2.0 COMPLEX NUMBER SYSTEM. Bakiss Hiyana bt Abu Bakar JKE, POLISAS BHAB 1

Transcription

1 2.0 COMPLEX NUMBER SYSTEM Bakiss Hiyana bt Abu Bakar JKE, POLISAS BHAB 1

2 COURSE LEARNING OUTCOME 1. 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. BHAB 2

3 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 BHAB 3

4 2.1 UNDERSTAND THE COMPLEX PLANE LABEL POSITIVE AND NEGATIVE NUMBERS BHAB 4

5 2.1 UNDERSTAND THE COMPLEX PLANE 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 BHAB 5

6 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 BHAB 6

7 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. BHAB 7

8 2.1.3 DRAW THE ANGULAR POSITION ON THE COMPLEX PLANE - Complex numbers can also be expressed in polar coordinates as r θ. BHAB 8

9 2.2 UNDERSTAND REAL AND IMAGINARY NUMBERS 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 BHAB 9

10 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 BHAB 10

11 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. BHAB 11

12 2.2.2 DETERMINE A VALUE OF J The imaginary unit, j, is defined as j = Therefore, j 2 = -1 we can notice that: j 3 = j 2 x j = -1 x j = -j j 4 = j 2 x j 2 = -1 x -1 = 1 Example: Simplify j 12 By what we saw above we can simply write j 12 = (j 4 ) 3 = ( j 2 x j 2 ) 3 = ( 1 ) 3 Therefore, j 12 = 1 BHAB 12

13 2.3 UNDERSTAND PHASOR QUANTITIES IN BOTH RECTANGULAR AND POLAR FORMS 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 BHAB 13

14 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 BHAB 14

15 2.3.1 EXPRESS PHASOR QUANTITIES IN BOTH RECTANGULAR AND POLAR FORMS - 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 BHAB 15

16 2.4 UNDERSTAND RECTANGULAR FORM AND POLAR FORMS CONVERT BETWEEN RECTANGULAR AND POLAR FORMS - Example: BHAB 16

17 TRANSFORM A POLAR TO RECTANGULAR FORM - Example: Express = 5.19 in rectangular form Therefore; Z = j 3 BHAB 17

18 2.5 UNDERSTAND ARITHMETIC OPERATIONS WITH COMPLEX NUMBER ADD: PERFORM OPERATION WITH COMPLEX NUMBER - 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 ) BHAB 18

19 Example: BHAB 19

20 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 + bdj 2 = ac + bdj 2 + ( 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 = a 2 + b 2 BHAB 20

21 Example: Given Z 1 = 2 2j, Z 2 = 3 + 4j Find Z 1.Z2 Z 1.Z 2 = ( 2 2j ).(3 + 4j ) = 6 + 8j 6j 8j 2 = 6 + 2j 8(-1) ; j 2 = -1 = j Example: Given Z = 3 2j, find if Z = 3 2j, then the conjugate is = ( 3 2j ).(3 + 2j ) = 9 + 6j 6j 4j 2 = = 13 = 3 + 2j, therefore, BHAB 21

22 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: Z 1 = 2 + 9j, Z 2 = 5 2j, find BHAB 22

23 MULTIPLICATION & DIVISION IN POLAR FORM MULTIPLICATION: DIVISION: BHAB 23

24 EXAMPLE: MULTIPLICATION & DIVISION IN POLAR FORM Multiplying together 6 30 o and 8 45 o 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. BHAB 24

25 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 multiplication of j by j gives j 2 = -1 the same as real numbers. 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. BHAB 25

26 Rectangular Form Polar Form BHAB 26

27 ADD SUBTRACT MULTIPLICATION DIVISION BHAB 27