2.
0 COMPLEX
NUMBER SYSTEM
Bakiss Hiyana bt Abu Bakar
JKE, POLISAS
BHAB
COURSE LEARNING OUTCOME
1.
2.
Explain AC circuit concept and their analysis using AC circuit law.
Apply the knowledge of AC circuit in solving problem related to
AC electrical circuit.
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CHAPTER CONTENT
Understand the complex plane
Understand real and imaginary
numbers
COMPLEX NUMBER SYSTEM
Understand phasor quantities in both
rectangular and polar forms
Understand rectangular form and polar
form
Understand arithmetic operations with
complex numbers
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2.1 UNDERSTAND THE COMPLEX PLANE
2.1.1 LABEL POSITIVE AND NEGATIVE NUMBERS
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2.1 UNDERSTAND THE COMPLEX PLANE
2.1.1 LABEL POSITIVE AND NEGATIVE NUMBERS
On the Argand diagram, the
horizontal axis represents all
positive real numbers to the right
of the vertical imaginary axis and
all negative real numbers to the
left of the vertical imaginary axis.
All positive imaginary numbers
are represented above the
horizontal axis while all the
negative imaginary numbers are
below the horizontal real axis.
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This then produces a two
dimensional complex plane with
four distinct quadrants labelled, QI,
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QII, QIII, and QIV
2.1.2 CONSTRUCT A COMPLEX PLANE
- A two-dimensional graph where the horizontal axis maps is the real part
and the vertical axis maps is the imaginary part of any complex number
or function.
- The complex plane is sometimes called the Argand plane because it is
used in Argand diagrams.
- A complex number can be viewed as a point or position vector in a twodimensional Cartesian coordinate system called the complex plane or
Argand diagram
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2.1.3 DRAW THE ANGULAR POSITION ON THE COMPLEX PLANE
- A complex number can be visually represented as a pair of
numbers (a,b) forming a vector on a diagram called an Argand
diagram, representing the complex plane.
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2.1.3 DRAW THE ANGULAR POSITION ON THE COMPLEX PLANE
- Complex numbers can also be expressed in polar coordinates
as r.
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2.2 UNDERSTAND REAL AND IMAGINARY
NUMBERS
2.2.1 DEFINE A REAL NUMBER AND IMAGINARY NUMBER
- A complex number has a real part & imaginary part.
- Standard form is:
a + bj
Real part
Imaginary part
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2.2.1 DEFINE A REAL NUMBER AND IMAGINARY NUMBER
Z = x + jy
Where:
Z = is the complex number representing the vector
x = is the Real part or the active component
y = is the Imaginary part or the reactive component
j = is define by -1
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2.2.1 DEFINE A REAL NUMBER AND IMAGINARY NUMBER
A complex number is an expression in the form: a + bj where
a and b are real numbers.
The symbol j is defined as j = -1 : j is the imaginary unit.
a is the real part of the complex number, and b is the complex
part of the complex number.
Then a + bj is called complex number.
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2.2.2 DETERMINE A VALUE OF J
The imaginary unit, j, is defined as j =
Therefore, j2 = -1
we can notice that: j3 = j2 x j = -1 x j = -j
j4 = j2 x j2 = -1 x -1 = 1
Example: Simplify j12
By what we saw above we can simply write j12 = (j4)3
= ( j2 x j2 ) 3
= ( 1 )3
Therefore, j12 = 1
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2.3 UNDERSTAND PHASOR QUANTITIES IN
BOTH RECTANGULAR AND POLAR FORMS
2.3.1 EXPRESS PHASOR QUANTITIES IN BOTH RECTANGULAR
AND POLAR FORMS
- THEORY: Rectangular coordinates & polar coordinates are 2 different
ways of using 2 numbers to locate a point on a plane.
- Rectangular: coordinates are in the form (x,y), where x and y are the
horizontal & vertical distances from the origin.
