Unit 3:
Introduction to Alternating Current and Voltage
In EET 150 you learned how to analyze circuits containing DC (directcurrent)
sources. In this course you'll learn how to analyze circuits containing AC (alternatingcurrent)
sources. AC electricity is more complicated than DC, so this unit
will introduce some concepts and terms that we'll use for the rest
of the course.
Actually, much of the AC material in this unit
should be a review of things that you studied in EET 114. But we will
pick up some new ideas, such as angular frequency and a way to write
mathematical expressions for sinusoidal waves. For instance, here's
a mathematical expression that describes a particular sinusoidal
voltage:
v = 4.62 V
sin(968t + 50°).
At first sight, this may look complicated, but with a bit of practice
you'll be able to interpret and use expressions of this kind.
In this unit we'll get a start on analyzing AC circuits. Fortunately, resistors in
AC circuits behave pretty much the same way they behave in DC circuits.
Therefore, you can use circuit rules that you already knowsuch as
Ohm's Law, Kirchhoff's Voltage Law, and Kirchoff's Current Lawin pretty
much the same way that you used them in DC circuits. (But as we'll see
in later units, inductors and capacitors behave one way in DC
circuits, and they behave entirely differently in AC circuits.)
First, read the following
material in Thomas Floyd's Principles
of Electric Circuits (8th edition):
 Introduction to Alternating Current and Voltage (Chapter
11)
Then work through the eLesson below.
After you finish the eLesson, you'll be ready to take Quiz #3,
perform Lab #3, and do Homework
#3.
Unit 2 Review
 This unit will build on material that you studied in Unit
2. So let's begin by taking this selftest to review what you
learned in that unit.

DC and AC
 Direct current (DC) is current that flows in one
direction only.
 A DC voltage source is a voltage source that produces
direct current.
 Examples: Batteries and dc power supplies (such as the power
supply built into the trainer that you use in lab) are DC voltage
sources.
 Alternating current (AC) is current whose direction
periodically reverses.
 An AC voltage source is a voltage source that
produces alternating current.
 Examples: Electrical outlets in the walls of your home provide
alternating current. The trainer that you use in lab also contains
an AC voltage source called a function generator.

Waveform
 In most DC circuits, current and voltage remain
constant as time passes. But in AC circuits the
voltage and current change as time passes.
 In Unit 2 of this course we'll see how to write down
mathematical expressions that describe how the values change in time.
But a simpler and more common way is to draw diagrams showing how
the voltage or current changes in time.
 Such a graph of a current or voltage versus time is called
a waveform. Below are several examples of voltage
waveforms. Notice that each of these diagrams plots voltage (measured
in volts or millivolts) on the vertical axis, and time (measured in
microseconds or milliseconds) on the horizontal axis.
 First, here is an example
of a triangle
waveform. In a triangle waveform, voltage increases gradually
along a straight line to its maximum value, and then gradually decreases
along a straight line to its lowest value, and then starts increasing
again.
 Next, here is an example of a square waveform. In
a square waveform, voltage remains constant at a high value for a while,
then suddenly drops to a low value, where it stays for a while before
suddenly jumping back to its high value.
AC versus Pulsating DC
 Both of the examples just shown are AC waveforms, because
the voltage actually changes polarity. (In other words, the voltage
is positive sometimes and negative sometimes.)
 On the other hand, you could have a waveform whose polarity does
not change, even though its value does change as time passes. This
would be a pulsating DC waveform. Here is an example:
Sine Wave
 The waveform studied most frequently in electrical circuit theory
is the sine wave. Here's an example:
 It's called a sine wave because this is the same shape that you get
if you make a plot of the mathematical function y = sin(x),
with x plotted on the horizontal axis and y on the vertical axis.
Sinusoid
 Strictly speaking, a sine wave must pass through the origin (the
point where the xaxis crosses the yaxis).
