Alternating Current Circuits

1
Chapter 21

• Alternating Current Circuits
• and Electromagnetic Waves

Conceptual questions 2,3,4,5, 8,11,15 Quick
quizzes 1,2,4,5 Problems 2a,11,23,48
2
AC Circuit
3
Resistor in an AC Circuit
4
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

5
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

6
Ohms Law in an AC Circuit

• Ohms Law for a resistor, R, in an AC circuit
• ?Vrms Irms R

7
Problem 21.2.a
What is the resistance of a lightbulb that uses
an average power of 75 W when connected to a 60
Hz power source with a peak voltage of 170V?
8
QUICK QUIZ 21.1
Which of the following statements might be true
for a resistor connected to an AC generator? (a)
Paverage 0 and iaverage 0 (b) Pav 0 and
iav gt 0 (c) Pav gt 0 and iav 0 (d) Pav gt 0
and iav gt 0.
9
Capacitors in an AC Circuit

• The current starts out at a large value and
  charges the plates of the capacitor, initially
  there is no resistance
• As the charge on the plates increases, the
  voltage across the plates increases and the
  current flowing in the circuit decreases

10
Capacitors in an AC Circuit

• The voltage lags the current by 90o

11
Capacitive Reactance and Ohms Law

• Ohms Law for a capacitor in an AC circuit
• ?Vrms Irms XC

12
Inductors in an AC Circuit

• The current in the circuit is impeded by the emf
  of the inductor
• The voltage across the inductor always leads the
  current by 90

13
Inductive Reactance, XL, and Ohms Law

• Ohms Law for the inductor
• ?Vrms Irms XL

• XL 2?ƒL

14
The RLC Series Circuit

• The current in the circuit is the same at any
  time and varies sinusoidally with time

15
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

16
Phasor Diagrams

• Represent the voltage across each element as a
  rotating vector, called a phasor
• Its projection on the y-axis represents the
  varying instantaneous voltage in the circuit
• The diagram is called a phasor diagram

17
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

18
Phasor Diagram

• The phasors are added as vectors to account for
  the phase differences in the voltages
• ?VL and ?VC are on the same line and so the net y
  component is ?VL - ?VC

19
?Vmax 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
• ? is the phase angle between the current and the
  maximum voltage

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QUICK QUIZ 21.2
For the circuit of the figure below, is the
voltage of the source equal to (a) the sum of the
maximum voltages across the elements, (b) the sum
of the instantaneous voltages across the
elements, or (c) the sum of the rms voltages
across the elements?
21
Impedance and Ohms Law

• Ohms Law can be applied to the impedance
• ?Vmax Imax Z

22
Problems 11 and 23
Problem 21.11. What value of capacitor must be
inserted in a 60 Hz circuit in series with a
generator of 170 V maximum voltage to produce an
rms current output of 0.75 A? Problem 21.31. A
60.0 Q resistor, a 3.00 mF capacitor, and a 0.400
H inductor are connected in series to a 90.0 V 60
Hz source. Find the voltage across the LC
combination. Repeat for the RC combination.
23
Questions
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Power in an AC Circuit

• No power losses are associated with capacitors
  and pure inductors in an AC circuit
• The average power delivered by the generator is
  converted to internal energy in the resistor
• Pav Irms?VR Irms?Vrms cos ?
• cos ? is called the power factor of the circuit
• We may maximize P by adjusting the power factor

V
I
rms
V
L
rms
V
R
V
C
26
Resonance in an AC Circuit

• Resonance occurs at the frequency, ƒo, where the
  current has its maximum value
• To achieve maximum current, Z must be minimum
• This occurs when XL XC

27
Transformer
The use of iron core results in a the same
magnetic flux in both Primary and secondary
windings
The power input into the primary equals the power
output at the secondary I1?V1 I2?V2
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Transformer
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Conceptual questions

• 2. What is the impedance of an RLC circuit at
  the resonance frequency?
• 3. When a dc voltage is applied to a
  transformer, the primary coil sometimes overheats
  and burns. Why?
• Why are the primary and secondary coils of a
  transformer wrapped on an iron core that passes
  through both coils?

30
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

31
EM Waves by an Antenna
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EM Waves by an Antenna

• 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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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

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Properties of EM Waves

• Speed of ALL electromagnetic waves is
• 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

36
QUICK QUIZ 21.4
In an apparatus such as that in the figure below,
suppose the black disk is replaced by one with
half the radius. Which of the following are
different after the disk is replaced? (a)
radiation pressure on the disk (b) radiation
force on the disk (c) radiation momentum
delivered to the disk in a given time interval.
37
The Spectrum of EM Waves

• c ƒ?
• 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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Questions

• 15. Does a wire connected to a battery emit an EM
  wave?
• When light (or any EM wave) travels across a
  given region, what is that moves?
• 11. Suppose a creature from another planet had
  eyes that were sensitive to infrared radiation.
  Describe what he would see if he looked around
  the room you are in now. What would be bright
  and what would be dim?

39
Question 21.5
Receiving radio antennas can be in the form of
conducting lines or loops. What should the
orientation of each of these antennas be,
relative to a broadcasting antenna that is
perpendicular to the Earth?
40
Problem 21.48. Assume that the solar radiation
incident on the Earth is 1340 W/m2. Calculate
the power radiated by the Sun. The average
Sun-Earth separation is 1.49 1011 m.
41
MCAD
A time varying magnetic field 1. Produces an
electric field 2. Provides power to rotate an
ac generator 3. Can induce currents to flow in a
conductive loop 4. Both 1 and 3

• A transformer is used to change
• Voltage
• Power
• Current
• Voltage and current

42
Which of the following most accurately describes
light?

• An electric and magnetic wave parallel to each
  other and perpendicular to the direction of
  propagation
• An electric and magnetic wave parallel to each
  other and parallel to the direction of
  propagation
• An electric and magnetic wave perpendicular to
  each other and perpendicular to the direction of
  propagation
• An electric and magnetic wave perpendicular to
  each other and parallel to the direction of
  propagation