Alternating-Current Circuits

Chapter 22

Section 22.2

AC Circuit Notation

PhasorsAC circuits can be

analyzed graphicallyAn arrow has a length

Vmax The arrow’s tail is at

the originThe 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

circuitThe arrow is called a phasorA phasor is not a vectorA phasor diagram provides a convenient way to

illustrate and think about the time dependence in an AC circuit

Section 22.2

Phasors, finalThe 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 phaseFor a circuit with only

resistorsSection 22.2

AC Circuits with CapacitorsAssume an AC circuit

containing a single capacitor

The instantaneous charge is

q = C V

= C Vmax sin (2 πƒt)

The capacitor’s voltage and charge are in phase with each other

Section 22.3

Current in CapacitorsThe 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 functionsI = Imax sin (2πƒt + Φ) where Φ = π/2

Section 22.3

Capacitor Phasor DiagramThe current is out of

phase with the voltageThe 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 90o

Section 22.3

Current Value for a CapacitorThe peak value of the current is

The factor Xc is called the reactance of the capacitorUnits of reactance are Ohms Reactance and resistance are different because the

reactance of a capacitor depends on the frequencyIf the frequency is increased, the charge oscillated more

rapidly and Δt is smaller, giving a larger currentAt high frequencies, the peak current is larger and the

reactance is smaller

maxmax C

C

VI where X

X πƒC

1

2

Section 22.3

Power In A CapacitorFor an AC circuit with a

capacitor, P = VI = Vmax Imax sin (2πƒt) cos (2πƒt)

The average value of the power over many oscillations is 0Energy 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 InductorsAssume an AC circuit

containing 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 InductorsThe instantaneous

current oscillates in time according to a cosine function

I = -Imax cos (2πƒt)

A plot of the current 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 DiagramThe current is out of

phase with the voltageFor this circuit, the

current and voltage are out of phase by -90o Remember, for a

capacitor, the phase difference was +90o

Section 22.4

Current Value for an InductorThe peak value of the current is

The factor XL is called the reactance of the inductor

Units of inductive reactance are Ohms As with the capacitor, inductive reactance depends

on the frequencyAs the frequency is increased, the inductive reactance

increases

maxmax L

L

VI where X πƒL

X 2

Section 22.4

Quiz!If the frequency in a

circuit with a Capacitor is halved.

The Reactance is?

A) DoubleB) The SameC) HalfD) ZeroE) Disproportionate

Section 22.4

Properties of AC Circuits

Power In An InductorFor an AC circuit with

an inductor, P = VI = -Vmax Imax sin (2πƒt) cos (2πƒt)

The average value of the power over many oscillations is 0Energy 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

LC Circuit

Most useful circuits contain multiple circuit elementsWill start with an LC circuit, containing just an

inductor and a capacitorNo AC generator is included, but some excess

charge is placed on the capacitor at t = 0Section 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 again 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, finalThe voltage and current in the circuit oscillate

between positive and negative valuesThe circuit behaves as a simple harmonic oscillatorThe charge is q = qmax cos (2πƒt)

The current is I = Imax sin (2πƒt)

Section 22.5

Quiz!If the frequency in a

circuit with an Inductor is doubled.

The Reactance is?

A) DoubleB) The SameC) HalfD) ZeroE) 42

Energy in an LC CircuitCapacitors and inductors

store energyA 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 CalculationsFor 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

maxcap

qqPE cos πƒt

C C

2221 1

22 2

ind maxPE LI LI sin πƒt 2 2 21 12

2 2

Section 22.5

Energy, finalThe 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 CircuitIn an LC circuit, the instantaneous voltage across

the capacitor and inductor are always equalTherefore, |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 frequencyThis frequency is the resonance frequency

resƒπ L C

1

2

Section 22.5

LRC CircuitsLet the circuit contain a

generator, resistor, inductor and capacitor in seriesLRC 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 DiagramAll the elements are in

series, so the current is the same through each one

All the current phasors are in the same direction

Resistor: current and voltage are in phase

Capacitor and inductor: current and voltage are 90o 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 ResonanceTuning 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 can be used to construct devices that are frequency dependent

