CHAPTER 6 Frequency Response, Bode Plots, and Resonance

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1 ELECTRICAL CHAPTER 6 Frequency Response, Bode Plots, and Resonance 1. State the fundamental concepts of Fourier analysis. 2. Determine the output of a filter for a given input consisting of sinusoidal components using the filter s transfer function.

2 3. Use circuit analysis to determine the transfer functions of simple circuits. 4. Draw first-order lowpass or highpass filter circuits and sketch their transfer functions. 5. Understand decibels, logarithmic frequency scales, and Bode plots. ELECTRICAL

3 6. Draw the Bode plots for transfer functions of first-order filters. ELECTRICAL

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5 Fourier Analysis All real-world signals are sums of sinusoidal components having various frequencies, amplitudes, and phases. ELECTRICAL

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8 Filters Filters process the sinusoid components of an input signal differently depending of the frequency of each component. Often, the goal of the filter is to retain the components in certain frequency ranges and to reject components in other ranges. ELECTRICAL

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10 Transfer Functions The transfer function H(f ) of the two-port filter is defined to be the ratio of the phasor output voltage to the phasor input voltage as a function of frequency: H ( f ) = V V out in ELECTRICAL

11 H ( f ) = V V out in The magnitude of the transfer function shows how the amplitude of each frequency component is affected by the filter. Similarly, the phase of the transfer function shows how the phase of each frequency component is affected by the filter. ELECTRICAL

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13 Determining the output of a filter for an input with multiple components: 1. Determine the frequency and phasor representation for each input component. 2. Determine the (complex) value of the transfer function for each component. ELECTRICAL

14 3. Obtain the phasor for each output component by multiplying the phasor for each input component by the corresponding transfer-function value. 4. Convert the phasors for the output components into time functions of various frequencies. Add these time functions to produce the output. ELECTRICAL

15 Linear circuits behave as if they: ELECTRICAL 1. Separate the input signal into components having various frequencies. 2. Alter the amplitude and phase of each component depending on its frequency. 3. Add the altered components to produce the output signal.

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20 FIRST-ORDER LOWPASS FILTERS 1 f = B 2πRC H ( f ) = ( f ) 2 f B H ( f ) 1 = 1 + j ( f ) f B H ( f ) = arctan f B f ELECTRICAL

21 H ( f ) = ( f ) 2 f B ELECTRICAL

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26 DECIBELS, THE CASCADE CONNECTION, AND LOGARITHMIC FREQUENCY SCALES H ( f ) log H ( f ) db = 20 ELECTRICAL

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30 Cascaded Two-Port Networks H( f ) = H ( f ) H ( f ) 1 2 ( f ) H ( f ) H ( f ) db 1 db 2 db H = + ELECTRICAL

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32 Logarithmic Frequency Scales On a logarithmic scale, the variable is multiplied by a given factor for equal increments of length along the axis. ELECTRICAL

33 A decade is a range of frequencies for which the ratio of the highest frequency to the lowest is 10. ELECTRICAL number of decades = log An octave is a two-to-one change in frequency. f f 2 1 ( f f ) f log () number of octaves = log 2 = f log 2 1

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35 BODE PLOTS A Bode plot shows the magnitude of a network function in decibels versus frequency using a logarithmic scale for frequency. H ( f ) 1 = 1 + j ( f ) f B ELECTRICAL H ( f ) = 10log db 1 + f f B 2

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37 1. A horizontal line at zero for f < f B / A sloping line from zero phase at f B /10 to 90 at 10f B. 3. A horizontal line at 90 for f > 10f B. ELECTRICAL

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41 FIRST-ORDER HIGHPASS FILTERS H ( f ) = V V out in = 1 j + ( f f ) B j( f f ) B 1 f = B 2πRC ELECTRICAL

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46 SERIES RESONANCE Resonance is a phenomenon that can be observed in mechanical systems and electrical circuits. ELECTRICAL

47 f 0 = 2π 1 LC 1 Q = s 2πf CR 0 2πf L Q = 0 s R Z s ( ) f f f = R 1 + jq 0 s f f 0 ELECTRICAL

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50 Series Resonant Circuit as a Bandpass Filter V V R s = 1 + jq s 1 ( f f f f ) 0 0 ELECTRICAL

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52 B = f f H L f f + H 0 B 2 f B = 0 Q s f L f 0 B 2 ELECTRICAL

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56 PARALLEL RESONANCE Z p = 1 j2πfc j 1 2πfL ( ) + ( ) 1 R f 0 = 2π 1 LC R = Q f CR 2πf L p = 2π 0 Q p 0 Z p = R 1 + jq p ( f f f f ) 0 0 ELECTRICAL

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60 Ideal Filters ELECTRICAL

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63 Second-Order Lowpass Filter H ( f ) = V V out in = ( f s 0 f ) ( f f f f ) jq 1 + jq s 0 0 ELECTRICAL

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71 DIGITAL SIGNAL PROCESSING ELECTRICAL

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73 Conversion of Signals from Analog to Digital Form If a signal contains no components with frequencies higher than f H, the signal can be exactly reconstructed from its samples, provided that the sampling rate f s is selected to be more than twice f H. ELECTRICAL

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76 Digital Lowpass Filter y () n = ay( n 1) + ( 1 a)() x n a τ T = 1 + τ T ELECTRICAL

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