1. MESB374 System Modeling and Analysis Chapter 11 Frequency Domain Design - Bode

2. Bode Magnitude Plot: plots the magnitude of G(j  ) in decibels w.r.t. logarithmic frequency, i.e., Bode Phase Plot: plots the phase angle of G(j  ) w.r.t. logarithmic frequency, i.e., A better way to graphically display the frequency response! Bode Plots Benefits: –Display the dependence of magnitude of the frequency response on the input frequency better, especially for magnitude approaching zero –Log axis converts the multiplications and divisions into additions and subtractions, which are easier to handle graphically –Allow straight-line approximations for quick sketch

3. Bode Plots Ex: 1.0049 0.5098 0.2233 0.1407 0.1096 0.0912 0.0711 0.0448 0.0196 0.0100 0.0428 -5.8520 -13.0211 -17.0329 -19.2012 -20.7988 -22.9671 -26.9789 -34.1480 -40.0428 -83.8623 -79.8358 -66.2974 -50.7016 -37.8750 -50.7106 -66.2974 -79.8358 -84.8623

4. Bode Plots of LTI Systems Transfer Function Frequency Response Bode Magnitude Plot Bode Phase Plot

5. Example Ex:Find the magnitude and the phase of the following transfer function:

6. Bode Plots of 1 st Order Poles Standard Form of Transfer Function: Frequency Response: Q:By just looking at the Bode diagram, can you determine the time constant and the steady state gain of the system ? break frequency

7. Example 1st Order Real Poles Transfer Function: Plot the straight line approximation of G(s)’s Bode diagram: Frequency (rad/sec) Phase (deg); Magnitude (dB) 0 5 10 15 20 -90 -45 0 10 10 0 1 2

8. Bode Plots of 1 st Order Zeros Standard Form of Transfer Function Frequency Response 0 Gjj Gj Gj z z z 1 1 22 1 1 1 1 1 () () ()(,) tan        , atan2

9. Example 1st Order Real Zeros Transfer Function: Plot the straight line approximation of G(s)’s Bode diagram: Phase (deg); Magnitude (dB) Frequency (rad/sec) 0 5 10 15 20 0 45 90 10 10 0 1 2 -3

10. Example Lead Compensator Transfer Function: Plot the straight line approximation of G(s)’s Bode diagram: Phase (deg); Magnitude (dB) -10 0 10 20 30 2 Frequency (rad/sec) -90 0 90 10 10 0 1 17

11. 1st Order Bode Plots Summary 1st Order Poles –Break Frequency –Mag. Plot Approximation 0 dB from DC to  b and a straight line with  20 dB/decade slope after  b –Phase Plot Approximation 0 deg from DC to. Between and 10  b, a straight line from 0 deg to  90 deg (passing  45 deg at  b ). For frequency higher than 10  b, straight line on  90 deg. 1st Order Zeros –Break Frequency –Mag. Plot Approximation 0 dB from DC to  b and a straight line with 20 dB/decade slope after  b –Phase Plot Approximation 0 deg from DC to. Between and 10  b, a straight line from 0 deg to 90 deg (passing 45 deg at  b ). For frequency higher than 10  b, straight line on 90 deg. Note:By looking at Bode plots you should be able to determine the relative order of the system, its break frequency, and DC (steady-state) gain. This process should also be reversible, i.e., given a transfer function, be able to plot a straight line approximation of Bode plots.

12. Bode Plots of Integrators Standard Form of Transfer Function Frequency Response 1 Gj j Gj Gj p p p 0 0 0 1 90 () () ()         Phase (deg); Magnitude (dB) Frequency (rad/sec) -60 -40 -20 0 20 0.11101001000 -135 -90 -45 0

13. Bode Plots of Differentiators Standard Form of Transfer Function Frequency Response Gjj Gj Gj z z z 0 0 0 90 () () ()       Frequency (rad/sec) Phase (deg); Magnitude (dB) 0.11101001000 -20 0 20 40 60 0 45 90 135

14. Example Combination of Systems Transfer Function: Plot the straight line approximation of G(s)’s Bode plots: Phase (deg); Magnitude (dB) -10 0 10 20 30 2 Frequency (rad/sec) -90 0 90 10 10 0 1 SUM THEM

15. Example Combination of Systems Transfer Function: Plot the straight line approximation of G(s)’s Bode plots: Phase (deg); Magnitude (dB) -120 -80 -40 0 40 Frequency (rad/sec) 10 10 0 1 2 3 -270 -180 -90 0

16. Bode Plots of Complex Poles Standard Form of Transfer Function Frequency Response  n  n nn  n  n Peak (Resonant) Frequency and Magnitude for

17. 2nd Order System Frequency Response Phase (deg); Magnitude (dB) Frequency (rad/sec) -60 -40 -20 0 20 40 -180 -135 -90 -45 0 0.1  n nn 10  n

18. 2nd Order System Frequency Response A Few Observations: Three different characteristic frequencies: –Natural Frequency (  n ) –Damped Natural Frequency (  d ): –Resonant (Peak) Frequency (  r ): When the damping ratio , there is no peak in the Bode magnitude plot. DO NOT confuse this with the condition for over-damped and under- damped systems: when  the system is under-damped (has overshoot) and when  the system is over-damped (no overshoot). As  ® 0,  r ®  n and  G(j  )    increases; also the phase transition from 0 deg to  180 deg becomes sharper.

19. Example Second-Order System Transfer Function: Plot the straight line approximation of G(s)’s Bode diagram: Phase (deg); Magnitude (dB) -120 -80 -40 0 40 Frequency (rad/sec) 10 0 1 2 3 4 -180 -90 0

20. Bode Plots of Complex Zeros Standard Form of Transfer Function Frequency Response Frequency (rad/sec)

21. Bode Plots of Poles and Zeros Bode plots of zeros are the mirror images of the Bode plots of the identical poles w.r.t. the 0 dB line and the 0 deg line, respectively: LetGs Gs Gj Gj GjGj GjGj GjGj p z p z pz pz pz () () () () ()() log() () ()()     |    | | 1 1 20 10      -40 -20 0 20 40 -180 0 180 Frequency (rad/sec) Phase (deg) Magnitude (dB)  n  n nn

22. 2nd Order Bode Diagram Summary 2nd Order Complex Poles –Break Frequency –Mag. Plot Approximation 0 dB from DC to  n and a straight line with  40 dB/decade slope after  n. Peak value occurs at: –Phase Plot Approximation 0 deg from DC to. Between and    n, a straight line from 0 deg to  180 deg (passing  90 deg at  n ). For frequency higher than    n, straight line on  180 deg. 2nd Order Complex Zeros –Break Frequency –Mag. Plot Approximation 0 dB from DC to  n and a straight line with 40 dB/decade slope after  n. –Phase Plot Approximation 0 deg from DC to. Between and    n, a straight line from 0 deg to 180 deg (passing 90 deg at  n ). For frequency higher than    n, straight line on 180 deg.

23. Combination of Systems Transfer Function: Plot the straight line approximation of G(s)’s Bode diagram: Example