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?
Goals for Chapter 31
• Use phasors to describe sinusoidally quantities
• Define reactance to describe voltage in a circuit
• Analyze an L-R-C series circuit
• Determine power in ac circuits
• See how an L-R-C circuit responds to frequency
• Learn how transformers work
Introduction
• How does a radio tune to a
particular station?
• Use a variable capacitor in
concert with inductors and
resistors!
Alternating currents
• AC source is device that supplies a sinusoidally varying voltage
V(t) = Vmaxcos wt
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
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”
Alternating currents across a resistor…
How do Resistors affect an AC
circuit?
What does VR (t) = Va – Vb look like in time?
Alternating current across a resistor…
• When resistor connected with AC source, voltage &
current amplitudes are related by Ohm’s law:
• Current and Voltage are in phase across resistors
•
VR(t) = Vmaxcos wt
&
iR (t) = imax cos wt
Alternating currents across a resistor…
• Current and Voltage are in phase across resistors
•
VR(t) = Vmaxcos wt
&
iR (t) = imax cos wt
Alternating currents across a capacitor…
How do CAPACITORS affect
an AC circuit?
What does VC (t) = Va – Vb look like in time?
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
• Current flows but charge builds up (so does V = Q/C!)
• Voltage across capacitor LAGS behind current!
Alternating currents across a capacitor…
CAPACITORS
• VOLTAGE lags CURRENT
• “Emf” behind “i”
• CURRENT leads Voltage
• “i” ahead of “Emf”
I C E
Alternating currents across a capacitor…
• Current and Voltage are out of phase across capacitors
•
VC(t) = Vmaxcos (wt - f) &
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 left plate reaches max AFTER current already decreasing
(but still positive)
•
Note! Va > Vb as charge builds from + current
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”
Alternating currents across a capacitor…
• Current and Voltage are out of phase across capacitors
•
VC(t) = Vmaxcos (wt - f) &
iC (t) = imax cos wt
Alternating currents across a inductor…
How do INDUCTORS affect an
AC circuit?
What does VL (t) = Va – Vb look like in time?
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 fastest
(from – to +, or from + to - )
Alternating currents across an inductor…
• Current &Voltage are out of phase across inductors
•
VL(t) = Vmaxcos (wt+f)
&
iL (t) = imax cos wt
• At t = 0, current @ max, but rate of change = 0,
inductor is just a wire (with no EMF)
• Va - Vb = 0!
Alternating currents across an inductor…
Example!
Say current is going + direction (from a to b) but
about to change direction and go from b to a…
THEN
Inductor flux is decreasing to zero, and changing fast!
Inductor generates EMF in the positive direction
EMF generated to
keep current
Current changing
Direction!
Positive to Negative
Alternating currents across an inductor…
Example!
Say current is going + direction (from a to b) but
about to change direction and go from b to a…
THEN
Inductor generates EMF so Vb > Va!
Va – Vb will be negative..
EMF generated to
keep current
Current changing
Direction!
Positive to Negative
Alternating currents across an inductor…
• Current &Voltage are out of phase across inductors
•
VL(t) = Vmaxcos (wt+f)
&
iL (t) = imax cos wt
• When current is 0 and changing fastest from + to -,
di/dt is maximum, and so is inductors + EMF
• Vb > Va
• Va-Vb = max -
Alternating currents across an inductor…
• Current &Voltage are out of phase across inductors
•
VL(t) = Vmaxcos (wt+f)
&
iL (t) = imax cos wt
• By the time the current has peaked in the negative
direction, change in flux is again 0, voltage Va – Vb
will already be zero.
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…
Alternating currents across an inductor…
INDUCTORS
• CURRENT lags VOLTAGE
• Voltage leads Current
• Emf ahead of “i”
E L I
Alternating currents across an inductor…
• Current and Voltage are out of phase across inductors
•
VL(t) = Vmaxcos (wt+f) &
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!
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
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!
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
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!)
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
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!
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
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?
How can we mathematically model AC circuits
and the complex relationships of voltage and
current through all components?
Phasors!
No, not PHASERS!
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
The “real” portion of a Phasor!
• Projection of vector onto horizontal axis
The “real” portion of a Phasor!
• Consider four different current phasors:
IB
IA
w
IC
ID
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?
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
I
I
B
A
w
I
C
ID
Resistor in an ac circuit
• VR = IR; VR in phase with I
Phasors for Voltage/Current across Resistor
• VR(t) = Vmaxcos (wt) &
iR (t) = imax cos wt
Capacitors in an ac circuit
• VC(t) = Vmaxcos (wt-f)
VC out of phase with I
Phasors for Voltage/Current across Capacitor
• VC(t) = Vmaxcos (wt-f)
&
iC (t) = imax cos wt
I - C- E: Current Leads Voltage Across Capcitor
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?
Inductors in AC circuits
• VL(t) = Vmaxcos (wt+f)
• VL out of phase with I
Phasors for Voltage/Current across Inductor
• VL(t) = Vmaxcos (wt+f)
&
iL (t) = imax cos wt
E-L-I: Voltage Leads Current Across Inductor
Inductor in an ac circuit
• The voltage amplitude across the inductor is
VL = IXL
• XL = “inductive reactance” = wL
• XL increases as frequency increases!
• WHY?
Comparing ac circuit elements
• Combining resistor, inductor, & capacitor in AC circuit.
Comparing ac circuit elements
• Combining resistor, inductor, & capacitor in AC circuit.
• When is OVERALL resistance in series least???
•
R is fixed
•
In series, all impedances add!
•
So minimize point where XC and XL intersect!
Comparing ac circuit elements
• Combining resistor, inductor, & capacitor in AC circuit.
• When is OVERALL resistance in series least???
•
R is fixed
•
In series, all impedances add!
•
So minimize point where XC and XL intersect!
Resonant
Frequency
Minimizes
Impedance!
Resonance in ac circuits
•
At the resonance angular frequency w0, the inductive reactance equals the
capacitive reactance and the current amplitude is greatest.
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
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.
The L-R-C series circuit
• Suppose inductive reactance > capacitive reactance?
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.
The L-R-C series circuit
• Suppose capacitive reactance > inductive reactance?
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)
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
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!
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
=
=
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
= 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.
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?
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?
Root-mean-square values
For sinusoidal ac sources, rms current & voltage:
Wall socket has voltage amplitude of V = 170 V, meaning
that voltage alternates
between +170 V & −170 V.
Rms voltage Vrms = 120 V.
Root-mean-square values
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?
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!
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
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
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.
Tuning a radio
• RMS voltage of 1.0V; what is resonance frequency?
At that frequency what are XL and XC and Z?
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.
Real transformers
• Real transformers always have some power losses