Lesson#28Lesson#28Topic: AC CircuitsTopic: AC Circuits

Objectives: Objectives: (After this class I will be (After this class I will be able to)able to)

1.1. Explain the difference between AC Explain the difference between AC and DCand DC

2.2. Describe how alternating current is Describe how alternating current is sinusoidal sinusoidal

3.3. Describe the three ways to measure Describe the three ways to measure voltage in an AC circuitvoltage in an AC circuit

12/7/06

Assignment: Packet p675 #11, 13, 14, 15, 17

Warm Up: What is the difference between AC and DC? Which is more commonly used?

Alternating CurrentAlternating Current Many of our most useful electronic Many of our most useful electronic

devices rely on AC voltage.devices rely on AC voltage. The current is continually changing The current is continually changing

direction.direction. The current and voltage change is The current and voltage change is

sinusoidal with time.sinusoidal with time. We use frequency to represent how We use frequency to represent how

quickly the voltage oscillates.quickly the voltage oscillates. AC electricity is a form of simple AC electricity is a form of simple

harmonic motion.harmonic motion.

Amplitude of Current and Amplitude of Current and VoltageVoltage

Measuring actual current or voltage is Measuring actual current or voltage is difficult because it is constantly difficult because it is constantly changing in magnitude and direction.changing in magnitude and direction.

AC voltage varies from VAC voltage varies from Vmaxmax to –V to –Vmaxmax

You can use VYou can use Vmaxmax=I=ImaxmaxR to solve for IR to solve for Imaxmax

Can we use VCan we use Vavgavg=I=IavgavgR to solve for IR to solve for Iavgavg??

What would VWhat would Vavgavg equal? equal? Instead of using max values or average Instead of using max values or average

values, we use RMS values. values, we use RMS values.

RMS ValuesRMS Values Since the peak voltage only lasts for an Since the peak voltage only lasts for an

instant and the average value is zero, instant and the average value is zero, we need to use RMS values.we need to use RMS values.

RMS = Root mean squaredRMS = Root mean squared First square all values to make them all First square all values to make them all

positivepositive Take an average of these squared Take an average of these squared

values.values. Then square root this average.Then square root this average. RMS values are the values closest to the RMS values are the values closest to the

actual voltage or current of the circuit.actual voltage or current of the circuit.

AC PracticeAC Practice1.1. A voltage of peak value 10V A voltage of peak value 10V

oscillates with a period of 1ms. oscillates with a period of 1ms. What is the frequency and angular What is the frequency and angular frequency of the signal?frequency of the signal?

2.2. A signal generator is set to produce A signal generator is set to produce a voltage with a period of 1s. With a voltage with a period of 1s. With what frequency does a light bulb what frequency does a light bulb wired in this circuit blink?wired in this circuit blink?

3.3. The amplitude of a sinusoidal signal The amplitude of a sinusoidal signal is 3V What is the RMS value?is 3V What is the RMS value?

Lesson#29Lesson#29Topic: RLC CircuitsTopic: RLC Circuits

Objectives: Objectives: (After this class I will be (After this class I will be able to)able to)

1.1. Calculate the power dissipated by an AC Calculate the power dissipated by an AC circuitcircuit

2.2. Define ImpedanceDefine Impedance

3.3. Describe new elements that may be found Describe new elements that may be found in AC circuitsin AC circuits

4.4. Explain how resistors, inductors, and Explain how resistors, inductors, and capacitors affect the overall impedance of capacitors affect the overall impedance of an AC circuit.an AC circuit.

12/8/06

Assignment: Packet p675 #12, 16, 18, 19, 20

Warm Up: What would be the calculated power dissipated by an AC circuit if you used the average voltage times the average current?

Power in AC circuitsPower in AC circuits To solve for resistance in a circuit you To solve for resistance in a circuit you

could use Vcould use Vmaxmax with I with Imaxmax or V or Vrmsrms with I with Irmsrms Either way will work.Either way will work. But when dealing with energy But when dealing with energy

consumption, or average power consumption, or average power dissipated, you need to use RMS values. dissipated, you need to use RMS values.

