Download Chapter 31: AC Circuits

Chapter 31
Alternating Current
PowerPoint® Lectures for
University Physics, Thirteenth Edition
– Hugh D. Young and Roger A. Freedman
Lectures by Scott Hildreth – Chabot College
Questions about AC Circuits
• How do AC circuits work, compared with DC?
• Advantages? Disadvantages?
• Westinghouse vs. Edison?
• What roles do inductors, capacitors, and resistors
play in AC circuits?
• How can we mathematically model AC circuits
and the complex relationships of voltage and
current through all components?
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Goals for Chapter 31
• To use phasors to describe sinusoidally varying
quantities
• To use reactance to describe voltage in a circuit
• To analyze an L-R-C series circuit
• To determine power in ac circuits
• To see how an L-R-C circuit responds to
frequency
• To learn how transformers work
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Introduction
• How does a radio tune to a
particular station?
• Use a variable capacitor in
concert with inductors and
resistors!
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Alternating currents
•
Voltage (supply) is a sinusoidal function of time
V(t) = Vmaxcos wt
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Alternating currents
•
Voltage (supply) is a sinusoidal function of time
V(t) = Vmaxcos wt
•
Resulting current is ALSO a sinusoidal function in time
i(t) = imax cos wt
•
But … phases of these are not necessarily the same through the circuit!
• When Voltage is maximum, Current may not be!
• V(t) = Vmaxcos (wt +/-f)
but i(t) = imax cos (wt)
• If f = 0 , Voltage and Current are described as “in phase”
• If f  0 , Voltage and Current are described as “out of phase”
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Alternating currents across a resistor…
How do Resistors affect an AC
circuit?
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Alternating current across a resistor…
• Current and Voltage are in phase across resistors
•
VR(t) = Vmaxcos wt
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&
iR (t) = imax cos wt
Alternating currents across a resistor…
• Current and Voltage are in phase across resistors
•
VR(t) = Vmaxcos wt
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&
iR (t) = imax cos wt
Alternating currents across a capacitor…
How do CAPACITORS affect an
AC circuit?
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Alternating currents across a capacitor…
• Current and Voltage are out of phase across capacitors
•
VC(t) = Vmaxcos (wt - f) &
iC (t) = imax cos wt
• Capacitors take time to reach maximum voltage
• Voltage across capacitor LAGS behind current!
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Alternating currents across a capacitor…
CAPACITORS
• VOLTAGE lags CURRENT
• CURRENT leads Voltage
I C E
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Alternating currents across a capacitor…
• Current and Voltage are out of phase across capacitors
•
VC(t) = Vmaxcos (wt - f) &
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iC (t) = imax cos wt
Alternating currents across a capacitor…
• Current and Voltage are out of phase across capacitors
•
VC(t) = Vmaxcos (wt - f) &
iC (t) = imax cos wt
•
Note current is max at time t = 0
•
But charge on capacitor is not yet built up to a maximum!
•
Charge on plates max AFTER current already decreasing (but still
positive)
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Alternating currents across a capacitor…
• Current and Voltage are out of phase across capacitors
•
VC(t) = Vmaxcos (wt - f) &
iC (t) = imax cos wt
•
Note current is max at time t = 0
•
Voltage isn’t maximum until some time t = + f/w later!
•
Voltage E will “lag” current I across a capacitor C
•
Remember “I – C – E”
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Alternating currents across a capacitor…
• Current and Voltage are out of phase across capacitors
•
VC(t) = Vmaxcos (wt - f) &
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iC (t) = imax cos wt
Alternating currents across a inductor…
How do INDUCTORS affect an
AC circuit?
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Alternating currents across an inductor…
• Current &Voltage are out of phase across inductors
•
VL(t) = Vmaxcos (wt+f)
&
iL (t) = imax cos wt
• Inductors “fight” current change, and push hardest in
the opposite direction when current changes from – to
+ or + to -
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Alternating currents across an inductor…
• Current &Voltage are out of phase across inductors
•
VL(t) = Vmaxcos (wt+f)
&
iL (t) = imax cos wt
• So voltage across the inductor will reach maximum
BEFORE the current through it builds to max…
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Alternating currents across an inductor…
INDUCTORS
• CURRENT lags VOLTAGE
• Voltage leads Current
E L I
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Alternating currents across an inductor…
• Current and Voltage are out of phase across inductors
•
VL(t) = Vmaxcos (wt+f) &
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iL (t) = imax cos wt
Alternating currents across an inductor…
• Current and Voltage are out of phase across inductors
•
VL(t) = Vmaxcos (wt+f)
&
iL (t) = imax cos wt
Consider cases: t = 0
•
Note current is max, and rate of change di/dt = 0
•
Voltage across inductor ONLY depends upon L di/dt!
