Transcript Chapter 18

Chapter 28
Alternating Current Circuits
AC Circuit
• An AC circuit consists of a combination of circuit
elements and an AC generator or source
• The output of an AC power source is sinusoidal and
varies with time according to the following equation
Δv = ΔVmax sin ωt
• Δv: instantaneous voltage
• ΔVmax is the maximum voltage (amplitude) of the
generator
2π
ω  2π ƒ 
T
• ω is the angular frequency of the AC voltage
Resistors in an AC Circuit
• Consider a circuit consisting of
an AC source and a resistor
ΔvR = ΔVmax sin ωt
• ΔvR is the instantaneous
voltage across the resistor
• The instantaneous current in the resistor is
vR Vmax
iR 

sin ωt  I max sin ωt
R
R
• The instantaneous voltage across the resistor is also
given as ΔvR = ImaxR sin ωt
Resistors in an AC Circuit
• 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
• The direction of the current has no effect on the
behavior of the resistor
Resistors in an AC Circuit
• The rate at which electrical
energy is dissipated in the circuit
is given by
  i 2R
• i: 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
I rms 
Imax
2
 0.707 Imax
• Alternating voltages can also be discussed in terms of
rms values
V
 Vrms 
max
2
 0.707  Vmax
• The average power dissipated in resistor in an AC
circuit carrying a current I is
2
rms
av  I
R
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 Law for a resistor, R, in an AC circuit
ΔVR,rms = Irms R
• The same formula applies to the maximum values of v
and i
Capacitors in an AC Circuit
• Consider a circuit containing a
capacitor and an AC source
• Kirchhoff’s loop rule gives:
 v   vC  0
q
v   0
C
• ΔvC: instantaneous voltage
across the capacitor
q ( t )  C  V max sin  t
dq


  C  V max cos  t   C  V max sin   t  
iC 
dt
2

 V max
I max 
1 / C
Capacitors in an AC Circuit
• The voltage across the capacitor
lags behind the current by 90°
• The impeding effect of a capacitor
on the current in an AC circuit is
called the capacitive reactance
(measured in ohms):
1
XC 
C
 V max  I max X C
q ( t )  C  V max sin  t
dq


  C  V max cos  t   C  V max sin   t  
iC 
dt
2

 V max
I max 
1 / C
Chapter 28
Problem 22
A capacitor and a 1.8-kΩ resistor pass the same current
when connected across 60-Hz power. Find the
capacitance.
Inductors in an AC Circuit
• Consider an AC circuit with a
source and an inductor
• Kirchhoff’s loop rule gives:
v  vL  0
di
v  L  0
dt
• ΔvL: instantaneous voltage
across the inductor
diL
  V max sin  t
v  L
dt
 V max
 V max
iL 
sin  tdt  
cos  t

 V max


L

L

sin   t  
 V max
I max 
L
2

L
Inductors in an AC Circuit
• The voltage across the inductor
always leads the current by 90°
• The effective resistance of a coil
in an AC circuit is called its
inductive reactance (measured in
ohms): X   L
L
diL
  V max sin  t
v  L
 V max  I max X L
dt
 V max
 V max
iL 
sin  tdt  
cos  t

 V max


L

L

sin   t  
 V max
I max 
L
2

L
LC Circuit
• A capacitor is connected to an inductor in an LC
circuit
• Assume the capacitor is initially charged and then the
switch is closed
• Assume no resistance and no energy losses to
radiation
• The current in the circuit and the
charge on the capacitor oscillate
between maximum positive and
negative values
LC Circuit
• With zero resistance, no energy is transformed into
internal energy
• Ideally, the oscillations in the circuit persist
indefinitely (assuming no resistance and no radiation)
• The capacitor is fully charged and the energy in the
circuit is stored in the electric field of the capacitor
Q2max / 2C
• No energy is stored in the inductor
• The current in the circuit is zero
LC Circuit
• The switch is then closed
• The current is equal to the rate at which the charge
changes on the capacitor
• As the capacitor discharges, the energy stored in the
electric field decreases
• Since there is now a current, some
energy is stored in the magnetic
field of the inductor
• Energy is transferred from the
electric field to the magnetic field
LC Circuit
• Eventually, the capacitor becomes fully discharged
and it stores no energy
• All of the energy is stored in the magnetic field of the
inductor and the current reaches its maximum value
• The current now decreases in magnitude, recharging
the capacitor with its plates having opposite their
initial polarity
• The capacitor becomes fully
charged and the cycle repeats
• The energy continues to oscillate
between the inductor and the capacitor
LC Circuit
• The total energy stored in the LC circuit remains
constant in time
2
2
Q
LI
U 

2C
2
Q dQ
dI
 LI
0
C dt
dt
2
dU d Q
d LI


0
dt dt 2C dt 2
2
Q
d Q
L 2 0
C
dt
• Solution:
Q (t )  Q max
2
 t

cos 
 
 LC

2
d Q
Q

2
dt
LC
LC Circuit
• The angular frequency, ω, of the circuit depends on
the inductance and the capacitance ω  1
LC
• It is the natural frequency of oscillation of the circuit
• The current can be expressed as a function of time:
dQ d
I (t ) 
 Q max cos  t       Q max sin  t   
dt dt
I ( t )   I max sin  t   
Q (t )  Q max
 t