PPavgavg = V = VrmsrmsIIrmsrms

Other previously derived equations for Other previously derived equations for power can also be used with rms voltage power can also be used with rms voltage and current.and current.

Power in AC circuitsPower in AC circuits The average power dissipated in a stereo The average power dissipated in a stereo

speaker is 55W. Assuming that the speaker is 55W. Assuming that the speaker can be treated as a resistor with speaker can be treated as a resistor with 4ohms resistance, find:4ohms resistance, find:

The The RMSRMS value of the AC voltage applied value of the AC voltage applied to the speaker and the to the speaker and the RMSRMS value of the value of the AC current through the speaker.AC current through the speaker.

The The peakpeak value of the AC voltage applied value of the AC voltage applied to the speaker and the to the speaker and the peakpeak value of the value of the AC current through the speaker. AC current through the speaker.

RLC CircuitsRLC Circuits With DC circuits we saw that greater With DC circuits we saw that greater

voltage caused greater current to flow.voltage caused greater current to flow. The same happens in AC circuits.The same happens in AC circuits. Certain elements in the circuit cause the Certain elements in the circuit cause the

current to also depend on the frequency current to also depend on the frequency of the applied voltage.of the applied voltage.

We can categorize these elements into We can categorize these elements into three types: Resistors, Capacitors, and three types: Resistors, Capacitors, and Inductors.Inductors.

Each element has a different dependence Each element has a different dependence on frequency. on frequency.

ImpedanceImpedance Impedance is the same thing as resistance.Impedance is the same thing as resistance. The resistance of a resistor is the resistor’s The resistance of a resistor is the resistor’s

impedance.impedance. ResistorsResistors are independent of frequency. are independent of frequency. Impedance is represented by the symbol Z.Impedance is represented by the symbol Z. ZZR R = R= R CapacitorsCapacitors have impedance that is have impedance that is

inversely proportional to frequencyinversely proportional to frequency ZZCC ~ 1 / f ~ 1 / f InductorsInductors have impedance that is directly have impedance that is directly

proportional to frequencyproportional to frequency ZZLL~ f~ f

AC PracticeAC Practice1.1. A resistor has an impedance of A resistor has an impedance of

100ohms when V100ohms when Vmaxmax = 10V and f= = 10V and f= 100Hz100Hz

a. What is Z if f = 1000Hz?a. What is Z if f = 1000Hz?

b. What is Z if f = 0Hz?b. What is Z if f = 0Hz?

c. What is Z if f = infinity?c. What is Z if f = infinity?

AC PracticeAC Practice2.2. When a rms voltage of 15V is applied When a rms voltage of 15V is applied

to a circuit containing only a to a circuit containing only a capacitor, an rms current of 3.7A is capacitor, an rms current of 3.7A is produced. produced.

a. What is Za. What is Zcc ? ?

b. What is Zb. What is Zcc if f is doubled? if f is doubled?

c. What is the current if f is doubled?c. What is the current if f is doubled?

d. What is Zd. What is Zcc when f = 0Hz? when f = 0Hz?

e. What is Ze. What is Zcc when f = infinity? when f = infinity?

AC PracticeAC Practice3.3. When a rms voltage of 15V is applied When a rms voltage of 15V is applied

to a circuit containing only an to a circuit containing only an inductor, an rms current of 3.7A is inductor, an rms current of 3.7A is produced. produced.

a. What is Za. What is ZLL ? ?

b. What is Zb. What is ZLL if f is doubled? if f is doubled?

c. What is the current if f is doubled?c. What is the current if f is doubled?

d. What is Zd. What is ZLL when f = 0Hz? when f = 0Hz?

e. What is Ze. What is ZLL when f = infinity? when f = infinity?

