ECE 202 Lesson 5 - the positive and negative half-cycles...

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Circuit Theory II Lesson 5 Objective: To introduce the Fourier series and to present the various types of symmetry in periodic signals. Sections 16.1 to 16.3, pp. 758-767
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2 Fourier Series ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 + + + = = = = = = + + = ± = T t t k T t t k T t t n n n n dt t k t f T b dt t k t f T a dt t f T a T t n b t n a a t f nT t f t f 0 0 0 0 0 0 0 0 0 1 0 1 0 sin 2 cos 2 1 2 sin cos ϖ ϖ π ϖ ϖ ϖ υ υ (16.1) (16.3) (16.4) (16.5) (16.2)
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3 Types of Symmetry Even-Function Symmetry ( 29 ( 29 t f t f - = Odd-Function Symmetry ( 29 ( 29 t f t f - - = No sine components present in Fourier Series No cosine components present in Fourier Series
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4 Types of Symmetry Half-Wave Symmetry ( 29 ( 29 2 T t f t f - - = When the function is multiplied by –1 and shifted one-half period, the same function is obtained. No d.c. term and no even-harmonics present in Fourier Series
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5 Types of Symmetry Quarter-Wave Symmetry 1. The function has half-wave Symmetry 2. There is symmetry about the midpoint of
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Unformatted text preview: the positive and negative half-cycles Need only integrate only over a quarter period 6 Triangle Wave Even symmetry Odd symmetry Half-wave symmetry Quarter-wave symmetry yes no + + +- 2 2 2 5 5 cos 3 3 cos 1 cos 4 2 x x x π Triangle Wave 7 ( 29 2 f x π-8 Right Triangular Wave Even symmetry Odd symmetry Half-wave symmetry Quarter-wave symmetry yes no -+- 3 3 sin 2 2 sin 1 sin 2 x x x Right Triangular Wave 9 ( 29 2 f x T--10 Saw Tooth Wave Even symmetry Odd symmetry Half-wave symmetry Quarter-wave symmetry yes no + + +- 3 3 sin 2 2 sin 1 sin 2 x x x π Sawtooth Wave 11 ( 29 2 f x T--...
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