alternating current circuits chapter 28. ac circuit an ac circuit consists of a combination of...
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Alternating Current Circuits
Chapter 28
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
• ω is the angular frequency of the AC voltage
22 ƒ
πω π
T
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
• The instantaneous voltage across the resistor is also given as ΔvR = ImaxR sin ωt
sin sin maxmaxIR
R
v Vi ωt ωt
R R
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: 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
2i R
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
• The average power dissipated in resistor in an AC circuit carrying a current I is
maxmax0.707
2rms
II I
maxmax0.707
2rms
VV V
2av rmsI 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:
• ΔvC: instantaneous voltage across the capacitor
tVCtq sin)( max
0 Cvv 0C
qv
dt
dqiC tVC cosmax
2sinmax
tVC
C
VI
/1max
max
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):
tVCtq sin)( max
dt
dqiC tVC cosmax
2sinmax
tVC
C
VI
/1max
max
CX C
1
CXIV maxmax
Chapter 28Problem 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:
• ΔvL: instantaneous voltage across the inductor
0 Lvv 0dt
diLv
tdtL
ViL sinmax t
L
V
cosmax
tV sinmax
L
VI
max
max
dt
diLv L
2sinmax
t
L
V
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): LX L
tdtL
ViL sinmax t
L
V
cosmax
tV sinmax
L
VI
max
max
dt
diLv L
2sinmax
t
L
V
LXIV maxmax
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 thecharge on the capacitor oscillatebetween maximum positive andnegative 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, someenergy is stored in the magneticfield of the inductor
• Energy is transferred from theelectric 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 fullycharged and the cycle repeats
• The energy continues to oscillatebetween the inductor and the capacitor
LC Circuit• The total energy stored in the LC circuit remains
constant in time
• Solution:
22
22 LI
C
QU 0
22
22
LI
dt
d
C
Q
dt
d
dt
dU
0dt
dILI
dt
dQ
C
Q0
2
2
dt
QdL
C
Q
LC
Q
dt
Qd
2
2
LC
tQtQ cos)( max
LC Circuit• The angular frequency, ω, of the circuit depends on
the inductance and the capacitance
• It is the natural frequency of oscillation of the circuit
• The current can be expressed as a function of time:
LC
tQtQ cos)( max tQtQ cos)( max
1ωLC
tQdt
d
dt
dQtI cos)( max tQ sinmax
tItI sin)( max
LC Circuit• Q and I are 90° out of phase with each other, so when
Q is a maximum, I is zero, etc.
tQtQ cos)( max
tItI sin)( max
Energy in LC Circuits• The total energy can be expressed as a function of
time
• The energy continually oscillatesbetween the energy stored in theelectric and magnetic fields
• When the total energy is stored inone field, the energy stored in theother field is zero
22
22 LI
C
QU t
LIt
C
Q 22
max22
max sin2
cos2
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 28Problem 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°
max
max
max
sin sin
sin cos 2
sin cos 2
R R
L L L
C C C
v I R ωt V ωt
πv I X ωt V ωt
πv I X ωt V ωt
Phasor Diagrams• Because of the different phase
relationships with the current, the voltagescannot 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
is the phase angle between the current and the maximum voltage
• The equations also apply to rms values
2 2max ( )
tan
R L C
L C
R
V V V V
V VV
Phasor Diagrams
ΔVR = Imax RΔVL = Imax XL
ΔVC = Imax XC
2 2max ( )
tan
R L C
L C
R
V V V V
V VV
22maxmax )( CL XXRIV
Impedance of a Circuit• The impedance, Z, can also be represented in a phasor
diagram
• 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
2 2( )
tan
L C
L C
Z R X X
X XR
Summary of Circuit Elements, Impedance and Phase Angles
Chapter 28Problem 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
• To achieve maximum current, the impedance must have a minimum value
• This occurs when XL = XC and
22 )( CL
rmsrms
XXR
VI
CL
00
1
LC
10
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 28Problem 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:
0 iRdt
diL
C
q
02
2
Rdt
dq
dt
qdL
C
q
Damped LC Oscillations
• Solution:
• Analogous to a damped harmonic oscillator
• When R = 0, the circuit reduces to an LC circuit (no damping in an oscillator)
teQtq dL
Rt
cos)( 2max
2
2
1
L
R
LCd
02
2
Rdt
dq
dt
qdL
C
q
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
Transformers
• The rate of change of the flux is the same for both coils, so the voltages are related by
• 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
tNV B
11 tNV B
22
11
22 V
N
NV
2211 VIVI
Chapter 28Problem 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