Nicholas J. Giordano
www.cengage.com/physics/giordano
Chapter 22
Alternating-Current Circuits
and Machines
Marilyn Akins, PhD • Broome Community College
DC Circuit Summary
• DC circuits
• DC stands for direct current
• Source of electrical energy is generally a battery
• If only resistors are in the circuit, the current is
independent of time
• If the circuit contains capacitors and resistors, the
current can vary with time but always approaches a
constant value a long time after closing the switch
Introduction
AC Circuit Introduction
• AC stands for alternating current
• The power source is a device that produces an electric
potential that varies with time
• There will be a frequency and peak voltage associated
with the potential
• Household electrical energy is supplied by an AC
source
• Standard frequency is 60 Hz
• AC power has numerous advantages over DC power
Introduction
DC vs. AC Sources
Introduction
Generating AC Voltages
• Most sources of AC voltage employ a generator
•
•
•
•
•
based on magnetic induction
A shaft holds a coil with many loops of wire
The coil is positioned between the poles of a
permanent magnet
The magnetic flux through the coil varies with time
as the shaft turns
This changing flux induces a voltage in the coil
This induced voltage is the generator’s output
Section 22.1
Generators
• Generators of electrical energy convert the
mechanical energy of the rotating shaft into electrical
energy
• The principle of conservation of energy still applies
• The source of electrical energy in a circuit enables
the transfer of electrical energy from a generator to
an attached circuit
Section 22.1
AC Circuits and Simple Harmonic Motion
• The voltage variation of an AC circuit is reminiscent
of a simple harmonic oscillator
• There is also a close connection between circuits
with capacitors and inductors and simple harmonic
motion
Section 22.1
Resistors in AC Circuits
• Assume a circuit
consisting of an AC
generator and a resistor
• The voltage across the
output of the AC source
varies with time
according to
V = Vmax sin (2 π ƒ t)
• V is the instantaneous
potential difference
• Vmax is the amplitude of
the AC voltage
Section 22.2
Resistors, cont.
• Applying Ohm’s Law:
I =V
R
• Since the voltage varies
sinusoidally, so does the
current
Vmax
sin ( 2π ƒt ) or
R
I = Imax sin ( 2π ƒt ) where
I=
Imax
Vmax
=
R
Section 22.2
RMS Voltage
• To specify current and
voltage values when
they vary with time, rms
values were adopted
• RMS stands for Root
Mean Square
• For the voltage
=
Vrms
( )
Vmax
=
V
avg
2
2
Section 22.2
RMS Current
• The root-mean-square value can be defined for any
quantity that varies with time
• For the current
Imax
1 2
Irms =
Imax ) =
≈ 0.71× Imax
(
2
2
• The root-mean-square values of the voltage and
current are typically used to specify the properties of
an AC circuit
Section 22.2
Power
• The instantaneous
power is the product of
the instantaneous
voltage and
instantaneous current
• P=IV
• Since both I and V vary
with time, the power
also varies with time:
P = Vmax Imax sin2 (2πƒt)
Section 22.2
Power, cont.
• The instantaneous power varies between Vmax Imax
and 0
• The average power is ½ the maximum power
• Pavg = ½ (Vmax Imax ) = Vrms Irms
•
This has the same mathematical form as the power in a DC
circuit
• Ohm’s Law can again be used to express the power
in different ways
2
Vrms
2
P
=
=
I
ave
rms R
R
Section 22.2
AC Circuit Notation
• It is important to distinguish between instantaneous
and average values of voltage, current and power
• The amplitudes of the AC variations and the rms
values are also important
Section 22.2
Phasors
• AC circuits can be
analyzed graphically
• An arrow has a length
Vmax
• The arrow’s tail is tied to
the origin
• Its tip moves along a circle
• The arrow makes an angle
of θ with the horizontal
• The angle varies with
time according to θ =
2πƒt
Section 22.2
Phasors, cont.