 The more general term sinusoid is used to describe any waveform
that has the same shape as a sine wave but that may be shifted to the
right or to the left along the xaxis. The waveform shown just above
is an example of a sinusoid. So is the following waveform, which has
the same shape but is shifted horizontally so that it does not pass
through the origin:
Sinusoids In, Sinusoids Out
 Here's a remarkable feature of sinusoidal waveforms that makes them
particularly easy to work with. Suppose you connect a function generator
to any circuit containing resistors, inductors, and capacitors. If
the function generator is set to produce a sinusoidal waveform, then
every voltage drop and every current in the circuit will also be a
sinusoid.
 The same thing is not true for waveforms of other
shapes. For instance, if the function generator is set to produce a
triangle waveform, and if the circuit contains any inductors or capacitors,
then the voltage drops across the components will be complicated waveforms,
not simple triangle waveforms.
 Fortunately, it turns out that sinusoids are not only the easiest
waveforms to work with, they're also the most useful. Therefore we'll
concentrate primarily on analyzing AC circuits that have sinusoidal
voltage sources, rather than triangle or squarewave or other voltage
sources.
Periodic Waveform
 A periodic waveform is a waveform whose values are repeated
at regular intervals.
 All of the waveforms shown above are periodic waveforms.
Waveform Parameters
 Important parameters associated with periodic waveforms include:
 Period
 Frequency
 Instantaneous Value
 Peak Value
 PeaktoPeak Value
 RMS Value (also called effective value)
 Average Value
 Each of these terms is explained below.
Cycle
 The plot of a periodic waveform shows a regularly repeating pattern
of values, each of which is called a cycle.
 Example: In the picture of the sine wave shown below,
we see a little less than two full cycles. The first cycle extends
from 0 ms to 50 ms, and the second (incomplete) cycle extends from
50 ms to the edge of the chart, where it is cut off:

Period
 The time required for the values to rise and fall through one complete
cycle is called the period of the waveform.
 The symbol for period is T.
 Period is measured in units of seconds, abbreviated s.
 Example: The sine wave shown above has
a period of 50 ms.
Frequency
 The frequency of an ac waveform is the number of cycles that
occur in one second.
 The symbol for frequency is f.
 Frequency is measured in units of cycles per second, or Hertz, abbreviated
Hz.
How Period and Frequency Are Related
 Period and frequency are the reciprocal of each other:
f = 1 ÷ T
and
T = 1 ÷ f
 Example: The sine wave shown earlier has
a period of 50 ms. Therefore, its frequency is 20 Hz.
Instantaneous Value
 The instantaneous value of an ac waveform is its value at
a specific instant of time.
 We'll see below how you can use
the mathematical expression for a waveform to find the waveform's
instantaneous values at specific times.
 You can read off approximate instantaneous values from the graph
of a waveform.
 Example: At
20 ms, the sine wave shown earlier has
an instantaneous value of about 300 mV.
Peak Value
 The maximum value reached by an ac waveform is called its peak
value.
 If the waveform is a voltage waveform, then its peak value
is also called its peak voltage, abbreviated V_{p}.
 If the waveform is a current waveform, then its peak value
is also called its peak current , abbreviated I_{p}.
 The peak value of a waveform is sometimes also called its amplitude,
but the term “peak value” is more descriptive.
 Example:The sine wave shown earlier has
a peak value of 500 mV_{p}. Notice that I write a "p" after
the unit to show that I'm talking about a peak value.
PeaktoPeak Value
 The peaktopeak value is the difference between a waveform's
positive peak value and its negative peak value.
 If the waveform is a voltage waveform, then its peaktopeak
value is also called its peaktopeak voltage,
abbreviated V_{pp}.
 If the waveform is a current waveform, then its peaktopeak
value is also called its peaktopeak current ,
abbreviated I_{pp}.
 If the waveform is symmetrical about the time axis, then the peaktopeak
value equals twice the peak value.
