Unit 4:
Capacitors and Inductors in AC Circuits
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In thi unit we'll turn our attention to the behavior of capacitors
and inductors in AC circuits. You saw in Unit 3 that resistors
behave pretty much the same way in AC circuits as they do in DC circuits.
But this is not true for capacitors and inductors. We use one set of
rules (which you learned in Unit 2) for finding a capacitor's voltage
and current in DC circuits, and a different set of rules (which you'll
learn in this unit) for finding a capacitor's voltage and current in
AC circuits.
By the way, the rules that you learned in Unit 2 for finding the total
capacitance and total inductance do hold for all circuits, DC or AC.
But when you're trying to figure out a capacitor's current or voltage,
you'll use different techniques depending on whether the circuit is
DC or AC.
AC circuit analysis relies heavily on trigonometry, which is the branch
of mathematics dealing with angles. The last part of this eLesson
covers some basic trigonometry, which you may have studied in a math
class. But even if you've never studied trigonometry, you should be
able to pick up this material without too much trouble.
First, you should read the following sections of Thomas Floyd's Principles
of Electric Circuits (8th edition):
 Capacitors in AC Circuits (Section 126)
 Capacitor Applications (Section 127)
 Inductors in AC Circuits (Section 135)
 Inductor Applications (Section 136)
Then work through the eLesson and SelfTest questions below.
After you finish the eLesson, you'll be ready to take Quiz #4,
perform Lab #4, and do Homework #4.
Unit 3 Review
 This unit will build on material that you studied in Unit
3. So let's
begin by taking this selftest to review what you learned in that unit.

Capacitors in AC Circuits
 In Unit 2 you learned some rules and equations that let you compute
initial values, transient values, and steadystate values in circuits
with capacitors. These rules and equations apply only to DC circuits;
now we'll learn entirely different rules and equations that
you use if you have capacitors in an AC circuit.
Comparing Capacitors to Resistors
 Recall from Unit 3 that resistors in AC circuits can be analyzed
using the same techniques (such as Ohm's law) that you use for resistors
in DC circuits. Recall also that in an AC circuit, a resistor's voltage
and current are in phase with each other, as shown here:
 In AC circuits, there are two crucial differences between resistors
and capacitors:
 Current and voltage in a capacitor are not in phase with each
other.
 Instead of having a resistance, a capacitor in an AC circuit
has something called a capacitive reactance, which is similar
to a resistance.
Phase Relationships in a Capacitor
 Current and voltage in a capacitor are not in phase with each other.
 For sinusoidal waves, the current through a capacitor leads the
voltage across it by 90º. (In other words, the voltage
lags the current by 90º.)
 For instance, in the diagram below, the short blue waveform represents
the voltage across a capacitor, and the tall purple waveform represents
the current through the capacitor. Do you agree that the current waveform
leads the voltage waveform by 90º?
A Memory Aid: ICE
 To remember that current leads voltage in a capacitor, remember
the word ICE.
 Since I comes before E in the word "ICE," this reminds
you that the current (I) in a capacitor (C) leads the voltage (E).
 Many textbooks use the letter E as the symbol for voltage,
and that's what E stands for in this memory aid. Our textbook uses V for
voltage, but you don't get a very easytoremember word if you write
ICV. So if you want to use this memory aid you'll have to think of
E as the symbol for voltage.
Capacitive Reactance
 AC current through a capacitor depends not only on the size of the
voltage across it, but also on the frequency of that voltage.
 A capacitor's opposition to AC current is called capacitive reactance,
abbreviated X_{C}. Like resistance, its
unit is the ohm. But unlike resistance, its value changes as frequency
changes.
 You can calculate a capacitor's capacitive reactance if you know
the capacitor's size (in farads) and the frequency (or angular frequency)
of the capacitor's current or voltage. Capacitive reactance is given
by the equation
X_{C} = 1 ÷ (2pfC)
Since 2pf is equal to ω (the
angular frequency), we could also write this equation as
X_{C} = 1 ÷ (ωC)
 You can see from these equations that as frequency increases,
capacitive reactance decreases. This means that a capacitor
opposes lowfrequency current more than it opposes highfrequency
current.

Ohm's Law for Capacitors
Analyzing AC Capacitive Circuits
 Now that you know how to calculate a capacitor's reactance, and
now that you know how to use Ohm's law with capacitors, you can analyze
circuits that contain an AC voltage source connected to any combination
of capacitors, such as the one shown below:
 To analyze such a circuit, you use the same basic techniques that
you used in EET 150 to analyze DC resistive circuits.
 The only difference is that first you have to calculate each
capacitor's reactance; but from that point onward, you can treat
reactances in the same way you're used to treating resistances.
For example, if you know a capacitor's reactance and its current,
then you can use Ohm's law to find its voltage.
 One thing to be aware of is that reactances in series and parallel
combine just like resistors in series and parallel. In other words:
 To find the total reactance of reactances connected
in series, add the reactances.
 To find the total reactance of reactances connected
in parallel, use the reciprocal formula.

