EGR 2201 Unit 11
Sinusoids and Phasors



Read Alexander & Sadiku, Chapter 9
and Appendix B.
Homework #11 and Lab #11 due next
week.
Quiz next week.
DC Versus AC

In a direct-current (DC) circuit,
current flows in one direction only.


The textbook’s Chapters 1 through 8
cover DC circuits.
In an alternating-current (AC) circuit,
current periodically reverses
direction.

The book’s Chapters 9 through 11 cover
AC circuits.
The Math Used in AC Circuits

Our study of AC circuits will rely
heavily on two areas of math:



Sine and cosine functions
Complex numbers
We’ll review the math after
introducing some terminology used in
discussing AC voltages and currents.
Waveforms

The graph of a current or voltage
versus time is called a waveform.


Example:
Note that this is an AC waveform:
negative values of voltage mean the
opposite polarity (and therefore opposite
direction of current flow) from positive
values.
Periodic Waveforms


Often the graph of a voltage or current
versus time repeats itself. We call this a
periodic waveform.
Common shapes for periodic waveforms
include:






Sinusoid
Square
Triangle
Sawtooth
Image from http://en.wikipedia.org/wiki/Sinusoid
Sinusoids are the most important of these.
Cycle


In a periodic signal, each repetition
is called a cycle.
How many cycles are shown in the
diagram below?
Waveform Parameters

Important parameters associated
with periodic waveforms include:
 Period T
 Frequency f
 Angular Frequency 
 Amplitude Vm (or Peak Value Vp)
 Peak-to-Peak Value
 Instantaneous Values
Period




The time required for one cycle is
called the waveform’s period.
The symbol for period is T.
Period is measured in seconds,
abbreviated s.
Example: If a waveform repeats
itself every 3 seconds, we’d write
T=3s
Frequency



A waveform’s frequency is the
number of cycles that occur in one
second.
The symbol for frequency is f.
Frequency is measured in hertz,
abbreviated Hz.


Some old-timers say “cycles per second”
instead of “hertz.”
Example: If a signal repeats itself 20
times every second, we’d write
f = 20 Hz
Period and Frequency


Period and frequency are the
reciprocal of each other:
f=1/T
T=1/f
For practice with these relationships, play
my Frequency-Period game.
Radians



Recall that the radian (rad) is the SI
unit for measuring angle.
It is related to degrees by
 radians = 180
We’ll often need to convert between
radians and degrees:

To convert radians to degrees, multiply
180
by
.
𝜋

To convert degrees to radians, multiply
𝜋
by
.
180
Angular Frequency


The quantity 2f, which appears in
many equations, is called the
angular frequency.
Its symbol is , and its unit is
rad/s:
 = 2f
One Question, Three Answers
So we have three ways of answering
the question, “How fast is the voltage
(or current) changing?”
1. Period, T, unit = seconds (s)


2.
Frequency, f, unit = hertz (Hz)

3.
Tells how many seconds for one cycle.
Tells how many cycles per second.
Angular frequency, , unit = rad/s

Tells size of angle covered per second, where
one complete cycle corresponds to an angle of
2 radians (which is 360).
Relating T, f, and 


If you know any one of these three
(period, frequency, angular
frequency), you can easily compute
the other two.
The key equations that you must
memorize are:
T = 1/f
 = 2f = 2/T
Amplitude (or Peak Value)

The maximum value reached by an
ac waveform is called its amplitude
or peak value.
Peak-to-Peak Value


A waveform’s peak-to-peak value
is its total height from its lowest
value to its highest value.
Many waveforms are symmetric
about the horizontal axis. In such
cases, the peak-to-peak value is
equal to twice the amplitude.
Instantaneous Value


The waveform’s instantaneous
value is its value at a specific time.
A waveform’s instantaneous value
constantly changes, unlike the
previous parameters (period,
frequency, angular frequency,
amplitude, peak-to-peak value),
which usually remain constant.
Lead and Lag


When two waveforms have the same
frequency but are not “in phase” with
each other, we say that the one
shifted to left leads the other one.
And we say that the one shifted to
the right lags the other one.
Phase Angle

To quantify the idea of how far a
waveform is shifted left or right
relative to a reference point, we
assign each waveform a phase
angle .



