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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Massachusetts Institute of Technology Department of Mechanical Engineering 2.004 Dynamics and Control II Spring Term 2008 Lecture 33 1 Reading: Nise: 10.1 Class Handout: Sinusoidal Frequency Response 1 Bode Plots (continued 1.1 Logarithmic Amplitude and Frequency Scales: 1.1.1 Logarithmic Amplitude Scale: The Decibel Bode magnitude plots are frequently plotted using the decibel logarithmic scale to display the function H ( j ) . The Bel, named after Alexander Graham Bell, is defined as the logarithm   to base 10 of the ratio of two power levels. In practice the Bel is too large a unit, and the decibel (abbreviated dB), defined to be one tenth of a Bel, has become the standard unit of logarithmic power ratio. The power ow P into any element in a system, may be expressed in terms of a logarithmic ratio Q to a reference power level P ref : Q = log 10 P Bel or Q = 10 log 10 P dB. (1) P ref P ref Because the power dissipated in a Dtype element is proportional to the square of the amplitude of a system variable applied to it, when the ratio of across or through variables is computed the definition becomes A 2 A Q = 10 log 10 = 20 log 10 dB. (2) A ref A ref where A and A ref are amplitudes of variables. Note: This definition is only strictly correct when the two amplitude quantities are measured across a common Dtype (dissipative) element. Through common usage, however, the decibel has been effectively redefined to be simply a convenient loga rithmic measure of amplitude ratio of any two variables. This practice is widespread in texts and references on system dynamics and control system theory. The table below expresses some commonly used decibel values in terms of the power and amplitude ratios. 1 copyright c D.Rowell 2008 331 Decibels Power Ratio Amplitude Ratio40 0.0001 0.0120 0.01 0.110 0.1 0.31626 0.25 0.53 0.5 0.7071 1.0 1.0 3 2.0 1.414 6 4.0 2.0 10 10.0 3.162 20 100.0 10.0 40 10000.0 100.0 The magnitude of the frequency response function H ( j ) is defined as the ratio of the   amplitude of a sinusoidal output variable to the amplitude of a sinusoidal input variable. This ratio is expressed in decibels, that is Y ( j ) 20log 10  H ( j )  = 20 log 10  U ( j )  dB.   As noted this usage is not strictly correct because the frequency response function does not define a power ratio, and the decibel is a dimensionless unit whereas H ( j ) may have   physical units. Example 1 An amplifier has a gain of 28. Express this gain in decibels....
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This note was uploaded on 02/23/2012 for the course MECHANICAL 2.004 taught by Professor Derekrowell during the Spring '08 term at MIT.
 Spring '08
 DerekRowell
 Mechanical Engineering

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