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Coordinate System and
Complex Number
Course Material
Gyeongsang National University
Dept. of Information & Communication Engineering
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Cartesian Coordinate System – 2 Dimension
• Cartesian Coordinate System
– Coordinates are represented by x-axis and y-axis – P(x-axis, y-axis)
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Cartesian Coordinate System – 2 Dimension
• Cartesian Coordinate System
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Cartesian Coordinate System – 3 Dimension
• Cartesian Coordinate System – 3 dimension
– Coordinates are represented by x-axis, y-axis and z-axis – P(x-axis, y-axis, z-axis)
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• Cartesian Coordinate System – 3 dimension
– Coordinates of the points A, B, and C?
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Polar Coordinates
• Polar coordinates
– Coordinates are represented by length r and angle θ – P(x, y) is represented by P(r, θ)? Vice versa?
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Polar Coordinates
• Polar coordinates
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Polar Coordinates
• Polar coordinates curve
– Line from the origin – y=mx -> θ=θc – Circle from the origin – r=rc, 0°≤θ≤360°
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Polar Coordinates
• Example
– Describe r=3, 0°≤θ≤180°.
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Cylindrical Polar Coordinate System
• Cylindrical polar coordinate system
– Coordinates are represented by length r, angle θ and z-axis – P(r, θ, z-axis)
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Cylindrical Polar Coordinate System
• Example
– Describe 1≤r≤2, θ=60°, 0≤z≤1.
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Spherical Polar Coordinates
• Spherical polar coordinates
– Coordinates are represented by length R, angle θ and angle φ – P(x, y, z) is represented by P(R, θ, φ)? Vice versa?
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Spherical Polar Coordinates
• Radiation pattern of half-wave dipole antenna
– Electric field – 𝑅 = 𝐾 cos
𝜋
2 cos 𝜑
sin 𝜑
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Complex Number
• Complex numbers
– Imaginary
– Complex number
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Argand Diagram
• Argand diagram
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Argand Diagram
• Argand diagram
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Operations of Complex Number
• Addition and subtraction
– For 𝑧1 = 3 − 4𝑗, 𝑧2 = 4 + 2𝑗, find 𝑧1 + 𝑧2 and 𝑧1−𝑧2.
• Multiplication
– For 𝑧1 = 2 − 2𝑗, 𝑧2 = 3 + 4𝑗, find 𝑧1𝑧2.
– Complex conjugate
• Complex conjugate of 𝑧 = 𝑎 + 𝑏𝑗 is ҧ𝑧 = 𝑎 − 𝑏𝑗
• Multiplication of complex number and its complex conjugate is always real number.
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Operations of Complex Number
• Division
– Using complex conjugate
– For 𝑧1 = 2 + 9𝑗, 𝑧2 = 5 − 2𝑗, find 𝑧1
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Operations of Complex Number
• Multiply j to complex number
– Complex number rotates 𝜋
2 radian counterclockwise
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Polar Form of Complex Number
• Polar form
– Represent 𝑧 = 𝑎 + 𝑏𝑗 by polar form 𝑟 and 𝜃 – Polar form of 𝑧?
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Polar Form of Complex Number
• Euler’s formula
– Alternatives of Euler’s formula
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de Moivre’s Theorem
• de Moivre’s Theorem
– cos 𝜃 + 𝑗sin𝜃 𝑛 = cos(𝑛𝜃) + 𝑗sin(𝑛𝜃)
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de Moivre’s Theorem
• de Moivre’s Theorem
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