Recall that one complete cycle corresponds to
an angle of 2 radians, or 360.
A positive phase angle causes the
waveform to shift left along the xaxis.
A negative phase angle causes it to
shift right.
Sinusoids



A sinusoid is a sine wave or a cosine
wave or any wave with the same shape,
shifted to the left or right.
Sinusoids arise in many areas of
engineering and science. They are the
waveform used most frequently in
electrical circuit theory.
The waveform we’ve been looking at is a
sinusoid.
Amplitude, Frequency, Phase
Angle

Any two sinusoids must have the same
shape, but can vary in three ways:




Amplitude (maximum value)
Angular frequency (how fast the values change)
Phase angle (how far shifted to the left or right)
We’ll use mathematical expressions for
sinusoids that specify these three things….
Mathematical Expression For a
Sinusoid

The mathematical expression for a
sinusoid looks like this:
v(t) = Vmcos(t + )

where Vm is the amplitude,  is the
angular frequency, and  is the
phase angle (relative to some
reference).
Example:
v(t) = 20 cos(180t + 30) V
Calculator’s Radian Mode and
Degree Mode

Recall that when using
your calculator’s trig
buttons (such as cos),
you must pay attention
to whether the calculator is in radian
mode or degree mode.


Example: If the calculator is in radian
mode, then cos(90) returns 0.448,
which is the cosine of 90 radians.
But if the calculator is in degree mode,
then cos(90) returns 0, which is the
cosine of 90 degrees.
Caution: Radians and Degrees

In the expression for a sinusoid,
v(t) = Vmcos(t + )
 is usually given in degrees, but 

is always given in radians per
second.
Recommendation: To compute a
sinusoid’s instantaneous value,
leave your calculator in radian
mode, and convert  to radians.
Schematic Symbols for
Independent Voltage Sources

Several different symbols are
commonly used for voltage sources:
Type of Voltage
Source
Generic voltage source
(may be DC or AC)
DC voltage source
AC sinusoidal voltage
source
Symbol Used in Our Symbol Used in
Textbook
Multisim Software
Function Generator

We use a function generator to
produce periodic waveforms.
Trainer’s Function Generator
Regular Output,
controlled by all four
knobs. In this course
we’ll always use this
one.
No matter which one
of these you use, you
must also use the
GROUND connection.
TTL Mode Output, controlled
only by the FREQUENCY and
RANGE knobs. Used in
Digital Electronics courses.
Oscilloscope


We use an oscilloscope to display
waveforms.
Using it, we
can measure
amplitude,
period, and
phase angle
of ac
waveforms,
as well as
dc values, transients, and more.
Oscilloscope Challenge Game


The oscilloscope is a complex
instrument that you must learn to use.
To learn the basics, play my
Oscilloscope Challenge game at
http://people.sinclair.edu/nickreeder/flashgames.htm.
Sine or Cosine?



Any sinusoidal waveform can be
expressed mathematically using
either the sine function or the
cosine function.
Example: these two expressions
describe the same waveform:
v(t) = 20 sin(300t + 30)
v(t) = 20 cos(300t  60)
In a problem where you’re given a
mixture of sines and cosines, your
first step should be to convert all of
the sines to cosines.
Trigonometric Identities Relating
Sine and Cosine


You can convert from sine to cosine
(or vice versa) using the trig
identities
sin(x + 90) = cos(x)
sin(x  90) = cos(x)
cos(x + 90) = sin(x)
cos (x  90) = sin(x)
These identities reflect the fact that
the cosine function leads the sine
function by 90.
A Graphical Method Instead of Trig
Identities