• The rotating arrow represents the voltage in an AC
circuit
• The arrow is called a phasor
• A phasor is not a vector
• A phasor diagram provides a convenient way to
illustrate and think about the time dependence in an
AC circuit
Section 22.2
Phasors, final
• The current in an AC circuit can also be represented by a
phasor
• The two phasors always make the same angle with the
horizontal axis as time passes
• The current and voltage are in phase
• For a circuit with only resistors
Section 22.2
AC Circuits with Capacitors
• Assume an AC circuit
containing a single
capacitor
• The instantaneous
charge is q = C V
= C Vmax sin (2 πƒ t)
• The capacitor’s voltage
and charge are in phase
with each other
Section 22.3
Current in Capacitors
• The instantaneous
current is the rate at
which charge flows onto
the capacitor plates in a
short time interval
• The current is the slope
of the q-t plot
• A plot of the current as
a function of time can
be obtained from these
slopes
Section 22.3
Current in Capacitors, cont.
• The current is a cosine function
I = Imax cos (2πƒt)
• Equivalently, due to the relationship between sine
and cosine functions
I = Imax sin (2πƒt + ϕ) where ϕ = π/2
Section 22.3
Capacitor Phasor Diagram
• The current is out of
phase with the voltage
• The angle π/2 is called
the phase angle, ϕ,
between V and I
• For this circuit, the
current and voltage are
out of phase by 90°
Section 22.3
Current Value for a Capacitor
• The peak value of the current is
Vmax
1
Imax =
where X C
2 π ƒC
XC
• The factor Xc is called the reactance of the capacitor
• SI unit of reactance is Ohms
• Reactance and resistance are different because the
reactance of a capacitor depends on the frequency
• If the frequency is increased, the charge oscillates
more rapidly and Δt is smaller, giving a larger current
• At high frequencies, the peak current is larger and the
reactance is smaller
Section 22.3
Power In A Capacitor
• For an AC circuit with a
capacitor, P = VI = Vmax
Imax sin (2πƒt) cos (2πƒt)
• The average value of the
power over many
oscillations is 0
• Energy is transferred from
the generator during part of
the cycle and from the
capacitor in other parts
• Energy is stored in the
capacitor as electric
potential energy and not
dissipated by the circuit
Section 22.3
AC Circuits with Inductors
• Assume an AC circuit
containing an AC
generator and a single
inductor
• The voltage drop is
V = L (ΔI / Δt)
= Vmax sin (2 πƒt)
• The inductor’s voltage
is proportional to the
slope of the current-time
relationship
Section 22.4
Current in Inductors
• The instantaneous
current oscillates in time
according to a cosine
function
• I = -Imax cos (2πƒt)
• A plot of the currenttime relationship is
shown
Section 22.4
Current in Inductors, cont.
• The current equation can be rewritten as
I = Imax sin (2πƒt – π/2)
• Equivalently,
I = Imax sin (2πƒt + Φ) where Φ = -π/2
Section 22.4
Inductor Phasor Diagram
• The current is out of
phase with the voltage
• For this circuit, the
current and voltage are
out of phase by -90°
• Remember, for a
capacitor, the phase
difference was +90°
Section 22.4
Current Value for an Inductor
• The peak value of the current is
Imax
Vmax
=
where X L 2 π ƒL
XL
• The factor XL is called the reactance of the inductor
• SI unit of inductive reactance is Ohms
• As with the capacitor, inductive reactance depends
on the frequency
• As the frequency is increased, the inductive reactance
increases
Section 22.4
Power in an Inductor
• For an AC circuit with an
inductor, P = VI = -Vmax
Imax sin (2πƒt) cos (2πƒt)
• The average value of the
power over many
oscillations is 0
• Energy is transferred from
the generator during part of
the cycle and from the
inductor in other parts of the
cycle
• Energy is stored in the
inductor as magnetic
potential energy
Section 22.4
Properties of AC Circuits
Section 22.4
LC Circuit
• Most useful circuits contain multiple circuit elements
• Will start with an LC circuit, containing just an
inductor and a capacitor
• No AC generator is included, but some excess
charge is placed on the capacitor at t = 0
Section 22.5
LC Circuit, cont.
