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2.0 COMPLEX NUMBER SYSTEM

Bakiss Hiyana bt Abu BakarJKE, POLISAS

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COURSE LEARNING OUTCOME1. Explain AC circuit concept and their analysis using AC circuit law.

2. Apply the knowledge of AC circuit in solving problem related to AC electrical circuit.

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CHAPTER CONTENT

COMPLEX NUMBER SYSTEM

Understand the complex plane

Understand real and imaginary numbers

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 PLANE2.1.1 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.

•This then produces a two dimensional complex plane with four distinct quadrants labelled, QI, QII, QIII, and QIV

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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 two-dimensional 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 Arganddiagram, 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∠θ.

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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

Where: Z = is the complex number representing the vectorx = is the Real part or the active componenty = is the Imaginary part or the reactive componentj = is define by √-1

Z = x + jy

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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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- Polar : Coordinates are in the form ( r, θ ) where r is the distance from the

origin to the point and θ is the angle measured from the positive x axis to the point.

- Polar form :

x

2.3.1 EXPRESS PHASOR QUANTITIES IN BOTH RECTANGULAR AND POLAR FORMS

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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

Therefore; Z = 5.19 + j 3

= 5.19

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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 ( j2 ) 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