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Lesson#28
Topic: AC Circuits
12/7/06
Objectives: (After this class I will be able to)
1. Explain the difference between AC and DC
2. Describe how alternating current is
sinusoidal
3. Describe the three ways to measure
voltage in an AC circuit
Warm Up: What is the difference between AC and DC?
Which is more commonly used?
Assignment: Packet p675 #11, 13, 14, 15, 17
Alternating Current
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Many of our most useful electronic
devices rely on AC voltage.
The current is continually changing
direction.
The current and voltage change is
sinusoidal with time.
We use frequency to represent how
quickly the voltage oscillates.
AC electricity is a form of simple
harmonic motion.
Amplitude of Current and Voltage
Measuring actual current or voltage is
difficult because it is constantly changing
in magnitude and direction.
 AC voltage varies from Vmax to –Vmax
 You can use Vmax=ImaxR to solve for Imax
 Can we use Vavg=IavgR to solve for Iavg?
 What would Vavg equal?
 Instead of using max values or average
values, we use RMS values.
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RMS Values
Since the peak voltage only lasts for an
instant and the average value is zero, we
need to use RMS values.
 RMS = Root mean squared
 First square all values to make them all
positive
 Take an average of these squared values.
 Then square root this average.
 RMS values are the values closest to the
actual voltage or current of the circuit.
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AC Practice
1.
2.
3.
A voltage of peak value 10V oscillates
with a period of 1ms. What is the
frequency and angular frequency of the
signal?
A signal generator is set to produce a
voltage with a period of 1s. With what
frequency does a light bulb wired in this
circuit blink?
The amplitude of a sinusoidal signal is 3V
What is the RMS value?
Lesson#29
Topic: RLC Circuits
12/8/06
Objectives: (After this class I will be able to)
1.
2.
3.
4.
Calculate the power dissipated by an AC circuit
Define Impedance
Describe new elements that may be found in AC
circuits
Explain how resistors, inductors, and capacitors
affect the overall impedance of an AC circuit.
Warm Up: What would be the calculated power
dissipated by an AC circuit if you used the average voltage
times the average current?
Assignment: Packet p675 #12, 16, 18, 19, 20
Power in AC circuits
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To solve for resistance in a circuit you could use
Vmax with Imax or Vrms with Irms
Either way will work.
But when dealing with energy consumption, or
average power dissipated, you need to use RMS
values.
Pavg = VrmsIrms
Other previously derived equations for power
can also be used with rms voltage and current.
Power in AC circuits
The average power dissipated in a stereo
speaker is 55W. Assuming that the speaker can
be treated as a resistor with 4ohms resistance,
find:
 The RMS value of the AC voltage applied to the
speaker and the RMS value of the AC current
through the speaker.
 The peak value of the AC voltage applied to the
speaker and the peak value of the AC current
through the speaker.
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RLC Circuits
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With DC circuits we saw that greater voltage
caused greater current to flow.
The same happens in AC circuits.
Certain elements in the circuit cause the
current to also depend on the frequency of the
applied voltage.
We can categorize these elements into three
types: Resistors, Capacitors, and Inductors.
Each element has a different dependence on
frequency.
Impedance
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Impedance is the same thing as resistance.
The resistance of a resistor is the resistor’s
impedance.
Resistors are independent of frequency.
Impedance is represented by the symbol Z.
ZR = R
Capacitors have impedance that is inversely
proportional to frequency
ZC ~ 1 / f
Inductors have impedance that is directly
proportional to frequency
ZL~ f
AC Practice
1.
A resistor has an impedance of 100ohms
when Vmax = 10V and f= 100Hz
a. What is Z if f = 1000Hz?
b. What is Z if f = 0Hz?
c. What is Z if f = infinity?
AC Practice
2. When a rms voltage of 15V is applied to
a circuit containing only a capacitor, an
rms current of 3.7A is produced.
a. What is Zc ?
b. What is Zc if f is doubled?
c. What is the current if f is doubled?
d. What is Zc when f = 0Hz?
e. What is Zc when f = infinity?
AC Practice
3. When a rms voltage of 15V is applied to
a circuit containing only an inductor, an
rms current of 3.7A is produced.
a. What is ZL ?
b. What is ZL if f is doubled?
c. What is the current if f is doubled?
d. What is ZL when f = 0Hz?
e. What is ZL when f = infinity?
Lesson#30
Topic: Capacitors
12/11/06
Objectives: (After this class I will be able to)
1.