•
So at that time, VL = 0!
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Alternating currents
• Current and Voltage are out of phase across inductors
•
VL(t) = Vmaxcos (wt+f) &
At t = 0,
current max,
voltage
across L = 0
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iL (t) = imax cos wt
Alternating currents across an inductor…
• Current and Voltage are out of phase across inductors
•
VL(t) = Vmaxcos (wt+f)
&
iL (t) = imax cos wt
Consider cases: t >0
•
Note current is positive but decreasing,
and rate of change di/dt <0
•
Voltage across inductor depends upon L di/dt!
•
Inductor reacts to decreasing current by
continuing to provide EMF from a to b
•
So at that time, VL = Va - Vb <0!
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Alternating currents
• Current and Voltage are out of phase across inductors
•
VL(t) = Vmaxcos (wt+f) &
At t > 0,
current +,
decreasing,
voltage
across L <0
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iL (t) = imax cos wt
Alternating currents across an inductor…
• Current and Voltage are out of phase across inductors
•
VL(t) = Vmaxcos (wt+f)
&
Consider cases: t = ¼ of period…
•
Note current is 0 at some time wt = + /2
•
At that time, current is changing from + to –
(large change in B field flux!)
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iL (t) = imax cos wt
Alternating currents
• Current and Voltage are out of phase across inductors
•
VL(t) = Vmaxcos (wt+f) &
At wt = + /2
current 0,
decreasing,
voltage
across L
max
negative
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iL (t) = imax cos wt
Alternating currents across an inductor…
• Current and Voltage are out of phase across inductors
•
VL(t) = Vmaxcos (wt+f)
&
iL (t) = imax cos wt
•
Note current is 0 and increasing at some time wt = 3/2
•
At that time, current is changing from - to +
(large change in B field flux!)
•
Inductor reacts to this change,
generating E to oppose this change
•
VL will be largest, positive
(Va > Vb) pushing the other way!
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Alternating currents
• Current and Voltage are out of phase across inductors
•
VL(t) = Vmaxcos (wt+f) &
At t,
wt = +3/2 ,
current 0,
increasing,
voltage
across L
max
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iL (t) = imax cos wt
How can we mathematically model AC circuits
and the complex relationships of voltage and
current, and power through all components?
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How can we mathematically model AC circuits
and the complex relationships of voltage and
current through all components?
Phasors!
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No, not PHASERS!
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Phasors
• Graphical representation of current/voltage in AC circuits
• Takes into account relative phases of different voltages
• Example: current phasor graphs i (t) = imax cos wt
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The “real” portion of a Phasor!
• Projection of vector onto horizontal axis
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The “real” portion of a Phasor!
• Consider four different current phasors:
IB
IA
w
IC
ID
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The “real” portion of a Phasor!
• Which phasor represents
• Positive current becoming
more positive?
• Positive current decreasing
to zero?
• Negative current becoming
more negative?
• Negative current decreasing
in magnitude?
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I
I
B
A
w
I
C
ID
The “real” portion of a Phasor!
• Which phasor represents
• Positive current becoming
ID
more positive?
• Positive current decreasing
IA
to zero?
• Negative current becoming
IB
more negative?
• Negative current decreasing
in magnitude?
IC
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I
I
B
A
w
I
C
ID
Resistor in an ac circuit
• VR = IR; VR in phase with I
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Phasors for Voltage/Current across Resistor
• VR(t) = Vmaxcos (wt) &
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iR (t) = imax cos wt
Capacitors in an ac circuit
• VC(t) = Vmaxcos (wt-f)
VC out of phase with I
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Phasors for Voltage/Current across Capacitor
• VC(t) = Vmaxcos (wt-f)
&
iC (t) = imax cos wt
I - C- E: Current Leads Voltage Across Capcitor
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Capacitance in an ac circuit
• The voltage amplitude across the capacitor is
VC = IXC
• Xc = “capacitive reactance” = 1/wC
• Xc = DECREASES as angular frequency increases
• WHY?
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Inductors in AC circuits
• VL(t) = Vmaxcos (wt+f)
• VL out of phase with I
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Phasors for Voltage/Current across Inductor
• VL(t) = Vmaxcos (wt+f)
&
iL (t) = imax cos wt
E-L-I: Voltage Leads Current Across Inductor
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Inductor in an ac circuit
• The voltage amplitude across the inductor is
VL = IXL
• XL = “inductive reactance” = wL
• XL increases as frequency increases!
• WHY?
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Comparing ac circuit elements
• Table 31.1 summarizes the characteristics of a resistor, an
inductor, and a capacitor in an ac circuit.
