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9.BodePlotsx4

# 9.BodePlotsx4 - Overview of Bode Plots Prerequisites and...

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Transfer Function Review H ( s ) x ( t ) y ( t ) Recall that if H ( s ) is known and x ( t ) = A cos( ωt + φ ) , then we can find the steady-state solution for y ( t ) : y ss ( t ) = A | H ( ) | cos ( ωt + φ + H ( )) J. McNames Portland State University ECE 222 Bode Plots Ver. 1.21 3 Overview of Bode Plots Review of transfer functions and bode plots Piece-wise linear approximations First-order terms Second-order terms (complex poles & zeros) J. McNames Portland State University ECE 222 Bode Plots Ver. 1.21 1 Bode Plots H ( s ) x ( t ) y ( t ) Bode plots are standard method of plotting the magnitude and phase of H ( s ) Both plots use a logarithmic scale for the x -axis Frequency is in units of radians/second (rad/s) The phase is plotted on a linear scale in degrees Magnitude is plotted on a linear scale in decibels H dB ( ) 20 log 10 | H ( ) | J. McNames Portland State University ECE 222 Bode Plots Ver. 1.21 4 Prerequisites and New Knowledge Prerequisite knowledge Ability to use transfer functions for steady-state sinusoidal circuit analysis Ability to generate and interpret Bode plots with tool such as Matlab New knowledge Knowledge of how poles and zeros affect transfer function frequency response J. McNames Portland State University ECE 222 Bode Plots Ver. 1.21 2

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Magnitude Components Consider the expression for the transfer function magnitude: | H dB ( ) | = 20 log 10 | H ( ) | = 20 log 10 k s ± (1 s z 1 ) . . . (1 s z m ) (1 s p 1 ) . . . (1 s p n ) s = = 20 log 10 | k | · | | ± | 1 z 1 | . . . | 1 z m | | 1 p 1 | . . . | 1 p n | = 20 log 10 | k | ± 20 log 10 ω +20 log 10 1 z 1 + · · · + 20 log 10 1 z m 20 log 10 1 p 1 − · · · − 20 log 10 1 p n J. McNames Portland State University ECE 222 Bode Plots Ver. 1.21 7 Bode Plot Approximations Until recently (late 1980’s) bode plots were drawn by hand There were many rules-of-thumb, tables, and template plots to help Today engineers primarily use MATLAB, or the equivalent Why discuss the old method of plotting by hand? It is still important to understand how the poles, zeros, and gain inﬂuence the Bode plot These ideas are used for transfer function synthesis, analog circuit design, and control systems We will discuss simplified methods of generating Bode plots Based on asymptotic approximations J. McNames Portland State University ECE 222 Bode Plots Ver. 1.21 5 Magnitude Components Comments | H dB ( ω ) | = 20 log 10 | k | ± 20 log 10 ω +20 log 10 1 z 1 + · · · + 20 log 10 1 z m 20 log 10 1 p 1 − · · · − 20 log 10 1 p n Thus, | H dB ( ω ) | can be written as a sum of simple functions This is similar like using basis functions { δ ( t ) , u ( t ) ,& r ( t ) } to write an expression for a piecewise linear signal We will use this approach to generate our piecewise linear approximations of the bode plot Note that there are four types of components in this expression – Constant – Linear term – Zeros – Poles J. McNames Portland State University ECE 222 Bode Plots Ver. 1.21 8 Alternate Transfer Function Expressions There are many equivalent expressions for transfer functions.
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