Nicholas J. Giordano

Marilyn Akins, PhD • Broome Community College

www.cengage.com/physics/giordano

Chapter 22

Alternating-Current Circuits and Machines

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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:

• Since the voltage varies sinusoidally, so does the current

VI R=

( )

( )

max

max

maxmax

VI sin ƒt orR

I I sin ƒt whereVI

R

π

π

=

=

=

2

2

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

( ) maxrms avgVV V= =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

• The root-mean-square values of the voltage and current are typically used to specify the properties of an AC circuit

( ) maxrms max maxII I . I= = ≈ ×21 0 712 2

Section 22.2

Power

• The instantaneous power is the product of the instantaneous voltage and instantaneous current • P = I V

• 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

rmsave rms

VP I RR

= =2

2

Section 22.2

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

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

• 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

maxmax C

C

VI where XX ƒCπ

= =1

2

Section 22.3

• 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

Power In A Capacitor

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 current-

time 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

• 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

maxmax L

L

VI where X ƒLX

π= = 2

Section 22.4

• 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

Power in an Inductor

Section 22.4

Properties of AC Circuits

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,

• For the inductor,

• 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

( )maxcapqqPE cos ƒt

C Cπ= =

2221 1 2

2 2

( )ind maxPE LI LI sin ƒtπ= =2 2 21 1 22 2

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

• This shows how the amplitudes of the current and charge oscillations in the LC circuits are related

max maxI qLC=

1

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

resƒ L Cπ=

12

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

( )total R L LV V V I R X or

I R ƒLπ

= + = +

= +

2 2 2 2

22 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 • 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

( )Z R ƒLπ= + 22 2

Section 22.7

Impedance, LCR Circuit

• The current phasor is on the horizontal axis

• The total voltage is

• The impedance is

( ) ( )total L CV IR I X X

I R ƒLƒC

ππ

= + −

= + −

2 22

22 12

2

Z R ƒLƒC

ππ

= + −

22 12

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

Section 22.8

Elements at Various Frequencies – Summary

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,

resƒ LCπ=

12

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

• If the input coil has Nin coils and the output coil has Nout turns, the flux in the coils is related by

• The voltages are related by

outinin outV and Vt t

ΦΦ= =∆ ∆

outout in

in

NN

Φ = Φ

outout in

in

NV VN

=

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

Chapter 22DC Circuit SummaryAC Circuit IntroductionDC vs. AC SourcesGenerating AC VoltagesGenerators AC Circuits and Simple Harmonic MotionResistors in AC Circuits Resistors, cont.RMS VoltageRMS CurrentPowerPower, cont.AC Circuit NotationPhasorsPhasors, cont.Phasors, finalAC Circuits with CapacitorsCurrent in CapacitorsCurrent in Capacitors, cont.Capacitor Phasor DiagramCurrent Value for a CapacitorPower In A CapacitorAC Circuits with InductorsCurrent in InductorsCurrent in Inductors, cont.Inductor Phasor DiagramCurrent Value for an InductorPower in an InductorProperties of AC CircuitsLC CircuitLC Circuit, cont.LC Circuit, finalEnergy in an LC CircuitEnergy CalculationsEnergy, finalFrequency Oscillations – LC CircuitLRC CircuitsLRC Circuit – Phasor DiagramResonance Applications of ResonanceReal Inductors in AC CircuitsReal Inductor, cont.ImpedanceImpedance, LCR CircuitElements and Frequencies in AC CircuitsElements at Various Frequencies – Summary High-Pass Filter (LR Circuit)Low-Pass Filter (RC Circuit)Application of a Low-Pass FilterFrequency Limits, RL CircuitFrequency Limits, RC CircuitFrequency Limits, LC CircuitFilter Application – Stereo SpeakersTransformersTransformers, cont.Transformers, EquationsTransformers, finalPractical TransformersApplications of TransformersTransformers and PowerPower, cont.MotorsMotors, cont.Advantages of AC vs. DCAdvantages, cont.Advantages, final