1. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 1 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Engineering 43 Fourier Transfer Fcn

2. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 2 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Fourier Transform  A Fourier Transform is an integral transform that re- expresses a function in terms of different Sine/Cosine waves of varying amplitudes, wavelengths, and phases.  A Conceptual Example This Irregular Signal Is the SAME as the Sum of these Sinusoids

3. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 3 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Fourier Transform  John Baptiste Joseph Fourier investigated Time Varying Heat-Flow in a Metal Bar  His great Insight: ANY Periodic Function Could be Expressed as the sum of Sinusoidal Functions  For a Given, arbitrary Periodic Function, f(t), The Fourier Equivalents

4. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 4 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example: Square Wave  The SquareWave Shown at Bottom-Lt can be described by a sum-of-sines

5. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 5 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Transfer Fuction, H(f)  Consider a “Black Box” that takes Input Power, v in & i in Transforms this Power into an Output, v out & i out A typical transformation would be to “Filter- Out” certain electrical frequencies.  For Phasor Voltages V in & V out Define the voltage Transfer Function as

6. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 6 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Transfer Function  Note that the Transfer Function Is a Function of FREQENCY ONLY Can Change (and usually does change) the Magnitude and Phase-Angle of many of the incoming, frequency-dependent, electrical signals  Measuring an Unknown “Black Box” Apply Sinusoidal Vin (V in  0°), Measure Vout (V out  φ°) and Plot: V out / V in and φ

7. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 7 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example Transfer Function

8. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 8 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example Transfer Function  Find v out for v in = 1.35Vcos(40∙2πt+65°) −25

9. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 9 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example Transfer Function  Then at 40 Hz (40∙2π rads/sec)  Recall v in  In Phasor for  Thus  Using the Values Taken from the H(f) Mag & Phase Graphs  Or in the Time Domain

10. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 10 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis MultiFrequency Example 6.2  Note the THREE Frequencies 0 Hz 1000 Hz –1000∙2π rad/sec 2000 Hz –2000∙2π rad/sec

11. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 11 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Ex6.2 Transfer Function  Apply to v in the Transfer Function  From the Transfer Function find Apply To components of v in

12. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 12 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example 6.2  Using This H(f) Set find  Note that the above Phasors CanNOT be added as they have DIFFERENT Frequencies.

13. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 13 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example 6.2  Because of Differing Frequencies MUST add TIME-DOMAIN Voltages  Then v out (t) is simply the SUM of the above

14. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 14 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis 1st Order Lo-Pass Filter  Consider the RC Ckt Shown below  In the Frequency Domain the Cap Impedance, Z c  Notice the Limits of Behavior  A cap is OPEN to Low-Freq SHORT to Hi-Freq

15. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 15 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis 1st Order Lo-Pass Filter  Thus the Behavior of a Cap-Based Impedance At LO-Frequencies a Cap acts as an OPEN Circuit At HI-Frequencies a Cap Acts as a SHORT Circuit  Now use Phasor V- Divider on RC ckt  Multiplying Top&Bot by j2πfC

16. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 16 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis 1st Order Lo-Pass Filter  Then the Transfer Function  ReWriting  Where  f B is the “Break point” Frequency at which H(f) falls to 70.7% of its Original Magnitude Value.  Note The Mag & Ph of H(f) in terms of f B :

17. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 17 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Lo-Pass Filter  The LoPass Filter Transfer Function  f B : is also call the Half-Power-Frequency Recall Full Power to a Resistor: Then HALF Power:

18. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 18 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis LR (LowPass) Filter  Find the Transfer Function for LR Ckt  Use Ohm Find The Single Loop Current  Then also by Ohm  ReWriting  Arrive at Xfer Fcn very similar to RC Ckt

19. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 19 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis The deciBel (dB)  Named after Alexander Graham Bell, the deciBel (dB) relates two Power Levels  SomeTimes The Power Level is Referenced to a Standard Value, P 0  In this case  ReCall a Current or Voltage delivering Power to a Resistor  Then the dB in Current or Voltage Ratios

20. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 20 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis The deciBel (dB)  dB In Terms of Voltage Ratios  Or dB for Currents  Now we Defined  Since |H(f)| is a Voltage Ratio, define

21. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 21 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis dB Plots (SemiLog) Plot  Plotting H(f) on the logarithmic dB Scale makes it easier to distinguish Very Large (10 4 vs 10 5 ) or Very Small (10 −4 vs 10 −5 ) Points on the Plots

22. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 22 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Cascaded NetWork Gain  Consider the Transfer Function of the “BlackBox” at Right  Looking inside the BlackBox find  Note that with V out1 = V in2  Or in dB form

23. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 23 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Reading Logarithmic Scales  Tools Needed Ruler Scientific Calculator  To Find a Value of a Pt Between Decades m & n Use Ruler to Measure –Decade Distance, d d –Distance from Pt to Lower Decade (Decade m), d p Then The Value at the Pt

24. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 24 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

25. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 25 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Octave  An octave is the interval between two points where the frequency at the second point is twice the frequency of the first.  Given Frequencies f 1 & f 2 MUSICAL Octaves

26. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 26 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis WhiteBoard Work  Let’s This Nice Problem   Find the OutPut Voltage for For this Input

27. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 27 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis All Done for Today 79.5 MHz Notch Filter

28. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 28 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Engineering 43 Appendix

29. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 29 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Logarithm Change of Base Proof

30. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 30 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

31. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 31 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

32. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 32 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

33. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 33 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis White Board RL Filter Problem

34. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 34 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis LR Filter Transfer Function f = 0:10:20e3 HfB = 1./sqrt(1+(f/fB).^2); plot(f,HfB,'LineWidth',3), grid, xlabel('f (Hz)'), ylabel('|H(f)') disp('showing fB plot - hit ANY KEY to continue') pause fB = 2700/(2*pi*68e-3) Hf = abs(2700./(2700 + j*2*pi*f*68e-3)); plot(f,Hf,'LineWidth',3), grid, xlabel('f (Hz)'), ylabel('|H(f)')

35. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 35 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis P5.57 Graphics

36. BMayer@ChabotCollege.edu ENGR-43_Lec-06a_Fourier_XferFcn.pptx 36 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis P5.81 Graphics