Post on 21-Feb-2018
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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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Complex Numbers for ACCircuits
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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Complex Numbers for ACCircuitsitive and Negative Numbers
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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Complex Numbers for ACCircuitsitive and Negative Numbers
%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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Complex Numbers for ACCircuitsitive and Negative Numbers
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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Complex Numbers for ACCircuits
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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Complex Numbers for ACCircuits
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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Complex Numbers for ACCircuits
#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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Complex Numbers for ACCircuits
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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Complex Numbers for ACCircuits
.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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Complex Numbers for ACCircuits
#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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Complex Numbers for ACCircuits
%hese characteristics of the j operator aresummari3ed as follows:
C l N b f AC
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Complex Numbers for ACCircuits
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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Complex Numbers for ACCircuits
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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Complex Numbers for ACCircuits
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:
Complex Numbers for AC
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Complex Numbers for ACCircuits
How Complex Numbers Are Applied to AC
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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Complex Numbers for ACCircuits
How Complex Numbers Are Applied to AC
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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Complex Numbers for ACCircuits
How Complex Numbers Are Applied to AC
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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Complex Numbers for ACCircuits
How Complex Numbers Are Applied to AC
Circuits
#n 0ig! 45)a and b, 'j is used for inductivebranch current I/, and j is used for capacitive
branch current IC!
Complex Numbers for AC
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Complex Numbers for ACCircuits
Impedance in Complex Form
Complex Numbers for AC
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Complex Numbers for ACCircuits
Impedance in Complex Form
Complex Numbers for AC
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Complex Numbers for ACCircuits
Impedance in Complex Form
Complex Numbers for AC
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Complex Numbers for ACCircuits
Impedance in Complex Form
Complex Numbers for AC
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Complex Numbers for ACCircuitsImpedance in Complex Form
Complex Numbers for AC
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Complex Numbers for ACCircuitsOperations with Complex Numbers
Complex Numbers for AC
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Complex Numbers for ACCircuitsOperations with Complex Numbers
Complex Numbers for AC
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Complex Numbers for ACCircuitsOperations with Complex Numbers
Complex Numbers for AC
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Complex Numbers for ACCircuitsOperations with Complex Numbers
Complex Numbers for AC
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Complex Numbers for ACCircuitsOperations with Complex Numbers
Complex Numbers for AC
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Complex Numbers for ACCircuitsOperations with Complex Numbers
Complex Numbers for AC
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Complex Numbers for ACCircuitsOperations with Complex Numbers
Complex Numbers for AC
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Complex Numbers for ACCircuitsOperations with Complex Numbers
Complex Numbers for AC
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Complex Numbers for ACCircuitsOperations with Complex Numbers
Complex Numbers for AC
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Complex Numbers for ACCircuitsgnitude and #ngle o! a "omplex Number
Complex Numbers for AC
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Complex Numbers for ACCircuitsgnitude and #ngle o! a "omplex Number
Complex Numbers for AC
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Complex Numbers for ACCircuits
gnitude and #ngle o! a "omplex Number
Complex Numbers for AC
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Complex Numbers for ACCircuits
gnitude and #ngle o! a "omplex Number
Complex Numbers for AC
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Complex Numbers for ACCircuits
gnitude and #ngle o! a "omplex Number
Complex Numbers for AC
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Complex Numbers for ACCircuits
gnitude and #ngle o! a "omplex Number
@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!
Complex Numbers for AC
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Complex Numbers for ACCircuitsolar %orm o! "omplex Numbers
Complex Numbers for AC
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Complex Numbers for ACCircuitsolar %orm o! "omplex Numbers
Complex Numbers for AC
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Complex Numbers for ACCircuits
lication and Division in Polar %orm o! "omplex Num
Complex Numbers for AC
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Complex Numbers for ACCircuits
tiplication in Polar %orm o! "omplex Numbers
Complex Numbers for AC
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Complex Numbers for ACCircuits
tiplication in Polar %orm o! "omplex Numbers
Complex Numbers for AC
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Complex Numbers for ACCircuits
ision in Polar %orm o! "omplex Numbers
Complex Numbers for AC
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Complex Numbers for ACCircuits
ision in Polar %orm o! "omplex Numbers
Complex Numbers for AC
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Complex Numbers for ACCircuits
"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!
Complex Numbers for AC
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Co p e u be s o CCircuits"onverting Polar to &ectangular %orm
Complex Numbers for AC
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pCircuits"onverting Polar to &ectangular %orm
Complex Numbers for AC
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pCircuits"onverting Polar to &ectangular %orm
Complex Numbers for AC
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pCircuits"onverting Polar to &ectangular %orm
Complex Numbers for AC
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pCircuits"onverting Polar to &ectangular %orm
Complex Numbers for AC
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pCircuits"onverting Polar to &ectangular %orm
Complex Numbers for AC
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pCircuits"onverting Polar to &ectangular %orm
Complex Numbers for AC
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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!
Complex Numbers for AC
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pCircuits"omplex Numbers in 'eries #" "ircuits
Complex Numbers for AC
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pCircuits"omplex Numbers in 'eries #" "ircuits
Complex Numbers for AC
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pCircuits"omplex Numbers in 'eries #" "ircuits
Complex Numbers for AC
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pCircuits"omplex Numbers in 'eries #" "ircuits
Complex Numbers for AC
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pCircuits"omplex Numbers in 'eries #" "ircuits
Complex Numbers for AC
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Circuits"omplex Numbers in 'eries #" "ircuits
Complex Numbers for AC
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Circuits"omplex Numbers in 'eries #" "ircuits
Complex Numbers for AC
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Circuits"omplex Numbers in 'eries #" "ircuits
Complex Numbers for AC
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Circuits"omplex Numbers in 'eries #" "ircuits
Complex Numbers for AC
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Circuits"omplex Numbers in 'eries #" "ircuits
Complex Numbers for AC
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Circuits"omplex Numbers in Parallel #" "ircuits
Complex Numbers for AC
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Circuits"omplex Numbers in Parallel #" "ircuits
Complex Numbers for AC
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Circuits"ombining T(o "omplex )ranch Impedances
Complex Numbers for AC
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Circuits"ombining T(o "omplex )ranch Impedances
Complex Numbers for AC
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Circuits"ombining T(o "omplex )ranch Impedances
Complex Numbers for AC
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Circuits"ombining T(o "omplex )ranch Impedances
Complex Numbers for AC
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Circuits"ombining "omplex )ranch "urrents
Complex Numbers for AC
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Circuits"ombining "omplex )ranch "urrents
Complex Numbers for AC
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CircuitsParallel "ircuit (ith Three "omplex )ranches
Complex Numbers for AC
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CircuitsParallel "ircuit (ith Three "omplex )ranches
Complex Numbers for ACi i
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Complex Numbers for ACCi i
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Complex Numbers for ACCi i
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Complex Numbers for ACCi i
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CircuitsParallel "ircuit (ith Three "omplex )ranches