Not Ready Yet for AC Circuits with Capacitors and Resistors
 So now you should be able to handle AC circuits with capacitors.
 You're still not ready, though, to analyze AC circuits that contain
capacitors and resistors, such as the one shown below:
 Circuits like this are trickier because of the complicated phase
relationships between the various currents and voltages in the circuit.
We'll get to circuits like this one in a few weeks.
Power in a Capacitor
 Recall from Unit 3 that you can use any of the following equations
to find the power dissipated by a resistor in an AC circuit:
P = V_{rms}^{ }× I_{rms}
P = V_{rms}^{2} ÷ R
P = I_{rms}^{2} × R
 Recall also that power dissipation in a resistor means that the resistor
is converting some of the circuit's electrical energy to heat energy.
This energy is lost from the circuit as the air around the resistor
is heated.
 Ideally, a capacitor does not dissipate any power. What this means
is that an ideal capacitor does not convert any of the circuit's electrical
energy to heat energy. Rather, an ideal capacitor stores electrical
energy, which can be returned to the circuit at a later time.
 In reality, no capacitor is perfectly ideal, so a real capacitor
will lose some electrical energy as heat.
 Because energy storage and loss is more complicated in capacitors
than it is in resistors, we distinguish two kinds of power: true power
and reactive power.
True Power
 From earlier courses you're familiar with the concept of power. Up
to now you've probably only studied one kind of power (which we call
true power). But as we study capacitors and inductors, you'll see that
power is more complicated than you may have realized.
 True power is a measure of the rate at which
a component or circuit loses energy. This energy loss is usually due
to dissipation of heat (as in a resistor) or conversion to some other
form of energy (as in a motor that converts electrical energy to
motion). True power is the type of power that you've studied in earlier
courses.
 We'll use the symbol P_{true} for
true power.
 Just like power in resistors, true power is measured
in watts (W).

True Power in a Capacitor
 In an ideal capacitor, true power is zero, since ideal capacitors
don't lose any energy as heat.
 In a real capacitor, true power will not be zero,
but generally it is small. There's no easy way to compute true power
in a real capacitor, since it depends on the capacitor's physical construction.
Because true power in capacitors is small, we'll usually
just assume that it's zero.
Reactive Power
 Reactive power is a measure of the rate at which
a component is storing energy or returning energy
to the circuit.
 We use the symbol P_{r} for reactive
power.
 You might think that reactive power would be measured in watts (W),
just like true power. But it is not. Instead,
we use a unit called the VAR, which stands for voltampere
reactive. We do this to make it clear that we're talking about
energy that is stored and can be returned to the circuit, rather than
energy that is lost from the circuit as heat.

Reactive Power in a Capacitor
 To compute a capacitor's reactive power, we use equations that look
very similar to the equations for a resistor's power:
P_{r} = V_{rms}^{ }× I_{rms}
P_{r} = V_{rms}^{2} ÷ X_{C}
P_{r} = I_{rms}^{2} × X_{C}
 Notice that these are the same as the equations for a resistor's
power, except in the second and third equations we replace resistance
(R) with capacitive reactance (X_{C}).
Inductors in AC Circuits
 In Unit 2 you learned some rules and equations that let you compute
initial values, transient values, and steadystate values in circuits
with inductors. These rules and equations apply only to DC circuits;
now we'll start learning entirely different rules and equations that
you use if you have inductors in an AC circuit.