Remembering and applying trig
identities may be difficult.
The book describes a graphical
method that relies on the following
diagram:
To use it, recall that we measure
positive angles counterclockwise, and
negative angles clockwise.
Mathematical Review: Complex
Numbers


The system of complex numbers is based
on the so-called imaginary unit, which is
equal to the square root of 1.
Mathematicians use the symbol i for this
number, but electrical engineers use j:
i  1
or
j  1
Rectangular versus Polar Form

Any complex number can be expressed in
three forms:

Rectangular form

Example: 3 + j 4

Polar form

Example: 5  53.1

Exponential form

Example: 5e j 53.1 or 5e j 0.927
Rectangular Form

In rectangular form, a complex number z is
written as the sum of a real part x and an
imaginary part y:
z = x + jy
The Complex Plane

We often represent complex numbers as
points in the complex
plane, with the real part
plotted along the
horizontal axis (or
“real axis”) and the
imaginary part plotted
along the vertical axis
(or “imaginary axis”).
Polar Form

In polar form, a complex number z is
written as a magnitude r at an angle :
z = r

The angle  is
measured from the
positive real axis.
Converting Between Rectangular
and Polar Forms


We will very often have to convert from
rectangular form to polar form, or vice
versa.
This is easy to do if
you remember a bit
of right-angle
trigonometry.
Converting from Rectangular Form
to Polar Form

Given a complex number z with real part x
and imaginary part y, its magnitude is
given by
r x y
2
2
and its angle is given by
 y
  tan  
x
1
Inverse Tangent Button on Your
Calculator
When using your calculator’s
tan1 (inverse tangent) button,
pay attention to whether the
calculator is in degree mode or radian mode.
Also recall that the calculator’s answer may
be in the wrong quadrant, and that you may
need to adjust the answer by 180.



The tan1 button always returns an angle in
Quadrants I or IV, even if you want an answer in
Quadrants II or III.
Converting from Polar Form to
Rectangular Form

Given a complex number z with
magnitude r and angle , its real part is
given by
x  r cos 
and its imaginary part is
given by
y  r sin 
Exponential Form

Complex numbers may also be written in
exponential form. Think of this as a
mathematically respectable version of polar
form.
Polar form
Exponential Form
r
Example:

330
rej


3ej/6
In exponential form,  should be in radians.
Euler’s Identity

The exponential form is based on Euler’s
identity, which says that, for any ,
e
j
 cos   j sin 
Mathematical Operations

You must be able to perform the following
operations on complex numbers:





Addition
Subtraction
Multiplication
Division
Complex Conjugate
Addition


Adding complex numbers is easiest if the
numbers are in rectangular form.
Suppose z1 = x1+jy1 and z2 = x2+jy2
Then z1 + z2 = (x1+x2) + j(y1+y2)

In words: to add two complex numbers in
rectangular form, add their real parts to get
the real part of the sum, and add their
imaginary parts to get the imaginary part of
the sum.
Subtraction


Subtracting complex numbers is also easiest
if the numbers are in rectangular form.
Suppose z1 = x1+jy1 and z2 = x2+jy2
Then z1  z2 = (x1x2) + j(y1y2)

In words: to subtract two complex numbers
in rectangular form, subtract their real parts
to get the real part of the result, and subtract
their imaginary parts to get the imaginary
part of the result.
Multiplication


Multiplying complex numbers is easiest if the
numbers are in polar form.
Suppose z1 = r1 1 and z2 = r2 2
Then z1  z2 = (r1r2)  (1+ 2)

In words: to multiply two complex numbers
in polar form, multiply their magnitudes to
get the magnitude of the result, and add
their angles to get the angle of the result.
Division


Dividing complex numbers is also easiest if
the numbers are in polar form.
Suppose z1 = r1 1 and z2 = r2 2
Then z1 ÷ z2 = (r1 ÷ r2)  (1  2)

In words: to divide two complex numbers in
polar form, divide their magnitudes to get the
magnitude of the result, and subtract their
angles to get the angle of the result.
Complex Conjugate


Given a complex number in rectangular
form,
z = x + jy
its complex conjugate is simply
z* = x  jy
Given a complex number in polar form,
z = r 
its complex conjugate is simply
z* = r 
Performing Complicated Operations
on Complex Numbers
Solving a problem may require us to
perform many operations on complex
numbers.




