Circuits 2_Complex Numbers.pptx

7/24/2019 Circuits 2_Complex Numbers.pptx

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Review of Complex

Numbers

Complex Numbers and its Operations

2nd 4th Week


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ObjectivesAt the end of this topic, we should be able to:

Understand that real numbers and imaginar

numbers are subsets of complex numbers!

Understand how to eliminate imaginar numbersin denominators!

Explain thej operator!

Defne a complex number!

Add, subtract, multiply, and divide complexnumbers!

Explain the di"erence between the rectangularand polar forms of a complex number!

Convert a complex number from polar torectangular form and vice versa!

Explain how to use complex numbers to solve

series and parallel ac circuits containing


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Complex Numbershe Imaginary Unit, i

#n studing e$uations, solutions sometimesinvolve the s$uare roots of negative numbers!

%o evaluate the s$uare root of a negative

number, li&e '(), we said *it is not a realnumber,+ because there is no real number suchthat x- '()!

%o include such roots in the number sstem,mathematicians created a new expanded set ofnumbers, called the complex numbers.

%he foundation of this new set of numbers is theimaginary unit i.


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Complex Numbers


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Complex Numbers


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Complex Numbers


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Complex Numbers


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Complex Numbers


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Complex Numbers


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Complex Numbers


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Complex Numbers


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Complex Numbers


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Complex Numbers


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Complex Numbers

C l b


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Complex Numbers

C l N b


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Complex Numbers


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Complex Numbers


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Complex Numbers


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Complex Numbers


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Complex Numbers

.ummar


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Complex Numbers for ACCircuits

Complex numbers refer to a numerical sstem

that includes the phase angle of a $uantit withits magnitude!

%herefore, complex numbers are useful in ac

circuits when the reactance of X/or XCma&es itnecessar to consider the phase angle!

0or instance, complex notation explains wh 12 isnegative withX

Cand 1# is negative with I

/!


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An tpe of ac circuit can be anal3ed with

complex numbers! %he are especiallconvenient for solving series'parallel circuits thathave both resistance and reactance in one ormore branches!

Although graphical analsis with phasor arrowscan be used, the method of complex numbers isprobabl the best wa to anal3e ac circuits withseries'parallel impedances!


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Complex Numbers for ACCircuitsitive and Negative Numbers

Our common use of numbers as either positiveor negative represents onl two special cases!

#n their more general form, numbers have both

$uantit and phase angle!

#n 0ig! 45(, positive and negative numbers areshown corresponding to the phase angles of 67

and (8 6, respectivel!7

l b f


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0or example, the numbers , 4, and ) representunits along the hori3ontal or x axis, extendingtoward the right along the line of 3ero phaseangle!

%herefore, positive numbers represent unitshaving the phase angle of 6, or this phase7angle corresponds to the factor of (!

C l b f C


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%o indicate ) units with 3ero phase angle, then, )is multiplied b ( as a factor for the positivenumber )!

%he sign is often omitted, as it is assumedunless indicated otherwise!

#n the opposite direction, negative numberscorrespond to (8 6, or this phase angle7

C l N b f AC


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Actuall, ') represents the same $uantit as )but rotated through the phase angle of (8 6!7

%he angle of rotation is the operator for the

number!

%he operator for '( is (87 9 the operator for6( is 761!6

C l N b f AC


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The j Operator

%he operator for a number can be an angle

between and ;)7 !76 6

.ince the angle of < 6 is important in ac7circuits, the factorj is used to indicate


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The j Operator

=ere, the number > means > units at 7 , the6

number '> is at (87 , and6 j> indicates thenumber > at the


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C l N b f AC


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#n mathematics, numbers on the hori3ontal axis

are real numbers, including positive andnegative values!

Numbers on the j axis are called imaginary

numbers because the are not on the real axis!

#n mathematics, the abbreviation i is used toindicate imaginar numbers!

#n electricit, however, j is used to avoidconfusion with i as the smbol for current!

0urthermore, there is nothing imaginar about

electrical $uantities on thej axis!

C l N b f AC


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An electric shoc& from j>77'? is just as

dangerous as >77'? positive or negative!

@ore features of the j operator are shown in 0ig!45;!

C l N b f AC


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Complex Numbers for ACCircuits %he angle of (87 corresponds to the6 j

operation of


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.ince jmeans (87 , which corresponds to the6

factor of '(, we can sa that jis the same as '(!

C l N b f AC


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#n short, the operator j for a number means tomultipl b '(!

0or instance,j8 is '8!