Comparing Inductors to Resistors and Capacitors
 You learned in Unit 3 that in an AC circuit,
a resistor's voltage and current are in phase with each other, as
shown in this example:
 You learned above that a capacitor has something called a capacitive
reactance, which is similar to a resistance and can be substituted
for resistance in Ohm's law. You also learned that, for sinusoidal
waves, the current through a capacitor leads the voltage across it
by 90º, as shown in this example:
 Like capacitors, inductors in AC circuits differ from resistors in
two crucial ways:
 Current and voltage in an inductor are not in phase with each
other.
 Instead of having a resistance, an inductor in an AC circuit
has something called an inductive reactance.
Phase Relationships in an Inductor
 Current and voltage in an inductor are not in phase with each other.
 For sinusoidal waves, the voltage across an inductor leads the
current through it by 90º. (In other words, the current
lags the voltage by 90º.) This is the opposite of what we saw
above for capacitors.
 For instance, in the diagram below, the tall blue waveform represents
the voltage across an inductor, and the short purple waveform represents
the current through the inductor. Do you agree that the voltage waveform
leads the current waveform by 90º?
A Memory Aid
 To remember whether current leads or lags voltage in a capacitor
or inductor, remember the phrase
“ELI the ICEman”
 "ELI" reminds you that the voltage (E) in an inductor
(L) leads the current (I), and "ICE" reminds you that the
current (I) in a capacitor (C) leads the voltage (E).
Inductive Reactance
 AC current through an inductor depends not only on the size of the
voltage across it, but also on the frequency of that voltage.
 An inductor's opposition to AC current is called inductive reactance,
abbreviated X_{L}. Its unit is the ohm. Like
capacitive reactance, and unlike resistance, inductive reactance changes
as frequency changes.
 Inductive reactance is given by the equation
X_{L} = 2pfL
Since 2pf is equal to ω (the
angular frequency), we can also write this equation as
X_{L} = ωL
 You can see from these equations that as frequency increases,
inductive reactance also increases. This is the opposite
of what we saw earlier for capacitors.

Ohm's Law for Inductors
Analyzing AC Inductive Circuits
 Now that you know how to calculate an inductor's reactance, and
now that you know how to use Ohm's law with inductors, you can analyze
circuits that contain an AC voltage source connected to any combination
of inductors, such as the one shown below:
 To analyze such a circuit, you use the same basic techniques that
you used in EET 150 to analyze DC resistive circuits.
 The only difference is that first you have to calculate each
inductor's reactance; but from that point onward, you can treat
reactances in the same way you're used to treating resistances.
For example, if you know an inductor's reactance and its voltage
drop, then you can use Ohm's law to find its current.

Not Ready Yet for AC Circuits with Inductors and Other Components
 So now you should be able to handle AC circuits with inductors.
 You're still not ready, though, to analyze AC circuits that contain
inductors and resistors or capacitors, such as the
one shown below:
 Circuits like this are trickier because of the complicated phase
relationships between the various currents and voltages in the circuit.
We'll get to circuits like this one in a few weeks.
Power in an Inductor
 Recall from above that for capacitors in AC circuits we distinguish
two kinds of power: true power and reactive power.
 We saw that true power is zero for an ideal
capacitor.
 We also saw that reactive power is calculated
using the same formulas used to calculate power in a resistor,
except that we replace resistance (R) in these formulas
with capacitive reactance (X_{C}).
 As we'll now see, similar comments apply to inductors in AC circuits.
True Power in an Inductor
Reactive Power in an Inductor
 Reactive power is a measure of the rate at which
an inductor is storing energy or returning energy to the circuit.
 To compute an inductor's reactive power, we use equations that look
very similar to the equations for a resistor's power:
P_{r} = V_{rms}^{ }× I_{rms}
P_{r} = V_{rms}^{2} ÷ X_{L}
P_{r} = I_{rms}^{2} × X_{L}
 Notice that these are the same as the equations for a resistor's
power, except in the second and third equations we replace resistance
(R) with inductive reactance (X_{L}).
 Recall that reactive power is measured in voltamperes reactive (VAR),
not in watts (W).
Quality Factor of an Inductor
Review of Electrical Quantities
 It's time to update our table of electrical quantities. 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 
capacitive reactance 
X_{C} 
ohm 
Ω 
true power 
P_{true} 
watt 
W 
reactive power 
P_{r} 
voltampere reactive 
VAR 
inductive reactance 
X_{L} 
ohm 
Ω 
quality factor 
Q 