• After t = 0, the charge moves from one capacitor
•
•
•
•
plate to the other and current passes through the
inductor
Eventually, the charge on each capacitor plate falls
to zero
The inductor opposes change in the current, so the
induced emf now acts to maintain the current at a
nonzero value
This current continues to transport charge from one
capacitor plate to the other, causing the capacitor’s
charge and voltage to reverse sign
Eventually the charge on the capacitor returns to its
original value
Section 22.5
LC Circuit, final
• The voltage and current in the circuit oscillate
between positive and negative values
• The circuit behaves as a simple harmonic oscillator
• The charge is q = qmax cos (2πƒt)
• The current is I = Imax sin (2πƒt)
Section 22.5
Energy in an LC Circuit
• Capacitors and
inductors store energy
• A capacitor stores energy
in its electric field and
depends on the charge
• An inductor stores energy
in its magnetic field and
depends on the current
• As the charge and
current oscillate, the
energies stored also
oscillate
Section 22.5
Energy Calculations
• For the capacitor,
2
1 q 2 1 qmax
2
=
=
PE
cos
( 2π ƒt )
cap
2C 2 C
• For the inductor,
1 2 1 2
2
PE
=
=
LI
LI
sin
( 2π ƒt )
ind
max
2
2
• The energy oscillates back and forth between the
capacitor and its electric field and the inductor and
its magnetic field
• The total energy must remain constant
Section 22.5
Energy, final
• The maximum energy in the capacitor must equal
the maximum energy in the inductor
• From energy considerations, the maximum value of
the current can be calculated
1
Imax =
qmax
LC
• This shows how the amplitudes of the current and
charge oscillations in the LC circuits are related
Section 22.5
Frequency Oscillations – LC Circuit
• In an LC circuit, the instantaneous voltage across
the capacitor and inductor are always equal
• Therefore, |VC| = |I XC| = |VL| = |I XL|
• Simplifying, XC = XL
• This assumed the current in the LC circuit is oscillating
and hence applies only at the oscillation frequency
• This frequency is the resonant frequency
ƒres
1
=
2π LC
Section 22.5
LRC Circuits
• Let the circuit contain a
generator, resistor,
inductor and capacitor
in series
• LRC circuit
• From Kirchhoff’s Loop
Rule,
VAC = VL + VC + VR
• But the voltages are not
all in phase, so the
phase angles must also
be taken into account
Section 22.6
LRC Circuit – Phasor Diagram
• All the elements are in
series, so the current is
the same through each
one
• All the current phasors
have the same orientation
• Resistor: current and
voltage are in phase
• Capacitor and inductor:
current and voltage are
90° out of phase, in
opposite directions
Section 22.6
Resonance
• The VC and VR values are
the same at the
resonance frequency
• Only the resistor is left to
“resist” the flow of the
current
• This cancellation between
the voltages occurs only
at the resonance
frequency
• The resonance frequency
corresponds to the
highest current
Section 22.6
Applications of Resonance
• Resonance is used in radios, cell phones and other
similar applications
• Tuning a radio
• Changes the value of the capacitance in the LCR
circuit so the resonance frequency matches the
frequency of the station you want to listen to
• LCR circuits are used to construct devices that are
frequency dependent
Section 22.6
Real Inductors in AC Circuits
• A typical inductor includes a nonzero resistance
• Due to the wire itself
• The inductor can be modeled as an ideal inductor in
series with a resistor
• The current can be calculated using phasors
Section 22.7
Real Inductor, cont.