 Example: The sine wave shown earlier has
a peaktopeak value of 1 V_{pp}. That's the difference
between the maximum positive value (which is 500 mV) and the maximum
negative value (which is 500 mV). Notice that I wrote a "pp" after
the unit to show that I'm talking about a peaktopeak value.
p and pp
 As mentioned in the two examples above, we write p as
the subscript of a quantity or unit to show that we're talking about
a peak value, and we write pp as the subscript of
a quantity or unit when we're talking about a peaktopeak value.
 Example: For the sine wave above,
we could write
V_{p} = 500 mV_{p}
or
V_{pp} = 1 V_{pp}.
 Similarly, if we were dealing with a current waveform whose peak
value is 20 mA and whose peaktopeak value is 40 mA, we could write
I_{p} = 20 mA_{p}
or
I_{pp} = 40 mA_{pp}.
 Some other textbooks use pk (instead of p)
as the abbreviation for peak values, and pp (instead
of pp) as the abbreviation for peaktopeak values.
So you may see these other abbreviations used from time to time.
Oscilloscope
 The oscilloscope is an instrument designed to display waveforms.
Using it, you can measure period, frequency, peak values, peaktopeak
values, and other important quantities.
 Shown here is a Tektronix 2213, one of the types of oscilloscopes
that we have in Sinclair's electronics labs. At the left is the screen
on which waveforms are displayed. To the right are the knobs and switches
that you can adjust to control the waveform's appearance.
 Below are three separate photos showing how a triangle wave, a square
wave, and a sine wave look on the oscilloscope screen.
Using the Oscilloscope to Measure Voltage
 The oscilloscope displays a graph of voltage versus time, with voltage
plotted on the vertical axis and time plotted on the horizontal
axis.
 To measure a waveform's peaktopeak voltage, you count how many
vertical divisions (squares) the waveform covers on the oscilloscope's
screen, and then you multiply this number times the setting of the
oscilloscope VOLTSPERDIVISION knob.
 This learning object will show you how to do it:

Using the Oscilloscope to Measure Period and Frequency
 Remember, the oscilloscope displays a graph of voltage versus time,
with voltage plotted on the vertical axis and time plotted
on the horizontal axis.
 To measure a waveform's period, you count how many horizontal divisions
(squares) the waveform covers on the oscilloscope's screen, and then
you multiply this number times the setting of the oscilloscope SECONDSPERDIVISION
knob.
 Once you know the waveform's period, you can use the formula f =
1 ÷ T to find its frequency.
 This learning object will show you how to do it:

Oscilloscope Challenge Game
 The oscilloscope is a complicated piece of equipment. You'll need
plenty of practice to learn how to use it correctly.
 To brush up on your oscilloscope skills, take some time right now
to play Oscilloscope
Challenge. In particular, work through the game's "Study" section,
which is a tutorial containing several pages of notes to help you identify
and use the oscilloscope's controls. This will be a good preparation
for Lab #2, in which you'll use a real oscilloscope to make measurements.
RMS Value (or Effective Value)
 The rootmeansquare (rms) value or effective value
of an ac waveform is a measure of how effective the waveform is in
producing heat in a resistance.
 Example: If you connect a 5 V_{rms} source across
a resistor, it will produce the same amount of heat as you would get
if you connected a 5 V dc source across that same resistor. On the
other hand, if you connect a 5 V peak source or a 5 V peaktopeak source
across that resistor, it will not produce the same amount of
heat as a 5 V dc source.
 That's why rms (or effective) values are useful: they give us a way
to compare ac voltages to dc voltages.
 To show that a voltage or current is an rms value, we write rms after
the unit: for example, V_{rms} = 25 V rms.
 A multimeter set to AC mode measures rms values.
Relationship Between Peak Values & RMS Values
 You need to be able to convert peak values to rms values, and
vice versa. There are some standard conversion factors that let you
do this.
 For a sine wave, to convert from peak values to rms values,
use these equations:
For voltages, V_{rms} ≈ 0.707 × V_{p}
For currents, I_{rms} ≈ 0.707 × I_{p}
To convert in the other direction (from rms values to
peak values), use these equations:
For voltages, V_{p} ≈ 1.414 × V_{rms}
For currents, I_{p} ≈ 1.414 × I_{rms}
 Note: the "squiggly" equals sign means approximately equal.