Lesson#30Lesson#30Topic: CapacitorsTopic: Capacitors

Objectives: Objectives: (After this class I will be (After this class I will be able to)able to)

1.1. Describe how a capacitor worksDescribe how a capacitor works2.2. Define and explain capacitanceDefine and explain capacitance3.3. Explain how frequency and capacitance Explain how frequency and capacitance

affect the impedance of a capacitoraffect the impedance of a capacitor4.4. Describe how the surface area and Describe how the surface area and

distance between plates of a capacitor will distance between plates of a capacitor will affect its capacitance.affect its capacitance.

12/11/06

Assignment: Packet p691 #1, 2, 4 p675 #30, 31

Warm Up: What would be the current in a DC circuit with a capacitor wired in series with a 12V battery? Should capacitors be used in AC or DC circuits?

CapacitorCapacitor Two conducting plates separated by a Two conducting plates separated by a

thin insulating material.thin insulating material. The insulator creates a “gap” in the The insulator creates a “gap” in the

circuit.circuit. Current cannot flow through the gap.Current cannot flow through the gap. Current can flow through the wires for a Current can flow through the wires for a

short time while one plate is “sucked” short time while one plate is “sucked” dry of electrons, and the other plate is dry of electrons, and the other plate is being saturated with electrons.being saturated with electrons.

Positive charge will accumulate on one Positive charge will accumulate on one side and negative charge on the other.side and negative charge on the other.

The overall capacitor remains neutral.The overall capacitor remains neutral.

CapacitorCapacitor

I

N

S

U

L

A

T

O

R

+

_

+

+

+

+

+

+

+

_

_

_

_

_

_

_

_ _ _

_

_

_ _ _

+

++

_+

Electron Flow Electron Flow

Current Current

CapacitanceCapacitance Capacitance is the quantity of how Capacitance is the quantity of how

much charge can be “stored” on much charge can be “stored” on each plate.each plate.

The larger the capacitance of a The larger the capacitance of a capacitor, the larger the “capacity” it capacitor, the larger the “capacity” it has to hold charge.has to hold charge.

The charge found on one of the The charge found on one of the plates (plates (QQ) is directly proportional to ) is directly proportional to the capacitance of a capacitor (the capacitance of a capacitor (C C ). ).

Q ~ CQ ~ C

CapacitanceCapacitance q q is the charge of a point or particle.is the charge of a point or particle. QQ is the charge of an object (like a is the charge of an object (like a

capacitor plate)capacitor plate) The amount of charge also depends The amount of charge also depends

on the amount of voltage that is on the amount of voltage that is applied.applied.

The larger the voltage, the more The larger the voltage, the more charge that will accumulate.charge that will accumulate.

Q ~ VQ ~ V

CapacitanceCapacitance Charge will accumulate and current Charge will accumulate and current

will flow until the voltage across the will flow until the voltage across the capacitor is the same as that across capacitor is the same as that across the battery.the battery.

Though these two proportionalities we Though these two proportionalities we can use the equation: can use the equation: Q = CVQ = CV

Q = charge stored on one plate of the Q = charge stored on one plate of the capacitorcapacitor

V = Voltage across the capacitorV = Voltage across the capacitor C = Capacitance of the capacitorC = Capacitance of the capacitor Capacitance has units of Coulomb per Capacitance has units of Coulomb per

Volt or a “Farad” (F). Volt or a “Farad” (F).

Impedance of a capacitorImpedance of a capacitor In a DC circuit, a capacitor will quickly fill and In a DC circuit, a capacitor will quickly fill and

then no current will flow (infinite impedance).then no current will flow (infinite impedance). In an AC circuit, it is possible to oscillate the In an AC circuit, it is possible to oscillate the

direction of the current fast enough that the direction of the current fast enough that the capacitor never fills up.capacitor never fills up.

At very high frequencies it is as if the At very high frequencies it is as if the capacitor isn’t even there (zero impedance).capacitor isn’t even there (zero impedance).

