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Introduction to Laplace Transform and Complex Numbers

This lecture introduces the Laplace transform and its applications in solving linear, time-invariant differential equations. It also covers complex numbers and their operations.

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Introduction to Laplace Transform and Complex Numbers

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  1. 動態系統建模分析與控制 Lecture2The Laplace Transform Instructor: Chen-Hsiung Yang Dynamic System Modeling Analysis and Control

  2. Outline 2-1 Introduction 2-2 Complex numbers, complex variables, and complex functions 2-3 Laplace transformation 2-4 Inverse laplace transformation 2-5 Solving linear, time-invariant differential equations

  3. 2-1 Introduction • Section 2-2 reviews complex numbers • Section 2-3 defines the Laplace transformation and gives Laplace transforms of several common functions of time. Also examined are some of the most important Laplace transform theorems that apply to linear system analysis. • Section 2-4 deals with the inverse Laplace transformation . • Section 2-5 presents the Laplace transform approach to the linear, time-invariant differential equation.

  4. 2-2 Complex Numbers, Complex Variables, and Complex Functions • Complex number z = x + jy • In converting complex numbers to polar form rectangular • To convert complex numbers to rectangular form from polar • Complex number Complex conjugate

  5. 2-2 Complex Numbers, Complex Variables, and Complex Functions(Cont.) • Euler’s theorem →

  6. 2-2 Complex Numbers, Complex Variables, and Complex Functions(Cont.) • Complex algebra • Equality of complex numbers • Addition • Subtraction • Power and roots • Comments： &

  7. 2-2 Complex Numbers, Complex Variables, and Complex Functions(Cont.) • Complex algebra • Equality of complex numbers • Multiplication • ∵ Counterclockwise rotation by 90 ° ∴ • ∵ ∴ • In polar form ∵ if ∴ ∵ if ∴

  8. 2-2 Complex Numbers, Complex Variables, and Complex Functions(Cont.) • Complex algebra • Equality of complex numbers • Division • ∵ ∴ • ∵ ∴ • Counterclockwise rotation by 90 ° • ∴ or

  9. 2-3 Laplace Transformation • Laplace Transformation • Define • Laplace transform： • Inverse laplace transform ：

  10. 2-3 Laplace Transformation(Cont.) • Laplace Transformation-Example • Exponential function • Step function • Unit Step function

  11. 2-3 Laplace Transformation(Cont.) • Laplace Transformation-Example • Ramp function • Sinusoidal function • <note>

  12. 2-3 Laplace Transformation(Cont.) • Laplace Transformation-Example • Pulse function • Impulse function

  13. 2-3 Laplace Transformation(Cont.) • Laplace Transformation-Theorem • Differentiation theorem • Integration theorem

  14. 2-3 Laplace Transformation(Cont.) • Laplace Transformation-Theorem • Final-value theorem • Initial-value theorem

  15. 2-4 Inverse Laplace Transformation • Partial-fraction expansion • F(s) involves distinct poles only

  16. 2-4 Inverse Laplace Transformation (Cont.) • Partial-fraction expansion • F(s) involves distinct poles only <Note>

  17. 2-4 Inverse Laplace Transformation (Cont.) • Laplace Transformation • F(s) involves multiple poles

  18. 2-5 Solving linear, time-invariant differential equations

  19. 2-5 Solving linear, time-invariant differential equations(Cont.)

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