These approximate conversion factors are close enough for our purposes.
 These equations are valid only for sinusoidal waveforms. For
other wave shapes, there are other numbers (which we won't need in
this course) that you would use to convert between peak values and
rms values.
True rms Meter
 Some inexpensive multimeters measure peak values of AC waves, and
then use the equations given above to compute rms values. Since
these equations hold only for sine waves, these meters give
incorrect rms values for nonsinusoidal waves.
 A true rms meter gives correct rms value for any AC wave.
The multimeters in Sinclair's EET labs are true rms meters.
Average Value
 The average value of a waveform is the average of its values
over a time period.
 Any waveform that is symmetrical about the time axis has zero average
value over a complete cycle.
 For example, consider the sine wave
shown below. Over one cycle, this waveform's positive values exactly
cancel out its negative values, so its average value over
a complete cycle is zero.
 Sometimes, though, it's useful to refer to the waveform's average value
over a half cycle. So, just looking at the positive
"hump" of the sine wave below, what is its average value? Using calculus,
it can be shown that a sine wave's average value over a half ccyle
is equal to 0.636 times its peak value.
 The average value of a waveform is also called its DC value.
 When a waveform is measured with a multimeter set to DC mode,
the meter indicates the waveform's average value over a complete cycle,
which would be zero for the sine wave pictured above.

Phase of a Sine Wave
 The pictures of sinusoidal waveforms shown above had voltage on
the vertical axis and time on the horizontal axis. That's a useful
picture when you're interested in knowing how the voltage changes
as time goes by.
 Recalling that one complete cycle of a sine wave is generated when
a generator's rotor rotates through an angle of 360° (a full rotation),
here's another way of plotting a sine wave:
 Here we're plotting voltage on the vertical axis, just as before,
but now we're plotting degrees of the rotor's rotation on the horizontal
axis. Notice that one complete cycle of the sine
wave corresponds to 360°.
 The quantity on the horizontal axis in this sort of picture is called
the phase of the sine wave. For instance, we can see
that this particular sine wave has a voltage of 0 V when its
phase is 0°, and a voltage of 300 V when its
phase is 90°, and so on.

Radians
 Often we measure angles in degrees, but the radian (rad) is
another unit for measuring angles. Since both units are widely
used, it's important to be comfortable with both units and to be able
to convert from degrees to radians (or vice versa).
 A full circle is equal to 360°, and it's also equal to 2p radians.
And since p is approximately equal to 3.14,
this means that 360° is approximately equal to 6.28 radians. As
an equation:
360° = 2p rad ≈ 6.28
rad
 Dividing both sides of that equation by 2, we can also see that
180° = p rad ≈ 3.14
rad
 Dividing both sides again by 2, we can also see
that
90° = p/2 rad ≈ 1.57
rad
 Using these equalities, we can say that the sine wave pictured above
has a voltage of
300 V when its phase is p/2 rad, and
a voltage of 0 V when its phase is p rad, and
so on.
Converting Between Degrees and Radians
 To convert any angle from radians to degrees, multiply
by 180 ÷ p.
 In the other direction, to convert from degrees to radians, multiply
by p ÷ 180.
Phase Shift
 Often we'll find ourselves dealing with two or more sinusoids that
have the same frequency, one of which is shifted to the right or the
left of the other one. For example:
 In the case pictured here, notice that the pink sinusoid is
rising through zero (the horizontal axis) at the same time that the
blue sinusoid is reaching its maximum value. Therefore the pink sinusoid
has a phase of 0° at the same time that the blue sinusoid has a phase
of 90°.
 We express this by saying that there's a 90° phase
shift (also called a 90° phase
angle) between
the two waveforms.
Lead and Lag
 In cases like the one shown above, we're interested not only
in how far apart the two waveforms are shifted, but which one "comes
before" the other one.
 The waveform shifted farther to the left is said to lead the
other waveform.