The larger the Capacitance, the less likely The larger the Capacitance, the less likely that it will ever fill up and cause impedance.that it will ever fill up and cause impedance.

ororC

Zc 1

fCZc 2

1

Parallel Plate CapacitorsParallel Plate Capacitors The capacitance of a capacitor depends on The capacitance of a capacitor depends on

its structure.its structure. If the plates are large in area, then more If the plates are large in area, then more

charge is able to accumulate on the plates. charge is able to accumulate on the plates. Capacitance is directly proportional to the Capacitance is directly proportional to the

area of the plates.area of the plates. If the distance between the plates is large, If the distance between the plates is large,

then the electric force causing charges to then the electric force causing charges to separate is weak and little charge will separate is weak and little charge will accumulate.accumulate.

Capacitance is inversely proportional to the Capacitance is inversely proportional to the distance between the plates.distance between the plates.

d

AC

Capacitor PracticeCapacitor Practice1.1. When a rms voltage of 15V and When a rms voltage of 15V and

10,000Hz is applied to a circuit 10,000Hz is applied to a circuit containing only a capacitor, an rms containing only a capacitor, an rms current of 3.7A is produced. How current of 3.7A is produced. How much charge can be stored on the much charge can be stored on the capacitor when a 15V steady potential capacitor when a 15V steady potential difference is applied across it? difference is applied across it?

2.2. A capacitor has a capacitance of 6A capacitor has a capacitance of 6μμF. F. If the width of the gap is doubled, If the width of the gap is doubled, what happens to the capacitance? what happens to the capacitance? What if the area of the plates double What if the area of the plates double also?also?

Lesson#31Lesson#31Topic: Total Capacitance and DielectricsTopic: Total Capacitance and Dielectrics

Objectives: Objectives: (After this class I will be (After this class I will be able to)able to)

1.1. Add capacitors that are wired in series and Add capacitors that are wired in series and in parallelin parallel

2.2. Define dielectrics Define dielectrics

3.3. Describe dielectric constantDescribe dielectric constant

4.4. Explain how the material placed between a Explain how the material placed between a parallel plate capacitor affects capacitance. parallel plate capacitor affects capacitance.

12/12/06

Assignment: Packet

Warm Up: A signal generator is wired in series with a capacitor and has a frequency of 2500Hz, a Vrms = 120V and an Irms= 1.5A What is the capacitance of the capacitor?

Equivalent CapacitanceEquivalent Capacitance

Wiring multiple capacitors in parallel is Wiring multiple capacitors in parallel is similar to adding surface area to one big similar to adding surface area to one big capacitor.capacitor.

Additional surface area increases Additional surface area increases capacitance.capacitance.

Capacitors wired in parallel add directly.Capacitors wired in parallel add directly.

nparallel CCCC ....21

Equivalent CapacitanceEquivalent Capacitance

Wiring multiple capacitors in series is Wiring multiple capacitors in series is similar to increasing the gap between the similar to increasing the gap between the plates of one large capacitor.plates of one large capacitor.

Greater distance between plates will Greater distance between plates will decrease capacitance.decrease capacitance.

Capacitors wired in series add inversely.Capacitors wired in series add inversely.

nseries CCCC

1....

111

21

DielectricsDielectrics The type of insulator placed between the The type of insulator placed between the

gap of a parallel plate capacitor will gap of a parallel plate capacitor will affect its capacitance.affect its capacitance.

A vacuum would work as an insulator A vacuum would work as an insulator that would provide the lowest that would provide the lowest capacitance. capacitance.

Any other insulator would make the Any other insulator would make the capacitance capacitance increaseincrease..

We will refer to the capacitance of a We will refer to the capacitance of a vacuum as vacuum as CCoo..

CC ≥ ≥ CCoo

DielectricsDielectrics A vacuum allows charges to easily “feel A vacuum allows charges to easily “feel

the presence” of the charges on the the presence” of the charges on the opposite plate.opposite plate.

An insulator will “dull” this feeling.An insulator will “dull” this feeling. These different insulators are called These different insulators are called

dielectricsdielectrics.. Capacitance can be found using: Capacitance can be found using: C = C = κκCCoo

κκ is known as the is known as the dielectric constantdielectric constant that that is different for each insulator.is different for each insulator.