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Root-mean-square values
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Current in a personal computer
• Suppose you have a device that draws 2.7 Amps from
a 120V, 60-Hz standard US power plug.
• What is the:
• AVERAGE current,
• Average of the current squared,
• Current amplitude?
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Current in a personal computer
• Suppose you have a device that draws 2.7 Amps from
a 120V, 60-Hz standard US power plug.
•What is the:
• AVERAGE current?
0 amps!
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Average over
1 period = 0!
Current in a personal computer
• Suppose you have a device that draws 2.7 Amps from
a 120V, 60-Hz standard US power plug.
•What is the:
•Average of current squared?
2.72 = 7.3 Amps2
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Current in a personal computer
• Suppose you have a device that draws 2.7 Amps from
a 120V, 60-Hz standard US power plug.
•What is the:
• Current amplitude?
Irms = .707 I
So I = 3.8 Amps
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The L-R-C series circuit
• Combine all three elements into simple series circuit
• The voltage amplitude across an ac circuit is V = IZ
• Overall effective resistance = Z (“impedance”)
• Z = [R2 + (XL - Xc)2] ½
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The L-R-C series circuit
• Suppose inductive reactance > capacitive reactance?
• XL > XC
• Inductor is dominating
• Current will be out of phase
with supply voltage
• “E – L – I “ reminds us that
current will LAG voltage.
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The L-R-C series circuit
• Suppose inductive reactance > capacitive reactance?
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The L-R-C series circuit
• Suppose capacitive reactance > inductive reactance?
• X C > XL
• Capacitor is dominating
• Current will be out of phase
with supply voltage
• “I – C – E ” reminds us that
current will LEAD voltage.
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The L-R-C series circuit
• Suppose capacitive reactance > inductive reactance?
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A resistor and a capacitor in an ac circuit
• 200 Ohm Resistor in series with 5 mF capacitor.
Voltage across resistor VR = 1.20V cos (2500 rad/sec) x t
• What is i(t)?
• What is the reactance?
• What is Vc(t)
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A resistor and a capacitor in an ac circuit
• 200 Ohm Resistor in series with 5 mF capacitor.
Voltage across resistor = 1.20V cos (2500 rad/sec) x t
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A resistor and a capacitor in an ac circuit
• 200 Ohm Resistor in series with 5 mF capacitor.
Voltage across resistor = 1.20V cos (2500 rad/sec) x t
Ohm’s Law applies (that’s why it is a LAW!  )
VR = IR
so
I = 0.006 A cos (2500 rad/sec) x t
Note current is in phase with the voltage across R!
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A resistor and a capacitor in an ac circuit
• 200 Ohm Resistor in series with 5 mF capacitor.
Voltage across resistor = 1.20V cos (2500 rad/sec) x t
Capacitive Reactance
XC = 1/wC
=
=
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1/(2500 rad/s) x 5.0 mF
80 W
A resistor and a capacitor in an ac circuit
• 200 Ohm Resistor in series with 5 mF capacitor.
Voltage across resistor = 1.20V cos (2500 rad/sec) x t
Voltage across Capacitor
VC = I Xc
VC = I Xc
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= 0.006 A x 80 W = 0.48 V
and
= 0.48 V cos (wt - )
A useful application: the loudspeaker
• The woofer (low tones) and
the tweeter (high tones) are
connected in parallel across
the amplifier output.
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An L-R-C series circuit
• R = 300 Ohms
• L = 60 mH
• C = 0.50 mF
• V = 50 V
• w = 10,000 rad/sec
• What are XL, Xc, Z, I,
Phase angle f, and VR,
Vc, VL?
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An L-R-C series circuit
• R = 300 Ohms
• L = 60 mH
• C = 0.50 mF
• V = 50 V
• w = 10,000 rad/sec
• What are XL, Xc, Z,
I, Phase angle f, and
VR, Vc, VL?
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Power in ac circuits
• Power = I x V
• Average Power = Irms Vrms cos f
• Note that the net energy transfer over one cycle is zero for an
inductor and a capacitor.
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Resonance in ac circuits
•
At the resonance angular frequency w0, the inductive reactance equals the
capacitive reactance and the current amplitude is greatest. (See Figure 31.18
below.)
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Tuning a radio
• RMS voltage of 1.0V; what is resonance frequency?
At that frequency what are XL and XC and Z?
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Transformers
•
Power is supplied to the
primary and delivered from the
secondary.
•
Terminal voltages:
V2/V1 = N2/N1.
•
Currents in primary and
secondary:
V1I1 = V2I2.
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Real transformers
• Real transformers always have some power losses
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