 If you haven't already played them, be sure to check out the ElectricalUnits
Matching Game and the ElectricalSymbols
Matching Game.
Trigonometry Review
 In the weeks ahead we'll need the following concepts from trigonometry:
 Angles and quadrants
 Trigonometric functions: sin, cos, tan
 Inverse trigonometric functions: sin^{1}, cos^{1},
tan^{1}
Angles & Quadrants
 In the xy plane, the positive xaxis is taken as the reference direction
for all angles; in other words, we regard the postive xaxis as lying
at an angle of 0°.
 Positive angles are measured counterclockwise from
the positive xaxis. Therefore:
 The positive yaxis lies at an angle of +90°.
 The negative xaxis lies at an angle of +180°.
 The negative yaxis lies at an angle of +270°.
 The xaxis and yaxis divide the xy plane into four quadrants, which
we label quadrants I, II, III, and IV.
 All of this is summarized in the following diagram:
Negative Angles
 As mentioned above, positive angles are measured counterclockwise from
the positive xaxis.
 Negative angles are measured clockwise from the
positive xaxis. Therefore:
 The negative yaxis lies at an angle of −90°.
 The negative xaxis lies at an angle of −180°.
 The positive yaxis lies at an angle of −270°.
 This is summarized in the following diagram:
 So we see that the same direction can be referred to in
more than one way. For example, the negative yaxis lies at an angle
of +270°,
and it also lies at an angle of −90°. In other words, +270° and −90° are
two different names for the same direction.
 Usually, we'll express angles in quadrants I and II as positive angles
between 0° and +180°, and we'll express angles in quadrants
III and IV as negative angles between 0° and 180°.
Vectors
 As you'll recall from your math classes, a vector is
a mathematical quantity that has both a magnitude (length) and a direction
(angle).
 We'll use the Greek letter θ (theta) to represent
the vector's angle.
 For example, the diagram below show a vector at an angle of 30°:
 Often when we're dealing with vectors, we're interested in the finding
the vector's xcomponent and its ycomponent. (Or conversely, in some
cases we start out knowing the vector's xcomponent and its ycomponent,
and we need to find the vector.) The following diagram shows a dotted
line segment dropped vertically down from the vector's tip to the xaxis.
The length of this dotted segement is the vector's ycomponent,
and the length of the red line segment is the vector's xcomponent.
 Notice that in this diagram a rightangle triangle is formed by the
vector, the dotted line segment, and the red line segment. Much of
trigonometry is devoted to the study of rightangle triangles. We'll
now briefly review some of the main results of this branch of mathematics.
RightAngle Triangle
 A rightangle triangle is a triangle in which two sides are
at right angles (at 90°) to each other.
 In diagrams of right triangles, a small square is drawn in one angle
to indicate that this is a 90° angle.
 In the diagram below, we've labeled the three sides so that we can
easily refer to them by their labels. The labels "A", "B",
and "C" have no special meaningany other letters would
serve just as well.
 In the diagram below we've gone one step further by labeling one
of the angles with the Greek letter theta, θ, so that we can
easily refer to that angle.
Three Kinds of Question
 In later units, we'll use rightangle triangles to represent electrical
quantities such as voltage. Very often we'll be in a situation where
we know some things about a particular triangle, and we wish to figure
out some other things.
 Here are the three sorts of question that we'll need to be able to
answer:
 Given the lengths of two of the sides, what is the length of
the third side? For example, in the triangle shown above, perhaps
we know the lengths of sides A and C, and we wish to find the
length of side B.
 Given the length of one side and the size of one angle, what
is the length of another side? For example, in the triangle shown
above, perhaps we know the length of side C and the size of angle θ,
and we wish to find the length of side B.
 Given the lengths of two sides, what is the size of an angle?
For example, in the triangle shown above, perhaps we know the
lengths of sides A and B, and we wish to find how big the angle θ is
(in degrees).
 Trigonometry is the key to being able to answer those questions.
Question #1: Finding Unknown Side From Two Known Sides
 The first question listed above was how to find the length of the
third side in a rightangle triangle if we know the lengths of the
other two sides.
 The solution to this problem is given by the Pythagorean
theorem.
Pythagorean Theorem
 The Pythagorean theorem says that in any rightangle triangle,
the length of the longest side squared is equal to the sum of the
lengths of the two shorter sides squared.
 So for the triangle shown below, we can write this in symbols as
C^{2} = A^{2 }+ B^{2}.
 Now a little algebra will let us solve for any one of the sides if
we know the other two sides. In particular,
 To find side C if you know sides A and B, use C = √ (A^{2} + B^{2}).
 To find side A if you know sides B and C, use A = √ (C^{2}  B^{2}).
 To find side B if you know sides A and C, use B = √ (C^{2}  A^{2}).
Fancy Names for the Sides of a RightAngle Triangle
 Until now we've been using the labels A, B, and C to refer to the
three sides of a right triangle. The three sides also have more formal
names, which we'll use below.
 The longest side of a rightangle triangle, which we've been calling
side C, is called the hypotenuse.
