EEEB123 - Chapter 9 Sinusoids and Phasors

7/24/2019 EEEB123 - Chapter 9 Sinusoids and Phasors

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Alexander

-

Sadiku

Fundamentals of Electric Circuits

Chapter 9

Sinusoids and Phasors

Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Instructor : Tan Ching

Sin

Universiti

Tenaga Nasional


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Dr. Tan

Chapter 9

Sinusoids and Phasors

9.1 Overview

9.2 Sinusoids

9.3 Phasors

9.4 Phasor Relationship for Circuit Elements

9.5 Impedance and Admittance

9.6 Impedance Combination

Read Alexander & Sadiku, Chapter 9 and Appendix B

(Complex Number).


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Dr. Tan

9.1 Overview

DC Versus AC

In a direct-current (DC) circuit,current flows in one direction only.

The textbooks Chapters 1 through 8

cover DC circuits.

In an alternating-current (AC) circuit,current periodically reversesdirection.

The books Chapters 9 through 14 coverAC circuits.


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Dr. Tan

The Math Used in AC Circuits

Our study of AC circuits will relyheavily on two areas of math:

Sine and cosine functions

Complex numbers

Well review the math afterintroducing some terminology used in

discussing AC voltages and currents. Applications of phasors and frequency

domain for passive elements

Concept of impedance and admittance


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Waveforms

The graph of a current or voltage versus timeis called a waveform.

Example:

o Note that this is an AC waveform: negativevalues of voltage mean the opposite polarity(and therefore opposite direction of current

flow) from positive values.


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Periodic Waveforms

Often the graph of a voltage or currentversus time repeats itself. We call this aperiodic waveform.

Common shapes for periodic waveformsinclude: Sinusoid

Square

Triangle Sawtooth

Image from http

://en.wikipedia.org/wiki/Sinusoid

Sinusoids are the most important of these.
http://en.wikipedia.org/wiki/Sinusoidhttp://en.wikipedia.org/wiki/Sinusoidhttp://en.wikipedia.org/wiki/Sinusoid


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Cycle

In a periodic signal, each repetitionis called a cycle.

How many cycles are shown in thediagram below?


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Waveform Parameters

Important parameters associatedwith periodic waveforms include:

Period T

Frequency f

Angular Frequency

Amplitude Vm (or Peak Value Vp)

Peak-to-Peak Value Instantaneous Values


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Period

The time required for one cycle iscalled the waveforms period.

The symbol for period is T.

Period is measured in seconds,abbreviated s.

Example: If a waveform repeats

itself every 3 seconds, wed writeT= 3 s


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Frequency

A waveforms frequency is thenumber of cycles that occur in onesecond.

The symbol for frequency is f.

Frequency is measured in hertz,abbreviated Hz.

Some old-timers say cycles per secondinstead of hertz.

Example: If a signal repeats itself 20times every second, wed write

f= 20 Hz


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Period and Frequency

Period and frequency are thereciprocal of each other:

f= 1 / TT= 1 / f


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Radians

Recall that the radian (rad) is the SIunit for measuring angle.

It is related to degrees by radians = 180

Well often need to convert betweenradians and degrees:

To convert radians to degrees, multiply

by

.

To convert degrees to radians, multiply

by

.


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Angular Frequency

The quantity 2f, which appears inmany equations, is called theangular frequency.

Its symbol is , and its unit israd/s:

= 2f


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One Question, Three Answers

So we have three ways of answeringthe question, How fast is the voltage(or current) changing?

1. Period, T, unit = seconds (s) Tells how many seconds for one cycle.

2. Frequency, f, unit = hertz (Hz) Tells how many cycles per second.

3. Angular frequency, , unit = rad/s Tells size of angle covered per second.


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Relating T, f, and

If you know any one of these three(period, frequency, angularfrequency), you can easily compute

the other two. The key equations that you must

memorize are:

T= 1/f

= 2f= 2/T


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Amplitude (or Peak Value)

The maximum value reached by anac waveform is called its amplitudeor peak value.