- Rectangular form : Z = a + bj
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2.3.1 EXPRESS PHASOR QUANTITIES IN BOTH RECTANGULAR
AND POLAR FORMS
- Rectangular form : Z = a + bj
- Example: identify 1 + 2j and 3 - j graphically
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2.3.1 EXPRESS PHASOR QUANTITIES IN BOTH RECTANGULAR
AND POLAR FORMS
- Polar : Coordinates are in the form ( r, ) where r is the distance from the
origin to the point and is the angle measured from the positive x axis to the point.
- Polar form :
x
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2.4 UNDERSTAND RECTANGULAR FORM AND
POLAR FORMS
2.4.1 CONVERT BETWEEN RECTANGULAR AND POLAR FORMS
- Example:
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TRANSFORM A POLAR TO RECTANGULAR FORM
- Example:
Express
in rectangular form
= 5.19
Therefore; Z = 5.19 + j 3
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2.5 UNDERSTAND ARITHMETIC OPERATIONS
WITH COMPLEX NUMBER
2.5.1 PERFORM OPERATION WITH COMPLEX NUMBER
ADD:
- Complex numbers are added by adding the real and imaginary parts of
the summands. That is to say:
( a + jb ) + ( c + jd ) = ( a + c ) + j( b + d )
SUBTRACT:
- To subtract the complex number from another, we subtract the
corresponding real parts & subtract the corresponding imaginary part.
- Subtraction is defined by;
( a + jb ) ( c + jd ) = ( a c ) + j( b d )
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Example:
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MULTIPLICATION:
- The multiplication of two complex numbers is defined by the following
formula:
( a + bj )( c + dj ) = ( ac bd ) + ( bc + ad )j
- The preceding definition of multiplication of general complex numbers
is the natural way of extending this fundamental property of the
imaginary unit. Indeed, treating j as a variable, the formula follows
from this
( a + bj )( c + dj ) = ac + bcj + adj + bdj2
= ac + bdj2 + ( bc + ad )j
= ac + bd (-1) + ( bc + ad )j
= ( ac bd ) + ( bc + ad )j
- In particular, the square of the imaginary ( j 2 ) unit is 1
- Whenever we multiply a complex number by it conjugate, the answer
is a real number
- If z = a + bj, then z = a2 + b2
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Example:
Given Z1 = 2 2j, Z2 = 3 + 4j
Find Z1.Z2
Z1.Z2 = ( 2 2j ).(3 + 4j )
= 6 + 8j 6j 8j2
= 6 + 2j 8(-1)
; j2 = -1
= 14 + 2j
Example:
Given Z = 3 2j, find
if Z = 3 2j, then the conjugate is
= 3 + 2j, therefore,
= ( 3 2j ).(3 + 2j )
= 9 + 6j 6j 4j2
=9+4
= 13
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DIVISION:
- To divide 2 complex number, it is necessary to make use of the
complex conjugate.
- We multiply both the numerator & denominator by the conjugate of
the denominator & then simplify the result.
- Example: Z1 = 2 + 9j, Z2 = 5 2j, find
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MULTIPLICATION & DIVISION IN POLAR FORM
MULTIPLICATION:
DIVISION:
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MULTIPLICATION & DIVISION IN POLAR FORM
EXAMPLE:
Multiplying together 6 30o and 8 45o in polar form gives us.
Likewise, to divide together two vectors in polar form, we must divide
the two modulus and then subtract their angles as shown.
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SUMMARY
Complex number consist of two distinct numbers, a real number
an imaginary number.
Imaginary number are distinguish from a real number by use of
the j operator.
A number with letter j in front of it identities is an imaginary
number in the complex plane.
By definition, the j operator = -1.
Imaginary number can be +, -, and the same as real numbers.
The multiplication of j by j gives j2 = -1
In rectangular form a complex number is represented by a point
in space on the complex plane.
In polar form a complex number is represented by a line whose
length is the amplitude and by the phase angle.
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Rectangular
Form
Polar Form
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ADD
SUBTRACT
MULTIPLICATION
DIVISION
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