DielectricsDielectrics

MaterialMaterial Dielectric Constant Dielectric Constant ((κκ))

VacuumVacuum 11

AirAir 1.0005361.000536

PaperPaper 22

RubberRubber 2.82.8

GlassGlass 3.83.8

WaterWater 80.480.4

Capacitor PracticeCapacitor Practice1.1. Is it possible to create a 1.5Is it possible to create a 1.5μμF capacitor from two F capacitor from two

capacitors of capacitance 1capacitors of capacitance 1μμF and 2F and 2μμF ?F ?2.2. Is it possible to create a 1.5Is it possible to create a 1.5μμF capacitor from two F capacitor from two

capacitors of capacitance 2capacitors of capacitance 2μμF each?F each?3.3. Two capacitors are used in series. Each capacitor Two capacitors are used in series. Each capacitor

has a C = 3has a C = 3μμF when used with a dielectric with F when used with a dielectric with κκ=2. If one is used with a dielectric with =2. If one is used with a dielectric with κκ=2 and =2 and the other is used with a dielectric with the other is used with a dielectric with κκ=4,=4,a) What is the equivalent capacitance of the a) What is the equivalent capacitance of the circuit?circuit?b) What is the equivalent impedance of the circuit?b) What is the equivalent impedance of the circuit?c) What is the equivalent capacitance if wired in c) What is the equivalent capacitance if wired in

parallel with each other?parallel with each other?d) What is the equivalent impedance if wired in d) What is the equivalent impedance if wired in parallel parallel with each other?with each other?

Lesson#32Lesson#32Topic: Inductors and Stored EnergyTopic: Inductors and Stored Energy

Objectives: Objectives: (After this class I will be (After this class I will be able to)able to)

1.1. Describe how inductors affect impedance Describe how inductors affect impedance of AC circuitsof AC circuits

2.2. Define inductanceDefine inductance

3.3. Explain how energy can be stored in a Explain how energy can be stored in a capacitor or an inductorcapacitor or an inductor

12/13/06

Assignment: Packet p691 #7, 8, 11 p676 #37, 40

Warm Up: What is the current through a DC circuit that has a 12V battery wired in series with an inductor?

InductorsInductors An inductor is a long wire wrapped in An inductor is a long wire wrapped in

the form of a coil.the form of a coil. For reasons discussed later in the For reasons discussed later in the

course, this coil resists the change in course, this coil resists the change in current.current.

Inductors cause high impedance at Inductors cause high impedance at high frequencies and have low high frequencies and have low impedance at low frequency.impedance at low frequency.

The proportionality constant between The proportionality constant between impedance and frequency is called impedance and frequency is called inductance.inductance.

InductanceInductance An inductor is a long wire wrapped in the An inductor is a long wire wrapped in the

form of a coil.form of a coil. For reasons discussed later in the course, For reasons discussed later in the course,

this coil resists the change in current.this coil resists the change in current. Inductors cause high impedance at high Inductors cause high impedance at high

frequencies and have low impedance at low frequencies and have low impedance at low frequency.frequency.

The proportionality constant between The proportionality constant between impedance and frequency is called impedance and frequency is called inductanceinductance ( (LL) and has units of Henry’s (H).) and has units of Henry’s (H).

fLZL 2

Energy stored in a capacitorEnergy stored in a capacitor

The amount of energy stored on a capacitor is The amount of energy stored on a capacitor is the same as the amount of work done to fill the the same as the amount of work done to fill the capacitor with charge.capacitor with charge.

If the voltage remained constant then E=QV If the voltage remained constant then E=QV where Q is the final charge and V is the where Q is the final charge and V is the voltage.voltage.

However V increases as Q increases, so we However V increases as Q increases, so we have to use an average V, so thereforehave to use an average V, so therefore

QVE 21

Energy stored in an inductorEnergy stored in an inductor Because inductors resist change in current, Because inductors resist change in current,

an inductor will keep current flowing for a an inductor will keep current flowing for a brief time after the voltage has been brief time after the voltage has been removed.removed.