2.
3.
4.
Describe how a capacitor works
Define and explain capacitance
Explain how frequency and capacitance affect the
impedance of a capacitor
Describe how the surface area and distance
between plates of a capacitor will affect its
capacitance.
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?
Assignment: Packet p691 #1, 2, 4 p675 #30, 31
Capacitor
Two conducting plates separated by a thin
insulating material.
 The insulator creates a “gap” in the circuit.
 Current cannot flow through the gap.
 Current can flow through the wires for a
short time while one plate is “sucked” dry
of electrons, and the other plate is being
saturated with electrons.
 Positive charge will accumulate on one
side and negative charge on the other.
 The overall capacitor remains neutral.
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Capacitor
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Electron Flow
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Current
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Electron Flow
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Current
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Capacitance
Capacitance is the quantity of how much
charge can be “stored” on each plate.
 The larger the capacitance of a capacitor,
the larger the “capacity” it has to hold
charge.
 The charge found on one of the plates (Q)
is directly proportional to the capacitance
of a capacitor (C ).
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Q~C
Capacitance
q is the charge of a point or particle.
 Q is the charge of an object (like a
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capacitor plate)
 The amount of charge also depends on
the amount of voltage that is applied.
 The larger the voltage, the more charge
that will accumulate.
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Q~V
Capacitance
Charge will accumulate and current will
flow until the voltage across the capacitor
is the same as that across the battery.
 Though these two proportionalities we can
use the equation: Q = CV
 Q = charge stored on one plate of the
capacitor
 V = Voltage across the capacitor
 C = Capacitance of the capacitor
 Capacitance has units of Coulomb per Volt
or a “Farad” (F).
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Impedance of a capacitor
In a DC circuit, a capacitor will quickly fill and
then no current will flow (infinite impedance).
 In an AC circuit, it is possible to oscillate the
direction of the current fast enough that the
capacitor never fills up.
 At very high frequencies it is as if the capacitor
isn’t even there (zero impedance).
 The larger the Capacitance, the less likely that it
will ever fill up and cause impedance.
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1
Zc 
C
or
1
Zc 
2fC
Parallel Plate Capacitors
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The capacitance of a capacitor depends on its
structure.
If the plates are large in area, then more charge
is able to accumulate on the plates.
Capacitance is directly proportional to the area of
the plates.
If the distance between the plates is large, then
the electric force causing charges to separate is
weak and little charge will accumulate.
Capacitance is inversely proportional to the
distance between the plates.
A
C
d
Capacitor Practice
1.
2.
When a rms voltage of 15V and
10,000Hz is applied to a circuit
containing only a capacitor, an rms
current of 3.7A is produced. How much
charge can be stored on the capacitor
when a 15V steady potential difference is
applied across it?
A capacitor has a capacitance of 6μF. If
the width of the gap is doubled, what
happens to the capacitance? What if the
area of the plates double also?
Lesson#31
Topic: Total Capacitance and Dielectrics
12/12/06
Objectives: (After this class I will be able to)
1.
2.
3.
4.
Add capacitors that are wired in series and in
parallel
Define dielectrics
Describe dielectric constant
Explain how the material placed between a
parallel plate capacitor affects capacitance.
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?
Assignment: Packet
Equivalent Capacitance
Wiring multiple capacitors in parallel is similar to
adding surface area to one big capacitor.
 Additional surface area increases capacitance.
 Capacitors wired in parallel add directly.
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C parallel  C1  C2  ....  Cn
Equivalent Capacitance
Wiring multiple capacitors in series is similar to
increasing the gap between the plates of one
large capacitor.
 Greater distance between plates will decrease
capacitance.
 Capacitors wired in series add inversely.
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1
Cseries
1
1
1
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 .... 
C1 C2
Cn
Dielectrics
The type of insulator placed between the
gap of a parallel plate capacitor will affect
its capacitance.
 A vacuum would work as an insulator that
would provide the lowest capacitance.
 Any other insulator would make the
capacitance increase.
 We will refer to the capacitance of a
vacuum as Co.
 C ≥ Co
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Dielectrics
A vacuum allows charges to easily “feel
the presence” of the charges on the
opposite plate.
 An insulator will “dull” this feeling.
 These different insulators are called
dielectrics.
 Capacitance can be found using: C = κCo
 κ is known as the dielectric constant that
is different for each insulator.