cos 
 
 LC

Q ( t )  Q max cos  t   
LC Circuit
• Q and I are 90° out of phase with each other, so when
Q is a maximum, I is zero, etc.
I ( t )   I max sin  t   
Q ( t )  Q max cos  t   
Energy in LC Circuits
• The total energy can be expressed as a function of
time
2
2
2
2
Q
LI
Q max
LI max
2
U 


cos  t    
sin 2  t   
2C
2
2C
2
• The energy continually oscillates
between the energy stored in the
electric and magnetic fields
• When the total energy is stored in
one field, the energy stored in the
other field is zero
Energy in LC Circuits
• In actual circuits, there is always some resistance
• Therefore, there is some energy transformed to
internal energy
• Radiation is also inevitable in this type of circuit
• The total energy in the circuit continuously decreases
as a result of these processes
Chapter 28
Problem 27
An LC circuit with a 20-µF capacitor oscillates with period 5.0 ms. The
peak current is 25 mA. Find (a) the inductance and (b) the peak
voltage.
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
The RLC Series 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°
v R  Imax R sin ωt  VR sin ωt
π

v L  Imax X L sin  ωt    VL cos ωt
2

π

v C  Imax X C sin  ωt    VC cos ωt
2

Phasor Diagrams
• Because of the different phase
relationships with the current, the voltages
cannot be added directly
• To simplify the analysis of AC circuits, a graphical
constructor called a phasor diagram can be used
• A phasor is a vector rotating CCW; its length is
proportional to the maximum value of the variable it
represents
• The vector rotates at an angular speed equal to the
angular frequency associated with the variable, and the
projection of the phasor onto the vertical axis
represents the instantaneous value of the quantity
Phasor Diagrams
• The voltage across the resistor is in phase with the
current
• The voltage across the inductor leads the current by 90°
• The voltage across the capacitor lags behind the current
by 90°
Phasor Diagrams
• 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
Phasor Diagrams
• 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
Vmax 
VR2  ( VL   VC )2
VL   VC
tan  
VR
•  is the phase angle between the
current and the maximum voltage
• The equations also apply to rms
values
Phasor Diagrams
ΔVR = Imax R
ΔVL = Imax XL
ΔVC = Imax XC
Vmax 
VR2  ( VL   VC )2
VL   VC
tan  
VR
Vmax  I max R  ( X L  X C )
2
2
Impedance of a Circuit
• The impedance, Z, can also be represented in a phasor
diagram
Z 
R2  ( X L  X C )2
XL  XC
tan  
R
• φ: phase angle
• Ohm’s Law can be applied to the impedance
ΔVmax = Imax Z
• This can be regarded as a generalized form of Ohm’s
Law applied to a series AC circuit
Summary of Circuit Elements,
Impedance and Phase Angles
Chapter 28
Problem 30
Find the impedance at 10 kHz of a circuit consisting of a 1.5-kΩ
resistor, 5.0-µF capacitor, and 50-mH inductor in series.
Power in an AC Circuit
• No power losses are associated with pure capacitors
and pure inductors in an AC circuit
• In a capacitor, during 1/2 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
• The average power delivered by the generator is
converted to internal energy in the resistor
Pav = Irms ΔVR,rms
ΔVR, rms = ΔVrms cos 
Pav = Irms ΔVrms cos 
•
cos  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,
ω0, where the current has its maximum
value
I rms 
Vrms
R2  ( X L  X C )2
• To achieve maximum current, the
impedance must have a minimum
value
• This occurs when XL = XC and
1
0 L 
0C
0 
1
LC
Resonance in an AC Circuit
• Theoretically, if R = 0 the current
would be infinite at resonance
• Real circuits always have some
resistance
• Tuning a radio: a varying capacitor
changes the resonance frequency of
the tuning circuit in your radio to
match the station to be received
Chapter 28
Problem 29
A series RLC circuit has R = 75 Ω, L = 20 mH, and resonates at 4.0
kHz. (a) What’s the capacitance? (b) Find the circuit’s impedance at
resonance and (c) at 3.0 kHz.
Damped LC Oscillations
• The total energy is not constant, since there is a
transformation to internal energy in the resistor at the
rate of dU/dt = – i2R
• Radiation losses are still ignored
• The circuit’s operation can be expressed as:
q
di
 L  iR  0
C
dt
2
q
d q dq
L 2  R0
C
dt
dt
Damped LC Oscillations
q (t )  Qmax e
• Solution:
d 
1  R 
 
LC  2 L 

Rt
2L
cos d t   
2
• Analogous to a damped harmonic oscillator
• When R = 0, the circuit reduces to an LC circuit (no
damping in an oscillator)
2
q
d q dq
L 2  R0
C
dt
dt
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
• 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
 B Transformers
 B
V1   N1
V2   N 2
t
t
• The rate of change of the flux is the same for both coils,
so the voltages are related by
N2
V2 
N1
V1
• When N2 > N1, the transformer is referred to as a step
up transformer and when N2 < N1, the transformer is
referred to as a step down transformer
• The power input into the primary
equals the power output at the
secondary
I1V1  I 2 V2
Chapter 28
Problem 37
You’re planning to study in Europe, and you want a transformer
designed to step 230-V European power down to 120 V needed to
operate your stereo. (a) If the transformer’s primary has 460 turns,
how many should the secondary have? (b) You can save money with
a transformer whose maximum primary current is 1.5 A. If your stereo
draws a maximum of 3.3 A, will this transformer work?
Answers to Even Numbered Problems
Chapter 28:
Problem 14
V = (325 V) sin[(314 s−1)t]
Answers to Even Numbered Problems
Chapter 28:
Problem 26
22 to 190 pF