transformer low voltage
(high current)
10
Step-up and step-down transformers

• Step-up transformer high voltage (low current)
  output (N2gtN1)
• Step-down transformer low voltage (high current)
  output (N2ltN1)

11
Example
A power transmission line has a resistance of
0.02 W/km. Calculate the I2R power loss if 200 kW
of power is transmitted from a power generator to
a city 10 km away at (a) 240 V and (b) 4.4
kV. (a) PIV ? IP/V200 kW/(240 V)833 A Power
loss I2R(833 A)2(0.2 W)1.4?105 W (b) I200
kW/(4.4 kV)45.5 A I2R(45.5 A)2(0.2 W)414 W At
240 V, 70 of the power is wasted. On the other
hand, at 4.4 kV, only about 0.2 is lost.
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Electrical Power Transmission

• When transmitting electric power over long
  distances, it is most economical to use high
  voltage and low current
• Minimizes I2R power losses
• In 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 customers utility pole

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21.8 Maxwells Predictions

• 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

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Maxwells 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 (Faradays Law)
• Magnetic fields are generated by moving charges
  or currents (Ampères Law)

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Maxwells 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 3x108
  m/s
• He concluded that visible light and all other
  electromagnetic (EM) waves consist of fluctuating
  electric and magnetic fields, with each varying
  field inducing the other (electromagnetic waves)

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21.9 Hertzs Confirmation of Maxwells Predictions

• Heinrich Hertz was the first to generate and
  detect electromagnetic waves in a laboratory
  setting

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Hertzs 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

18
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

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Hertzs 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

20
Hertzs Experiment

• Several meters away from the transmitter is the
  receiver
• The receiver consist 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

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Hertzs Conclusions

• Hertz hypothesized the energy transfer was in the
  form of waves
• These are now known to be electromagnetic waves
• Hertz confirmed Maxwells theory by showing the
  waves existed and had all the properties of light
  waves
• They are differed only in frequency and wavelength

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Hertzs 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 ?, v was found
• v was very close to 3x108 m/s, the known speed of
  light
• This provided first evidence in support of
  Maxwells theory

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21.10 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

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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)

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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

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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

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Electromagnetic Waves, Summary

• A changing magnetic field produces an electric
  field
• A changing electric field produces a magnetic
  field
• These fields are in phase (in air and vacuum)
• At any point, both fields reach their maximum
  value at the same time

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21.11 Properties of Electromagnetic 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 or plane
  waves

29
Properties of EM Waves, cont.

• 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

Free space permittivity
Vacuum permeability
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Properties of EM Waves, cont.

• 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

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Properties of EM Waves, cont.

• Energy carried by EM waves is shared equally by
  the electric and magnetic fields

Units
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Properties of EM Waves, final

• Electromagnetic waves transport linear momentum
  as well as energy
• For complete absorption of energy U, pU/c
• For complete reflection of energy U, p(2U)/c
• Radiation pressures can be determined
  experimentally

33
Radiation Pressure
The electric field of an electromagnetic wave
that strikes a surface acts on an electron,
giving it a velocity (v). The magnetic field then
exerts a force on the moving charge in the
direction of propagation of the incident light.
That is the origin of the light pressure.
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Determining Radiation Pressure

• This is an apparatus for measuring radiation
  pressure
• About 5?10-6 N/m2
• In practice, the system is contained in a vacuum
• The pressure is determined by the angle at which
  equilibrium occurs

35
21.11 The Spectrum of EM Waves

• Forms of electromagnetic waves exist that are
  distinguished by their frequencies and
  wavelengths
• c ? (light velocityfrequency?wavelength)
• 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

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21.12 The EMSpectrum

• Note the overlap between types of waves
• Visible light is a small portion of the spectrum
• Types are distinguished by frequency or wavelength

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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

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Notes on the EM Spectrum, cont.

• Infrared waves
• Incorrectly called heat waves
• Produced by hot objects and molecules
• Readily absorbed by most materials
• Visible light
• Part of the spectrum detected by the human eye
• Most sensitive at about 560 nm (yellow-green)

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Notes on the EM Spectrum, cont.

• 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 high-energy
  electrons bombarding a metal target
• Used as a diagnostic tool in medicine

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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

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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

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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

43
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