This is a temporary source of energy.This is a temporary source of energy. Exactly how an inductor works will be Exactly how an inductor works will be

explained in the next unit.explained in the next unit.

221 LIE

Inductor PracticeInductor Practice1.1. When a rms voltage of 15V is When a rms voltage of 15V is

applied to a circuit containing only applied to a circuit containing only an inductor, an rms current of 3.7A an inductor, an rms current of 3.7A is produced. If the frequency was is produced. If the frequency was originally 10,000Hz, what is the originally 10,000Hz, what is the value of the inductors inductance?value of the inductors inductance?

2.2. What is the energy needed to What is the energy needed to increase the current from zero up to increase the current from zero up to its maximum value?its maximum value?

Inductor PracticeInductor Practice3.3. When a rms voltage of 15V and When a rms voltage of 15V and

10,000Hz is applied to a circuit 10,000Hz is applied to a circuit containing only a capicitor, an rms containing only a capicitor, an rms current of 3.7A is produced. current of 3.7A is produced.

a) How work is required to charge a) How work is required to charge the capacitor?the capacitor?

b) How much energy is release b) How much energy is release when the capacitor discharges?when the capacitor discharges?

Lesson#33Lesson#33Topic: Phase Shifts and Total ImpedanceTopic: Phase Shifts and Total Impedance

Objectives: Objectives: (After this class I will be able to)(After this class I will be able to)1.1. Explain what it means for two waves to be “in Explain what it means for two waves to be “in

phase” with one anotherphase” with one another2.2. Describe the terms “leading” and “lagging” and Describe the terms “leading” and “lagging” and

how they apply to phase shifts between voltage how they apply to phase shifts between voltage and currentand current

3.3. Describe the phase shifts caused by resistors, Describe the phase shifts caused by resistors, capacitors, and inductorscapacitors, and inductors

4.4. Use vector diagrams to solve for total impedance Use vector diagrams to solve for total impedance and the angle of phase shiftand the angle of phase shift

5.5. Define Resonance FrequencyDefine Resonance Frequency

12/14/06

Assignment: Packet p691 #13, p692 #15, 17, 19 p677 #65

Warm Up: Explain how a radio uses a combination of capacitors and inductors to eliminate all but one set frequency (channel).

Current in an AC circuitCurrent in an AC circuit By wiring a capacitor and an inductor By wiring a capacitor and an inductor

in series with a resistor, we can in series with a resistor, we can control current.control current.

The capacitor will block out low The capacitor will block out low frequency and the inductor will block frequency and the inductor will block out high frequency.out high frequency.

This will only allow a small range of This will only allow a small range of frequencies.frequencies.

This is very useful for many common This is very useful for many common devices (like radios and TV’s)devices (like radios and TV’s)

PhasePhase A capacitor wired with an inductor in A capacitor wired with an inductor in

series can have lower impedance series can have lower impedance than just a capacitor or inductor than just a capacitor or inductor wired alone.wired alone.

You cannot just add the impedances You cannot just add the impedances as you would with resistors wired in as you would with resistors wired in series.series.

One element cancels out the effect of One element cancels out the effect of the other.the other.

PhasePhase As we’ve mentioned, in an AC circuit, As we’ve mentioned, in an AC circuit,

current will oscillate at the same current will oscillate at the same frequency as voltage.frequency as voltage.

However, the peak voltages don’t However, the peak voltages don’t always occur at the same always occur at the same timetime as the as the peak current.peak current.

Current doesn’t oscillate with the Current doesn’t oscillate with the same same phasephase as the voltage. as the voltage.

ResistorsResistors Resistors are independent of Resistors are independent of

frequencyfrequency Therefore, the voltage peaks occur at Therefore, the voltage peaks occur at

the same time as the current peaks the same time as the current peaks through a resistor.through a resistor.

Voltage across a resistor is “in phase” Voltage across a resistor is “in phase” with the current through the resistor.with the current through the resistor.