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Peak-to-Peak Value

A waveforms peak-to-peak valueis its total height from its lowestvalue to its highest value.

Many waveforms are symmetricabout the horizontal axis. In suchcases, the peak-to-peak value is

equal to twice the amplitude.


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Instantaneous Value

The waveforms instantaneousvalue is its value at a specific time.

A waveforms instantaneous valueconstantly changes, unlike theprevious parameters (period,frequency, angular frequency,amplitude, peak-to-peak value),

which usually remain constant.


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Lead and Lag

Generally, phase shift/angle isintroduce to account for relative timingof one wave versus another.

Consider the two sinusoids below

v2 leads v1 by

1 2sin and sinm mv t V t v t V t


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Phase Angle

To quantify the idea of how far awaveform is shifted left or rightrelative to a reference point, we

assign each waveform a phaseangle .

Apositive phase angle causes the

waveform to shift leftalong the x-axis.

A negative phase angle causes it toshift right.


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9.2 Sinusoids

A sinusoid is a sine wave or a cosinewave or any wave with the same shape,shifted to the left or right.

Sinusoids arise in many areas ofengineering and science. They are thewaveform used most frequently inelectrical circuit theory.

The waveform weve been looking at is asinusoid.


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Mathematical Expression For a

Sinusoid

The mathematical expression for asinusoid looks like this:

v(t) = Vmcos(t+ )

where Vm is the amplitude, is theangular frequency, and is the

phase angle. Example:

v(t) = 20 cos(180t+ 30) V


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Calculators Radian Mode and

Degree Mode

Recall that when usingyour calculators trigbuttons (such as cos),

you must pay attentionto whether the calculator is in radianmode 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 thecosine of 90 degrees.


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Caution: Radians and Degrees

In the expression for a sinusoid,

v(t) = Vmcos(t + )

is usually given in degrees, but isalways given in radians per second.


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Type of Voltage

Source

Symbol Used in Our

Textbook

Symbol Used in

Multisim Software

Generic voltage source(may be DC or AC)

DC voltage source

AC sinusoidal voltagesource

Schematic Symbols for

Independent Voltage Sources

Several different symbols arecommonly used for voltage sources:


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Sine or Cosine?

A sinusoidal waveform can beexpressed mathematically usingeither 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 youre given a

mixture of sines and cosines, yourfirst step should be to convert all ofthe sines to cosines.


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Trigonometric Identities Relating

Sine and Cosine

You can convert from sine to cosine(or vice versa) using the trigidentities

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 thatthe cosine function leads the sinefunction by 90.


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A Graphical Method Instead of Trig

Identities

Remembering and applying trigidentities may be difficult.

The book describes a graphical

method that relies on the followingdiagram:

To use it, we measure positive anglescounterclockwise, and negative

angles clockwise.

sin(x + 90) = cos(x)

sin(x 90) = cos(x)

cos(x

+ 90) =

sin(

x

)cos (x 90) = sin(x)


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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 ifor this

number, but electrical engineers usej:

or1i 1j


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Rectangular versus Polar Form

Any complex number can be expressed in

three forms:

Rectangular form; Example: 3 +j 4

Polar or Exponential form;

Example: 5 53.1 Example: 5e j 53.1 or 5e j 0.927

z x jy

jz r re


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Rectangular Form

In rectangular form, a complex number zis

written as the sum of a real partxand an

imaginary part y:

The Complex Plane

z x jy


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Polar Form

In polar form, a complex number zis

written as a magnitude rat an angle :

z = r

The angle ismeasured from the

positive real axis.


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Given a complex number zwith real partx

and imaginary part y, its magnitude is

given by

and its angle is given by

Converting from Rectangular Form

to Polar Form

22yxr

x

y1tan


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Dr. Tan

Inverse Tangent Button on Your

Calculator

When using your calculators

tan1 (inverse tangent) button,

pay attention to whether the

calculator is in degree mode or radian mode.