6∠30°+5−𝑗3
Example:
2+𝑗4
With a powerful calculator such as the TI89, you can do this quickly and easily.
With other calculators it’s more tedious,
since you must repeatedly convert
between rectangular and polar forms.
Another option is to use MATLAB.
Entering Complex Numbers in
MATLAB

Entering a number in rectangular form:
>> z1 = 3 + j  4

Entering a number in polar (actually,
exponential) form:
>> z5 = 3  exp(j  −30  pi / 180)


You must give the angle in radians, not degrees.
MATLAB always displays complex
numbers in rectangular form, no matter
how you enter them.
Operating on Complex Numbers in
MATLAB

Use the usual mathematical operators for
addition, subtraction, multiplication,
division:
>> z8 = z1 + z5

>> z9 = z5 / z6
and so on.
Built-In Complex Functions in
MATLAB
Useful MATLAB functions:







real(z1) gives z1’s real part
imag(z1) gives z1’s imaginary part
abs(z1) gives z1’s magnitude
angle(z1) gives z1’s angle in radians
conj(z1) gives z1’s complex conjugate
Since angle(z1) gives you the angle in
radians, you must multiply this by 180/pi if
you want the angle in degrees.
Online Alternative to MATLAB

Recall that wolframalpha.com is a free
online math tool.
An earlier example in MATLAB:
>>
>>
>>
>>
>>
Same example in WolframAlpha:
z1 = 3 + j  4
z5 = 3  exp(j  −30  pi / 180)
z8 = z1 + z5
abs(z8)
angle(z8)  180 / pi
Note that you must use i rather
than j for the imaginary unit.
Next slide shows results of this
command.
Online Alternative to MATLAB
(cont'd.)

Rect.
form
Polar
form
Here is part of the result of the previous
command in wolframalpha.com:
Reminder About Calculators



In this course I’ll let you use any
calculator or MATLAB on the exams.
But the Fundamentals of Engineering
exam and the Principles and Practice
of Engineering exam have a
restrictive calculator policy.
To succeed on those exams, you
must be able to do complex-number
math with a “bare-bones” calculator.
Useful Properties of j
j is the only number whose reciprocal is
equal to its negation: 1

j

Therefore, for example,
Also, 𝑗


= 190 = 𝑒
j
1
j
 
j C
C
𝜋
𝑗
2
Therefore multiplication by j is equivalent to a
counterclockwise rotation of 90 in the complex
plane.
Sinusoids Everywhere


If you connect a sinusoidal voltage
source or current source to a circuit
made up of resistors, capacitors, and
inductors, then all voltages and
currents in the circuit will be
sinusoids.
You can’t make the same statement
for triangle waves, square waves,
sawtooth waves, or other waveshapes.
Kirchhoff’s Laws in AC Circuits


KCL and KVL hold in AC circuits.
But to apply these laws, we must add
(or subtract) sinusoids instead of
adding (or subtracting) numbers.

Example: In the
circuit shown,
KVL tells us that
v = v1 + v2.
Suppose that
v1 = 10 cos(200t + 30) V and
v2 = 12 cos(200t + 45) V
How can we add those to find v?
Adding Sinusoids



We often need to find the sum of two
or more sinusoids.
A unique property of sinusoids: the
sum of sinusoids of the same
frequency is always another
sinusoid of that frequency.
You can’t make the same statement
for triangle waves, square waves,
sawtooth waves, or other waveshapes.
Adding Sinusoids (Continued)

For example, if we add
v1 = 10 cos(200t + 30) V and
v2 = 12 cos(200t + 45) V
we’ll get another sinusoid of the same
angular frequency, 200 rad/s:
v1 + v2 = Vm cos(200t + ) V