• The elements are in series, so the current is the
same through both elements
• Voltages are VR = I R and VL = I XL
• The voltages must be added as phasors
• The phase differences must be included
• The total voltage has an amplitude of
Vtotal =
VR2 + VL2 = I R 2 + X L2 or
= I R + ( 2π ƒL )
2
2
Section 22.7
Impedance
• The impedance, Z, is a measure of how strongly a
circuit “impedes” current in a circuit
• The impedance is defined as Vtotal = I Z where
=
Z
R + ( 2π ƒL )
2
2
• This is the impedance for an RL circuit only
• The impedance for a circuit containing other
elements can also be calculated using phasors
• The angle between the current and the impedance
can also be calculated
Section 22.7
Impedance, LCR Circuit
• The current phasor is
on the horizontal axis
• The total voltage is
2
2
Vtotal = ( IR ) + I 2 ( X L − XC )

1 
=I R 2 +  2π ƒL −

ƒC
2
π


2
• The impedance is

1 
Z =R 2 +  2π ƒL −

2
π
ƒC


2
Section 22.7
Elements and Frequencies in AC Circuits
• Resistor
• Resistors in an AC circuit behave very much like
resistors in a DC circuit
• The current is always in phase with the voltage
• Capacitor or inductor
• Both are frequency dependent
• Due to the frequency dependence of the reactants
• XC is largest at low frequencies, so the current through
a capacitor is smallest at low frequencies
• XL is largest at high frequencies, so the current
through an inductor is smallest at high frequencies
Section 22.8
Elements at Various Frequencies –
Summary
Section 22.8
High-Pass Filter (LR Circuit)
• When the input frequency
is very low, the reactance
of the inductor is small
• The inductor acts as a wire
• Voltage drop will be 0
• At very high frequencies,
the inductor acts as an
open circuit
• No current is passed
• The output voltage is equal
to the input voltage
• This circuit acts as a
high-pass filter
Section 22.8
Low-Pass Filter (RC Circuit)
• When the input frequency
is very low, the reactance
of the capacitor is large
• The current is very small
• The capacitor acts as an
open circuit
• The output voltage is equal
to the input voltage
• At high frequencies, the
capacitor acts as a short
circuit
• The inductor acts as a wire
• The output voltage is 0
• This circuit acts as a low-
pass filter
Section 22.8
Application of a Low-Pass Filter
• A low-pass filter is used in radios and MP3 players
• A music signal often contains static
• Static comes from unwanted high-frequency
components in the music
• These high frequencies can be filtered out by using a
low-pass filter
Section 22.8
Frequency Limits, RL Circuit
• For an RL circuit, the input frequency is compared to
the RL time constant
• The time constant is τRL = L / R
• Define a corresponding frequency as ƒRL = 1/ τRL =
R/L
• The high-frequency limit applies when the input
frequency is much greater than ƒRL
• A frequency higher than ~10 x ƒRL falls into the high-
frequency limit
• The low-frequency limit applies when the input
frequency is much less than ƒRL
• A frequency lower than ~ƒRL / 10 falls into the low-
frequency limit
Section 22.8
Frequency Limits, RC Circuit
• For an RC circuit, the input frequency is compared to
the RC time constant
• The time constant is τRC = R C
• Define a corresponding frequency as ƒRC = 1/ τRC =
1 / RC
• The high-frequency limit applies when the input
frequency is much greater than ƒRC
• A frequency higher than ~10 x ƒRC falls into the high-
frequency limit
• The low-frequency limit applies when the input
frequency is much less than ƒRC
• A frequency lower than ~ƒRC / 10 falls into the low-
frequency limit
Section 22.8
Frequency Limits, LC Circuit
• The resonant frequency determines the boundary
between high- and low-frequency limits
• Remember,
ƒres
1
=
2π LC
Section 22.8
Filter Application – Stereo Speakers
• Many stereo speakers actually contain two separate
speakers
• A tweeter is designed to perform well at high
frequencies
• A woofer is designed to perform well at low
frequencies
• The AC signal passes through a crossover network
• A combination of low-pass and high-pass filters
• The outputs of the filter are sent to the speaker
which is most efficient at that frequency
Section 22.8
Transformers
• Transformers are devices that can increase or decrease
the amplitude of an applied AC voltage
• A simple transformer consists of two solenoid coils with
the loops arranged so that all or most of the magnetic field
lines and flux generated by one coil pass through the
other coil
Section 22.9
Transformers, cont.