We can then compare the voltage We can then compare the voltage graph across the entire circuit to the graph across the entire circuit to the voltage graph across just the resistor.voltage graph across just the resistor.

This will tell us how much the voltage This will tell us how much the voltage is “out of phase” with the current. is “out of phase” with the current.

Capacitors Capacitors Current through a capacitor Current through a capacitor leadsleads the the

voltage across the capacitor. voltage across the capacitor. When a capacitor is uncharged there is no When a capacitor is uncharged there is no

potential difference across it.potential difference across it. Current needs to flow and charge needs to Current needs to flow and charge needs to

build up on the capacitor before a maximum build up on the capacitor before a maximum potential difference is obtained.potential difference is obtained.

Max voltage occurs Max voltage occurs afterafter max current flows. max current flows. In fact, max voltage always occurs when In fact, max voltage always occurs when

current is at zero (analyze capacitor current is at zero (analyze capacitor diagram).diagram).

Max I occurs ¼ of a cycle before max V. Max I occurs ¼ of a cycle before max V.

Inductors Inductors Current through an inductor Current through an inductor lagslags the the

voltage across the inductor. voltage across the inductor. Inductors resist change in current.Inductors resist change in current. If you change an applied voltage, an If you change an applied voltage, an

inductor will resist that change and inductor will resist that change and therefore delay the corresponding change therefore delay the corresponding change in current.in current.

Max voltage occurs Max voltage occurs beforebefore max current max current flows. flows.

In fact, max voltage always occurs when In fact, max voltage always occurs when current is at zero.current is at zero.

Max I occurs ¼ of a cycle after max V.Max I occurs ¼ of a cycle after max V.

Phase PracticePhase Practice1.1. An oscillating voltage is applied to An oscillating voltage is applied to

an RLC circuit in series. At which of an RLC circuit in series. At which of the following frequencies would you the following frequencies would you expect the current through the expect the current through the circuit to lead the voltage across circuit to lead the voltage across the circuit?the circuit?

a) At low frequenciesa) At low frequencies

b) At high frequenciesb) At high frequencies

c) At some frequency in between c) At some frequency in between

Phase PracticePhase Practice2.2. When a rms voltage of 15V is applied When a rms voltage of 15V is applied

to a circuit containing only a to a circuit containing only a capacitor, an rms current of 3.7A is capacitor, an rms current of 3.7A is produced. produced.

a) Which comes first, the max a) Which comes first, the max current or the max voltage?current or the max voltage?

b) Suppose the frequency is b) Suppose the frequency is 10,000Hz what is the time difference 10,000Hz what is the time difference between max current and max between max current and max voltage? voltage?

Phase PracticePhase Practice3.3. When a rms voltage of 15V is applied When a rms voltage of 15V is applied

to a circuit containing only an to a circuit containing only an inductor, an rms current of 3.7A is inductor, an rms current of 3.7A is produced. produced.

a) Which comes first, the max a) Which comes first, the max current or the max voltage?current or the max voltage?

b) Suppose the frequency is b) Suppose the frequency is 10,000Hz what is the time difference 10,000Hz what is the time difference between max current and max between max current and max voltage? voltage?

Phase Angles Phase Angles Phase can be represented with angles.Phase can be represented with angles. A full period is 2A full period is 2ππ radians. radians. A ¼ period is A ¼ period is ππ/2 radians./2 radians. The phase shift for the voltage across a The phase shift for the voltage across a

resistor is 0 radians.resistor is 0 radians. The phase shift for the voltage across a The phase shift for the voltage across a

capacitor is capacitor is ππ/2 radians./2 radians. The phase shift for the voltage across an The phase shift for the voltage across an

inductor is -inductor is -ππ/2 radians./2 radians. All three waves need to be combined to All three waves need to be combined to

find the total phase shift across the entire find the total phase shift across the entire circuit.circuit.