 Once you've picked an angle of interest, the side farthest from that
angle is called the opposite. For example, relative
to angle θ, the side that we've been calling side A is the opposite,
since it's on the opposite side of the triangle from θ.
 The remaining side, which is next to (or "adjacent to")
the angle of interest, is called the adjacent. For
example, relative to angle θ, the side that we've been calling
side B is the adjacent.
Question #2: Finding Unknown Side From Known Side and Known Angle
 The second question listed above was
how to find the length of a side in a rightangle triangle if we know
the length of one other side and the size of an angle.
 The solution to this problem is given by the trigonometric
functions.
Trigonometric Functions: sin, cos, tan
 The three basic trigonometric functions are the sine function, the
cosine function, and the tangent function. These are abbreviated sin,
cos, and tan.
 In a right triangle:
 sin θ = opposite ÷ hypotenuse;
 cos θ = adjacent ÷ hypotenuse;
 tan θ = opposite ÷ adjacent.
 A handy way to remember this is to remember the nonsense word "Sohcahtoa."
Each letter in this word stands for one of the words in
the three equations listed above. So if you can remember "Sohcahtoa,"
then you can remember those three equations.
 Your calculator has keys that give you the sine, cosine, or tangent
of any angle.
 As you know, angles can be measured either in degrees or in
radians. Scientific calculators can handle either unit of measurement,
but you must be sure to put your calculator in the proper mode (degree
mode or radian mode) whenever you're doing a calculation
that involves sines, cosines, or tangents.
Answering Question #2
 Remember that the second type of question we wanted to be able to
answer was to find the length of an unknown side in a right triangle
if we know the length of any other side and the size of one angle.
The equations above for sin θ, cos θ, and tan θ give
us the key to answering this question.
 For example, suppose we know the length of the hypotenuse and the
size of the angle θ, and we wish to find the length of the opposite.
Then we can take the equation above for sin θ, which says
sin θ = opposite ÷ hypotenuse
and multiply both sides by the hypotenuse, giving us
sin θ × hypotenuse = opposite.
So to find the opposite (which was our goal) we'll use our calculator to
find the sine of θ and then multiply by the length of the hypotenuse.
Question #3: Finding Unknown Angle From Two Known Sides
 The third question listed above was
how to find the size of an angle in a rightangle triangle if we know
the lengths of two of the sides.
 The solution to this problem is given by the inverse trigonometric
functions.
Inverse Trigonometric Functions
 The three inverse trigonometric functions are the inverse sine function,
the inverse cosine function, and the inverse tangent function. These
are abbreviated sin^{1}, cos^{1}, and tan^{1}.
 As mentioned above, angles can be measured either in degrees or in
radians. You must be sure to put your calculator in the proper mode
(degree mode or radian mode) whenever you're doing a
calculation that involves inverse sines, inverse cosines, or inverse
tangents.
Inverse Sine: sin^{1}
 The inverse sine of a number is the angle whose sine is equal to
that number:
θ = sin^{1} k means
that sin θ = k.
 This is not as confusing as it may sound. For example, since the
sine of 60° is 0.866, it follows that 60° is the inverse sine
of 0.866. Those are really just two different ways of saying the same
thing:
the sine of 60° is 0.866 (or, in symbols,
sin 60° = 0.866)
means the same thing as
60° is the inverse sine of 0.866 (in symbols, 60° = sin^{1} 0.866)
Inverse Cosine and Inverse Tangent: cos^{1} and tan^{1}
!!!Caution!!!
 When you use a calculator to compute an inverse trigonometric function,
the calculator's answer may be in the wrong quadrant.
 Draw a sketch so that you can see which quadrant the answer should
be in; then, if necessary, correct the calculator's answer.
 Most often this will happen when you're working with an angle in
quadrants II or III.
 When you take the inverse tangent of an angle in quadrant
II, the calculator will give an answer in quadrant IV, which
you must adjust by adding 180°.
 Similarly, when you take the inverse tangent of an angle in
quadrant III, the calculator will give an answer in quadrant
I, which you must adjust by subtracting 180°.
 This is the end of our trigonometry review.
Unit 4 Review
 This eLesson has covered several important topics, including:
 capacitive reactance in AC circuits
 phase relationships in capacitors in AC circuits
 true power and reactive power in a capacitor
 inductive reactance in AC circuits
 phase relationships in inductors in AC circuits
 true power and reactive power in an inductor
 trigonometric functions
 inverse trigonometric functions.
 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 #4.
 Perform Lab #4 and turn in a typed
short lab report. (You may wish to review my instructions on
writing short reports.) Also,
the lab requires you to make some graphs. If you've never used Microsoft
Word or Excel to make graphs, now's the time to learn
how.
 Do Homework #4.
 Keep practicing your skills by playing the games on the Games page.
 Prepare for Unit 5 by reading Chapter 14 and Section 151
of Thomas Floyd's Principles of Electric Circuits (8th
edition).
Then you'll be ready to go on to Unit 5 .
Nick Reeder
 Electronics Engineering Technology  Sinclair Community College
Send comments to nick.reeder@sinclair.edu 