0urthermore, the angle of 6 is the same as7'< 6, which corresponds to the operator '7 j!

C l N b f AC


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%hese characteristics of the j operator aresummari3ed as follows:

C l N b f AC


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Denition o! a "omplex Number in #" "ircuits

%he combination of a real and an imaginar termis called a complex number!

Usuall, the real number is written Brst!

As an example, ; j4 is a complex numberincluding ; units on the real axis added to 4 units


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Complex Numbers for ACCircuitsDenition o! a "omplex Number in #" "ircuits

hasors for complex numbers shown in 0ig! 454are tpical examples!

%he j phasor is up for


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Complex Numbers for ACCircuitsDenition o! a "omplex Number in #" "ircuits

Draphicall, the sum is the hpotenuse of theright triangle formed b the two phasors!

.ince a number li&e ; j4 speciBes the phasors

in rectangular coordinates, this sstem is therectangular orm of complex numbers!

Ee careful to distinguish a number li&e j, where

is a coeFcient, from j

, where is theexponent!

%he numberj means units up on the j axis of


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Denition o! a "omplex Number in #" "ircuits

Another comparison to note is betweenj; andj;!

%he numberj; is ; units up on thej axis, andj;is

the same as the 'j operator, which is down on the'69 when thej term is smaller, the angle is lessthan 4>6!

Complex Numbers for AC


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How Complex Numbers Are Applied to AC

Circuits

Applications of complex numbers are a $uestionof using a real term for 76, j for illustrates the following rules:



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Circuits

An angle o 76 or a real number without an joperator is used for resistance R! 0or instance, ;'

H of R is stated as ;'H !

An angle o


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Circuits

%his rule alwas applies to X/, whether it is in

series or parallel with R!

%he reason is the fact that IX/represents voltage

across an inductance, which alwas leads thecurrent in the inductance b


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Circuits

%he reason is that IXC is the voltage across a

capacitor, which alwas lags the capacitorIs

charge and discharge current b '


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Circuits

#n 0ig! 45)a and b, 'j is used for inductivebranch current I/, and j is used for capacitive

branch current IC!



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Impedance in Complex Form



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Complex Numbers for ACCircuitsImpedance in Complex Form



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Complex Numbers for ACCircuitsOperations with Complex Numbers



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Complex Numbers for ACCircuitsgnitude and #ngle o! a "omplex Number



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Complex Numbers for ACCircuitsgnitude and #ngle o! a "omplex Number



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gnitude and #ngle o! a "omplex Number



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@an scientiBc calculators have &es that canconvert from rectangular coordinates to themagnitude5phase angle form Jcalled polarcoordinatesK directl!

#f our calculator does not have these &es, theproblem can be done in two separate parts:

J(Kthe magnitude as the s$uare root of the sumof two s$uares and

JKthe angle as the arctan e$ual to the j termdivided b the real term!



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Complex Numbers for ACCircuitsolar %orm o! "omplex Numbers



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Complex Numbers for ACCircuitsolar %orm o! "omplex Numbers



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lication and Division in Polar %orm o! "omplex Num



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tiplication in Polar %orm o! "omplex Numbers



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tiplication in Polar %orm o! "omplex Numbers



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ision in Polar %orm o! "omplex Numbers



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ision in Polar %orm o! "omplex Numbers



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"onverting Polar to &ectangular %orm

Complex numbers in polar form are convenientfor multiplication and division, but the cannotbe added or subtracted if their angles are

di"erent because the real and imaginar partsthat ma&e up the magnitude are di"erent!

Ghen complex numbers in polar form are to beadded or subtracted, therefore, the must be

converted into rectangular form!



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Co p e u be s o CCircuits"onverting Polar to &ectangular %orm



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pCircuits"onverting Polar to &ectangular %orm



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pCircuits"omplex Numbers in 'eries #" "ircuits

Refer to the circuit, although a circuit li&e thiswith onl series resistances and reactances canbe solved graphicall with phasor arrows, thecomplex numbers show more details of the

phase angles!



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Circuits"omplex Numbers in 'eries #" "ircuits



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Circuits"omplex Numbers in Parallel #" "ircuits



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Circuits"omplex Numbers in Parallel #" "ircuits



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Circuits"ombining T(o "omplex )ranch Impedances



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Circuits"ombining "omplex )ranch "urrents



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Circuits"ombining "omplex )ranch "urrents



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CircuitsParallel "ircuit (ith Three "omplex )ranches



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Complex Numbers for ACi i


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Complex Numbers for ACCi i


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