But how do we figure out the resulting
sinusoid’s amplitude Vm and phase
angle ?
Using MATLAB to Plot the
Sinusoids We’re Adding


We have
v1 = 10 cos(200t + 30) V and
v2 = 12 cos(200t + 45) V
and we’re trying to find v1 + v2.
In MATLAB:
>> fplot('10 * sin(200 * t + pi/6)', [0, 0.1])
>> hold on
>> fplot('12 * sin(200 * t + pi/4)', [0, 0.1], 'r')
>> fplot('10*sin(200*t + pi/6) + 12*sin(200*t + pi/4)', [0, 0.1], 'g')
Complex Numbers to the Rescue!


One method for adding sinusoids
relies on trig identities.
But we’ll use a simpler method, which
relies on complex numbers.

In fact, the only reason we’re interested in
complex numbers (in this course) is that
they give us a simple way to add and
subtract sinusoids.
Phasors



A phasor is a complex number that
represents the amplitude and phase
angle of a sinusoidal voltage or
current.
The phasor’s magnitude r is equal
to the sinusoid’s amplitude.
The phasor’s angle  is equal to the
sinusoid’s phase angle.

Example: We use the phasor
V = 1030 V to represent the sinusoid
v(t) = 10 cos(200t + 30) V.
Time Domain and Phasor Domain



Some fancy terms:
We call an expression like
10 cos(200t + 30) V the timedomain representation of a
sinusoid.
We call 1030 V the phasordomain representation of the
same sinusoid. (It’s also called the
frequency-domain representation.)
Using Phasors to Add Sinusoids
To add sinusoids of the same
frequency:
1. If any of your sinusoids are
expressed using sine, convert them
all to cosine.
2. Write the phasor-domain version of
each sinusoid.
3. Add the phasors (which are just
complex numbers).
4. Write the time-domain version of
the resulting phasor.

Example of Using Phasors to Add
Sinusoids

v1 = 10 cos(200t + 30) V and
v2 = 12 cos(200t + 45) V:
Transform from time domain to phasor
domain:
V1 = 1030 V and V2 = 1245 V .

Add the phasors:
1030 V + 1245 V = 21.838.2 V


Transform from phasor domain back
to time domain:
v1 + v2 = 21.8 cos(200t + 38.2) V
Phasor Relationships for Circuit
Elements


We’ve seen how we can use phasors
to add sinusoids.
Next we’ll revisit the voltagecurrent relationships for resistors,
inductors, and capacitors, assuming
that their voltages and current are
sinusoids.
Phasor Relationship for Resistors

For resistors we have, in the time
domain:
v = iR



Example:
If i = 2 cos(200t + 30) A and R = 5 ,
then v = 10 cos(200t + 30) V
For this same example, in the phasor
domain we have:
If I = 230 A and R = 5 , then
V = 1030 V
So we can write V = IR.
What This Means



For resistors, if i is a sinusoid, then
v will be a sinusoid with the same
frequency and phase angle as i.
Therefore i and v
reach their peak
values at the
same instant.
We say that a
resistor’s voltage
and current are in phase.
Summary for Resistors
In the time domain:
In the phasor domain:
Phasor Relationship for Inductors

For inductors we have, in the time
domain:
𝑣=



𝑑𝑖
𝐿
𝑑𝑡
Example:
If i = 2 cos(200t + 30) A and L = 5 H,
then v = 2000 cos(200t + 120) V
For this same example, in the phasor
domain we have:
If I = 230 A and L = 5 H, then
V = 2000120 V
So we can write V = jLI.
What This Means

For inductors, if i is a sinusoid, then
v will be a sinusoid with the same
frequency as i, but i will lag v by
90.
Summary for Inductors
In the time domain:
In the phasor domain:
Phasor Relationship for Capacitors

