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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

Engineering 43. Fourier Transfer Fcn. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. Fourier Transform.

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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

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  1. Engineering 43 FourierTransfer Fcn Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. 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. 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. Example: Square Wave • The SquareWave Shown at Bottom-Lt can be described by a sum-of-sines

  5. Transfer Fuction, H(f) • Consider a “Black Box” that takes Input Power, vin & iin Transforms this Power into an Output, vout & iout • A typical transformation would be to “Filter-Out” certain electrical frequencies. • For Phasor Voltages Vin & Vout Define the voltage Transfer Function as

  6. 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 (Vin0°), Measure Vout (Voutφ°) and Plot: Vout/ Vin and φ

  7. Example Transfer Function

  8. Example Transfer Function • Find vout for vin = 1.35Vcos(40∙2πt+65°) −25

  9. Example Transfer Function • Then at 40 Hz (40∙2πrads/sec) • Recall vin • In Phasor for • Thus • Using the Values Taken from the H(f) Mag & Phase Graphs • Or in the Time Domain

  10. MultiFrequency Example 6.2 • Note the THREE Frequencies • 0 Hz • 1000 Hz • 1000∙2πrad/sec • 2000 Hz • 2000∙2πrad/sec

  11. Ex6.2 Transfer Function • Apply to vin the Transfer Function • From the Transfer Function find • Apply To components of vin

  12. Example 6.2 • Using This H(f) Set find • Note that the above PhasorsCanNOT be added as they have DIFFERENTFrequencies.

  13. Example 6.2 • Because of Differing Frequencies MUST add TIME-DOMAIN Voltages • Then vout(t) is simply the SUM of the above

  14. 1st Order Lo-Pass Filter • Consider the RC Ckt Shown below • In the Frequency Domain the Cap Impedance, Zc • Notice the Limits of Behavior • A cap is • OPEN to Low-Freq • SHORT to Hi-Freq

  15. 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. 1st Order Lo-Pass Filter • Then the Transfer Function • ReWriting • Where • fB 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 fB:

  17. Lo-Pass Filter • The LoPass Filter Transfer Function • fB: is also call the Half-Power-Frequency • Recall Full Power to a Resistor: • Then HALF Power:

  18. LR (LowPass) Filter • Find the Transfer Function for LR Ckt • Use Ohm Find The Single Loop Current • Then also by Ohm • ReWriting • Arrive at XferFcn very similar to RC Ckt

  19. 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, P0 • In this case • ReCall a Current or Voltage delivering Power to a Resistor • Then the dB in Current or Voltage Ratios

  20. 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. dB Plots (SemiLog) Plot • Plotting H(f) on the logarithmic dB Scale makes it easier to distinguish Very Large (104vs 105) or Very Small (10−4vs 10−5) Points on the Plots

  22. Cascaded NetWork Gain • Consider the Transfer Function of the “BlackBox” at Right • Looking inside the BlackBox find • Note that with Vout1 = Vin2 • Or in dB form

  23. Tools Needed Ruler Scientific Calculator To Find a Value of a Pt Between Decades m & n Use Ruler to Measure Decade Distance, dd Distance from Pt to Lower Decade (Decade m), dp Then The Value at the Pt Reading Logarithmic Scales

  24. 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 f1 & f2 MUSICAL Octaves

  25. WhiteBoard Work • Let’s This Nice Problem  • Find the OutPut Voltage for For this Input

  26. All Done for Today 79.5 MHzNotchFilter

  27. Engineering 43 Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  28. Logarithm Change of Base Proof

  29. White Board RL Filter Problem

  30. 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)')

  31. P5.57 Graphics

  32. P5.81 Graphics

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