Section 22.6

Real Inductors in AC Circuits

A typical inductor includes a nonzero resistanceDue 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 elementsVoltages are VR = I R and VL = I XL

The voltages must be added as phasorsThe phase differences must be included

The total voltage has an amplitude of

total R L LV V V I R X or

I R πƒL

2 2 2 2

22 2

Section 22.7

ImpedanceThe impedance, Z, is a measure of how

strongly a circuit “impedes” current in a circuitThe impedance is defined as Vtotal = I Z where

This is the impedance for an RL circuit onlyThe impedance for a circuit containing other

elements can also be calculated using phasorsThe angle between the current and the

impedance can also be calculated

Z R πƒL 22 2

Section 22.7

Impedance, LCR CircuitThe current phasor is

on the horizontal axisThe total voltage is

The impedance is

total L CV IR I X X

I R πƒLπƒC

2 22

2

2 12

2

Z R πƒLπƒC

2

2 12

2

Section 22.7

Resonance in an LCR CircuitThe current depends on the impedance,

Imax = Vmax /Z

Since the impedance depends on the frequency, the current amplitude also varies with frequency

For the maximum current, the impedance must be a minimum

The minimum impedance occurs when

πƒLπƒC

1

2 02

Section 22.7

Resonance, cont.Solving for the frequency gives

This is the same result as was found for the LC circuitThe maximum current occurs at the resonant

frequencyThis is the frequency at which the LCR circuit

responds most strongly to an applied AC circuit

ƒπ LC

1

2

Section 22.7

Behavior of Elements at Various Frequencies

Section 22.8

Elements and Frequencies, cont.Resistor

Resistors in an AC circuit behave very much like resistors in a DC circuit

The current is always in phase with the voltageCapacitor or inductor

Both are frequency dependentDue to the frequency dependence of the reactancesXC 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

RL Circuit ExampleWhen the input

frequency is very low, the reactance of the inductor is smallThe inductor acts as a

wire Voltage drop will be 0

At high frequencies, the inductor acts as an open circuitNo current is passedThe output voltage is

equal to the input voltage

This circuit acts as a high-pass filter

Section 22.8

RC Circuit ExampleWhen the input frequency

is very low, the reactance of the capacitor is largeThe current is very smallThe capacitor acts as an

open circuitThe output voltage is

equal to the input voltageAt high frequencies, the

capacitor acts as a short circuitThe inductor acts as a

wire The output voltage is 0

This circuit acts as a low-pass filter

Section 22.8

Quiz!

A) 2.0B) 0.5C) 1.0D) 0.1

Vac= 9VC1=10 uFC2= 2 uFF=2kHzHow many Amps?

Application of a Low-Pass FilterA low-pass filter is used in radios and MP3 playersA 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 CircuitFor an RL circuit, the input frequency is compared

to the RL time constantThe time constant is τRL = L / RDefine a corresponding frequency as ƒRL = 1/ τRL = R / LThe 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

limitThe low-frequency limit applies when the input frequency is

much less than ƒRL A frequency lower than ~ƒRL / 10 falls into the low-frequency

limitSection 22.8

Frequency Limits, RC CircuitFor an RL circuit, the input frequency is

compared to the RL time constantThe time constant is τRL = R CDefine a corresponding frequency as ƒRC = 1/ τRC =

1 / RCThe 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 limitThe low-frequency limit applies when the input frequency

is much less than ƒRC A frequency lower than ~ƒRC / 10 falls into the low-

frequency limitSection 22.8

Filter Application – Stereo SpeakersMany stereo speakers actually contain two separate

speakersA tweeter is designed to perform well at high

frequenciesA woofer is designed to perform well at low

frequenciesThe AC signal passes through a crossover network

A combination of low-pass and high-pass filtersThe outputs of the filter are sent to the speaker

which is most efficient at that frequency

Section 22.8