For capacitors we have, in the time
domain:
𝑖=



𝑑𝑣
𝐶
𝑑𝑡
Example:
If v = 2 cos(200t + 30) V and C = 5 F,
then i = 2000 cos(200t + 120) A
For this same example, in the phasor
domain we have:
If V = 230 V and C = 5 F, then
I = 2000120 A
So we can write I = jCV.
What This Means

For capacitors, if i is a sinusoid,
then v will be a sinusoid with the
same frequency as i, but i will lead
v by 90.
Summary for Capacitors
In the time domain:
In the phasor domain:
Summary: Textbook’s Table 9.2
A Memory Aid

To remember whether current leads
or lags voltage in a capacitor or
inductor, remember the phrase
“ELI the ICEman”

(For this to make sense, you must
know that E is sometimes used as
the abbreviation for voltage.)
Impedance



The impedance Z of an element or a
circuit is the ratio of its phasor
voltage V to its phasor current I:
𝐕
𝐙=
𝐈
Impedance is measured in ohms.
Like resistance, impedance
represents opposition to current: for
a fixed voltage, greater impedance
results in less current.
A Resistor’s Impedance
For resistors, V = IR, so a resistor’s
impedance is:
𝐕
𝐙= =𝑅
𝐈
 So a resistor’s impedance is a pure
real number (no imaginary part), and
is simply equal to its resistance.
 To emphasize this, we could write
𝐙 = 𝑅 + 𝑗0
or
𝐙 = 𝑅∠0°

Resistors and Frequency


A resistor’s impedance does not
depend on frequency, since Z=R for
a resistor.
Therefore, a resistor doesn’t oppose
high-frequency current any more or
less than it opposes low-frequency
current.
An Inductor’s Impedance
For inductors, V = jLI, so an
inductor’s impedance is:
𝐕
𝐙 = = 𝑗𝜔𝐿
𝐈
 So an inductor’s impedance is a pure
imaginary number (no real part).
 To emphasize this, we could write
𝐙 = 0 + 𝑗𝜔𝐿
or
𝐙 = 𝜔𝐿∠90°

Inductors and Frequency



The magnitude of an inductor’s
impedance is directly proportional
to frequency, since Z=jL for an
inductor.
Therefore, an inductor opposes
high-frequency current more than it
opposes low-frequency current.
Also, as 0, Z0, which is why
inductors act like short circuits in dc
circuits.
A Capacitor’s Impedance
For capacitors, I = jCV, so an
inductor’s impedance is:
𝐕
1
𝑗
𝐙= =
=−
𝐈 𝑗𝜔𝐶
𝜔𝐶
 So a capacitor’s impedance is a pure
imaginary number (no real part).
 To emphasize this, we could write
𝑗
𝐙=0−
𝜔𝐶
or
1
𝐙=
∠ − 90°
𝜔𝐶

Capacitors and Frequency



The magnitude of a capacitor’s
impedance is inversely proportional
𝑗
to frequency, since 𝐙 = −
for a
𝜔𝐶
capacitor.
Therefore, a capacitor opposes lowfrequency current more than it
opposes high-frequency current.
Also, as 0, Z, which is why
capacitors act like open circuits in dc
circuits.
Summary: Impedances of the
Basic Elements
Ignore this
column for
now.
Ohm’s Law Generalized



We know that in DC circuits, Ohm’s
law applies only to resistors, and
says:
𝑣 = 𝑖𝑅
In AC circuits, a generalized form of
Ohm’s law replaces resistance R with
impedance Z, and applies to all
elements:
𝐕 = 𝐈𝐙
In this generalized form of Ohm’s
law, V, I, and Z are complex
numbers.
A Typical AC Circuit Problem


Suppose we want to find
i(t) in this circuit.
Here are the steps:
1.
2.
3.
4.
Transform to the phasor domain (i.e.,
write the voltage source in phasor form
Vs and find the resistor’s impedance ZR
and the capacitor’s impedance ZC).
Combine ZR and ZC to find the circuit’s
total impedance Zeq. (See next slide.)
Apply Ohm’s law: I = Vs ÷ Zeq.
Transform back to the time domain by
converting I to i(t).
Combining Impedances in Series