Also recall that the calculators answer may

be in the wrong quadrant, and that you may

need to adjust the answer by 180.

The tan1 button always returns an angle in

Quadrants I or IV, even if you want an answer in

Quadrants II or III.


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Given a complex number zwith

magnitude rand angle , its real part is

given by

and its imaginary part isgiven by

Converting from Polar Form to

Rectangular Form

cosrx

sinry


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Complex numbers may also be written in

exponential form. Think of this as a

mathematically respectable version of polarform.

In exponential form, should be in radians.

Example: 330 3ej/6

Exponential Form

Polar form Exponential Form

r

rej


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Mathematical Operations

You must be able to perform the following

operations on complex numbers:

Addition Subtraction

Multiplication

Division

Complex Conjugate


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Addition

Adding complex numbers is easiest if thenumbers are in rectangular form.

Suppose z1

=x1+jy

1and z

2=x

2+jy

2

Then z1 + z2 = (x1+x2) +j(y1+y2)

In words: to add two complex numbers inrectangular form, add their real parts to getthe real part of the sum, and add theirimaginary parts to get the imaginary part of

the sum.


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Subtraction

Subtracting complex numbers is also easiestif the numbers are in rectangular form.

Suppose z1

=x1+jy

1and z

2=x

2+jy

2

Then z1 z2 = (x1x2) +j(y1y2)

In words: to subtract two complex numbersin rectangular form, subtract their real partsto get the real part of the result, and subtracttheir imaginary parts to get the imaginary

part of the result.


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Multiplication

Multiplying complex numbers is easiest if the

numbers are in polar form.

Suppose z1 = r1 1 and z2 = r2 2

Then z1 z2 = (r1r2) (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.


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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.


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Complex Conjugate

Given a complex number in rectangular

form,

z = x + jyits complex conjugate is simply

z* = x jy

Given a complex number in polar form,

z = rits complex conjugate is simply

z* = r


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Dr. Tan

Scientific Calculator

Casio Fx-991ms/Fx-570msExample : (4+3i) (5 -20)=?1. Set your calculator and display it in CMPLX

MODE , and Deg Mode

2. Then press key a number of times until

you reach the setup screen press

3. Toggle to select r setting (i.e., all solutionswill be presented in polar)

Disp r = 25

Disp =16.87

Disp x = 23.924

Disp y =7.255i

To get convert polar form Rec Form


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Performing Complicated Operations

on Complex Numbers

Solving a problem may require us to

perform many operations on complex

numbers.

Example:3+3

+4

Using Scientific Calculator, you can do this

quickly and easily. With normal calculatorits more tedious, since you must

repeatedly convert between rectangular

and polar forms.

Another option is to use MATLAB.


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Useful Properties ofj

jis the only number whose reciprocal is

equal to its negation:

Therefore, for example,

Also, =

Therefore multiplication byjis equivalent to a

counterclockwise rotation of 90 in the complex

plane.

j

j

1

C

j

Cj

1


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Kirchhoffs Laws in AC Circuits

KCL and KVL hold in AC circuits. But to apply these laws, we must add

(or subtract) sinusoids instead ofadding (or subtracting) numbers.

Example: In thecircuit shown,KVL tells us that

v= v1 + v2.But suppose

v1 = 10 cos(200t+ 30) V andv2 = 12 cos(200t+ 45) V

How can we add those to find v?


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Adding Sinusoids (Continued)

For example, if we add

v1 = 10 cos(200t+ 30) V andv2 = 12 cos(200t+ 45) V

well get another sinusoid of the sameangular frequency, 200 rad/s:

v1 + v2 = Vm cos(200t+ ) V

But how do we figure out the resultingsinusoids amplitude Vm and phaseangle ?


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Complex Numbers to the Rescue!

