1. Department of EECS University of California, Berkeley EECS 105 Fall 2003, Lecture 2 Lecture 3: Bode Plots Prof. Niknejad

2. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Get to know your logs! Engineers are very conservative. A “margin” of 3dB is a factor of 2 (power)! Knowing a few logs by memory can help you calculate logs of different ratios by employing properties of log. For instance, knowing that the ratio of 2 is 3 dB, what’s the ratio of 4?

3. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Bode Plot Overview Technique for estimating a complicated transfer function (several poles and zeros) quickly Break frequencies :

4. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Summary of Individual Factors Simple Pole: Simple Zero: DC Zero: DC Pole:

5. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Example Consider the following transfer function Break frequencies: invert time constants

6. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Breaking Down the Magnitude Recall log of products is sum of logs Let’s plot each factor separately and add them graphically

7. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Breaking Down the Phase Since Let’s plot each factor separately and add them graphically

8. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Magnitude Bode Plot: DC Zero 80 20 60 40 -20 -60 -80 -40 10 4 10 5 10 6 10 7 10 8 10 910 10 11 0 dB

9. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Phase Bode Plot: DC Zero 180 45 135 90 -45 -135 -180 -90 10 4 10 5 10 6 10 7 10 8 10 910 10 11

10. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Magnitude Bode Plot: Add First Pole 80 20 60 40 -20 -60 -80 -40 10 4 10 5 10 6 10 7 10 8 10 910 10 11

11. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Phase Bode Plot: Add First Pole 180 45 135 90 -45 -135 -180 -90 10 4 10 5 10 6 10 7 10 8 10 910 10 11

12. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Magnitude Bode Plot: Add 2 nd Zero 80 20 60 40 -20 -60 -80 -40 10 4 10 5 10 6 10 7 10 8 10 910 10 11

13. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Phase Bode Plot: Add 2 nd Zero 180 45 135 90 -45 -135 -180 -90 10 4 10 5 10 6 10 7 10 8 10 910 10 11

14. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Magnitude Bode Plot: Add 2 nd Pole 80 20 60 40 -20 -60 -80 -40 10 4 10 5 10 6 10 7 10 8 10 910 10 11

15. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Phase Bode Plot: Add 2 nd Pole 180 45 135 90 -45 -135 -180 -90 10 4 10 5 10 6 10 7 10 8 10 910 10 11

16. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Comparison to “Actual” Mag Plot

17. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Comparison to “Actual” Phase Plot

18. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Why do I say “actual”? I plotted the transfer characteristics with Mathematica The range of frequency for the plot is 6 orders of magnitude. The program has to find the “hot spots” in order to plot the function. Near the hot spots, more points are plotted. In between hot spots, the function is interpolated. If you pick the wrong points, you’ll end up with the wrong plot: mag = LogLinearPlot[20*Log[10, Abs[H[x]]], {x, 10^4, 10^11},PlotPoints -> 10000, Frame -> True,PlotStyle -> Thickness[.005], ImageSize -> 600,GridLines -> Automatic, PlotRange -> {{10^4, 10^11}, {-20, 100}} ]

19. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Don’t always believe a computer!

20. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Second Order Transfer Function The series resonant circuit is one of the most important elementary circuits: The physics describes not only physical LCR circuits, but also approximates mechanical resonance (mass-spring, pendulum, molecular resonance, microwave cavities, transmission lines, buildings, bridges, …)

21. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Series LCR Analysis With phasor analysis, this circuit is readily analyzed +Vo−+Vo−

22. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Second Order Transfer Function So we have: To find the poles/zeros, let’s put the H in canonical form: One zero at DC frequency  can’t conduct DC due to capacitor +Vo−+Vo−

23. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Poles of 2 nd Order Transfer Function Denominator is a quadratic polynomial:

24. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Finding the poles… Let’s factor the denominator: Poles are complex conjugate frequencies The Q parameter is called the “quality-factor” or Q-factor This parameters is an important parameter: Re Im

25. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Resonance without Loss The transfer function can parameterized in terms of loss. First, take the lossless case, R=0: When the circuit is lossless, the poles are at real frequencies, so the transfer function blows up! At this resonance frequency, the circuit has zero imaginary impedance Even if we set the source equal to zero, the circuit can have a steady-state response Re Im

26. EECS 105 Fall 2003, Lecture 3Prof. A. Niknejad Department of EECS University of California, Berkeley Magnitude Response The response peakiness depends on Q