 So in the picture shown above, the blue waveform leads the pink waveform
by 90°.
 As another way of saying the same thing, the waveform shifted farther
to the right is said to lag the
other waveform.
 So in the picture shown above, the pink waveform lags the blue
waveform by 90°.
Using the Oscilloscope to Measure Phase Shifts
 By displaying two waveforms simultaneously and then measuring the
time interval between corresponding points on the two waveforms, we
can determine the phase shift between them.
 In particular, if T is the period of the waveforms, and t is
the time interval between corresponding points on the two waveforms,
then the phase shift φ is given by the equation:
φ = (t ÷ T) × 360°
 The following animated learning object also shows how to measure
phase angles. Their explanation is a little different from
mine, but you should be able to see it's the same idea.
They first count the number of "spaces" (by which they mean
hashmarks) in a complete cycle, then divide that by 360° to find
the number of degrees per "space," and then multiply that
times the number of "spaces"
between the two waveforms. The end result will be the same that you'd
get using the formula φ = (t ÷ T) × 360°.

Lowercase and Uppercase
 From EET 150 you know that uppercase letters are used to represent
most DC quantities.
 For example, V represents DC voltage,
and I represents DC current.
 On the other hand, most AC quantities are represented by lowercase
letters.
 For example, v represents AC voltage,
and i represents AC current.
 Within AC, uppercase letters
are also used to represent constants that don't change with time.
 For example, V_{p} represents peak voltage,
and I_{p} represents peak current. Peak voltage and
peak current don't continually change as time passes, so that's why
they're represented by uppercase letters.
Mathematical Expression for a Sine Wave
Mathematical Expression for a PhaseShifted Wave
 Above you learned that a phaseshifted sinusoid is one that is shifted
to the right or to the left along the horizontal axis, so that the
wave does not pass through the origin.
 Mathematically, this is shown by having a fixed angle added
to θ in the expression for the waveform. In other words, instead
of having an expression of the form
v = V_{p} sin(θ)
we'll have an expression that looks like this:
v = V_{p} sin(θ + φ)
where φ is a constant angle called (not surprisingly) the phase shift.
 Note: φ is the Greek letter phi, pronounced "fie."
 For
example, here is the graph of v = 300 V sin(θ + 90°):
 Notice that this graph has the same shape as the sine wave shown
just above, but this one is shifted 90° (onequarter of
a cycle) to the left.
 A positive phase shift causes the waveform to shift left along
the xaxis, and a negative phase shift causes it to shift right.
 As another example, here is the graph of v = 300 V sin(θ −
90°):
 Notice that this graph has the same shape as the sinusoids shown
above, but this one is shifted 90° (onequarter of a cycle)
to the right of the origin.
 In each of these diagrams, we're looking at only one cycle of the
waveform. You should imagine the wave extending indefinitely to the
left and to the right.
Phasors
 A phasor is a vector that represents an AC electrical quantity,
such as a voltage waveform or a current waveform.
 The phasor's length represents the voltage's or current's peak value.
 The phasor's angle represents the voltage's or current's phase.
 For example, in the following diagram, think of the phasor as representing
a particular voltage waveform that has, let's say, a peak voltage
of 4 V p. If you wanted to draw on this diagram another
phasor representing a voltage waveform with a peak voltage of 8 V p,
then you would draw a new phasor that's twice as long as the original
one.
Rotating Phasors
 Phasors have two main uses in studying AC circuits. In the first
use, we imagine a phasor rotating counterclockwise around the origin
at a speed that depends on the waveform's frequency. (Higherfrequency
waveforms rotate more quickly than lowerfrequency waveforms.)
 As the phasor rotates, the waveform's instantaneous value at any
time is equal to the phasor's ycomponent.
 For example, looking at our sample diagram again, imagine the phasor
to be rotating counterclockwise around the origin, and think of this
diagram as a "snapshot" of the phasor at one instant in
time. As we know from above, the phasor's length represents the waveform's
peak voltage, V_{p}. The phasor's ycomponent, which
is equal to V_{p} sin(θ), represents the
waveform's instantaneous voltage.