Phase Angles Phase Angles Waves can be transformed into vectors Waves can be transformed into vectors

and combined exactly like vectors.and combined exactly like vectors. The phase angles can be used to draw the The phase angles can be used to draw the

direction of the voltage vector for each direction of the voltage vector for each element on a set of x y axis's. element on a set of x y axis's.

VC

VL

VR

Phase Angles Phase Angles Combine the vectors and then plot the Combine the vectors and then plot the

resultant voltage.resultant voltage. The angle between the resultant voltage The angle between the resultant voltage

and the voltage across the resistor is the and the voltage across the resistor is the phase shiftphase shift of the circuit of the circuit ((θ)..

Vtotal

θ

Resonance Resonance There are frequencies where the phase shift is zero.There are frequencies where the phase shift is zero. This occurs when the effect of the capacitor cancels out This occurs when the effect of the capacitor cancels out

the effect of the inductor and vice versa (because they the effect of the inductor and vice versa (because they are are ππ radians out of phase with each other). radians out of phase with each other).

VVc c = V= VLL; this is referred to as the ; this is referred to as the resonance frequencyresonance frequency. .

VR= Vtotal

VC

VL

VR

Phase Angle PracticePhase Angle Practice1.1. Derive an equation for VDerive an equation for Vtotaltotal given V given VLL, V, Vcc, ,

and Vand VRR..

2.2. Derive an equation for Derive an equation for θ given the same three variables..

3.3. An 18V alternating voltage is applied to An 18V alternating voltage is applied to an RLC circuit containing a 8an RLC circuit containing a 8ΩΩ resistor, a resistor, a 12mH inductor and a 1212mH inductor and a 12μμF capacitor in F capacitor in series.series.

Is the potential difference across the Is the potential difference across the voltage generator equal to the sum of voltage generator equal to the sum of the individual voltages across each the individual voltages across each element?element?

Phase Angle PracticePhase Angle Practice4.4. Derive an equation for ZDerive an equation for Ztotal total from your from your

previously derived equation for Vpreviously derived equation for Vtotaltotal and the and the definition of impedance.definition of impedance.

5.5. Derive an equation for Derive an equation for θ given the impedance of each element in an RLC circuit..

6.6. An 18V, 75Hz alternating voltage is applied An 18V, 75Hz alternating voltage is applied to an RLC circuit containing a 8to an RLC circuit containing a 8ΩΩ resistor, a resistor, a 12mH inductor and a 1212mH inductor and a 12μμF capacitor in F capacitor in series.series.a) What is the phase shift of the circuit?a) What is the phase shift of the circuit?b) Does the maximum current come before b) Does the maximum current come before or after the maximum voltage?or after the maximum voltage?

Phase Angle PracticePhase Angle Practice7.7. An 18V, 75Hz alternating voltage is applied An 18V, 75Hz alternating voltage is applied

to an RLC circuit containing a 8to an RLC circuit containing a 8ΩΩ resistor, a resistor, a 12mH inductor and a 1212mH inductor and a 12μμF capacitor in F capacitor in series.series.

a) Which is greater, the impedance of the a) Which is greater, the impedance of the resistor, capacitor, or inductor?resistor, capacitor, or inductor?

b) What is the impedance of the circuit?b) What is the impedance of the circuit?

c) Is it possible to find a frequency at which c) Is it possible to find a frequency at which the total impedance is less than 8the total impedance is less than 8ΩΩ? ?

8.8. Derive an equation for resonant angular Derive an equation for resonant angular frequency from Zfrequency from ZC C = Z= ZLL..

Phase Angle PracticePhase Angle Practice9.9. A circuit contains a 1.4A circuit contains a 1.4μμH inductor, H inductor,

a 1.82pF capacitor and a 12a 1.82pF capacitor and a 12ΩΩ resistor in series.resistor in series.

a) What is the resonance a) What is the resonance frequency?frequency?

b) If such a circuit is used in a radio, b) If such a circuit is used in a radio, does the radio pick up AM signals or does the radio pick up AM signals or FM signals? How do you know?FM signals? How do you know?