• The wires are covered with a non-conducting layer
so that current cannot flow directly from one coil to
the other
• An AC current in one coil will induce an AC voltage
across the other coil
• An AC voltage source is typically attached to one of
the coils called the input coil
• The other coil is called the output coil
Transformers, Equations
• Faraday’s Law applies to both coils
Φ out
Φ in
Vin =
and Vout
∆t
∆t
• If the input coil has Nin coils and the output coil has
Nout turns, the flux in the coils is related by
Nout
Φ out=
Φ in
Nin
• The voltages are related by
Vout
Nout
=
Vin
Nin
Section 22.9
Transformers, final
• The ratio of the turns can be greater than or less
than one
• Therefore, the input voltage can be transformed to a
different value
• Transformers cannot change DC voltages
• Since they are based on Faraday’s Law
Section 22.9
Practical Transformers
• Most practical
transformers have
central regions filled
with a magnetic material
• This produces a larger
flux, resulting in a larger
voltage at both the input
and output coils
• The ratio Vout / Vin is not
affected by the
presence of the
magnetic material
Section 22.9
Applications of Transformers
• Transformers are used in the transmission of electric
power over long distances
• Many household appliances use transformers to
convert the AC voltage at a wall socket to the smaller
voltages needed in many devices
• Two steps are needed – converting 120 V to 9 V then
AC to DC
Section 22.9
Transformers and Power
• The output voltage of a transformer can be made
much larger by arranging the number of coils
• According to the principle of conservation of energy,
the energy delivered through the input coil must
either be stored in the transformer’s magnetic field
or transferred to the output circuit
• Over many cycles, the stored energy is constant
• The power delivered to the input coil must equal the
output power
Section 22.9
Power, cont.
• Since P = V I, if Vout is greater than Vin, then Iout must
be smaller than Iin
• Pin = Pout only in an ideal transformer
• In real transformers, the coils always have a small
electrical resistance
• This causes some power dissipation
• For a real transformer, the output power is always
less than the input power
• Usually by only a small amount
Section 22.9
Motors
• An AC voltage source can be use to power a motor
• The AC source is connected to a coil wound around a
horseshoe magnet
• Called the input coil
• The input coil induces a magnetic field that circulates
through the horseshoe magnet
Section 22.10
Motors, cont.
• A second coil is mounted between the poles of the
horseshoe magnet and attached to a rotating shaft
• The forces acting on the second coil produce a
torque on the coil
• This causes the shaft to rotate
• As the AC current in the input coil changes direction,
so do the forces
• The torques continue to produce a rotation that is
always in the same direction
• The oscillations of the AC current and field make the
shaft rotate
Section 22.10
Advantages of AC vs. DC
• Biggest advantage is in
the systems that
distribute electric power
across long distances
• The power generated at
a power plant must be
distributed to distance
places
• The power plant acts as
an AC generator
Section 22.11
Advantages, cont.
• There is power dissipated in the power lines
• Pave = (Irms )2 Rline
• The power company wants to minimize these power
losses, so they want to make Irms as small as
possible
• The voltage is increased by using a transformer
• The increase in voltage is done in order to decrease
the current
• A transformer is used to drop the high voltages in the
power lines to the lower voltages at the house
Section 22.11
Advantages, final
• The power lines have typical voltages of 500,000 V
or higher
• The transformer reduces the voltage to a maximum
voltage of 170 V
• Typically 5% to 10% of the energy that leaves the
power plant is dissipated in the resistance of the
power lines
Section 22.11