One method for adding sinusoidsrelies on trig identities.

But well use a simpler method, whichrelies on complex numbers. In fact, the main reason were interested

in complex numbers (in this course) isthat they give us a simple way to addsinusoids.


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9.3 Phasors

A phasor is a complex number thatrepresents the amplitude and phaseangle of a sinusoidal voltage or

current. The phasors magnitude ris equal

to the sinusoids amplitude.

The phasors angle is equal to the

sinusoids phase angle. Example: We use the phasor

V =

30 V to represent the sinusoid

v(t) = 10 cos(200t+ 30) V.


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Time Domain and Phasor Domain

Some fancy terms:

We call an expression like10 cos(200t+ 30) V the time-

domain representation of asinusoid.

We call

30 V the phasor-

domain representation of thesame sinusoid. (Its also called thefrequency-domain representation.)


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Dr. Tan

Using Phasors to Add Sinusoids

To add sinusoids of the samefrequency:

1. If any of your sinusoids are

expressed using sine, convert themall to cosine.

2. Write the phasor-domain version ofeach sinusoid.

3. Add the phasors (which are justcomplex numbers).

4. Write the time-domain version of

the resulting phasor.

E l f U i Ph t Add


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Example of Using Phasors to Add

Sinusoids

v1 = 10 cos(200t+ 30) V andv2 = 12 cos(200t+ 45) V:

Transform from time domain to phasordomain:V1 =

30 V and V2 =

45 V .

Add the phasors:

30 V +

45 V =.

38.2 V

Transform from phasor domain backto time domain:v1 + v2 = 21.8 cos(200t+ 38.2) V

9 4 Ph R l ti hi f


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9.4 Phasor Relationships for

Circuit Elements

Weve seen how we can use phasorsto add sinusoids.

Next well revisit the voltage-current relationships for resistors,inductors, and capacitors, assumingthat their voltages and current are

sinusoids.


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Phasor Relationship for Resistors

For resistors we have, in the timedomain: 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 phasordomain we have:

If I =

30 A and R = 5 , then

V =

30 V

So we can write V = IR.


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Summary for Resistors

In the time domain:

In the phasor domain:


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Phasor Relationship for Inductors

For inductors we have, in the time

domain: =

Example:If i= 2 cos(200t+ 30) A and L = 5 H,then v= 2000 cos(200t+ 120) V


If I =

30 A and L = 5 H, then

V =

120 V

So we can write V =jLI.


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Summary for Inductors

In the time domain:



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Phasor Relationship for Capacitors

For capacitors we have, in the time

domain: =

Example:If v= 2 cos(200t+ 30) V and C= 5 F,then i= 2000 cos(200t+ 120) A


If V =

30 V and C= 5 F, then

I =

120 A

So we can write I =jCV.


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Summary for Capacitors

In the time domain:



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Summary: Textbooks Table 9.2

9 5 Impedance and Admittance


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9.5 Impedance and Admittance

Impedance

The impedance Z of an element or acircuit is the ratio of its phasorvoltage V to its phasor current I:

=

Impedance is measured in ohms.

Like resistance, impedance

represents opposition to current: fora fixed voltage, greater impedanceresults in less current.


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A Resistors Impedance

For resistors, V = IR, so a resistorsimpedance is:

=

=

So a resistors impedance is a purereal number (no imaginary part), andis simply equal to its resistance.

To emphasize this, we could write = 0

or = 0


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Resistors and Frequency

A resistors impedance does notdepend on frequency, since Z=R fora resistor.

Therefore, a resistor doesnt opposehigh-frequency current any more orless than it opposes low-frequencycurrent.


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An Inductors Impedance

For inductors, V =jLI, so aninductors impedance is:

=

=

So an inductors impedance is a pureimaginary number (no real part).

To emphasize this, we could write

= 0 or

= 90


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Inductors and Frequency

The magnitude of an inductorsimpedance is directly proportionalto frequency, since Z=jL for an

inductor. As , Z, inductors act like

open circuit.