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Dielectrics
Material
Dielectric Constant (κ)
Vacuum
1
Air
1.000536
Paper
2
Rubber
2.8
Glass
3.8
Water
80.4
Capacitor Practice
1.
2.
3.
Is it possible to create a 1.5μF capacitor from two
capacitors of capacitance 1μF and 2μF ?
Is it possible to create a 1.5μF capacitor from two
capacitors of capacitance 2μF each?
Two capacitors are used in series. Each capacitor has a
C = 3μF when used with a dielectric with κ=2. If one
is used with a dielectric with κ=2 and the other is used
with a dielectric with κ=4,
a) What is the equivalent capacitance of the circuit?
b) What is the equivalent impedance of the circuit?
c) What is the equivalent capacitance if wired in
parallel with each other?
d) What is the equivalent impedance if wired in parallel
with each other?
Lesson#32
Topic: Inductors and Stored Energy
12/13/06
Objectives: (After this class I will be able to)
1.
2.
3.
Describe how inductors affect impedance of AC
circuits
Define inductance
Explain how energy can be stored in a capacitor
or an inductor
Warm Up:
What is the current through a DC circuit that has a
12V battery wired in series with an inductor?
Assignment: Packet p691 #7, 8, 11 p676 #37, 40
Inductors
An inductor is a long wire wrapped in the
form of a coil.
 For reasons discussed later in the course,
this coil resists the change in current.
 Inductors cause high impedance at high
frequencies and have low impedance at
low frequency.
 The proportionality constant between
impedance and frequency is called
inductance.
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Inductance
An inductor is a long wire wrapped in the form of
a coil.
 For reasons discussed later in the course, this coil
resists the change in current.
 Inductors cause high impedance at high
frequencies and have low impedance at low
frequency.
 The proportionality constant between impedance
and frequency is called inductance (L) and has
units of Henry’s (H).
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Z L  2fL
Energy stored in a capacitor
The amount of energy stored on a capacitor is
the same as the amount of work done to fill the
capacitor with charge.
 If the voltage remained constant then E=QV
where Q is the final charge and V is the voltage.
 However V increases as Q increases, so we have
to use an average V, so therefore
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E  QV
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Energy stored in an inductor
Because inductors resist change in current, an
inductor will keep current flowing for a brief time
after the voltage has been removed.
 This is a temporary source of energy.
 Exactly how an inductor works will be explained
in the next unit.
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E  LI
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Inductor Practice
1.
2.
When a rms voltage of 15V is applied to
a circuit containing only an inductor, an
rms current of 3.7A is produced. If the
frequency was originally 10,000Hz, what
is the value of the inductors inductance?
What is the energy needed to increase
the current from zero up to its maximum
value?
Inductor Practice
3. When a rms voltage of 15V and
10,000Hz is applied to a circuit
containing only a capicitor, an rms
current of 3.7A is produced.
a) How work is required to charge the
capacitor?
b) How much energy is release when the
capacitor discharges?
Lesson#33
Topic: Phase Shifts and Total Impedance
12/14/06
Objectives: (After this class I will be able to)
1.
2.
3.
4.
5.
Explain what it means for two waves to be “in phase”
with one another
Describe the terms “leading” and “lagging” and how they
apply to phase shifts between voltage and current
Describe the phase shifts caused by resistors, capacitors,
and inductors
Use vector diagrams to solve for total impedance and the
angle of phase shift
Define Resonance Frequency
Warm Up: Explain how a radio uses a combination of capacitors
and inductors to eliminate all but one set frequency (channel).
Assignment: Packet p691 #13, p692 #15, 17, 19 p677 #65
Current in an AC circuit
By wiring a capacitor and an inductor in
series with a resistor, we can control
current.
 The capacitor will block out low frequency
and the inductor will block out high
frequency.
 This will only allow a small range of
frequencies.
 This is very useful for many common
devices (like radios and TV’s)
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Phase
A capacitor wired with an inductor in
series can have lower impedance than just
a capacitor or inductor wired alone.
 You cannot just add the impedances as
you would with resistors wired in series.
 One element cancels out the effect of the
other.
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Phase
As we’ve mentioned, in an AC circuit,
current will oscillate at the same
frequency as voltage.
 However, the peak voltages don’t always
occur at the same time as the peak
current.
 Current doesn’t oscillate with the same
phase as the voltage.
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Resistors
Resistors are independent of frequency
 Therefore, the voltage peaks occur at the
same time as the current peaks through a
resistor.