The equivalent
impedance of
series-connected
impedances is the
sum of the
individual impedances:
𝐙𝑒𝑞 = 𝐙1 + 𝐙2 + ⋯ + 𝐙𝑁
Thus, series-connected impedances
combine like series-connected
resistors.
Combining Impedances in Parallel
The equivalent impedance
of parallel-connected
impedances is given by
the reciprocal formula:
1
𝐙𝑒𝑞 =
1
1
1
+ + ⋯+
𝐙1 𝐙2
𝐙𝑁
 For two impedances in parallel we can also
use the product-over-sum formula:
𝐙1 𝐙2
𝐙𝑒𝑞 =
𝐙1 + 𝐙2
 Thus, parallel-connected impedances
combine like parallel-connected resistors.

Voltage-Divider Rule


As in dc circuits, the
voltage-divider rule
lets us find the
voltage across an
element in a series
combination if we
know the voltage across the
entire series combination.
Example: In the circuit shown,
𝐕1 =
𝐙1
𝐕
𝐙1 +𝐙2
and
𝐕2 =
𝐙2
𝐕
𝐙1 +𝐙2
Current-Divider Rule


As in dc circuits, the
current-divider rule
lets us find the
current through an
element in a parallel
combination if we
know the current through the entire
parallel combination.
Example: In the circuit shown,
𝐈1 =
𝐙2
𝐈
𝐙1 +𝐙2
and
𝐈2 =
𝐙1
𝐈
𝐙1 +𝐙2
Summary of Chapter 9

We’ve seen that we can apply these
familiar techniques to sinusoidal ac
circuits in the phasor domain:






Ohm’s law (𝐕 = 𝐈𝐙)
Kirchhoff’s laws (KVL and KCL)
Series and parallel combinations
Voltage-divider rule
Current-divider rule
In each case, we must use complex
numbers (phasors) instead of real
numbers.
Steps to Analyze AC Circuits
1.
2.
3.
Transform the circuit from the time
domain to the phasor domain.
Solve the problem using circuit
techniques (Ohm’s law, Kirchhoff’s
laws, voltage-divider rule, etc.)
Transform the resulting phasor to
the time domain.
Terminology: Impedance,
Resistance, and Reactance

Since impedance Z is a complex
number, we can write it in
rectangular form:
𝐙 = 𝑅 + 𝑗𝑋



We call the real part (R) the
resistance.
We call the imaginary part (X) the
reactance.
Impedance, resistance, and
reactance are measured in ohms.
Impedance, Resistance, and
Reactance of Single Elements
Element
Impedance Resistance
Reactance
Resistor
R
R

Inductor
jL

L
Capacitor
−𝑗
𝜔𝐶

−1
𝜔𝐶

Inductors and capacitors are called
reactive elements because they
have reactance but no resistance.
Admittance



Recall that conductance, measured in
siemens (S), is the reciprocal of
resistance:
G=1/R
The reciprocal of impedance is called
admittance, abbreviated Y:
Y=1/Z
The unit of admittance is the
siemens.
More Terminology: Admittance,
Conductance, and Susceptance

Since admittance Y is a complex
number, we can write it in
rectangular form:
𝐘 = 𝐺 + 𝑗𝐵



We call the real part (G) the
conductance.
We call the imaginary part (B) the
susceptance.
Admittance, conductance, and
susceptance are measured in
siemens.
Admittance, Conductance, and
Susceptance of Single Elements
Element
Resistor
Inductor
Capacitor
Admittance Conductance Susceptance
G
−𝑗
𝜔𝐿
jC
G



−1
𝜔𝐿
C
What’s Next?

In Chapter 10 we’ll see that we can
also apply these other familiar
techniques in the phasor domain:






Nodal analysis
Mesh analysis
Superposition
Source transformation
Thevenin’s theorem
Norton’s theorem