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# lecture_33 - MIT OpenCourseWare http/ocw.mit.edu 2.004...

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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 .

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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 ( ) . 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 ﬂow 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 D–type 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 D–type (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 33–1
Decibels Power Ratio Amplitude Ratio -40 0.0001 0.01 -20 0.01 0.1 -10 0.1 0.3162 -6 0.25 0.5 -3 0.5 0.7071 0 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 ( ) 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 ( ) 20 log 10 | H ( ) | = 20 log 10 | U ( ) | 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 ( ) may have | | physical units. Example 1 An amplifier has a gain of 28. Express this gain in decibels. We note that 28 = 10 × 2 × 1 . 4 10 × 2 2. The gain in dB is therefore 20 log 10 10 + 20 log 10 2 + 20 log 10 2, or × Gain(dB) = 20 + 6 + 3 = 29 dB. The advantages of a logarithmic amplitude scale include: Compression of a large dynamic range.

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lecture_33 - MIT OpenCourseWare http/ocw.mit.edu 2.004...

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