 Voltage across a resistor is “in phase” with
the current through the resistor.
 We can then compare the voltage graph
across the entire circuit to the voltage
graph across just the resistor.
 This will tell us how much the voltage is
“out of phase” with the current.
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Capacitors
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Current through a capacitor leads the voltage
across the capacitor.
When a capacitor is uncharged there is no
potential difference across it.
Current needs to flow and charge needs to build
up on the capacitor before a maximum potential
difference is obtained.
Max voltage occurs after max current flows.
In fact, max voltage always occurs when current
is at zero (analyze capacitor diagram).
Max I occurs ¼ of a cycle before max V.
Inductors
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Current through an inductor lags the voltage
across the inductor.
Inductors resist change in current.
If you change an applied voltage, an inductor
will resist that change and therefore delay the
corresponding change in current.
Max voltage occurs before max current flows.
In fact, max voltage always occurs when current
is at zero.
Max I occurs ¼ of a cycle after max V.
Phase Practice
1.
An oscillating voltage is applied to an
RLC circuit in series. At which of the
following frequencies would you expect
the current through the circuit to lead
the voltage across the circuit?
a) At low frequencies
b) At high frequencies
c) At some frequency in between
Phase Practice
2. When a rms voltage of 15V is applied to
a circuit containing only a capacitor, an
rms current of 3.7A is produced.
a) Which comes first, the max current or
the max voltage?
b) Suppose the frequency is 10,000Hz
what is the time difference between max
current and max voltage?
Phase Practice
3. When a rms voltage of 15V is applied to
a circuit containing only an inductor, an
rms current of 3.7A is produced.
a) Which comes first, the max current or
the max voltage?
b) Suppose the frequency is 10,000Hz
what is the time difference between max
current and max voltage?
Phase Angles
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Phase can be represented with angles.
A full period is 2π radians.
A ¼ period is π/2 radians.
The phase shift for the voltage across a resistor
is 0 radians.
The phase shift for the voltage across a
capacitor is π/2 radians.
The phase shift for the voltage across an
inductor is -π/2 radians.
All three waves need to be combined to find the
total phase shift across the entire circuit.
Phase Angles
Waves can be transformed into vectors and
combined exactly like vectors.
 The phase angles can be used to draw the
direction of the voltage vector for each element
on a set of x y axis's.
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VC
VL
VR
Phase Angles
Combine the vectors and then plot the resultant
voltage.
 The angle between the resultant voltage and the
voltage across the resistor is the phase shift of
the circuit (θ).
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Vtotal
θ
Resonance
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There are frequencies where the phase shift is zero.
This occurs when the effect of the capacitor cancels out
the effect of the inductor and vice versa (because they
are π radians out of phase with each other).
Vc = VL; this is referred to as the resonance frequency.
VC
VL
VR
VR= Vtotal
Phase Angle Practice
1.
2.
3.
Derive an equation for Vtotal given VL, Vc, and
VR.
Derive an equation for θ given the same three
variables.
An 18V alternating voltage is applied to an RLC
circuit containing a 8Ω resistor, a 12mH
inductor and a 12μF capacitor in series.
Is the potential difference across the voltage
generator equal to the sum of the individual
voltages across each element?
Phase Angle Practice
4. Derive an equation for Ztotal from your
previously derived equation for Vtotal and the
definition of impedance.
5. Derive an equation for θ given the impedance
of each element in an RLC circuit.
6. An 18V, 75Hz alternating voltage is applied to
an RLC circuit containing a 8Ω resistor, a 12mH
inductor and a 12μF capacitor in series.
a) What is the phase shift of the circuit?
b) Does the maximum current come before or
after the maximum voltage?
Phase Angle Practice
7.
An 18V, 75Hz alternating voltage is applied to
an RLC circuit containing a 8Ω resistor, a 12mH
inductor and a 12μF capacitor in series.
a) Which is greater, the impedance of the
resistor, capacitor, or inductor?
b) What is the impedance of the circuit?
c) Is it possible to find a frequency at which
the total impedance is less than 8Ω?
8. Derive an equation for resonant angular
frequency from ZC = ZL.
Phase Angle Practice
9.
A circuit contains a 1.4μH inductor, a
1.82pF capacitor and a 12Ω resistor in
series.
a) What is the resonance frequency?
b) If such a circuit is used in a radio,
does the radio pick up AM signals or FM
signals? How do you know?