Phasor Diagrams
 The second, and more important, use of phasors is to represent the
relationship between two or more waveforms with the same frequency.
 For example, consider the following diagram, which shows two phasors
labeled v_{1} and v_{2}.
 Phasor v_{1} is drawn at an angle of 0°,
and it has a length of 10 units.
 Phasor v_{2} is drawn at an angle of 45°,
and it's half as long as v_{1}. (Measure
it with a ruler if you don't believe me.)
 Such a diagram might represent two voltage waveforms in a circuit.
From the diagram we can see that v_{1}'s peak voltage
is twice as great as v_{2}'s peak voltage. (Assuming
that the units are V, then v_{1} has a peak voltage
of 10 V p, which means that v_{2} must
have a peak voltage of 5 V p.) We can also see that v_{2} leads v_{1 }by
a phase shift of 45°.
 In terms of the equations for sinusoidal waveforms that you studied
in the previous unit, this diagram would then be a pictorial representation
of the equations
v_{1} = 10 V sin(θ)
v_{2} = 5 V sin(θ + 45°)
 The equations and the phasor diagram convey the same information,
but the diagram can be easier to understand and interpret, especially
in cases where you're dealing with half a dozen waveforms instead
of just two.
 The same information can also be conveyed using a sinusoidal diagram
such as the following:
 Carefully compare the phasor diagram, the equations, and the sinusoidal
diagram given above, until you're convinced that they all describe
the same pair of waveforms.
 So we can use sinusoidal diagrams, mathematical equations, or phasor
diagrams to describe the relationships among waveforms in a circuit.
Of these three representations, phasor diagrams are the probably the
easiest to use and to understand, so they are widely used in AC circuit
analysis.
Angular Frequency
 As mentioned above, when you imagine a phasor rotating about the
origin, the waveform's frequency determines the speed of the phasor's
rotation. In particular, if the waveform's frequency is f,
then the phasor will rotate with an angular speed of 2pf.
 This quantity 2pf, which appears
in many equations, is called the waveform's angular frequency.
 Its symbol is ω, and its unit is radians per second (rad/s):
ω = 2pf
 Of course, since a waveform's frequency is equal to the reciprocal
of its period (f = 1 ÷ T),
we can also write
ω = 2p ÷ T
 So if you know a waveform's frequency or period, you can easily
compute its angular frequency (or vice versa).
 Note that ω is the Greek letter omega; it's not a w.
V_{p} sin(ωt)
 Until now, we've been writing the mathematical expressions for voltage
waveforms as
v = V_{p} sin(θ)
 Recall that here V_{p} is a constant equal to the
waveform's peak voltage. On the other hand, θ is a variable
that changes continually as time passes. It can also be shown that θ=ωt,
where ω is the waveform's angular frequency and t is
time (measured in seconds).
 Making this subsitution for θ, the expression for a voltage
sine wave with peak value V_{p} and angular frequency ω becomes
v = V_{p} sin(ωt)
 Similarly, for a current sine wave with peak value I_{p} and
angular frequency ω,
i = I_{p} sin(ωt)
Instantaneous Value
General Form of a Sinusoid
 We've just seen that the mathematical expression for a voltage or
current sine wave contains two important pieces of information: the
wave's peak value (V_{p} or I_{p})
and its angular frequency (ω).
 If we now add in the possibility that the wave may also have a phase
shift (φ), we find that the general expression for a sinusoidal
voltage or current is
v = V_{p} sin (ωt + φ)
or
i = I_{p} sin (ωt + φ)
 Often the phase angle φ is given in degrees, but the
angular frequency ω is almost always given in radians per
second. So, before you can use the calculator to do a computation
involving these quantities, you must convert φ from degrees to
radians. If you don't do this, you'll be mixing degrees with radians,
and you'll get the wrong answer. (That would be like trying to add
2 inches plus 10 centimeters without first converting one of those
quantities so that they both have the same unit.)
