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Bode_Tutorial

# Bode_Tutorial - Bode Plot Tutorial Contents 1 Introduction...

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Bode Plot Tutorial Contents 1 Introduction 1 2 Bode Plots Basics 1 2.1 Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.2 Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 Combining Poles and Zeroes 4 1 Introduction Although you should have learned about Bode plots in previous courses, this tutorial will give you a brief review of the material in case your memory is fuzzy. 2 Bode Plots Basics Making the Bode plots for a transfer function involves drawing both the magnitude and phase plots. The magnitude is plotted in decibels (dB) while the phase is plotted in degrees ( ). For both plots, the horizontal axis is either frequency ( f ) or angular frequency ( ω ), measured in Hz and rad / s respectively. In addition, the horizontal axis should be logarithmic (i.e. increasing by factors of 10). Most of the transfer functions we will encounter in this lab manual can be rearranged into the general form: H ( ) = A · jω/ω z 1 (1 + jω/ω z 2 ) (1 + jω/ω z 3 ) ... jω/ω p 1 (1 + jω/ω p 2 ) (1 + jω/ω p 3 ) ... , (1) where A is an arbitrary constant and j is 1. Besides the exception of jω/ω c , the basic component of this transfer function is 1 + jω/ω c , where ω c is some numerical constant. Let us analyze this basic component first before we analyze the transfer function as a whole. 2.1 Magnitude Recall the definition of magnitude (measured in dB): | H ( ) | dB = 20 log | H ( ) | = 20 log radicalBig ( [ H ( )]) 2 + ( [ H ( )]) 2 (2) Let us apply this definition to our basic component (1 + jω/ω c ), which is also called a zero when it appears in the numerator of the transfer function: | 1 + jω/ω c | dB = 20 log | 1 + jω/ω c | = 20 log radicalBig 1 + ( ω/ω c ) 2 (3) For small values of ω , we have 20 log | 1 + jω/ω c | ≈ 0 dB. For large values of ω , 20 log | 1 + jω/ω c | → ∞ . When ω = ω c , the magnitude of the transfer function is approximately 3 dB. Since there is little change in the magnitude of the transfer function from ω = 0 to ω = ω c , we can approximate the magnitude expression as equal to 0 dB within this interval. As for the ω > ω c interval, the ( ω/ω c

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Bode_Tutorial - Bode Plot Tutorial Contents 1 Introduction...

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