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Unformatted text preview: EE 435 Lecture 29 Spectral Performance – Tool Use and Validation Spectral Analysis T 2 π ω = ( ) ∑ ∞ = + + = 1 k k k θ t k ω sin A A f(t) alternately ( ) ( ) ∑ ∑ ∞ = ∞ = + + = 1 k k 1 k k t k ω cos b t k ω sin a A f(t) 2 k 2 k k b a A + = If f(t) is periodic Termed the Fourier Series Representation of f(t) .• • • • • Review from last lecture .• • • • • Distortion Analysis Total Harmonic Distortion, THD l fundamenta of voltage RMS harmonics in voltage RMS THD = 2 A ... 2 A 2 A 2 A THD 1 2 4 2 3 2 2 + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 1 2 k 2 k A A THD ∑ ∞ = = .• • • • • Review from last lecture .• • • • • Distortion Analysis • Often noise is present at other nonharmonic frequencies • At higher frequencies the harmonics are often buried in the noise k A .• • • • • Review from last lecture .• • • • • Distortion Analysis Theorem: In a fully differential symmetric circuit, all even harmonics are absent in the differential output for symmetric differential excitations ! Proof: ( ) ∑ ∞ = = = k k ID k ID OD V h V f V Expanding in a Taylor’s series around V ID =0, we obtain Assume V ID =Ksin( ω t) W.L.O.G. assume K=1 V ID V OD + + V 1 V 2 ( ) [ ] ∑ ∞ = = k k k O1 t ω sin h V ( ) [ ] ∑ ∞ = = k k k O2 t ω sin h V ( ) [ ] ( ) [ ] ( ) ( ) [ ] ( ) ( ) [ ] ( ) ∑ ∑ ∞ = ∞ = − − = − = − = k k k k k k k k k O2 O1 OD t ω sin 1 t ω sin h t ω sin t ω sin h V V V Observe the evenordered harmonics are absent in this last sum .• • • • • Review from last lecture .• • • • • Distortion Analysis ( ) 1 h m 1 mN Χ N 2 A P m ≤ ≤ + = ( ) k Χ = THEOREM: If N P is an integer and x(t) is band limited to f MAX , then and for all k not defined above where is the DFT of the sequence f = 1/T, and ( ) 1 N k k Χ − = ( ) 1 N k S kT x − = MAX P f N f = • 2 N ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ .• • • • • Review from last lecture .• • • • • Distortion Analysis ( ) k Χ If the hypothesis of the theorem are satisfied, we thus have FFT is a computationally efficient way of calculating the DFT, particularly when N is a power of 2 .• • • • • Review from last lecture .• • • • • FFT Examples Recall the theorem that provided for the relationship between the DFT terms and the Fourier Series Coefficients required 1. The sampling window be an integral number of periods 2. max P SIGNAL 2 f N > N f Considerations for Spectral Characterization •Tool Validation •FFT Length •Importance of Satisfying Hypothesis •Windowing Considerations for Spectral Characterization •Tool Validation •FFT Length •Importance of Satisfying Hypothesis •Windowing FFT Examples Recall the theorem that provided for the relationship between the DFT terms and the Fourier Series Coefficients required 1. The sampling window be an integral number of periods 2....
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 Fall '11
 GEIGER
 Fourier Series, Total harmonic distortion, Spectral Characterization

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