Also, as 0, Z0, which is why

inductors act like short circuits in dccircuits.


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A Capacitors Impedance

For capacitors, I =jCV, so aninductors impedance is:

=

=

1

=

So a capacitors impedance is a pure

imaginary number (no real part).

To emphasize this, we could write

= 0

or

=

1

9 0


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Capacitors and Frequency

The magnitude of a capacitorsimpedance is inversely proportional

to frequency, since =

for a

capacitor. As , Z0, inductors act like

short circuit.

Also, as 0, Z, which is whycapacitors act like open circuits in dccircuits.

Impedance Resistance and


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Impedance, Resistance, and

Reactance

Since impedance Z is a complexnumber, we can write it inrectangular form:

= We call the real part (R) the

resistance.

We call the imaginary part (X) the

reactance. Impedance, resistance, and

reactance are measured in ohms.


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Admittance

Conductance, measured in siemens(S), is the reciprocal of resistance:

G = 1 / R

The reciprocal of impedance is calledadmittance, abbreviated Y:

Y = 1 / Z

The unit of admittance is thesiemens.

Admittance Conductance and


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Admittance, Conductance, and

Susceptance

Since admittance Y is a complexnumber, we can write it inrectangular form:

= We call the real part (G) the

conductance.

We call the imaginary part (B) the

susceptance. Admittance, conductance, and

susceptance are measured insiemens.

9 6 Impedance Combinations


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9.6 Impedance Combinations

Combining Impedances in Series

The equivalentimpedance ofseries-connected

impedances is thesum of theindividual impedances:

=

Thus, series-connected impedancescombine like series-connected

resistors.


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Combining Impedances in Parallel

The equivalent impedanceof parallel-connectedimpedances is given bythe reciprocal formula:

= 11

1

1

For two impedances in parallel we can alsouse the product-over-sum formula:

=

Thus, parallel-connected impedancescombine like parallel-connected resistors.


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Voltage-Divider Rule

As in dc circuits, thevoltage-divider rulelets us find the

voltage across anelement in a seriescombination if weknow the voltage across theentire series combination.

Example: In the circuit shown,

=

+ and =

+


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Current-Divider Rule

As in dc circuits, thecurrent-divider rulelets us find the

current through anelement in a parallelcombination if weknow the current through the entireparallel combination.

Example: In the circuit shown,

=

+ and =

+


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Impedance Transformation

The Delta-Wye transformation is:

1

2

3

b c

a b c

c a

a b c

a b

a b c

Z ZZ

Z Z Z

Z ZZ

Z Z Z

Z ZZ

Z Z Z

1 2 2 3 3 1

1

1 2 2 3 3 1

2

1 2 2 3 3 1

3

a

b

c

Z Z Z Z Z ZZ

Z

Z Z Z Z Z ZZ

Z

Z Z Z Z Z ZZ

Z

Delta-Wye Wye-Delta


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Summary of Chapter 9

Weve seen that we can apply thesefamiliar techniques to sinusoidal accircuits in the phasor domain:

Ohms law ( = )

Kirchhoffs laws (KVL and KCL)

Series and parallel combinations

Voltage-divider rule

Current-divider rule Impedance Transformation

In each case, we must use complexnumbers (phasors) instead of real

numbers.


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Steps to Analyze AC Circuits

1. Transform the circuit from the timedomain to the phasor domain.

2. Solve the problem using circuit

techniques (Ohms law, Kirchhoffslaws, voltage-divider rule, etc.)

3. Transform the resulting phasor tothe time domain.


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Whats Next?

In Chapter 10 well see that we canalso apply these other familiartechniques in the phasor domain:

Nodal analysis Mesh analysis

Superposition

Source transformation

Thevenins theorem

Nortons theorem


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Thank You

Q & A

Documents

EEEB123 - Chapter 9 Sinusoids and Phasors