 Of course, since we know that ω = 2pf,
we could also write these two equations as
v = V_{p} sin (2pft + φ)
i = I_{p} sin (2pft + φ)
Analyzing AC Circuits
 In later units we'll find that analyzing an AC circuit can get pretty
tricky if the circuit contains capacitors or inductors. But if the
circuit contains only resistors, then you can analyze it using the
same techniques you've learned for DC circuits, with a couple of slight
changes mentioned below. Once you understand how to apply Ohm's law,
Kirchhoff's Laws, and the power laws to DC circuits, you already know
almost everything you need to analyze resistive AC circuits.
Sinusoids In, Sinusoids Out
 Here's a remarkable feature of sinusoidal waveforms that makes them
particularly easy to work with. Suppose you connect a function generator
to any circuit containing resistors, inductors, and capacitors. If
the function generator is set to produce a sinusoidal waveform, then
every voltage drop and every current in the circuit will also be a
sinusoid.
 The same thing is not true for waveforms of other
shapes. For instance, if the function generator is set to produce
a triangle waveform, and if the circuit contains any inductors or
capacitors, then the voltage drops across the components will be complicated
waveforms, not simple triangle waveforms.
 Fortunately, it turns out that sinusoids are not only the easiest
waveforms to work with, they're also the most useful. Therefore we'll
concentrate primarily on analyzing AC circuits that have sinusoidal
voltage sources, rather than triangle or squarewave or other voltage
sources.
Phase in Resistors
 The voltage across any resistor and the current through that resistor
have the same phase angle. They reach their peak values at the
same instant.
 We say that the resistor's voltage and current are in phase with
each other.
 For instance, in the diagram below, suppose the tall blue waveform
represents the voltage across a resistor, and the short purple waveform
represents the current through that resistor. These two waveforms
are in phase with each other, since they reach their peak values at
the same instant.
 This means that in a phasor diagram, the phasor for a resistor's
voltage must point in the same direction as the phasor for that resistor's
current.
Ohm's Law for Resistors
 Ohm's law can be applied to resistors in AC circuits:
I_{p} = V_{p} ÷ R
 In words, this says that a resistor's peak current is equal to the
resistor's peak voltage drop divided by the resistor's resistance.
 The same law also applies to peaktopeak values and rms values:
I_{pp} = V_{pp} ÷ R
and
I_{rms} = V_{rms} ÷ R
 Of course, you can also rearrange any of these equations algebraically
if you wish to solve for voltage or resistance. For example, if you
wish to calculate peak voltage based on known values of resistance
and peak current, you would rearrange the first equation to the form V_{p} = I_{p} × R.
KVL and KCL for AC Circuits
 Recall from your studies of DC circuits that Kirchhoff's Voltage
Law (KVL) says that the sum of the voltage drops around any closed
loop in a circuit equals the sum of the voltage rises around that
loop.
 Recall also that Kirchhoff's Current Law (KCL) says that the
sum of all currents entering a point is equal to the sum of all
currents leaving that point.
 You can also apply KVL and KCL to AC circuits that contain just
resistors, as long as you're careful to use all peak values, or all
rms values, or all peaktopeak values.
Power in a Resistor
 Recall from EET 150 that, in DC circuits, you can
use any one of the following equations to find the power dissipated
in a resistor
P = I^{2} × R
P = V^{2} ÷ R
P = V^{ }× I
In these equations, V is the DC voltage drop across the
resistor, and I is the DC current through the resistor.
 In AC circuits, similar equations apply, but you
must be sure to use the resistor's rms voltage and rms current.:
P = I_{rms}^{2} × R
P = V_{rms}^{2} ÷ R
P = V_{rms}^{ }× I_{rms}
In these equations, V_{rms} is the rms voltage
drop across the resistor, and I_{rms} is the rms
current through the resistor. If you use peak values or
peaktopeak values instead of rms values, you'll get the wrong
answer for the power.
Superimposed DC and AC Voltages
 Many circuits that you'll encounter in the field contain not just
AC voltages sources or just DC voltage sources, but a combination
of AC and DC voltage sources.
 As a very simple example of this, the following diagram shows a
series circuit containing a DC voltage source, and AC voltage source,
and a single resistor.
 In this case, the voltage across the resistor is a 10 V pp
sine wave superimposed on 8 V DC. What this means is that
we'll have a 10 V pp sine wave that is "moved up" 8 V
from where it would normally be, as in the following diagram.
 Notice the scale on the vertical axis. We have a 10 V pp
sine wave, but instead of ranging from −5 V to +5 V,
the sine wave's voltage ranges from +3 V to +13 V.
 When you use a multimeter or oscilloscope to make measurements in
such a circuit, you must pay close attention to whether your equipment
is set to measure AC values or DC values (or both). Unit 9 of EET
114 was devoted to this topic, so you should go back and review
the eLesson for that unit, along with its SelfTest questions,
before continuing with this lesson.
Review of Electrical Quantities
 The table below summarizes the electrical quantities that we've
studied so far. The table shows the abbreviation for each quantity,
along with the standard unit for measuring the quantity and the abbreviation
for the unit.
 You studied the first eleven quantities (through inductance) in
EET 150. The others have been added in this course.
Quantity 
Abbreviation 
Unit 
Abbreviation for the
Unit 
charge 
Q 
coulomb 
C 
current 
I 
ampere 
A 
voltage (or emf) 
V (or E) 
volt 
V 
resistance 
R 
ohm 
Ω 
conductance 
G 
siemens 
S 
energy (or work) 
W 
joule 
J 
power 
P 
watt 
W 
efficiency 
η 


capacitance 
C 
farad 
F 
time constant 
τ 
second 
s 
inductance 
L 
henry 
H 
period 
T 
second 
s 
frequency 
f 
hertz 
Hz 
peak voltage 
V_{p} 
volts peak 
V p 
peak current 
I_{p} 
amps peak 
A p 
peaktopeak voltage 
V_{pp} 
volts peaktopeak 
V pp 
peaktopeak current 
I_{pp} 
amps peaktopeak 
A pp 
rms voltage 
V_{rms} 
volts rms 
V rms 
rms current 
I_{rms} 
amps rms 
A rms 
time 
t 
second 
s 
varying voltage 
v 
volt 
V 
varying current 
i 
ampere 
A 
angle 
θ 
radian or degree 
rad or ° 
phase angle 
φ 
radian or degree 
rad or ° 
angular frequency 
ω 
radians per second 
rad/s 
 Notice that the abbreviations for quantities are usually written
with italicized letters, while the abbreviations for units are usually
written with plain, nonitalicized letters.
 We'll continue to learn about more new quantities throughout this
course. Unit 4 will include an expanded
version of the table.
Unit 3 Review
 This eLesson has covered several important topics, including:
 waveforms
 waveform parameters, including:
 period
 frequency
 instantaneous value
 peak value
 peaktopeak value
 rms value (also called effective value)
 average value
 the oscilloscope
 mathematical expressions for waveforms
 phasors
 mathematical expressions for waveforms
 resistors in AC circuits
 average power
 measuring phase shifts.
 To finish the eLesson, take this selftest to check your understanding
of these topics.

Congratulations! You've completed the eLesson for this unit. What's
next?
 Take Quiz #3.
 Perform Lab #3 and turn in a typed short lab report.
Read my instructions on how to write Short
Lab Reports.
 Do Homework #3.
 Keep practicing your skills by playing the games on the Games page.
 For more practice with the material from this unit, visit the textbook's Chapter
11 web page and take the multiplechoice, true/false, circuitanalysis,
and fillintheblank quizzes provided there.
 Prepare for Unit 4 by reading Sections 126, 127, 135, and 136
of Thomas Floyd's Principles of Electric Circuits (8th edition).
Then you'll be ready to go on to Unit 4.
Nick Reeder
 Electronics Engineering Technology  Sinclair Community College
Send comments to nick.reeder@sinclair.edu
